NONLINEAR PROBLEMS: PRESENT A N D FUTURE
This Page Intentionally Left Blank
NORTH-HOLLAND MATHEMATICS STUDIES
61
Nonlinear Problems: Present and Future Proceedings of the First Los Alamos Conference on Nonlinear Problems, Los Ala mos, NM, U.S.A., March 2-6,1981
Edited by
ALAN BISHOP DAVID CAMPBELL BASIL NICOLAENKO Los Alarnos National Laboratory LosAlarnos, New Mexico, U.S.A.
1982 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
NEW YORK
OXFORD
lil North- Holland
Publishing Company 1982
All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted. in any form or by any means, electronic, mechanical, photocopying. recording o r otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 86395 8
Publish ers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017
Library of Congress Cataloging in Publication Data
Los Alamos Conference on N o n l i n e a r Problems (1st : 1981) N o n l i n e a r problems. (North-Holland mathematics s t u d i e s ; 61) 1. N o n l i n e a r t h e o r i e s - - C o n g r e s s e s . 2 . MatheI. B i s h o p , A.R. m a t i c a l hysics--Congresses. (Alan R.f 11. Campbell, David, 1944 J u l y 23111. Nicolaenko. B a s i l . 1947IV. T i t l e . V. S e r i e s . 530.1'5 82 -3504 QC20.7.N6L67 1981 ISBN 0-444-86395-8 AACF2
.
PRINTED IN T H E NETHERLANDS
V
PREFACE
Recognizing the growing awareness o f common problems and answers i n t h e nonlinear sciences, t h e Los Alamos Center f o r Nonlinear Studies (CNLS) was established by the Laboratory D i r e c t o r i n October 1980, t o coordinate i n t e r d i s c i p l i n a r y studies, t o strengthen t i e s among Laboratory researchers and w i t h the academic community, and t o i n t e r f a c e between basic and a p p l i e d areas. The breadth o f research problems a t Los Alamos and the v a r i e t y o f technical e x p e r t i s e r e f l e c t e d i n i t s s t a f f o f f e r s a f e r t i l e environment f o r the CNLS. But from i t s inception, the Center was intended n o t f o r Los Alamos alone, b u t a l s o as an i n t e r n a t i o n a l resource f o r the e n t i r e nonl inear communi ty. To r e a l i z e t h i s i n t e n t i o n , the CNLS, through i t s Chairman, Alwyn C. Scott, has developed several continuing modes o f operation: *
i d e n t i f y i n g broad themes f o r i n t e n s i v e i n t e r d i s c i p l i n a r y research ( c u r r e n t l y , these themes are nonlinear phenomena i n r e a c t i v e flows and chaos and coherence i n physical systems); coordinating an a c t i v e v i s i t o r program, i n c l u d i n g b o t h guest l e c t u r e series and longterm c o l l a b o r a t i v e v i s i t s ;
*
-
sponsoring technical workshops aimed a t b r i n g i n g experts together t o expound recent r e s u l t s and d e l i n e a t e f u t u r e d i r e c t i o n s i n s p e c i f i c nonlinear problems (recent workshops have included "Adaptive G r i d Methods,'' "Coupled Nonlinear Osci 1 l a t o r s ,I' and "Sol itons and t h e Bethe Ansatz") ; and hosting an annual i n t e r n a t i o n a l conference on some t o p i c a l area o f nonlinear science.
I n the l a s t category, the major event f o r the CNLS i n i t s f i r s t year was an i n t e r n a t i o n a l conference on 'Nonlinear Problems: Present and Future' h e l d a t t h e Los Alamos National Security and Resources Study Center, Los Alamos, March 2-6, 1981, chaired by Mark Kac and Stanislav Ulam. This volume contains the e d i t e d proceedings o f t h a t conference. We are honored t o dedicate t h e proceedings t o Fermi, Pasta, and Ulam, distinguished Los Alamos alumni who have s e t a t r a d i t i o n o f excellence t o which t h e CNLS must aspire. [We have a l l been saddened by t h e death o f John Pasta since the Conference took place and hope t h a t t h i s volume w i l l be accepted as our small t r i b u t e t o h i s imaginative career.] As b e f i t s an inaugral conference, a very wide spectrum o f t o p i c s were represented ranging from pure mathematics, through numerical methods, t o sophist i c a t e d experiments on f l u i d s and s o l i d s , More s p e c i a l i z e d meetings are a n t i c ipated i n f u t u r e years i n c l u d i n g some o f t h e many t o p i c s t h a t i t was impossible t o cover i n 1981. However, the d e l i b e r a t e l y i n t e r d i s c i p l i n a r y atmosphere o f t h e inaugral meeting t r u l y r e f l e c t e d an e x c i t i n g stage o f development o f nonlinear science as a u n i f i e d subject. The provocative t i t l e a t t r a c t e d w e l l over two hundred p a r t i c i p a n t s and a distinguished l i s t o f speakers, many o f whom succeeded admirably i n overcoming s c i e n t i f i c language b a r r i e r s and generating a broad i n t e r e s t i n t h e i r fundamental problems. Four major t o p i c s were reperesented through survey l e c t u r e s and workshop a c t i v i t i e s : turbulence i n plasmas and f l u i d s (both the onset and fully-developed turbulence); n o n l i n e a r i t y i n f i e l d theory and i n low-dimensional s o l i d s ; reaction- d i f f u s i o n processes; and new methods i n nonl i n e a r mathematics. As described i n d e t a i l i n t h e Contents, we have preserved these d i v i s i o n s i n the Proceedings, supplementing i n v i t e d papers w i t h a small number o f r e l e v a n t c o n t r i b u t e d ones. I t i s our f i r m impression t h a t these art i c l e s , through survey and o r i g i n a l work, represent the c u t t i n g edges o f several important areas o f n o n l i n e a r i t y . We hope they w i l l be valuable reading f o r novi c e s and experts a l i k e .
vi
PREFACE
We t h i n k t h a t a l l those who attended t h e Conference w i l l remember i t f o r i t s stimulation unobtrusive organization. T h i s c r u c i a l combination c o u l d n o t have been achieved w i t h o u t the advice and support o f our colleagues i n t h e CNLS. The D i r e c t o r and h i s e f f i c i e n t s t a f f provided every conceivable help i n c o o r d i n a t i n g the splendid Laboratory f a c i l i t i e s . Mark Kac and S t a n i s l a v Ulam were supportive conference chairman and Stanislav Ulam g r a c i o u s l y agreed t o g i v e a n o s t a l g i c a f t e r - d i n n e r speech on the FPU problem which was admired by a l l . No conference i s b e t t e r than i t s secretary. I n Janet Gerwin we had a secretary whose competence was apparent a t every stage before, during, and a f t e r t h e conference. A l l t h r e e organizers are immeasurably indebted t o her f o r her s k i l l i n t h e face o f t h e continual crises! We are a l s o happy t o thank Janet, Chris Davis, Frankie Gomez, Mary Plehn, agd Kate Procknow f o r t h e i r s k i l l f u l assistance i n preparing these proceedings. Last b u t n o t l e a s t we must thank our p u b l i s h e r s f o r t h e i r e x c e l l e n t cooperation and every conference attendee f o r j o i n i n g us i n t h i s c e l e b r a t i o n o f nonlinear physics. Los Alamos
A. R. Bishop D. K. Campbell B. Nicolaenko
vii
DEDICATION
It started with an experiment--the new kind in which the same instrument, the new computer, both creates and probes an idealization of the real world. The purpose, quite modest, was to test what seemed beyond doubt, namely, that in a nonlinear discretized string the energy initially concentrated in one vibrational mode ultimately distributes itself among all modes. The result, first announced in a Los Alamos report, was however, startlingly different: after an initial tendency toward equipartition, the energy flowed back to the initial mode. Thus a new chapter of nonlinear science began and much of its spectacular growth in the past quarter of a century is directly or indirectly traceable to the pioneering experiment with the nonlinear string. It is therefore fitting that this Conference marking the creation of the Los Alamos Center for Nonlinear Studies be dedicated to the authors of that historic 1955 report: Fermi, Pasta, and Ulam.
Los Alamos
M. Kac N. Metropol is
This Page Intentionally Left Blank
ix
TABLE OF CONTENTS
PREFACE
V
vii
DEDICATION PART I :
NEW METHODS AND RESULTS I N NONLINEAR ANALYSIS
Optimal C o n t r o l o f Non-Well-Posed D i s t r i b u t e d Systems and R e l a t e d N o n l i n e a r P a r t i a l D i f f e r e n t i a l Equations J.-L. LIONS
3
Compactness and T o p o l o g i c a l Methods f o r Some N o n l i n e a r V a r i a t i o n a l Problems o f Mathematical Physics P.L. LIONS
17
On Yang-Mills F i e l d s I . M. SINGER
35
Gauge Theories f o r S o l i t o n Problems D. H SATTI NGER
51
The I n v e r s e Monodromy Transform i s a Canonical T r a n s f o r m a t i o n H. FLASCHKA and A.C. NEWELL
65
Numerical Methods f o r N o n l i n e a r D i f f e r e n t i a l Equations J.M. HYMAN
91
.
L i m i t A n a l y s i s o f P h y s i c a l Models M. FREMOND, A. FRIAA, and M. RAUPP
109
V i s c o s i t y S o l u t i o n s o f Hamil ton-Jacobi Equations M.G. CRANDALL
117
B i f u r c a t i o n o f S t a t i o n a r y Vortex C o n f i g u r a t i o n s J.I. PALMORE
127
Exact I n v a r i a n t s f o r Time-Dependent N o n l i n e a r Hami 1t o n i a n S y s t e m H.R. LEWIS and P.G.L. LEACH
133
I s o l a t i n g I n t e g r a l s i n G a l a c t i c Dynamics and t h e C h a r a c t e r o f S t e l 1a r O r b i t s W . I . NEWMAN
147
PART I 1
:
NONLINEARITY I N FIELD THEORIES AND LOW DIMENSIONAL SOLIDS
Physics i n Few Dimensions V.J. EMERY
159
S o l u t i o n o f t h e Kondo Problem N. ANDRE1
169
Kinks o f F r a c t i o n a l Charge i n Quasi-One Dimensional Systems J.R. SCHRIEFEER
179
TABLE OF CONTENTS
X
T h e o r e t i c a l l y Predicted Drude Absorption by a Conducting Charged S o l i t o n i n Doped-Polyacetylene M.J. R I C E
189
Pol arons in Polyacetylene A.R. BISHOP and D.K. CAMPBELL
195
S. ETEMAD and A.J.
L i g h t S c a t t e r i n g and Absorption i n Polyacetylene HEEGER
209
Quasi-Solitons: A Case Study o f the Double Sine-Gordon Equation P. KUMAR gnd R.R. HOLLAND
229
Classical F i e l d Theory w i t h Z(3) Symmetry H.M. RUCK
237
PART 111 :
PART
REACTION-DIFFUSION PROCESSES
Some C h a r a c t e r i s t i c N o n l i n e a r i t i e s o f Chemical Reaction Engineering R. A R I S
247
Propagating Fronts i n Reactive Media P.C. FIFE
267
IV :
NONLINEAR PHENOMENA I N FLUIDS AND PLASMAS
Regularity Results f o r t h e Equations o f Incompressible F l u i d s Mechanic a t the B r i n k o f Turbulence C. BARDOS
289
F i n i t e Parameter Approximative S t r u c t u r e o f Actual Flows C. FOIAS and R. TEMAM
317
The Role o f C h a r a c t e r i s t i c Boundaries i n the Asymptotic S o l u t i o n o f the Navier-Stokes Equations F.A. HOWES
329
Some Approaches t o the Turbulence Problem J. MATHIEU, 0. JEANDEL, and C.M. BRAUNER
335
I n s t a b i l i t y o f Pipe A.T. PATERA and S.A.
Flow ORSZAG
367
Some Formalism and Predictions o f the Period-Doubling Onset o f Chaos M.J. FEIGENBAUM
379
T r i c r i t i c a l Points and B i f u r c a t i o n s i n a Q u a r t i c Map S.-J. CHANG, M. WORTIS, and J.A. WRIGHT
395
Recent Experiments on Convective Turbulence J.P. GOLLUB
403
Experimental Observations o f Complex Dynamics i n a Chemical Reaction J.C. ROUX, J.S. TURNER, W.D. McCORMICK, and H.L. SWINNEY
409
Nonl inear P1asma Dynamics Be1ow the Cyclotron Frequency J.M. GREENE
423
TABLE OF CONTENTS
xi
Turbulence and S e l f - c o n s i s t e n t F i e l d s i n Plasmas D. PESME and D.F. DUBOIS
435
S e l f - F o c u s i n g Tendencies o f t h e N o n l i n e a r S c h r o d i n g e r and Zakharov Equations H.A. ROSE and D.F. DUBOIS
465
C h a o t i c O s c i l l a t i o n s i n a S i m p l i f i e d Model f o r Langmuir Waves P.K.C. WANG
479
INDEX OF CONTRIBUTORS
483
This Page Intentionally Left Blank
PART I New Methodsand ResuIts in Nonlinear Analysis
This Page Intentionally Left Blank
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.1 0 North-Holland Publishing Company, 1982
3
OPTIMAL CONTROL OF NON WELL POSED DISTRIBUTED SYSTEMS AND RELATED NON LINEAR PARTIAL DIFFERENTIAL EQUATIONS Jacques-Louis LIONS College de France and I N R I A
Various p r a c t i c a l problems l e a d t o the question o f optimal cont r o l o f non w e l l posed systems such as non l i n e a r unstable systems. We begin w i t h simple l i n e a r quadratic examples o f non..well posed e v o l u t i o n systems ; the o p t i m a l i t y system leads t o new non l i n e a r P a r t i a l D i f f e r e n t i a l Equations o f R i c a t t i ' s type. We study then non l i n e a r parabolic unstable systems, w i t h a d i s t r i b u t e d o r a boundary c o n t r o l , w i t h o r without const r a i n t s . The o p t i m a l i t y system i s given.
-
-
INTRODUCTION
1. L e t d be a p a r t i a l d i f f e r e n t i a l operator, l i n e a r o r not, o f e v o l u t i o n o r o f s t a t i o n a r y type. I n the usual theory o f optimal c o n t r o l o f d i s t r i b u t e d systems, the s t a t e equation i s given, i n a formal manner, by
(1)
ffy = B v
where v denotes the c o n t r o l v a r i a b l e ( o r f u n c t i o n ) and is an operator which can be thought o f as g i v i n g boundary c o n d i t i o m ( 1 ) one has t o add i n i t i a l condit i o n s i f ff i s o f e v o l u t i o n type. I n the usual theory one assumes t h a t , given v i n a s u i t a b l e Banach s ace U , equation (1) subject t o appropriate boundary and i n i t i a d i t s a u n i ue s o l u t i o n denoted by y ( v ) ; y ( v ) i s t h e s t a t e o f t h e system ; y ( v ) m o n g s t o a space w en v spans U.
+
Then the cost f u n c t i o n i s given by J(v)
Cp(Y(V))
+
uJ( I I V l l " )
(2)
where Cp i s a continuous f u n c t i o n a l from Y * R , where l l v l l v i n U and where X + $ ( A ) i s continuous f o r A t 0, $(O)
=\,
denotes the norm o f $ ( A ) * +EC as X + i-.
IfUad denotes a ( s u i t a b l e ) subset o f U, then the problem i s t o f i n d
i n f J(v), v
E
uad.
(3)
If( 3 ) admits a s o l u t i o n u, one o f the main questions i s then t o f i n d a s e t o f necessary ( o r necessary and s u f f i c i e n t ) conditions f o r c h a r a c t e r i z i n g u, i.e. t o f i n d the o p t i m a l i t y system. For these questions we r e f e r t o J.L. LIONS [1][2][3] and t o the Bibliography therein. 2. A s l i g h t l y d i f f e r e n t s i t u a t i o n can occur i f the f u n c t i o n a l 4 i n ( 2 ) i s not def i n e d on t h e whole space Y . Then one has t o introduce new f u n c t i o n a l spaces.
4
J.-L. LIONS
L e t us g i v e an example. Let R be a bounded open s e t o f R3 w i t h boundary consider the s t a t e equation
a t y
-
By = v ( t ) b ( x - b ) i n Q = R
x
r
; we
IO,TC
0 on C = T x 10,TC,
(4)
Y(XY0) = 0 where 6(x-b) denotes the Dirac measure a t p o i n t b c R . Given V E LL(O,T), probl m (4) admits a unique weak s o l u t i o n ( c f . J.L. E. MAGENES C11) y ( v ) E L ( Q ) .
5
1
LIONS and
L e t us consider now the c o s t f u n c t i o n J(v) = [y(X,T;V)-Zd12 dx + N V2 d t ,
IT
R
(5)
0
2 where Zd i s given i n L ~ ( R )and N > 0. I n general, f o r v E L (o,T), y ( * ,T;v) i s d e f i n d b u t as an element o f H-l(Q) (Sobolev space o f order -1) and n o t an e l e m $R). Therefore ( 5 ) does n o t make sense f o r v E Lz(0,T). One h a s X e n t o r e s t r i c t J t o those v ' s such t h a t V E
L2 (0,T) and y(*,T;v)
E
L2(Q).
(6)
This defines a H i l b e r t space (when provided w i t h the norm ( say U, and i f
we consider again problem ( 3 ) . I n order t o proceed i t i s necessary t o study U, n o t only t o make things more precise b u t a l s o because the dual U ' o f U i s needed f o r w r i t i n g the o p t i m a l i t y system. One v e r i f i e s t h a t U coincides w i t h the s e t o f those v ' s i n LZ(0,T) such t h a t
l',:
( 2 T - ( t t ~ ) ) - ~ " v ( t ) v ( x ) d t ds
O. We consider the "no-constraints" problem, namely i n f J(v,z)
, v,z
subject t o (1.1) ...( 1.4).
1.2. O p t i m a l i t y system. I t i s a simple matter t o check t h a t problem (1.7) admits a unique s o l u t i o n u,y. We want t o characterize u,y ; t h i s c h a r a c t e r i z a t i o n i s given by the f o l l o w i n g r e s u l t : there e x i s t s a unique s e t {u,y,pl o f f u n c t i o n s such t h a t
OPTIMAL CONTROL OF NON-WELL-POSED DISTRIBUTED SYSTEMS
m(t)Ay = u
t
,-
3
t
m(t)Ap = y-zd
7
,
(1.8)
y = p = 0 on 1,
(1.9) (1.10)
Y(x,O) = 0, p(x,T) = 0 YyPE
(1.11)
L2(Q)
ptNu = 0
(1.12)
One can o f course e l i m i n a t e u by (1.8) (1.12). The system i n u,y,p o p t i m a l i t y system ; p i s a Lagrange m u l t i p l i e r . .
i s called the
The proof o f t h i s r e s u l t can be obtained as f o l l o w s ; one considers the penalized problem
(1.13) where where
2 2 L (Q), Z E L (Q), > 0 i s "small".
V E
E
$
t
2 m(t)AzE L ( Q ) and z(x,O)
= 0 ; z = 0 i n 1, and
Then (1.14)
= J,(u,,Y,).
i n f J,(v,z)
One defines p, by 1 P, = (Tt mAy,
-
-
u),
.
(1.15)
One v e r i f i e s t h a t
I-
t
p,(x,T)
m(t)Ap, = 0
, p,
y,-zd
i n Q,
(1.16) = 0 on 1
and t h a t pE
+ NU,
(1.17)
= 0 i n Q.
It follows from (1.13) and (1.17) t h a t
uEyyE,pE remain
, as E + O , i n a bounded s e t o f ( L2 ( Q ) )3 ,
(1.18)
and t h a t
T e r e f o r one can pass t o the l i m i t i n E -+ 0. One v e r i f i e s t h a t u,,yE L! (9)! x $L (9) and t h a t u,y,p s a t i s f y the o p t i m a l i t y system.
+
u,y i n
the
One can w r i t e a v a r i a t i o n a l formulation f o r the problem, as f o l l o w s : p i s solution of 1 (1.20) t m(t)AP, t m(t)Aq) t (psq) = -(zds t mAq) vq (2 2 (where (p,q) = p q dx d t ) where q E L ( Q ) , t m(t)Aq E L ( Q ) , q(x,T) = 0,
1,
-2
a
8
-#
8
J.-L. LIONS
1.3. We are now going t o show how t o "uncoup1e"the o p t i m a l i t y system.
The technique i s s i m i l a r t o the one i n J.L. LIONS C 11 b u t one deals now w i t h s o l u t i o n s , due t o the f a c t t h a t the " s t a t e equation" i s n o t w e l l posed.
weak
One considers a problem analogous t o (1.8) ...( 1.12) b u t i n t h e i n t e r v a l Is,lC O<s 0).
operties of regularity o f the l i m i t of PE depends on the sign H - l ( R ) ; Hk(R)) ( resp. P ~ l e ( H - l ( n ); H - l ( n ) ) ) i f v < O (resp.
T-t )6(~-6). If u=O, P(X,S,t) = fl ( t h -
Remark 1.3 : One can also consider the equation
z t A z = v
(1.33)
at
where Az = - a ( x az ) on R = R , with z(x,O) = 0 ( t h i s i s again a non well posed problem). The Zorresbonding Riccati's type equation will be : tm
P(x,S,t) 2. 2.1.
-
=
P(S4,t)
.
.
-m
Control of unstable systems. Distributed control w i t h o u t c o n s t r a m .
We are given v,z such t h a t
v,z
L ~ ( Q x)
L?Q) ,
(1.34)
10
J:L. LIONS
(z(x,o)
o
i n R, z =
o
on C . 2 Remark 2.1 : Given v i n L ( Q ) , problem ( 2 . 2 ) does n o t necessarily adm t a globa s o l u t i o n i n time. The s o l u t i o n , defined l o c a l l y , can blow up. =
I
Remark 2.2 : I f Z E L 6 ( Q ) then i t f o l l o w s from (2.2) t h a t az
- AZ
3
2
= V t Z E L (9) d
which imp1ies , w i t h the boundary conditions i n (2.2)
, that
n
The c o s t f u n c t i o n J(v,z)
i s given by
6 where zd i s given i n L ( Q ) and where N>O. I n the case w i t h o u t c o n s t r a i n t we want t o f i n d i n f J(v,z),
v,z subject t o ( 2 . 1 ) ( 2 . 2 ) .
One has f i r s t problem (2 . 6 ) admits a s o l u t i o n {u,y}. (There i s no reason why t h i s s o l u t i o n should be unique). For the proof, we consider a minimizin sequence vn,z5. By v i r t u e o f (2.5) a VnsZn r e a i n i n a bounded set o f L 2 ( Q ) x L ( Q ) . Then Vntzn remains i n a bounded s e t o f L ( Q ) SO t h a t Zn remains i n a bounded s e t of H ~ Y ~ ( Q Ther ). f o r e one can e x t r a c t a subsequence, ) ) s t i l l denoted by VnaZny such t h t v -P u i n L ( Q ) weakly, zq -t y i n H ~ Y ~ (nQL6(Q weakly. Since, i n p a r t i c u l a r , H$.1(1) + L ( 9 ) i s compact, Zn * y 3 i n L2(Q ) weakly and therefore
i
?
5
8-
AY
-
y3 = u, y(x,o)
One can then show t h a t p 6 L 2(Q ) and
-
Ay
-
y3 = u,
~ ( ~ 3 0 0, ) p(x,T)
=
if u,y
-3-
Ap
o , y=o on c.
rn
i s a s o l u t i o n o f (2.6) then t h e r e e x i s t s p such t h a t
-
2 5 3y p = (y-zd)
,
= 0,
(2.9)
y = p = 0 on X,
ptNu = 0 i n Q For the proof, one considers, as i n Section 1.2,
(2.10) the penalized problem
OPTIMAL CONTROL OF NON-WELL-POSEDDISTRIBUTED SYSTEMS
for
L2 ( Q ) ,
Ve
Zc
L6 ( Q ) , Z c H’J(Q),
11
z ( o ) = 0, z l C = 0.
We c ons ider i n f J€(V,Z) and we denote b y v,,yE t h i s p e n a l i z e d problem. We s e t
a s o l u t i o n (which e x i s t s ! ) o f (2.12)
We have 2 ~Y,P,
- at -
AP,
-
p,(x,T)
= 0
, p,
= (Y,-zd)5 = 0
on
i n Q, (2.13)
C,
and p,t
NU, = 0.
(2.14)
Therefore pE =-NU, it follows t h a t 3y,p,2
(y,-zd)5
t
s o t h a t p,
i s bounded ( a s E + O )
i n L2 (Q) ; s i n c e y,
i s bounded i n L6 (Q),
i s bounded i n L 6 l 5 ( Q ) hence t h e r e s u l t w i l l f o l l o w
i s bounded i n W 291;6/5(Q),
.b
2.2. D i s t r i b u t e d c o n t r o l w i t h c o n s t r a i n t s . L e t us t a k e a gain t h e s i t u a t i o n ( 2 . 1 ) ( 2 . 2 ) traints VE
Uad
, Uad
= c l o s e d convex subset o f L
o f S e c t i o n 2.1 ; we add now t h e cons2 (Q),
(2.15)
and we make t h e ( s t r o n g ) assumption : uad has a non empty i n t e r i o r .
(2.16)
We a l s o assume t h a t t h e s e t o f { v , z l n o t empty.
such t h a t (2. 1)(2. 2)(2. 15)
take place
,-&
By a s u i t a b l e a d a p t a t i o n of an i d e a o f
P. RIVERA C11 one c a n a g a i n show t h e e x i s tence o f a Lagrange m u l t i p l i e r p f o r a s o l u t i o n u,y o f J(u,y)
= i n f J(v,z),
s a t i s f y (2.1)(2.2)(2.15). 2 More p r e c i s e l y , t h e r e e x i s t s P E L (9) s a t i s f y i n g (2. 8) (2.9) and (ptNu,v-u)
’0
v,z
v v 6 uads
E
uad
.
(2.17)
(2.18)
2.3. Boundary c o n t r o l . L e t us assume now t h a t v and z s a t i s f y v E ~ ~ ( , 2z E) L ~ ( Q )
(2.19)
12
J.-L. LIONS
_ a~ - az at az
--i-
av
z3 =
o
i n Q,
v on 1,
(2.20)
z(x,O) = 0 i n R . If z
E
L6(Q) then
_ a' - A Z at
= z
3 E
2 L (Q)
so t h a t (we use :fractional"
(2.21) Sobolev spaces, as, f o r instance, i n J.L. LIONS,
E. MAGENES C 11) 'ZE
L2 ( o , T ; H ~ / ~ ( R ) );
(2.22)
then (2.21) gives az 2 T
L~ ( 0 ,T;H-~/* (0)
(2.23)
so t h a t z i s (a.e. equal t o a function) continuous from C O Y T I therefore
+
H112(.Q)
, and (2.24)
z 6 L~(o,T;H~/~(R)). The cost f u n c t i o n i s given by
(2.25) We want now t o f i n d
I
V E
, for
v,z subject t o (2.19)(2.20) 2 uad = closed convex subset o f L ( C ) ;
i n f J(v,z)
and (2.26)
we assume t h a t the s e t o f v,z s a t i s f y i n g these c o n s t r a i n t s i s n o t empty. This problem admits a s o l u t i o n Cu,y) , n o t necessarily unique.
I n order t o o b t a i n an o p t i m a l i t y system (and a Lagrange m u l t i p l i e r ) we s h a l l assume ( f o r technical reasons explained below) t h a t 2 RcR ,
(2.27)
and t h a t Zd E L10(o~T;t.6(fi))(1)
(2.28)
We have then t h e existence o f p such t h a t p c L ~ ( o , T ; L ~ ( R )n) L ~ ( o , T ; H ~ ( R ) )
(2.29)
( * ) I t would be s u f f i c i e n t t o assume t h a t z d c L6(Q) and such t h a t t h e r e e x i s t s B > o Such t h a t Zd6 L10(OJ;L~'~(R)).
OPTIMAL CONTROL OF NON-WELL-POSED DISTRIBUTED SYSTEMS
- AY - y 3 = o , - 8 - AP - 3y2p
13
(y-zd)5 i n Q, (2.30)
\y(x,o)
c
0, ~ ( x , T ) =
=
(p+NU)(V-U) dc
o 0
i n R, Vve uad
U E
uad.
(2.31)
L e t us sketch t h e proof. We s t a r t from t h e p e n a l i z e d problem : (2.32) VE
L2 ( I ) , Z
z(x,O)
= 0 and
where
Then
E
L6
az
(a), $ - Azc L2 (Q), = v on 1.
z s a t i s f i e s (2.22)(2.23)(2.24).
The problem
, VE
i n f J,(v,z)
uad and v,z as above
admits ( a t l e a s t ) a s o l u t i o n uEYyE. As
2 remain i n a bounded s e t o f L ( C )
u, ,y 1
E
(K 5% -
- y,)3
Ay,
(2.33) +
x
0 we have
6 L (9) ,
remains i n a bounded s e t o f L2
(2.34)
(a).
(2.35)
We i n t r o d u c e (2.36) We v e r i f y t h a t
12
=
0 on C,
P,(x,T)
=
o
in
(2.37)
n
and (2.38) I t f o l l o w s from (2.35)(2.34)
-
Ayf = f f
that
, fE bounded
2
i n L (Q),
(2.39)
and we have
av = u,
on C, y,(x,O)
= 0 i n R.
Therefore ( c f . (2.22) (2.23) (2.24) )
(2.40)
14
J.-L. LIONS
y,
remains i n a bounded s e t o f L2 (0,T;H3/2(R)
The re f o re , f o r e v e r y 0 such t h a t 0 s 9 y,
nLm(0,T;H1/2(R)).
(2.41)
1, one has 1/2+0 remains i n a bounded s e t o f L2le(O,T;H (0)). 5
B u t - s i n c e t h e space dimension e q u a l s 2 H1'2+9(Q)cL 6 ( Q ) i f 0 = 1
-
(2.42)
one has
.
(2.43)
Then (2 . 4 2) shows t h a t y,
remains
i6
a bounded s e t o f L
We s h a l l v e r i f y below t h a t p, p,
12
6
(0,T;L
(2.44)
(R)).
satisfies
remains i n a bounded s e t of Lm(O,T;L
2
(a)) n L2 (O,T,H 1(0)).
(2.45) 2
(F)
xL6(Q)x Th5n one fan pass t o t h e l i m i t . One s h o w s ' t h a t uE,YE,PE * u,y,p i n L X L (0,T;H ( Q ) ) weakly ( a n d a l s o i n weak t o p o l o g i e s o r weak s t a r t o p o l o g i e s c o r r e s ponding t o (2 . 4 1)(2 . 4 4 ) ( 2 . 4 5 ) ) , where u,y i s a s o l u t i o n o f (2.26) and where (2.29) (2.30) (2.31) h o l d t r u e .
I n o r d e r t o v e r i f y (2.45) l e t us change t i n t o T - t and l e t us s e t $ ( x y t ) = p,(x,T-t) (y,(X,T-t)-Zd(X,T-t))
42- y,(xyT-t) 5 = h,
= o,(xyt)
.
,
We have
I
(2.46)
Y! av
=
o
We s e t :
on C, ~ ( X , O )=
o
1$12
,ll~111~ = J
$2 dx
R
IV$12dx +
R
R
1$12
dx.
Tak ing t h e s c a l a r p r o d u c t o f (2.46) w i t h 9 , we o b t a i n I d Ta l W 2 + I l 9 ( t ) 1 I 2 - l W l 2 - (n$N (where ( f ,I)) =
1
R
f 9 dx) -
.
= (h&
(2. 47)
But
so t h a t (and t h i s i s v a l i d i n dimension n s 3 )
(2.48)
(where t h e C's denote v a r i o u s c o n s t a n t s ) . We have a l s o
OPTIMAL CONTROL OF NON-WELL-POSED DISTRIEUTED SYSTEMS
15
so t h a t
(2.49)
(2.50)
1 remains i n a bounded s e t o f L (0,T)
, i t follows L6W from (2.51) and Gronwall’s i n e q u a l i t y t h a t J, remains i n a bounded s e t o f Lm(O,T; i.e. (2.45). m L2(R)) n L2 (O,T;Hl(R)) and since i n particularllQ,ll
2.4. Various remarks. Remark 2.3 : By estimates analogous t o those o f S e c t i o n 2.3, one sees t h a t t h e Section 2.2 i s v a l i d w i t h uad n o t n e c e s s a r i l y s a t i s f y i n g (2.16) i f Remark 2.4 : L e t us consider the equation
at -
AyE
-
y,
3
+
EY,
ayE
av = 0 on I, y,(x,O) For
E
5
= v in
Q, (2.52)
0 i n R.
> 0 t h i s problem admits a unique s o l u t i o n y,(v).
One can then d e f i n e (2.53)
Then one can v e r i f y t h a t :
l i m ( i n f J E ( v ) ) = i n f J(v,z), E+O
v,z s u b j e c t t o (2.1)(2.2)(2.15).
(2.54)
VEUad
Remark 2.5 : One can o b t a i n s i m i l a r r e s u l t s f o r o p t i m a l i t y systems connected w i t h s t a t i o n a r y problems such as -AZ
- z’
= v in
R, (2.55)
and
18
J.-L. LIONS
BIBLIOGRAPHY
A.V.
FOURSIKOV C11 On some problems o f c o n t r o l . Doklady Akad. Nauk, 1980, pp. 1066-1070.
J.P. KERNEVEZ, J.L. LIONS and D. THOMAS C 11 To appear. J.L. LIONS 111 Sur l e c o n t r a l e optimal des systemes gouvernes par des equations aux derivees p a r t i e l l e s . Paris, Dunod-Gauthier-Villars, 1968 ( tn g l i.s h t r a n s l a t i o n by S.K. MITTER, Springer, 1971). C21 3ome aspects of the optimal c o n t r o l o f d i s t r i b u t e d parameter sysReg. Conf. Ser. i n Applied Math., S I A M 1 6 , 1972.
tems.
C31 Remarks on the theory o f optimal c o n t r o l o f d i s t r i b u t e d systems, i n Control Theory of Systems Governed by P a r t i a l D i f f e r e n t i a l tquations, ed. by A.K. A Z I Z , J.W. WINGATE, M.J. BALAS, Acad. Press, pp. 1-103. C41 Function spaces and optimal, c o n t r o l o f d i s t r i b u t e d systems. U.F.R. J . Lecture Notes, Rio de Janeiro, 1980. C51 Some methods i n t h e Mathematical Analysis o f systems and t h e i r conScience Pub. Company, BEIJING, 1981.
trol.
J.L. LIONS and E . MAGENES C11 Problemes aux l i m i t e s non homogenes e t a p p l i c a t i o n s , Vol 1,2. Dunod, Paris, 1968 English t r a n s l a t i o n by P. K t N " H , Springer Verlag , 1972.
.
.
P. RIVERA C11 To appear
L. SCHWARTZ C11 Theorie des noyaux. Proc. I n t . Congress o f Math. 1 (1950) 230.
J . SIMON C11 To appear.
, pp.
220-
Nonlinear Problems: Present and Future
17
A.R. Bishop, D.K. Campbell, 8. Nicolaenko Ids.) 0 North-Holland Publishing Company, 1982
COMPACTNtSS AND TOPOLOGICAL METHODS FOR SOME NONLINEAR
VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS P i e r r e L.
LIONS
Laboratoi r e d ' Analyse Numerique U n i v e r s i t e P. e t M. Curie 4, place Jussieu, 75230 Paris Cedex 0.5 FRANCE A general method t o solve some nonlinear v a r i a t i o n a l problems o f Mathematical Physics i s i l l u s t r a t e d on two examples : the so-called Hartree and Choquard equations. This method i s based upon t h e use o f , f i r s t , c r i t i c a l p o i n t theory and, second, symmetries i n order t o gain compactness. INTRODUCTION
We want t o present here some general methods t o solve some nonlinear v a r i a t i o n a l problems o f Mathematical Physics. I n general terms, we emphasize the use o f c r i t i c a l p o i n t theory i n order t o prove existence and m u l t i p l i c i t y r e s u l t s . But, since c r i t i c a l p o i n t theory needs some form o f compactness ( P a l a i s - h a l e c o n d i t i o n f o r example) and since i n many problems o f Mathematical Physics the N problem i s s e t i n an unbounded domain ( l i k e IR f o r example) compactness may be lacking : t o avoid t h i s d i f f i c u l t y we w i l l show below t h a t using t h e symmetries o f the problem, we may o b t a i n some form o f compactness. These general (and vague) coments w i l l be i l l u s t r a t e d here on two problems : i n section I below, we consider the general Hartree equation (and r e l a t e d questions)
u of :
t h a t i s we look f o r s o l u t i o n s
- Au -
-,$,.
u
t Xu
+
(I
2
dy)u = 0
u (x-y) IR3
u
E
H1( R3) ( l ) , u f 0
( o f course we want n o n - t r i v i a l s o l u t i o n s : the equation). Here X
E
IR, z
>
u
f 0, since u i 0 i s a s o l u t i o n o f
0 ( z i s the so-called t o t a l charge). As i t w i l l be
indicated below i n s e c t i o n I,we c a n t r e a t a s w e l l general Coulomb p o t e n t i a l s 3 zi w i t h xi E R , zi > 0 instead o f ), systems, Hartree-Fock equa-
'f
Ix-xilr
-&
t i o n s and r e l a t e d questions, I n section I below, we g i v e general r e s u l t s ( t h a t we
18
P. L. LIONS
b e l i e v e a r e o p t i m a l ) concerning e x i s t e n c e of m u l t i p l e s o l u t i o n s : those r e s u l t s a r e obtained by a simple use o f c r i t i c a l p o i n t theory. Section I 1 i s devoted t o the study o f t h e s o - c a l l e d Choquard e q u a t i o n , t h a t i s we look f o r solutions u o f
where
X E
IR. I n s e c t i o n 11, we g i v e optimal e x i s t e n c e and m u l t i p l i c i t y r e s u l t s
which a r e o b t a i n e d u s i n g , i n p a r t i c u l a r , c r i t i c a l p o i n t theory. A t t h i s p o i n t , l e t us make a remark on ( 2 ) : i f u i s a s o l u t i o n of ( 2 ) c l e a r l y un(x) = u ( x t nE)
1, 151 = l ) l i s a l s o a s o l u t i o n o f ( 2 ) , b u t un converges weakly ( f o r 2 o r i n H ) t o 0 and t h i s convergence cannot be s t r o n g ( s i n c e f o r 2 2 e x a m p l e t h e L n o r i n o f u n i s e q u a l t o t h e L normofI;). I n o t h e r w o r d s we have a se(with n
2
example i n L
quence of s o l u t i o n s ( w i t h t h e same norms i n every,say,
Sobolev space)
that i s not
compact ( i n any Sobolev space) : t h i s prevents t h e d i r e c t a p p l i c a t i o n o f c r i t i c a l p o i n t t h e o r y ( l a c k o f compactness). As i t w i l l be seen i n s e c t i o n 11, i n o r d e r t o be a b l e t o use c r i t i c a l p o i n t t h e o r y we w i l l use t h e symmetries o f problem ( 2 ) - t h a t i s o f course t h e s p h e r i c a l symmetry
-
t o o b t a i n some form of compactness.
F i n a l l y i n s e c t i o n 111, we g i v e b r i e f l y a l i s t o f o t h e r problems t h a t can be t r e a t e d by s i m i l a r methods ( i n c l u d i n g N o n l i n e a r S c a l a r F i e l d equations, R o t a t i n g Stars, Thomas-Fermi Problems, Vortex Rings, ...). We a l s o p r e s e n t some nonexistence r e s u l t s showing t h a t t h e symmetries of ( i n p a r t i c u l a r ) problem ( 2 ) a r e n o t o n l y e s s e n t i a l f o r t h e proof of e x i s t e n c e r e s u l t s b u t d i r e c t l y f o r e x i s t e n c e purposes. L e t us f i n a l l y mention t h a t t h e r e s u l t s presented i n s e c t i o n I a r e taken from P. L. L i o n s [26],
w h i l e those o f s e c t i o n I11 a r e from P. L. L i o n s C271. F i n a l l y
t h e s o l u t i o n o f a l l the problems l i s t e d i n s e c t i o n I 1 1 can be found i n H. B e r e s t y c k i and P. L. L i o n s [8],[91,[10], [20],[21]
P.L.
L i o n s [281, M.J.
Esteban and P. L. L i o n s
; and a systematic account o f a l l these problems i s g i v e n i n P. L. L i o n s
C291.
I. THE HARTREE EQUATION AND RELATED QUESTIONS L e t us r e c a l l t h e problem : f i n d s o l u t i o n s u o f
SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS
19
t h i s problem i s some k i n d o f model problem f o r general Hartree o r Hartree-Fock equations (see, f o r example, E. H. Lieb and B. Simon C251 f o r the relevance of such problems t o Physics) : these equations were introduced by D. Hartree C231 (see also J . C. S l a t e r C361) as an approximation method t o describe the Coulomb Hamiltonian o f electrons i n t e r a c t i n g w i t h s t a t i c
nucleii.
REMARK 1.1 : We could t r e a t as w e l l general Hartree o r Hartree-Fock systems (see
P. L. Lions [26],[29] f o r more d e t a i l s ) : i n p a r t i c u l a r we could replace the
Coulomb po t e n t i a 1
- -,$,-
by very general p o t e n t i a l s V(x) i n c l u d i n g f o r example Z
x.I (Xi E IR3 , zi E R , zi > 0 ) . i+ We w i l l i n d i c a t e below how our main r e s u l t s modify i n t h i s case (see P. L. Lions the general CbuTamb p o t e n t i a l : V(x) =
-
Z
C26IyC291 f o r more d e t a i l s ) . F i n a l l y we could consider t h e general Thomas-Fermil Von Weizacker equations (see R. Benguria, H. Brezis and E.H. L i e b [7 I f o r the i n t r o d u c t i o n o f these equations) :
{u
E
*i
-;I
Au
H1( R 3 ) ,
2 * "u1
u t B(u) t Au t ( u
= 0
in I R ~
u $ 0 ;
where x E R , zi > 0, xi E R 3 and B i s some increasing f u n c t i o n s a t i s f y i n g B ( 0 ) = B ' ( 0 ) = 0. The r e s u l t s are t o t a l l y s i m i l a r t o those concerning (1). ( 2 ) . REMARK I . 2 : An important r o l e w i l l be played by theeigenvalues o f the l i n e a -
r i z e d problem (around 0) namely by the r e a l (s) A such t h a t there e x i s t s u f 0 solution o f :
- Au -
z
,-$ t A u
= 0,
u
E
H1( R3).
(4)
I t i s well-known (see f o r example E.C.Titchmarsh C401) t h a t t h i s occurs i f a n d o n l y
2 i f A = x
k= ?
(k
2
1) and f o r such A = A k there e x i s t s o n l y one
m u l t i p l i c a t i v e constant) s o l u t i o n of (4). We may now s t a t e our main r e s u l t :
Vk
(up t o a
P.L. LIONS
20
THEOREM 1.1 :
i) ii)
If
X < 0
If, xktl
d
5
x
S(v) =
for a l l
2
l ) , there e x i s t a t l e a s t
< S(U2)
5
... 5 S(Uk)
IR3 - VI 1 1DvI2
1 3 H (R )
v
i i i ) j ~ -x = 0
k
.
distinct pairs
Al
< hk ( k
(+u.)
S(Ul)
f
of
e x i s t s a secquenceof d i s t i n c t u a i r s ( i Uk)kr
that (5)
S(u2)
lktrtl , one has i n f a c t f o r
REMARK 1.5 --
, zi
R3
i
- A t V
X
< Xk
Xk-l
> X k = Xktl
=
...
( k t r ) d i s t i n c t pairs o f solutions)
.
: Hartree equations have been studied
M. Reeken C341
(If
by many authors (see f o r example
, N. Bazley and R.Seyde1 C51 , N. Bazley and B. Zwahlen C61 ,
21
SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS
K. Gustafson and D. Sather [21],
C. A. S t u a r t [38],[39],
J. Wolkowisky [41]
b u t a l l these works e i t h e r g i v e l o c a l r e s u l t s (A near Al) o r use h e a v i l y t h e
-
spherical symmetry and reduce the problem t o some O.D.E. question in p a r t i c u l a r do n o t y i e l d any information i n t h e case o f a general Coulomb potential.
The most general known r e s u l t s
-
symmetry
-
i n absence o f spherical
were those o f E. H . Lieb and B..Simon [25] b u t were o n l y concerned
w i t h the existence o f ground s t a t e s o l u t i o n s ( e s s e n t i a l l y those s a t i s f y i n g (5)). We w i l l only sketch the proof o f Theorem 1-1, t h e d e t a i l e d p r o o f may be found i n [26].
REMARK
1-6:
that i f 0
5
Using a method due t o N. Bazley and R. Seyde [5], A < A1,
min
one can prove
t h e s o l u t i o n o f the minimization problem: S(v)
vE H i s unique (up t o a change o f sign) and i s thus u.l I n a d d i t i o n i t i s obvious t o deduce t h a t u,l
0 < A 5 A1 and t h a t as A
+
{ u E L6(J?),
and weakly i n
Du E L2($))
0, ul(A)
as a f u n c t i o n o f A, i s C 1 f o r
converges t o ul(0)
i n norm i n t h e space
H1(d).
We now i l l u s t r a t e Theorem 1-1 by a few " b i f u r c a t i o n diagrams": emphasize t h a t these diagrams are more o r l e s s formal. introduce some notations: t h a t Du E L 2 ( d ) , u E L6(& Du(x).Dv(x)dx.
0
....&-- A-J
let
X
l e t us
F i r s t we need t o
be t h e H i l b e r t space o f f u n c t i o n s
u
equipped w i t h t h e s c a l a r product ( ( u , ~ ) )
=
such
The b i f u r c a t i o n diagram i n X looks l i k e Fig. 1. below:
%gure
1.
But now; i f we look a t t h e b i f u r c a t i o n diagram i n H1 ( o r L2) we f i n d the f o l l o w i n g Fig. 2:
22
P.L. LIONS
Figure 2 The d i f f e r e n c e between these two b i f u r c a t i o n diagrams i s due t o the f a c t t h a t f o r A = 0, w h i l e uk
3 0 ,
Iu I
i s bounded away from 0.
L2 2z
We have proved t h a t :
'I2
(V k
2
1) and we conjecture t h a t f o r
A = 0 we have:
lu I
>
z1'2
(V k
1) and
L2 F i n a l l y l e t us mention t h a t i n [25], proved t h a t l u
lim lu I = z 1/2 k+@ L2 [7] (by two d i f f e r e n t methods) i t i s
> z 1/2.
I 12
L
REMARK 1-7: C l e a r l y t h e r e i s b i f u r c a t i o n from each eigenvalue Ak (which i s t o be expected i n view o f the general r e s u l t s o f M. G. Grandall and P. H. Rabino-
[la]).
A n i c e phenomenon i s what happens f o r t h e branches near A = 0: i n branches stop when they h i t the continuous spectrum o f - A ( ( A 5 0)). More p r e c i s e l y i t can be proved (P. L. Lions [29]) t h a t 1 u1 (=ul(A)) i s a C curve i n H1 and t h a t f o r a l l 0 < A < A1 the l i n e a r l i z e d oberator AA around ul(A) i.e. :
witz
some
sense t h e
H 1 ~ V + - A V - & V + A V + ( U 2~
1 * T;;T'v
+ 2
((up) *
i s coercive and thus i s an isomorphism from H1 onto H-'.
But f o r A = 0,
SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS
23
A. i s o n l y i n j e c t i v e and i s n o t o n t o and t h i s e x p l a i n s why t h e branch cannot be "extended i n t h e s e t { h < 0)".
We now t u r n t o t h e p r o o f o f Theorem 1-1: we w i l l o n l y g i v e some s k e t c h o f t h e p r o o f , f o r a d e t a i l e d p r o o f t h e r e a d e r i s r e f e r r e d t o P.L. L i o n s C261. Sketch o f t h e p r o o f o f Theorem 1-1 : We w i l l c o n s i d e r o n l y t h e case when 0 < h < h l ( t h e non e x i s t e n c e p a r t i s q u i t e easy and t h e case when 1 = 0 i s q u i t e t e c h n i c a l ) . As i t was s a i d b e f o r e , we w i l l a p p l y c r i t i c a l p o i n t t heory; and we w i l l j u s t i n d i c a t e what a r e t h e p r o p e r t i e s s a t i s f i e d b y S here which
enable us t o appl y
c r i t i c a l p o i n t theory.
S E C 1( H 1) s a t i s f i e s :
Therefore, we c l a i m t h a t
1) S(0)
, S i s even (and t h u s S '
0
=
we have on
3 ) S(u) 4) S
.+
t
my
BE : S(u)
,
s a t i s f i e s the Palais- h a l e condition : If (
u
~
E
)H'
~i s ~such~ t h a t : S(un) isbounded,
S'(un)
-1' 0 H
t h en t h e r e e x i s t s a subsequence
of (u,)
which converges i n
H1
.
llull
is
small t h en S(u) behaves e s s e n t i a l l y as i t s q u a d r a t i c p a r t which i s j u s t "
the
Property
1) i s o b v i o u s ; t o understand 2 ) l e t us j u s t remark t h a t i f
q u a d r a t i c f o rm a s s o c i a t e d w i t h ( - A
+A)
and t h u s 2 ) i s c l e a r ( r e c a l l t h a t
m zT) .
i s the j - t h eigenfunction o f - A To e x p l a i n 3), l e t us j u s t i n d i c a v j t h e n t h e main d i f f i c u t y t o check 3) l i e s w i t h t h e t e t h a t i f 1 > hj+l j
w~
subspace Q IR v 21 o f H1 which i s f i n i t e d i mensional. B u t on t h i s subspace i=l a l l norms a r e e q u i v a l e n t and t h u s t h e t e r m R3 d x ' d y w i l l be
jR3x
predominant a t i n f i n i t y . F i n a l l y 4), which i s t h e fundamental s t r a i g h t f o r w a r d t o check u s i n g 3 ) .
c o n d i t i o n i n C r i t i c a l P o i n t Theory, i s
To conclude we j u s t need t o a p p l y a r e s u l t due t o L j u s t e r n i k and
24
P.L. LIONS
Schnirelman [ 3 l ] ( f o r the precise statement, see P.H.
Rabinowitz 1331, o r Krasnoleskii [24] ) : conditions 1 ) - 4 ) imply t h e existence o f a t l e a s t k
t i n c t p a i r s o f . c r i t i c a l p o i n t s o f S,that i s s o l u t i o n s of (1) f o r X
c
dis-
Xk ,
REMARK 1-7 : I f one c o n s i d e r s t h e time dependent H a r t r e e e q u a t i o n t h a t i s :
where 0 are complex valued functions. Then solutions o f (1) provide s o l i t a r y waves
f o r (7) : 0(x,t)
= e- i h t uk(x)
.
I n T. Cazenave and P.L. Lions C171,it i s proved t h a t the o r b i t (,-iht 0 5 X < X1) i s stable. Ul(X) ) t 2 0
(for
11. THE CHOQUARD EQUATION AND RELATED QUESTIONS We now look f o r s o l u t i o n s
-
Au t Xu
-
u of
1 (u2 *m)u
1 3 u c H (R )
,
= 0
in
R3
u 8 0 .
We w i l l also consider t h e "normalized" problem : f i n d (A,u) s o l u t i o n o f
r where M
-
Au t Xu
-
(u 2 * m1 ) u = 0
i s given > 0 ( f o r example M
Equations ( 2 )
-
R3
in
1)
.
( 2 ' ) were introduced by Choquard and t h e existence o f a
p o s i t i v e s o l u t i o n d f ( 2 ' ) was proved by
E.H.Lieb C241
. Problem (2')
a l s o i n the so-called mean f i e l d theory (see M. Donsker and S.R .S.
Some other references where such problems are considered are given i n P.L. Lions C271. Our main r e s u l t s concerning (2) o r ( 2 ' ) a r e :
occurs
Varadhan [19]).
SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS
26
THEOREM 11.1 :
i) ii)
If
x
2
x
> 0
there i s no s o l u t i o n o f (1)
0
of
there e x i s t s a sequence o f d i s t i n c t p a i r s ( ' u ~ ) ~ ~
solutions o f (2) such t h a t = ~ ~ ( 1 x >1 )0 ; u1
ul(x)
0 < S(ul)
S(v)
5
i s decreasing i n
for a l l
k(x) = u k ( l x l )
v
(8)
1x1
solution o f ( 2 )
(9)
( V k 2 1)
(10)
THEOREM 11.2 :
There e x i s t s a sequence
, (10)
such t h a t ( 8 )
J(ul)
(xk,
+uk)
( f o r any M > 0)
o f d i s t i n c t solutions o f ( 2 ' )
hold and : = min{J(v)/vE H
1
, IvI
(9'1
= M} < 0
L
-1 4
0 (Yk
(11')
v2(y)lx-yl-' l),xk
2
2 L
fi
Duk-pO
k+O ++a dx dy
k z m O
,
(VV
E
H1)
uk-p 0
;
2
Lp
.
REMARK 11-1 : L e t us remark t h a t i n Theorem 11-1,s i s n o t bounded from below
t'qmm if v #
(and surely n o t from above) since
S(tv)
REMARK 11-2 : The existence of u1
i n Theorem 11.2.
ved by of :
E.H.
Lieb [24]
IvI 2 = My v L
E
H
E
H
.
( w i t h M = 1) has been pro-
. I n a d d i t i o n i t i s proved i n [24]
J ( v ) = minCJ(w)/w
0
t h a t any s o l u t i o n v
1 IWIL2 =
1
i s necessarily o f the form : v ( x ) =
some xo R 3 and where E = +1. can be found i n P.L. Lions C271.
, for
EU~(X-X~)
The d e t a i l e d proofs o f Theorems 11.1, 11.2
26
P . L LIONS
REMARK 11-3 : let
I f one c o n s i d e r s more g e n e r a l e q u a t i o n s t h a n ( 2 ) of t h e f o r m :
N> 2 -Au
where
V
t XU
-
(U
2
*V(lXl))u
satisfying :
f o r example i s some p o t e n t i a
v
Efl,(RN)
V
E
MP(RN)
t t
RN
in
V(x) = v ( l x 1 ) z 0
l s q < t my if N M q ( R N ) ( 3 ) f o r some T < p < q < t
t h e n Theorem 11-1 s t i l l h o l d s ( s e e P.L.
N
f o r some
Lq(RN)
5
m y
if
3 Nr4
(14)
L i o n s C271, i n C271 a r e a l s o g i v e n
c o r r e s p o n d i n g e x t e n s i o n s of ( 2 ' ) ) . I n p a r t i c u l a r , l e t us remark t h a t assumptions ( 1 3 ) , (14)
V =6,
in
R'
( t h e D i r a c mass a t 0) s a t i s f i e s
and we o b t a i n e x i s t e n c e o f m u l t i p l e s o l u t i o n s o f
-AutXu=u3
in
R3
,
1 3 u c H ( R ) ,
u
$
0 ;
t h i s had a l r e a d y been proved by s e v e r a l a u t h o r s (M.S. Berger C161, Z. N e h a r i C321, G.H.
Ryder C351
, W.
S t r a u s s C271). A l s o remark t h a t t h e above e q u a t i o n a r i s e s
when l o o k i n g f o r S o l i t a r y waves i n c u b i c n o n l i n e a r S c h r 8 d i n g e r e q u a t i o n s a r i s i n g i n o p t i c s ( s t u d y o f l a s e r beams). Before going i n t o
t h e p r o o f s o f Theorems 11-1,
11-2, l e t us g i v e t h e
b i f u r c a t i o n diagram c o r r e s p o n d i n g t o Theorem 11-1 ( w h i c h can be made r i g o r o u s ) :
a,
>
A
F i g u r e 3. I n p a r t i c u l a r we see t h a t t h e r e i s b i f u r c a t i o n f r o m (0,O) o f a c o u n t a b l e number o f branches
Gk
= {(X,uk(A))1 : t h i s phenomena occurs i n some n o n l i n e a r problems
when t h e r e i s some c o n t i n u o u s spectrum ( s e e a l s o H. B e r e s t y c k i and P . L . L i o n s Clll f o r some o t h e r example).
27
SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS
We now t u r n t o t h e p r o o f s ofTheorems 11.2, 11.3 : a g a i n we w i l l o n l y s k e t c h it.
Sketch o f t h e p r o o f o f Theorem 11.1 : We w i l l o n l y c o n s i d e r t h e case when X
>
0
( t h e non e x i s t e n c e p a r t i s q u i t e easy). Again we w i l l j u s t i n d i c a t e ( w i t h o u t p r o o f ) what a r e t h e p r o p e r t i e s s a t i s f i e d by S which e n a b l e u s t o use c r i t i c a l p o i n t t h e o ry.
We c l a i m t h a t S
E
1 1
C (H ) s a t i s f i e s :
1 ) S(0) = 0, S i s even ( a n d t h u s S ' ( 0 ) = 0 ) . 2)
]E
, 6 > 0 such
ifI I v l k l s E, v # 0
t h a t S(v) > 0
3 ) For any f i n i t e dimensional subspace X o f H1, we have : S(u)
+
-
m,
if u
E
X, I J u11 H
-+
+
m.
4) S s a t i s f i e s t h e Palais-Smale c o n d i t i o n i n H:
= (v
E
H1,
v ( x ) = v ( ( x 1)
I.
P r o p e r t i e s 1) and 2) a r e c l e a r ; and p r o p e r t y 3) i s e a s i l y deduced f r o m t h e f a c t t h a t on X a l l norms a r e e q u i v a l e n t . Now 4) i s s m e k i n d o f compactness p r o p e r t y t h a t we w i l l e x p l a i n below. I f one admits t h e s e p r o p e r t i e s , t h e n a p p l y i n g a c r i t i c a l p o i n t theorem i n t h e 1 space Hr due t o A. Ambrosetti and P.H. R a b i n o w i t z C11, m e o b t a i n s ( e s s e n t i a l l y )
t h e e x i s t e n c e o f s o l u t i o n s as i n Theorem 11.1.
To conclude t h i s s k e t c h o f p r o o f , we would l i k e t o emphasize t h e f a c t t h a t S does n o t s a t i s f y t h e Palais-Smale c o n d i t i o n i n H 1 ( i n p a r t i c u l a r because o f t h e d i f f i c u l t y due t o t r a n s l a t i o n s as mentioned i n t h e I n t r o d u c t i o n ) : b u t r e s t r i c t i n g o u r s e l v e s t o t h e f u n c t i o n s having a l l t h e symmetries o f t h e problem ( h e r e t h e s p e r i c a l s y m e t r y ) , we o b t a i n some compactness. A key lemma i n t h e p r o o f o f p r o p e r t y 4) i s t h e f o l l o w i n g L e m a which i s a consequence o f r e s u l t s due t o C371 (see a l s o [81, C291) :
1 LEMMA 11.1 : The r e s t r i c t i o n t o Hr o f t h e Sobolev embedding f r o m
i s compact f o r 2
< p
i I D u l 2dx =
and m i n i m i z i n g ,
1Du12 = q. t h e f u n c t i o n a l :
-
wr2
R2, r
IT
k)+)dx
01, w
(22)
> 0, k 2 0, q > 0 a r e g i v e n ; F i s a f u n c t i o n
s a t i s f y i n g : F ( 0 ) = 0. Using t h e symmetry ( w i t h r e s p e c t t o z ) o f t h e problem, i n H. B e r e s t y c k i and P.L. Lions 1101, t h i s problem i s s o l v e d under o p t i m a l c o n d i t i o n s .
5) General n o n l o c a l problems : Here, one wants t o f i n d t h e s o l u t i o n s o f : K
*
u = f ( u ) i n l RN
where K ( x ) = K ( l x l ) , f i s a g i v e n n o n l i n e a r i t y s a t i s f y i n g : f ( 0 ) = 0. T h i s problem i s s o l v e d i n P.L. L i o n s C291. Footnotes : (1) H
1
m3
= {U
E
L 2 P 3 ) , Du
E
L2(R3)}
( 2 ) The a u t h o r would l i k e t o thank E.H.
Lieb f o r having pointed o u t t o h i s a t t e n t i o n
equ at io n ( 3 ) . ( 3 ) MP(!RN) denotes t h e u s u a l M a r c i n k i e w i c z space.
References C11
Ambrosetti, A. and Rabinowitz, P.H'.
, Dual v a r i a t i o n a l methods i n c r i t i c a l
p o i n t t h e o r y and a p p l i c a t i o n s , J. Funct. Anal. 14 (1973) 349-381. C21
Auchmuty, J.F.G.,
E x i s t e n c e o f axisymmetric e q u i l i b r i u m f i q u r e s , Arch. Rat.
Mech. Anal. 65 (1977) 249-261. C31
Auchmuty, J.F.G.
and Beals, R.,
V a r i a t i o n a l s o l u t i o n s o f some n o n l i n e a r
32
P.L. LIONS
free boundary problems, Arch. Rat. Mech. Anal. 43 (1971) 255-271. C43
Bader, P., Variational method for the Hartree equation of the Helium atom, Proc. Roy. SOC. Edim. 82 A (1978) 27-39.
C51
Bazley, N. and Seydel, R . , Existence and bounds for critical energies of the Hartree operator, Chem. Phys. Letters 24 (1974) 128-132.
C61
Bazley, N and Zwahlen, B., A branch of positive solutions of nonlinear eigenvalue problems, Manuscripta Math. 2 (1970) 365-374.
C71
Benguria R., Brezis, H. and Lieb E.H., The Thomas-Fermi-von Weiz'dcker theory o f atoms and molecules, to appear in Comm. Maths. Phys.
C81
Berestycki, H. and Lions, P.L., Existence of solutions for nonlinear scalar field equations. I. The ground state, to appear in Arch. Rat. Mech. Anal.
C93
Berestycki, H. and Lions, P.L., Existence of solutions for nonlinear scalar field equations. 11. Existence of infinitely many bound states, to appear in Arch. Rat. Mech. Anal.
I101
Berestycki, H. and Lions, P.L., to appear.
C111
Berestycki, H. and Lions, P.L., Existence of stationary states of nonlinear scalar field equations. In Bifurcation phenomena in Mathematical physics (Reidel, New-York, 1979).
C123
Berestycki, H. and Lions, P.L., Existence d'ondes solitaires dans des problemes nonlineaires du type Klein-Gordon, Cmptes-Rendus Paris 288 (1979) 395-398.
C131
Berestycki, H. and Lions, P.L., Existence of a ground state in nonlinear equations o f the type Klein-Gordon. In Variational Inequalities (J. Wiley, New-York, 1979).
C141
Berestycki, H. and Lions, P.L.., Une methode locale pour l'existence de solutions positives de problemes semi-lineaires elliptiques dans RN, J. Anal. Math. 38 (1980) 144-187.
C151
Berestycki, H., Lions, P.L., and Peletier, L.A., An O.D.E. approach to the existence of positive solutions for semi-linear problems in lRN, Ind. Univ. Math. J. 30 (1981) 141-157.
33
SOME NONLINEAR VARIATIONAL PROBLEMS OF MATHEMATICAL PHYSICS
C 161
Berger, M.S.,
On t h e existence and s t r u c t u r e o f s t a t i o n a r y states f o r a
nonlinear Klein-Gordon equation, J. Funct. Anal. 9 (1972) 249-261. C 171
c 181
Cazenave, T.,
and Lions, P.L.,
Crandall, M.G.,
t o appear.
and Rabinowitz, P.H.,
B i f u r c a t i o n from simple eigenvalues,
J . Funct. Anal. 8 (1971) 321-340. C 191
Donsker, M.,
and Varadhan, S.R.S.
c 201
Esteban, M.J.,
and Lions, P.L.,
, to
appear.
Non existence de s o l u t i o n s non t r i v i a l e s
pour des problemes semilineaires dans des ouverts non born&.
Comptes-
Rendus Paris 290 (1980) 1083-1085. c211
Esteban , M. J
.
and Lions, P.L.,
Existence and non existence r e s u l t s f o r
semi1 inear e l i p t i c problems i n unbounded danains, t o appear i n Proc-Boy. SOC. Edim.
c221
Gustafson, K.
and Sather, D.,
Branching analysis o f the Hartree equations,
Rend. d i Mat. 4 (1971) 723-734. C231
Hartree, D . , The wave mechanics o f an atom w i t h a non-coulomb c e n t r a l f i e l d . Part 1. Theory,and methods, Proc. Camb. P h i l . SOC. 24 (1928) 89-132.
C241
K r a s n o s e l ' s k i i , M.A.,
Topological methods i n t h e theory o f nonlinear i n -
t e g r a l equations (Mac M i l l a n , New-York, 1964). C251
Lieb, E.H.,
and Simon, B., The Hartree-Fock theory f o r Coulomb systems,
Comn. Math. Phys. 53 (1977) 185-194. C261
Lions, P.L.,
Sane remarks on Hartree equation, t o appear i n Nonlinear Anal.
T.M.A.
C271
Lions, P.L., The Choquard eouation and r e l a t e d questions, Nonlinear Anal. T.M.A. 4 (1980) 1063-1073.
C281
Lions, P.L., 236-275.
Minimization problems i n L'm'),
C291
Lions, P.L.,
Topological and canpactness methods f o r some nonlinear v a r i a -
J . Funct. Anal. 41 (1981)
t i o n a l problems o f Mathematical Physics, t o appear.
34
C30]
P.L. LIONS
1 3 Minimization problems i n L (IR ) and a p p l i c a t i o n s t o some f r e e
Lions, P.L.,
boundary problems. I n Free boundary problems, Pavia, 1979 (Roma, 1980).
1311
L j u s t e r n i k , L.A.,
and Schnirelman, L.G.,
Topological methods i n t h e c a l -
culus o f v a r i a t i o n s (Hermann, Paris, 1934).
C321
Nehari, Z.,
On a non l i n e a r d i f f e r e n t i a l equation a r i s i n g i n nuclear phy-
s i c s , Proc. Roy. I r i s h Acad. 62 (1963) 117-135.
C331
Rabinowitz, P.H.
, Variational
methods f o r nonlinear eigenvalue problems.
I n Eigenvalues o f nonlinear problems (Cremonese, Rome, 1974).
C341
Reeken, M.,
General theorem on b i f u r c a t i o n and i t s a p p l i c a t i o n t o t h e
Hartree equation o f the helium atom, J. Math. Phys. 11 (1970) 2505-2512.
1351
Rvder, G.H.,
Boundary value problems f o r a class o f nonlinear d i f f e r e n t i a l
equations, P a c i f i c J . Math. 22 (1967) 477-503.
C361
S l a t e r , J.C., A note on H a r t r e e ’ s method. Phys. Rev. 35 (1930) 210-211.
C37]
Strauss, W.A.,
Existence o f s o l i t a r y waves i n higher dimensions, Cmm.
Math. Phys. 55 (1977) 149-162.
[38]
Stuart, C.A.,
Existence theory f o r the Hartree equation, Arch. Rat. Mech.
Anal. 51 (1973) 60-69.
C391
Stuart, C.A.,
An example i n nonlinear f u n c t i o n a l a n a l y s i s : t h e Hartree
equation, J. Math. Anal. Appl. 49 (1975) 725-733.
C403
Titchmarsh,
E.C.,
Eigenfunctions expansions, P a r t I 1 (Oxford Univ. Press,
London , 1962).
C411
Wolkowisky, J . , Existence o f s o l u t i o n s o f the Hartree equations f o r N electrons. An a p p l i c a t i o n o f t h e Schauder-Tychonoff theorem. Ind. Univ. Math. J . 22 (1972) 551-558.
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North-HollandPublishing Company, 1982
35
ON YANG-MILLS FIELDS
I.M. Singer Department o f Mathematics U n i v e r s i t y o f C a l i f o r n i a ] Berkeley Berkeley, C a l i f o r n i a 94720 U. S. A.
INTRODUCTION For t h i s conference on nonlinear problems, I was asked t o survey the developments i n global nonlinear p a r t i a l d i f f e r e n t i a l equations a r i s i n g i n quantum f i e l d theory. I w i l l , i n f a c t , t r y t o g i v e an expository account o f the most s i g n i f i c a n t recent development: t h a t o f t h e c l a s s i f i c a t i o n o f t h e s e l f dual Yang-Mills equations o f motion. I propose n o t t o be technical because o f t h e diverse backgrounds o f the p a r t i c i p a n t s . I n f a c t , as a l i s t e n e r , I o f t e n g e t l o s t e a r l y i n a t a l k because the speaker assumes I know more than I do. While he i s exposing technical d e t a i l s , I am sadly f r u s t r a t e d s t r u g g l i n g w i t h some basic questions whose answers I ' m supposed t o know. So I p l a n t o be simple. 1owi ng questions:
For the t o p i c I ' v e chosen I ' d l i k e t o address t h e f o l -
(1) What s c i e n t i f i c problem l e d t o the nonlinear equation? (2) What methods have provided t h e s o l u t i o n ? ( 3 ) Are these methods applicable t o other f i e l d s ? Our general experience i n mathematics shows t h a t new i n s i g h t s i n one area o f t e n spread and have other applications. So i t i s q u i t e useful t o understand what the breakthroughs are i n a f i e l d using n o n l i n e a r i t y ; t h e methods occasionally have other uses.
(4)
How does n o n l i n e a r i t y show i t s e l f i n t h e special p r o p e r t i e s o f t h e solutions?
and f i n a l l y ,
(5)
Do the s o l u t i o n s r e a l l y shed l i g h t on t h e o r i g i n a l s c i e n t i f i c question?
Often a nonlinear problem, o r any mathematical problem ends up having a l i f e o f i t s own f a r removed from i t s place o f o r i g i n . And one r a r e l y asks i f i t s solut i o n s are r e l e v a n t t o the o r i g i n a l m o t i v a t i n g question. OUTLINE My t o p i c i s gauge t h e o r i e s i n quantum f i e l d theory.
Iw i l l
Review the basic setup f o r gauge f i e l d s , I n d i c a t e where t h e nonlinear problem comes from, Describe t h e problem t h a t has been solved, Point t o t h e new methods t h a t lead t o the s o l u t i o n s , E x h i b i t t h e s o l u t i o n s and some new features they r e f l e c t , L i s t some remaining problems, Comment on the usefulness o f the s o l u t i o n s i n quantum f i e l d theory. Rather, I should say, comment on t h e K u s e f u l n e s s i n quantum f i e l d theory t o date.
I.M. SINGER
36
The new features e x h i b i t e d are (a) corn a c t i f i c a t i o n ; t h e s o l u t i o n s , which appear t o have s i n g u l a r i t i e s (sometimes a-yl' l i v e on compact manifolds and e x h i b i t topological-geometrical p r o p e r t i e s . Thus a t o p o l o g i c a l discreteness o r quantization appears i n t h e form o f t h e Pontryagin charge. Another new feat u r e i s (b) complexification; t h e i n s i g h t t h a t l e d t o the s o l u t i o n s stems from the complex p r o j e c t i v e " t w i s t o r " program o r i g i n a t e d by Roger Penrose. F i n a l l y , t h e r e i s (c) e x c i t a t i o n , e s p e c i a l l y o f some o f us geometers and t o p o l o g i s t s n o t accustomed t o f i n d i n g our research a p p l i c a b l e t o h i g h energy physics. GAUGE FIELDS
L e t me f i r s t b r i e f l y review gauge f i e l d s over Euclidean 4-space R4. Euclidean gauge theory on R4 i s e a s i e r than t h e theory over Minkowski (3 + 1) space. There i s a theorem (Osterwalder-Schrader [18]) t h a t says c e r t a i n c o n s t r u c t i v e f i e l d t h e o r i e s i n the Euclidean domain w i t h t h e c o r r e c t p o s i t i v i t y assumptions can be a n a l y t i c a l l y continued back t o r e a l time. We are s t i l l f a r away from a c o n s t r u c t i v e gauge theory i n 4 dimensions. S t i l l considerable i n s i g h t i s gained studying quantum f i e l d theory i n t h e Euclidean domain. Let A be a vector p o t e n t i a l - - o r connection, as i t i s c a l l e d i n mathematics--with values i n a simple L i e algebra--say, su(N), t h e skew-adjoint complex matrices o f t r a c e 0. [When I r e f e r t o electromagnetism, t h e L i e algebra i s u ( l ) , t h e pure imaginary numbers.] So A has components A and t h e A have values i n t h e L i e algebra su(N): CI' c1
A = A dx
CIP
and
A
= AaT
IJ
P a
'
where Ta i s an orthonormal basis o f su(N). Associated w i t h A i s i t s f i e l d o r curvature F, a two form w i t h values i n su(N):
-
Note t h a t the bracket [A A 3 = A A A A makes F a nonlinear f u n c t i o n o f A. A l l the t r o u b l e and a l l p t h e u i n t e k t t st8mPfrom t h e nonlinear term. I t i s a feature o f t h e non-Abelian L i e algebra. The a c t i o n S i s t h e L2 norm o f t h e f i e l d
We are i n t e r e s t e d i n vector p o t e n t i a l s w i t h f i n i t e a c t i o n ; so l e t A be t h e s e t o f a l l vector p o t e n t i a l s w i t h f i n i t e action:
Before we can proceed we s h a l l need a few o t h e r concepts. d i f f e r e n t i a l , DA i s t h e operator
F i r s t the covariant
37
ON YANG-MILLS FIELDS
1 .
(5)
Next, a gauge transformation i s a f u n c t i o n $ ( X I w i t h values i n t h e group G--say, SU(N), the group o f N x N u n i t a r y matrices o f determinant 1. L e t G denote t h e group o f a l l gauge transformations. G acts on A: a gauge transformation $ ( X I acts on a vector p o t e n t i a l A t o g i v e t h e vector p o t e n t i a l $-A where
or, e q u i v a l e n t l y
It i s easy t o check t h a t
Since $ has values i n a compact group--here SU(N)--formula (8) i m p l i e s t h a t t h e a c t i o n i s i n v a r i a n t under G:
F i n a l l y , we need some geometry o f 4-space. L e t R4: 12 -+ 34, 13 + -24, 14 + 23, and * 2 = + l . and (*F)14 = F23. Also l e t
*
denote d u a l i t y o f two forms i n So (*F)12 = F34, (*F)13 = -Fe4
D u a l i t y decomposes t h e f i e l d F i n t o i t s s e l f dual F+ and a n t i - s e l f dual Fcomponents; these components are the (+1)and (-1) eigenvalues under *. Ordinary two forms--not j u s t two forms w i t h values i n SU(N)--split t h e same way. Two forms a r e denoted by A 2 , the space o f skew symmetric 2-tensors. The s p l i t t i n g gives
where A2 are the self-dual and a n t i - s e l f - d u a l o r d i n a r y forms respectively. since t h e Formula 411) gives the decomposition o f sO(4) i n t o two sD(3)'s--or, L i e algebras are t h e same--into two sU(2)'s.
I.M. SINGER
38
Note t h a t
For l a t e r use we define
Observe t h a t * i s conformally i n v a r i a n t i n 4 dimensions, so t h a t S(A) i s a conformal i n v a r i a n t . S h o r t l y we s h a l l see how t h e 4-sphere S 4 , the conformal comp a c t i f i c a t i o n o f R4, enters. THE PROBLEM We can now s t a t e the mathematical question: What a r e t h e s t a t i o n a r y p o i n t s o f S(A)? The m o t i v a t i o n f o r the problem stems from t h e saddle p o i n t method o f e v a l u a t i n g the Feynman-Kac path i n t e g r a l
where
F(A)
i s a gauge i n v a r i a n t q u a n t i t y : f(g.A) = f ( A ) .
I t i s a fundamental problem o f quantum f i e l d theory t o "make sense" o f t h e i n t e g r a l above--one r e a l l y doesn't know what DA, " i n t e g r a t i o n over a l l f i e l d s " means. I n the saddle p o i n t method one f i x e s on t h e s t a t i o n a r y p o i n t s . Then one w r i t e s t h e a c t i o n as a quadratic term i n A p l u s a remainder, and uses a perturbat i v e expansion f o r the remainder term. One emulates quantum electrodynamics (QED) where t h e method has been spectacularly successful. It i s essential, then, t o know the s t a t i o n a r y solutions. It i s easy t o compute the Euler-Lagrange equations f o r t h e s t a t i o n a r y points:
where + i n d i c a t e s the a d j o i n t operator; i n components,
We seek, then, t h e s o l u t i o n s A t o (16) w i t h f i n i t e a c t i o n on R4. EXTENSION TO S4 One o f t h e p o i n t s I wish t o emphasize i n t h i s l e c t u r e i s t h a t c o n s i d e r a t i o n o f
ON YANG-MILLS FIELDS
30
the problem above leads n a t u r a l l y t o t h e 4-sphere, t o topology and f i b r e bundles; these concepts are n o t introduced a r t i f i c a l l y . To t h i s end I c i t e a theorem o f Karen Uhlenbeck [20]. Theorem:
Suppose
(i)S(A) < m, (ii) A s a t i s f i e s the equation o f motion, i . e . , (iii)A has an i s o l a t e d (point) s i n g u l a r i t y .
i s a s t a t i o n a r y p o i n t , and
Then the s i n g u l a r i t y i s removable by a gauge transformation i n t h e f o l l o w i n g sense: i f t h e s i n g u l a r i t y i s a t p, t h e r e i s a b a l l (D ) around p and a gauge transformation I$ defined on D -p so t h a t @ * A i s e x t e n d h l e t o a l l o f D [See Fig. 1.1 P P'
0 * A EXTENDABLE
Figure 1: The domain D around a p o i n t p used i n extending t h e v e c t o r p o t e n t i a l A t o remove 8 p o i n t s i n g u l a r i t y . We have s p l i t R4 i n t o two parts: the B a l l D , and i t s complement; t h e i r common boundary i s t h e 3-sphere S3. On D , t h e v e c t b p o t e n t i a l @ * A i s nonsingular and there i s a mismatch between A
[email protected] on the common boundary S3. We use i t t o b u i l d a new object--a f i b r e bundle. Before describing the f i b r e bundle i t i s worth n o t i n g t h a t we can a l l o w t h e s i n g u l a r i t y t o be a t i n f i n i t y . Because o f conformal invariance, i n f i n i t y i s no d i f f e r e n t than any other p o i n t i n R4; t h a t i s , a conformal transformation brings i n f i n i t y back t o the o r i g i n . So a f i n i t e a c t i o n , s t a t i o n a r y solution-w i t h no s i n g u l a r i t i e s on R 4 , b u t n o t defined a t i n f i n i t y - - c a n be considered t o have an i s o l a t e d s i n g u l a r p o i n t a t a. By a conformal transformation, b r i n g m back t o the o r i g i n and apply Uhlenbeck's theorem t o remove t h e s i n g u l a r i t y . Figure 1 i s now modified and becomes Fig. 2. The b a l l D i s t h e upper hemisi t s compliment i s the lower hemisphere D-, andPtheir common boundary phere D,, i s the equator S3. Now we are ready t o b u i l d a p r i n c i p a l bundle over t h e 4-sphere, p o t e n t i a l associated w i t h it.
and a vector
40
I.M. SINGER
D, x G
I
\
s3
D-xC
Figure 2:
The c o n s t r u c t i o n o f a f i b r e bundle over t h e 4-sphere.
The simplest p r i n c i p a l bundle over t h e 4-sphere i s S4 x G. The t r o u b l e i s t h e vector p o t e n t i a l A i s s i n g u l a r . To remove t h e s i n g u l a r i t y , we break A up i n t o $ * A i n t h e upper hemisphere and A i n t h e lower hemisphere producing a mismatch between A and 4 - A along the equator. We compensate f o r t h i s mismatch by b u i l d i n g a new bundle. On t h e upper hemisphere i t i s 0, x G. We patch o r paste along the equator by i d e n t i f y i n g the s e t x x G o f D, x G w i t h x x G o f D- x G by l e f t m u l t i p l i c a t i o n w i t h $(x). We g e t a f i b r e bundle, t h a t i s , a c o l l e c t i o n o f f i b r e s , groups i n t h i s case, one f o r each p o i n t o f S4. But t h e f i b r e s have been sheared along t h e equator by t h e gauge transformation $(x), x e 53. Since t h e gauge transformation $(x) on S3 w i t h values i n G can be t o p o l o g i c a l l y nont r i v i a l , i . e . , can wind around G, the newly constructed p r i n c i p a l bundle P can be complicated. The p o i n t o f i t s c o n s t r u c t i o n i s t h a t i t compensates f o r t h e mismatch o f A and $.A on t h e equator 9. We have produced a new vector potent i a l AuI$.A on P w i t h o u t s i n g u l a r i t i e s . s3
Uhlenbeck's theorem says, i n short, t h a t any s t a t i o n a r y p o i n t o f t h e a c t i o n on R4 w i t h i s o l a t e d s i n g u l a r i t i e s can be promoted t o a nonsingular v e c t o r potent i a l on a p r i n c i p a l bundle over S4. The new vector p o t e n t i a l i s a s t a t i o n a r y p o i n t f o r the a c t i o n on t h i s p r i n c i p a l bundle. One can recapture t h e o r i g i n a l p i c t u r e on R4 by stereographic p r o j e c t i o n . A DIVERSION F i b r e bundles have become so common i n h i g h energy physics (grand u n i f i c a t i o n schemes, symmetry breaking, and dimensional reduction), t h a t perhaps a s h o r t digression i n t o t h e i r geometry i s i n order. A few p i c t u r e s i n t h e simplest cases are worth many formulas, i n my view. Consider, then Table I and Fig. 3.
ON YANG-MILLS FIELDS
ICI
Figure 3: F i b r e bundles over the c i r c l e S1: (a) the torus; (b) the K l e i n b o t t l e ; (c) an a l t e r n a t i v e d e s c r i p t i o n o f the F i b r e bundle leading t o a K l e i n bottle. TABLE I FIBRE BUNDLES 1.
C i r c l e bundles over a c i r c l e S1:
torus, K l e i n b o t t l e
2.
C i r c l e bundles over a 2-sphere 52:
The Chern-Gauss-Bonnet formula and Dirac Magnetic Monopoles
3.
P r i n c i p a l bundles over a 4-sphere S4: .
Our previous examples; c l a s s i f i c a t i o n i n terms o f t h e Pontryagin charge o r second Chern class
41
1. M. SINGER
42
On the c i r c l e (Sl) t h e r e are two c i r c l e bundles, one o f which i s t r i v i a l and one t h a t i s n ' t . You have seen them both. The f i r s t i s a torus. Take a c i r c l e as shown i n Fig. 3a, and b r i n g i t around t h e base c i r c l e . The second i s a K l e i n b o t t l e (Fig. 3b): when t h e f i b r e c i r c l e i s brought around t h e base c i r c l e t o the s t a r t i n g p o i n t , the i d e n t i f i c a t i o n i s made w i t h t h e i n i t i a l f i b r e by 0 -0 instead o f 0 0 . A s l i g h t l y d i f f e r e n t way o f l o o k i n g a t t h i s c o n s t r u c t i o n generalizes t o higher dimensions. Take t h e base S1 and break i t up i n t o two pieces, so t h e r e i s an upper hermisphere (arc) and lower hemisphere. Each hemisphere i s an i n t e r v a l and the "equator" c o n s i s t s o f two points. The o n l y issue i s how t o patch the f i b r e c i r c l e s a t t h e equator. A t one end patch by the i d e n t i t y and a t t h e other end patch by 0 -0 g i v i n g t h e K l e i n b o t t l e (see Fig. 3c).
-
-
-
Next consider t h e two sphere ( S 2 ) (Fig. 4), decomposed i n t o t h e upper hemisphere (D,) and the bottom hemisphere (D-). We want t o c o n s t r u c t a bundle o f c i r c l e s by patching o r p a s t i n g D, x S1 and D- x S1 along the equator. To do so r e q u i r e s a map Q f r o m t h e equator (a c i r c l e ) i n t o t h e c i r c l e S1. We know these maps Q are determined t o p o l o g i c a l l y by t h e i r winding number. Patching by I$ a t x on the equator means i d e n t i f y i n g the c i r c l e from t h e northern hemisphere w i t h t h e c i r c l e from t h e southern hemisphere by a r o t a t i o n through angle Q(x). The topology o f the bundle i s completely determined by t h e winding number o f Q which has the formula
1J-f 2n . 1
As an example, consider the c i r c l e bundle o f u n i t tangent vectors o f S2 (Fig. 4a). The f i b r e a t each p o i n t i s the c i r c l e o f u n i t tangent vectors emanating from t h a t p o i n t . What i s t h e winding number f o r t h i s bundle? We can e a s i l y i d e n t i f y a l l the f i b r e s i n the upper hemisphere by adopting a r e l a t i v i s t i c p o i n t o f view. An observer a t the n o r t h p o l e c a r r i e s a c l o c k (a u n i t tangent vector) along w i t h him as he moves along a g r e a t c i r c l e p a t h (geodesic) t o any o t h e r p o i n t i n the northern hemisphere. Since the u n i t tangent v e c t o r o f a geodesic are a u t o - p a r a l l e l , the d i r e c t i o n i s carried i n t o the direction , determining the i d e n t i f i c a t i o n o f t h e c i r c l e l a b e l e d C ' w i t h the one l a b e l l e d C (Fig. 4b). Do the same f o r an observer a t the south pole; a l i t t l e surface theory, t e l l s you the shear i s t w i c e the s o l i d angle R. Thus, t h e winding number i s two.
$'
Figure 4: Construction o f a f i b r e bundle on t h e 2-sphere, S2.
43
ON YANG-MILLS FIELDS
Here i s another geometric example t o help b u i l d your i n t u i t i o n f o r t h e f o u r dimensional case w e ' l l need l a t e r . Think o f t h e 3-spherel S3, as the s e t o f p a i r s o f complex numbers (Zl,Z2) subject t o
(18) Map the 3-sphere t o the two sphere by sending a p a i r (Zl,Z2)
s3 .I
n
= [(Z,,Z2)
SI
S2
:
.IQ
lZllL
i n t o Z1/Z2.
n
I Z 2 f = 11
+
(eiezl,eiez2)
z1/z2
because Z1/Z, c'f8 For our present purposes, S2 equals t h e complex plane p l u s be = i f Z, = 0 and lZll = 1. Since 1Z112 + 1Z2I2 m y b t i p l y i n g by e Z2. Moreovph i f r o t a t e s vectors i n S3 b u t preserves t h e r a t i o : Z1/Z2 = e' Zl/e Z1/Z2 = W1/W, and (Z1/Z2) i s o f the same l e n g t h as (Wl/W2)l then Z . = e W., j = 1,2. So we have e x h i b i t e d S3 as a f a m i l y o f c i r c l e s over S2:J a c i r c l e bundle where each c i r c l e p r o j e c t s i n t o the same r a t i o o r p o i n t o f S 2 . This bundle has winding number one, though t h a t ' s n o t easy t o see. I t ' s computable by the Gauss-Bonnet formula which we now discuss.
=.d,
L e t ' s again t h i n k o f the two sphere as u n i t vectors i n t h r e e space. Dirac found a l l c i r c l e bundles over S2, motivated by magnetic monopoles [lo]. He s t a r t e d o u t w i t h Maxwell's equations f o r an electromagnetic f i e l d F. I n concise mathematical notation, F s a t i s f i e s dF = 0 d*F = j,
the current
.
Dirac wanted t o reverse t h e two equations above t o o b t a i n dF = pM(0), d*F = 0
a magnetic pole a t t h e o r i g i n
.
He was i n t e r e s t e d i n t h e s t a t i c case, so t h e equations reduce t o t h r e e space. Away from t h e o r i g i n , say on a sphere o f f i x e d p o s i t i v e radius, dF = 0. On a region t h a t i s c o n t r a c t i b l e , one could conclude t h a t F f dA. But the sphere i s I n fact, we know t h a t t h e not c o n t r a c t i b l e so F = dA on the e n t i r e sphere. i n t e g r a l o f F over the two sphere has t o be the f l u x o f the magnetic f i e l d and i t i s not zero. But the upper hemisphere, D+ i s c o n t r a c t i b l e and we can w r i t e
44
1. M. SINGER
S i m i l a r l y , on the lower hemisphere, F I D = dA-. along t h e equator; t h e r e i s a mismatcl. dA+ = F = dA- so d(A+
-
However, A+ and A - w i l l n o t agree
I n a l i t t l e band about t h e equator
.
A_) = 0
One can show t h a t
A+
-
A-
=
=I-
9 1$
on t h e equator
-
f o r some f u n c t i o n t$ t h a t doesn't vanish. [ I n general J(A A 1 around t h e c i r c l e could be any number. But i f matter f i e l d s a r e introauced-and A and Aare t o be vector p o t e n t i a l s , J A+ A- i s an i n t e g e r and Eq. (24) holas, thus q u a n t i z i n g the magnetic pole a t the orig'in.] The f l u x w i l l be (up t o a u n i versal f a c t o r ) equal t o the winding number o f t$. That i s
-
-- c
A+
-
A- =
C
J
* (0
= c winding no. o f $
.
The f u n c t i o n $ can be used t o b u i l d a c i r c l e bundle as I have already described. That i s , c i r c l e s from the t o p hemisphere a r e patched w i t h c i r c l e s from t h e bottom hemisphere by $ / 1 $ 1 . Now A and A- f i t together t o g i v e a v e c t o r potent i a l on t h i s c i r c l e bundle over 9. The c i r c l e bundle i s determined by t h e winding number which we can compute i n two d i f f e r e n t ways. F i r s t , we can c a l c u l a t e i t as t h e f l u x ; t h i s i s the generalized Chern, Gauss-Bonnet theorem, and gives t h e winding numbers as an i n t e g r a l formula, t h e averaged curvature f i e l d . Second, given t h e vector p o t e n t i a l , by p a r a l l e l t r a n s p o r t along geodesics from the n o r t h and south poles (as we d i d w i t h t h e tangent c i r c l e bundle), we can compute the mismatch a t t h e equator and thus compute t h e winding number. Now l e t ' s r e t u r n t o t h e 4-sphere, S4, and reconsider Fig. 2. We had two hemisspheres, D+ and 0, and the equator 5 3 . The SU(N) bundles are determined by maps o f S3 i n t o SU(N). When N = 2, SU(2) i s the same as t h e u n i t quaternions, i . e . , S3. The maps o f S3 -t S3 are determined up t o homotopy by an i n t e g e r , t h e winding number k. (This remains t r u e f o r a r b i t r a r y N.) We have already seen t h a t t h e v e c t o r p o t e n t i a l s on R4 w i t h i s o l a t e d s i n g u l a r i t i e s s a t i s f y i n g t h e e uations of motion can be promoted t o vector p o t e n t i a l s on these bundles over S What i s t h e corresponding formula f o r k i n terms o f an i n t e g r a l formula fqr the f i e l d ? The ansEer i s again t h e Chern-Gauss-Bonnet formula: i n terms o f F , the self-dual, and F , t h e a n t i - s e l f - d u a l p a r t o f t h e f i e l d
9.
45
ON YANG-MILLS FIELDS
F i n a l l y , l e t me end the geometric discussion w i t h an example o f an SU(2) bundle over S4 t h a t generalizes (19). We use quaternions i n s t e a d o f complex numbers.
.L s3
Consider S7 as p a i r s o f quaternions with t h e sum o f absolute values squared = 1. Since q1 and q, are f o u r dimensional, t h e r e s t r i c t i o n 1q1l2 + 1q,I2 = 1 defines a sphepq i n eight-dimensional space, i . e . , S7. Map S7 t o S4 by mapping (ql,q2) i n t o q, ql, a l s o a quaternion. i s t o be i n t e r p r e t e d as quaternions p l u s i n f i n i t y , because when q, = 0, q ql-equals i_nfinity. As before, i f I q l = 1, then (qq,,qq,) i s also on S7 and (qq,) lqql = q,'ql. Furthermore, i f two p o i n t s on S7 g i v e the same r a t i o , they d i f f e r by a u n i t quaternion r a t i o . Hence we g e t the 3-sphere bundle, S7 over S4.
-s4
One i n t e r e s t i n g f a c t t h a t emerges from our discussion i s t h a t t h e t o p o l o g i c a l quantization i s inherent i n the problem. We have shown t h a t c e r t a i n s t a t i o n a r y solutions could be promoted t o S4 as vector p o t e n t i a l s on bundles over S4. A vector p o t e n t i a l w i t h f i n i t e a c t i o n may o s c i l l a t e near i n f i n i t y , and may n o t be promotable t o S4. Nevertheless, the right-hand s i d e o f Eq. (26) makes sense. I t ' s a r e a l number and n o t i n f i n i t y because i t i s l e s s than t h e action. One can ask whether i t ' s always an i n t e g e r as i n t h e s t a t i o n a r y case. The answer i s yes, i f m i l d decay conditions are assumed f o r A as x + m. [Then AWI- $ la $ and $, as a f u n c t i o n o f rays, has a winding number.] So t h e s e t o f most" Y i n i t e a c t i o n vector p o t e n t i a l s s p l i t s i n t o a d i s c r e t e s e t (quantized by the i n t e g e r k) c a l l e d Pontryagin sectors. THE SOLUTION I have taken considerable time describing our problem and i t s promotion t o a geometric-topological context. Now I want t o t a l k about s o l u t i o n s . We s t i l l do It i s n o t know a l l the s t a t i o n a r y s o l u t i o n s (up t o a gauge transformation). known [7] t h a t a l o c a l minimum, i n a given Pontryagin sector, i s a minimum. One can now say a great deal about the minima. By s u b t r a c t i n g Eq. (26) from Eq. (12), we get
We
can assume k > 0, since $ change o f o r i e n t a t i o n changes the s i g n o f k. Thus equal t o 0, i . e . , F = F , assures us a minimum. Such F are c a l l e d " s e l f dual" f i e l d s . What do these self-dual . f i e l d s l o o k l i k e ? Note t h a t , whereas being s t a t i o n a r y leads t o a second-order equation, t h e c o n d i t i o n f o r a s e l f dual minimum gives a f i r s t order equation; i t i s much easier t o deal with.
F
Theorem.[6]
Self-dual s o l u t i o n s e x i s t f o r N = 2 and f o r a l l k
There's an Ansatz f o r them:
I f f i s a harmonic function,
2
0.
1. M. SINGER
46
i s self-dual. The m a t r i x q - - I ' m using physics n o t a t i o n here--is j u s t t h e e onto A: = su(2) so t h a t AO! i s a L i e algebrap r o j e c t i o n o f t h e two-form valued form. The s p e c i f i c h a h o n i k f u n c t i o n s used a r e
cup-
and g i v e the " m u l t i - i n s t a n t i o n " s o l u t i o n s [13,14]. They have i s o l a t e d singul a r i t i e s a t p.. By Uhlenbeck's theorem, t h e s i n g u l a r i t i e s can be removed; these vector p d t e n t i a l s have Pontryagin charge k. One can count t h e number o f degrees o f freedom f o r t h e s o l u t i o n s above. finds:
One
TABLE 2 k
Number o f parameters
0 1 2 k > 2
unique s o l u t i o n 5 degrees o f freedom 13 degrees o f freedom 5k + 4
(N = 2)
One can a l s o count the number o f degrees o f frqedom by l i n e a r i z i n g the d i f f e r e n t i a l equation f o r s e l f d u a l i t y . The l i n e a r equation i s a standard e l l i p t i c one and the index theorem counts the number o f degrees o f freedom. For SU(2), i t gives 8k 3. For SU(N) we have t h e
-
Theorem:[3]
The s e l f - d u a l s o l u t i o n s a r e a maniform o f dim 4Nk
-
-
N2
+ 1.
For N = 2 we f i n d 8k 3 > 5k + 4 when k > 3, so t h e r e must be a d d i t i o n a l solut i o n s f o r k 2 3. What do they look like?- S u r p r i s i n g l y , t h e answer comes from algebraic geometry. And i t i s the i n s i g h t o f the Penrose t w i s t o r program t h a t t r a n s l a t e s the s e l f - d u a l problem t o a holomorphic one [4,19,21]. B r i e f l y B what happens i s t h i s . The sphere S7 i n (28) i s acted upon by t h e c i r c l e e (see Eq. (31). CP3 denotes complex p r o j e c t i v e space, t h e complex l i n e s i n complex 4-space; t h a t i s [(z1,z2,z3,z4) (Az1,Az2,Az ,Az4); A & f, and A # 01. T h i s space i s a f i b r e bundle over S4 w i t h f i b r e S2. ?he map i s given by (z1,z2,z3,z4) + z1 + ~ 2 j / z 3 + z 4 j .
-
s4
(zl
+
z2j/z3 + z 4 j )
47
ON YANG-MILLS FIELDS
The SU(N) bundle P over S4 p u l l s up t o a bundle over CP3, and t h e f i b r e can be complexified t o SL(N,c) [see Eq. (3211.
That i s . the SWN) f i b r e over a D o i n t D o f S4 i s l i f t e d t o a l l t h e S2 p o i n t s i n CP3 over p. The-group SU(N) i s enlakged t o t h e complex special 1inear group SL(N,C), constructing a new p r i n c i p a l buedle Pc over CP3. The v e c t o r p o t e i t i a i A can be promoted t o a vector p o t e n t i a l A on P . S e l f d u a l i t y f o r A t r y s l a t e s immediately d n t o the Nirenberg-Newlander conditions [17] f o r making P , with connection A , i n t o a holomorphic bundle. That i s t h e i n s i g h t provided by t h e Penrose Twister Program. Put more concretely, the r a t i o s z1/z2, z2/z4, 23/24 are coordinates f o r CP3 (when 24 # 0). Two r a t i o s would s u f f i c e t o c o o r d i n a t i z e S4 l o c a l l y ; b u t a l l three complex coordinates are needed t o t r a n s l a t e s e l f dual i n t o holomorphic (even though the t h i r d coordinate i s redundant and a c t s somewhat l i k e gauge freedom). There i s a s i m i l a r e f f e c t i n two-dimensional a b e l i a n Yang-Mills/Higgs theory [151. There the a c t i o n S(A,$) depends on an a b e l i a n ( G = U(1)) v e c t o r p o t e n t i a l A and a complex scalar f i e l d 0:
C r i t i c a l p o i n t s are complicated.
DI;F = Imag.
$D~+,
The equations o f motion are:
(D~DI; + DI;D~)$ = 1-1l2 4 .
(34)
.+
Note t h a t t o have f i n i t e action, 141 + 1 as r + 01 and 110 $ 1 1 + 0 as r + 00. So A, = $ %,d + O ( l / r 2 ) . Again the f i n i t e a c t i o n c o n f i g u r a t i o n s f a l l i n t o classes 1 determined by the winding number k = 1 F e d2x. ;,I$'/$ = 6 A, = ,,v ,,v
rn
For A = 1, t h e minima are much easier t o determine.
DA$ = i*DAd,
rn h2
The equations are:
*F =
Equation (35) says t h a t
(35)
;i[(Al+iAZ)/2]
($)
= 0.
That i s e-W$ i s holomorphic
where aw = (Al+iA2)/2. The s o l u t i o n s @ are o f t h e form $ = h(z) ll w i t h the winding number k = ha. a
(z
- z a ) nQ
The reduction t o a holomorphic problem i s again p r e d i c t a b l e from t h e t w i s t e r program, which also leads t o new non-Abel i a n magnetic monopoles [151.
I.M. SINGER
48
Once the self-dual problem i s t r a n s l a t e d t o a holomorphic one on CP3, i t becomes a problem i n algebraic geometry. The a l g e b r a i c geometers have been studying t h i s matter extensively; t h a t i s , studying t h e c l a s s i f i c a t i o n o f holomorphic bundles over CP3 [5,121. Those t h a t come from t h e s e l f - d u a l v e c t o r p o t e n t i a l s on t h e f o u r spheres can be c l a s s i f i e d . We describe them i n two ways [2]. F i r s t , t h e a b s t r a c t d e s c r i p t i o n f o r the c f m p l e t % k q t o f s o l u t i o n s i s t h i s : consider l i n e a r maps, T(z), z c C4, from C t o C ; t h e s + $ r e 2k + 2 by k has a symplecmatrices, depending l i n e a r l y on z. Assume the second space C t r i c s t r u c t u r e given by a skew-adjoint m a t r i x , J. Also assumf t h a t T(z) s a t i s f&$2these conditions: (1) I t ' s o f maximad rank so i t maps C i n j e c t i v e l y i n t o C ; t h a t i s , f o r any vector v i n T(z)(C ), (Jv, v> = 0. Then
k i s two dimensional. For T(z)(C ) i s i s o t r o p i c and l i e s i n s i d e the J- orthogonal complement which i s o f dimension k + 2. The two-dimensional space E(z) i s independent o f the scale and t h i s c o n s t r u c t i o n assigns t o every l i n e i n C4 o r every p o i n t o f CP3 a two-dimensional space. The SU(2) bundle i s t h e s e t o f orthonormal frames i n each E(z). Theorem. -this
Every SU(21-bundle w i t h s e l f - d u a l v e c t o r p o t e n t i a l can be repreway.
For a thorough discussion o f t h i s theorem and i t s g e n e r a l i z a t i o n t o SU(N) see [l]. Despite the a b s t r a c t construction, t h e s e l f - d u a l v e c t o r p o t e n t i a l s can be r e a l i z e d q u i t e concretely, as i l l u s t r a t e d by t h e Ansatz I describe next. F i r s t l e t me remark t h a t any vector p o t e n t i a l can be w r i t t e n as
A
CI
= T
+ aT
K ,
(37)
P
where T i s an L by N m a t r i x f u n c t i o n w i t h T+T = IN (see Narasiman and Varadan [16]). These A w i l l be self-dual i f T s a t i s f i e s c e r t a i n conditions. F i r s t , f $ r the k t h P o k r y a g i n sector, L i s N + 2k. Next, note th$g2khe c o n d i t i o n T T = IN mean3+@e columns o f T are N orthogonal vectors i n C . Choose 2k o r t j o g o n a l t o the columns o f T. Denote t h i s N + 2k x 2k vectors i n C m a t r i x by A. Hence, T A = 0. Self duality f o r A
P
A(x) = a + bx AA '
i s given by t h e f o l l o w i n g equations i n A:
,
and
(38a)
commutes w i t h m u l t i p l i c a t i o n by quaternions.
I n Eq. (38a), t h e p o i n t x i n i s , x = xo + xli + x 2 j + x3k matrices. Also, a = (al,a2) matrices. Equation (38b) says
(38b)
R4 stands f o r quaternionic m u l t i p l i c a t i o n . That w i t h i, j, k represented by the usual 2 x 2 and b 7 (blrb2) w i t h a., b. constant N + 2k x k t h a t A A i s a k x k m a t i i x ti I2 [a].
49
ON YANG-MILLS FIELDS
I t i s easy t o check t h a t vector p o t e n t i a l s s a t i s f y i n g t h i s Ansatz are s e l f dual. One can also check t h a t t h e number o f degrees o f freedom i s 8k 3 f o r N = 2. The algebraic geometry assures us t h a t a l l s o l u t i o n s are gauge equivalent t o t h e exhibited solutions. I t i s amusing t o o t e t h a t i f one could prove d i r e c t l y t h a t the space o f a l l solutions i s connected, then the detour through geometry would be unnecessary. For the e x h i b i t e d solutions form a complete manifold o f the proper dimension.
-
OPEN PROBLEMS We close w i t h a l i s t o f open problems. (a)
Are there i r r e d u c i b l e s t a t i o n a r y s o l u t i o n s which are n o t s e l f ( a n t i ) dual? See [11,23] f o r a t w i s t o r d e s c r i p t i o n o f the equations o f mot ion.
(b)
What are the complex s o l u t i o n s o f the Yang-Mills equations o f motion, i . e . , s o l u t i o n s f o r C4 w i t h G = SJ?(N,C)?
(c)
I s the space o f self-dual s o l u t i o n s connected f o r k
(d)
I f no decay assumptions are mage, does 3 vector p o t e n t i a l A with I I F I t 2 ) an i n t e g e r ? f i n i t e a c t i o n have k = 1/8n2 ( I I F 1 1 2
(e)
What are the a p p l i c a t i o n s t o quantum gauge theories?
2 3?
-
So f a r , the existence o f solutions o f d i f f e r e n t t o p o l o g i c a l type, i.e., t h e existence o f d i s t i n c t Pontryagin sectors, describes t h e vacuum s t r u c t u r e o f t h e The s p e c i f i c forms f o r t h e s o l u t i o n as displayed by theory ( a l l a tunnelling). Eqs. (37) and (38) imply a n i c e closed form [9] f o r t h e Green's f u n c t i o n of t h e Dirac operator $A coupled t o A:
and t h i s form has been useful i n computing determinants i n t h e weak coupling limit. But attempts t o i n t e r p r e t o r compute Feynman namics using the s e l f - d u a l s o l u t i o n s have n o t " d i l u t e gas approximation'' doesn't work and, as r a t i o n s o f s e l f - and anti-dual s o l u t i o n s t h a t been found.
i n t e g r a l s i n Quantum Chromodybeen successful t o date. The f a r as I can t e l l , t h e configudominate the i n t e g r a l have n o t
REFERENCES
[l]Atiyah, M. F., Geometry o f Yang-Mills Fields, Fermi Lectures 1979 (Pisa). [2]
Atiyah, M. F., D r i n f e l d , V. G., H i t c h i n , N. J., and Manin, Yu. I . , Con(1978). pp. 185-187. s t r u c t i o n o f instantons, Phys. L e t t .
[3]
Atiyah, M. F. , H i t c h i n , N. J . , and Singer, dimensional Riemannian geometry, Proc. Roy. pp. 425-461.
[4]
e
I. M., S e l f - d u a l i t y i n f o u r SOC.
London,
A362 (19781,
Atiyah, M. R. and Ward, R. S., Instantons and algebraic geometry, Commun. Math. Phys. 55 (1977), pp. 117-124.
I.M. SINGER
Earth, W., Some p r o p e r t i e s o f s t a b l e rank-2 vector bundles on Pn, Math. Ann. 226 (19771, pp. 125-150. Belavin, A., Polyakov, A., Schwartz, A . , and Tyupkin, Y., Pseudoparticle s o l u t i o n s o f the Yang-Mills equations, Phys. L e t t . 598 (1975), pp. 85-87. Bourguignon, J.-P., Lawson, H. B., and Simons, J., S t a b i l i t y and gap phenomena f o r Yang-Mills f i e l d s , Proc. Nat. Acad. Sci. U.S.A. 76 (1979), pp. 1550-1553. C h r i s t , N., Stanton, N., and Weinberg, E. J., General s e l f - d u a l Yang-Mills s o l u t i o n s , Columbia U n i v e r s i t y p r e p r i n t , 1978. Corrigan, E., F a i r l i e , 0. B., Goddard, P., and Templeton, S., A Green's Paris). f u n c t i o n f o r the general s e l f - d u a l gauge f i e l d , p r e p r i n t (E.N.S., Dirac, P.A.M.,
Proc. Roy. SOC.
s, 60 (1931).
Green, P. S., Isenberg, J., and Yasskin, P., Phys. L e t t . 788 (19781, pp. 462-464.
Non-self-dual
gauge
Horrocks, G., Vector bundles on the punctured spectrum o f a l o c a Proc. London Math. SOC. (19641, pp. 684-713.
14
ields, ring,
Jackiw, R., Nohl, C . , and Rebbi, C., C l a s s i c a l and semi-classica s o l ut i o n s t o Yang-Mills theory, Proceedings 1977 B a n f f School, Plenum Press. Jackiw, R., Nohl, C., and Rebbi, C., Conformal p r o p e r t i e s o f pseudop a r t i c l e configurations, Phys. Rev. 150 (19771, pp. 1642-1646. J a f f e , A. and Taubes, C.,
Vortices and Monopoles, Birkhauser (1980) Boston.
Narasimhan, M.S. and Ramanan, S., Existence o f universal connections, Amer. J. Math. 83 (19611, pp. 563-572. Newlander, A. and Nirenberg, L., Complex a n a l y s t i c coordinates i n almost complex manifolds, Ann. o f Math. 65 (1957), pp. 391-404. Osterwalder, K. and Schrader, R., Axioms f o r Euclidean Green's f u n c t i o n s I, 11, Comm. Math. Phys. 31 (1973); 42 (1975). Penrose, R., pp. 65-76.
The Twistor programme, Rep. Mathematical Phys.
2
(1977),
Uhlenbeck, K., Removable s i n g u l a r i t i e s i n Yang-Mills f i e l d s , B u l l . Amer. Math. SOC. 1 (19791, pp. 579-581. Ward, R.S., Ward, R.S.,
79 (1981),
On s e l f - d u a l gauge f i e l d s , Phys. L e t t .
61A (19771, pp.
81-82.
A Yang-Mills-Higgs monopole o f charge 2, Commun. Math. Phys. pp. 317-325.
Witten, E., An i n t e r p r e t a t i o n o f c l a s s i c a l Yang-Mills theory, Phys. L e t t . 778 (1978), pp. 394-398.
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 9. Nicolaenko (eds.) 0 North-Holland Publishing Company, 1982
51
GAUGE THEORIES FOR SOLITON PROBLEMS
D.H. S a t t i n g e r
*
Department o f Mathematics University of Minnesota Minneapolis, Minn. U.S.A.
Lax
1.
equations
and i s o s p e c t r a l deformations.
Gardner, Greene, Kruskal, and Miura [41 showed t h a t t h e eigenvalues of t h e Schrodinger operator
L=b
evolves according t o t h e
2
x
+ -61
KdV
u
are invariant i f the potential
equation
.
u +u +uu = O t xxx x
u=u(x,t)
P. Lax [ 5 ] gave
i s a linear
t h i s observation t h e following g e n e r a l formulation: Suppose L(u) operator depending on t h e f u n c t i o n u i s u n i t a r i l y equivalent when
evolves according t o a nonlinear e v o l u t i o n equa-
u
.
t i o n &(u,ux,uxx,ut,uxt, . . ) = O ily of u n i t a r y operators
U(t)
such t h a t t h e family of o p e r a t o r s [ L ( u ( t ) ) ]
This means t h a t t h e r e i s a one parameter fam-
such t h a t
U-'(t)L(u(t))U(t)
.
U-kU
Differentiating
with r e s p e c t t o
t
"he u n i t
U = B U , where
a r y operators must s a t i s f y a d i f f e r e n t i a l equation of t h e type B*=-B
.
=L(u(O))
t
we o b t a i n t h e equation
atL = [B,Ll
(1)
I n t h e context of s o l i t o n t h e o r i e s t h i s equation has sometimes been c a l l e d t h e
Lax equation, e=O
.
and t h e operators B
I n t h e case of t h e
1 B=-b(b 3 + x 4
x
+ 81 "XI
check t h a t t h e commutator Thus
L =[B,L]
t
s o l u t i o n of t h e
The KdV
*
and
KdV
.
a Lax p a i r for t h e e v o l u t i o n equation
equation Lax gave f o r
Then
[B,L]
L
b L= t
g1 ut
B
the operator
( m u l t i p l i c a t i o n ) and one can
i s t h e m u l t i p l i c a t i o n operator
1 - g(uxxx+
i s an a p e r a t o r equation which i s s a t i s f i e d whenever KdV
u
uux)
is a
equation.
equation can be solved e x p l i c i t l y by t h e method of i n v e r s e s c a t t e r i n g
This research was supported i n p a r t by NSF Grant #MCS 78-00415.
52
D. H. SA TTINGER
applied to the scattering problem (L- A ) v = O
.
Some of the other striking
properties of the K d V equation are its Hamiltonian structure,an infinite number of conservation laws, an associated invariant (isospectral) scattering problem, and multi-soliton solutions.
Since the first investigations of the K d V
equation it has become apparent
that the K d V equation is not alone in possessing these properties. Other evolution equations with many of the same properties are the nonlinear Schrodinger equation i Yt + Yxx + I Y I
2
Y =0
, which was shown to share similar properties by
Zakharov and Shabat [ 101; and the Sine-Gordon equation, which was investigated by Ablowitz, Kaup, Newell, and Segur [l].
These three equations, as well as others, can all be obtained in the context of a gauge theory with gauge group SL(2.R) will be discussed in $4 ; and in
.
The gauge theory of these equations
$5 we discuss isospectral problems from the
general point of view of gauge theories on fiber bundles. In [81 we derived a generalization of the nonlinear Schrodinger equations for the gauge group
sC(3,C)
.
In $6 we derive multi-component analogues of the sine/sinh-Gordon
equations for a general semi-simple Lie algebra. We illustrate these equations explicitly for the case of algebras of rank two, namely A2,B2
and Gp
.
Similar equations have been obtained by Fordy and Gibbons [ 3 ] and Mikhailov
c.&1.
[61.
In the gauge theory point of view the Lax equation (1) arises as the condition of zero curvature of connection [D D ) x' t the form [Dx,Dt]= O
.
on a principal bundle: thus (1) takes
The role of the scattering operator L is then played
by Dx and that of the infinitesimal generator of the unitary transformations by Dt
.
We begin in $52,3 with a discussion of Wahlquist and Estabrook's notion of
prolongation albegras and show how the isospectral problems and the connection may be introduced from their point of view.
GAUGE THEORIES FOR SOLITON PROBLEMS
53
I wish t o thank Leon Green and P e t e r Olver f o r many u s e f u l d i s c u s s i o n s concerning t h e work i n t h i s paper.
f i o l o n g a t i o n Albegras [ 9 1
2.
The f i r s t s t e p i n t h e Wahlquist-Estabroak program i s t o r e p l a c e t h e evolution This
equation by an i d e a l of 2-forms closed under e x t e r i o r d i f f e r e n t i a t i o n .
procedure i s b e s t i l l u s t r a t e d by an example, say t h e sine-Gordon equation
.
u =sin u xt then, i n
We w r i t e t h i s as a f i r s t order system
x,t,u,p
, Cr2=dp A
d x + s i n u dx A d t
The f o u r dimensional space with coordinate
IR = [u,p]
over
(x,t,u(x,t)
,
X
space we introduce t h e two forms
a l = d u A d t - p dx A d t
2
u = p , p t = s i n u : and
R 2 = [ x , t]
.
p(x,t))
.
x,t,u,p
.
i s regarded as a bundle:
A s e c t i o n of t h i s bundle i s a graph of t h e form
When Crl
and
a r e r e s t r i c t e d t o such a s e c t i o n
Q12
they t a k e t h e form E1=(ux-p)dx A d t Thus,
Crl
B2=(sin u-pt)dx A d t
.
and a2 vanish on s o l u t i o n manifolds of t h e sine-Gordon equations.
fi1=E2 = O a r e equivalent t o t h e
According t o C a r t a n ' s theory, t h e equations p a r t i a l d i f f e r e n t i a l equation
u =sin u xt
i s closed under e x t e r i o r d i f f e r e n t i a t i o n . d Z 2 = c o s u dx A C r
clC2 = d t A Cr2
1
i f f t h e i d e a l generated by
(al,Cr2]
A simple computation shows t h a t
1 '
s o indeed t h i s i d e a l i s c l o s e d .
The Cartan i d e a l ( t h e i d e a l generated by t h e two forms; i n t h e c a s e above, t h a t generated by variables
Cr
1
and
) i s now prolonged by introducing a d d i t i o n a l
Q2
; one-forms
yl,...y,
(uk=dyk+Fk d x + G k d t 1 n and r e q u i r i n g t h a t [(u , . .(u ,
.
iation. variables
The functions u,p
Fk
(21
; Cx1,Cr2,.
and Gk
..]
be c l o s e d under e x t e r i o r d i f f e r e n t -
a r e assumed t o depend only on t h e dependent
( a s t h e case may b e ) and t h e
k k h =dF
A
k dx+dG A d t
y
1
. ..y n
.
I n t h e c a s e a t hand,
54
D. H. SA TTINGER
=
#
5 k
du R d x + #
U
P
dp A d x +
= G k du A d t +Gk dp A d t
P
du A d t = ’ Y l + p dx A d t
.
and
CX1
k
+
bY and
From (2) d y j A d x = o j A d x - G j d t A dx
Also, from t h e d e f i n i t i o n of
d y j A dx
aY
d y j A d t = d wj A d t - $ d x
,
(r2
.
dyj A d t
j
dp A d x = ’ Y 2
- sin
A
.
dt
u dx A d t
and
The prolongation equations a r e obtained by s u b s t i t u t -
i n g t h e s e i d e n t i t i e s i n t h e expression above f o r modulo t h e i d e a l generated by
(r1,a2 , and ok
and then s e t t i n g
duk
.
duk
I
0
I n t h e c a s e of t h e Sine-Gordon
equation we o b t a i n G =O
G =O
P sin u F = O P Prolongation equations f o r o t h e r w e l l known e v o l u t i o n equations have a l s o been U
[G,F] +flu-
derived. the
Wahlquist and Estabrook have derived t h e p r o l o n g a t i o n equations for and n o n l i n e a r Schrodinger equations. The problem now i s t o solve t h e
KdV
prolongation equations f o r v e c t o r f i e l d s
F
by Wahlquist and Estabrook f o r t h e
equation
KdV
.
and G
This was o r i g i n a l l y done
[ g l . The c a s e of t h e Sine-
Gordon equation i s much simpler ( s e e [ 8 1 ) .
3. I s o s p e c t r a l Problems Let us t u r n now t o t h e problem of f i n d i n g a s s o c i a t e d i s o s p e c t r a l problems f o r a given e v o l u t i o n equation.
We suppose t h a t v e c t o r f i e l d s
F
and
G
have been
found which s a t i s f y t h e prolongation equations and a r e l i n e a r i n t h e y - v a r i a b l e s : F=yjFjk
and
G =yjGjk
hY
3
.
This g i v e s us an isomorphism between t h e L i e
a l g e b r a oF-matrices and of v e c t o r f i e l d s . matrices
F=Fjk
and
G=G
k
3
by
L=bx-G
A Lax p a i r i s given i n terms of t h e
and
The a s s o c i a t e d
j
i s o s p e c t r a l s c a t t e r i n g problem i s given by
Ly = 0
Let us v e r i f y t h i s f o r t h e sine-Gordon equations. equations we have L =Fu +Fp =Fp t u t p t p t and
[B,L] = -[G,L] = -[G,ax
.
B=-G
- F]
or
bx
-F2yk = 0
.
From t h e p r o l o n g a t i o n
GAUGE THEORIES FOR SOLITONPROBLEMS
=
axG - [F,Gl
=
flu(flusin u F 1 =sin u F P P
55
The sine-Gordon equation thus comes from the Lax equation Lt = [ B , L ]
.
We can
get explicit realizations of the isospectral problems by taking specific representations of the Lie algebra so ( 3 ) .
For example, by choosing the representation of
so ( 3 ) given by the Pauli spin matrices
one is led to the scattering problem introduced by Ablowitz et. al. [11. (See [81).
4. Connections on Fiber Bundles; s t ( 2 , E ) The Lax equations can be interpreted from the point of view of gauge theories, as follows. Consider the case of the sine-Gordon equation, and let bx
, bt
be
the total derivatives on the jet bundle x,t,u,p
+... + p b bu + p x h p +... b = A+ u - b t bt t bu +Pthp Putting Dx = b, - F , Dt = at G , we have, from equations (3) b =- b
x
bx
-
[Dx,Dtl=[bx-F, bt-Gl =btF-bxG+[F,GI
-
=F
- sin u
p Gup +flu P t = (p, sin u)F
-
.
P
F
P
Thus the curvature [Dx,Dt] of the connection vanishes on an integral manifold
of the sine-Gordon equation.
It is useful to compare this formalism with the gauge theories of force fields. For example, in electromagnetic theory one introduces the connection D =b +eA +
P
P
where ~ = 0 , 1 , 2 , 3 ; the b
P
are the derivatives
, and the bxp
electromagnetic &-potentials. In electromagnetic theory the A
P
A
P
are the
are scalars,
for the gauge group of electromagnetic theory is the abelian group U(1) non-abelian gauge theories the A
LL
; but in
take their values in a Lie algebra. The
D.H. SATTINGER
56
commutator [D , D v ] = F I-1
F
=e(b A
CIV
P V
-b
.
A ) V P
gives the field strengths. In electromagnetic theory
I.Lv
Thus the condition of zero curvature [D , D V ] = O
means
CI
zero external field. The Lax equations may be interpreted as the vanishing of the curvature of a connection, o r , as we shall see below, as an integrability condition.
To see how to proceed let us take sL(2,lR)
as gauge group; sL(2,R) gauge
theories lead to the sine-Gordon, K d V and nonlinear Schrodinger equations. As basis for sC(2,IR) we may take
These matrices have the commutation relations [o1,a21 =2u2 [u~,,J~] =al
D,
.
, [u,,~~]=-20 3 '
Form the connection
= bx + i@,
- qu2 - ra3
Dt = b t - k 1 - b 2 - C b
3 -
Setting the curvature [Dx,D 1 = O we are led to the system of equations t -hxA + qC - rB = 0 qt - bxB - ?(qA
+ i@)
=0
r - b C+Z'(rA+iGC) = O t x which were obtained by Ablowitz &.
(5)
g.
[2].
The coefficients A,B,C depend
only on q,r and their derivatives. Ablowitz
e.&. construct solutions of
( 5 ) by expanding N
N N A=C CnAn , B = C , C = Z GnCn ; n=O n=O n=O these series may be substituted into ( 5 ) and the coefficients An'Bn.Cn for recursively.
6'
=
me
evolution equations are obtained as the coefficents of the
1 terms.
Examples: (1.1 Nonlinear Schrodinger equation r = 1 , q=V 2
2
A=-2i6 -ilYI B=26Y
+iYx
c=2p, E ( k - l ) i A r g g < + l 0
and Rk as Argy= :(k-l). asymptotic behavior c a y i n g whereas q.-(;)e-'
$
L e t Q=(J,,$) b e a fundamental s o l u t i o n m a t r i x o f ( 4 . 1 ) w i t h 0 i n S1, The s o l u t i o n $-(l)e i s r e c e s s i v e ( a s y m p t o t i c a l l y dei s dominant ( i t i s u n i q u e as i t i s d e f i n e d on t h e i n i t i a l
r a y c=O,where 8 i s i m a g i n a r y ) .
C o n t i n u i n g t o s e c t o r S2, t h e r e c e s s i v e s o l u t i o n $
becomes t h e dominant s o l u t i o n T2 i n t h a t s e c t o r ; however, i n s o l u t i o n J, i n S1 become t h e r e c e s s i v e s o l u t i o n $ i n S, we f a c t o r ( t h e stokes m u l t i p l i e r ) t i m e s t h e r e c e s s i 6 e s o l u t i o n mental s o l u t i o n m a t r i x Q. v h i c + has a s y m p t o t i c b e h a v i o r T Jtl preceding neighbor b y Qjtl
=
tJjMj,
o r d e r t h a t t h e dominant must add a c o n s t a n t $. I n g e n e r a l , t h e fundai n Sjtl i s r e l a t e d t o i t s
(4.5)
$. dominant, t h e n t h e nonzero o f f - d i a g o n a l element o f M . occupies t h e (1,Z) p o s i t i o n . J
J
77
THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION
We c a l l t h e s e t o f Stokes m u l t i p l i e r s { s j } &
t h e monodromy d a t a M f o r (4.1).
Since
t h e t o t a l monodromy around c=m i s t h e i d e n t i t y (as i t equals t h a t about c=O, an ord i n a r y p o i n t ) , o n l y (211-2) o f t h e s e t V a r e independent, F o r reasons o f symmetry we w i l l choose t o work w i t h t h e s e t i s k } f o r ksZ={l ,Z,---n-l , n + l Y - - 2 n - 1 ~ . One o f t h e c e n t r a l r e s u l t s o f o u r p r e v i o u s papers i s : g i v e n 0 j = l ,--2n w i t h asympjy t o t i c behav iors 3 which s a t i s f y ( 3 . l a ) , ( 3 . l b ) , ( 3 . 1 ~ )and ~ V . ( S.) d e f i n e d by (4.5),
J
J
then t h e stokes m u l t i p l i e r s a r e c o n s t a n t s , independent o f to,x,---t,-2. i s again pro v ed i n S e c t i o n 5.
T his r e s u l t
We t h e r e f o r e have a map f r o m o l d v a r i a b l e s
t o new v a r i a b l e s
new v a r i a b l e s t h e e q u a t i o n s a r e
k = 0,1,--n-2,tl
sj, tk = 0,
je2,
H j, t k =
j = 0,--n-2,
tjytk
0, = 6
jk
= x.
k = 0,1,--n-2.
j , k = 0,---n-2.
Our n e x t goal i s t o show t h a t t h i s map i s c a n o nical. I n o r d e r t o do t h i s , i t i s necessary t o express t h e i n f i n i t e s i m a l v a r i a t i o n s 6Q, 6P i n terms o f 6q, 6p. 5.
5a.
IYFINITESIMAL VARIATIONS AND THE t j DEPENDENCE o f Infinitesimal variations:
Sk.
Take t h e i n f i n i t e s i m a l v a r i a t i o n o f
and s o l v e b y v a r i a t i o n o f parameters. I n t e g r a t e t h e r e s u l t between cl, which l i e s n e a r
(5.18)
i n terms o f t h e i n f i n i t e s i m a l w r i a t i o n s o f t h e new c o o r d i n a t e s
Sk,6x,6t
First, recall that
6R
n- 2 dR-JVH 6 t -JVH16X- C JVHk6tk. 0 0 2
6Hl,6H.
jy
J'
(5.19)
Second, observe t h a t s i n c e
(5.20)
6T. = 6t' - t ( = [Hk,Hol appears anywhere).
partial
a r e o f t h e form t h e terms p r o p o r t i o n a l t o &to
= 0 ( s i n c e Ho i s independent o f tk and to never e x p l i c i t l y
I n these e q u a t i o n s we have a l s o r e p l a c e d VA,,
Hkx s i n c e t h es e a r e r e s p e c t i v e l y e q u a l .
= -. w =
d e r i v a t i v e o f H1 w i t h r e s p e c t t o
Akx
by VHk and
T A l s o we r e c a l l t h a t s i n c e J =J,<Ju,v>=
A f t e r a l i t t l e c a l c u l a t i o n we now f i n d , n-2 + 6x"6+H1 + 6t.A~H j* J
5
The terms p r o p o r t i o n a l t o 8 ( A 6 t . and 6 t . A 6 t k have as c o e f f i c i e n t s CHj,H1l J J r e s p e c t i v e l y , w hic h a r e zero.
(5.22) and CH H k'
3
83
THE INVERSE MONODROMY TRANSFORM IS A CANONICAL TRANSFORMATION
But from (5.13), - =
-
Nj,,G1SL62~j from
(5.91,
= ($S)T =
where P is t h e m a t r i x o f Poisson brackets.
By a j u d i c i o u s choice o f b a s i s U guch
t h a t <JUkyU,,> = 6 k , R - 1 , k 0 since U. - 0 at these mesh points. This implies that the wave can never piopagate to the right of zero, no matter how fine a mesh is used. Thus, we have a stable consistent approximation that can never converge to the true solution. If Eq. ( 5 0 ) is differenced directly using Eq. (6) with G = U3 then the resulting numerical solution will converge to the correct answer. Nonlinear Method Example Consider the unidirectional wave equation
with the solution u(x,t) = u(x+t,O). After discretizing, we approximate the space derivative using a nonlinear transformation method (Space Discretization, Grid Resolution). The transform interpolates the discrete solution with a monotonicity preserving interpolant The derivative of this that has a continuous first derivative 1131. interpolant is then used at the grid points to define the ODES [Eq. (3)].
We know that the true solution to Eq. (52) is monotone and we might expect that an interpolant that preserves this monotonicity will greatly improve the accuracy of the calculation. However, since the interpolant is monotone, U (xi) = 0, we have Ut (x i) = 0 at all the grid points. The solution never cKanges!
SUMMARY We have used a modular approach to develop accurate and robust methods for the numerical solution of PDEs. The methods to discretize the spatial operator, the boundary conditions, and the time variable, and solve any algebraic system that may arise are combined when writing a code to solve the PDE system. As the last two examples show, special care must be taken when solving a nonlinear equation or when using a nonlinear method. This means that the code must be field tested.
105
J.M. HYMAN
106
The field test is to check the reliability of the method on a particular nonlinear system of PDEs. The numerical results should be insensitive to reformulations of the equations, small changes in the initial conditions, the mesh orientation and refinement, and the choice of a stable accurate discretization method. Another excellent analysis tool is verification that any auxiliary relationships (such as conservation laws) hold for the numerically generated solution. These checks should be made even if one is absolutely, positively sure that the numerical solution and coding are correct.
-
When the above checks are made, the methods given here can be combined to give reliable efficient methods for solving a fairly large class of nonlinear PDEs. Also, by using the modular approach presented here, these codes can evolve efficiently since new methods can be quickly incorporated into the program. ACKNOWLEDGEMENT
I am grateful to Joe Dendy, Blair Swartz, and Burton Wendroff for providing much welcome advice in our many discussions during this work. REFERENCES [l] Wachspress, E. L., Iterative solution of elliptic systems (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1966).
Leventhal, S., and Weinberg, B., "The operator implicit method for parabolic equations," J. Comp. Phys.
[2] Ciment, M.,
compact (1978)
135-166. [3] Hyman, J. M.,
The method of lines solution of partial differential equations, Courant Institute of Mathematical Sciences report COO-3077-139 (1976).
[ 4 ] Hyman, J. M., A method of lines approach to the numerical solution of
-
conservation laws, Adv. in Comp. Methods for PDEs 111, Vichnevetsky, R., and Stepleman, R. S., (eds) Publ. IMACS (1979) 313-321. [5] Hyman, J. M., "The numerical solution of time dependent PDEs on an adaptive mesh,'' Los Alamos Scientific Laboratory LA-UR-80-3702 (1980). [6] Gelinas, R. J., Doss, S. K., and Miller, K., "The moving finite element
method: applications to general partial differential equations with multiple large gradients," J. Comp. Phys. 40 (1981) 202-249. [7] Berger, M., Gropp, W., and Oliger, J., Grid generation for time dependent
problems: criteria and methods, numerical grid generation techniques, NASA Conf. Proc. Publication 2166 (1980) 181-188. [8]
Hyman, J. M., Explicit A-stable methods for the solution of differential equations, Los Alamos Scientific Laboratory LA-UR-79-29 (1979).
[9] Young, D. M., Iterative solution of large linear systems (Academic Press, New York, NY, 1971). [lo] Brandt, A., Multi-level adaptive solutions to boundary value problems, Math. Comp. 2 (1977) 333-390.
NUMERICAL METHODS FOR NONLINEAR DIFFERENTIAL EOUATIONS
[ll] Concus, P., Golub, G. H., and O'Leary, D. P., Numerical solution of nonlinear elliptic partial differential equations by the generalized conjugate gradient method, Computing 19 (1978) 321-339.
[12] Manteuffel, T. A . , The Tchebychev iteration systems, Numer. Math. (1977) 307-327.
a
for nonsymmetric linear
[13] Hyman, J. M., Accurate monotonicity preserving cubic interpolation, Los Alamos National Laboratory report LA-8796-MS (1981).
107
This Page Intentionally Left Blank
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) @ North-Holland Publishing Company, 1982
109
LIMIT ANALYSIS OF PHYSICAL MODELS M. FREMOND Laboratoire Central des Ponts et Chaussees 5 8 bld Lefebvre 75732 PARIS Cedex 15 - France
-
A. F R I A ~ Ecole Nationale d'IngEnieurs de Tunis B.P. n"37, BelvBdSre TUNIS - Tunisie
-
-
-
M. RAUPP Centro Brasileiro de Pesquisas Fisicas avenida Wenceslau-Braz 71, CEP 2000, RIO DE JANEIRO
- Bresil -
Let us consider a physical model where some internal variables must belong to a given set in order for the model to be valid or not to breakdown. Our aim is to determine the external actions which achieve this condition. We present two examples. The first one describes a simple case in limit analysis of structures in mechanics. The second describes the breakdown analysis of an electrical circuit. In conclusion a common formalism is presented which can be applied to other situations [2]. AN EXAMPLE IN MECHANICS [2]
Let us consider the beam described on figure 1,
A
h
m A
n
n
n
+I
X
D
Figure The beam loaded by
he torque Af.
Two equal and opposite torques, with intensity A , are applied at its ends. The interior forces of the beam are the bending moment at any point x of the beam. We assume the beam is made of a material which cannot support moments greater than k. This threshold k can have several physical meanings, elastic limit, yielding, ... We look for the external loads, caracterized by the number A , such that there exist a moment which equilibrate them and which are lower than the threshold k. Let us recall several notions, most of them being classical in the strengh of material theory. The set of kinematically admissible displacements v(x)
is
110
M. FREMOND, A. F R I A ~M. RAUPP
v = {v I vE"**'(-R,+!2)
I
; v(-i) =v(O) -v(i)
=o}
+!?.
with the norm IlvII=
Iv"(x)ldx.
-!2
We use throughout this text the Sobolev spaces W Z s 1 , H 1, H2 , regularity of the functions involved in the equations [ I ] .
... which
describe the
The set of the external loads is V'the dual space of V. The applied torques Af are, indeed, elements of V' because *Af,v%*
:1
X(V'(&)-V'(-~))
=X
v"(x)dx,
(the double brackets denote the duality application between V and V'). The set of the strain rates D and the set of the interior forces S are respectively
s =.D' = L-(-I,+L).
D = ~'(-i,+i), The deformation operator ned by
E,
which is linear and continuous from V into D, is defiE(V)
Let us call
t
E
= v",
the transposed operator of
E.
The mechanical power developped by the moment M and the strain rate d is
If,
-<M,d>=-
M(x)d(x)dx.
The set 1 of statically admissible bending moments, i.e. the set of M which equilibrate an exterior load Xf, is I = { M E S 13XElR ; W v E V , -<M,~(v)>+*Af,v*= = { M Es
I M"=O
on ] - ~ , o [ u ]o,+L[
0 ; or
, M is continuous at o ; M(-E) =M(L)
Let us remark that W M E S , V v E V , =*Af,v*= This relation defines two linear applications
-L(M)
-
t ~(V)=hf}
-
A},
A(v'(i)-v'(-i)},
zE.C(lJR) , LEL(VJR)
by
-
X and L(v) = v ' ( i ) v' ( 4 ) .
The set G of the moments which can be supported is G={Ml MES
j
IM(x)l k or to 0 if Q G k . In K1 general $ is the indicator function of a convex set K1. Under reasonable assumpK1 tions K equals K1 [ 2 ] , [3] , [5].
Let us notice that KlC{Q
I Q*L(v)
-nG(~(v))
0, x
E
R , (0.7
X E
R.
Clearly, u I 0 i s a s o l u t i o n o f (0.7). Another s o l u t i o n , i n t h e almost everywhere sense, i s u = 0 f o r 1x1 > t and u = t 1x1 f o r t 1x1. can one d i s t i n g u i s h between these solutions?
-
HOW
Let us motivate t h e answer t o t h i s question given herein. To "solve" (0.6) one frequently uses t h e approximation o f i t by t h e regularized problem
119
VISCOSITY SOLUTIONS OF HAMlL TONJACOB1 EOUA TIONS
vt
H(vX)
t
V(X,O)
-
t
c v X X = 0,
uo(x),
)
X E
0, x
e
R ,
R ,
i n which E > 0 and c o n s i d e r s t h e l i m i t o f t h e s o l u t i o n o f (0.6) as, Assum6 T h i s i s an example o f t h e method o f " v a n i s h i n g v i s c o s i t y " . on
-
Hb,)
V t +
R 0 =
EVXX
1
7 (VVt)
U(R x ( O , C - ) ) ~
IP E
( t h e nonnegative
+
0.
C i functions
We have, on t h e s u p p o r t o f 7 ,
(0,m)).
x
Let
= 0.
E
1
+ H(;IPV~)
-
1
E
7qVXx
and (9v)t
9Vt =
9vx = ( 4 ,
- 9tV
-
9vxx = (9Vlxx
'pXV
-
I
y
2PXVX
- VXXV , 0
= (9Vlxx
29;
v - 9xxv - 76 ( 9 V l x + 9 *@
X
so t h a t
and (vv),,(x0,t0) t o conclude
- -9t 9
at E
+
V
+ H(-
2": - V ) - "(-2. v - -V ) 9X 9
9
qxx 9
(xo,tO). Assuming t h a t t h e s o l u t i o n 0 we t h e r e f o r e expect t h a t
- -" u 9
t
H(-
@X u) 9
on t h e s e t where
qu
< 0,
so we can
< 0 v
o f (0.6), t e n d s t o a l i m i t
< 0
u
as
(0.8)
assumes i t s maximum value.
I n t h e n e x t s e c t i o n t h e n o t i o n o f s o l u t i o n suggested by t h e s e c o n s i d e r a t ons i s presented i n g e n e r a l i t y t o g e t h e r with a few b a s i c r e s u l t s . Subsequently, n S e c t i o n 2, t h e r e l a t e d e x i s t e n c e and uniqueness t h e o r y i s i l l u s t r a t e d i n mode problems. D e t a i l s and p r o o f s a r e n o t g i v e n i n t h i s b r i e f i n t r o d u c t o r y e x p o s i t i o n . They may be found i n t h e paper [l] by t h e a u t h o r and P. L. L i o n s which i s t h e o r i g i n o f t h i s t h e o r y . We a l s o r e l y on [l] and P. L. L i o n s [6] o r t h e i r r e f e r e n c e s t o t h e v e r y s u b s t a n t i a l p r e v i o u s l i t e r a t u r e on t h e s u b j e c t o f Hamilton-Jacobi equations. Here we o n l y mention t h a t much o f t h i s l i t e r a t u r e i s concerned w i t h convex H a m i l t o n i a n s and o u r n o t i o n o f s o l u t i o n i n t h e g e n e r a l nonconvex case appears c o m p l e t e l y new. There i s some p a r a l l e l , however, w i t h t h e t h e o r y o f a s i n g l e c o n s e r v a t i o n law, e.g. [5], [7]. We s h o u l d a l s o m e n t i o n t h e works o f Fleming [S] and Friedman [4] i n t h e nonconvex case, wherein i t i s shown t h a t t h e v a n i s h i n g v i s c o s i t y method converges. When t h i s method converges, one
120
M.G. CRANDALL
has s i n g l e d o u t a p a r t i c u l a r s o l u t i o n and t h i s can be adopted as a ( n o n i n t r i n s i c ) d e f i n i t i o n o f a s o l u t i o n . F i n a l l y we mention [2] i n which e r r o r e s t i m a t e s a r e g i v e n which e s t a b l i s h t h e convergence o f t h e v a n i s h i n g v i s c o s i t y method as w e l l as a c l a s s o f d i f f e r e n c e a p p r o x i m a t i o n s t o o u r v i s c o s i t y s o l u t i o n s . SECTION 1.
VISCOSITY SOLUTIONS
L e t 0 be an open subset o f RM and F : 0 x R x RM .+ R be c o n t i n u o u s . u ). The e q u a t i o n We denote p o i n t s o f 0 by y and Du = (uyl. YM
...,
F(y,u,Du)
= 0
(1.1)
includes t h e equations
H(x,u,Du)
= 0
and
ut
H(x,t,u,Du)
t
= 0
by t a k i n g
M x R x R ) M denote t h e c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s on 0 and on 0 x R x R r e s p e c t i v e l y and convergence i n t h e s e spaces means u n i f o r m convergence on compact
y = x
i n t h e f i r s t case and
i n t h e second. C(0)
y = (x,t)
C"
s e t s . U(0)' denotes t h e nonnegative on compact subsets o f 0. If
Q
C(0)
E
C(O
0 which a r e s u p p o r t e d
we s e t
= Iy
E
0 : Q(y) = max J, 0
>
01
E-(Q) = { y
E
0 : ~ ( y )= min
0. I f u E C1(0), t h e n
-
121
VISCOSITY SOLUTIONS OF HAMIL TONJACOBI EOUATIONS
- k))
D(v(u
-
v
= 0
on
E+(v(u
np =
nu
on
-
so
k))
,
~ + ( v ( u- k))
e t c . It f o l l o w s t h a t c l a s s i c a l s o l u t i o n s o f F < 0, F > 0 and F = 0 a r e a l s o v i s c o s i t y s o l u t i o n s . It i s a l s o t r u e t h a t smooth v i s c o s i t y s o l u t i o n s a r e c l a s s i c a l s o l u t i o n s , as we see l a t e r . To s e t t h e stage f o r t h e next r e s u l t , c o n s i d e r t h e f o l l o w i n g problem: (ux)2
-
1 = 0,
- l < x < l
~ ( - 1 )= ~ ( i= )n
.
(1.4)
-
T h i s problem has a l a r g e s t L i p s c h i t z c o n t i n u o u s s o l u t i o n u = 1 1x1 and a s m a l l e s t L i p s c h i t z continuous s o l u t i o n u = 1x1 1 which s a t i s f y t h e equation I t has many o t h e r s o l u t i o n s , e.g. a.e.
-
un(-l) =
n,
u;(x)
= (-1)j
-
j = 0,...,4n
defines a s o l u t i o n
un
1
on
-
~ ( j2n)/2n,
(j
t
-
1
2n)/2n[,
,
such t h a t
0 < un < 1/2n
for
un + 0 u n i f o r m l y , b u t u z 0 i s n o t a s o l u t i o n o f c o n t r a s t , f o r v i s c o s i t y s o l u t i o n s we have:
n = 1,2,...
.
Clearly
(u,)~ = 1 anywhere.
In
( S t a b i l i t y o f t h e c l a s s o f v i s c o s i t y s o l u t i o n s ) . L e t {F,) be a M sequence i n C(O x R x R ) and u, be a v i s c o s i t y s o l u t i o n o f FQ = 0. L e t F, F i n C(O x R x R M ) and u, + u i n C(O x R x RM ). Then u i s a v i s c o s i t y s o l u t i o n o f F = 0.
Theorem 1.
+
once.
We p r o v e t h i s r e s u l t t o i l l u s t r a t e t h e n a t u r e o f D e f i n i t i o n 1 a t l e a s t For p r o o f s of subsequent r e s u l t s t h e reader i s r e f e r r e d t o [ll.
-
L e t q E -D(O)+, k E R and 9 E E+(v(u k)). Then v(Y)(u(Y) - k ) > 0 v(j)(u,(y) k) > 0 f o r l a r g e R. Thus E+(v(u, k ) ) f d f o r large and, by assumption, t h e r e e x i s t s yQ such t h a t
Proof.
-
ando
Passing t o a subsequence i f necessary, we assume y + i n (1.5) and (1.6) shows t h a t E Et(v(u -'k)) an% F(jsu(j),
-
Dv($)) < 0 p(i)
-
?E
0.
Taking t h e l i m i t s
.
Hence u i s a v i s c o s i t y s o l u t i o n o f F < 0. S i m i l a r l y , u i s a v i s c o s i t y s o l u t i o n o f F > 0, and hence a v i s c o s i t y s o l u t i o n o f F = 0. I n t h e above pcoof we used t h e m i l d requirements o f D e f i n i t i o n 1 i n t h a t k)) # @ we concluded o n l y t h e e x i s t e n c e o f from E+(v(u E E (v(u k)) f o r which t h e d e s i r e d i n e q u a l i t y held. However, one can prove D e f i n i t t o n 1 c o u l d
-
-
122
M.G. CRANDALL
be s t r e n g t h e n e d a g r e a t deal w i t h o u t a l t e r i n g t h e n o t i o n d e f i n e d . The n e x t r e s u l t o u t l i n e s t h e e x t e n t t o which t h i s i s t r u e . F o r $ E C(0) we l e t d($) = {y where
0 : D$(y)
E
"D$(y)
Theorem 2. F(Y,U,
exists)
e x i s t s " means
Let
-
IP
-
Ip
U,IP,$
II, i s ( F r e c h e t ) d i f f e r e n t i a b l e a t y.
C(0)
E
and
u
be a v i s c o s i t y s o l u t i o n o f
DIP + D$) c 0 everywhere on
F = 0.
Then
E,(V(U
-
$))
E-(IP(u
-
$ ) ) n d(lp) n d ( $ )
d(V) n d($)
and F(Y,u,
DIP
+
D$) > 0
everywhere on
.
Using Theorem 2 (and t h e p r o o f o f Theorem 2) one can p r o v e : Theorem 3.
If
(Consistency).
F(Y,u(Y),Du(Y))
=
u
i s a viscosity solution o f
F = 0,
then
0 on d ( u )
I n p a r t i c u l a r , a L i p s c h i t z continuous v i s c o s i t y s o l u t i o n o f s a t i s f i e s t h e e q u a t i o n p o i n t w i s e almost everywhere.
F = 0
As a f i n a l example o f t h e many r e s u l t s o f [l, S e c t i o n I], we c o n s i d e r t h e s i t u a t i o n i n which 0 i s d i v i d e d i n t o two open p a r t s 0- and 0, by a s u r f a c e
0.,
r.
i ( y ) be t h e normal t o r a t y u E C(0) be u- i n 0- and u,
Let Let
continuously d i f f e r e n t i a b l e i n
0,
E
r in
and assume i ( y ) p o i n t s i n t o 0, and assume u, is
respectively.
Let
u+
be c l a s s i c a l
F = 0 in 0 We ask how t h e jump i n nu -across r i s f' r e s t r i c t e d i n o r d e r t h a t u be a v i s c o s i t y s o l u t i o n o f F = 0 i n 0 ( i t i s assumed t h a t Du, extend c o n t i n u o u s l y t o r). The answer i s t h a t u i s a
solutions o f
viscosity solution i n
where
I
0
i f and o n l y i f :
i s t h e i n t e r v a l w i t h endpoints
means t h e i n n e r - p r o d u c t o f
a
and
b.
Du,(y)*;(y) and
pTDu*(y)
and
Du-(y)*i(y),
a-b
means t h e ( e q u a l ) p r o j e c -
t i o n o f Du,(y) and Du-(y) on t h e t a n g e n t space t o r a t y. T h i s c r i t e r i o n 1x1) o f (0.7) i n t h e i n t r o d u c t i o n and a l l s o l u r u l e s o u t t h e s o l u t i o n max(0,t 1x1, as t h e r e a d e r can v e r i f y . t i o n s o f (1.4) mentioned i n t h e t e x t except 1
-
SECTION 2.
-
MODEL EXISTENCE AND UNIQUENESS RESULTS
We now i n d i c a t e , i n model problems, some o f t h e v e r y g e n e r a l s t a t e m e n t s c o n c e r n i n g e x i s t e n c e and uniqueness o f s o l u t i o n s o f (DP) and (CP) w h i c h a r e v a l i d f o r v i s c o s i t y s o l u t i o n s . L e t us c o n s i d e r t h e Cauchy problem ut t H(Du) = 0, u(x,O)
t
>O,
X E
RN
= un(X)
where H E C(RN). L e t BUC(n) f u n c t i o n s on n. We have:
denote t h e bounded and u n i f o r m l y c o n t i n u o u s
VISCOSITY SOLUTIONS OF HAMIL TONJACOBI EOUATIONS
Theorem 4. u
E
C(RN
x
Let
(iii)
BUC(RN),
E
i s a viscosity solution of
u ( x , ~ ) = uo(x) u
Then t h e r e i s e x a c t l y one f u n c t i o n
with the following properties:
LO,-))
(i)u
(ii)
uo
BUC(RN
E
123
for
H ( D ~ )=
o
on
RN
H z 0 and
uo
x
(o,-).
x e R ~ .
[O,T])
x
+
ut
f o r each
T h i s r e s u l t i s c o m p l e t e l y new.
T
>
0.
I f , e.g.,
i s nowhere
d i f f e r e n t i a b l e , then u(x,t) u o ( x ) i s nowhere d i f f e r e n t i a b l e i n x. The a b i l i t y t o f o r m u l a t e an e x i s t e n c e and uniqueness r e s u l t f o r (2.1) i n o u r g e n e r a l i t y thus requires t h e a b i l i t y t o discuss n o n d i f f e r e n t i a b l e s o l u t i o n s o f t h e equation, and t h i s has n o t been p o s s i b l e before. However, t h e e x i s t e n c e a s s e r t i o n s can be proved by expanding on t h e arguments i n t h e i n t r o d u c t i o n concerning t h e r e l a t i o n s h i p o f t h e v a n i s h i n g v i s c o s i t y method and t h e n o t i o n o f v i s c o s i t y s o l u t i o n s , so we can adapt known methods here. The uniqueness i s t h e n t h e main new p o i n t . Theorem 4 a l l o w s one t o d e f i n e a f a m i l y o f mappings S ( t ) : BUC(RN)
+
BUC(RN) by l e t t i n g
S(t)S(T) = S ( t +
IS(t)uo(x
+
Y)
S(t)uo(*)
One can a l s o show t h a t f o r
s o l u t i o n o f (2.1).
be t,t
u(*,t)
where
> 0 and u o y v o
u f
i s the BUC(RN)
,
T)
- S(t)uoo()l
O
Here $,(x) operator.
i s an e l e c t r o n f i e l d with s p i n components a = 24 and
7:
i s a spin
The Hamiltonian represents a gas o f electrons i n t e r a c t i n g v i a s p i n exchange w i t h an i m p u r i t y l o c a l i z e d a t x=O, and described by t h e s p i n operator
3.
It has been extensively studied i n the p a s t [l] and many o f i t s p r o p e r t i e s were
understood.
Let me summarize those aspects t h a t a r e r e l e v a n t t o our problem.
To s t a r t w i t h , we s h a l l concentrate o n l y on q u a n t i t i e s t h a t are independent o f the c u t - o f f D.
I n other words, as i s t h e case with every renormalizable f i e l d
theory, requires a c u t - o f f i n order t o be p r o p e r l y defined. There are v a r i ous ways o f i n t r o d u c i n g c u t - o f f s and as long as they are kept f i n i t e , t h e r e a r e p r o p e r t i e s t h a t do depend on them and there are p r o p e r t i e s t h a t do not.
We
s h a l l i n v e s t i g a t e only t h e l a t t e r which are’ c a l l e d universal q u a n t i t i e s .
To e f f e c t t h i s we s h a l l r e s t r i c t the temperature T and t h e magnetic f i e l d H t o be small compared w i t h t h e c u t - o f f 0, (T, H>TH, i t approaches the f r e e value
creased.
and we have determined a new scale TH by absorbing t h e (log)-’ ( l o g ) - l term.
term i n t o the
When H+O, on t h e o t h e r hand, we see t h a t Mi approaches zero w i t h
a f i n i t e slope ( i . e . ,
susceptibility)
i n d i c a t i n g screening o f the i m p u r i t y spin.
The r a t i o between TH and To i s a uni-
versal number and i s given by
We have, then, achieved our goal t o c a l c u l a t e a u n i v e r s a l number and may have rested here had n o t Wilson chosen t o c a l c u l a t e a d i f f e r e n t r a t i o TK/To; b u t i t i s a c t u a l l y easy t o convert t o h i s number as b o t h TK and TH c h a r a c t e r i z e regions where p e r t u r b a t i o n theory i s v a l i d so t h a t t h e r a t i o can be e a s i l y calculated. 1 We f i n d TK where fin p = J dx(l-x)2x(n2csc2nx-x-2) and any = E u l e r ’ s
Ti-
0
constant.
Combining i t w i t h (13) we f i n a l l y have
TK
-
2g&2e-9/4
w=5-
= 0.102676 x 471
177
SOLUTION OF THE KONDO PROBLEM
which i s i n agreement w i t h the value found numerically by Wilson = 0.1032+0.0005
.
To summarize:
The Kondo model can be completely diagonalized and i t s essential physics exposed. One can c a l c u l a t e various universal numbers a n a l y t i c a l l y using our c o n s t r u c t i o n and f i n d them t o be i n agreement w i t h other ways o f c o n s t r u c t i n g t h e model [l]. O f course, using the exact s o l u t i o n one has access t o new and even more i n t e r e s t i n g physics.
I would l i k e t o thank J. H. Lowenstein f o r a close and f r u i t f u l c o l l a b o r a t i o n . REFERENCES:
[l] Wilson, K., Rev. Mod. Phys. [2] Anderson, P. W., [3] Gross H. 0.
47, 773 (1975).
Yuval, G.,
and Hamann, D. R.,
Phys. Rev.
E,
4464 (1970).
D., and Wilczek, F., Phys. Rev. L e t t . 30, 1343 (1973); P o l i t z e r , Phys. Rev. L e t t . 30, 1346 (1973).
[4] Gross O . , and Neveu, A., Phys. Rev. D
lo, 3235 (1979).
[5] Andre , N . , and Lowenstein, J. H., Phys. Rev. L e t t . [6] Andre , N., Phys. Rev. L e t t .
46, 356
(1981).
5 , 379 (1980).
[7] Wiegmann, P. B., Pis'ma Zh. Eksp. Theor. Fiz. 3 l , 392 (1980). [8] Bethe, H., Z. Phys.
c,205 (1931).
[9] Yang, C. N., Phys. Rev. L e t t .
[lo] Gaudin, M., Phys. L e t t .
19, 1312
e,55 (1967).
(1967).
This Page Intentionally Left Blank
Nonlinear Problems: Present and Future
179
A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.)
0 North-Holland Publishing Company, 1982
KINKS OF FRACTIONAL CHARGE I N QUASI-ONE-DIMENSIONAL SYSTEMS J. R. S c h r i e f f e r
Department o f Physics and I n s t i t u t e f o r Theoretical Physics University o f California USA Santa Barbara. CA 93106 Recent t h e o r e t i c a l studies have shown t h a t i n quasi-one-dimensional conductors having a P e i e r l s d i s t o r t i o n o f commensurability index n ( r a t i o o f the d i s t o r t i o n p e r i o d t o l a t t i c e spacing), t h e r e e x i s t excit a t i o n s whose charge i s Q = f 2e/n o r f 2e/n f e. These e x c i t a t i o n s are kinks i n the order parameter I$ describing t h e l a t t i c e d i s t o r t i o n . I n trans polyacetylene, n = 2 and Q = 0, f e with s p i n S = 4 0 respectively. For n = 3 (e.g., TTF-TCNQ a t 19Kb) Q = f 8 e, f e, f 4 e w i t h s p i n S = 0, 3, 0 respectively. Electronic states localized a t the k i n k have energies i n t h e P e i e r l s gaps. Properties o f these stable f r a c t i o n a l l y charged objects are discussed.
4
INTRODUCTION
F r a c t i o n a l l y charged e x c i t a t i o n s have o f t e n been considered i n t h e past. Rec e n t l y , however, i t has been shown how a theory c o n t a i n i n g o n l y fundamental p a r t i c l e s o f integer charge Q can lead t o e x c i t a t i o n s having charge q which i s a f r a c t i o n o f Q. I n a continuum model o f charge d e n s i t y waves i n one-dimensional metals i t was shown by Rice, Bishop, Krumhansl, and T r u l l i n g e r ' t h a t commensurability e f f e c t s can s t a b i l i z e kinks which c a r r y a charge q = Ae/n, where A0 = 2n/n, w i t h n being the commensurability, e.g., n = 2 f o r a dimerized system, n = 3 f o r a t r i m e r i z e d system. Independently, Jackiw and Rebbi2 studied a r e l a t i v i s t i c f i e l d theory model i n one dimension i n which a spinless Dirac f i e l d i s coupled t o a s e l f - i n t e r a c t i n g Bose f i e l d having two degenerate mean f i e l d ground states. They found t h a t the model possessed s o l i t o n e x c i t a t i o n s which correspond t o a change i n t h e number o f Dirac fermions i n the v i c i n i t y o f t h e k i n k being 4. A c-number Fermion s t a t e a t zero energy was found whose wavefunction was l o c a l i z e d about t h e s o l i ton. Depending on whether t h i s s t a t e i s occupied o r not, t h e s o l i t o n s would c a r r y Fermion number f 4, corresponding t o charge k e/2. Independently, J. Hubbard3 studied a one-dimensional t i g h t b i n d i n g chain w i t h very l a r g e o n - s i t e repulsions and weaker nearest neighbor i n t e r a c t i o n s . In t h i s l i m i t , the problem s p l i t s i n t o a s e t o f spinless Fermions and a s e t o f Heisensberg spins. Hubbard showed t h a t f o r one e l e c t r o n p e r two s i t e s on average (quarter f i l l e d band), an i n j e c t e d e l e c t r o n would s p l i t i n t o two kinks, each o f charge - e/2. A r e l a t e d problem w i l l be discussed by M. J. Rice i n t h e f o l l o w i n g paper. Motivated by experiments on trans (CH) , Su, S c h r i e f f e r , and Heeger (SSH)4 studied a one-dimensional model o f e l e c t r o n s hopping along a chain, w i t h t h e hopping i n t e g r a l l i n e a r l y modulated by l a t t i c e displacements. S i m i l a r studies were c a r r i e d o u t by Rice.6 Because o f t h e P e i e r l s d i s t o r t i o n , t h e mean f i e l d ground s t a t e i s degenerate. For the h a l f - f i l l e d band (one e l e c t r o n p e r s i t e on average) appropriate t o undoped (CH) , the ground s t a t e i s t w o f o l d degenerate. I n t h i s model, the symmetry b r e a k i n g ( l o s s o f i n v e r s i o n symmetry i n t h e phonon coordinates) i s dynamically generated by the electron-phonon coup1 i n g as opposed t o the imposed broken symmetry i n the r e l a t i v i s t i c model o f Jackiw and Rebbi.
J. R. SCHRIEFFER
180
For t h e (CH) model, i t was found t h a t t h e r e are low energy e x c i t e d s t a t e s correspondingXto s o l i t o n s which a c t as moving w a l l s separating domains having d i f f e r e n t ground states, A and 13. For each i s o l a t e d s o l i t o n , an e l e c t r o n i c s t a t e occurs i n the center o f the P e i e r l s gap w i t h a wavefunction centered about the s o l i t o n , as i n the Jackiw and Rebbi mode1.316 An important d i f f e r ence, however, i s t h a t i n (CH) e l e c t r o n s o f both s p i n o r i e n t a t i o n enter symm e t r i c a l l y i n t h e Fermi sea so & a t t h e charge q = f e/2 i s " s p i n masked," being doubled t o q = f e. These s t a t e s a r e o f s p i n zero, w h i l e f o r (CH) t h e r e i s an added charge s t a t e o f the s o l i t o n , q = 0 w i t h s p i n 4. Thus, i n s b a d o f f r a c t i o n a l charge being d i r e c t l y observable i n (CHI , t h e e f f e c t appears as an i n terchange o f t h e conventional charge-spin r e l a t i o n s o f e l e c t r o n i c e x c i t a t i o n s i n s o l i d s ; instead o f q = f e, S = 4 f o r conventional e l e c t r o n s and holes, one has q = f e, S = 0 f o r charged s o l i t o n s and q = 0, S = 4 f o r n e u t r a l s o l i t o n s . This breaking o f the charge-spin r e l a t i o n s o f t h e u n d e r l y i n g e l e c t r o n f i e l d i s c h a r a c t e r i s t i c o f the s o l i t o n e x c i t a t i o n s i n a degenerate ground s t a t e system. The o n e - t h i r d f i l l e d band ( t r i m e r i z e d chain) was s t u d i e d by Su and S c h r i e f f e r ' using t h e SSH hamiltonian. A consequence o f t h e t h r e e f o l d degeneracy o f the ground s t a t e i s t h e existence o f two types o f kinks, K1 and K, each having t h r e e charge states: 2e/3, -e/3, -4e/3 f o r K1 and -2e/3, e/3, and 4e/3 f o r K., The s p i n o f t h e f e/3 kinks i s S = 4 w h i l e t h e o t h e r kinks have S = 0. Thus, s p i n no longer masks f r a c t i o n a l charge f o r t r i m e r i z e d chains. Below I discuss the o r i g i n o f these r e s u l t s derived i n c o l l a b o r a t i o n w i t h W. P. Su.
TRIMERIZED CHAIN We consider the model Hamiltonian
H =
-
-u ) I ( c ~ + ~ , ~ c ~ , ~ + H . c+. )'2 K 2 (Un+l-un)2 n
t [to-u(un+l ns
+
'K2
2 un -2
(1)
wbere u i s t h e displacement o f the nth u n i t tfirom i t s e q u i l i b r i u m p o s i t i o n , un t and M i s the mass o f one i s &e n e l e c t r o n c r e a t i o n operator on t h e n c uRSt o f the chain. K i s an e f f e c t i v e s p r i n g constant d e s c r i b i n g an harmonic approximation t o the u bond energy. For t h e p e r f e c t l y t r i m e r i z e d chain, we w r i t e i n t h e ( a d i a b a t i c ) mean f i e l d approximation t h e most general f u n c t i o n having t h e t r a n s l a t i o n symmetry An = 3,
un = u
2n
C O S ( ~n
-e)
I n Fig. 1 the energy p e r s i t e i s p l o t t e d as a f u n c t i o n o f 0 f o r t h r e e values o f the amplitude u = 0, 0.0411 and 0.07x where, as an example, we have chosen t h e (CH) parameters t = 2.5 eV, 01 = 4.81 eV/h, K = 17.4 eV/X2. The energy i s minAs one can ' 0, f 2 d 3 , f 4n/3, mod f 271,. and uo 0.078. imizgd by 0 n/6 % see, t h e minimum energy path t o go between these minima i s by changing t h e phase angle e w i t h t h e amplitude e s s e n t i a l l y f i x e d . This leads t o a small b a r r i e r ( 2 t /A)Za. For t h e dimerized height o f order A 3 / t 2 and a s o l i t o n w i d t h E3 system, o n l y t h e ampfitude v a r i a b l e e x i s t s , and t h e eondensation energy passes through zero a t the center o f the k i n k , g i v i n g a b a r r i e r h e i g h t of order A2/t per s i t e , and a s o l i t o n w i d t h t2 ( 2 t /A) a. Hence, the w i d t h o f t h e t r i m e r e i z e d s o l i t o n i s l a r g e r than t h a t o? t h e dimerized system by t h e f a c t o r (2to/A) >> 1.
-
..
-
-
-
KINKS OF FRACTIONAL CHARGE IN QUASI-ONE DIMENSIONAL SYSTEMS
181
-2.75
-2
5Y
W
-2.76
-2.76
Figure 1: Total energy per s i t e p l o t t e d as a f u n c t i o n o f the phase angle 6 f o r t h r e e d i f f e r e n t values o f the amplitude o f t r i m e r i z a t i o n u.
To determine t h e charge and s p i n o f the kinks i n the o n e - t h i r d f i l l e d band case, one cannot use the charge conjugation symmetry d e r i v a t i o n which has been used f o r the dimerized system since t h i s symmetry gives no s i g n i f i c a n t i n f o r m a t i o n i n t h i s case. Rather, one can use 1) a Green f u n c t i o n method t o determine the charge and s p i n density o f the s o l i t o n or, 2) a simple b u t general argument based on t r a n s l a t i o n a l invariance and charge conservation. It i s t h i s l a t t e r argument we present below.
-
-
Consider an i n f i n i t e l y long chain, < n < m, w i t h 0 n/6 = 0, i . e . , the A ground state. Suppose t h a t 0 i s a d i a b a t i c a l l y defopmed so t h a t the chain remains i n the A s t a t e f o r an< n < nl, i s i n t h e B phase (0 n/6 = 2n/3) between nl and n2, i s i n the C phase (e n/6 = Zn). The change o f e near nl, n2, and n, need n o t be abrupt, b u t the t r a n s i t i o n region ( s o l i t o n wid!h) is assumed t o be small compared t o the spacings n2 nl , and n, n2. There w i l l be a c e r t a i n w i d t h 5, which w i l l minimize t h e energy. As a r e s u l t o f t h i s phase deformation, one has created 3 type I k i n k s ( i . e . , A-43, B-4, C+A).
-
-
-
-
-
To determine the charge and spin o f t h e kinks, suppose t h a t one constructs two surfaces, one a t n f a r t o the l e f t o f nl and another a t n f a r t o the r i g h t o f n,. From global c8nservation o f charge, we know t h a t t h e r t o t a l charge passing through these surfaces when 0 i s slowly deformed i s the negative o f t h e sum o f the charges o f the three s o l i t a n s , AQ, m
and j are the e l e c t r i c c u r r e n t d e n s i t i e s a t t h e two surfaces. Since n/& remaiis zero near n , no c u r r e n t flows a t t h i s surface, w h i l e t h e
where j 8
-
cRarge passing the right-hand %urface i s the e l e c t r o n i c charge i n one u n i t c e l l of s i z e 3a o f the t r i m e r i z e d chain, i.e., -2e and
AQ = 2e
.
(4)
182
J.R. SCHRIEFFER
However, from the invariance o f H under t r a n s l a t i o n s f. a, i t f o l l o w s t h a t a l l p r o p e r t i e s o f the t h r e e kinks must be t h e same. Thus, t h e fundamental charge o f type I kinks KO i s
AQ = 39
= 2e
(5)
KO
or
QKo
_- -2e 3
a
A s i m i l a r argument holds i f t h e phase angle 0 i s decreased as one moves t o t h e r i g h t so t h a t one passes from A t o C t o B t o A moving from l e f t t o r i g h t , w i t h 0 n/6 = 0, - 2n/3, 4n/3, 2n. The e l e c t r i c c u r r e n t i s no_w i n t h e reverse d i r e c t i o n , and the fundamental charge o f these type I1 kinks KO ( a n t i k i n k s ) i s
-
-
-
Since s p i n up and s p i n down s t a t e s a r e i d e n t i c a l l y occupied d u r i n g t h e a d i a b a t i c deformation, t h e s p i n o f KO and KO i s zero,
s
=s- = o
.
KO
Figure 2: The three degenerate ground s t a t e s o f a p e r f e c t l y t r i m e r i z e d chain.
KINKS OF FRACTIONAL CHARGE IN OUASI-ONEDIMENSIONAL SYSTEMS
183
As i n the dimerized case, l o c a l i z e d s t a t e s ar e expected i n the e l e c t r o n i c energy gap connected w i t h the s o l i t o n s since st at es are missing from the valence band, leading t o t h e f r a c t i o n a l charge. Using a Green f u n c t i o n method one can determine t h e electronic-spectrum i n t h e presence o f kinks. I f one imposes a sharp kinks K o r a n t i k i n k K on the system, one f i n d s t h e spectrum shown i n Fig. 3. For the kink, there are two gap states Q and Q l ocate d i n the upper h a l f o f t he lower gap and symmetrically i n the l%wer ha# o f the upper gap. D i r e c t c a l c u l a t i o n o f t h e f r a c t i o n a l parentage coe f f i c i e n t s o f these st at es shows t h a t f o r K, Q i s derived 1/3 from t h e bottom band and 2/3 from t he middle band, w h i l e I) conks 1/3 from t h e t o p band and 2/3 f r o m the middle band. Since 1/3 o f a s t a t d i s removed from the bottom band by K and a l l valence band st at es remain f i l l e d as K i s created, i t f o l l o w s t h a t i f I) and I) are unoccupied (K ), then the charge o f K i s Q = 2/3 e, i n agreem%nt w i d (6) and (8). Howher, i f I), i s s i n g l y octupied &en
and i f 6, i s doubly occupied, then 4e
Q K 2 = - 3
'
K2 = O
Figure 3: Gap sta tes associated w i t h (a) a sharp t y p e - I k i n k and (b) a sharp type- II kink.
J. R. SCHRIEFFER
184
S i m i l a r l y , the a n t i k i n k i corresponds t o having no holes i n I),, trons, so t h a t i t s c h a r g e y s
i.e.,
2 elec-
I n (ll), the f a c t o r o f two m u l t i p l y i n g 2e/3 a r i s e s from i n agreement w i t h (7). two s p i n o r i e n t a t i o n s i n accounting f o r the missing charge i n t h e valence band due t o the removal o f 2/3 o f a s t a t e per gpin by an a n t i k i n k from t h e lowest band. Since a l l s t a t e s are s p i n p a i r e d f o r KO, i t f o l l o w s t h a t
F i n a l l y , i f I), i s s i n g l y occupied i n
kI, then
and i f I), i s empty, then
~ i = + : e 2 We see t h a t K, charge.
,s- = o K2 and i(, a c t as a n t i p a r t i c l e s , having t h e same s p i n b u t reversed
As i n t h e dimerized case, t h e r e are c o n s t r a i n t s on c r e a t i n g s o l i t o n s . I f the chain i s t o be unaffected a t i n f i n i t y , t o p o l o g i c a l constrai-Ecs r e q u i r e t h a t kinks must be created i n p a i r s (KK o r KK), t r i p l e t s (KKK o r KKK), o r combinat i o n o f these allowed sets. From t h e k i n k quantum numbers (Q,S),
one can see t h a t f r a c t i o n a l charge and s p i n cannot be created g l o b a l l y b u t o n l y l o c a l l y . Hence, one would n o t be able t o observe t h i s type o f f r a c t i o n a l charge by measurements sampling the t o t a l charge o f t h e system. Rather, t h e f r a c t i o n a l charge r e f l e c t s an i n t e r n a l d i s t r i b u t i o n o f an i n t e g e r t o t a l charge o f t h e system. One method f o r observing f r a c t i o n a l charge i n polymers i s through t h e shot noise voltage f l u c t u a t i o n spectrum. For nth order commensurability, t h e p r i m a t i v e k i n k quantum numbers a r e
KINKS OF FRACTIONAL CHARGE IN QUASI-ONE DIMENSIONAL SYSTEMS
QK,
-- 2- e = _ n
Q and i0 S
As above, KO and states.
= S-
KO
KO
=
0
185
.
KO
can be decorated by electrons and holes i n the . J c a l i z e d
REAL OR APPARENT FRACTIONAL CHARGE? The question has been r a i s e d whether t h e f r a c t i o n a l charge discussed above i s a quantum mechanically sharp observable o r simply t h e quantum average o f two o r more i n t e g e r charges, t h e r e s u l t o f which i s f r a c t i o n a l . 8 I n other words, i f one can d e f i n e a quantum operator Q which measures t h e charge o f the s o l i t o n , then i s the p r o b a b i l i t y d i s t r i b u t i o W ) P(Q) o f t h i s operator i n t h e one s o l i t o n sector a d e l t a f u n c t i o n a t the f r a c t i o n a l k i n k charge QK
o r i s i t given by
where
91, Q,,
.. .
are integers,
and o n l y the expected charge i s f r a c t i o n a l
?
= 1 aiQi = QK i
(19)
An exapple of apparent f r a c t i o n a l charge, (18) and (19), i s given by considering the H, ion. Suppose the e l e c t r o n i s i n the bonding (even p a r i t y ) molecular o r b i t a l JI+ and the i n t e r n u c l e a r spacing i s a d i a b a t i c a l l y increased t o a very I n t h i s l i m i t $+ reduces t o a l i n e a r combination o f 1s atomic l a r g e value. o r b i t a l s JI centered on t h e two protons 2 and r, 1 1 *+=-$g+-$r
J Z J Z
*
I f Qop = n2 i s the occupation number f o r $,,
then
has nonvanishing m a t r i x elements between $+ and t h e n e a r l y However, because Q degenerate antibonafng s t a t e $-
J.R. SCHRIEFFER
186
i t f o l l o w s t h a t JI+ i s n o t an eigenfunction o f Q and the mean square f l u c t u a t i o n i s given by OP
For t h i s simple example,
The usual quantum theory o f measurement i n t e r p r e t a t i o n o f t h i s f i c t i c i o u s f r a c t i o n a l charge i s t h a t w h i l e t h e average (expected) charge i s f r a c t i o n a l , o n l y integer charge w i l l be measured i n any s i n g l e experiment. How i s t h i s r e l a t e d t o the kinks? The e s s e n t i a l p o i n t i s t h a t the o f f diagonal elements o f the k i n k charge operator vanish e x p o n e n t i a l l y i n the l i m i t o f w i d e l y spaced kinks, r a t h e r than appoaching a constant [4, see Eq. (23)] f o r t h e H, example. Therefore t h e one s o l i t o n s t a t e approaches an eigenfunction o f Q f o r widely spaced kinks. This c r u c i a l d i f f p r e n c e a r i s e s from t h e f a c t t h a t thepe i s a unique e l e c t r o n i c ground s t a t e o f KK f o r any spacing, w i t h no low l y i n g exc i t e d states, a l l e x c i t e d s t a t e s having an energy o f order o r g r e a t e r than A . Furthermore, even these s t a t e s are n o t e x c i t e d by a very long wavelength potent i a l since a slowly varying p o t e n t i a l o n l y couples t o the t o t a l charge o f the system i n the v i c i n i t y o f t h e kink. Therefore, t h e fundamental f r a c t i o n a l charge o f the k i n k i s sharp and a r i s e s t o t a l l y from t h e d e p l e t i o n o f s t a t e s from the f i l l e d valence band when a s o l i t o n i s formed.
I f Qop measures the up s p i n charge on the s o l i t o n , then f o r KO,
A t o t a l l y separate b u t somewhat confusing question i s the spacial d i s t r i b u t i o n o f charge located i n t h e gap states. The discussion i n t h e previous paragraph concerns c o n f i g u r a t i o n s i n which there are no e l e c t r o n s i n t h e gap center s t a t e s o r when these s t a t e s are completely f i l l e d . I n t h i s case t h e r e i s no degeneracy, and a slowly vgrying f i e l d cannot admix e x c i t e d states. Suppose, however, t h a t we consider KIK so t h a t t h e r e i s one s p i n up d e c t r o n i n t h e gap center s t a t e o f K. This con9iguration i s degenerate w i t h K K1; however, t o r f i n i t e spacing there i s mixing which produces the analog o f t h 8 $ + s t a t e o f H
Jz
K
ii l o
1
-
+ - KoK1
Q
.
I n t h i s case, t h e gap center e l e c t r o n f l u c t u a t e s between K and f l u c t u a t i n g K charge whose p r o b a b i l i t y d i s t r i b u t i o n i s
k
leading t o a
187
KINKS OF FRACTIONAL CHARGE IN QUASI-ONE DIMENSIONAL SYSTEMS
which i s t o be compared w i t h (24) showing a sharp d i s t r i b u t i o n a t Q = +. Clearly, the f r a c t i o n a l charge associated w i t h t h e vacuum d e f i c i t ( o r vacuum p o l a r i z a t i o n ) i s a sharp quantum number, and i t i s t h i s we r e f e r t o when d i s A f i c t i c i o u s f r a c t i o n a l charge can e n t e r from a cussing f r a c t i o n a l charge. t o t a l l y d i f f e r e n t and u n i n t e r e s t i n g e f f e c t , namely t h a t analogous t o charge f l u c t u a t i o n s i n H,. CONCLUSIONS While the noninteger charge e f f e c t s discussed above w i l l be screened by the d i e l e c t r i c constant o f the medium, as i n any s o l i d the n e t charge macroscopically c a r r i e d i s the unscreened charge. The shot noise experiment should measure t h i s charge. F r a c t i o n a l i z e d quantum numbers may be observable i n other systems, such as domain w a l l s i n magnets and 3He texture^.^ Recently, Goldstone and WilczeklO have shown how these ideas can be generalized t o r e l a t i v i s t i c f i e l d t h e o r i e s i n higher dimensions. ACKNOWLEDGMENTS The author i s g r a t e f u l t o Drs. S. Kivelson and W. P. Su f o r h e l p f u l discussions. This work was supported i n p a r t by the National Science Foundation under Grant No. DMR80-07432, and Grant No. PHY77-27084. REFERENCES
1.
M. J. Rice, A. R. Bishop, J . A. Krumhansl, and S. E. T r u l l i n g e r , Phys. Rev. L e t t . 2 , 432 (1976).
2.
R. Jackiw and C. Rebbi, Phys. Rev. D.
3.
J. Hubbard, Phys. Rev. B.
4.
W. P. Su, J. R. S c h r i e f f e r , and A. J. Heeger, Phys. Rev. L e t t . (1979); Phys. Rev. B 11, 2099 (1980).
5.
M. J. Rice, Phys. L e t t .
6.
For a review o f the r e l a t i o n between f r a c t i o n a l l y charged kinds i n condensed matter physics and r e l a t i v i s t i c f i e l d theory, see R. Jackiw and J. R. S c h r i e f f e r , Nucl. Phys. @ [FS 31, 253 (1981).
7.
W. P. Su and J. R. S c h r i e f f e r , Phys. Rev. L e t t . 4 6 , 738 (1981).
8.
The author i s indebted t o D r . A. K. Kerman and V. Weisskopf f o r r a i s i n g t h i s question. The discussion below r e l i e s on work o f D r . S. A. Kivelson and the author, t o be published.
9.
Jason Ho, p r i v a t e communication.
10.
F. Wilczek and J. Goldstone, t o be published.
2,3398
(1976).
lJ, 494 (1978).
42,
1698
E,152 (1979).
This Page Intentionally Left Blank
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (4s.) @ North-Holland PublishingCompany, 1982
189
THEORETICALLY PREDICTED DRUDE ABSORPTION BY A CONDUCTING CHARGED SOLITON I N DOPED-POLYACETYLENE M. J . Rice Xerox Webster Research Center Webster , NY 14580, USA With the a i d o f a novel quantum theory o f t h e charged s o l i t o n we conclude t h a t the observation o f a " d i s c r e t e Drude absorption'' would c o n s t i t u t e an experimental v e r i f i c a t i o n o f charged-sol i t o n t r a n s p o r t i n doped-polyacetylene. The suggestion [1,2] t h a t charged s o l i t o n s may be generated i n trans-polyacet y l e n e [ ( C H I ] by l i g h t acceptor o r donor doping has stimulated much subsequent experimental 'and t h e o r e t i c a l i n t e r e s t i n t h i s subject. [3] A recent development o f considerable i n t e r e s t has been the r e p o r t by the Penn group [4] o f a range o f dopant concentration f o r which the homogeneously doped polymer e x h i b i t s metal 1i c c o n d u c t i v i t y w i t h an e s s e n t i a l l y zero P a u l i s p i n s u s c e p t i b i l i t y . Although elect r o n t r a n s p o r t a r i s i n g from a small b u t f i n i t e Fermi-level d e n s i t y o f l o c a l i z e d one-electron states cannot be r u l e d out, [ 5 ] t h e conclusion t h a t the m e t a l l i c l i k e d.c. c o n d u c t i v i t y r e s u l t s from t h e t r a n s p o r t o f charged-solitons i s s e r i ously suggested f o r the f i r s t time. I n order t o conceive o f an experiment t h a t could d i s t i n g u i s h between s o l i t o n and e l e c t r o n t r a n s p o r t we have t h e o r e t i c a l l y investigated the Drude absorption o f a conducting charged-soliton i n (CH) . Remarkably, we f i n d t h a t i n a d d i t i o n t o the usual Drude absorption expected fir a d i f f u s i n g f r e e - c a r r i e r , the charged s o l i t o n ' s Drude absorption possesses a v i b r a t i o n a l component which leads t o a d i s c r e t e absorption band a t t h e frequency w. o f the s o l i t o n ' s i n t e r n a l breathing mode.[6,7] This band i s a consequence of tAe coupling o f t h e i n t e r n a l and t r a n s l a t i o n a l motions o f t h e s o l i t o n which endows the mobile charged s o l i t o n with an o s c i l l a t i n g e l e c t r i c d i p o l e moment proportional t o i t s mean v e l o c i t y o f t r a n s l a t i o n . The observation o f t h i s " d i s c r e t e Drude absorption'' would c o n s t i t u t e an experimental v e r i f i c a t i o n of charged-sol i t o n t r a n s p o r t i n doped-(CH),. I n order t o c a l c u l a t e t h e charged s o l i t o n ' s Drude absorption we employ t h e r e s u l t s o f a quantum theory which we have developed f o r the charged s o l i t o n . The l a t t e r theory, which w i l l be published i n an independent a r t i c l e , [ 8 ] i s a s t r a i g h t f o r w a r d extension t o quantum mechanics o f a c l a s s i c a l Hami l t o n i a n theory o f the s o l i t o n r e c e n t l y introduced by Rice and Mele.[6] I n t h e quantum theory the s o l i t o n i s described by the wavefunction +(x,a) where ( $ ( ~ , ! 2 ) ( dxda ~ determines the p r o b a b i l i t y t h a t the center o f the s o l i t o n be found between t h e spac i a l p o i n t s x and x + dx w i t h a l e n g t h between 2 and Q + &.. The frequencydependent c o n d u c t i v i t y o f the free-charged s o l i t o n , u(w), i s
where IE,) denote the energy eigenvalues o f the f r e e s o l i t o n , R denotes t h e volume o f t h e system, P(Ea) = exp(-pE,)/Z,exp(-pE,) a b i l i t y o f f i n d i n g the s o l i t o n i n t h e s t a t e with energy E, T = 3/kBp, and v (4) i s t h e m a t r i x element YO! c a m
v
Y,
(9) = i
J dZ! J dx V* (x,!L)v(q,!Z)V,(x,ll) 0
-00
Y
= Ey - ,E, YP i s t h e proba t temperature w
190
M.J. RICE
I n (3) M (Q)denotes the c l a s s i c a l length-dependent t r a n s l a t i o n a l mass o f t h e s o l i t o n f6] and p, = iMVx i s the s o l i t o n ' s t r a n s l a t i o n a l momentum operator. The energy eigenvalues o f t h e f r e e s o l i t o n are determined by t h e Schrodinger equation Wa(x,2)
= EaVa(x,Q)
(4)
where t h e Hamiltonian operator H i s H = (1/4) (Mi(!2)-lp;
+
pa Mi(2) -1)
Here, Mi(a) denotes the c l a s s i c a l length-dependent i n t e r n a l i n e r t i a l mass-of the s o l i t o n , [ 6 ] p = i Va i t s i n t e r n a l momentum operator, and Vi(!2) = A!2 + Be i t s i n t e r n a l po?ential oenergy. A and B are p o s i t i v e constants which determine and e q u i l i b r i u m dength 2! o f the c l a s s i c a l s t a t i c s o l the formation energy E i t o n according t o the f a k i l i a r r e l a t i o n s E = 2dAB and 2 = JA/B. I f we denote by MS and M. the t r a n s l a t i o n a l and interna? i n e r t i a l magses o f t h e l a t t e r s o l i t o n , we haJe according t o the c l a s s i c a l Hamiltonian theory, [ 6 ] M.(11) = M.!2 /Q and M ( a ) = M Q /a. An extension o f (4) and (5) t o provide a q u a h a 1 des&r?pt i o n %f a chdrled s o l i t o n bound t o an i o n i c dopant molecule w i l l be discussed elsewhere. [8] The eigenspectrum defined by Eqs. (4) and (5) and t h e l a t t e r s t a t e d Q-dependences o f the c l a s s i c a l masses, may be solved f o r e x a c t l y t o y i e l d t h e f r e e s o l i t o n eigenvalues
2 4 2 Zk2)1/2 = [l + ( Z / K ) (n + 1/21] (M sc + co
.
where n = 0,1,2,. . , i s an i n t e r n a l v i b r a t i o n quantum number and k = (A/L)s, w i t h s = f l,&Z,..., s p e c i f i e s t h e q u a n t i z a t i o n o f t& & o l i t o n ' s t r a n s l a t i o n a l and ~ / 2denotes t h e momentum along a chain o f l e n g t h L (L +$). M c20 dimensionless parameter ~ / 2= (2M.Q A/ 2 ) . Thd l a t t e r ' $ magnitude s p e c i f i e s the e x t e n t t o which t h e nature of'tt?e i n t e r n a l motion o f t h e s o l i t o n i s quantum mechanical. C l e a r l y , i n the l i m i t K + 01, (6) y i e l d s t h e c l a s s i c a l energy spectrum o f the s o l i t o n . For (CH)x we estimate ~ / 2 3. The corresponding eigenfunctions are
where
191
THEORETICALLY PREDICTED DRUDE ABSORPTION
i n which, f o r j # 0, the c o e f f i c i e n t s a . are determined by t h e recurrence r e l a t i o n aj+l = - a . ( n - j ) / [ ( j + l ) ( j + l + ~ ) ] andJao by the normalization o f y k(x,!2). I n (7a) Ak denotes J the c h a r a c t e r i s t i c l e n g t h Ak = Q , ~ - l [ l + (
~ / M , C ~ ) ~4 ] . With
the use o f t h e r e l a t i o n K = 2Eo / w., where IN. = 2(B/2M.2 )JI i s t h e c l a s s i c a l b r e a t h i n d mode o f t'h8 s t a t i c s o l i t o n , (6) harmonic frequency[6] o f the $rnaf may be expanded f o r 2k2/MS = 2 (K+2n+l)/K f o r the band n. This increase i n w i t h v i b r a t i o n a l e x c i t a t i o n i s rgsponsible f o r the corresponding decrease i n M The remarkable c o n t r a s t between t h e simp l e harmonic nature o f t h e v i b r a t f o n a l eigenspectrum o f (8) and t h e decidedly anharmonic nature o f the associated wavefunctions (7b) w i l l be commented on i n an independent publication. [a] With the use o f (81, (7) and the assumption t h a t u(w) may be evaluated t o y i e l d ~ ( w )= uo ( l - i w / r ) - '
EoS
and wi>>kBT,
eq. (1) for
+ (kgT/ wi)
where w lnnl
=
I
dwn(e)e
*,I
(2)
0
= -(l+~)-"*. which, from eq. (7), gives 2 / 2 I n evaluating (1) we have i n troduced a phenomenological Ed)ns#nt Drude l i f e t i m e T = Iand a n a t u r a l w i d t h r. f o r - t h e i n t e r n a l breathing mode o f t h e s o l i t o n . u i s t h e d.c. c o n d u c t i v i t y u' = R le2r/M and the f i r s t term o f (9) c o r r e s p o n d t o t h e Drude absorption o p d i n a r i l y exbgcted f o r a d i f f u s i n g f r e e - c a r r i e r . The second term i s t h e d i s c r e t e Drude absorption a t wi a r i s i n g from the coupling o f the s o l i t o n ' s i n t e r n a l and t r a n s l a t i o n a l motions. I t s o r i g i n i s c l e a r l y apparent from a c l a s s i c a l i n t e r p r e t a t i o n o f the s o l i t o n ' s v e l o c i t y operator (3). Since, c l a s s i c a l l y , $ i s o s c i l l a t i n g about i t s e q u i l i b r i u m value 2 , the v e l o c i t y v = p /MS(2) o f t h e s o l i t o n i n a s t a t e o f constant t r a n s l a t i 8 n a l momentum p has %n o s c i l l a t o r y component. As the s o l i t o n i s charged i t consequently i s etdowed w i t h an o s c i l l a t i n g e l e c t r i c d i p o l e moment which i s p r o p o r t i o n a l t o i t s mean v e l o c i t y o f translation. The i n t e g r a t e d o s c i l l a t o r strength o f the d i s c r e t e Drude absorpt i o n , S. = S ( k T/ w.) ( l + ~ ) - l , i s t h q e f o r e , understandably, p r o p o r t i o n a l t o the absblute' t e d e r a t l r e T S = ( a e2/2MSo) i s t h e i n t e g r a t e d o s c i l l a t o r strength o f t h e ordinary Drude abkorption.
M.J. RICE
192
I
0
0
I
I
1
2
3
w/w,
Figure 1: The r e a l and imaginary p a r t s o f o(w)/ao vs w/wi f o r the parameter values s t a t e d i n t h e t e x t .
computed from eq. (9)
I n Fig. 1 we have p l o t t e d t h e r e a l and imaginary p a r t s o f a(w)/a as a f u n c t i o n o f w/w. f o r t h e reprgsentative choice o f parameter values r =%., r . = w./5, k T = u] /5 and K = 2E /w. = 6. To date, an accurate microscopit c a l c u l a t i o n o! wi ?or the chargedSso\iton i n (CH) has n o t been undertaken. Following a phenomen-qlogical approach, Rice and A l e [ 6 ] have established t h e estimate 1280 cm , whereas Su e t a l . , [7] have a r r i v e d a t t h e r e s u l t wi 780cm on the basis o f a s i m p l i f i e d microscopic model o f (CH),. An experimental search f o r t h e d i s c r e t e Drude absorption would be made d i f f i c u l t i f the value o f w . l i e s close t o e i t h e r o f t h e two dopant-dependent i n t e n s e l y I R activ_e modes albeady observed [9] f o r inhomogeneously doped-(CH) a t 900 and 1370 cm l , r e s p e c t i v e l y . The l a t t e r modes have been i d e n t i f i e d flO] as spec i f i c i n t r i n s i c s t r u c t u r a l v i b r a t i o n a l e x c i t a t i o n s o f t h e charged s o l i t o n s . It i s hoped t h a t t h i s paper w i l l s t i m u l a t e an experimental search f o r t h i s poss i b l e absorption band.
F i n a l l y , i t i s i n t e r e s t i n g t o note t h a t the r e l a t i v e l y small value o f ~ / 2f o r (CH) i m p l i e s t h a t t h e bond a l t e r n a t i o n amplitude o f n a t u r e ' s simplest polymer has \he character o f a quantum f i e l d . The relevance o f t h e quantum f i e l d theory l i t e r a t u r e [ll] t o t h e present problem i s t h e r e f o r e c a l l e d t o a t t e n t i o n . We thank G i l l Stengle, John Bardeen and Eugene Mele f o r several u s e f u l d i s cussi ons. REFERENCES
[l] Rice, M. J., Phys. L e t t . 7 l A , 152 (1979).
[2] Su, W. P., S c h r i e f f e r , J. R . , and Heeger, A. J., Phys. Rev. L e t t . (1979). (31 See, e.g.,
2, 1698
Proceedings o f I n t e r n a t i o n a l Conference on Low-Dimensional Synt h e t i c Metals (Helsingor, Denmark, August 10-15, 19801, t o be published i n Chemica S c r i p t a (1981).
193
THEORETICALLY PREDICTED DRUDEABSORPTION
[4] Ikekata, S., Kaufer, J., Woener, T.,
Pron, A., Druy, M. A., Sirak, A., Heeger, A. J., and MacDiarmid, A. G., Phys. Rev. L e t t . 45, 1123 (1980): see, also, Peo, M., Roth, S. and Hocker, J., Proc. I n t e r . Conf. on LowDimensional Synthetic Metals (Helsingor, Denmark, Aug. 10-15, 1980) t o be published i n Chemica S c r i p t a (1981); Epstein, A. J., Rommelmann, H., Druy, M. A., Heeger, A. J., .and MacDiarmid, A. G., p r e p r i n t 1980.
[5] Mele, E. J., and Rice, M. J., Phys. Rev. B (submitted). [6]
Rice, M. J., and Mele, E. J., S o l i d State Commun.
[7] Su, W. P., (1980).
S c h r i e f f e r , J. R.,
and Heeger, A.
2, 487
(1980).
J., Phys. Rev. B
22,
2099
[8] Rice, M. J., and Mele, E. J., t o be published. [9] Fincher, C. R., Jr., Ozaki, M., Heeger, A. J., and MacDiarmid, A. G., Phys. Rev. B 3, 4140 (1979). [lo] Mele, E. J., and Rice, M. J., Phys. Rev. L e t t 45, 926 (1980). [ll] Jackiw, Rev. Mod. Phys.
49,
681 (1977).
This Page Intentionally Left Blank
Nonlinear Problems: Present and Future 195
A.R. Bishop, D.K. Campbell, B. Nicolaenko (4s.)
0 North-Holland Publishing Company, 1982
POLARONS I N POLYACETYLENE
Alan R. Bishop and David K. Campbell Theoretical D i v i s i o n and Center f o r Nonlinear Studies Los Alamos National Laboratory Los Alamos, NM 87545 USA Recent t h e o r e t i c a l studies o f polyacetylene, (CH) have focussed on k i n k - l i k e s o l i t o n s i n the trans isomer. I t i s stown t h a t t h e same t h e o r e t i c a l models a l s o p r e m o l a r o n - l i k e s o l i t o n s i n both c i s - and W - ( C H ) . The e x p l i c i t f o r m h e s e polarons, t h e i r r e l a t i o n t o kinks, p o & i b l e i m p l i c a t i o n s f o r experiment, and open questions are discussed. INTRODUCTION During the l a s t few years the extensive t h e o r e t i c a l and experimental i n t e r e s t i n nonlinear e x c i t a t i o n s i n polyacetylene ((CH) ) has focused almost e x c l u s i v e l y on k i n k - l i k e s o l i t o n states. [l-101 This i s x h a r d l y s u r p r i s i n g , f o r a p a r t from t h e i r possible d i r e c t experimental i m p l i c a t i o n s [2,11] f o r t r a n s p o r t properties, doping mechanisms, and t h e observed semi-conductor metal t r a n s i t i o n i n (CHI , these k i n k s o l i t o n s , w i t h t h e i r b i z a r r e spin/charge assignments, [3] have s t i m b l a t e d t h e o r e t i c a l work [12,13,14] on t h e existence and r o l e o f " f r a c t i o n a l charge'' i n both s o l i d s t a t e systems and f i e l d theory models. Recently, however, i t has been discovered [15,16,17] t h a t t h e k i n k s o l i t o n s are not the only nonlinear e x c i t a t i o n s p r e d i c t e d by t h e o r e t i c a l models o f (CH)x. Indeed, i n both (numerical studies o f ) l a t t i c e models [15] and ( a n a l y t i c studies o f ) continuum theories, [16,17] a nonlinear "polaron" e x c i t a t i o n has been found. Although more conventional i n i t s p r o p e r t i e s than the k i n k s o l i t o n , the polaron, as we s h a l l see, i s the lowest energy e x c i t a t i o n a v a i l a b l e t o a s i n g l e electron. Experimentally, the polaron may thus p l a y an important r o l e i n t h e doping process o r i n e l e c t r o n i n j e c t i o n and, t h e o r e t i c a l l y , i t s presence f u r t h e r embelOne p a r t i c u l a r l y s i g n i l i s h e s the already r i c h s t r u c t u r e p r e d i c t e d f o r (CH) f i c a n t feature o f the polaron i s t h a t it, u n l i k e t h e k i n k s o l i t o n which can e x i s t only i n the trans isomer o f ( C H I , i s p r e d i c t e d t o occur i n both c i s and trans isomers. The possible i m p l i c a t i h s o f t h i s f o r comparative e x p e r x n t a l s t u d i e s o f cis and trans a r e o f great c u r r e n t i n t e r e s t . Thus i n t h i s a r t i c l e we s h a l l focus on t h i s o r e conventional" b u t nonetheless i n t e r e s t i n g nonlinear e x c i t a t i o n , describing i t s predicted nature i n d e t a i l and discussing p o t e n t i a l experimental imp1 i c a t i o n s b r i e f l y .
.
POLARONS I N m - ( C H ) , From the l a t t i c e model [3] f o r trans-(CH) , a standard sequence o f approximat i o n s [6] leads i n the c o n t i n u u m m i t 61an e f f e c t i v e a d i a b a t i c mean-field Hamiltonian i n terms o f t h e ( r e a l ) gap parameter A and e l e c t r o n f i e l d Y. The result i s
A.R. BISHOP, D.K. CAMPBELL
196
where w2/g2 i s t h e n e t e f f e c t i v e electron-phonon coupling constant, [3,6] ai i s
Q
the i t h Pauli matrix, V(y) =
(!$;),
and vF i s t h e Fermi v e l o c i t y .
[18]
For
l a t e r purposes we note t h a t i n e r i v i n g t h i s mean f i e l d Hamiltonian, t h e l a t t i c e which would add a term p r o p o r t i o n a l t o A2(y) i n (1) has been k i n e t i c energy e x p l i c i t l y ignored. Obviously, t h i s w i l l have no e f f e c t on the s t a t i c e x c i t a t i o n s we discuss below, b u t i t . w i l l prove s i g n i f i c a n t f o r on-going studies i n v o l v i n g the dynamics o f s o l i t o n s i n (CH),.
-
-
V a r i a t i o n o f H leads t o the s i n g l e p a r t i c l e e l e c t r o n wave f u n c t i o n equations
and the s e l f - c o n s i s t e n t gap equation
The summation i n (3) i s over occupied e l e c t r o n s t a t e s and s i s a s p i n l a b e l (suppressed i n (1) and (2)). Equations (2) and (3) are t h e continuum e l e c t r o n phonon equations f o r t r a n s (CH) , [5,6] and remarkably one can f i n d a n a l y t i c , closed form e x p r e s s i o f i r seveval n o n t r i v i a l solutions. [5,6,16,17] Unsurprisingly, t h i s i s r e l a t e d t o the ( p a r t i a l ) s o l i t o n features o f these equations. [16,17] Although we wish t o focus on the polaron s o l u t i o n s t o (2) and (3), f o r c l a r i t y and completeness we s h a l l b r i e f l y review t h e other solutions. F i r s t , the ground s t a t e o f t h e i n f i n i t e chain o f --(CH) e r a t e (see Figure l), and i s described by A(y) = + A.
A,
= W exp (-A -1)
i s t w o - f o l d degeno r Afy) = A -,, w i t h [5,6]
(4. a)
where
The parameters i n (4) include W, t h e f u l l one-electron bandwidth (e 10 eV i s trans-(CHI 1; v , t h e Fermi v e l o c i t y (vF = aW/2, [ l a ] where a i s t h e underi n =-(CH),); and, as i n d i c a t e d p r e v i o u s l y , l a t d c e sFacing (= 1.22
POLARONS IN POL YACETYLENE
197
u2/g2, the n e t e f f e c t i v e electron-phonon coupling constant (whose value i s i n 0.7 eV, eqn. (4.a) eqfect determined by (4)). Since experimentally A. 0.4. implies the dimensionless coupling A
.. H
H
H I
H I
...c A+ /;\ /\ /\ c.. .
Figure 1: (a) two (c) gap A =
I
CI
CI
CI
I
H
H
H
H
H
-
and (b) The two schematic bond s t r u c t u r e s corresponding t o t h e degenerate ground states f o r =-(CH) A p l o t f o r =-(CH) o f the energy ber u n i t l e n g t h versus band parameter A f o r spatialTy constant A. Note the degeneracy between f A and t h e l o c a l maximum ( i n d i c a t i n g i n s t a b i l i t y ) a t A = 0.
-
.
0
the s i n g l e e l e c t r o n energy spectrum i s given by c(k) = Fig. 2) and the corresponding e l e c t r o n wave f u n c t i o n s a r e simply plan8 waves, the e x p l i c i t forms o f which are given i n r e f s . [5,6, and 171. Physically, A # 0 implies a gap (= 2A , see Fig. 2) around the fermi surface i n t h e s i n g f e e l e c t r o n spectrum; t h a t q h i s gap should e x i s t i n t h e continuum model o f ( C H I i s a d i r e c t consequence o f t h e well-known P e i e r l s i n s t a I n chemical b i l i t y [19] o f one-'bimensional coupled electron-phonon systems. terms, A, # 0 indicates a "bond a l t e r n a t i o n " between s i n g l e and double bonds (see Fig. l), and i t s value i s p r o p o r t i o n a l t o t h e lengthening (shortening) o f a s i n g l e (double) bond w i t h respect t o the i d e a l i z e d uniform bond l e n g t h which would occur were A, = 0. For
-4
f
constant A
(s8;
198
A. R. BISHOP, D. K. CAMPBELL
a) -A0 0
-2
2
Figure 2: The s p a t i a l s t r u c t u r e o f t h e gap parameter (A(y)) and t h e associated e l e c t r o n i c l e v e l s f o r --(CH) f o r (a) the ground s t a t e ; (b) the k i n k s o l i t o n ; (c) t h e polaron. foblid l i n e s i n t h e l e f t - h a n d drawings i n d i c a t e A(y); dashed l i n e s show e l e c t r o n d e n s i t i e s f o r l o c a l i z e d states. The right-hand drawings show the e l e c t r o n i c l e v e l s i n the conduction band ( s i n g l e shading), valence band (crossed shading), and gap states. Q = charge and S = spin. The two-fold degeneracy o f t h e ground s t a t e o f trans-(CH) has the very import a n t consequence t h a t a l o c a l i z e d , f i n i t e e n e r f i n k sditon corresponding t o a l o c a l i z e d l a t t i c e "defect" can e x i s t ; t h i s k i n k (K) i n t e r p o l a t e s between the degenerate ground s t a t e s (see Fig. 2) and has t h e e x p l i c i t form
-
AK(y) = Aotanh VFy/Ao
.
-
(5)
There i s a l s o an a n t i - k i n k (k) s o l i t o n with Ak(y) = -A (y). The associated s i n g l e e l e c t r o n energy l e v e l s c o n s i s t o f extended c o n d u c t i h (E(k) = + 4 and valence (E(k) = J v q band s t a t e s [5,6,17] which a r e modffied' ( e s s e n t i a l l y "phase s h i f t e d ) fr8m t h e plane waves o f t h e ground s t a t e and an a d d i t i o n a l l o c a l i z e d "mid-gap" s t a t e ( a t E = 0, see Fig. 2). The e x p l i c i t form o f the wave f u n c t i o n f o r t h i s l o c a l i z e d h a t e i s
-
POLARONS IN POL YACETYLENE
Uo(y) = No sech A0y/vF
and Vo(y) = - i N o sech Aoy/vF
w i t h No =
For e i t h e r K o r i,t h i s "mid-gap" leading t o l o c a l i z e d e x c i t a t i o n s 6,121 i n d i c a t e d i n Fig. 2. It i n t e r e s t i n " f r a c t i o n a l charge."
s t a t e can be occupied by 0, 1, o r 2 electrons, w i t h t h e b i z a r r e spin/charge assignments [3,5, i s these states t h a t have e x c i t e d the recent [12-141
One v i t a l consequence o f t h e k i n k s o l i t o n ' s i n t e r p o l a t i n g between t h e degenerate ground states i s t h a t t h e r e i s a r e s t r i c t i o n u s u a l l y c a l l e d a " t o p o l o g i c a l c o n s t r a i n t " [12,17] on t h e production o f kinks. S p e c i f i c a l l y , i n the i n f i n i t e chain polymer, kinks can be excited from the ground s t a t e o n l y i n p a i r s o f k i n k and a n t i - k i n k . The i n t e r e s t e d reader should convince himself o f t h i s by showing t h a t the production o f a s i n g l e k i n k from t h e ground s t a t e t h a t i s , going from requires overcoming an the A(y) c o n f i g u r a t i o n i n Fig. 2a t o t h a t i n Fig. 2b i n f i n i t e ( f o r an i n f i n i t e l e n g t h polymer) energy b a r r i e r .
-
-
-
-
F i n a l l y , we come t o the polaron s o l u t i o n t o equations (2) and ( 3 ) . The r e c e n t l y discovered [16,17] a n a l y t i c form f o r the gap parameter o f t h e polaron can be w r i t t e n i n t h e revealing form (c.f., eq. (5))
where tanh 2Koyo = KovF/Ao
.
(7. b)
This c o n f i g u r a t i o n f o r A(y), which corresponds t o a s p a t i a l l y l o c a l i z e d deviat i o n from t h e ground s t a t e value (here chosen t o be + A ), i s sketched i n Fig. 2c. Again t h e s i n l e e l e c t r o n states include extended s l a t e s i n t h e conduction (&(k) = + and valence (E(k) = q-4 bands, which s t a t e s are phase-shifted by th% polaron "defect"; t h e expl i c f t ?orms o f t h e corresponding wave functions are given i n r e f . 17. I n addition, two l o c a l i z e d e l e c t r o n i c states, w i t h energies symmetrically placed, form i n t h e gap a t &+ 5 f uo where
-
-
ug
-J
=
The e l e c t r o n wave functions f o r these l o c a l i z e d s t a t e s are, f o r Uo(Y) = No
and
I ( l - i ) s e c h Ko(y+yo)
+
(l+i)sech Ko(y-yo)]
E
= +wo
A. R. BISHOP, 0.K. CAMPBEL L
200
V,(Y)
= No
where No = 4&, 1
{ ( l + i ) s e c h Ko(y+yo) + (1-i)sech Ko(y-yo)]
and f o r &=-wO
u-o
5
= +iVo and V-,
UIE=-,,,
(9b) :VIE=-u
0
= -iUo. 0
-
-
I t i s important t o note t h a t the polaron c o n f i g u r a t i o n f o r A(y) eq. (6) and the associated e l e c t r o n wave functions s a t i s f y the e l e c t r o n p a r t eq. (2) of the continuum equations f o r 9 KovF i n t h e allowed range 0 5 KovF 5 A., It i s
-
-
-
-
the s e l f - c o n s i s t e n t gap c o n d i t i o n eq. ( 3 ) which determines t h e s p e c i f i c value o f K v f o r an actual s o l u t i o n t o t h e coupled equations. Using some aspects o f %.o!iton theory, [17] one can, i n e f f e c t , convert t h e gap equation t o a s t r a i g h t f o r w a r d minimization problem. Apart from s i m p l i f y i n g t h e problem t e c h n i c a l l y , t h i s i s very appealing i n t u i t i v e l y , f o r the q u a n t i t y being m i n i mized i s e s s e n t i a l l y t h e ener o f t h e f u l l i n t e r a c t i n g electron-phonon system. I n p a r t i c u l a r , one f i n d s d e polaron-type c o n f i g u r a t i o n i n =-(CH), that t Ep(n++' n-, KO) = (n+
-
n- + 2)w0 +
4
KovF
-
-n4 wo t a n
-
where n are r e s p e c t i v e l y t h e occupation numbers which can be o n l y 0, 1, o r 2 o f $he l o c a l i z e d s t a t e s E+ = +UJ and e- = -w kntroducing 8 such t h a t KovF = Aosin8 and (see (8)) wo = A0co& and minimizpng E w i t h respect t o 8 g i v e P
-
.
From eq. (11) i t f o l l o w s d i r e c t l y t h a t f o r a stable, l o c a l i z e d polaron s o l u t i o n the electrons must be d i s t r i b u t e d i n one o f two c o n f i g u r a t i o n s : (1) n+ = 1, n- = 2, the " e l e c t r o n polaron" s t a t e ; or (2)
n+ = 0, n- = 1, the "hole polaron" s t a t e .
Before describing these s t a t e s i n d e t a i l , we note t h a t studying o t h e r p o s s i b l e configurations reveals some i n t e r e s t i n g phenomena. When n = n- --- f o r n+ = 0, 1, o r 2 (11) i m p l i e s 6 = n/2, so Kov = A , w = 0 fcorresponding t o t h e "mid-gap" state), and by (7.b) y + a. henceothe& p u t a t i v e trans-(CH), "bipolarons" separate i n t o an i n f i R i t e l y separated k i n k / a n t i - k w p a i r s , w i t h charges determined by the value o f nt = n-. When n+ = 0, n- = 2, 6 = 0, so w = A , K v = 0, and the "unoccupied polaron collapses t o the ground s t a t e , A'. O!herockoices o f n+ and n- lead simply t o combinations o f the above e x c i t a t?ons ( i n f i n i t e l y separated i n space); f o r example, n+ = 1, n- = 0 leads t o a c o n f i g u r a t i o n o f kink, a n t i - k i n k , and polaron.
---
Returning t o t h e t r u e , s t a b l e polaron states, we see t h a t f o r n+ = 1, n- = 2 ( e l e c t r o n polaron) o r n+ = 0, n- = 1 (hole polaron) eq. (11) y i e l d s 6 = n/4 o r
POLARONS IN POL YACETYLENE
KovF =
201
A0/G= wo
(12)
From eq. (10) the energy o f t h i s e x c i t a t i o n i s seen t o be
E =@A P n o
0.90Ao
(13)
From e ns. (61, (7.b), and (12) one sees t h a t the s p a t i a l e x t e n t o f t h e polaron ( = 11 f o r trans-(CH) ) i s s l i g h t l y l a r g e r than t h a t o f the k i n k (= 9 A f o r W - ( C H ) );this compirison i s made v i s u a l i n Fig. 2. Although general arguments sug&sting the existence o f polarons i n systems l i k e trans-(CH) that i s , (quasi)-one-dimensional P e i e r l s dimerized chains can b e v a n c e # , [20] , i t i s encouraging t o f i n d t h a t the continuum model, w i t h o u t any ad hoc addit i o n s , e x p l i c i t l y contains these e x c i t a t i o n s . I t i s a l s o heartening t h a t numeri c a l simulations o f the l a t t i c e model [15] have revealed a polaron e x c i t a t i o n whose s t r u c t u r e i s i n good agreement w i t h t h a t p r e d i c t e d by eq. (6), (7), and (12).
1
-
-
Another aspect o f the polaron f i e l d c o n f i g u r a t i o n (eq. (7)) and corresponding energy expression (eq. (10)) i s worth mentioning: namely, since t h e e l e c t r o n equations eqns. (2) are s a t i s f i e d f o r a l l K i n t h e range 0 < K 5 A /v , eq. (10) can be viewed as a s i n g l e parameter (K expression f o r tfie Rink/&!Thus, f o r example, k i n k i n t e r a c t i o n energy as a f u n c t i o n o f separ%tion, 5,. w i t h n+ = n- one f i n d s , f o r yo >> E l
-
-
9
with I K i l l where K changes w i t h y according t o eq. (7.5). T h i s expression represents a s l i g h t fhprovement on ppevious continuum estimates o f k i n k / a n t i k i n k forces and i s i n good agreement w i t h d i s c r e t e simulations. [21] Thus f a r we have t r e a t e d the polaron as a s t a t i c c o n f i g u r a t i o n , a s o l u t i o n t o the a d i a b a t i c mean f i e l d Hamiltonian, H i n eq. (1). For many experimental applications, i t i s v i t a l t o understand t h e dynamics o f k i n k / a n t i - k i n k and kinkA f i r s t step i n t h i s d i r e c t i o n i s t o r e i n s t a t e t h e polaron i n t e r a c t i o n s . l a t t i c e k i n e t i c energy term, AHw e x p l i c i t l y dropped i n d e r i v i n g H. I n t h e i t i o n a l term o f t h e form continuum l i m i t t h i s leads t o an
where M i s the mass o f a s i n g l e CH u n i t (" 13 m , w i t h m t h e p r o t o n ' s mass), a is i s the lattice,, spacing, and CL i s a constant &hose v a h e , i n trans-(CH) roughly 4.1 eV/A. [3] Unfortunately, t h e (time-dependent) e q u a t i o n s a t fd low from H z H + AH have not been solved f o r n o n - t r i v i a l l y time-dependent A, and thus dhe can asL&t say nothing d e f i n i t i v e about k i n k and polaron dynamics i n the continuum e l e c t r o n phonon models. However, some i n t r i g u i n g p a r t i a l r e s u l t s i n t h i s area4can be obtained i n two ways. F i r s t , one can r e f e r t o the phenomenological @ models [8,9] o f trans-(CH)x t o discuss dynamics. Here one f i n d s
202
A.R. BISHOP, D.K. CAMPBELL
t h a t c o l l i s i o n s between g4 kinks (and polarons [17,22]) have a f a s c i n a t i n g and complex s t r u c t u r e which suggests p o s s i b l y i n t e r e s t i n g e f f e c t s i n t r a n s p o r t and recombination processes i n ( C H I Much o f t h i s s t r u c t u r e i n o4 appears due t o a k i n k shape o s c i l l a t i o n mode. [yi3] Since arguments f o r a s i m i l a r mode i n t h e theory have r e c e n t l y been advanced, [24] a s i m i l a r complex continuum (CH) dynamics might'be expected. Second, f o r small v e l o c i t y kinks and polarons, one can crudely estimate kinematic e f f e c t s by simply making a G a l i l e a n boost o f t h e s t a t i c s o l u t i o n and r e t a i n i n g o n l y lowest order, nonvanishing terms i n the v e l o c i t y ; i n e f f e c t , t h i s gives an approximation t o the s o l i t o n ' s i n e r t i a l mass and thus gives some idea o f i t s r o l e i n t r a n s p o r t phenomena. For k i n k s o l i t o n s , For t h e polaron, r e p l a c i n g y by the r e s u l t s are given i n r e f s . 3 and 6. = y-v t i n (6) and e v a l u a t i n g the expression i n (14) gives P
y
m
J
4 { sech4 KO(;
-m
a
4 + yo) + sech KO(;
2 2 -2 sech KO(? + yo) sech Ko(y
-
-
yo)
yo))
.
Evaluating the i n t e g r a l s (using (7.b) c r u c i a l l y ) and d e f i n i n g M as the t o t a l P c o e f f i c i e n t o f v2 i n (15) leads t o
P
2
(Kl)~ ( OK Fv )
M =- ( K v ) 16a M2 o F
where
f(KoVF)
=3 8
-
2
tn
moAo3 (KoVF 1
' t VF}
($+KovF)
(16.b)
'O-~O'F
.9 m , which shows t h a t t h e polaron has an Using eq. (12), we f i n d t h a t m even smaller e f f e c t i v e mass t h a n k h e k i n t , f o r which mK 5me. [3,6] POLARONS I N =-(CH),
From the perspective o f the nonlinear e x c i t a t i o n s we are discussing, t h e most important d i f f e r e n c e between t h e and Cis isomers o f (CH) i s that for cis-(CHIx, the two chemical s t r u c t u r e s corresponding t o t h e p h a t i v e ground s t a t e do not have the same energy. Thus t h e r e i s a unique, nondegenerate ground s t a t e for-=-(CH) This s i t u a t i o n i s shown i n Figure 3. An immediate consequence i s t h a t si#ce t h e r e are n o t two degenerate ground s t a t e s f o r k i n k s o l i tons t o connect, there are no k i n k s o l i t o n s o l u t i o n s i n G - ( C H ) ! A p h y s i c a l l y i n t u i t i v e understanding o f t h i s s i t u a t i o n f o l l o w s by consideving what would happen i f one t r i e d t o create a k i n k / a n t i - k i n k p a i r from t h e ground s t a t e . Cons i d e r t h e case r e l e v a n t t o &-(CH)x where 6E/2 i s small (see Fig. 3) so t h e
trans
.
-
-
POLARONS IN POLYACETYLENE
203
Figure 3: (a) and (b) - The two schematic band s t r u c t u r e s corresponding t o t h e actual cis-(CH) ground s t a t e ((a), the c i s - t r a n s o i d c o n f i g u r a t i o n ) and t h e m e t a s A b l e s t a t e ((b), the &-'configuration.) (c) A p l o t f o r *-(CH) o f t h e energy per u n i t l e n g t h versus band gap parameter_ A f o r s p a d a l l y constant A. Note t h e unique ground s t a t e a t A = A. and the metastable s t a t e o f A = Ams.
-
metastable and ground s t a t e have almost the same energy per u n i t length, One could then imagine c o n s t r u c t i n g a c o n f i g u r a t i o n t h a t i n t e r p o l a t e d between A and A f o r a s p a t i a l distance d, andothen (an a n t i - k i n k , say), remalned i n A i l t e r p o l a t e d between A and A (a kin@. This c o n f i g u r a t i o n would thus l o o k l i k e eq. (6), w i t h d =mfy Fo% f i n i t e d, the c o n f i g u r a t i o n would have f i n i t e f o r d + 0)) i t s energy would become i n f i n i t e l i k e energy, b u t since bE/E (6E/E).d. I n t h i s sense, one can say t h a t the p u t a t i v e l i n k s i n c&-(CH) are "confined." O f course, r e f e r r i n g t o eg. (6) we see t h a t permanently confingd KK only polaronp a i r s have t h e same s t r u c t u r e as polarons: Thus, i n &-(CH),, nonlinear e x c i t a t i o n s w i l l e x i s t .
. >8,
204
A. R. BISHOP, D.K. CAMPBELL
To make t h i s precise, i t i s very useful t o study an e x p l i c i t model o f e - ( C H ) (due t o Brazovskii and Kirova [16]) which i s both a n a l y t i c a l l y solvable an8 r e l a t e d , i n a c e r t a i n l i m i t , t o the model we have examined-for trans-(CH) . One assumes [16] t h a t the gap parameter can be w r i t t e n as A(y) + Ax, w h e r e m i s s e n s i t i v e t o e l e c t r o n feedback (as f o r t r a n s ) b u t A 'is a co8s t a n t , J x t r i n s i c component.[25] This ansatz can be motivated i n t%rms o f the e f f e c t s a r i s i n g from molecular o r b i t a l s o t h e r than t h e one included e x p l i c i t l y i n eq. (11, (21, and (3). [16]
=m)
The mean f i e l d Hamiltonian f o r @-(CHIx
then becomes [16]
Although i n t h e i n t e r e s t o f s i m p l i c i t y we. have n o t i n d i c a t e d t h i s e x p l i c i t l y , i t i s very important t o note t h a t a l l the physical parameters i n cis-(CH) l a t t i c e spacing, e f f e c t i v e e l e c t r o n phonon coupling, band width, fer?iirvelocfty, band gap - can have d i f f e r e n t values from those_ i n trans. Comparing (1) and (17) we see t h a t w i t h the replacement A(y) + A(y) m y ) + A t h e e l e c t r o n equations f o r t r a n s are the same as f o r c i s . Thus, m u t a t j s mutandis, t h e s t r u c t u r e o f the e m o n spectrum and the e i g e n f u n c t i o n s b e m s a m e . The gap equation, however, i s modified t o read
-
which leads t o d i f f e r e n t c o n s t r a i n t s on s o l u i i o n s afid on bound s t a t e eigenvalues. S p e c i f i c a l l y , the ground s t a t e has the form A(y) = A. = Ai + Ae where
A:
5
-1
(19. a)
Wc exp(-Ac )
and
(19. b)
Here f o r c l a r i t y we have i n d i c a t e d e x p l i c i t l y t h a t f o r c & - ( C H I t h e dimensionsee eq. (4.b) and f u l l band width, W , a#e n o t necessarl e s s coupling, A, i l y the same as f o r Z - ( C H ) It can be shown d i r e c t b t h a t t h e r e a r e no k i n k s o l u t i o n s t o the gap equatfon (la), i n c o n f i r m a t i o n o f our e a r l i e r i n t u i t i v e arguments. For polaron c o n f i g u r a t i o n s , however, we expect s o l u t i o n s , and indeed we f i n d t h a t e x a c t l y t h e same f u n c t i o n a l form given by equations (7) can s a t i s f y t h e continuum equations f o r c&-(CH) , provided t h a t K ( o r u ) i s chosen appropriately. To determine K we f o l l c h our previous pr8cedur8 and again convert t h e gap equation t o a miRimization problem for the energy o f the f u l l i n t e r a c t i n g e l e c t r o n phonon system. For fi-(CH), t h e energy expression becomes
-
.
-
-
-
POLARONS IN POL YACETYLENE
~oy[tanh-l(Ko~F~o)-KovFfio]
+
.
where y = A /A Again i n t r o d u c i n g we can w r i t 8 tfie'equation minimizing E
o
=
205
sine
(n+
-
n- + 2)
-e
N
P
-
such t h a t K v = A s i n 0 and wo = iocosO, w i t h respect €0 8 9s
3.
y tane
, o 5 0 5 n/2
.
I n the l i m i t o f zero e x t r i n s i c gap, A = 0, y = 0, and as expected eq. (21) reduces t o the trans-(CH) r e s u l t o f 8s. (11). For y > 0, eq. (21) unlike equation (11) - h a s s o l u t f o n s f o r a l l combinations o f n+ and n-. These are summarized i n Table I f o r the s p e c i f i c c a s e o f y = 1.
-
Table I
-
-
n-
1
4
!
0
0
2
+2
0.71
1.43
BIPOLARON (h,h)
0
1
1
+1
0.38
0.98
POLARON (h)
0
2
0
0
0
0
GROUND STATE
1
0
3
+1
0.95
1.96
TRIPOLARON (h, h,e)
1
1
2
0
0.71
1.43
BIPOLARON (e-h)
1
2
1
-1
0.38
0.98
POLARON (e)
2
0
4
0
1.11
2.36
QUADRIPOLARON
2
1
3
-1
0.95
1.96
TRIPOLARON (e,e,h)
2
2
2
-2
0.71
1.43
BIPOLARON (e,e)
n+
g
o
INTERPRETATION
(e,e,h,h)
-
The c&-(CH) polaron states f o r various values o f N = n+ n- + 2 f o r t h e case o f y = 1. f = 2 nn+ i s the e l e c t r i c charge o f t h e state, 0 i s the angle defined i n t h e t e x t (given here i n radians), and E fi gives the f u l l energy o f the e x c i t a t i o n . The i n t e r p r e t a t i o n column gives t h g g h e r a l nature o f t h e e x c i t a t i o n (polaron, bipolaron, ...) and t h e type ( e l e c t r o n 5 e, hole 5 h).
-
-
trans-
O f p a r t i c u l a r i n t e r e s t are the "bipolaron" s o l u t i o n s i n which n+ = n-. I n ( C H I the analogous s o l u t i o n s were i n f i n i t e l y separated KK p a i r s . Indeed, one can %ee from eq. (20) t h a t y e s s e n t i a l l y plays the r o l e o f a confinement parame t e r , leading t o a term i n the energy which increases l i n e a r l y ( r e c a l l eq. (7.b)) w i t h the k i n k / a n t i - k i n k "separation," 2y0. This i s the e x p l i c i t r e a l i z a t i o n o f our e a r l i e r h e u r i s t i c argument about k i n k confinement.
206
A.R. BISHOP, D.K. CAMPBELL
From Table I we see t h a t both the polaron w i d t h (2yo) and depth increase w i t h increasing occupation number, N = n+ + 2 n-.
-
As i n t h e case o f trans, the problem o f polaron dynamics i n &-(CHI i s largely open. One can, o f u r s e , estimate t h e i n e r t i a l mass as before, ht the more i n t e r e s t i n g possible e f f e c t s o f c o l l i s i o n dynamics - recombination, long-1 i v e d oscillatory states have thus f a r eluded q u a n t i t a t i v e d e s c r i p t i o n .
-
IMPLICATIONS AND DISCUSSION Since polarons have o n l y r e c e n t l y been discovered [13,16,17] i n the t h e o r e t i c a l models f o r (CH),, the p o s s i b l e i m p l i c a t i o n s i n p a r t i c u l a r , f o r experiment - o f t h e i r existence are n o t y e t completely understood. Obviously the existence o f polarons and kinks i n trans-(CH) , contrasted w i t h t h e existence o f ( m u l t i - ) provrdes a n a t u r a l d i s t i n c t i o n between these two polarons only i n isomers. To understand t i e q u a n t i t a t i v e i m p l i c a t i o n s o f t h i s d i s t i n c t i o n f o r experiment, however, f u r t h e r t h e o r e t i c a l studies s t r e s s i n g t h e dynamics o f k i n k and polaron i n t e r a c t i o n s are needed. Two aspects o f dynamics seem p a r t i c u l a r l y crucial. F i r s t , w i t h i n the time-de endent mean f i e l d approximation (see eqns. (1) and (14)), the nature o h n d kink-polaron i n t e r a c t i o n s must be understood. Second, one must go beyond the mean f i e l d approximation t o study the f u l l quantum dynamics o f (CH) . Although t h e r e i s , as y e t , l i t t l e quantit a t i v e progress i n e i t h e r o f the& areas, t h e r e c e n t l y noted connection between model r e l a t i v i s t i c f i e l d t h e o r i e s and the continuum theory o f (CH) o f f e r some q u a l i t a t i v e suggestions. [17] F i r s t , t h i s connection suggests that! l o n g - l i v e d o s c i l l a t o r y s t a t e s may e x i s t i n (CH) and t h a t these may be important i n (CH) dynamics. Second, known r e s u l t s on 4uantum dynamics i n f i e l d theory r a i s e t h g p o s s i b i l i t y t h a t quantum f l u c t u a t i o n s can d e s t a b i l i z e t h e polaron i n (CHI , causing i t t o disappear from the theory. Although p r e l i m i n a r y quantum Monte Carlo studies [26] do show l a r g e quantum f l u c t u a t i o n s i n (CHI , a n a l y t i c e s t i mates [24] o f the f i r s t order quantum c o r r e c t i o n s i n d i c a t e t h a t ( t o t h i s order, a t l e a s t ) the polaron i s n o t d e s t a b i l i z e d . C l e a r l y , f u r t h e r q u a n t i t a t i v e work on these problem i s required.
-
&-(m
Despite the absence o f c r i t i c a l q u a n t i t a t i v e d e t a i l , i t i s p o s s i b l e t o i n d i cate a number o f i n t e r e s t i n g , p o t e n t i a l experimental i m p l i c a t i o n s o f polarons. F i r s t , on energy grounds [16,17] one knows t h a t s i n g l e e l e c t r o n s added by doping o r by i n j e c t i o n - t o c i s - o r trans-(CH) should form polarons. When many electrons are added, i t i e n e r g e t m l y fa6orable f o r them t o form k i n k / anti-kink pairs ( i n o r multi-polarons ( i n Thus f o r --(CH) , the very l i g h t l y doped m a t e r i a l an average o f 5 one e l e c t r o n ( o r hole) pgr (CH) chain might r e f l e c t polaron t r a n s p o r t p r o p e r t i e s , whereas the more h e a v f l y q e d material would e x h i b i t (charged) k i n k t r a n s p o r t behavior. In Unfortunately t h i s simple cis-(CH),, one would n o t expect t h i s d i f f e r e n c e . p i c t u r e w i l l be complicated by, among o t h e r features, t h e presence o f n e u t r a l samples and the u n c e r t a i n t i e s over kinks "quenched" i n t o undoped --(CHI isomerization from c i s t o t r a n s upon dopifig. Nonetheless, i f polarons are pre, then t h e i r o p t i c a l absorption [27] sent i n l i g h t l y dop=e-and=-(CH) i s s u f f i c i e n t l y d i f f e r e n t from t h a t o f d n k s [28] t h a t experiments comparing "mid-gap" and i n f r a r e d absorption should be able t o d e t e c t t h e d i f f e r e n c e . [27]
-
trans)
-
-
cis).
7
The d i s t i n c t i o n between allowed nonlinear e x c i t a t i o n s i n c i s and trans-(CH) suggests [16] a p l a u s i b l e scenario f o r t h e experimental l y o b s e r v e d m conX t r a s t s between these isomers upon p h o t o i n j e c t i o n ( e x c i t a t i o n o f an electron-hole p a i r by intense r a d i a t i o n ) . I n trans-(CH) , p h o t o i n j e c t i o n leads t o a photoc u r r e n t which has been i n t e r p r e t e n 6 , 2 9 l X a s r e s u l t i n g from a charged kink/ anti-kink pair. I n cis-(CH) , one observes instead a photoluminescence, suggesting the r e c o m b i n m o n o? t h e e-h p a i r which c o u l d f o l l o w because o f t h e e f f e c t i v e confinement o f the p u t a t i v e kinks ( t h e l o c a l i z a t i o n o f m u l t i - p o l a r o n
POLARONS IN POL YACETYLENNE
207
states i n c i s aids recombination). P r e c i s e l y the p r e v i o u s l y discussed theoreti c a l dynamical studies are what i s r e q u i r e d t o make t h i s q u a l i t a t i v e p i c t u r e more precise. I n conclusion, we r e i t e r a t e t h a t recent t h e o r e t i c a l developments [13,16,17] have shown t h a t t h e nature o f nonlinear e x c i t a t i o n s i n polyacetylene may be even more i n t r i g u i n g and complex than previously believed. The c e n t r a l remaining challenge i s t o e s t a b l i s h - o r disprove d e f i n i t i v e l y t h e relevance o f t h i s elegant t h e o r e t i c a l s t r u c t u r e t o t h e r e a l material.
-
ACKNOWLEDGEMENTS We are g r a t e f u l t o many colleagues f o r t h e i r advice and encouragement. Special thanks are due t o J. L. Bredas, J. R. S c h r i e f f e r , and W. P. Su f o r enlightening discussions o f t h e i r numerical simulations and t o S. Etemad f o r h i s shared i n s i g h t s i n t o the experimental s i t u a t i o n . REFERENCES See f o r example, D. Bloor, Proc. Amer. Chem. S O C . , Houston, A p r i l 1980; t o appear i n J. Chem. Phys. For recent reviews see a r t i c l e s i n Proc. I n t . Conf. on Low-Dimensional Synt h e t i c Metals (Helsingor, Denmark, August 1980) i n Chemica S c r i p t a 2 (1981); Physics i n One Dimension, eds. J. Bernasconi and T. Schneider (Springer Verlag, 1981); and A. J. Heeger and A. G. MacDiarmid, p. 353-391 i n The Physics and Chemistry o f Low Dimensional Solids, ed. L. Alchcer (Reidel , 1980). Su, W. P., S c h r i e f f e r , J. R., and Heeger, A. J., Phys. Rev. L e t t . (1979); Phys. Rev. B 2 , 2099 (1980).
42,
1698
42, 408 and 416 (1977). Brazovskii, S. A., JETP L e t t e r s 28, 606 (1978) (trans. of Pisma Zh. Eksp. Teor. F i z . 28, 656 (1978)); and Soviet Phys. JETP 51, 342 (1980) (trans. o f Zh. Eksp. Teor. Fiz. 'a, 677 (1980)). Kotani, A.,
J. Phys. SOC. Japan
Takayama, H., Lin-Liu, Y. R. and Maki, K., Phys. Rev. B 3,2388 (1980); Krumhansl, J. A., Horovitz, B. , and Heeger, A. J., S o l i d State Commun. 2 , 945 (1980); Horovitz, B., S o l i d State Commun. 2 , 61 (1980). Horovitz, B.
,
Phys. Rev. L e t t .
Rice, M. J., Phys. L e t t . 7lJ,
5 , 742
(1981).
152 (1979).
Rice, M. J., and Timonen, J., Phys. L e t t .
El 368
(1979).
[lo] Mele, E. J., and Rice, M. J., i n Chemica S c r i p t a IJ, 21 (1981). [ll] See a r t i c l e s by Etemad S., and Rice, M. J., these proceedings. [12] Jackiw, R. and Rebbi, C.,
Phys. Rev. D 13, 3398 (1976); Jackiw, R., S c h r i e f f e r , J . R., Nuc. Phys. B 190, 253 (n81).
[13] Su, W. P., and S c h r i e f f e r , J. R., Phys. Rev. L e t t .
5 , 738
and
(1981).
[14] Rice, M. J., and Mele, E. J., " P o s s i b i l i t y o f S o l i t o n s w i t h charge * e / 2 i n Highly Correlated 1:2 Salts o f TCNQ," Xerox Webster p r e p r i n t , 1981.
A.R. BISHOP, D.K. CAMPBELL
208
[15] Su, W. P., and S c h r i e f f e r , J. R., Proc. Nat. Acad. Sci. 77, 5526 (Physics) (1980); Bredas, J. L. , Chance, R. R., and Silbey, R., Proceedings o f I n t e r national Conference on Low-Dimensional Conductors, Molecular C r y s t a l s and L i q u i d Crystals (Gordon Breach), t o be published.
[16] Brazovskii, S. , and Kirova, N.,
Pisma Zh. Eksp. Teor. Fiz.
[17] Campbell, D. K. and Bishop, A. R . , Phys. Rev. B
24,
2,
6 (1981).
4859 (1981); Nuc. Phys.
B ( i n press).
[18] Following the standard conventions i n t h e continuum model, we use u n i t s w i t h h = 1. [19] P e i e r l s , R. E., Quantum Theory o f S o l i d s (Clarendon Press, Oxford, 1955) p. 108; Allender, D. Bray, J. W., and Bardeen, J . , Phys. Rev. B. 9, 119 (1974). [20] Bishop, A. R.,
S o l i d State Commun.
Y. R., unpublished.
[21] Lin-Liu,
and Maki, K.,
2 , 955
(1980).
Phys. Rev.
822
5754 (1980); Kivelson, S.
[22] The phenomenological
$4 theories, i f coupled t o a phenomenological e l e c t r o n f i e l d describing o n l y the l o c a l i z e d , "gap" s t a t e s , do c o n t a i n b o t h the k i n k s o l i t o n s ( o f a l l charges) and t h e polaron.
[23] Campbell, 0. K., and Wingate, C., [24] Nakahara, N.
, and
Maki, K.,
i n preparation.
p r e p r i n t (1981).
[25] I n another context t h i s ansatz has been discussed by Rice; see Rice, M. S., Phys. Rev. L e t t . 37, 36 (1976). [26] Su, W. P.,
unpublished.
[27] Fesser, K., Bishop, A. R . , and Campbell, 0. K.
,
i n preparation.
1281 Gammel, 3. T. and Krumhansl, J. A. Phys. Rev. B 24, 1035 (1981). Maki, K. and Nakahara, N., Phys. Rev. B 23, 5005 (1981E Horovitz, B., p r e p r i n t (1981); Kirelson, S. , Lee, T.-Y. , T i n - L i u , Y. R., Peschel, I. , and Yu, L . , p r e p r i n t (1981). [29] Etemad, S. , these Proceedings.
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North-Holland Publishing Company, 1082
209
LIGHT SCATTERING AND ABSORPTION IN POLYACETYLENE Shahab Etemad and Alan J. Heeger Department of Physics University of Pennsylvania Philadelphia, PA 19104, U.S.A.
Recent experimental results on the lattice dynamics, band structure and photoexcitations of polyacetylene are reviewed, and interpreted in terms of the soliton model. The data indicate that injection of an electron-electron (hole-hole) pair by chemical doping or photoinjection of an electronhole pair invariably distort the lattice leading to formation of a soliton-antisoliton pair. It is concluded that unlike traditional semiconductors, the polyacetylene lattice is inherently unstable in the presence of electron and/or hole excitations. INTRODUCTION It consists of weakly Polyacetylene is the simplest conjugated polymer (1). coupled chains of CH units forming a pseud -one-dimensional lattice. Three of the four carbon valence electrons are in sp8 hybridized orbitals; two of the 6-type bonds construct the 1-d lattice while the third forms a bond with the hydrogen side group. The 120° bond angle between these three electrons can be and c&-(CH& satisfied by two possible arrangements of the carbons; trans-(CH with two and four CH monomer per unit cell respecively -Fig.$). -
.
-
-
L
C
C
C
/P\ -/a\-/p\ -/p\ -?)\-/p\ -/p\ -/p R The lattice structure of trans-(CH), showing - - - - - - allthepossible arrangeL a/(?\a/m\a/m\/m\ L ments of the distortion E F T ? pattern for a polymer chain. .- ! - - The lattice structure t of a cis-(CH), showing L /( ? / \I /p\ /p\ /p\ R the distortion pattern m m m for the nondegenerate & i - - cis-transoid ground Figure 1
R
( ? m ( ? ( ? m ( ? ( ? ( ?
/(?\ /(?\ /(?\
4
-
e
/(?\
state.
-
-
e
-
/(?\
-
-
I
O I Q )
TRANS-(CH),
(?
0
(?
210
S. ETEMAD, A.J. HEEGER
I n e i t h e r isomer the remaining valence e l e c t r o n has t h e symmetry o f a 24 o r b i t a l w i t h i t s charge densitv lobes perpendicular t o t h e plane defined by the other three. I n terms o f an energy-band description, the c-bonds form low-lying completely f i l l e d bands, w h i l e the T - b o n d corresponds t o a h a l f f i l l e d conduction band provided a l l t h e bond lengths are equivalent. As a r e s u l t of P e i e r l s i n s t a b i l i t y of the 1-d metal (21, t h e bondlengths are not equivalent and a l t e r n a t e i n s i z e r e s u l t i n g i n opening o f an associated P e i e r l s gap a t t h e Fermi level. Simple estimates lead t o a p i c t u r e o f polyacetylene as a broad band pseudo-onedimensional semiconductor. I n order t o describe t h e P e i e r l s d i s t o r t e d s t a t e we consider, as an i n i t i a l approximation, a model i n which theTT-electrons o f trans-(CH), are t r e a t e d i n a t i g h t b i n d i n g approximation and t h e C - e l e c t r o n s areassumed t o move a d i a b a t i c a l l y w i t h the nuclei. L e t u, be a c o n f i g u r a t i o n coordinate f o r displacement o f the nth CH group along the molecular s,ynmetry a x i s (x), where u= ,O f o r t h e undimerized chain. The Hamiltonian i s
where t o f i r s t order i n the u's, t
n+l,n = t-(3(un+1-un) o
[*I
M i s t h e mass o f the CH u n i t , K i s the s p r i n g constant f o r t h e CF-energy when expanded t o second order about the e q u i l i b r i u m undimerized systems and c k (ens) creates ( a n n i h i l a t e s ) a 7 - e l e c t r o n o f s p i n s on t h e nri, CH group. The band s t r u c t u r e o f the p e r f e c t i n f i n i t e dimerized trans s t r u c t u r e i s shown i n
Figure 2 The band s t r u c t u r e o f =-(CH),.
LIGHT SCATTERING AND ABSORPTION IN POL YACETYLENE
Fig.
2. u
21 1
I n t h i s p e r f e c t s t r u c t u r e , t h e displacements a r e o f t h e f o r m =
-r
131
( - l ) n ug
where f corresponds t o t h e two p o s s i b l e degenerate s t r u c t u r e s ( d o u b l e bonds p o i n t i n g " l e f t " and double bonds p o i n t i n g " r i g h t " ) . The t r a n s f e r i n t e g r a l s f o r the p e r f e c t chain are
(
tn+l,n=
[
to-tl to+tl
" s i n g l e " bond 143
"double" bond
The o v e r a l l bandwidth i s 4 b - 8-10 eV; whereas t h e energy gap 2 A = 4 t , depends on t h e magnitude o f t h e d i s t o r t i o n . F o r example, i n trans-(CH), f o r 2A = 4 t , - 1.5 eV and B E 6 . 9 e V / l , as i n f e r r e d f r o m t h e a n a m o f t h e o p t i c a l a b s o r p t i o n c o e f f i c i e n t and t h e l a t t i c e dynamic r e s u l t s ( s e e below), uo which m i n i m i z e s t h e ground s t a t e energy can be o b t a i n e d t h e value o f from
a
= 4pu,
t o be u, .-.025 4 compared w i t h an i n t e r s i t e s e p a r a t i o n o f a = 1.28 a l o n g t h e c h a i n d i r e c t i o n . As a r e s u l t o f t h e l a r g e bandwidth and u n s a t u r a t e d r - s y s t e m , (CH), i s analoguous t o t h e t r a d i t i o n a l i n o r g a n i c semiconductors. The t r a n s v e r s e The l a r g e n e a r e s t t r a n s f e r i n t e g r a l t, i s , however, much l e s s t h a n t,. neighbor i n t e r c h a i n s e p a r a t i o n o f c.4A ( 4 ) i m p l i e s N tl . l e v compared t o t,- 2.5 eV; t h u s j u s t i f y i n g t h e pseudo-one-dimensionality o f t h e system. There has been a growing i n t e r e s t i n t h e s t u d y o f t h e p h y s i c a l p r o p e r t i e s o f (CH), due t o t h e f a c t t h a t t h e s t r u c t u r e o f trans-(CHI, e x h i b i t s a broken symmetry and has a t w o - f o l d degenerate g r o u n m t e . ( s e e F i g . 1 ) As a r e s u l t , one expects s o l i t o n - l i k e e x c i t a t i o n s , i n t h e f o r m o f bond a l t e r n a t i o n domain w a l l s , c o n n e c t i n g t h e two degenerate phases. The p r o p e r t i e s o f s o l i t o n s i n I CH), have been e x p l o r e d t h e o r e t i c a l l y i n s e v e r a l r e c e n t papers (5-8), which show t h a t t h e c o u p l i n g o f t h e s e c o n f o r m a t i o n a l e x c i t a t i o n s t o t h e T - e l e c t r o n s l e a d s t o unusual e l e c t r i c a l and magnetic p r o p e r t i e s . The p o s s i b i l i t y o f e x p e r i m e n t a l s t u d i e s o f such s o l i t o n s i n p o l y a c e t y l e n e , t h e r e f o r e , r e p r e s e n t s a unique o p p o r t u n i t y t o e x p l o r e n o n l i n e a r phenomena i n condensed m a t t e r physics. I n an e f f o r t t o c h a r a c t e r i z e t h e s o l i t o n e x c i t a t i o n s o f p o l y a c e t y l e n e , Su, S c h e i e f f e r , and Heeger (SSH) c o n s i d e r e d t h e two domains w i t h L domain t o t h e l e f t and R domain t o t h e r i g h t o f a domain w a l l o r s o l i t o n as shown i n F i g (1CI. They determined t h e p r o p e r t i e s o f t h e s o l i t o n ( c r e a t i o n energy, w i d t h , mass, spin, e t c . ) i n terms o f t h e m i c r o s c o p i c parameters o f e q u a t i o n ( 1 ) b y m i n i m i z i n g t h e g r o u n d - s t a t e energy o f t h e system f o r a displacement p a t t e r n which reduces t o t h e L and R phases as one moves t o t h e f a r l e f t o r r i g h t . Using a staggered o r d e r parameter t r i a l f u n c t i o n W , ,
they with in f i.e., they bond
f i n d t h a t due t o t h e c o m p e t i t i o n between t h e e l a s t i c t e r m ( i n c r e a s e s a decrease i n {)and t h e condensation energy ( i n c r e a s e s w i t h an i n c r e a s e ) t h e s o l i t o n i n trans-(CHI, i s spread o v e r a p p r o x i m a t e l y 15 l a t t i c e s i t e s , 2 f = 15 U s i n g x a d i a b a t i c a p p r o x i m a t i o n f o r t h e e l e c t r o n i c motion, e s t i m a t e t h e t r a n s l a t i o n a l mass o f s o l i t o n ; i.e., t h e i n e r t i a o f t h e a l t e r n a t i o n p a t t e r n o f e q u a t i o n 6 t o m o t i o n a l o n g t h e chain, t o be
.
212
S. ETEMAD. A.J. HEEGER
4
2
c71
m 9 - M( .%) s 3 a
.07 a t which t i m e t h e V: and 'V! c u r v e s s e p a r a t e . Vc r e a c h e s a maximum f o r a=.2 and t h e n d e c r e a s e s f o r h i g h e r a . Energy c o n s e r v a t i o n r e q u i r e s f o r t h e a d e p e n d e n c e of V,':
The o b s e r v e d v a l u e s of V ': to t h e i n e l a s t i c effects.
are s l i q t t l y d i f f e r e n t from Eq.(8) d u e T h e s e Vc r a p i d l y a p p r o a c h o n e so t h a t
QUASI-SOLITONS: THE DOUBLE SINE-GORDON EQUATION
Q
W
~0.3
pout
-
-
-
- 0.5
0.5
*
-
*
m
-
e
*-•
0.3
0.1
-
-
-
- 0.1
. '
W
I
I
vout
0.7
/ /
c
-
0.1
-
0.I
0.5
/
:*
-
0.3
0.3
F i g . 3. The scattering matrix f o r a = . 3 . The l e f t hand s c a l e r e f e r s to t h e frequency while t h e r i g h t hand s c a l e r e f e r s to t h e v e l o c i t y of outgoing SS p a i r s . The v e r t i c a l arrows indicate the thresholds f o r incoming v e l o c i t i e s .
I
a = 0.6
1.
233
0.3
0.I
F i g . 4 . The scattering matrix The l e f t f o r a=.6. hand scale refers t o the frequency while t h e r i g h t hand scale r e f e r s to t h e v e l o c i t y of outgoing SS p a i r s . The v e r t i c a l arrows indicate the thresholds for incoming v e l o c i t i e s .
P. KUMAR, R.R. HOLLAND
234
Fig. 5. The critical velocities as a function of a . Th solid line shows VL' if the collisions were energy conserving. are the calculated while (01 are calculated Vc.
"C
0.9
0.7
0.5
0 0
0
0
0
0
1
1
0.3
0.I 1
0.1
1
1
0.3
1
1
0.5
1
0.7
1
0.9 U
for a2.3, the range of velocities for particle conversion becomes negligibly small. The final velocity after particle conversion, for an energy conserving collision is given by
which describes the data surprisingly well provided that we use the observed value of V, (instead of Eq. (8)). The effects of damping are more important at small a , as described earlier. Yet another measure of inelasticity is the rate of decay of the bound state amplitude. It decays in time with a power law as t-". The exponent n varies from n=1.5 for a=.l to n=.2 for a = . 9 . The characteristic time scale also increases with a . In all cases however, the frequency of the bound state remains well defined. The damping remains a weak effect and the solitons can be described as quasi-solitons.
OUASI-SOLITONS: THE DOUBLE SINE-GORDON EQUATION
235
We thank A.C. Scott for encouraging discussions. This work was supported by NSF grant no. DMR 8006311 and by the Research Corporation. REFERENCES Maki, K. and Kumar, P., Phys. Rev. B14 (1976) 3920. Kitchenside, P.W. Bullough, R.K. and Candrey, P.J., Creation of Spin Waves in h e B , in: Bishop, A . and Schneider, T. (eds), Solitons and Condensed Matter Physics (Springer-Verlag, Berlin 1978). Shiefman, J. and Kumar, P., Physica Scripta 20 (1979) 435. Kaup, D.J. and Newell, A.C., Phys. Rev. B18 (1978) 5162. McLaughlin, D.W. and Scott, A.C., Phys. Rev. A 18 (1978) 1652. Hopfinger, A.J. Lewanski, A . J . , Sluckin, T.J. and Taylor, P.L., Solitary Wave Propagation as a Model for poling in PvF2, in: Bishop and Schneider (red.) Solitons and Condensed Matter Physics (Springer-Verlag, Berlin 1978). Bullough, R.K., Solitons in: Interaction of Radiation with Condensed Matter Vol. I, IAEA, Vienna 1977. Makhankov, V., Comp. Phys. Comm. 21 (1980) 1.
This Page Intentionally Left Blank
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 8. Nicolaenko (%is.) @ North-Holland Publishing Company. 1982
237
CLASSICAL FIELD THEORY WITH Z ( 3 ) SYMMETRY H e r b e r t M. Ruck Nuclear Science D i v i s i o n Lawrence B e r k e l e y L a b o r a t o r y University o f California Berkeley, CA 94720
ABSTRACT We p r e s e n t s o l u t i o n s and some o f t h e i r p r o p e r t i e s o f a c l a s s i c a l v e c t o r f i e l d model i n two-dimensional Minkowski space w i t h i n t e r n a l symmetry Z(3)--the c y c l i c group o f o r d e r t h r e e .
1.
INTRODUCTION
C y c l i c symmetric f i e l d t h e o r i e s a r e o f i n t e r e s t i n s o l i d s t a t e p h y s i c s where t h e symmetry f o l l o w s f r o m t h e l a t t i c e s t r u c t u r e o f t h e c r y s t a l s 1 9 2 and i n elementary p a r t i c l e p h y s i c s where t h e c o l o r t h e o r y o f quarks and gluons imposes c y c l i c t r a n s f o r m a t i o n s o f t h e quantum numbers3. I n t h i s s h o r t communication we a r e d e a l i n g w i t h t h e mathematical s t r u c t u r e o f a system o f n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s t h a t i s i n t e r e s t i n g i n i t s e l f . Some p h y s i c a l a p p l i c a t i o n s a r e g i v e n elsewhere4.
2.
THE Z ( 3 ) MODEL
A. The Lagrangian d e n s i t y o f t h e model5 i s a f u n c t i o n a l o f a two-component v e c t o r f i e l d 3 = [ b l ( t , x ) , b 2 ( t , x ) ] E Rp i n one time, one space dimensions xu = ( t , x ) E Rp, p , =~ 0 , l :
V(b1,62) i s t h e p o t e n t i a l p a r t o f t h e Lagrangian t h a t c o n t a i n s t h e polynomial s e l f - i n t e r a c t i o n o f the s c a l a r f i e l d s :
The m e t r i c i s gpV = d i a g ( + l , - 1 ) , p , v = 0 , l ; ap = alaxp, a$ = a! - a t . The parameters x, II a r e p o s i t i v e x > 0, u > 0. The c o n s t a n t y s d e ermined such t h a t t h e p o t e n t i a l ( 2 ) i s p o s i t i v e d e f i n i t e V 2 0. The parameter v i s chosen t o be p o s i t i v e v > 0. The group t r a n s f o r m a t i o n s o f t h e f i e l d s t h a t l e a v e t h e Lagrangian d e n s i t y ( 1 ) i n v a r i a n t a r e ( i ) a r e f l e c t i o n o f 62: 61
+
61
, 62 + -62
and ( i i ) a Z ( 3 ) d i s c r e t e r o t a t i o n by 120' o f t h e f i e l d s :
(3)
238
H. M. RUCK
6, + b1cos(2nn/3) - b 2 s i n ( 2 n n / 3 )
,
(4a)
b2 +
,
(4b)
b1sin(2nn/3) + b2cos(2nn/3)
w i t h n = 0,1,2. The d i s c r e t e symmetry Z ( 3 ) Eq. ( 4 ) i s imposed by t h e c u b i c t e r m i n t h e p o t e n t i a l ( 2 ) . When v i s s e t z e r o t h e Lagrangian o b v i o u s l y has f u l l SO(2) r o t a t i o n a l symmetry. There e x i s t t h r e e more r e p r e s e n t a t i o n s o f t h e L a g r a n g i a n (1): i n terms o f complex f i e l d s , i n p o l a r c o o r d i n a t e s , and i n terms o f a m a t r i x f i e l d . 6. With t h e complex f i e l d s 6 = 41 + ib2, becomes:
6*
=
61 - i d 2 t h e L a g r a n g i a n
The r e f l e c t i o n symmetry ( 3 ) becomes a complex c o n j u g a t i o n :
6 + 6* and b* + b ,
(61
and t h e d i s c r e t e r o t a t i o n ( 4 ) becomes a phase t r a n s f o r m a t i o n :
b + bexp(i2nn/3)
, 6* +
6*exp(-i2~n/3)
.
(7)
C.
I n p o l a r c o o r d i n a t e s t h e Z ( 3 ) i n v a r i a n c e i s e a s i l y seen. The f i e l d s a r e = p s i n e . The p o t e n t i a l p a r t ( 2 ) o f . t h e Cagrangian becomes:
41 = Pcose, and 62
V(p,e)
= xp4
- vp3
cos(3e) -
pp
2
-
.
y
(8)
The r e f l e c t i o n t r a n s f o r m a t i o n ( 3 ) becomes a r e f l e c t i o n o f t h e a n g l e 8,-e
8:
,
(9)
and t h e r o t a t i o n a l t r a n s f o r m a t i o n ( 4 ) a t r a n s l a t i o n a l o p e r a t i o n :
e
+e
+ 2rn/3
,n
=
0,1,2
.
(10)
The o n l y e-dependent f a c t o r i n (8) c o s ( 3 e ) i s e v i d e n t l y i n v a r i a n t under t h e mappings ( 9 ) and (10).
D. The l a s t f o r m i s i n terms o f a m a t r i x f i e l d t h a t l i n k s t h e model t o a S U ( 2 ) representation. Define t h e m a t r i x f i e l d :
where 01, a2, 63 a r e t h e P a u l i m a t r i c e s . d e n s i t y (1) as a t r a c e ( T r ) e x p r e s s i o n :
Then we can w r i t e t h e L a g r a n g i a n
CLASSICAL FIELD THEORY WITH Z(3) SYMMETRY
239
I i s t h e 2 x 2 dimensional u n i t m a t r i x . The r e f l e c t i o n symmetry ( 3 ) becomes t h e mapping:
;3
03;03
Y
and t h e Z ( 3 ) t r a n s f o r m a t i o n t h e mapping:
i+o+;o
,
w i t h t h e operator:
o
, 03
= exp(-inno2/3)
=
I
,
n = 0,1,2
.
T h i s m a t r i x r e p r e s e n t a t i o n i s a s c a l a r v e r s i o n o f t h e SU(2) Yang-Mills f i e l d 6 : 3
A
= I.1
A:aa
,
a=I
w i t h t h e i m a g i n a r y component set t o z e r o A$ = 0. The r e p r e s e n t a t i o n (11) i s s u i t a b l e f o r g e n e r a l i z a t i o n t o more t h a n two f i e l d components b y an expansion i n terms o f t h e g e n e r a t o r s o f t h e SU(3) group.
3.
SOLITON SOLUTIONS The Euler-Lagrange f i e l d equations i n two dimensions are:
F i r s t we a r e l o o k i n g f o r t i m e i n d e endent s o l u t i o n s o f t h e f i e l d e q u a t i o n s 7 . One dimensional s o l u t i o n s of t h e 7 t h e o r y have found a p p l i c a t i o n s i n p o l y a c e t y l e n e 8 . For f u r t h e r d i s c u s s i o n o f s o l i t o n s i n (CH), see Ref. 9. I n a d d i t i o n , r e f e r e n c e s t o t h i s problem can be found i n t h i s volume o f t h e proceed ings.
9
Plane wave s o l u t i o n s i n h i g h e r d i m e n s i o n a l space can be o b t a i n e d f r o m t h e one dimensional s o l u t i o n by t h e mapping10 x 3 k,x,/k,k,. I n order t o solve t h e f i e l d equations, i t i s useful t o consider t h e t o p o l o g y o f t h e p o t e n t i a l V(61,62) as a s u r f a c e d e f i n e d on t h e 61,62 space. There a r e t h r e e minima-also c a l l e d vacua-denoted by QlyR2,R3, where t h e p o t e n t i a l i s zero, l o c a t e d on a c i r c l e o f r a d i u s 6~ a t an a n g l e o f 120' f r o m each o t h e r . The c o o r d i n a t e s o f t h e minima are:
bv i s t h e v a l u e o f t h e vacuum f i e l d determined as t h e p o s i t i v e r o o t o f t h e e q u a t i o n aV(p,O)/ap = 0. I n t h e c e n t e r o f t h e 61.62.plane t h e p o t e n t i a l i s e l e v a t e d V(0,O) = -y > 0. For l a r g e v a l u e s o f t h e f i e l d s 6f + 63 >> 66 t h e p o t e n t i a l has steep w a l l s due t o t h e p o s i t i v i t y o f A. here are t h r e e The d i s t a n c e f r o m t h e c e n t e r o f saddle p o i n t s located a t 60", 180° and 240'. t h e saddle p o i n t s i s g i v e n by t h e p o s i t i v e r o o t o f t h e e q u a t i o n a V ( p Y n / 3 ) / a p = 0.
H.M. RUCK
240
S o l u t i o n s t u n n e l i n g f r o m one vacuum t o t h e o t h e r a r e o b t a i n e d by t h e method o f t r a j e c t o r i e s i n f i e l d spacell. When t h e parameters A,
V,
u,
Y
a r e r e l a t e d t o each o t h e r i n t h e f o l l o w i n g
way:
t h e geometry o f t h e p o t e n t i a l arranges i t s e l f such t h a t t h e saddle p o i n t between two vacua l i e s on t h e s t r a i g h t l i n e c o n n e c t i n g t h e s e two vacua. The t u n n e l i n g f r o m one vacuum t o t h e o t h e r w i l l now o c c u r a l o n g t h i s s t r a i g h t l i n e t h r o u g h t h e sadd l e p o i n t
.
There a r e s i x p a i r s o f s o l u t i o n s t o t h e f i e l d e q u a t i o n s c o n n e c t i n g t h e minima p a i r w i s e . Three p a i r s o f s o l u t i o n s a r e generated by t h e mapping ( 4 ) ; t h e o t h e r t h r e e obtained by t h e r e f l e c t i o n ( 3 ) . We use t h e n o t a t i o n a = ( 3 ~ / 2 ) 1 / 2 6 v . The b a s i c set o f s o l u t i o n s i s : .
1.
The s o l u t i o n g o i n g f r o m 01 t o Q2, i.e.
B1(x) = 1 b v [ l
+
3tanh(ax)];
b2(x) =
+
b(+m)
7 6 bV[1 -
E
4,
tanh(ax)l
+
f ~ ( - ~E)
9 is
.
(19)
The t r a j e c t o r y i s t h e a l g e b r a i c r e l a t i o n independent o f x:
b1
+
J5 6,
=
B,
.
(20)
The next two p a i r s a r e o b t a i n e d f r o m ( 1 9 ) by r o t a t i o n o f 120" and 240'.
2.
$(+-I E
The s o l u t i o n t u n n e l i n g f r o m R2 t o R3 w i t h t h e a s y m p t o t i c v a l u e s ~ 2 , E a3 i s
d(-m)
The t r a j e c t o r y i s
1
61=-28, 3.
.
(22)
The s o l i t o n t u n n e l i n g f r o m Q3 t o
B1(x) =
1
bv[l - 3tanh(ax)l
: 62(x)
=
a1
&(+a) E
J5 b,[1 -7
+
sl3,
&(-m)
tanh(ax)]
E
R1 i s
,
(23)
with the trajectory
Q1- 43
b2
=
6,
.
A l l t r a j e c t o r i e s t o g e t h e r [Eqs. (20),(22), and ( 2 4 ) ] f o r m an e q u i l a t e r a l t r i a n g l e w i t h t h e c o r n e r s i n t h e minima. o f t h e p o t e n t i a l .
4.
PROPERTIES OF THE SOLITON SOLUTIONS
The energy-momentum t e n s o r i s a f u n c t i o n a l o f t h e f i e l d s and t h e i r d e r i v a t i v e s d e f i n e d as:
(24)
CLASSICAL FIELD THEORY WITHZ(3) SYMMETRY
241
The k i n e t i c and p o t e n t i a l energy o f t h e f i e l d c o n f i g u r a t i o n g i v e n b y any o f t h e Eqs. (19),(21), o r ( 2 3 ) a r e equal. T h e i r sum i s t h e energy d e n s i t y :
.
(x) = 3 9 IB,[COS~(~X)]-~ 4
T
00
(26)
The i n t e g r a t e d f i e l d energy g i v e s t h e mass o f t h e s o l i t o n :
J
M =
.
dx TOO(x) = aB, 3
-03
The o t h e r components o f t h e f i e l d t e n s o r - t h e p r e s s u r e T11-van i s h .
f i e l d momentum To1 and t h e f i e l d
The l i n e a r e x t e n s i o n ( a n a l o g o f t h e r a d i u s i n one d i m e n s i o n ) o f t h e energy d i s t r i b u t i o n i n c o n f i g u r a t i o n space i s d e f i n e d by:
R
dx x
= [3f
TOO(x)I
p
0
dx T O O ( x ) 1 l /2
,
(28)
0
[Too(x) = T m ( - x ) ] , such t h a t a b o x - l i k e energy d i s t r i b u t i o n o f a r b i t r a r y h e i g h t and l e n g t h 2s g i v e s t h e c o r r e c t answer R = s. The dependence on a and bv i s e x p l i c i t :
R = (0.80305 ...)aWith B = radius:
-y
1 / 2 -1
6,
.
> 0 t h e mass o f t h e s o l i t o n can be w r i t t e n i n terms o f t h e l i n e a r
M = (1.1438 ...)B(2R)
5.
(29)
.
(30)
RESONANCES
E x c i t e d s t a t e s o f t h e s c a l a r f i e l d s a r e g i v e n b y f l u c t u a t i o n s i n t i m e about t h e c l a s s i c a l f i e l d s 61 and 62. Again we c o n s i d e r t h e s o l u t i o n Eq. ( 1 9 ) . The p e r t u r b a t i v e expansion o f t h e f i e l d s 1 2 (Q2 and 93 a r e n e g l e c t e d i n t h e f i e l d equations) : 6;(t,x)
= 61(x)
+
Q(t,x)
; b;(t,x)
1
= 62(x) --9P(t,x)
s
4-
( 3 11
i s chosen such t h a t t h e new f i e l d s 6; and 6; s a t i s f y t h e same a l g e b r a i c r e l a t i o n ( 2 0 ) as t h e c l a s s i c a l f i e l d s . W i t h t h i s expansion t h e e n e r g y spectrum o f t h e resonance has two l e v e l s :
where t h e e n e r g i e s and t h e p e r t u r b a t i o n s are:
,
( t , x ) = Acos(Et) sech(ax) t a n h ( a x )
t,x)
= Acos(Et)[l
-
sech2(ax)]
.
(33)
(34)
H.M. RUCK
242
6.
TWO SOLITON INTERACTIONS
I n physical applications t h e behavior o f m u l t i c e n t e r s o l u t i o n s i s i m p o r t a n t . I have n o t found any e x a c t m u l t i c e n t e r s o l u t i o n i n t h i s model.
I w i l l s h o r t l y a n a l y z e an ansatz f o r a f i e l d c o n f i g u r a t i o n w i t h two c e n t e r s a t x and x?. There i s an o r d e r i n of t h e c e n t e r s i m p l i e d x2 Q 71. The t r i a l f u n c t i o n c o n t a i n s l i n e a r l 3 9 1 l and q u a d r a t i c forms o f t h e s i n g l e s o l i t o n s o l u t i o n s Eq. ( 1 9 ) l o c a t e d a t x i and t h e s o l u t i o n Eq. ( 2 3 ) w i t h t h e s i g n o f 62 changed [symmetry Eq. ( 3 ) ] l o c a t e d a t x2:
61( 2 ) (x,x1,x2) +
sin
2
= cos
~1 6 , [ 1
(E)
6, ( 2 ) (X,X 1 ,X 2 ) +
E
2
(E)
+
T1 4,fi
3[1
- [l -
tanh(a(x
+
3 t a n h ( a ( x - xl))][-l
= cos
s i n2 ( E )J38 6,[1
+
+
+
-
x,))
i s a v a r i a t i o n a l parameter w i t h v a l u e s
+
E
-
3tanh(a(x
t a n h ( a ( x - x,))
t a n h ( a ( x - xl))][l
-tanh
-
x ) 2
tanh
t a n h ( a ( x - x,))]
E [O,n/2].
I f one o f t h e c e n t e r s i s removed t o i n f i n i t y , t h e f i e l d s (35,36) reduce t o exact s i n g l e s o l i t o n s o l u t i o n s . T h i s l i m i t process i s independent o f t h e v a l u e o f E. I f x2 + -m, t h e f i e l d s (35,36) reduce t o Eq. ( 1 9 ) and f o r x 1 + + - t h e f i e l d s reduce t o Eq. (23) w i t h t h e s i g n o f 62 i n v e r t e d .
The s t a t i c p o t e n t i a l energy between two s o l i t o n s a t a d i s t a n c e d = x 1 i s t h e d i f f e r e n c e between t h e t o t a l f i e l d energy and t w i c e t h e mass o f an isolated soliton:
-
x2
rn
P(d,E) = /.dx
Thi)(x,d,c)
-
2M
.
(37)
-w
A n u m e r i c a l c a l c u l a t i o n shows t h a t t h e p o t e n t i a l i s n e g a t i v e , i.e. t h e two s o l i t o n s a t t r a c t each o t h e r when t h e y come c l o s e t o g e t h e r . F o r E = 0 ( p u r e l i n e a r c o m b i n a t i o n ) we o b t a i n a lower bound and f o r E = K/2 ( p u r e q u a d r a t i c f o r m ) an upper bound f o r t h e p o t e n t i a l energy. P(d,O)
Q P(d,E)
Q P(d,n/2)
;
Q0
E
.
E (O,n/2)
(38)
The p o t e n t i a l converges f a s t t o z e r o when t h e d i s t a n c e d becomes s l i g h t l y l a r g e r t h a n t h e l i n e a r e x t e n s i o n o f t h e p a r t i c l e s d > 2R. The q u a l i t y o f t h e ansatz Eqs. (35,36) can be checked by computing t h e a c t i o n f o r t h i s f i e l d c o n f i g u r a t i o n . The a c t i o n p e r u n i t t i m e i s : m
S(d,E)
= -
J
dx Thi)(x,d,E)
.
(39)
-W
F o r l a r g e d i s t a n c e s d >> R t h e a c t i o n i s -2M independent o f E . F o r f i x e d d i s t a n c e s i n t h e i n t e r a c t i o n r e g i o n , t h e a c t i o n has a minimum f o r E = K / 2 and a maximum f o r E = 0. -2M Q S ( d , n / 2 )
GS(d,e)
GS(d,O)
QO
;
E
E (O,n/2)
.
(40)
CLASSICAL FIELD THEORY WITHZ(3) SYMMETRY
I n t h e s t a t i c t r i a l f u n c t i o n s ( 3 5 , 3 6 ) t h e q u a d r a t i c f o r m o f two s i n g l e s o l i t o n s o l u t i o n s m i n i m i z e s t h e a c t i o n a t any d i s t a n c e and seems t h e r e f o r e t o be p r e f e r r e d t o t h e l i n e a r form. The net e f f e c t i s t h a t t h e s t a t i c s o l i t o n s a t t r a c t each o t h e r . T h i s i s c o n s i s t e n t w i t h t h e f i n d i n g s i n t h e v4 model a t low v e l o c i t i e s . H i g h v e l o c i t y s o l i t o n s c o l l i d i n q w i t h ea h. o t h e r may s c a t t e r backward showing a r e p u l s i v e p o t e n t i a l at high ~ p e e d l ~ . ~ ~ . CONCLUSIONS We analyzed a f i e l d t h e o r y model w i t h Z ( 3 ) i n t e r n a l symmetry i n 1 + 1 dimensions. Exact s o l i t o n s o l u t i o n s a r e found i n one space dimension. An ansatz f o r t h e two-center f i e l d shows t h a t t h e s t a t i c p o t e n t i a l between two solitons i s attractive. ACKNOWLEDGMENTS
I am g r a t e f u l t o David K. Campbell f o r many h e l p f u l and i n t e r e s t i n g d i s c u s s ions. T h i s work was supported by t h e D i r e c t o r , O f f i c e o f Energy Research, D i v i s i o n o f N u c l e a r P h v s i c s o f t h e O f f i c e o f H i g h Energy and N u c l e a r P h y s i c s o f t h e U.S. Department o f Energy under C o n t r a c t W-7405-ENG-48.
243
244
H.M. RUCK
REFERENCES R.8. Potts, Some generalized order-disorder t r a n s f o r m a t i o n s , Proc. Cambr. P h i l . SOC. 4s (1952) 106-109. 2. A.N. Berker, S. Ostlund and F.A. Putnam, Renormalization-group treatment o f a P o t t s l a t t i c e gas f o r k r y p t o n adsorbed o n t o graphite, Phys. Rev. 617 (1978) 3650-3665. 3. A. de Rujula, H. Georgi and S.L. Glashow, A t h e o r y o f f l a v o r mixing, Ann. o f Phys. 109 (1977) 258-313. H.M. R u c k T o l y n o m i a l chromodynamics i n 1+1dimensions I . Confinement o f 4. quarks and t h e s t r u c t u r e o f c m p o s i t e p a r t i c l e s , Lawrence Berkeley Laboratory p r e p r i n t LBL-12316 (February 1981). 5. F. Constantinescu and H.M. Ruck, Quantum f i e l d t h e o r y P o t t s model, J. Math. Phys. 19 (1978) 2359-2361; Phase t r a n s i t i o n s i n a continuous t h r e e s t a t e s model X t h d i s c r e t e gauge symmetry, Ann. o f Phys. 115 (1978) 474-495. 6. C.N. Yang and R. M i l l s , Conservation o f i s o t o p i c s f i and i s o t o p i c gauge invariance, Phys. Rev. 96 (1954) 191-195. 7. H.M. Ruck, S o l i t o n s i n c y c l i c symmetry f i e l d t h e o r i e s , Nucl. Phys. 8167 (1980) 320-326. 8. M.J. Rice, Charged n-phase k i n k s i n l i g h t l y doped polyacetylene, Phys. L e t t . 71A (1979) 152-154. W.P. SCJ.R. S c h r i e f f e r and A.J. Heeger, S o l i t o n s i n polyacetylene, Phys. 9. Rev. L e t t . 42 (1979) 1698-1701; S o l i t o n e x c i t a t i o n s i n polyacetylene, Phys. Rev. 822 (1x0) 2099-2111. 10. P.6. F t , S o l i t a r y waves i n n o n l i n e a r f i e l d t h e o r i e s , Phys. Rev. L e t t . 32 (1974) 1080-1081. 11. R. Rajaraman, S o l i t o n s o f coupled s c a l a r f i e l d t h e o r i e s i n two dimensions, Phys. Rev. L e t t . 42 (1979) 200-204. 12. H.M. Ruck, Quark confinement i n f i e l d t h e o r i e s w i t h d i s c r e t e gauge symmetry Z(3), i n : K.B. Wolf (ed.), Group t h e o r e t i c a l methods i n physics, L e c t u r e Notes i n Physics, 135 (Springer Verlag, B e r l i n , 1980). 13. A.E. Kudryavtsev, m i t o n l i k e s o l u t i o n s f o r a Higgs s c a l a r f i e l d , JETP L e t t . 22 (1975) 82-83. 14. V.G. M a a n k o v , Dynamics o f c l a s s i c a l s o l i t o n s (In non-integrable systems), Phys. Rep. C35 (1978) 1-128. p r i v a t e communication. 15. D.K. Campbe1.
PART 111 Reaction-Diffusion Processes
This Page Intentionally Left Blank
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 8.Nicolaenko (eds.) 0 North-Holland PublishingCompany, 1982
247
SOME CHARACTERISTIC NONLINEARITIES OF CHEMICAL REACTION ENGINEERING
Rutherford A r i s Sherman Fai r c h i 1d Distinguished Scholar C a l i f o r n i a I n s t i t u t e o f Technology A b r i e f survey i s made o f the d i f f e r e n t nonlinear forms which a r i s e as expressions o f chemical r e a c t i o n r a t e i n the general problem o f d i f f u sion and r e a c t i o n i n a s o l i d c a t a l y s t . These may be c l a s s i f i e d i n order o f increasing complexity as powers and polynomials, r a t i o n a l functions, and transcendentals. The c h a r a c t e r i s t i c s and some o f the t y p i c a l problems associated w i t h each o f these classes w i l l be surveyed.
INTRODUCTION I n attempting t o survey the c h a r a c t e r i s t i c types o f n o n l i n e a r i t y o f chemical r e a c t i o n engineering i t may be w e l l t o take examples from a general class o f problems which i s o f c e n t r a l importance i n the theory o f chemical r e a c t i o n processes. The complexity o f a c a t a l y t i c system requires a very thorough a n a l y s i s both o f the physical and o f the chemical processes t h a t are a t work. The porous catal y s t p e l l e t i s a network o f pore space o f various and varying diameters o f t e n l i n k e d t o a subsystem o f even f i n e r pores through a manufacturing process which involves the pressing together o f a f i n e powder. No exact model o f d i f f u s i o n i n so complicated a geometry i s possible and, i n e v i t a b l y , a s i m p l i f i e d model o f the geometrical s t r u c t u r e must be made. Even t h i s i s g e n e r a l l y n o t taken t o the second stage o f being t r e a t e d as a d i f f u s i o n model w i t h the r e a c t i o n as i n the boundary condition, b u t r a t h e r serves t o define an e f f e c t i v e d i f f u s i v i t y i n which both t h e geometry o f the pore s t r u c t u r e and physical p r o p e r t i e s o f t h e material which i s d i f f u s i n g are bound up together i n an e f f e c t i v e d i f f u s i v i t y . This d i f f u s i v i t y w i l l generally be dependent on the concentration o f the d i f f u s i n g species b u t f o r many p r a c t i c a l purposes t h i s dependence i s ignored. Jackson's monograph [lo] gives an e x c e l l e n t treatment o f t h i s t o p i c . But, w h i l e i t i s o f t e n possible t o make the assumption o f constant d i f f u s i v i t y and c o n d u c t i v i t y , i t i s n o t u s u a l l y f e a s i b l e , o r sensible, t o make any s i m p l i f y i n g assumption as t o the r e a c t i o n r a t e . This i s g e n e r a l l y present as a nonl i n e a r f u n c t i o n o f the dependent v a r i a b l e s i n a p a i r o f p a r a b o l i c o r e l l i p t i c equations. The f u l l d e r i v a t i o n o f these w i l l n o t be c a r r i e d o u t i n d e t a i l here f o r i t i s the burden o f t h e second chapter o f my t r e a t i s e on the "Mathematical Theory o f D i f f u s i o n and Reaction i n Permeable Catalyst.'' ( [ 4 ] ; t h i s w i l l be referred t o as MT, followed by a page o r equation number.) L e t i t here s u f f i c e t o say t h a t t h e equations are obtained by the usual mass o r energy balances and i n t h e i r time-dependent form, g i v e a p a i r o f equations f o r the dimensionless concentration, u, and the dimensionless temperature, v. L e t us f u r t h e r s i m p l i f y the assumptions by assuming t h a t only a s i n g l e r e a c t i o n i s t a k i n g place and t h a t t h i s may be characterized by the concentration o f a s i n g l e component o r by a v a r i a b l e which represents t h e extent o f reaction. The equation f o r the concentration then i s
240
R. A RIS
I n t h i s equation u i s the concentration, made dimensionless by t h e surface concentration u , t i s the dimensionless time equal t o a d i f f u s i o n c o e f f i c i e n t times t h e r e a l %ime d i v i d e d by the square o f some l e n g t h parameter; $2 i s t h e r a t i o o f the maximum r a t e o f r e a c t i o n t o t h e corresponding d i f f u s i o n f l u x and i s a measure o f t h e r e l a t i v e preponderance o f r e a c t i o n r a t e . R(u) i s t h e dimensionless r e a c t i o n r a t e which a t u = 1 has been normalized t o have the value R(1) = 1. Again, f o r s i m p l i c i t y , t h i s equation w i l l be used w i t h t h e O i r i c h l e t c o n d i t i o n t h a t u = 1 on the surface o f t h e r e g i o n f o r which i t i s t o be solved. I n t h e equation i t s e l f the r a t e o f change o f concentration i n any element i s necessarily due t o the n e t f l u x by d i f f u s i o n (the f i r s t term on t h e right-hand side) minus t h e r a t e a t which t h a t substance i s disappearing by t h e reaction. ( C f . MT. pp. 51-9.) When t h e r e are several r e a c t i o n s i t i s necessary t o i n s e r t more variables and, o f course, t o have more than one r e a c t i o n r a t e . When t h e r e a c t i o n i s n o t isothermal, i . e . , the temperature o f t h e c a t a l y s t p e l l e t i s n o t uniform o r kept c o n s t a n t ' b y some o t h e r means, i t i s necessary t o make R a funct i o n o f u and v.
A s i m i l a r equation can be obtained by an energy balance i n t h e p a r t i c l e and i t y i e l d s a second p a r t i a l d i f f e r e n t i a l equation f o r v, the dimensionless temperature. Again, t h i s i s a q u a s i l i n e a r p a r a b o l i c equation i n which t h e temperat u r e change i s the sum o f a c o n t r i b u t i o n from conduction d i f f u s i o n and one from reaction:
2 (2) a t = v2v + p$ ~ ( u , v ) i n R, v = l on aR . Here p i s a heat o f r e a c t i o n parameter and L t h e r a t i o o f the thermal and L
material d i f f u s i v i t i e s . We w i l l again keep t h i n g s simple by using t h e D i r i c h l e t conditions v = 1 on t h e boundary o f t h e region. The p a r t i a l d i f f e r e n t i a l equat i o n s o b t a i n i n a connected region o f t h r e e dimensional space, R , w i t h t h e boundary conditions asserted on t h e piece-wise smooth boundary o f t h i s region, an. ( C f . Fig. 1.)
b! 2
Figure 1: The c a t a l y s t p e l l e t
NONLINEARITIES OF CHEMICAL REACTION ENGINEERING
249
The theme o f the paper w i l l , therefore, concern t h e behavior o f t h i s sytem when R(u,v) takes on various forms. For t h e most p a r t we s h a l l be d e a l i n g w i t h t h e steady s t a t e where the parabolic nature o f the d i f f e r e n t i a l equations degenerates i n t o e l l i p t i c i t y . Thus we w i l l n o t be concerned w i t h t h e i n i t i a l condit i o n s although there s u r e l y are such. I n the steady s t a t e there i s a most important f u n c t i o n a l o f the s o l u t i o n which represents t h e effectiveness o f the c a t a l y s t p e l l e t under t h e conditions o f operation. I t i s , i n f a c t , the average r a t e o f r e a c t i o n throughout the p e l l e t d i v i d e d by the r a t e o f r e a c t i o n which w i l l o b t a i n i n t h e absence o f any d i f f u sion o r conduction l i m i t a t i o n whatsoever. B r i e f l y , i t can be given by t h e integral
By i t s d e f i n i t i o n , t h i s f u n c t i o n a l w i l l tend t o value 1 when t h e r e a c t i o n i s very slow compared w i t h t h e speed o f t h e d i f f u s i v e processes and w i l l tend t o zero as the r e a c t i o n r a t e dominates the possible r a t e o f d i f f u s i o n . The measure o f t h i s r a t i o o f r e a c t i o n t o d i f f u s i o n r a t e i s the parameter 0, which i s o f t e n r e f e r r e d t o as the T h i e l e modulus i n honor o f the American engineer who, almost simultaneously w i t h Zeldowich i n Russia and Damklihler i n Germany, recognized i t s importance and devised t h e effectiveness as a way o f expressing i t s consequences. (For a f u l l e r treatment o f the h i s t o r y see MT pp. 37-42 and [ 3 ] . ) Powers and Polynomials Let us assume t h a t the p a r t i c l e i s isothermal and t h a t we, t h e r e f o r e , have no need f o r t h e second equation derived from an energy balance. The simplest assumption i s , o f course, t h a t the r e a c t i o n i s o f t h e f i r s t order, which i s expressed mathematically by making R(u) = u. This gives a thoroughly l i n e a r equation and i t would be out o f order i n t h e present context t o discuss i t i n any d e t a i l . One remark may be made i n passing however, i t concerns the simplest case and i s almost t r i v i a l l y obvious. I f t h e r e g i o n R i s a p a r a l l e l - s i d e d s l a b o f i n f i n i t e e x t e n t o r sealed edges, then the equation becomes
The immediate s o l u t i o n o f t h i s i s
A graph o f q against Q i n l o g a r i t h m i c coordinates shows t h e features t h a t we should expect; namely, i t i s asymptotic t o 1 as I$ goes t o 0 and t o 0 as $ goes t o i n f i n i t y . This i s a basic feature of the problem and these asymptotic r e l a t i o n s h i p s h o l d i n a l a r g e number o f o t h e r cases. I n f a c t , t h e product o f r l and I$ tends t o a constant value and t h i s i s merely because a t i n c r e a s i n g l y high r e a c t i o n r a t e s the d i f f u s i n g and r e a c t i n g substance disappears by r e a c t i o n w i t h i n a very s h o r t distance o f the surface. The effectiveness o f the p e l l e t thus becomes proportional t o i t s e x t e r i o r surface area r a t h e r than t o i t s volume. We remark i n passing t h a t there are some important generalizations o f t h i s l i n e a r case t o more recondite shapes o f R and t o simultaneous f i r s t order reaction. The l a t t e r gives a r a t h e r s i m i l a r formulation i n terms o f vectors o f concentrat i o n and matrices representing d i f f u s i v i t i e s and r e a c t i o n r a t e constants.
R. ARlS
250
I f the f i r s t - o r d e r r e a c t i o n corresponds t o t h e l i n e a r case and may be dismissed a t a conference dedicated t o nonlinear problems, the next e a s i e s t form o f non1i n e a r i t y i s s u r e l y t h a t o f t h e pth-order r e a c t i o n which i n symmetrical bodies (slab, c y l i n d e r o r sphere) gives a form o f t h e Emden-Fowler equation:
I n the geometry o f the slab w i t h q = 0 we see t h a t t h e equation may be i n t e g r a t e d by C l a i r a u t ' s transformation i . e . , by m u l t i p l y i n g both sides by t w i c e t h e f i r s t d e r i v a t i v e and i n t e g r a t i n g . I f M2 i s the f i r s t i n t e g r a l o f t h e r i g h t hand side o f t h i s equation, U
[ u ' ( p ) I 2 = $*
2R(w)dw = ~$~[M(u;u,)]~
(7)
uO
Then, f o r the pth-order r e a c t i o n we merely have some s t r a i g h t f o r w a r d i n t e g r a l s t o calculate.
Whence a second i n t e g r a t i o n gives
and, since u(1) = 1,
I t i s best t o t u r n t h i s problem arsey-versy and l e t u be t h e parameter i n terms o f which both t h e effectiveness f a c t o r and t h e Thieye modulus are calculated. Figure 2 shows t h e T h i e l e modulus as a f u n c t i o n o f t h e dimensionless center conc e n t r a t i o n and Figure 3 the e f f e c t i v e n e s s f a c t o r versus t h e normalized T h i e l e modulus. By normalized T h i e l e modulus we mean t h a t t h e T h i e l e modulus i s m u l t i p l i e d by a f a c t o r such t h a t t h e asymptotic value o f qQ i s 1. I t i s c l e a r from Eq. 9 t h a t t h i s i s t h e square r o o t o f (p+1)/2. An i n t e r e s t i n g aspect o f the second f i g u r e i s t h a t f o r values o f p l e s s than o r equal t o 1 t h e i n t e g r a l def i n i n g $I i n terms o f u does n o t diverge as u tends t o zero. It f o l l o w s t h a t f o r values o f $ greatep than a c e r t a i n c r i t i c a ? value ( i t s e l f dependent upon p) the o n l y way i n which a s o l u t i o n can be obtained i s f o r t h e r e t o be a dead region i n the center o f t h e p e l l e t i n which t h e concentration i s i d e n t i c a l l y zero. Then t h e boundary conditions
NONLINEARITIES OF CHEMICAL REACTION ENGINEERING
Dimensionless centre eonccnlration u.
Figure 2:
The r e l a t i o n between the dimensionless c e n t r e concentration, uo, and t h e Thiele modulus $ f o r the pG order reaction.
Figure 3:
The effectiveness f a c t o r f o r the p g order r e a c t i o n as a f u n c t i o n o f t h e normalized Thiele modulus.
25 1
252
R. A R K
u(p) = u ' ( p ) = 0 a t p = IN
(12)
u(1) = 1 serve t o determine w.as w e l l as t h e constants o f i n t e g r a t i o n . For an i n t e r e s t i n g g e n e r a l i z a t i o n o f t h i s f r e e boundary phenomenon see N i c h o l s ' papers [13,14,15]; the geometry o f a f i n i t e c y l i n d e r has been s t u d i e d by Stephanopoulos [18]. The reduction t o the hypergeometric fu?F+tlion may be i l l u s t r a t e d i n cases p > 1 f o r then the s u b s t i t u t i o n U = l-(uo/u) t u r n s t h e i n t e g r a l i n eq. (11) i n t o an incomplete beta f u n c t i o n and t h i s can be expressed as
Now the hypergeometric f u n c t i o n converges on t h e c i r c l e 11-u;'l I = 1, and i t follows t h a t since the exponent o f uo i s negative (0 w i l l range from i n f i n i t y t o zero as uo goes from 0 t o 1. Also
showing t h a t $ i s monotonic decreasing and t h a t t h e r e l a t i o n between $ and u For -1 < p < 1 t h e hypergeometric f u n c t i o n must be transformed bg i s unique. the r e l a t i o n F(a,b;c;z)
= (l-z)C-a-bF(c-a,c-b;c;z)
,
giving
The s o l u t i o n f o r p = -1may be obtained i n terms o f Dawson's i n t e g r a l 2
x
F(x) = e-'
et2 d t
,
0
and i s
Since F has a maximum o f 0.541 the value o f $* i s 0.765. The various cases can best be summarized i n t a b u l a r form
253
NONLINEARITIES OF CHEMICAL REACTION ENGINEERING
,
m = l/lp+ll
4 = (2m)*(l-U)
U4F(cr,b;c;U)
= -(2m)-4(l-U)cr'U-4F(crl
, ,bl ;cl ;U)
.
When p = -m/(m+l) o r -(2m+3)/2m+l) and m i s a p o s i t i v e i n t e g e r t h e hypergeomet r i c functions are polynomials o f degree m. Table 1 Range o f m P
U
a
C
a'
a'
bl
c'
3
4+m
7
-4
-4
m-4
4
1-u: +l0
1
1-m
3
-4
-4
m-4
4
3
-4
-4
m-4
4
1-m
1
1-m
4
--
--
--
--
0,4
l - u p0 + l
l,o
4,l 3
b
4
m,l
1
a
m-4
1,z
l - u P0 + l
0
1
1-m
-3,-1 7,"
3
l-uP+l
0
1
1-m
3
-1,-
1-u-p-l 0
4
1
4-m
4
O q 1
0
m,O
Rational Functions The next most complicated n o n l i n e a r i t y a r i s e s i n t h e case o f k i n e t i c s which are governed by so-called Langmuir-Hinshelwood o r Michael is-Menten k i n e t i c s (actua l l y the names o f Hougen and Watson can be r i g h t l y associated with t h e engineeri n g a p p l i c a t i o n s o f t h i s form o f k i n e t i c s , as can those o f Briggs and Haldane w i t h the biochemical use). When t h e k i n e t i c s o f t h e r e a c t i o n depend on adsorpt i o n on the s o l i d surface o f the c a t a l y s t p e l l e t i t i s n a t u r a l and proper t o make the r a t e o f r e a c t i o n a f u n c t i o n o f the adsorbed concentration r a t h e r than t h a t o f the gas and the pores. This can g i v e a dimensionless r a t e o f r e a c t i o n o f the form o f a q u o t i e n t o f two polynomials: U(E0+E1U+. R(u) =
Eo
+
El
+
. .+up-l) ... + 1
(20)
A simple reduction allows t h i s f u n c t i o n o f u t o vanish a t u = 0 as w e l l as t o s a t i s f y the c o n d i t i o n o f being 1 a t u = 1. I n t h e s l a b geometry t h i s y i e l d s a s i m i l a r normalization o f t h e T h i e l e modulus and leads t o such curves as are seen i n Figure 4 f o r various cases o f the parameters. (For references see MT p. 170, Table 3.8.)
254
R. A R E
5
,
1
I
I
, , , , ,
I,
2
I 9
0.5
0.2
0. I 0.1
Figure 4:
0.2
I
2
10
The normalized effectiveness f a c t o r p l o t f o r second-order LangmuirHinshelwood k i n e t i c s . Fixed LO=l and various K.
The simplest form o f t h e Langmuir isotherm, i n which the adsorbed species tend t o form a monolayer on the adsorbing surface, has some h a p p i l y simple propert i e s . I n p a r t i c u l a r , i n the simultaneous adsorbtion o f two s o l i d s from a flowi n g medium the equations o f balance are r e d u c i b l e and thus amenable t o t h e hodograph transformation. Under t h e i r transformation t h e c h a r a c t e r i s t i c s o f the system i n the plane o f the two concentrations c o n s i s t s o f a p a t t e r n o f s t r a i g h t l i n e s . This f o r t u i t o u s l i n e a r i t y o f a p a r t i c u l a r nonlinear system i s t h e secret o f the way i n which i t i s p o s s i b l e t o get some very f a r reaching r e s u l t s i n t h e theory o f chromography o f such substances. [17] Transcendental Functions Once t h e assumption o f i s o t h e r m a l i t y i s abandoned i t becomes i n e v i t a b l e t h a t t h e k i n d o f n o n l i n e a r i t y t h a t w i l l a r i s e i s a thoroughly transcendental one. T h i s i s because temperature influences t h e r e a c t i o n r a t e constants i n a h i g h l y nonl i n e a r fashion so t h a t they increase w i t h temperature according t o t h e law dank/dT = E/RT2. Thus, t o take t h e very simplest o f r e a c t i o n s , t h e i r r e v e r s i b l e pth-order disappearance o f the key substance, t h e r a t e o f r e a c t i o n w i l l be k(T) cp and k(T) = k e-E/RT. Reduction t o dimensionless v a r i a b l e s thus leads t o a p a i r o f equationsoof the form
u py a t = ~ ~ u - + ~exp
t-l)
These equations are n o t as objectionable as one might a t f i r s t suppose f o r t h e n o n l i n e a r i t y enters i n t o one term o n l y o f t h e right-hand s i d e and does so i n p r e c i s e l y the same fashion f o r each. Thus i f we form a combination o f temperat u r e and concentration @u+v=z
NONL INEA RITIES OF CHEMICAL REACTION ENGINEERING
255
and i f L = 1, t h i s combination s a t i s f i e s the heat equation. I n t h e steady s t a t e i t s a t i s f i e s Laplace's equation. But z i s constant and equal t o (l+f!) on t h e surface and hence, by the well-known p r o p e r t y o f a p o t e n t i a l f u n c t i o n , i s cons t a n t everywhere. Thus pu + v
E
1 + p and
or
I f we choose u as the v a r i a b l e the various forms o f G(u) w i l l , f o r constant p , depend on t h e two parameters p and y. The same applies t o t h e f u n c t i o n F(v) and may, perhaps, be i l l u s t r a t e d r a t h e r more e a s i l y f o r t h i s function. I t has been normalized so t h a t F ( l ) = 1 and since u vanishes when v = l + p t h e curve must take one o f t h e forms shown i n Figure 5. I n p a r t i c u l a r , i f t h e p o i n t fi, y l i e s beneath the hyperbola By = 1 ( i . e . , i n region A o f t h e By-plane) then t h e funct i o n F i s monotone decreasing i n the i n t e r v a l o f i n t e r e s t which i s (1, 1+p). I f (ply) l i e s i n the region p then F has a maximum, b u t no p o i n t o f i n f l e c t i o n . On the other hand i f p , y l i e s i n t h e region C between t h e two hyperbolae, By/(l+p) i s greater than 2 b u t less than 4, then t h e curve has an i n f l e c t i o n i s monotone decreasing as v passes from 1 t o 1 + 0. Howp o i n t , b u t F(v)/(v-1) ever, i f p,y l i e s above t h i s t h i r d hyperbola then F/v i s n o t monotonic and Luss has shown t h a t t h i s i s a necessary c o n d i t i o n f o r m u l t i p l i c i t y o f steady s t a t e s 1121.
*I-
Q
Figure 5:
The forms o f t h e f u n c t i o n F(v) and P(w) f o r p = l and several values o f BlY.
256
R. ARlS
L e t me t u r n aside t o note t h a t even here t h e nonlinear expression has a very i n t e r e s t i n g s i m i l a r i t y p r o p e r t y and one which allows every p o i n t o f t h e calcul a t i o n t o give some i n f o r m a t i o n f o r a f a m i l y o f q,~$curves. V i l l a d s e n and Michelsen [19] observed t h a t t h e f u n c t i o n G(u) was p a t i e n t o f the f o l l o w i n g s i m i l a r i t y transformation. I f we w r i t e
and s e t u = ulU,
then a f t e r some a l g e b r a i c manipulation we have
S i m i l a r l y , w i t h the temperature f u n c t i o n
we have f o r v = vVl
where
,
B = (1+ P-vl)/vl
r
= y/vl
(31)
.
As we should expect, t h e l o c i o f p o i n t s B,T t h a t are t r a c e d o u t i n t h e plane by varying u1 o r v1 are t h e same, namely
(1 + p)r-yB = y
.
(B,r)(32)
This represents a s t r a i g h t l i n e through (p,y) and (-1,O). This transformation implies t h a t every p o i n t i n t h e i n t e g r a t i o n o f t h e d i f f e r e n t i a l equations (25) f o r symmetrical geometries can be i n t e r p r e t e d t o g i v e t h e effectiveness f a c t o r f o r some problem. Consider, f o r example, i n t e g r a t i n g t h e I t i s convenient t o s e t f = $p so t h a t symmetrical equation (6).
and t o i n t e g r a t e t h i s as an i n i t i a l value problem w i th adu4 --0 ,
u = u0
atf=O
.
(34)
257
NONLINEARITIES OF CHEMICAL REACTION ENGINEERING
When the p o i n t a t which u = 1 i s reached, t h e value o f +1du
5
i s Q and
@ du = +1du
4sq
Q=>+=eaf
(35)
By keeping p and y f i x e d and varying u , the (q,t$)-curve can be t r a c e d out, f o r u i s a parameter along it. But i f %he i n t e g r a t i o n i s stopped a t the p o i n t wRere u = u a problem i s solved f o r t h e parameters B and r given by eqn. (28). To see t h i s b e s u b s t i t u t e
E' 2 -- p2Q2G(ul;B,Y)/ul
u = ulu,
(36)
i n eqn. (6) so t h a t , by eqn. (27), i t becomes
$+hdU - G(U;B,T) . I f t h i s i s i n t e g r a t e d from
5'
= 0 u n t i l U = 1 then t h e value o f
(37)
5"
Q 2 -- 02P2G(ul;B,Y)/ul
is (38)
and
Thus the i n t e g r a t i o n of eqn. (33) t o t h e p o i n t a t which u = u,l gives
f o r the B and
r
say
u1 = u(El),
given by eq. (28).
It has sometimes been f e l t t h a t the nature o f t h e n o n l i n e a r i t y i n the r e a c t i o n r a t e constant i s mathematically complicated and t h a t , since t h e usual range of temperatures i s i n t h e region where t h e exponential i s increasing r a p i d l y , i t should be p o s s i b l e t o replace the Arrhenius f u n c t i o n by a p o s i t i v e exponential i n temperature. When t h i s i s done, as i s o f t e n t h e case i n combustion studies, a very i n t e r e s t i n g equation r e s u l t s . I n engineering c i r c l e s t h i s i s associated w i t h Franck-Kamenetskii [ 5 ] and i n more mathematical w i t h t h e name o f Gelfand [7]. See a l s o Joseph and Lundgren [ll].
Though the j u s t i f i c a t i o n f o r considering the problem v2u+eU = u = o
o
in
R ,
on aR
258
R. A R A
may be r a t h e r slender i f i t i s t o be regarded as an approximation t o t h e zerothorder exothermic r e a c t i o n , t h e equation remains o f g r e a t i n t e r e s t . FrankKamenetskii (1939) was the f i r s t t o apply i t t o combustion problems. He wrote the equation as +
6eU =
o
,
.
u(*l) = 0
(42)
I f the s o l u t i o n i s computed i n the usual fashion using u as parameter t h e r e i s a maximum value o f 6, which Frank-Kamenetskii took t o % e t h e measure o f a c r i t i c a l s i z e above which spontaneous i g n i t i o n should take place. T h i s i s t r u e i f spontaneous i g n i t i o n i s understood t o mean t h a t s o l u t i o n s o f eqns. (42) o n l y e x i s t i f there i s a dead core w i t h
This was f i r s t solved by quadratures f o r t h e slab ( q = 0) and l a t e r Chambr6 (1952) found an a n a l y t i c a l s o l u t i o n f o r t h e c y l i n d e r ( q = 1). For the sphere no a n a l y t i c a l s o l u t i o n i s obtained, b u t t h e problem becomes p a r t i c u l a r l y i n t e r e s t i n g as t h e r e are i n f i n i t e l y many s o l u t i o n s f o r a c r i t i c a l value o f 6. The paper o f Hlavicek and Marek [ 8 ] gives the most complete treatment from t h e chemical engineering viewpoint.
(i) The i n f i n i t e f l a t p l a t e . t o eq. (42) namely
For the s l a b t h e r e i s an a n a l y t i c a l s o l u t i o n
,
eu = A sech2{p&A6/2))
(44)
and the boundary c o n d i t i o n can be matched i f
.
A = cosh2{,/(A6/2))
(45)
This provides an equation f o r A which has two s o l u t i o n s when 6
6, = 0.878
.
(46)
I f 6 = 6 there i s j u s t one s o l u t i o n w i t h u(0) = En A = 1.188 and hence a maximum temphature r i s e w i t h i n o f 1.2RTZ/E. This c r i t i c a l value o f 6 i s h e l d t o be a c r i t e r i o n f o r the c r i t i c a l s l a b thfckness above which no steady s t a t e s o l u t i o n can be found and i s i n t e r p r e t e d as the explosion l i m i t .
However t r u e t h i s may be, i n p r a c t i c e i t i s based on an incomplete a n a l y s i s o f t h e problem. Steady s t a t e s o l u t i o n s can be found f o r any value o f 6 b u t t h e To see t h i s we boundary conditions (43) w i l l have t o be invoked f o r 6 > 6., r e t u r n t o eqn. (42) and w r i t e t h e boundary c o n d i t i o n s as u(1) = 0
,
u(w) = uo,
u'(w) = 0
Then i f w = o
,
O,U,,BY
we have eqn. (42) and i f
I
.
(47)
259
NONLINEARITIES OF CHEMICAL REACTION ENGINEERING
l > w > O,
uo=py
we have eqn. (43).
M u l t i p l y i n g eqn. (42) f o r q = 0 by 2u'(p) and i n t e g r a t i n g we
have U
[U'(P)12 = 2*(euo-e )
,
(48)
and hence
-4Uo
= 2e
(49)
sech-'[ expi-+( uo-u(p)))]
The boundary c o n d i t i o n a t p = 1 i s s a t i s f i e d i f
4
-4u,
-4U
(26) (1-w) = 2e
'sech-'(e
)
,
which since A = euO i s j u s t a form o f eq. (48) when w = 0. written 6(1-w)'
= 2e-uo{sech-1(e -4u0)]2
and observe t h e form o f t h e curve r e l a t i n g S(1-w)
Figure 6:
L e t t h i s equation be
(51) 2
and uo as shown i n Fig. 6.
The s o l u t i o n o f eq. (51).
26a
R. A RIS
Now t h e equation as we have w r i t t e n i t i n eq. (42) takes no account o f the f a c t t h a t a r e a c t i o n cannot proceed when t h e r e a c t a n t bas been completely exhausted, i.e., when u exceeds py. Thus i n place o f e we should r e a l l y have e {l-H(u-py)), where H i s the Heaviside step f u n c t i o n . I f py > 1.188 the s i t u a t i o n i s as i n d i c a t e d by the upper h o r i z o n t a l l i n e i n Fig. 6 which i s drawn through the p o i n t . 6 ( 1 - ~ )=~ 6* = 2e-pY{sech-1(e -4Bu)]2
.
Since u cannot exceed py t h e r e i s o n l y one s o l u t i o n o f eqn. (51) when 6 < 6 * , namely 1 p o i n t on t h e p a r t OA o f t h e curve. There can be no s o l u t i o n w i t h w > 0 f o r 6 < 6*. I f 6* < 6 < 6 t h e r e a r e two s o l u t i o n s w i t h UFO on t h e branches AB and BC r e s p e c t i v e l y , ht t h e r e i s a t h i r d s o l u t i o n w i t h u = py and w > 0 on the l i n e CD. I f A > 6 then t h e r e i s no steady s t a t e s o l g t i o n w i t h UFO, b u t there i s always a s o l u t f o n on DE w i t h w > 0. I n t h e l a t t e r cases t h e value o f w i s given by
I f py < 1.188 then the h o r i z o n t a l l i n e i n Fig. 6 would have t o be drawn from a p o i n t n: the branch OB. I n t h i s case, t h e s o l u t i o n i s unique. Hlavlcek and Marek [ 8 ] appear t o be the f i r s t t o have recognized t h i s f e a t u r e o f r e a c t a n t exhaustion. The effectiveness f a c t o r
when t h e s o l u t i o n s a t i s f i e s t h e boundary c o n d i t i o n s w i t h w > 0 then
These curves are shown i n Fig. 7, which i s based on t h e c a l c u l a t i o n s o f H1avicek and Marek. [9]
./ ,A 2
I
Figure 7:
0.01
0.1
0.2
0.5
1
10
Effectiveness f a c t o r f o r Frank-Kamenetskii's equation i n t h e slab.
261
NONLINEARITIES OF CHEMICAL REACTION ENGINEERING
(ii)The i n f i n i t e c y l i n d e r . Chambr6 (1952) showed how t o o b t a i n an anaI n case w0 t h e l y t i c a l s o l u t i o n o f eq. (42) f o r the i n f i n i t e c y l i n d e r , q=l. substitutions P du q - -V ,
p2eu = w
(56)
are made, reducing the equation t o dv 6 ai+2+V=O
(57)
Since v = w = 0 a t the center, the immediate i n t e g r a l o f eq. (57) i s v2 + 4v + 26w = 0
(58)
or
But eq. (42) can be w r i t t e n
so t h a t by s u b t r a c t i o n
The equation can be i n t e g r a t e d t o give eU =
8B 6(Bp2+1)'
where B has t o be chosen t o s a t i s f y the boundary c o n d i t i o n a t the surface, i . e . S(B+l)'
= 88
.
(63)
This equation again has two s o l u t i o n s provided t h a t 6 < 6 = 2. The same type o f analysis goes through t h e r e l a t i o n between uo and 6 an% i s shown as one o f the three curves i n Fig. 8. eu = (p*/b)' w =
where b s a t i s f i e s
cosh2[1n{b+(b2-1)*]
- i},
+ a I n p]
a = b(6/2)
4
(64) (65)
262
R. A RIS
;
I
‘1
‘
\
u,
Figure 8:
Centre ‘temperature’, u as a f u n c t i o n o f 6(1-w)2 f o r s l a b c y l i n d e r (C), and spherg (5).
(L),
The effectiveness f a c t o r i s then
(iii) The s here. The i n t e g r a t i o n o f eq. (42) i n t h e sphere requires, as we have s h e previous section, a numerical i n t e g r a t i o n and leads t o the isothermal f u n c t i o n o r i t s generalizations. However, a very s i g n i f i c a n t d i f f e r e n c e a r i s e s when the ( u ,&) curve i s continued t o l a r g e r values o f u as i n the curve 5 o f Fig. 8. I t o i s seen t h a t t h e curve does n o t merely go thr8ugh a maximum o f 6, the so-called c r i t i c a l value 6 = 3.32, b u t t h a t i t does n o t come back a s y m p t o t i c a l l y t o t h e u - a x i s b u t o g c i l l a t e s about t h e l i n e 6=2. Steggerda has given t h e f i r s t e i g h t t u r n i n g p o i n t s :
6
3.3220
1.6641
2.1083
1.9666
2.0095
1.9957
2.0002
1.9992
uo
1.6075
6.7409
11.3769
16.1363
20.9159
25.3332
31.7950
35.6683
This means t h a t we can o b t a i n an a r b i t r a r i l y l a r g e number o f s o l u t i o n s by making py l a r g e enough and 6 close enough t o 2. For &=2 t h e r e i s a s i n g u l a r s o l u t i o n which we can o b t a i n as follows. Let
263
NONLINEARITIES OF CHEMICAL REACTION ENGINEERING
,
v = p -du+ 2 dP
w=u+21np
(68)
then n
dv = 2-v-6eL dw
v
This ordinary d i f f e r e n t i a l equation has a s i n g u l a r p o i n t a t v=O, w=Pn(2/6). w=ln(2/6) i s a c t u a l l y a s o l u t i o n o f
But
which though i t does n o t s a t i s f y the boundary c o n d i t i o n a t p=O, w i l l s a t i s f y u(1) = 0 i f 6 = 2. This i s Emden's s i n g u l a r s o l u t i o n , and t h e s o l u t i o n s f o r 6 = 2 w i t h u corresponding t o successive crossing p o i n t s o f the curve S o f Fig. 8 w i t h the v g r t i c a l l i n e 6 = 2 approaching more and more c l o s e l y t o the s i n g u l a r s o l u t i o n , departing only i n the neighborhood o f p = 0. Since i t i s the sphere t h a t f i r s t shows the phenomenon o f i n f i n i t e m u l t i p l i c i t y , i t i s i n t e r e s t i n g t o i n q u i r e what behavior the s o l u t i o n f o r a f i n i t e c y l i n d e r might e x h i b i t . A t one extreme the c o i n t h i s i s a f l a t p l a t e ; a t t h e other the c u r t a i n rod i t approaches the i n f i n i t e c y l i n d e r . On t h e other hand, a c y l i n d e r w i t h equal h e i g h t and diameter has the same surface t o volume r a t i o as a sphere and one i s n a t u r a l l y l e d t o ask i f i t gives r i s e t o an i n f i n i t y o f solutions when the parameter i s
-
v2u
+ heU =
-
-
-
o
has some c r i t i c a l value. The most complete r e s u l t s o f t h i s k i n d have been given by V i l l a d s e n and Ivanov They showed [20] who t r e a t e d the exponent q i n eq. (42) as a "geometry factor.'' t h a t an i n f i n i t e number o f steady s t a t e s p r e v a i l e d when 1 < q < 9. No more than three e x i s t f o r q < 1 o r q > 10.44 and o n l y a f i n i t e number f o r 9 5 q 5 10.44. Thus i t would seem-that when the p e l l e t i s the l e a s t b i t more s p h e r i c a l ' than i t i s ' c y l i n d r i c a l ' ( i . e . , q = 1+&), t h e r e can be an i n f i n i t e number o f solutions. An exact p r o o f t h a t the p o s i t i v e exponential approximation does n o t change t h e p i c t u r e q u a l i t a t i v e l y has n o t been given f o r the case o f d i f f u s i o n and r e a c t i o n though there are two d i r e c t i o n s from which i t may be approached f o r the w e l l mixed reactor. Since there are grounds f o r b e l i e v i n g t h a t the q u a l i t a t i v e behav i o r o f the c a t a l y s t p a r t i c l e has some a f f i n i t y w i t h t h a t o f t h e s t i r r e d tank, i t may be o f i n t e r e s t t o comment on t h i s here [2]. One such discussion i s t o be found i n Poorels paper i n the Archive f o r Rational Mechanics and Analysis [16]. He studies the phase plane o f the s t i r r e d tank and shows t h a t i t i s closed i n both f o r the Arrhenius f u n c t i o n and t h e p o s i t i v e exponential. Another approach has been provided by Golubitsky and K e y f i t z [7], i n t h e i r " q u a l i t a t i v e study o f the steady s t a t e s s o l u t i o n s f o r a continuous f l o w s t i r r e d tank chemical reactor." Their argument turns on a c a r e f u l a n a l y s i s o f t h e p r o p e r t i e s o f the organizing center o f t h i s p a r t i c u l a r problem and they o b t a i n formulae appropriate t o a s i n g u l a r i t y they c a l l t h e 'winged cusp. ' T h e i r arguments t u r n on the behavior o f the Arrhenius expression and t h e i n e q u a l i t i e s which i t s d e r i v a t i v e s s a t i s f y . The p o s i t i v e exponential approximation could be obtained from t h e i r f u n c t i o n by r e p l a c i n g yy by v and t a k i n g t h e l i m i t o f y
264
R. ARlS
going t o i n f i n i t y . The i n e q u a l i t i e s which they f i n d necessary t o impose are s a t i s f i e d equally w e l l by t h i s l i m i t as by t h e more complicated f u n c t i o n . I t would, therefore, seem c l e a r t h a t the q u a l i t a t i v e behavior o f the two systems i s identical. ACKNOWLEDGEMENT This paper was prepared w h i l s t the author had t h e p r i v i l e g e o f being a Sherman F a i r c h i l d Scholar a t t h e C a l i f o r n i a I n s t i t u t e o f Technology. Anyone who knows Caltech need o n l y be t o l d t h a t i n the generosity o f i t s s t r u c t u r e , t h i s program f u l l y r e f l e c t s t h e excellence o f t h a t i n s t i t u t i o n . References and Schilson, R. E. (1961) " I n t r a p a r t i c l e D i f f u s i o n and Conduction i n Porous Catalysts," Chem. Engrg. Sci. 13,226, 237.
[l] Amundson, N. R.,
[2] A r i s , R. (1969), "On S t a b i l i t y C r i t e r i a o f Chemical Reaction Engineering,'' Chem. Engrg. Sci. 25, 149. [3] A r i s , R. (1974), "The Theory o f D i f f u s i o n and Reaction neering Symphony," Chem. Engrg. Educ. 8, 19.
-
A Chemical Engi-
[4] A r i s , R.
(1975), "The Mathematical Theory o f D i f f u s i o n and Reaction i n Permeable Catalysts" (2 vols.), Clarendon Press, Oxford.
[5] Frank-Kamenetskii , D. A. (1939), " C a l c u l a t i o n o f Thermal Explosion Limits," Acta phys. - chim. URSS 0 , 365. [6] Gelfand, I. M. (1969), "Some Problems i n the Theory o f Q u a s i l i n e a r Equations," Usp. Mat. Nauk XV, No. 2, (86), 87. Trans. Amer. Math. SOC. (1963), (2), 9,295. [7] Golubitsky, M.,
and K e y f i t z , 6. L. (1980), "A Q u a l i t a t i v e Study o f t h e Steady-State Solutions f o r a Continuous S t i r r e d Tank Chemical Reactor," S I A M J. Math. Anal. 316.
11,
, and Marek, M. (1968), "Heat and Mass Transfer i n a Porous C a t a l y s t P a r t i c l e . On t h e M u l t i p l i c i t y o f Solutions f o r t h e Case o f an Exothermic Zeroth-order reaction. 'I C o l l Czech. Chem. Commun. (Eng. Edn. )
[8] Hlavikek, V.
.
33, 506.
[9] Hlavicek, V.,
and Marek, M. (1968), "Zum Entwurf k a t a l y t i s c h e r Rohrreakt o r e n bei r a d i a l e n Warmetransport," Proc. I n s t . Chem. Engrs. and V.T.G.V.D.I. Meeting, Brighton 1968.
[lo] Jackson, R.
, "Diffusion
i n Porous Media" (1977), E l s e v i e r , New York.
and Lundgren, T. S. (1973), " Q u a s i l i n e a r D i r i c h l e t Problems Driven by P o s i t i v e Sources," Arch. Rational Mech. Anal. 9, 241.
[ll] Joseph, D. D.,
[12] Luss, D. (1971), "Uniqueness c r i t e r i a f o r lumped and d i s t r i b u t e d parameter chemically r e a c t i n g systems , I ' Chem. Engrg. Sci. 26, 1713. [13] Nicolaenko, B. (1977), "A General Class o f Nonlinear B i f u r c a t i o n Problems from a P o i n t i n the Essential Spectrum. A p p l i c a t i o n t o Shock Wave Solut i o n s o f K i n e t i c Equations," A p p l i c a t i o n s o f B i f u r c a t i o n Theory, Academic Press, New York, 1977.
NONLINEARITIES OF CHEMICAL REACTION ENGINEERING
265
[14] Nicolaenko, B., and Brauner, C. M. (1979), "Singular Perturbations and Free Boundary Value Problems," Proc. I V t h I n t e r n a t i o n a l Symposium on Computing Methods i n Applied Sciences and Engineering. I.R.I.A. V e r s a i l l e s , 1979, North-Holland Publishing.
[15] Nicolaenko, B. and Brauner, C. M. (1979) "Nonlinear Eigenvalue Problems which Extend i n t o Free Boundary Problems," Proc. Conf. on Nonlinear Eigenvalues. Springer-Verlag Lecture Notes i n Mathematics No. 782.
[XI Poore, A. B. (1973), " M u l t i p l i c i t y , S t a b i l i t y and B i f u r c a t i o n o f P e r i o d i c Solutions i n Problems a r i s i n g from Chemical Reactor Theory," Arch. Rational Mech. Anal. 52, 358.
[17] Rhee, H-K.,
A r i s , R . , and Amundson, N. R. (1971), "Multicomponent Adsorpt i o n i n Continuous Countercurrent Exchangers," P h i l . Trans. Roy. SOC. A.
269, 187.
[18] Stephanopoulos, G. (1980), "Zero Concentration Surfaces i n a C y l i n d r i c a l Catalyst Pellet,'' Chem. Engrg. Sci. 3, 2345. [19] Villadsen, J., and Michelsen, M. Spherical Catalyst P e l l e t s : Chem. Engrg. Sci. 3 , 751.
L. (1972), " D i f f u s i o n and Reaction on Steady-state and Local S t a b i l i t y Analysis,''
[20] Villadsen, J., and Ivanov, E. (1978), "A note on t h e m u l t i p l i c i t y o f t h e Weisz-Hicks problem f o r an a r b i t r a r y geometry factor,"
33, 41.
Chem. Engrg. Sci.
This Page Intentionally Left Blank
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko Ids.) 0 North-Holland Publishing Company, 1982
267
PROPAGATING FRONTS I N REACTIVE MEDIA Paul C. F i f e Mathematics Department U n i v e r s i t y of Arizona Tucson, AZ 8 5 7 2 1
Wavelike phenomena of many d i f f e r e n t t y p e s can be modeled w i t h t h e a i d of systems of p a r t i a l d i f f e r e n t i a l e q u a t i o n s of convection-diffusion-reaction type. Combustion t h e o r y provides a t y p i c a l s e t t i n g f o r t h e s e e q u a t i o n s , but t h e y a r e used i n a g r e a t number of o t h e r c o n t e x t s as w e l l . I shall concentrate on onedimensional steady-state A review ( n e c e s s a r i l y t r a v e l i n g f r o n t solutions. incomplete) w i l l be made of a number of important advances i n (1) asymptotic and o t h e r methods f o r r e d u c i n g t h e complexity of given systems, and ( 2 ) more r i g o r o u s r e s u l t s on s y s t e m of one o r two equations. INTRODUCTION
The s u b j e c t of my t a l k , propagating f r o n t s , h a s t o do w i t h a phenomenon observed i n a wide v a r i e t y of c o n t e x t s p h y s i c a l , chemical, and b i o l o g i c a l . The analogy i s more t h a n s u p e r f i c i a l : t o a g r e a t e x t e n t , s i m i l a r concepts and t o o l s can be used both i n modeling and i n a n a l y z i n g wave phenomena i n a l l t h e s e disciplines. I s h a l l of c o u r s e touch on some, p o s s i b l y t h e most b a s i c , of t h e "common ground" i n t h i s l e c t u r e ; but nuch mst n e c e s s a r i l y be l e f t unsaid.
-
F i r s t , t h e general picture. The p r o t o t y p i c a l example of f r o n t s i n r e a c t i v e media i s a flame, and a couple of my c a s e s t u d i e s w i l l be drawn from flame theory. But more a b s t r a c t l y , one can simply envisage a medium continuously d i s t r i b u t e d i n space which can e x i s t , a t each p o i n t i n space, i n a v a r i e t y of p o s s i b l e s t a t e s , d e s c r i b e d by an n-dimensional vector. These s t a t e s a r e e assume t h e r e are dynamic, i n t h a t they can change a c c o r d i n g t o known r u l e s . W a t l e a s t two r e s t s t a t e s a v a i l a b l e ; t h e s e a r e states which w i l l not change u n l e s s p e r t u r b e d by some i n f l u e n c e beyond t h e l o c a l dynamics. Suppose t h e e n t i r e medium e x i s t s i n i t i a l l y i n one of t h e a v a i l a b l e rest s t a t e s , and t h a t a l o c a l " i g n i t i o n " event happens: i n a l i m i t e d r e g i o n of space, t h e medium i s taken i n t o t h e domain of a t t r a c t i o n of a n o t h e r rest s t a t e . This p e r t u r b a t i o n , and t h e r e s u l t a n t a t t r a c t i o n t o t h e o t h e r s t a t e , may a f f e c t nearby regions i n t h e medium through some t r a n s p o r t mechanism, and t h e s e nearby r e g i o n s a r e induced t o undergo a s i m i l a r t r a n s i t i o n between rest s t a t e s . This i s t h e beginning of a chain r e a c t i o n , or domino e f f e c t . L e t us c a l l i t a propagating front. I n t h e above p i c t u r e , a f r o n t w i l l propagate i n a l l d i r e c t i o n s from t h e i g n i t i o n s p o t , but I would l i k e t o l i m i t t h e concept t o t h e case of a u n i d i r e c t i o n a l f r o n t i n a o n e d i m e n s i o n a l medium. With x denoting p o s i t i o n t h e s t a t e of t h e system, t h e state dynamics and t r a n s p o r t and u ( x , t ) e Rn mechanism a r e assumed t o be d e s c r i b e d by a system of p a r t i a l d i f f e r e n t i a l e q u a t i o n s of t h e t y p e ut
+
(f(u)),
-
( D ( U ) U , ) ~+ g ( u )
(1)
268
P.C. FIFE
(D i s a matrix). D e t e r m i n i s t i c continuous space-time models of t h e s i t u a t i o n described above almost always may be put i n t o e i t h e r t h i s form o r t h a t of a n i n t e g r o - d i f f e r e n t i a l equation. The second term on t h e l e f t of (1) accounts f o r "convection", and those on t h e r i g h t are t h e " d i f f u s i v e t r a n s p o r t " and " r e a c t i o n " terms. We c a l l D t h e d i f f u s i o n matrix. Fronts a r e now defined t o be s o l u t i o n s of t h e form v e l o c i t y c, s a t i s f y i n g
-
u(+)
uf
u = u(x-ct) = u(z)
f o r some
(bounded d i s t i n c t l i m i t s )
The f r o n t phenomenon is e s s e n t i a l l y nonlinear, because no such s o l u t i o n s e x i s t when f and g a r e l i n e a r f u n c t i o n s of u ( a t l e a s t when D i s a c o n s t a n t ) . I s h a l l consider only smooth s o l u t i o n s (and discontinuous shocks). Front s o l u t i o n s s a t i s f y -cuZ
+
(f(u)IZ u(*)
so
will
exclude
strictly
( D ( U ) U ~+) ~ g(u)
(24
=
(2b)
Uf
The following b a s i c q u e s t i o n s about f r o n t s arise: a. b. c. d.
For which u+, u-, and c do t h e r e e x i s t f r o n t s ? What i s t h e s t r u c t u r e of t h e f r o n t ' s p r o f i l e ? Is t h e f r o n t s t a b l e , (I) i n one space dimension (11) when imbedded i n a h i g h e r dimensional space? If u n s t a b l e , what o t h e r s t a b l e s t r u c t u r e s may appear?
The emphasis h e r e w i l l be on ( a ) and ( b ) , and t o s6me e x t e n t ( c ) . be i n two p a r t s :
The t a l k w i l l
I. Techniques ( p r i m a r i l y asymptotic) t o reduce t h e cornplexity of systems. 11. Some of t h e more d e t a i l e d and r i g o r o u s r e s u l t s t h a t have been obtained f o r t h e simplest systems (one o r two equations). I have not a t t e n p t e d t o c o q i l e complete r e f e r e n c e l i s t s f o r r e s u l t s similar t o those described here, o r even t o t r a c e t h e o r i g i n of many of t h e i d e a s o u t l i n e d . Moreover, t h e v a s t amount of e f f o r t on numerical approaches w i l l n o t be mentioned f u r t h e r . There a r e c e r t a i n immediate necessary c o n d i t i o n s f o r t h e e x i s t e n c e of a f r o n t which should be made c l e a r i n t h e s e i n t r o d u c t o r y comments. I n t e g r a t i n g (2a) from to -, one o b t a i n s
-
m
-
-CAU
+ Af(u)
g(u(z))dz
(3)
4
-
where Au u u-. lhis i s a condition r e l a t i n g u+,u-, and c, but i t i s not t o o usefuf i f t h e integrand g ( u ( z ) ) i s not known. A t y p i c a l occurence i s t h a t some components of of u accordingly:
g
vanish.
I s h a l l , i n f a c t , s e g r e g a t e t h e cornponents
ui
i s a p h y s i c a l v a r i a b l e i f g,(u)
f
0;
ui
i s a chemical v a r i a b l e i f gi(u)
1
0.
(The number of each k i n d i s not an i n v a r i a n t w i t h r e s p e c t t o transformations of (1) i n t o equivalent systems.)The s e t of p h y s i c a l v a r i a b l e s I denote by , t h e chemical ones by ; then
PROPAGATING FRONTS IN REACTIVE MEDIA
u
-
-
Af(u)
From (3) we o b t a i n A
d u and
A
-
(us u),
and s i m i l a r l y
f
-
1
269
-
(f, f).
1
(Rankine-Hugoniot
(4)
relations)
(5)
g(u*) * 0 In all, these a r e
n
equations r e l a t i n g
u+
and
c.
Though n e c e s s a r y , t h e i r
s a t i s f a c t i o n is by no means s u f f i c i e n t f o r t h e e x i s t e n c e of a f r o n t . I. REDUCING A SYSTEM'S COMPLEXITY I s h a l l proceed by o u t l i n i n g f o u r examples of asymptotic methods by which l a r g e r systems may be reduced t o a combination of smaller systems. This n o t o n l y r e s u l t s i n a n e a s i e r a n a l y s i s of t h e o r i g i n a l system, b u t i t a l s o c l a r i f i e s some q u a l i t a t i v e f e a t u r e s of t h e f r o n t ' s p r o f i l e . These f o u r examples are meant t o exemplify f o u r c a t e g o r i e s of approaches, each r e a l l y c o n p r i s i n g many p o s s i b l e And indeed i n some v a r i a t i o n s o r r e l a t i v e s which w i l l n o t a l l be mentioned. c a s e s many s i g n i f i c a n t r e l a t e d t r e a t m e n t s have appeared i n t h e l i t e r a t u r e . Again, my i n t e n t i o n i s n o t t o g i v e a complete survey.
I s h a l l attempt t o d i s r e g a r d d e t a i l s which do not bear d i r e c t l y on t h e mathematical s t r u c t u r e of t h e argument being presented; t h i s is done i n t h e i n t e r e s t of s a v i n g time and of e l u c i d a t i n g t h a t b a s i c s t r u c t u r e i t s e l f by doing away w i t h confusing s i d e i s s u e s . For example, I s h a l l n o t p o i n t o u t t h e v a r i o u s dimensionless c o n s t a n t s u s u a l l y a s s o c i a t e d w i t h combustion problems. F i n a l l y , t h e s i t u a t i o n s d e s c r i b e d i n v o l v e a small parameter E, and t h e methods E + 0. can be c l a s s e d a s lowest o r d e r formal asymptotic methods as Approximations which a r e h i g h e r o r d e r i n E can be o b t a i n e d , though sometimes with difficulty. 1. ZND and C J d e t o n a t i o n a n a l y s i s .
The names a s s o c i a t e d w i t h t h e T h i s i s t h e o l d e s t and s i m p l e s t of t h e examples. i n i t i a l s are Zeldovi& von Neumann, Doering, Chapman, and Jouguet. The f i r s t t h r e e developed t h e i r approaches t o d e t o n a t i o n s independently of one a n o t h e r , The CJ models go back f u r t h e r . There a r e around t h e t i m e of World War 11. s e v e r a l p o s s i b l e avenues t o t h e study of ZND d e t o n a t i o n s (see Williams (1965), f o r example). The approach I t a k e i s simply t o assume t h e t r a n s p o r t m a t r i x D is small, and proceed by formal asymptotics. Therefore r e p l a c e D by ED i n (21, where E
0.
C l e a r l y , by (5) t h i s c o n s t a n t is a s o l u t i o n of (2a). 5
-
(2)
The shock has p r o f i l e determined by s t r e t c h i n g t h e v a r i a b l e u(z) = U ( C ) , so t h a t
Z/E,
2.
Set
270
P.C. FIFE
Neglecting t h e O(E) c o r r e c t i o n , one discovers t h a t t h e r e a c t i o n terms have vanished. The reduced problem is t h e r e f o r e l i k e a p h y s i c a l v a r i a b l e problem, and t h e Rankine-Hugoniot necessary c o n d i t i o n s apply: c(U+
- u-) + f(U+) -
f(UJ
-
0.
(7)
Since U+ = u+ is known, t h i s is a n equation r e l a t i n g U- t o c. Let u s assume t h a t t h e r e e x i s t s a t l e a s t one s o l u t i o n U- of ( 7 ) f o r each c i n some parameter range I, and moreover t h a t t h e r e e x i s t s an e x a c t s o l u t i o n of (6) connecting U+ t o U-. Then t h e r e w i l l be a one-parameter family of p o s s i b l e shock p r o f i l e s U(s) , with v e l o c i t y c being t h e parameter.
In cases of most p h y s i c a l i n t e r e s t , t h e system ( 7 ) can be reduced t o t h e w e l l known Rankine-Hugoniot system in t h e p h y s i c a l v a r i a b l e s d e n s i t y , momentum, and energy. In f a c t , assume t h a t t h e s e a r e t h z p h y s i c a l v a r i a b l e s i n q u e s t i o n , and t h a t t h e chemical v a r i a b l e s are given by U = pZ w i t h f = p a , where Z i s t h e vector of mass f r a c t i o n s of t h e c h e d x a l s p e c i e s , v A i s t h e f l u i d v e l o c i t y ; If p = U,, s a y , t h e n f l = pv. L e t us write and p i s t h e t o t a l d e n s i t y . h
f(U) as a f u n c t i o n of U and 2: f(^U,Z). TheJollowing f a c t can be e a s i l y v e r i f i e d : I f , f o r some constant Z+, t h e v e c t o r s U* s a t i s f y
then ( 7 ) is s a t i s f i e d w i t h t h e s e given
and w i t h
U+,
U* = p*Z+.
On t h e Rankine-Hugoniot l e v e l , then, t h e problem reduces t o (8) with Z+ given by t h e conditions ahead of t h e shock. This i s an elementary and well-studied problem. On t h e higher l e v e l of t h e f u l l equations ( 6 ) , l e t us simply assume t h e e x i s t e n c e of a s o l u t i o n connecting U+ and U-. If t h e r e is no s p e c i e s DUc d i f f u s i o n (more s p e c i f i c a l l y , i f t h e chemical components of t h e v e c t o r vanish whenever Z c o n s t a n t ) , then such an e x i s t e n c e r e s u l t follows from a r e s u l t of Gilbarg (1951), t o be mentioned l a t e r . A l l t h i s r e s u l t s in a onew i t h U(-) = U-, depending o n ' t h e parameter family of shock p r o f i l e s U(c) parameter c.
-
(3) Behind t h e shock, we r e v e r t t o t h e o r i g i n a l v a r i a b l e s , s o (6a) a p p l i e s w i t h D replaced by ED. This e q u a t i o n i s t o hold f o r z < 0 To match w i t h t h e shock p r o f i l e considered above, i t i s necessary t h a t U-. Now set E = 0 t o o b t a i n a n ordinary d i f f e r e n t i a l equation i n i t i a l u(0) value problem:
.
-
-cu
+
(f(u))2 = g(u), u(0) =
u-
.
2
> 1 but kE 0, w e r e v e r t t o (13). Since i n t h i s r e g i o n u is s t r i c t l y assumption (14) s a y s t h a t t h e r e a c t i o n term i n (13) i s between uand u+, for E. There r e s u l t s t h e f o l l o w i n g boundary v a l u e problem:
(DUz),
+ cuz
-
u ( 0 ) = u-, u(-) U,(O)
0,
-
z
> 0;
u+,
= YV,
(21a) (2 1b) (21c)
t h e l a s t coming from (20). There being t o o many boundary c o n d i t i o n s , t h i s problem is overdetermined and h a s a s o l u t i o n o n l y if a c o p p a t i b i l i t y c o n d i t i o n is s a t i s f i e d . To determine what t h a t c o n d i t i o n is, i n t e g r a t e ( 2 l a ) once t o o b t a i n Lhz
+ cu
= c o n s t = cu+.
274
At
P.C. FIFE
z = 0,
t h i s gives
I n view of ( 2 1 c) , t h e c o n d i t i o n i s t h e r e f o r e
- u-).
yDoV = c(u+
( 2 2)
Although t h i s i s a v e c t o r e q u a t i o n , i t t u r n s o u t t h a t t h e r e i s a s i n g l e v a l u e of c f o r which i t i s s a t i s f i e d . To see t h i s , t a k e t h e scalar p r o d u ct of (13) w i t h t h e two v e c t o r s Mi o r t h o g o n a l t o K. After i n t e g r a t i n g once, one f i n d s cMii u
The
v al u es
M .(u+-u-) 1
at = 0,
z = f
rmst
m
which means t h a t
c known, t h e p r o f i l e
-
u
for
+
u
-
.
The p o s i t i v e d e f i n i t e n e s s of s o l u t i o n w i t h u(-) u+
hz = co n st .
be
same,
the
T h e r e f o r e (22)
K.
>
z
0
-
uz
0
there,
so
K:
y = cb.
becomes
This
c an be determined by s o l v i n g
-1 cD ( u ) ( u u + )
0,
u(0) = u-
and p o s i t i v i t y of
D
and
i s a m l t i p l e of
(u+-u-)
D,V
Recall t h a t by d e f i n i t i o n , de t er m i n es t h e v e l o c i t y C. With
+ Mio
c
.
e n s u r e t h e e x i s t e n c e of a
A t t h i s p o i n t I would l i k e t o do two t h i n g s : go back and j u s t i f y t h e claim t h a t u- can be determined beforehand, and s e c o n d ly go t h r o u g h t h e above c o n s t r u c t i o n of J, f o r Ar r h en i u s te m p e r a t u r e dependence.
a l l w e need i s The f i r s t is eas y . From (23) i t i s c l e a r t h a t t o d et er m i n e u-, b. But t h i s can be determined from t h e c o n d i t i o n t h a t one of t h e two c o n c e n t r a t i o n s cA = u2 o r cB u3 is z e r o . Suppose t h a t u2+ < u3+. 'Ihen i t is
u2
-
-,
t h a t v an i s h e s a t
so t h e second component o f (23) s a y s
b =
and we are done.
-u2-/K2,
For Ar r h en i u s k i n e t i c s , t h e r e a c t i o n f u n c t i o n can be t a k e n a s A
~ ( E ) c ~ c ~ ~ x ~ [,- ~ / E T ]
6 where
T
is a b s o l u t e t e n p e r a t u r e , and
a
may depend on
L
t o acco u n t f o r i t s
A
p o s s i b l y b ei n g l a r g e o r small. and d e f i n e a s c a l e d t e m p e r a tu r e
Let
be t h e t em p er at u r e of t h e incoming g a s ,
T+ . . a
TI
T-- T+ A
--
ul.
T+
0
The e x p l i c i t dependence of from (24) by s e t t i n g
CA =
u2,
1
on
-
CB
u1 u2 CA ,=, E E
u3,
1
and
L;
ET+
-CB E
and t h e exponent e q u a l t o 1
(Ul/E)
-I--A
E
*
T+
1
+E(~~/E)
can be found
PROPAGATING FRONTS IN REACTIVE MEDIA
275
+
The s e p a r a t e d form (12) i s found by r e p l a c i n g ui by u E V ~ and keeping only t h e l e a d i n g o r d e r term as E + 0. Doing t h i s w i t h tf;, knowledge v2- = 0 yields
,.
- 2 $ 4 ~ ) = v2exp[vlT+/(T-) I and
Since v1 < 0 f o r u s t r i c t l y between (14) and (15) follow immediately.
u+
and
u-,
our e s s e n t i a l assunptions
There i s a n a l t e r n a t e form f o r (13) which, being c o n c i s e , may be convenient f o r some purposes. Assume D i s d i a g o n a l and c o n s t a n t . For small E , t h e q u a n t i t y 1 u-u- $ (7) i s important only f o r u n e a r u-, where w e have s e e n t h a t v = aV,
so t h a t
-
u-u-
aV =-
u 1- u 1-
"1 '
With t h i s s u b s t i t u t i o n made i n t h e argument of of t h a t e q u a t i o n becomes
JI
The l a s t term can i n some s e n s e be approximated by
i n (131, t h e f i r s t component
B6(T-T-),
where
and we have -cT
DITzz
+
B6(T-T-).
-
'Ihis form may be immediately g e n e r a l i z e d t o h i g h e r dimensional problems f o r which T-, i s n o t knawn. I f t h e flame s h e e t h a s c u r v a t u r e t h e flame f r o n t , where T >> E . then t h e meaning of t h e G-function i s t h a t t h e square of t h e normal T a t t h e flame has a jump d i s c o n t i n u i t y t h e r e e q u a l t o 2B/D1. d e r i v a t i v e of An e x p r e s s i o n u s i n g a G-function i n t h e space v a r i a b l e s was used by Margolis and Matkowsky (1981).
3.
Weak shocks inducing a r e a c t i o n
The f r o n t s c o n s t r u c t e d i n Secs. 1 and 2 were such t h a t t h e i n t e r a c t i o n between t h e physics and t h e chemistry w a s of a very simple n a t u r e : i n t h e f i r s t case, t h e d o d n a n t f e a t u r e was a p h y s i c a l shock i n which no chemical r e a c t i o n s occur, and i n t h e second, t h e physics was u n a f f e c t e d by t h e chemistry. There a r e important s i t u a t i o n s , however, when t h e i n t e r a c t i o n is e s s e n t i a l t o t h e problem, and t h i s s e c t i o n c o n t a i n s t h e asymptotic a n a l y s i s of such a s i t u a t i o n . Models of t h i s The p r e s e n t treatment i s s o r t were s t u d i e d by F i c k e t t (1979) and Majda (1980). very roughly p a t t e r n e d a f t e r t h e asymptotic development i n Rosales and Majda (1980), which c o n t a i n s many o t h e r r e s u l t s as w e l l . The t h r e e main assumptions have t o do w i t h t h e f u n c t i o n s D, f , and g. ( 1 ) The t r a n s p o r t i s weak ( r e p l a c e D by ED) and decoupled: D - ( D
0).
(26)
O D (2) With energy a p h y s i c a l v a r i a b l e , r e p l a c i n g temperature (which was a chemical v a r i a b l e i n t h e preceding s e c t i o n ) , t h e e f f e c t of t h e chemical v a r i a b l e s
P.C. FIFE
276
on t h e p h y s i c s is weak, in t h e s e n s e t h a t ,.
f
-
2f(U,E u ) . A , .
(3) The a c t i v a t i o n energy i s high:
f o r some
uo,
with
g
~
and i t s d e r i v a t i v e s O(1) as
o + 0.
(u*
The s o l u t i o n s of (2) I s h a l l c o n s t r u c t have t h e appearance o f weak shocks is Go) which i n t e r a c t w i t h t h e chemistry by m a n s of g ' s s t r o n g approximately dependence on The formal expansion is as f o l l o w s , up t o t e r m s of o r d e r E.
c.
A
,
.
+ EV;
u = u
-
u0
c o n s t = u+; v
-
+ E V ~ ; vo
v
0;
f
(27)
,.
The f u n c t i o n
f
can be expanded t o second o r d e r i n
f(U,E 2-u) = f(uo,O)
A , .
+E
fiv
E
as f o l l o w s :
+ E 2fAl uN + T1 E 2 -f i l v v ,
(29)
2 t h e arguments of from t h e symbol
f;
e t c . always b e i n g
S u b s t i t u t i n g (26-29)
Below, I s h a l l omit t h e c a r e t s
i n t o t h e p h y s i c a l p a r t of (2) y i e l d s ( w i t h
I
= d/dz)
+$
+ flvt + ~
-cvl
(uo,O).
f.
[ f ~ u ' ( f l l w ) ' ] = E(Dv')'.
(30)
The terms of O ( 1 ) a r e C'
v
0 0
' + flVo'
= 0.
'.
T h i s is a homogeneous system of a l g e b r a i c e q u a t i o n s f o r t h e v a r i a b l e s v Since a n o n t r i v i a l s o l u t i o n is d e s i r e d , i t is n e c e s s a r y t o choose co t o b? a n eigenvalue of t h e m a t r i x fl(uo,O). I n t h e s t a n d a r d case, t h i s is t h e sound epee d. It a l s o f o l l o w s t h a t yo1 Therefore t h e same i s t r u e of
is a m l t i p l e of t h e a s s o c i a t e d e i g e n v e c t o r vo
Y.
itself: vo(z) = u(z) Y .
O(E) in (30) are
The terms of 'C
v ' 0 1
+ flVll
-
-
c v
l o
' - f 2 U" '
- 1
claY
- f 2 U - Aou'
- -*1 (fllvovo) +
1
-
1
+ (Dvo')
(U'ODY)',
where A = fll\YY. For a s o l u t i o n v1 t o e x i s t , i t is n e c e s s a r y t h a t t h e r i g h t hand s i d e be o r t h o g o n a l t o t h e n u l l v e c t o r Y* of t h e a d j o i n t o p e r a t o r fl* coI. We can assume t h a t t h e r e is only one such n u l l v e c t o r , and may normalize i t so t h a t YW 1; t h e n t h e c o n d i t i o n i s
-
-
clu' where
B =
m a t r i x of
A*",
-D = -D ' Y a y * ,
- V * -U ' - BUU'
and
V
-
* Y f2,
+
(5~')' 0
,the factor
,
( 3 1) f2
being the Jacobian
w i t h r e s p e c t t o i t s second argument.
f
.
I n t h i s way, variable u
t h e s e t of p h y s i c a l v a r i a b l e s has been reduced Let u s p a s s on t o t h e chemical p o r t i o n of (2):
t o a single
277
PROPAGATING FRONTS IN REACTIVE MEDIA
-,.- - c(Du')' -..
.
+ f(u,u)'
-cu' Terms of O ( 1 ) :
-
-c u'
+
-
A
f(UO'U)' = g(aY,u).
..-
,. In t h e ( t y p i c a l ) case t h a t
-
+ g(v,u).
f = F(u)u, u'
-
t h i s becomes
h(u,u)
where
The problem has been reduced t o a s i n g l e wave f r o n t e q u a t i o n ( 3 1 ) f o r a p h y s i c a l variable a, coupled w i t h a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n ( 3 2 ) f o r t h e chemical v a r i a b l e s 'ii,
-
From (26), the where g(O,u+) = 0.
boundary conditions The Rankine-Hugoniot
provide necessary c o n d i t i o n s on
c1
and
g(o-Y,u-)
- -t i )
at +oo are u(-) = 0, u(-) u r e l a t i o n s ( 4 ) and t h e r e l a t i o n s f o r a solution t o exist:
a-,u-
-
0.
I f t h e l a t t e r e q u a t i o n could be s o l v e d f o r u- as a f u n c t i o n of u-, then t h e f i r s t equation might provide a one-parameter family of p o s s i b l e end s t a t e s 0-, c1 being t h e parameter i n some range of values. The e x i s t e n c e of a s o l u t i o n of ( 3 1 ) , ( 3 2 ) w i t h t h e s e boundary c o n d i t i o n s i s a h a r d e r q u e s t i o n ; i f i s one-dimensional, t h i s problem has e s s e n t i a l l y completely been solved by Majda (1981). Computations have a l s o been performed (losales-Majda 1980). In summary, t h e s e f r o n t s c o n s i s t of weak shocks coupled w i t h h i g h l y a c t i v a t e d chemical r e a c t i o n s , w i t h weak t r a n s p o r t . The v e l o c i t y comes o u t t o be a p e r t u r b a t i o n of sound speed. The mathematical problem reduces t o a boundary v a l u e problem f o r a s c a l a r f r o n t e q u a t i o n coupled w i t h a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n f o r t h e chemical v a r i a b l e s .
4. Sharp Front Asymptotics f o r Chemical and B i o l o g i c a l Problems The u s e of (1) i n connection w i t h biology and chemical r e a c t o r t h e o r y i s u s a l l y c h a r a c t e r i z e d by t h e l a c k of a convection term f , and a l l v a r i a b l e s being chemical. I n Sec. 2 of t h e t a l k , I spoke of a c l a s s of problems w i t h no convection f o r which asymptotic a n a l y s i s produces a f r o n t w i t h a s h a r p c o r n e r a t one point. There i s a n o t h e r wide c l a s s of models, i n v o l v i n g r e a c t i o n f u n c t i o n s g of "sigmoidal" type, whose s o l u t i o n s a r e c h a r a c t e r i z e d by t h e appearance of s h a r p t r a n s i t i o n s r a t h e r t h a n s h a r p corners. These s o l u t i o n s may be genuine wave f r o n t s i n t h e s e n s e used h e r e , o r dynamical s t r u c t u r e s which l o c a l l y appear t o be conposed of s h a r p f r o n t s , but do not conform g l o b a l l y t o t h e s t r i c t d e f i n i t i o n . 'Ihese sigmoidal models have been used i n many c o n t e x t s . A very few of t h e r e f e r e n c e s a r e i n connection with: impulse propagation a l o n g nerve axons (Cesten e t a1 (1975)); o t h e r waves i n e x c i t a b l e t i s s u e ( O s t r o v s k i i and Yakhno (1979) 1;
(1975),
Keener
P.C. FIFE
270
chemical waves and p a t t e r n s (Tyson and F i f e (1980)); geographic d i s t r i b u t i o n of populations (Miaura and Murray (19781, Mirmra and Nishiura (1978); p a t t e r n s i n developmental biology (Mimra and Nishiura (1978)); more a b s t r a c t s e l f -organization models ( F i f e (1976) ). Here I w i l l go through one example t o i l l u s t r a t e t h e c o n s t r u c t i o n of a wave f r o n t composed of a s h a r p t r a n s i t i o n s e p a r a t e d by two regions of The example c o n s i s t s of two equations, with t h e components variation. g2(u) vanishing on t h e curves i n d i c a t e d , and p o s i t i v e and negative regions indicated:
i s assumed t o be l a r g e (making
The component replace i t by t h a t u1 t h e form
g1 1
F
gl, E
0. Sfnce u ' I 0 t h e r e , (33b) would be v i o l a t e d . So t h i s attempt i s doom3 t o f a i l u r e , an$ i t i s necessary t o f i n d a way from u- t o u+ without following t h e sigmoidal curve a l l t h e way. W e can follow i t most of t h e way, however, i f a h o r i z o n t a l jump from one branch of i t t o another i s allowed a t some (as y e t unknown) value u2 = w a s i n d i c a t e d i n t h e f i g u r e . The d o t t e d l i n e i n t h a t f i g u r e , then, i s t h e proper " t r a j e c t o r y " . Such a jump n e c e s s i t a t e s a l o c a l r e s c a l i n g , so t h a t (33) w i l l s t i l l be satisfied. 'he f r o n t p r o f i l e experiences a sudden t r a n s i t i o n i n t h e u u2. As i n t h e examples i n Secs. 1 and 2, I s h a l l component, but not t h e c o n s t r u c t t h e f r o n t i n t h r e e parts: (a)
For
z
O
,
u(0) = uo
i s given. For the o n l y sake o f s i m p l i f y i n g the presentation i n the
sequkl we s h a l l assume t h a t
Pf = 0
, i.e.
t h a t the forces are p o t e n t i a l .
The f o l l o w i n g are w e l l known c l a s s i c a l r e s u l t s (see, f o r instance, [5], o r [ll]) :
(')
The problem (1.6) has a (weak) s o l u t i o n
u( .)
A c t u a l l y we s h a l l use o n l y the f a c t t h a t where c2 i s s u i t a b l y chosen, OO ,
and consequently i f (3.5) then
Let
to = t o ( r )
and choose a subset
M(m) o f
S(to)Cuo=V:l~uol~c r l maximal under the property
such t h a t
luol
2
< Eo
. Then
FINITE PARAMETER APPROXIMATIVE STRUCTURE
where
m
OF ACTUAL FLOWS
323
satisfies
(3.8) Then, by v i r t u e o f the squeezing property (see 5.2), we have t h a t t h e distance ( i n H) o f u(s+to) = S(to)u(s) t o M(m) i s bounded by
whenever
Ilu(s)ll 4 r
so t h a t i f to < 1
. But f o r
- 9 , then
~ { S C (t,t+l) :
any
, we
tat,
have obviously
by (3.6
t30 :
for all
\Iu(s)II 4 r } ] 3 1 - T / Z
-
to
.
I f we assume now t h a t
then i t f o l l o w s t h a t the s e t o f i s Q E:/' M(m)
has measure 3 1
-
T
t's
such t h a t the distance o f
on any i n t e r v a l
i s the graph o f the f u n c t i o n $,
: Pm M(m)
M(m) By (3.7),
u(t)
.
to
(t,ttl) c ( O p ) -(I-P,)H defined by
which moreover s a t i s f i e s the c o n d i t i o n (3.11)
lJI0(Pmu)
- $o(pmv)l
O i s f i x e d ,and the boundary problem (1.2) i s replaced by t h e p e r i o d i c c o n d i t i o n s (4.1)
u(O,..)
... , u(..,O)
= U(L,..),
= u(..,L)
.
S i n c e the Navier-Stokes equations are Galllean i n v a r i a n t , we can assume, by a change o f the reference frame, t h a t t h e s o l u t i o n s v e r i f y a l s o
.
I Q u dx = 0 Therefore
H and V become t h e closure i n L2 (9)n and H 1
o f the space o f a l l
IRn-valued trigonometric polynomials w(x)
;1
v.w
dx = 0 and
E
0
, respectively, such t h a t
.
A l l the other d e f i n i t i o n s and p r o p e r t i e s remain e s s e n t i a l l y unchanged. A c t u a l l y i n t h i s case much more concrete a l g e b r a i c s t r u c t u r e i s present. For instance the X,
are now o f the form 4a2 L-21k[
with
k c Zn
, k#O
and wAu
+
B(u,u)
has the f o l l o w i n g property. For any rn = 1, 2,
_.-
... t h e r e e x i s t s an i s o m e t r i c operator U
: H e(I-P,)H
such t h a t --
where X -
i s some adequate constant such t h a t
0
A,
, then if
and s e t Uw. wJ J J (Here some supplementary care i s necessary i n order t o avoid the complication due t o the f a c t t h a t the m u l t i p l i c i t y o f h o r XJ i s not simple. Since we i n t e n d J t o study t h i s s e l f s i m i l a r i t y phenomena elsewhere, we d o n ' t i n s i s t w i t h the d e t a i l s o f t h e proof.) F i n a l l y , f o r u0= V the s o l u t i o n s o f (1.6) s a t i s f y , i n case n=2 , besides the r e l a t i o n s (1.8), t h e r e l a t i o n s
FIN1TE PARAMETER APPROXIMA TI VE STRUCTURE OF ACTUAL FLOWS
--
o f (1.6). rf E1/Eo and v1/3 and El/Eo < 1/36 (4.6)
Proof _----
mv
(for instance
then
= dimension o f I$
--f
m
for
+
v
.
0
:
Mv
Let
=
M
where
J,,
= J,
J,V
I f u,v
in
are s u f f i c i e n t l y small
T
325
E
M
and t h e i r distance t o
H
enjoys the p r o p e r t i e s given i n Theorem 1.
$,
i s Q El
, then
there e x i s t
u*, vx
with
J,V
so t h a t
( u - v l & 6E:"
(4.7) Therefore i f satisfying v( . ) )
u(.)
t 21Pm ( u - v ) ) V
and v ( . )
luo19 l v o l \< Eo
.
are s o l u t i o n s o f (1.6) w i t h
i t f o l l o w s from (4.7) and (3.4)
u(0) = uo v(0) = vo ( f o r u ( . ) and
that I{tE ( 0 , l ) : l u ( t ) - v ( t ) I
.
0
ij are considered as simple functions, which assume that the memory of turbulent structures has faded enough. It is a crude assumption which can only be supported by its consequences. from the mean flow, for example,
Besides, if we admit that the dissipative structures are not directly strained by the mean field (the amount energy flux is transformed without alteration through the inertial range, but during these exchanges it can be conjectured that turbulence structures lose most of their orientation), the anisotropy is considered as globally taken into account through b (disregarding the mean ij’ velocity gradient which generates such an anisotropy). We can then write:
I = I [bij,
2 q
, -E] .
(1.16)
With a suitable choice of arguments, we can admit I to be an isotropic
iz and E . Pipkin and Rivlin have demonstrated that such a ij’ function can only depend on the invariants (with respect to the orthogonal
function of b
group) which are determined from b.. , IJ
sz and z.
For a small degree of anisotropy the previous function can be expanded with respect to the anisotropic parameter:
I = f[E,
q2]
+
g
[Z,
q2] 11 t
...
.
(1.17)
More precisely we find that:
(1.18)
SOME APPROACHES TO THE TURBULENCE PROBLEM
where I1 is the second invariant of b.. 1J
experimental data.
'
343
the constants must be adjusted from
The second function Pij can be determined in a similar but more complex way, the pressure term having linear and non-linear relations to the fluctuating field.
(1.19)
These methods have extensively been developed during the last decade in several forms by Launder, Jeandel,
....
Computing Methods for Complex Flows
1.2.2.
We will take advantage of
this section devoted to modelling methods in
physical space to indicate some progress in computing approaches for complex turbulent flows. The patterns of bounded flows under consideration are strongly non homogeneous in more or less extended areas; the rate equation for the mean flow has to be coupled with a more or less complex turbulence model. Where a first gradient approach is used with VT as a given function of space and time variables, we must deal with both the momentum equation for the mean field and the continuity equation. The plane channel flow simplifications result in the dimensionaless basic equation: (1.20)
with the following given characteristic scales (2D channel width), 1 dP u =[---D]) 0
4
PdX
(1.21)
and the initial and boundary conditions (1.22)
J. MATHIEU, D. JEANDEL, C.M. BRAUNER
344
i
(0,
t) =
i
(1, t) = 0
(1.23)
Ro stands for uo 2D/v. Two kinds of viscosity models are retained leading to available predictions, namely (1.24)
(model 1 )
in which v
T
is dimensionless parameter and
vT (y, t) = R: ky &(o,
t) [ l
- exp - Y Ro]
;y
E
[O;
(1.25)
0, 1)
aY
vT (y, t) = vT (0, 1; t) VT
(yt t) =
VT (1
- Y; t)
y
i
i
y
(1.26)
[ O , 1; 0, 51
E
E [ot
5 1 11
-
(1.27)
(model 2)
Where second order closures are concerned such as previously proposed by
...
Donaldson, Launder, Lumley, Jeandel treated.
a partial differential system must be
The momentum equations and the continuity equation are supplemented
by four equations related to the components of the Reynolds stress tensor. The dissipation rate must also be considered as an unknown function. Where a channel flow is concerned, the problem may be formulated in a rather simple manner. Using dimensionless variables for the Reynolds stresses and dissipation a model of this kind may be written in the general form:
(1.28)
U is the vector the components of which are:
-2 U
2 -
uO
; U g = - u2 2
uO
(1.29)
We must notice that inhomogeneities greatly complicate the modeling of the velocity pressure correlations. In the model of Launder," a sophisticated
345
SOME APPROACHES TO THE TURBULENCE PROBLEM
expression inversely proportional to the incorporated in the non-linear operator
distance from the wall is [;I. Consequently, boundary
conditions for the mean and turbulent fields must be given in the well known semilogarithmic region. We also notice that the Reynolds stress equations contain gradient terms in the form:
(1.30)
that reinforces the elliptic character of the equations governing the turbulent quantities. A
numerical solution may be
searched by means
of an optimal control
When non-linear operators are concerned, they can be split into two parts: a linear operator A and a non-linear one B; and formulation of the problem. l2
we write: A [U]
=
B[U]
.
(1.31)
The associated problem can be written: A
[UAl =
B [A]
.
(1.32)
In that case, A may be considered as a control parameter, and the last state equation is solved minimizing a cost function of the form (1.33)
1.2.3
Provisional Comments
Even though these methods are carried out in physical space, references are often made to the size of the turbulent structures which are responsible for a phenomenum; in fact, an equilibrium state is always supposed which reinforces the role played by the energy flux which travels without alteration across the inertial range.
If a deterministic phenomenon is superimposed on the random field (with vortex shedding, the spectrum exhibits a peak at a rather low wave number) the amount of energy flux can be significantly altered so that many of previous assumptions are spurious. Launder, Cousteix take into account this disturbance by introducing an additional parameter (as is made in the
346
J. MATHIEU, D. JEANDEL. C.M. BRAUNER
thermodynamics of i r r e v e r s i b l e phenomena i n o r d e r t o account f o r a d i s e q u i l i -
I t i s a l s o p o s s i b l e t o d i r e c t l y work i n s p e c t r a l space.
brium). 1.3. Random
S p e c t r a l Methods motion
is
analysed are
accounting
referenced
for
both
by
the
structures
which
Non-linear
e f f e c t s d i s p l a y a n energy t r a n s f e r ;
r o l e played
their
sizes
and
by
turbulent
orientations.
t h i s energy f l u x g e n e r a l l y
t r a v e l s from t h e l a r g e s t t o t h e s m a l l e s t s t r u c t u r e s , b u t a r e v e r s e cascade process has been brought t o l i g h t very c l e a r l y (Gence13). Fourier transform of t h e two components u. and u
.i
S t a r t i n g from t h e
of t h e f l u c t u a t i n g v e l o c i t y
+
i t i s p o s s i b l e t o w r i t e t h e equation of t i e c o r r e l a t i o n a t two p o i n t s k and
of
spectral
space (which corresponds t o t h r e e p o i n t s
I n t e g r a t i n g over t h e wave number spectral
equation a t
a
1
it i s p o s s i b l e t o
unique p a i n t
+
2
i n physical space).
find the classical
k of s p e c t r a l space.
The g e n e r a l
s t r u c t u r e of t h i s equation can be represented by
For t h i s open problem it i s a l s o p o s s i b l e t o write a n equation f o r t h e t r i p l e c o r r e l a t i o n s a t t h r e e p o i n t s i n s p e c t r a l space
+
... + ... = Sib
+ +
[PI
q1
k,
t]
.
(1.36)
SOME APPROACHES TO THE TURBULENCE PROBLEM
347
To close this open set of equations Millionshikov (1941) and Proudman and Reid (1954)14 assumed that the probability law
-+
-+
-+
p [Ui' UQ' Url
$1
(1.37)
+ + which is parametrized by k, p,
l, 3 and
t is very similar to a gaussian law
so
that they write:
(1.38)
This assumption leads to: %
%
2
,
= o i l l r
.
(1.39)
Disregarding some contradictions 1 ese authors admit that a closure method can be given in the symbolic form:
Thereby, the role played by the cumulant terms if neglected.
In the previous
equation no limitation is imposed on the evolution of the triple correlations which increase too much.
These terms being responsible for the energy
transfer across the spectrum, the amount of energy located in some regions becomes negative which is irrelevant.
With the quasi normal hypothesis the
solution of the equation for the triple correlations
(1.41)
is given by
(1.42)
348
J. MATHIEU, D. JEANDEL, C.M. BRAUNER
where Siam must be considered as a known quantity. It is obvious that the memory of the turbulence is overestimated, the exponential term being only controlled by the kinematic viscosity.
In order to improve the method,
Orszag15 supposed that the role played by the cumulant terms can be introduced through a linear relaxation which has the symbolic form
This artificial viscosity contributes to reduce the memory time of turbulence. Moreover the integration is drastically simplified by supposing that the triple correlations can be taken at time t (in place of t').
From this
Markovian approximation the integration with respect to t is only carried out over the exponential term. The artificial viscosity pT plays a role in spectral space which could be compared to the turbulent viscosity introduced by Boussinesq in physical space. Starting from experimental data, adjustments can be made, but it is also possible to evaluate the characteristic time by considering, in the manner of Kraichman, a supplementary problem which is the advection of a passive scalar by a random field with coupled variables (Test Field Model TFM). The eddy damped quasi normal markovian model (E.D.Q.N.M.) has been extended to homogeneous fields with constant mean velocity gradients by Cambon. l6 spectral equation has previously been given by Mitchner and Craya:
The
( 1 .MI
with
;+;=g.
(1.45)
The action of the mean velocity gradient is explicitly represented by I 2ji whereas Raji accounts for non-linear effects. We suppose Reji to be evaluated in the form: (1.46)
We find that
(1.47)
SOME APPROACHES TO THE TURBULENCE PROBLEM
349
Besides it is written: 2
v(k + p
The
2
2
+ q 1
aQji
incorporation of
the
typically non-isotropic term Yaji is a
diagonalized form has extensively been discussed. Two facts can support this assumption. At first, the variation of @ with time should be specially Qji controlled by Secondly we are only concerned with a part of the triple correlations for the behavior of the equations which determine the evolution
Cljyi.
of the double correlations.
In order to simplify computing methods, the evolution of the turbulent structures is only considered as a function of their sizes, not of their orientation. so
-+
Consequently an integration is made over a sphere of radius lkl
that suitable functions are introduced such as:
+ An equation in terms of functions of modulus of k is obtained.
An angular
parametrization of the second-order spectral tensor is introduced in order to integrate analytically all the directional terms over a spherical shell. Modelling methods similar to those used in physical space are introduced; for example, the spectral deviatoric tensor €Iij plays a similar role to b ij' Finally, the terms of the closed equation for $ij exhibit only a tensorial anisotropy. The specific anisotropy related to directional effects disappears by integrating over the sphere.
At this stage, two scalar parameters a(k, t) and q(k, t) are to be adjusted. Exact linear solutions can successfully be used to determine a(k, t).
Complex evolution of spectral curves can be
predicted, for instance a peak can be introduced in a local range of wave number and direct and reversed cascades of energy can be brought to light Fig. 1 and 2 ) . Many extensions of this method have been made to turbulent fields including buoyancy effects, inertial forces and so on.
350
J. MATHIEU. D. JEANDEL, C.M. BRAUNER
1.4.
Pseudo Deterministic Approach (large eddy simulation)
Starting from the Navier-Stokes equations, direct simulations of isotropic turbulent fields have been made.
So
far, this simulation is not adequately
supported by mathematical considerations.
Bifurcation points, if they exist,
are detected and treated through an algorithmic scheme which should introduce its own mark so that it should be difficult to interpret the final results which include both the properties of the initial equations and those of the numerical scheme. We can suppose that a large eddy simulation is more typically convenient for turbulent fields subjected to overall effects capable of influencing more or less extended groups of turbulent structures. Except for isotropic turbulence which is only subjected to its own mechanism, in most of the flow patterns, selected turbulent structures are subjected to straining mechanisms or driving forces such as mean velocity gradients, buoyancy effects, etc. Therefore, it is possible to
conjecture
the
existence of coherent structures whose
development could be more suitably tackled in a deterministic way.
These
coherent structures, the sizes of which are comparable with those of the overall
flow, develop
in the presence of smaller structures which are
accounted for in the statistical coefficients.
For example, the pairing
process which is responsible to a large extent for the growth of turbulent areas can be considered as a deterministic mechanism whose developments are somewhat hampered by the fine grained turbulence which plays a role similar t o a viscous effect. So
far, the definition of such coherent structures is not precisely supported In these approaches, Deadorff17 and Leonard 18
by mathematical theories.
introduce a typical splitting of the overall field.
The "filter function"
used by Leonard can be linked to the spectral approach more directly. Therefore we use it preferentially. -*
Leonard separates the velocity field U.(x, t) into a grid scale component -
-9
*
li(x, t) and a subgrid component u i ( x , t)
U. = 1
ii.1 +
u!
1 '
'
0
[;/;I]
l
&'
=
1
.
(1.50)
The overbar precisely denotes the filtering operation which is not to be confused with either the mean value of the velocity, which has a statistical
J. MATHIEU, D. JEANDEL, C.M. BRAUNER
351
meaning, or with any statistical averaging, indicated by angular brackets. A filtered quantity has to be considered as a typical event which can occur under any circumstance.
This event, when it happens, develops in an
environment the properties of which have to be considered in connection with the existence of this principal event.
This should lead to conditional
samplings supplemented by averaging operations. If the physical properties of the fluid are constant, the result of filtering the Navier-Stokes equation is:
(1.51)
This equation is supplemented by the continuity equation:
(1.52)
For the advection term it is found that:
- u . u = U . U . + u!u. + u . u ! + u'u' i j 1 1 1 1 1 1 i j
(1.53)
'
By using a series expansion it is found that:
--
a
a u- .u- $1
(1.54)
The Leonard term can be written
-L i j = uiuj
-
-UiUj
.
(1.55)
The product of filtered quantities is considered without additional filtering. The "cross term"
cij =
u.u! 1 1
+
upj
SOME APPROACHES TO THE TURBULENCE PROBLEM
352
represents interactions between the residual field and the filtered field. "A priori" this sort of interaction should not be located in spectral space.
On the other hand, the subgrid Reynolds tensor
emphasizes the role of small structures. However, i
can
?.
prove1
t
at such
interactions can occasionally generate structures far outside the subgrid scale from a triadic analysis. We set
Tij = R i j
t Cij
and introduce the deviatoric tensor:
r i j = Tij
- 21 Tkk b i j .
(1.56)
On the other hand we write:
Finally we have:
aii
at t
a -uiuj - = j
-
- 1P * - &j [ L i j ax
t
r..] =I
.
(1.58)
Subsequent workers have followed Smagorinsky19 and write
aui
%Ii
Tlj
=
- V N 'ax
+
%I
j
for the "subgrid term". Such an assumption gives rise to several remarks:
(1.59)
SOME APPROACHES TO THE TURBULENCE PROBLEM
363
No tractable information has first been extracted for the subgrid term from the basic equations. With such a splitting, the subgrid terms are not objective terms. Therefore, the application of representative theorems to these quantities becomes questionable.* For a long time the energy cascade was considered as a one way street, the small structures being fed by the larger ones. Moreover it was conjectured that
interactions
were
only
approximately the same sizes.
possible
between
turbulent
eddies
having
This opinion was more or less supported by
experimental data extracted from isotropic turbulent fields.
In fact, the
energy cascade is a two way street and interaction occurs between structures of very different sizes. Applications of successive plane strains to grid generated turbulence has been illuminating (Gence 13-20).
At first the isotropic turbulence is subjected to
a straining process and turbulent structures are oriented by the mean field.
A second straining, the axes of which are different from the first ones by an angle
LY
is suddenly applied so that the turbulence is reoriented by this
second straining.
Gence13 has shown that the term describing the exchange of
energy can be written, after the change in position of the principal axes of the strain, as
-D 2 [b22
Q
- b33][2 cos2 - I ] .
(1.60)
LY
Either positive or negative values can be obtained depending on the angle a , D stands for aii/ax.. 3
The assumption of Smagorinsky supposes the principal axes of the subgrid turbulence to be the same as those of the filtered quantities.
This
hypothesis is not supported by Gence's experiments and by direct simulations carried out by Ferziger21 who states that "the two sets of axes appear to be essentially random with respect to each other."
The mean relative position of
the two systems is of course a most important parameter. Occasionally such a coincidence could occur.
* At
first, representation theorems require the introduction of additional argument in connection with, for instance, the rotation of the filtered field.
354
J. MATHIEU, D. JEANDEL, C.M. BRAUNER
Comparisons have been made between the computational results coming from an extended
E.D.Q.N.M.
method
(Cambon16)
and
from
large eddy simulations
including Smagorinsky’s assumption (Courty, Bertoglio, private communication), (Fig. 3 ) . Great discrepancies are shown in Fig. 3 , probably originating from the subgrid With the large eddy simulation, the evolution of the correlation term
model.
seems to be rather slow, a fact which should be in close connection with the evolution of the transverse velocity component.
1.5 Probabilistic Approach far, the probabilistic
So
properties of
the
flow have been
treated
disregarding the precise behaviour of the Navier-Stokes equation which controls any motion of the complex turbulent medium.
The motion is split into
two parts, the properties of the fluctuating part are supposed not to be far from a normal law.
At any point in the theory, it is necessary to clarify
whether bifurcation points exist or do not.
In fact, all the pervious
treatments are supported by very pragmatic considerations not typically related to the nature of the basic equation.
However, large eddy simulations
start from the Wavier-Stokes equation for a selected group of structures, but the deterministic computation is carried out up to an arbitrary solution no averaging process is introduced in order to give a statistical value to this approach. Besides the evaluation of the subgrid terms is very rough. Simple experience confirms that some statistical properties of turbulent fields can behave in a very typical way; experimental data show that the departure from a normal law can be very significant, so that is is not possible to account for the cumulant terms through a linear relaxation as is done in the case of the E.D.Q.N.H. assumption. Turbulent fields with chemical reactions display such typical behaviour.
If two non-reactive chemical
species are mixed together, it is found that the statistical properties of the concentration fluctuations more or less relax to a gaussian curve, but if the two species exhibit very strong reactive properties A and B cannot coexist at the same point, so
that the probability density function for the two
fluctuating concentrations should be represented by two planes. Even though asymptotic situations such as flames could be easier to tackle, it would be interesting to follow the evolution of the probability density function by controlling the reaction rate.
Experimental investigations carried out by Bennanni” are in progress to seize intermediate situations. The Damkholer Number compares two characteristic scales: the reaction time, which is directly connected with chemical properties, and a characteristic time linked
SOME APPROACHES TO THE TURBULENCE PROBLEM
to dynamical properties of the turbulent field such as
355
At first, it
seems that the Damkholer Number does not act on the probability density function in a continuous way
so
that it should be possible to encounter a
probability distribution not far from a Gaussian curve for large ranges of small chemical reaction rate. It is probably interesting to seize the topology of the reactive domain.
For flames where the rate of reaction is
high, reactive zones are complex surfaces; where diluted media are concerned, reactive zones can become more or less extended volumes. For the treatment of chemical reactions the trend should be to consider a probability density equation, a means which has recently been used. In the cases under consideration, the theory of moments having failed, scientists have introduced a fine probability function. The solutions have been developed in very simple cases.
For our purpose, we must underline the
role played by the Hopf equation which can be considered as the starting point for this theory as it has been shown by Gence (private communication).
The
Hopf equation can be considered as a probabilistic form of the Navier-Stokes equation. It is established in phase space and supposes the uniqueness of the solution when convenient initial conditions are given.
This restriction is
introduced through a punctual correspondence between the sets of points which represent the state of the system at two selected times.
In the original
paper by Hopf, this difficulty is perfectly brought to light. For the sake of simplicity, we consider an isotropic turbulence which develops in a reactive medium, and we indicate the equation governing the probability density function: (1.61)
1 1 1 that f [r,, Y1 t] d TA dTB represents the probability of having the 9’ + t) and cB(xl, t) respectively located in a range concentration c ( x 1 1 11’ 1 TA, rA + dTA1 andATB, r B + drB1 at a given time t. Finally we obtain: so
J. MATHIEU, 0. JEANDEL, C.M. BRAUNER
356
where ( I .63)
is the probability density of finding given values of concentration at two
+
separate points r.
In this form, the closure problem is clearly brought to
light.
At the present time, this approach is an unweeded path. Experimental data are unavailable and information is required to suggest suitable closures.
11. MATHEMATICAL STUDY OF A TURBULENT VISCOSITY MODEL
11.1. Introduction In this part, we shall present a mathematical study of a nonlinear parabolic equation including one of the above turbulent viscosity models. We wish to focus our attention on the more original one, namely model 2 (see Section 1.2.2). For a mathematical study of model 1, see Laink.23'24 In this paper, the proofs will be only outlined. For further details, we refer the 25 reader to Lain&,23 Brauner and Lain&. Our mathematical notations will be as follows: Q is the open interval ] 0 , 1 [ , x the space variable.
Let ] O , S [
be the time interval, t or
s
the time
variable, and Q = h ] O , S [ . We shall use some Sobolev spaces (see e.g., H2(Q), H-'(Q) (3), as well as the spaces namely Hi(Q), 1 L2(0,S; HO(R)), etc. Without any further specification, we shall denote by (
,)
2
the inner product in L (Q) and 1 1
.
1 1 the associated norm.
For u regular enough, let us define the "turbulent viscosity"
(2.1)
where G
2:
C1(fi),
G(x)
2
v1 > 0
.
We consider the following mathematical problem.
I
u ( 0 , t) = u(1, t) = 0 u(x,o)
=
Uo(X),
x e Q
Find u = u(x,t)
solution of
(2.2)
SOME APPROACHES TO THE TURBULENCE PROBLEM
where
V
>>
V
1
357
and u (u) is given by (2.1).
T
11.2. Existence of a Solution The existence of a solution to problem (2.2) is obtained via a standard Galerkin method. Specifically, we establish the following theorem: Theorem 2 . 1 :
2 1 L e t f given i n L ( Q ) , uo i n HO(Q).
solution
u
in
the
space
Then problem ( 2 . 2 ) has a
H~(Q))n ~ " ' ( 0 , s ;H~(Q)).
~ ~ ( 0 , s ;
Furthermore,
Proof:
Introduce the eigenfunctions w. of J w.(x) = sin jnx, j = 1, 2, .... One has w'! + J J look for a solution um(t) = gjm(t) wj
jll
2 2 the operator -d /dx , namely 2 A w = 0 where A . = (jn) . We j j J of the nonlinear differential
system:
(2.4)
u
m
(0)
= uom = p r o j . of uo on
vm ( 4 )
System (2.4) has a solution on an interval ]O,Sm[.
. The following estimates
will in particular yield that Sm = S. Lema 2.1:
2
The sequence {urn]i s bounded i n t h e space L ( 0 , s ;
n',(Q))
n L"'(0,S; L 2 ( Q ) ) , and
Proof:
Simply multiply (2.4) by g . (t) and add for j Jm
1 to m.
It follows
& = (f(t), um (t))
The integrate over ]O,S[ and apply Young's formula to the R.H.S.
358
J. MATHIEU, 0. JEANDEL, C.M. BRAUNER
Lemma 2 . 2 :
The sequence rum] is bounded in the space L 2 ( 0 , S ; €?(ill) and
n
~"(0,s; H,1(R)),
I-
( 0 , .) If
ax
2
a2Um
Sketch of the proof: and sum.
is bounded in L2 ( Q ) .
ax
I t comes:
In (2.4), replace w . by -1/A ", then multiply by g . ( t ) J j Jm
Integrating over lO,S[ leads t o
The l a s t term i n the R.H.S. may be bounded by
2 IG'
which can b e s p l i t i n t o two parts using Young's formula.
Finally we get:
2 (0,s)
If 2 a Il 2 ax
L~ (PI
359
SOME APPROACHES TO THE TURBULENCE PROBLEM
1
a
hence the requested estimates (for the estimate in L (0,s; HO(R)), integrate over ] O , s [ , 0
0 Y t E ]O,S]. ax The idea of the proof is to extend a demonstration by Protter-Weinberger [31] to positive weak solutions.
Corollary 2.2:
See the details in [23][25].
Suppose
Then exists a constant
01
> 0 such that (2.12)
11.5
Uniqueness of the Solution
The demonstration of the uniqueness is based
on
Gronwall's lemma, with
Corollary 2.2 as basic tool. Theorem 2.4:
The hypotheses are that of Theorem 2.3. a unique positive solution.
Then problem (2.2) has
SOME APPROACHES TO THE TURBULENCEPROBLEM
and u2 be two positive solutions of (2.2). 1 Clearly, u verifies
Sketch of the proof: u = u1
-
u2.
u(0,t)
363
Let u
= u(l,t) = 0
U(X,O) = 0
Set
(2.13)
. n
Multiply ( 2 . 1 3 ) by
- a% 7: ax
(2.14)
We remark that
( a given by Corollary 2.2) and
&o,
t) I
5 JZI
I%(t)
364
J. MATHIEU. D. JEANDEL, C.M. BRAUNER
Plugging this estimate in the R.H.S. of (2.14), it comes out, after several computations based on Young's formula, a relation of the form: (2.15)
where M
E
L1(0,s). By Gronwall's lemma, 1Ig(t)II
2
0 , hence
in Q and the uniqueness.
Acknowledgements:
The authors are
indebted
to Professor M. Crandall who
brought J. Moser's results to their attention.
FOOTNOTES See C. Lain6 [23].
A weighting process is introduced with the contraction kaeaji which minimizes the role played by the large anisotropic structures. 2 Recall that ff(Q) is the subset of functions of L (a) whose partial 2 derivatives up to the order are also in L ( Q ) . H:(Q) is the closure of 1 its dual space. If the set of smooth functions D(Q) in H ( Q ) , and H-'(Q) X is a Banach space, LP (0,s; X) is the set of functions of ]O,S[ into X of power p integrable, 1 5 p < +cD (essentially bounded if p = + V m = finite dimensional space spanned by wl, ..., wm'
a).
REFERENCES Glowinski, R., Mantel, B., Periaux, J., and Pironneau, O., "H-'
least
squares method for the Navier-Stokes equations,'' 1st Int. Conf. on Numerical
Methods
in Laminar and Turbulent Flow, Eds. Taylor, C.,
Morgan, K. and Brebbia, C. A. (Pentech Press, Swansea, 1978). Bradshaw, P., Ferris, D. H., and Atwell, N. P., "Calculation of boundary development using the turbulent energy equation," J.F.M. p. 593-616.
28 (1967)
365
SOME APPROACHES TO THE TURBULENCE PROBLEM
Launder, B. E., and Spalding, D. B., Lecture in "Mathematical Models of Turbulence," (Academic Press, New-York, 1972). Jeandel, D., Brison, J. F . ,
and
Mathieu, J.,
"Modeling methods
in
physical and spectral space," Physics of Fluids, December 1977. Hopf, E., J. Ratl. Mech. Anal. (1952), p. 1-87. Monin, A. S., and Yaglom, A. M., "Statistical fluid mechanics: mechanics of turbulence," The M.I.T. Press, Vol. 2, 1967. Lundgreen, T. S., Physics of Fluids 10, (1967), p. 969. Dopazo, _., and O'Brien, E. E., Combusion Science and Technology, Vol. 19 (1975) p. 99-122.
Mathieu, J., "IdCes actuelles sur la turbulence," CANCAM 79, Sherbrooke (1979), p. 465-502. Lumley, J., "Von Karman Institute," J. 1975, Lectures series 76. Launder, B. E., Reece, G. J., and Rodi, W., "Progress in the development of a Reynolds Stress closure," J. Fluid Mech. 68, p. 537-566. Cea, J., and Geymonat, G . , "Une mithode de liniarisation l'optimisation,"
Instituto
Nazionale
di
Alta
Matematica,
via
Symposia
Mathematica, Vol. 10 (1972). Gence, J. N., and Mathieu, J., "On the application of successive plane strains to guid generated turbulence," J. Fluid Mech. (1979), Vol. 93, Part 3, p. 501-513. Proudman, I., and Reid, W. H., "On the decay of a normally distributed and homogeneous turbulent velocity field," Phil. Trans. Roy. SOC. A , (1954), p. 163-189. Orszag, S. A., "Analytical theories of turbulence," J. Fluid Mech. 41, (1970), p. 262-386.
Cambon, C.,
Jeandel, D.,
and
Mathieu, J.,
"Spectral
modeling
of
homogeneous non isotropic turbulence," Journal of Fluid Mechanics, Vol. 104 (1981), p. 247-262. Deardorff, J. W., "A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers," J. Fluid Mech. 41 (19701, p. 453. Leonard, A., "Energy cascade in large-eddy simulations of turbulent fluid flows," Adv. in Geophys. A 18 (1974), p. 237. Smagorinsky, J. S., "General circulation experiments with the primitive equations. I: the basic experiment," Mon. Weath. Rev. 91 (1963) p. 99. Gence, J. N., and Matheiu, J., "The return to isotropy of an homogeneous turbulence having been submitted to two successive plane strains,'' J.F.M., Vol. 101, Part 3 (1980), p. 555-556. Ferziger, J. R., "Higher-level simulations of turbulent flows," V.K.I. lecture series 5 (1981).
J. MATHIEU, D. JEANDEL, C.M. BRAUNER
366
(221 Bennani, A., Alcaraz, E., and Mathieu, J., "An experimental setup for
interaction turbulence-chemical reaction research," To be published in Int. J. of Heat ans Mass Transfer. [23] Lain;, C., Etude mathematique et numerique de modiles de turbulence en
conduite plane, Thise, Lyon (1980). [24] Lain;, C., C. R. Acad. Sc. Paris, Sirie B, 291 (1980), p. 63-66. [25] Brauner, C. M. and Lain;,
C.,
"Mathematical and numerical study of a
turbulent viscosity model," to appear. [26] Adams,
-.,
Sobolev spaces, Academic Press, New York (1975).
[27] Lions, J. L., "Quelques me'thodes de re'solution de prob1;mes
aux limites
non lineares," Dunod, Paris (1968). "Prob1;mes aux limites non homoge'nes et applications," Dunod, Paris (1968). [29] Moser, J., "A Harnack inequality for parabolic differential equations," Comm. Pure Appl. Math. 17 (1964), p. 101-134, and correction in 20 (1967), p. 231-236. [30] Moser, J., "On a pointwise estimate for parabolic differential equations," Comm. Pure Appl. Math. 24 (1971), p. 727-740. [31] Protter, M. H. and Weinberger, H. F., "Maximum principle in differential equations, Prentice Hall (1967).
[28] Lions, J. L. and Magenes, E.,
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, 8. Nicolaenko ( d o . ) @ North-Holland Publishing Company, 1982
367
INSTABILITY OF PIPE FLOW
Anthony T. P a t e r a and Steven A. Orszag Department o f Mathematics Massachusetts I n s t i t u t e o f Technology Cambridge, Mdss. 02139
The s t a b i l i t y of a x i s y m m e t r i c d i s t u r b a n c e s i n HagenP o i s e u i l l e ( p i p e ) f l o w t o i n f i n i t e s i m a l non-axisymmetric p e r t u r b a t i o n s i s s t u d i e d by numerical s o l u t i o n o f t h e Navier-Stokes e q u a t i o n s . I t i s shown t h a t , i f t h e axisymmetric flow i s allowed t o evolve according t o t h e Navier-Stokes e q u a t i o n s , t h e non-axisymmetric i n s t a b i l i t y i s s t r o n g f o r Reynolds numbers R 23000 b u t weak f o r RL1000, t h e c u t - o f f b e i n g due b o t h t o i n c r e a s e d decay o f t h e a x i s y m m e t r i c f l o w and reduced g r o w t h o f t h e nona x i s y m m e t r i c p e r t u r b a t i o n . The non-axisymmetrlc i n s t a b i l i t y can be i s o l a t e d from t h e v e r y f a s t a x i symmetric decay i f t h e axisyrnmetric f l o w i s f o r c e d i n such a way t h a t i t s energy i s m a i n t a i n e d c o n s t a n t , b u t phase r e l a t i o n s a r e a l l o w e d t o develop. I n t h i s case, i t i s found t h a t t h e Reynolds number a t which t r a n s l t l o n o c c u r s i n experiments, R 2 2000, d e l i m i t s t h e r e g i o n s o f i n v i s c i d (1.e. R-independent) growth and v i s c o s i t y dominated decay. INTRODUCTION Experiments i n d i c a t e t h a t , u n l e s s g r e a t c a r e i s t a k e n t o remove background d i s t u r b ances, H a g e n - P o i s e u i l l e ( p i p e ) f l o w undergoes t r a n s i t i o n t o t u r b u l e n c e a t Reynolds 2000 numbers R (based on p i p e r a d i u s and c e n t e r l i n e v e l o c i t y ) o f t h e o r d e r o f (Wygnanski 8 Champagne 1973). Experimental work has s u c c e s s f u l l y addressed t h e problem o f r e l a t i n g t r a n s i t i o n s t r u c t u r e s (e.g. p u f f s , s l u g s ) t o f u l l y - d e v e l o p e d t u r b u l e n t f l o w (Rubin e t . a l . 19791, b u t t h e r e has been l i t t l e p r o g r e s s experimenta l l y o r a n a l y t i c a l l y towards f i n d i n g and c h a r a c t e r i z i n g an i n s t a b i l i t y which i s s i g n i f i c a n t a t t h e low Reynolds numbers a t which t r a n s i t i o n o c c u r s n a t u r a l l y . I t i s now g e n e r a l l y accepted t h a t H a g e n - P o i s e u i l l e flow i s s t a b l e t o a l l l i n e a r axisymmetric and non-axisymmetric d i s t u r b a n c e s (Davey 8 D r a z i n 1969; M e t c a l f e 8 Orszag 1973; Salwen e t . a t . 1980). Thus n o n l i n e a r i n t e r a c t i o n s must be i n v o l v e d i f e x p e r i m e n t a l o b s e r v a t i o n s a r e t o be e x p l a i n e d . Most p r e v i o u s f i n i t e - a m p l i t u d e s t a b i l i t y work has been a x i s y m m e t r i c i n n a t u r e . The a m p l i t u d e expansion techniques o f S t u a r t (1971) [extended t o problems where no n e u t r a l c u r v e e x i s t s (Reynolds 8 P o t t e r (196711 have been used t o suggest t h e e x i s t e n c e o f a x l s y m m e t r i c e q u i l i b r i a and, hence, axisymmetric s u b c r i t i c a l i n s t a b i l i t y (Davey 8, Nguyen 1971; l t o h 1977). However, P a t e r a 8 Orszag (1981) have shown n u m e r i c a l l y t h a t t h e s e s o l u t l o n s do not exist. I t was found t h a t , i n t h e r e g i o n o f wavenumber ( a ) , Reynolds number ( R ) space i n which t h e search was conducted, p i p e flow i s s t a b l e t o a l l f i n i t e a m p l i t u d e axisymmetric d i s t u r b a n c e s . We suspect t h a t p i p e f l o w i s g l o b a l l y s t a b l e t o a l l l i n e a r and n o n l i n e a r a x i s y m m e t r i c d i s t u r b a n c e s a t a l l (a,R). Thus n o t o n l y f i n i t e - a m p l i t u d e b u t a l s o non-axisymmetric d i s t u r b a n c e s must be considered b e f o r e an l n s t a b l l i t y can be o b t a i n e d i n H a g e n - P o i s e u i l l e f l o w . We
A.T. PATERA, S.A. ORSZAG
368
d i s c u s s t h i s s i t u a t i o n b r i e f l y i n t h e c o n t e x t o f p l a n e channel f l o w s . I n plane P o i s e u l l l e f l o w , two-dimensional e q u i l i b r i a do e x i s t (Zahn e t . a l . 1974; H e r b e r t 1976); by themselves, t h e y can n o t e x p l a i n t r a n s i t i o n inasmuch as t h e y o n l y e x i s t f o r R 22900 ( w h i l e t r a n s i t i o n may o c c u r a t R = 1000) and t h e y r e s u l t i n s a t u r a t e d s t a t e s , n o t c h a o t i c m o t i o n . However, we have p r e v i o u s l y shown (Orszag 8, P a t e r a 1980, 1981), t h a t t h e two dimensional s t a t e s a r e s t r o n g l y u n s t a b l e t o three-dimensional p e r t u r b a t i o n s . Furthermore, v e r y slowly-decaying two-dimensional s t a t e s ( q u a s i - e q u i l i b r i a ) e x i s t down t o R "= 1000. To t h e e x t e n t t h a t three-dimensional f i n i t e - a m p l i t u d e f l o w s a r e t y p i c a l l y c h a o t i c i n shear f l o w s (Orszag 8, # e l l s 19801, t h i s i s o l a t e s t h e b a s i c mechanism o f t r a n s i t i o n I n terms o f f i n i t e - a m p l i t u d e two-dimensional d i s t u r b a n c e s and i n f i n i t e s i m a l three-dimensional p e r t u r b a t i o n s . I n c o n t r a s t t o plane P o i s e u i i l e flow, i n plane Couette f l o w t h e r e a r e n e i t h e r e q u i l i b r i a n o r q u a s i - e q u i l i b r i a . However, i t was found (Orszag & P a t e r a 19811 t h a t t h e same mechanism t h a t operates i n p l a n e P o i s e u i l l e f l o w a l s o h o l d s i n p l a n e C o u e t t e flow, i . a . twodimensional decaying f i n i t e - a m p l i t u d e s t a t e s a r e s t r o n g l y u n s t a b l e t o t h r e e dimensional i n f i n i t e s i m a l p e r t u r b a t i o n s . The three-dimensional i n s t a b i l i t y i s n o t y e t c o m p l e t e l y understood, a l t h o u g h i t i s c l e a r from numerical experiments t h a t i t i s i n v i s c i d i n n a t u r e , I n c o n t r a s t t o t h e two-dimensional e q u i l i b r i a (and t h e i r l i n e a r c o u n t e r p a r t , T o l l m i e n S c h l i c h t i n g waves), which r e q u i r e f i n i t e v i s c o s i t y . Note, however, t h a t t h i s i n v i s c i d three-dimensional I n s t a b i l i t y i s n o t a c l a s s i c a l i n f l e c t i o n a l I n s t a b i l i t y . For, i f t h e i n s t a n t a n e o u s l y i n f l e c t i o n a l two-dimensional p r o f i l e s (generated by t h e e q u i l i b r i a ) d r i v e t h e i n s t a b i l i t y , t h e n one would a l s o e x p e c t two-dimensional p e r t u r b a t i o n s t o t h e e q u i l i b r i a t o be u n s t a b l e , which i s n o t t h e case. The success o f t h e two-dimensional f inite-amp1 i t u d e / t h r e e - d i m e n s i o n a l i n f l n i t e simal s t r u c t u r e i n e x p l a i n i n g s u b c r i t i c a l t r a n s i t i o n i n p l a n e channel f l o w s suggests t h a t such a mechanism may a l s o prove r e l e v a n t i n p i p e P o i s e u i l l e f l o w , e s p e c i a l l y s i n c e p i p e P o i s e u i l l e f l o w and p l a n e C o u e t t e f l o w a r e v e r y s i m i l a r i n terms of b o t h l i n e a r t h e o r y and two-dimensional f i n i t e - a m p l i t u d e b e h a v i o r . I n t h i s paper, we show t h a t t h i s i s indeed t h e case. Furthermore, new i n s i g h t s a r e p r o v i d e d i n t o s u b c r i t i c a l 1 i n e a r t h r e e - d i m e n s i o n a l i n s t a b i I i t y , and new t o o l s a r e presented f o r c a r r y i n g o u t t h e s t a b i l i t y a n a l y s i s when two-dimensional e q u i l i b r i a do n o t e x i s t . THREE-DIMENSIONAL LINEAR PROBLEM The problem t o be s t u d i e d here i n v o l v e s t h e s t a b i l i t y o f H a g e n - P o i s u i l l e flow ( d e f i n e d as f l o w i n a c i r c u l a r p i p e o r r a d i u s I d r i v e n by t h e c o n s t a n t pressure g r a d i e n t -4/R) t o f i n i t e - a m p l i t u d e a x l s y m m e t r i c and i n f i n i t e s i m a l t h r e e dimensional (non-axisymmetric) p e r t u r b a t i o n s :
Here x,r, and 0 a r e t h e a x i a l , r a d i a l , and a z i m u t h a l c o o r d i n a t e s , r e s p e c t i v e l y , and E i s a ( s m a l l ) parameter r e l a t e d t o t h e a m p l i t u d e o f t h e three-dimensional component. W i t h t h e decomposition ( 1 ) (and E < < I ) , t h e Navier-Stokes equations a r e sepqrable i n 0 so t h a t 'it i s c o n s i s t e o t t o assume t h a t t h e If I s a l s o expanded i n a &dependence o f v i l ) i s o f t h e form e+ime F o u r i e r s e r i e s i n x (keeping a l l modes i n t h s d i r e c t i o n because o f n o n l i n e a r i n t e r a c t l o n s ) , t h e v e l o c i t y i s represented as
.
e
where
2a/a
i s the periodicity interval i n
i a n x Iqme e
he x - d i r e c t i o n .
R e a l i t y of t h e
INSTABILITY OF PIPE FLOW
369
v e l o c i t y f i e l d implies t h e f o l l o w i n g symmetry:
where * denotes complex conjugate. Note t h e assumption o f p e r i o d i c i t y i n x and 0 corresponds i n c l a s s i c a l stability theory t o a temporal s t a b i l i t y analysis. The incompressible Navier-Stok$s equations govern t h e e v o l u t i o n o f To O ( E ~ ) , t h e equations f o r v i 9 ) are
-f
v
i n time.
at
where 6 . i s t h e KroneckeSdeIta f u n c t i o n and t h e c a r a t denotes a u n l t v e c t o r The bounhdry c o n d i t i o n s on v are
;Aq)
(r = 0)
bounded
the l a t t e r being t h e n o - s l i p c o n d i t i o n . No boundary c o n d i t i o n s a r e r e q u i r e d i n t h e o t h e r d l r e c t i o n s due t o t h e assumption o f p e r l o d i c l t y . Note t h a t c l a s s i c a l l i n e a r s t a b i l i t y theory i s a s p e c i a l case o f t h e above formI n t h i s case, t h e r e l e v a n t parameter u l a t i o n i n which v ( o ) ( x , r , t ) = ( l - r * ) x . space i s (R,a,m), and I t i s g e n e r a l l y agreed t h a t a l l disturbances decay. The s o l u t i o n t o t h e l i n e a r c h a r a c t e r i s t i c - v a l u e problem w i l l be denoted as CR,a,ml. With n o n l i n e a r e f f e c t s allowed, t h e parameter space must be extended t o i n c l u d e a measure o f amplitude, such as t h e modal energies: (4a)
(The f a c t o r of 6 i s so t h a t ener l e z f o r n # 0 a r e measured r e l a t i v e t o t h a t ( I) represent t h e and o f the laminar p a r a l l e l flow ( I - r 1 x 1. E# axisymmetric and non-axlsymmetric p e r t u r b a t i o n energies>%spectlvely.
9
370
A.T. PATERA. S.A. ORSZAG
NUMERICAL METHOD The numerical technique used here f o r s o l v i n g t h e Navler-Stokes equations i s very s i m i l a r t o t h a t discussed i n d e t a i l i n Patera 8 Orszag (1981a) f o r s o l v i n g t h e f i n i t e - a m p l l t u d e axisymmetric problem. The e v a l u a t i o n o f t h e n o n l i n e a r terms, the time-stepping method, and t h e m a t r i x i n v e r s i o n techniques a l l c a r r y over d i r e c t l y t o t h e non-axlsymmetric problem. I t i s o n l y I n t h e treatment o f t h e a x i s , and, i n p a r t i c u l a r , i t s e f f e c t on t h e choice o f s u i t a b l e s p e c t r a l expansions and boundary c o n d i t i o n s i n which s p e c i a l care must be taken i n s o l v i n g t h e t h r e e dimensional problem. We l i m i t discussion here t o these more s u b t l e p o i n t s .
-+
is expanded i n As i n t h e axisymmetric problem, t h e r a d i h l dependence o f v a Chebyshev polynomial s e r i e s , where t h e p t h Chebyshev polynomial i s d e f i n e d
T
P
(cos 8 ) = cos p e
52
The f a c t t h a t t h e domain o f i n t e r e s t i s 0 r 2 I whereas t h e argument o f t h e Chebyshev polynomials, z, satisfies -I z O
,
in
;
I, Rnr
# I;
I , Rnr
r = 0 (restricting
m=O
m = I
(Batchelor 8 G i I I 1962; Orszag 1974). Consistent w i t h t h i s behavior t h e complete s p e c t r a l expansion o f t h e v e l o c i t y f i e l d can be w r i t t e n as
R
where
= 2p
+
b(k,m
191)
( 0 (r = 1 b(r,s) =
and
k
I
I (r = I
and
and
s
even)
or
( r = 2,3
and
s
odd)
and
s
odd 1
or
(r = 2,3
and
s
even)
r e f e r s t o d i r e c t i o n a l component.
R e a l i t y imposes t h e c o n d i t i o n
on t h e f i n i t e F o u r i e r s e r i e s . F i n a l l y , we discuss i m p o s i t i o n o f boundary c o n d i t i o n s a t r = 0. When t h e equation t o be solved can be regularized, no boundary c o n d i t i o n i s r e q u i r e d (1.e. t h e s i n g u l a r i t y i n t h e equation i s removed by r e j e c t i n g t h e unbounded s o l u t i o n ) . I f t h e equation cannot be regularized, a boundary c o n d i t i o n a t r = 0 i s r e q u l r e d A boundary c o n d i t i o n i s e a s i l y obtained from ( 7 ) f o r e i t h e r t h e s o l u t i o n o r i t s derivative. We w i l l comment b r i e f l y here on t h e s o l u t i o n o f t h e l i n e a r - theory c h a r a c t e r i s t i c value problem as l i n e a r modes provide t h e i n i t i a l c o n d i t i o n s f o r t h e f i n i t e
372
A.T. PATERA, SA. ORSZAG
amplitude runs. As i n t h e simulation, a pseudospectral Chebyshev method i s used t o s e t up t h e m a t r i x equations, and e i t h e r a c u b i c a l l y convergent ( i n v e r s e i t e r a t l o n / R a y l e i g h q u o t i e n t ) l o c a l method o r a g l o b a l (OR) a l g o r i t h m i s used t o s o h e t h e a l g e b r a i c eigenvalue problem. RESULTS Before presenting f i n i t e - a m p l i t u d e r e s u l t s , we comment t h a t t h e numerical code a c c u r a t e l y simulates l i n e a r behavior. Test runs i n d i c a t e t h a t a t R = 4000, a = 1.0, w i t h P = 32, A t = 0.025, t h e energy o f b o t h m = 0 modes, {R,a,m) = {4000, 1.0,O) , and m = I modes, 14000, I .O, I ) , decay a t a r a t e i n a d d i t i o n t o these l i n e a r which agrees t o w i t h i n about 0.1% o f l i n e a r theory. t e s t s , axlsymmetrlc f i n i t e amplitude runs were a l s o made t o c o n f i r m t h e c o n c l u s i o n s o f o u r previous work on p i p e f l o w (Patera & Orszag 1981). I n the f i n i t e - a m p l i t u d e a x l s y m m e t r i c / l l n e a r non-axisymmetric runs, i n i t i a l c o n d i t i o n s are o f t h e form
+
where t h e l i n e a r mode v I s chosen t o be t h e l e a s t s t a b l e w a l i mode a t t h e g i v e n i n i t i a l ehergy o f t h e axisymmetrlc d i s t u r b a n c e i s s p e c i f i e d as ( t = 0 ) . The i n i t i a l amplitude o f t h e non-axisymmetric component as t h e theory i s l i n e a r I n t h e azimuthal d i r e c t i o n . Note t h a t t h e r e s u l t s discussed below concern f l o w s developing from i n i t i a l c o n d i t i o n s c o n s i s t i n g o f w a l l modes w i t h m = I . The behavior o f c e n t e r modes and modes w i t h general m ( # I ) i s n o t important and w i l l be discussed b r i e f l y a t t h e end o f t h i s section. For t h e range o f R,a run here, t h e numerical parameters P = 32, N = 8, A t = 0.025 p r o v i d e f u l l y converged r e s u l t s as was confirmed by doubling ( o r h a l v i n g ) t h e values g i v e n f o r P,N, and A t above and comparing the resulting solutions. The n a t u r a l way t o proceed i n seeking a t r a n s i t i o n - r e l a t e d i n s t a b i i i t y i s t o e s t a b l i s h t h a t an i n s t a b i l i t y e x i s t s f o r Reynolds numbers B l l g h t l y l a r g e r than RT, t h e t r a n s i t i o n a l Reynolds number, and f o l l o w l n g t h e behavior o f t h e instabI n Fig. I we demonstrate t h a t a threei l i t y as R passes through RT. dimensional i n s t a b i l i t y o f t h e type found p r e v i o u s l y i n plane channel flows a c t u a l l y e x i s t s i n Hagen-Poiseuille flow; t h e wavenumber dependence o f t h e l n s t a b l I i t y a t R = 4000 i s i I l u s t r a t e d by a p l o t o f En E+& f o r q = 0, I vs = 0.04). For a = 0.5 t h e r e i s o n l y weak c o u p l i n g o f t h e time. (Here E(0) axlsymmetric a n ~ o b 8 - a x i s y m m e t r l cf i e l d s and three-dimensional growth i s less than f o r a = 1.0 o r a = 2.0. The growth o f non-axisymmetric disturbances a t a = 2.0 i s approximately t h e same as t h a t a t a = 1.0, b u t t h e axisymmetric s o l u t i o n s decay much more q u i c k l y . Thus, t a k i n g i n t o c o n s i d e r a t i o n both a x l symmetric decay and non-axlsymmetrlc growth, a = 1.0 appears t o be n e a r l y t h e most dangerous wavenumber. I n Fig. 2 we agaln p l o t En E# f o r q = 0,i vs t i m e b u t now a t a = 1.0 l e t t i n g t h e Reynolds number v a r y from 500 t o 4000. Although i t i s c l e a r from t h i s s e r i e s o f runs t h a t t h e I n s t a b i l i t y i s dangerous a t R = 3000 and I n s i g n i f f c a n t a t R = 500, a Reynolds number of 2000 ( o r s l i g h t l y l a r g e r ) I s n o t s i n g l e d i n t h e p l o t as d i s t i n g u i s h i n g growth from decay. T h i s problem o f i n t e r -
INSTABILITY OF PIPE FLOW
373
t
100.
C
a=i. 2.
In E .5
Fig. 1. A plot of the growth of non-axisymmetric (three-dimensional) perturbations on (decaying) axisymmetric states at R = 4000. When the axisymmetric flow is of sufficiently large amplitude, the nonaxisymetric perturbation grows exponentially. Observe that when (I 1.0 the pipe flow is most unstable. Lower wavenumbers significantly decrease the growth o f non-axisymmetric perturbations, while higher wavenumbers result in sharply increased axisymmetric decay.
-5c
t
0. 0.
% x R=4000
-
InE
'
-50.
Pig. 2. A plot of the growth of non-axisymmetrfc perturbations on decaying axisymmetric states with a = 1.0 for various R. The instability is strong for R 2 3 0 0 0 and weak for R S 1000. the cut-off due primarily to increased axisymnetric decay.
100.
374
A.T. PATERA, S.A. ORSZAG
p r e t a t i o n a r i s e s from t h e f a c t t h a t , I n p i p e flow, f i n i t e - a m p l i t u d e a x i s y m m e t r i c d i s t u r b a n c e s decay on a t i m e s c a l e even s h o r t e r t h a n t h o s e i n p l a n e C o u e t t e f l o w , and t h u s t h e d e t a i l s o f t h e i n s t a b i l l t y a r e obscured by t h e two-dimensional damp i ng
.
To i s o l a t e t h e three-dimensional i n s t a b i l i t y r e q u l r e s f o r c i n g o f t h e axisymmetric f l o w i n such a way t h a t i t e v o l v e s n a t u r a l l y w h i l e m a i n t a i n i n g a f i x e d a m p l i t u d e . (amplitude -1 frozen, We term such a f o r c i n g o f t h e two-dimensional f l o w Ameaning t h a t o v e r a l l a m p l i t u d e s a r e c o n s t r a i n e d , b u t phase r e l a t i o n s a r e n o t . To w i t , t h e f o l l o w i n g scheme i s used: a f t e r eachE ( 8 ) and E ( O ) are mode i s t h e n s c a l e d by and a l I n a r e ?he d e s i r e d ( c o n s t a n t i , where and F$gi and p e r t u r b a t i o n e n e r g i e s , r e s p e c t i v e l y . Note t h a t , w i t h t h i s scheme, harmonics can develop and waves can propagate. We a r e o n l y imposing a weak e n e r g e t l c c o n s t r a i n t on t h e a x l s y m m e t r i c f l o w .
$8)
%$I n F i g . 3 we show a p l o t s i m i l a r t o F i g . 2 e x c e p t h e r e t h e a x i s y m m e t r i c f i e l d i s A-frozen. Note f i r s t t h a t t h e growth o f three-dimensional modes i s v e r y c l o s e l y pure e x p o n e n t i a l as would be expected i f one a c t u a l l y had a x i s y m m e t r l c e q u i l i b r l a ; t h u s we have e f f e c t i v e l y separated t h e e f f e c t s o f axlsymmetric decay and non-axisymmetric growth. More i m p o r t a n t l y , n o t e t h a t i n t h i s p l o t R = 2000 delimits regions of i n v i s c i d (i.e. R-independent) growth ( R L 2 0 0 0 ) and viscous-dominated behaviour ( R $2000). T h i s i s r e f l e c t e d by t h e c l o s e spacing o f the- Rn E(;At c u r v e s f o r R 22000 and t h e a b r u p t change t o wide spacing a t R = 2000. The f a c t t h a t f o r c i n g i s r e q u l r e d t o h i g h l i g h t t h e most s i g n i f i c a n t aspects o f the i n s t a b i l i t y i n Hagen-Poiseuille f l o w b u t n o t i n plane P o i s e u i l l e f l o w i s c e r t a i n l y l i n k e d t o t h e d i f f e r e n t n a t u r e G f t r a n s i t i o n I n t h e s e two f l o w s . The appearance o f p u f f s and s l u g s I n H a g e n - P o i s e u i l l e f l o w experiments i s evidence t h a t e x p l o s i v e l y u n s t a b l e t r a n s i t i o n s t r u c t u r e s may u l t i m a t e l y decay ( b a r r i n g v e r y l a r g e d i s t u r b a n c e s ) , j u s t as t h e b e h a v i o r o f o u r I n s t a b i l i t y i n t h e u n f o r c e d case i s numerical evidence o f t h e same phenomenon. F u r t h e r i n s i g h t i n t o t h e p r e s e n t non-axlsymmetric i n s t a b i l i t y i s o b t a i n e d by i n v e s t i g a t i n g t h e requirements on t h e f o r c i n g p r o t o c o l i n o r d e r t h a t t h r e e dimensional I n s t a b i l i t y be maintained. I n p a r t i c u l a r , we d e f i n e an a x i s y m m e t r i c f i e l d t o be $-(phase - ) f r o z e n I f , a t every t i m e step, t h e axisymmetric f i e l d I s r e s e t t o i t s f r o z e n v a l u e i n t h e l a b o r a t o r y c o o r d i n a t e frame; note the d i f f e r e n c e between $ - f r o z e n (where phase r e l a t i o n s a r e n o t a l l o w e d t o e v o l v e f o r A-frozen and $ - f r o z e n n a t u r a l l y ) and A-frozen. I n F i g . 4 , ]In E ( I ) f i e l d s a r e compared. Note t h a t t h e r e i s v i r k b l l y no growth i n $ - f r o z e n case. Furthermore, i t i s n o t t h e l a c k o f harmonic g e n e r a t i o n i n t h e $ - f r o z e n f i e l d t h a t r u l e s o u t growth, f o r , i f t h e a x i s y m m e t r i c f i e l d i s $ - f r o z e n a t some t i m e T a f t e r i t has e v o l v e d n a t u r a l l y f o r 0 ( t < T , three-dimensional g r o w t h r a p i d l y t u r n s o f f . The d r a s t l c a l l y d i f f e r e n t r e s u l t s o b t a i n e d a r e no doubt due t o t h e s y n c h r o n i z a t i o n phenomenon found i n p l a n e P o i s e u i l l e f l o w ( P a t e r a 8 Orszag 1981b), where b o t h t h e two-dimensional wave and t h e most dangerous t h r e e dimensional p e r t u r b a t i o n t r a v e l a t t h e same phase speed (1.e. t h e temporal e l g e n v a l u e r~ i s r e a l ) . I n t h e $I - f r o z e n case t h i s s y n c h r o n i z a t i o n i s inhibited. We have n o t y e t t h o r o u g h l y i n v e s t i g a t e d t h e i n s t a b i l i t y o f a x i s y m m e t r i c d i s t u r b ances t o h i g h e r azimuthal-wavenumber (m> 1 ) p e r t u r b a t i o n s . However, i t appears (as i n l i n e a r t h e o r y ) t h a t t h e e f f e c t o f i n c r e a s i n g m Is s t a b i l i z i n g . Also, r u n s have been made i n which t h e i n i t i a l c o n d i t i o n s I n v o l v e c e n t e r modes r a t h e r t h a n w a l l modes. I t i s found t h a t , f o r t h e same I n i t i a l maximum a m p l i t u d e o f t h e axisymmetric d i s t u r b a n c e , w a l l modes a r e t h e more dangerous. (Note t h a t i f one compares non-axisymmetric growth due t o a w a l l and c e n t e r mode o f t h e same i n i t i a l energy, t h e l a t t e r appears more dangerous. However, t h i s r e s u l t i s an a r t i f a c t of t h e energy d e f i n i t i o n s ( 4 ) I n which t h e energy o f t h e
INSTABILITY OF PIPE FLOW
375
t
100.
0
1
/ R=4000 2-D , A - FROZEN
3000
2000
In E
1000
500
1
-5c
Pig. 3. A plot of the growth of non-axisynmetric perturbations on A-frozen axisymmetric states with a = 1.0 for various R. In A-freezing the twodimensional field we prevent axisymmetric decay but rllow non-axisymmetric qrowth, thus isolating the mechanism of three-dimensional instability. It is seen that at R G 2 0 0 0 there is a change from viscous behavior (strongly R-dependent) to inviscid growth (approximately R independent).
-
t
100.
0,
/
A- FROZEN
3-D
In E
50
?ig. 4. A comparison of the growth of non-axisymmetric perturbations driven by A-frozen and +-frozen axisynmetric fields. In the +-frozen case synchronization between the axisymnetric and non-axisymmetric flow is inhibited and there is no three-dimensional growth. These runs are at R 4000. a = 1.0.
-
A.T. PATERA. S.A. ORSZAG
376
d i s t u r b a n c e I s weighted w i t h d i s t a n c e from t h e a x i s . Thus v e r y l a r g e a m p l i t u d e c e n t e r modes r e s u l t i n small e n e r g i e s . F o r example, an i n i t i a l energy o f 0.04 corresponds t o r o u g h l y a 10%. maximum amp1 i t u d e f o r a wal I mode, b u t t o almost a 40% maximum a m p l i t u d e f o r a c e n t e r mode.) T h i s work was supported by t h e O f f i c e o f Naval Research under C o n t r a c t s No. NOOOI4-7FC-0138 and No. N00014-79-C-0478. Development o f t h e s t a b i l i t y codes was supported by NASA Langley Research Center under C o n t r c a t No. NASI-16237. REFERENCES: [I1
B a t c h e l o r , F.K. 8 G i l l , A . E. 1962 axlsymmetric j e t s . J . F l u i d Mech.
c21
Davey, A. 8 Drazin,P. G. a pipe. J. F l u i d Mech.
C31
1971 F i n i t e - a m p l i t u d e s t a b i l i t y o f p i p e Davey, A. 8 Nguyen, H.P.F. flow. J . F l u i d Mech. 701.
C41
H e r b e r t , T. 1976 P e r i o d i c secondary m o t i o n s i n a p l a n e channel. I n Proc. 5 t h I n t . Conf. on Numerlcal Methods i n F l u i d Dynamics (ed. A . I . van de Vooren and P. J . Zandbergen), p. 235, S p r i n g e r .
C5l
I t o h , N. 1977 N o n l i n e a r s t a b i l i t y o f p a r a l l e l f l o w s w l t h subc r i t i c a l Reynolds number. P a r t 2. S t a b i l i t y o f p i p e P o i s e u i l l e f l o w t o f i n i t e axisymmetric d i s t r u b a n c e s . J . F l u i d Mech. 82, 469.
c61
M e t c a l f e , R. W. 8 Orszag, S. A. l i n e a r s t a b i l i t y o f p i p e flows. Washington.
C7l
1969
2,209.
Analysis o f the s t a b i l i t y o f
14,529.
The s t a b i l i t y o f P o i s e u l l l e f l o w I n
45,
Orszag, S. A. 56.
1973 Numerical c a l c u l a t i o n s o f t h e Flow Research Report No. 25, Kent,
1974 F o u r i e r s e r i e s on spheres. Mon. Wea. Rev.
102, -
C8l
Orszag, S. A. 8 K e l l s , L. C. 1980 T r a n s i t i o n t o t u r b u l e n c e i n p l a n e P o i s e u i l l e and p l a n e C o u e t t e flow. J . F l u i d Mech. 96, 159.
C9l
1980 S u b c r i t i c a l t r a n s i t i o n t o Orszag, S. A . 8 Patera, A. T. t u r b u l e n c e i n p l a n e channel f l o w s , Phys. Rev. L e t t e r s , 989.
[I01
198J Three dimensional i n s t a b i l i t y Orszag, S. A. 8 Patera, A. T. o f p l a n e channel f l o w s . To be p u b l i s h e d .
C I I1
Patera, A. T. 8 Orszag, S. A. 1981 F i n i t e - a m p l i t u d e s t a b i l i t y of axisymmetric p i p e flow. To appear i n J . F l u i d Mech.
c121
Reynolds, W. C. 8 P o t t e r , M. C. 1967 F i n i t e - a m p l i t u d e I n s t a b i l i t y o f p a r a l l e l shear f l o w s . J . F l u i d Mech. 278 465.
[I31
Rubln, Y., Wygnanski, I., 8 H a r l t o n l d l s , J. H. 1979 F u r t h e r o b s e r v a t i o n s on t r a n s i t i o n i n a p i p e . I n Laminar-Turbulent T r a n s i t i o n , IUTAM Symposium (ed. R. E p p l e r 8 H. F a s e l ) , p. 17, Springer.
C141
Salwen, H., Cotton, F. W . , 8 Grosch, C. E. 1980 L i n e a r s t a b i l i t y o f P o i s e u i l l e f l o w i n a c i r c u l a r pipe, J . F l u i d Mech., 98, 273.
45,
INSTABILITY OF PIPE FLOW
[I51
S t u a r t , J . T. Mech. 3, 341.
[la]
Wygnanski, I . 8 Champagne, F. H. 1973 On t r a n s i t i o n I n a pipe, P a r t I . The o r i g i n o f puffs and s l u g s and t h e flow I n a t u r b u l e n t s l u g . J . F l u i d Mech. 2, 281.
[I71
Zahn, J.-P., Toomre, J . , S p i e g e l , E. A. 8 Gough, D. 0. 1974 N o n l i n e a r c e l l u l a r rnotlons i n P o i s e u i l l e channel flow, J . F l u i d Mech. 2, 319.
1971 N o n l i n e a r s t a b i l i t y t h e o r y , Ann. Rev. F l u i d
377
This Page Intentionally Left Blank
Nonlinear Problems: Present arid Future
379
A.R. Bishop, D.K. Campbell, 6. Nicolaenko (eds.) 0 North-HollandPublishing Company, 1982
SOME FORMALISM AN0 PREDICTIONS OF THE PERIOD-DOUBLING ONSET OF CHAOS
M i t c h e l l J. Feigenbaum Theoretical D i v i s i o n Los Alamos National Laboratory Los Alamos, New Mexico 87545 We i n i t i a l l y consider one-parameter f a m i l i e s o f maps fA w i t h t h e p r o p e r t y t h a t f o r a convergent sequence o f parameter values, the map possesses s t a b l e p e r i o d i c o r b i t s o f successively increasing period: An
+
A,
;
fA has a s t a b l e o r b i t o f p e r i o d 2n n
.
I n an i n t e r v a l o f parameter values around ,A,
a f i x e d - p o i n t theory provides uni-
versa1 l i m i t s f o r these maps when a p p r o p r i a t e l y i t e r a t e d and rescaled.
The
operator whose f i x e d p o i n t i s considered i s
where f has a quadratic extremum conventionally l o c a t e d a t t h e o r i g i n .
IY i s
determined by r e q u i r i n g T t o have a f i x e d p o i n t g: Tg = g; g(0) = 1, g(x) = $(x2), $ r e a l a n a l y t i c . About g, DT has a unique eignevalue, 6, i n excess o f 1. Numerically, 9 IY = 2.5029..
6 = 4.6692..
I n terms o f t h i s f i x e d p o i n t g,
where v i s an f-dependent magnification.
We assume f has been s u i t a b l y magni-
f i e d so t h a t we can take v = 1; the l i m i t i n (2) i s i n t h i s sense universal. (Observe t h a t the o b j e c t on the l e f t i s T n f
hat
.)
380
M.J. FEIGENBAUM
Consider
(An(p))
i s a sequence o f parameters f o r which fA has a period-doubling sequence
o f o r b i t s a l l o f s t a b i l i t y p. superstable values,
etc.).
(p = -1 determines b i f u r c a t i o n values, p = 0
For I p I < 1 xn(p)
i s an element o f a s t a b l e Zn-
cycle, which we choose as t h a t element o f smallest modulus. For l p l > 1 xn(p) i s the smallest element o f an unstable c y c l e which i s c o e x i s t e n t w i t h an a t t r a c t o r o f i n t e r e s t . For example, there i s a value o f p < -1 which i s t h e slope o f fZna t i t s smallest f i x e d p o i n t when t h e r e i s a superstable 2"'
cycle:
Figure 1: A schematic p l o t o f fZn showing t h e p o i n t a t which the slope equals p < -1.
F p r a more negative value o f p , we are considering an unstable f i x e d p o i n t o f f 2" when t h e r e i s a superstable 2"' such t h a t a Zn+'-cycle
cycle.
f Zn when period doubling has accumulated. A,
where
S i m i l a r l y , t h e r e i s a sequence o f pr < -1
i s superstable, u n t i l a t
fi
we consider t h e f i x e d p o i n t o f
Clearly,
381
PERIOD-DOUBLING ONSET OF CHAOS
(5) g(2) =
R ,
g'(2) =
;.
Thus, C c p c - 1 correspond t o stable p e r i o d i c behaviors below A,, an i n t e r v a l about the o r i g i n
while p
,A
a
l i e s w i t h i n a curve [5]
or
A t t h i s p o i n t we enlarge our scope from maps on an i n t e r v a l t o general m u l t i -
dimensional, d i s s i p a t i v e dynamical systems. For a wide class o f such systems i t turns o u t t h a t the mapping theory discussed above c a r r i e s over w i t h o u t modifications (see reference [9] f o r d e t a i l s and f u r t h e r general information).
How-
ever, beyond the usual 4.669 convergence r a t e , t h e dynamical consequences o f t h e theory are most n a t u r a l l y phrased i n the format o f a s c a l i n g theory. This scali n g formalism, w h i l e constructed from an "underlying" map, has t h e strong advantage o f being transparent t o any mention o f a map.
We proceed now i n a more
"physical" v e i n t o exposit t h i s formalism and determine i t s experimentally v e r i f i a b l e consequences. Simply put,
as a c o n t r o l parameter i s varied,
any dynamical v a r i a b l e demon-
s t r a t e s a systematic doubling o f i t s p e r i o d p r i o r t o t h e a p e r i o d i c l i m i t sign a l l i n g the onset o f chaos. However, since the p e r i o d doublings accumulate so q u i c k l y , gross features remain nearly constant, w h i l e the new temporal features c h a r a c t e r i z i n g t h e newly
306
M.J. FEIGENBAUM
doubled p e r i o d contain a small f r a c t i o n o f t h e energy.
Moreover, as period-
doubling i s p e r i o d i c i n terms o f l o g a r i t h m i c d e v i a t i o n s from the t r a n s i t i o n p o i n t , each new m o d i f i c a t i o n i s a constant f r a c t i o n o f i t s predecessor. T h i s e n t a i l s very d e f i n i t e consequences, whether viewed temporally o r through a Fourier transform, which we now explore. The basic t h e o r e t i c a l q u a n t i f i c a t i o n o f t h i s n o t i o n i s t h a t t h e d e v i a t i o n s from t h e o l d p e r i o d i c i t y t h a t determine t h e new and doubled p e r i o d s y s t e m a t i c a l l y (and u n i v e r s a l l y ) scale from one doub i n g t o t h e next.
where xn(t)
Forma ly, w r i t e
i s any coordinate when t h e parameter i s such t h a t t h e n - t h p e r i o d
doubling has occurred, f o r which t h e p e r i o d i s Tn = 2*Tn-1
.
(Actually, as t h e parameter v a r i e s Tn i s o n l y approximately constant. t h e parameter values accumulate, (22) i s a s y m p t o t i c a l l y c o r r e c t . ) shows dn t o be nonzero i f the p e r i o d o f xn i s r e a l l y Tn and
not
However,
Equation (21) Tn-l, and so dn
i s the d e v i a t i o n i n t h e t r a j e c t o r y associated w i t h t h e doubling period. t h e o r e t i c a l r e s u l t (which i s published elsewhere [4])
The
i s t h a t f o r l a r g e n,
so t h a t u has t h e symmetry
u(x +
5-)
= -u(x)
u(x + 1) = u(x)
.
Equation (23) i s a very strong p r e d i c t i o n : acquire t h e temporal data xn(t) and compute from i t the h a l f - p e r i o d d e v i a t i o n f u n c t i o n o f (21);
similarly obtain
dn+l and d i v i d e i t by dn; then f o r each n on an a p p r o p r i a t e l y scaled p l o t , t h e
same f u n c t i o n u(x) should be observed.
This i s already a s t r o n g s c a l i n g r e s u l t .
However, there i s an even stronger f e a t u r e being tested: known, a v a i l a b l e f u n c t i o n w i t h t h e approximate value
namely t h a t u(x) i s a
PERIODDOUBLING ONSET OF CHAOS
a-2 = 1/6.25
a(x) =
...
la-’ = 1/2.5029
A l l spectral
307
O < x < .25
...
.25 < x < .5
t e s t s f o r period-doubling u n i v e r s a l i t y a r e t e s t s o f (23) while, a t
present, the r e s u l t s f o l l o w i n g from t h e approximation (25) are as p r e c i s e as experiments can hope t o resolve. There are,
however,
d i f f i c u l t i e s i n employing (23)
as an experimental t e s t ,
which f o l l o w from t h e requirement o f making two sets o f measurements a t d i f f e r e n t parameter values.
The simpler problem i s the phasing o f the two time bases:
t o use (23) t h e time o r i g i n s must be t h e same. varying a delay, i n say dn(t), dent w i t h those o f dn+l(t).
This i s most e a s i l y handled by
u n t i l i t s zero crossings are most n e a r l y c o i n c i -
(There are indeed many n e a r l y c o i n c i d e n t crossings.
Also, the time scale o f dn(t) might r e q u i r e some adjustment so t h a t dn+l p r e c i s e l y t w i c e the period o f dn.)
has
More seriously, t h e parameter values must be This
chosen so t h a t the l i m i t cycles o f xn and x ~ + ~have , identical stability.
requires, f o r example, some convergence data which might be hard t o come by unless successive parameter values a t the b i f u r c a t i o n p o i n t s (when t h e p e r i o d doubles) are used.
I f t h i s d i f f i c u l t i e s are attended t o , however, (23) and (25)
then serve as a simple and very d i r e c t measurement o f t h e u n i v e r s a l i t y theory. (These same d i f f i c u l t i e s a f f l i c t the F o u r i e r measurements, except t h a t phasing i s obviously unimportant i f only amplitude spectra a r e determined.) I n f a c t , the universal f u n c t i o n u i s d i f f e r e n t f o r d i f f e r e n t s t a b i l i t i e s (e.g., a t superstable values as opposed t o b i f u r c a t i o n values o f the parameter).
In-
deed, f o r each s t a b i l i t y p , the appropriate u i s computed from t h e functions However, the differences show up o n l y i n the c o r r e c t i o n s t o t h e o f (8). gr,lJ approximation (25). Indeed there are o t h e r functions u agreeing t o t h e l e v e l o f (25) which determine d e v i a t i o n signals, d(t),
pertaining t o the y & parameter
value ( t y p i c a l l y t o be used a t t h e t r a n s i t i o n value) which then bypass t h e above d i f f i c u l t i e s .
We s h a l l explore these t e s t s a f t e r determining F o u r i e r
properties which s h a l l motivate them. Corresponding t o t h e d e v i a t i o n signal, the fundamental frequency o f xn i s 1 l / T n = 2(l/Tn-l), t h a t i s , a h a l f subharmonic o f t h e previous fundamental. Had dn(t) vanished, there would have been no components a t t h e odd m u l t i p l e s o f t h e new fundamental.
The even mu1t i p l e s represent t h e components a t t h e previous
fundamental ; since the parameter change becomes i n c r e a s i n g l y small, these components cannot s u f f e r s i g n i f i c a n t changes. Accordingly, t h e basic r e s u l t o f the doubled p e r i o d i s the s e t o f spectral l i n e s a t t h e odd m u l t i p l e s o f the new subharmonic fundamental.
These are determined n o t by
o f xn(t),
b u t o n l y by
M.J. FEIGENBAU M
388
i t s "subharmonic p a r t " dn(t).
Since these scale (by a), i t i s then c l e a r t h a t
these successive a d d i t i o n s t o t h e spectrum geometrically decrease i n a u n i v e r s a l way, as determined by the F o u r i e r a n a l y s i s o f u. vations q u a n t i t a t i v e .
L e t us now make these obser-
By d e f i n i t i o n ,
w i t h inverse,
Manipulating (26)
or
1
2(
(2p+l) =
Tn-l
J
d t dn(t) e2ni
n-1 0
n)
2p+l t 'n
w i t h inverse
dn(t) = 1 2
P (n)
(2p+l) e-2ni
t
Thus, t h e spectral components a t odd m u l t i p l e s o f t h e subharmonic fundamental are determined s o l e l y by the d e v i a t i o n signal dn(t). Since a l l the new subharmonic components a r i s e from t h e same time s i g n a l , they can be smoothly connected as an ensemble by i n t e r p o l a t i n g them.
The n a t u r a l
i n t e r p o l a t i o n t h a t r e f l e c t s t h e i r common source i s , then, t h e a n a l y t i c continua t i o n provided by (29):
389
PERIOD-DOUBLING ONSET OF CHAOS
(31) Tn-l I dt dn(t) eZniwt n-1 0
2n(w) = T 1
with
icn (-+I 2 + 1 - 2(n)(2P+l) n
Utilizing (301, this interpolation is 1
Tn-l
kn(w) = 1 2 (2p+l) T J n-1 0 P (n)
-2ni(Ep+l-uTn)t/Tn dt e
or niwT (1
an(w) =
"1
+
$ 1P
x^( ,)(ZP+l) 2p+l-uTn
(33)
*
We now ask how %n+l(w) is related to iCn(w).
gn+l(w) =
rn1 ITnO dt dn+l(t)
Be definition
Zniwt e
Employing the scaling formula for large n, we have
$,+l(u)
- 71 I dt dr)dn(t) t Tn n O
e2niwt
n
1 Tn
rn IO dt d
= t 2(,,)(2~+1)
P
= t2
P
(n)
(2p+l)
$- ;n-l
-2ni(2p+l-uTn)
~e )
-5
n
dtk(r)
n-1 0
-
n
niwtn e a(&-
n
+
-2ni(2p+l-wTn)t/T, ije
I f we now use the approximation o f (25), then
gn+l(w)
'
;(
-
niwT, e
1 t 2(,,)(2~+1) P
1 Tn-l -Zni(Zp+l-wT,)t/T, T I dt e n-1 0
M.J. FEIGENBAUM
390
by use o f the i n t e r p o l a t i o n formula (32). Equation (34) i s the basic r e s u l t o f t h i s paper regarding F o u r i e r analysis, and i s r i c h i n content [6].
The f i r s t conclusion t o be drawn i s t h a t r a t h e r than
s c a l i n g uniformly, d i f f e r e n t p a r t s o f t h e spectrum scale very d i f f e r e n t l y .
Thus
, the actual new spectral components c h a r a c t e r i z i n g the n + l - s t n+l l e v e l o f period doubling,
1.
At w =
or
That i s , the new spectral components a r e obtained by s c a l i n g t h e previous i n t e r p o l a t i o n a t these new p o s i t i o n s by a f a c t o r o f
or, dropped l o g a r i t h m i c a l l y by 10 loglO4.6
-"
6.6
(The approximation (35) i s t h e same as t h a t f o r the s c a l i n g o f successive RMS averages o f spectral l i n e s as obtained by Nauenberg [7].
2
, t h a t i s , a t those frequencies o f t h e n - t h l e v e l , one f i n d s n t h e n + l i n t e r p o l a t i o n scaled by
2.
At w =
391
PERIOD-DOUBLING ONSET OF CHAOS
That i s , i n the next generation o f doubling, the i n t e r p o l a t i o n scales by t h e anomalously small amount o f
or 10 1 0 g ~ ~ 3 .: 6 5 . 5 db
a t those frequencies t h a t had j u s t p r i o r l y come i n t o existence. 3.
2 At w = 9
, one has
T
'n
where
or 10 1 0 g ~ ~ 8 .: 3 9.2 db Thus,
.
i n the second and a l l f o l l o w i n g generations a f t e r a spectral l i n e has
appeared, the successive i n t e r p o l a t i o n s drop anomalously quickly.
(The approxi-
mation (37) i s e a s i l y seen t o be t h e same approximation o f Grossman [8] f o r t h e "leading" edge o f the spectrum). 4.
I n consequence o f 2. and 3.,
i f t h e n - t h i n t e r p o l a t i o n i s r a i s e d by x f o r
any 5.5 db < x < 9.2 db, these spectra w i l l a l l have regions o f overlap. In p a r t i c u l a r , the o r i g i n a l spectral p r e d i c t i o n [4] o f x = 8.2 db i s included i n t h i s range.
The present formula (34) i s the f u l l r e a l i z a t i o n o f the ideas o f
t h a t previous paper. 5.
Since d i f f e r e n t p a r t s o f the spectrum have d i f f e r e n t geometric scalings, a
geometric mean o f the n - t h l e v e l s p e c t r a l l i n e s i s a more s i g n i f i c a n t average than the mean o f the squares. Given (34), we can then compute t h e s c a l i n g o f the geometric mean, o r l o g a r i t h m i c a l l y , t h e average o f log-amplitudes:
M.J. FEIGENBAUM
392
Then,
However, t h e i n t e g r a l on t h e r i g h t i d e n t i c a l l y vanishes f o r A
Xn+l
-
.. Xn
l c r l > 1. Thus,
-2n(2cr)
or, the mean l o g amplitudes drop by 10 l 0 g ~ ~ 5 .=0 7.0 db
10 loglo(2a)
.
(This r e s u l t i s unchanged i f t h e averaging i s performed over
(39)
all t h e
n-th level
spectral l i n e s up t o any m u l t i p l e o f the o r i g i n a l fundamental, r a t h e r than j u s t the subharmonic p a r t . ) Equation (38) i s a new r e s u l t , and t h i s approximate value has been numerically v e r i f i e d t o be c o r r e c t t o t h e p r e c i s i o n o f (39).
A t t h i s p o i n t we want t o extend these r e s u l t s t o s p e c t r a l p r o p e r t i e s a t a f i x e d parameter value, r a t h e r than a t successive period-doubl i n g values. As prev i o u s l y demonstrated, as t h e parameter increases, t h e n - t h l e v e l l i n e s s l i g h t l y increase u n t i l they saturate a f t e r several f u r t h e r p e r i o d doublings.
This i n -
crease i s uniform f o r each l e v e l o f introduced s p e c t r a l l i n e s from which i t follows t h a t p r o p e r t i e s 1.-5.
a r e e q u a l l y v a l i d w i t h i n a spectrum a t f i x e d A f o r
a l l intermediate values o f n ( i . e . ,
a f t e r several p e r i o d doublings have occurred,
b u t n o t f o r t h e lowest l y i n g i n t e r p o l a t i o n s which have n o t y e t saturated.) Taking the fundamental p e r i o d t o be 1, UJ = 1 i s t h e o r i g i n a l fundamental, and the n - t h l e v e l o f spectral l i n e s are l o c a t e d a t t h e frequencies
w=2p+l 2n
.
(40)
393
PERIOD-DOUBLING ONSET OF CHAOS
Now, a l l the spectral p r o p e r t i e s are consequences o f (34) which i t s e l f i s a consequence o f (23).
Working backwards, i f we define, a t a f i x e d parameter value
An, d (t) as t h a t p a r t o f xn(t) constructed p u r e l y from t h e m-th l e v e l specn,m t r a l l i n e s , then (23) must again hold f o r a new s c a l i n g f u n c t i o n u which has t h e
same approximate value as (25).
That i s ,
where (41) i s v a l i d f o r a l l in such t h a t 1 0, b > 0, f ( x ) can a t most give rise t o a s t a b l e f i x e d point. For a < 0, b < 0, t h e f u n c t i o n f ( x ) has a s i n g l e q u a d r a t i c hump, and Feigenbaum u n i v e r s a l i t y holds. I n one parameter single-peaked maps, the limit point of i n f i n i t e l y b i f u r c a t e d ZN c y c l e s is a s i n g l e p o i n t . In two parameters single-peaked maps that point becomes a l i n e t h a t we r e f e r t o a s t h e Feigenbaum l i n e . For t h e regions a < 0, b > 0 and a > 0 , b < 0, f ( x ) has e i t h e r two peaks and one v a l l e y or v i c e versa. I n t h e following, we s h a l l refer t o the peak or v a l l e y a t x 0 as t h e c e n t r a l , peak, and t h e peaks or v a l l e y s at x # 0 a s t h e s i d e peaks. The e x i s t e n c e of both c e n t r a l and s i d e peaks implies t h a t f ( x ) may develop independent iterat i o n regions a s i n d i c a t e d i n regions I and I1 in Ng. 2.
-
Ngure 2 A map f o r a < 0, b > 0. Note t h a t regions I and I1 are independent i t e r a t i o n regions. I n region I, f ( x ) d e s c r i b e s a single-hump map with a q u a d r a t i c peak at x = 0. We s h a l l r e f e r t o region I a8 the region c o n t r o l l e d by the c e n t r a l peak. I n region 11, f ( x ) d e s c r i b e s an i n v e r t e d single-hump map with t h e peak a t x = m b . W e s h a l l r e f e r t o I1 as the r e g i o n c o n t r o l l e d by the s i d e peak. Since t h e two s i d e peaks have equal h e i g h t , they map i n t o t h e same x (=1Thus, we have a t a2/4b). Hence, they cannot both be members of an N-cycle. most one most s t a b l e region c o n t r o l l e d by the s i d e peaks. The e x i s t e n c e of two independent regions may a l s o appear i n the case a > 0, b < 0.
(B)
Duality transformation
For the study of cycles and t h e i r sequences, we only need t o consider t h e region c o n t r o l l e d by the c e n t r a l peak. The c y c l e s a s s o c i a t e d with t h e s i d e
397
398
S.-J. CHANG, M. WORTIS, J.A. WRIGHT
peaks can be obtained from those with t h e c e n t r a l peak by a d u a l i t y t r a n s f o r mation d e s c r i b e d below. Consider the q u a r t i c map (2.1).
W e can rewrite i t a s
~ ~ + ~ = l + a * + ,b 4b f O 2
= 1
-%+
b(%+
2 2 xn)
.
(2.2) Thus, we can f a c t o r the q u a r t i c map as the product of two q u a d r a t i c maps:
a2 + b 5,2 xn+1 a 1 W e can now v i s u a l i z e t h e o r i g i n a l mapping as
-
x1
+
c1
+
x2 +
c2
+ x3 +
,
* a * .
(2.5)
It is easy t o see t h a t i f the x-mapping has a s t a b l e N-cycle, so does t h e 6-mapping and v i c e versa. The mapping from to is simply
&+
-&+
.
En+1 = (1 b 58)2 After a r e s c a l i n g , we can put t h e S-mapping (2.6) = 1
+ a ' ST, + b'
(2.6) i n t o the s t a n d a r d form
F.4,
(2.7)
I t is easy t o v e r i f y t h a t the s i d e peaks of x a r e mapped i n t o t h e c e n t r a l peak of E , and t h e s i d e peaks of 5 a r e mapped i n t o the c e n t r a l peak of x. The d u a l i t y r e l a t i o n is r e c i p r o c a l : The d u a l i t y t r a n s f o r m a t i o n of ( a ' , b ' ) is t h e o r ig in a l (a,b). (C)
Most-stable c y c l e s and doubly-stable
cycles
For a q u a r t i c map, the most-stable N-cycles ( x l , x 2 . . . , ~ N )
a r e described by (2.10)
d N dx [f (x'],
- .
(2.11)
0
i
An N-cycle
is wst s t a b l e , i f e i t h e r t h e c e n t r a l peak (x = 0) o r one of t h e s i d e peaks ( x = * m b ) is a member of t h e c y c l e xi The polynomials a s s o c i a t e d with a s t a b l e N-cycle give rise t o a l i n e i n t h e a-b plane. In Fig. 3, the s o l i d l i n e s r e p r e s e n t th6 l o c i of most-stable 2-cycles. Fbnctions whose parameters s a t i s f y a + b + 1 0 have most-stable 2-cycles a s s o c i a t e d with the c e n t r a l peak. The curved l i n e s are the l o c i of p o i n t s of most-stable 2-cycle a s s o c i a t e d with the s i d e peaks. It is easy t o see t h a t , with the exception of t h e o r i g i n , the most s t a b l e N-cycle t r a j e c t o r i e s a s s o c i a t e d w i t h t h e c e n t r a l peak never i n t e r s e c t among themselves. Neither do t h e mst s t a b l e t r a j e c t o r i e s a s s o c i a t e d with t h e s i d e peaks. However, t h e t r a j e c t o r i e s assoc i a t e d w i t h t h e c e n t r a l peak do i n t e r s e c t those a s s o c i a t e d w i t h t h e s i d e peaks. Actually, they i n t e r s e c t i n two completely d i f f e r e n t ways. At i n t e r -
-
.
TRICRITICAL POINTS AND BIFURCATIONS IN A QUARTIC MAP
s e c t i o n s A and A' i n Mg. 3, the most s t a b l e 2-cycles pass through both t h e e c a l l t h i s kind of c y c l e a doublyc e n t r a l peak and one of the s i d e peaks. W s t a b l e cycle. Two t r a j e c t o r i e s which i n t e r s e c t a t a doubly-stable c y c l e a r e "dynamically c o u p l e d at t h e i n t e r s e c t i o n . A t i n t e r s e c t i o n s B and B ' , the 2c y c l e s a r e c o n t r o l l e d by d i f f e r e n t peaks which are not r e l a t e d : some regions of x a r e a t t r a c t e d t o one cycle and some t o the o t h e r . W e r e f e r t o t h i s kind of i n t e r s e c t i o n as "dynamically independent". Note t h a t doubly s t a b l e p o i n t s A and A' a r e dual t o each other. The concept of dynamical coupling is invari a n t under d u a l i t y transformation.
Figure 3 The s o l i d l i n e s r e p r e s e n t t h e most-stable 2 c y c l e s , t h e c r o s s e s denote t r i c r i t i c a l p o i n t s , and t h e dashed l i n e s d e s c r i b e t h e Feigenbaum l i n e s . 111. (A)
TRICRITICAL POINTS B i f u r c a t i o n along b-axis
I f we vary b ( n e g a t i v e ) , k e e p i n g 2 small and n e g a t i v e , we encounter t h e u s u a l b i f u r c a t i o n sequence, s t u d i e d by Feigenbaum. I f we vary b ( n e g a t i v e ) , keeping a small and p o s i t i v e , we f i n d t h a t the map goes through t h r e e d i f f e r e n t most These 2-cycles are cons t a b l e 2-cycles before b i f u r c a t i n g i n t o a 4-cycle. t r o l l e d by t h e t h r e e d i f f e r e n t peaks as shown i n f i g . 1. These phenomena The same phenomena r e p e a t themp e r s i s t no matter how small t h e value of s e l v e s at higher qN cycles. Thus, t h e r e are q u a l i t a t i v e d i f f e r e n c e s between b i f u r c a t i o n s on the tw d i f f e r e n t s i d e s of t h e b-axis. When a = 0 and b < 0 , t h e curve f ( x ) is single-humped with a q u a r t i c maximum. We f i n d a sequence of b i f u r c a t i o n s which always occur a t the doubly-stable 2N c y c l e s :
3Q9
400
S. J. CHANG, M. WORTIS, J.A. WRIGHT
bN
2N 2
-1
4
-1.503 93
8
-1.58225
16
-1.59316
32
-1.59466
...
...
The pN cycles have a l i m i t i n g point a t b, = -1.59490, i n c l o s e analogy t o t h e q u a d r a t i c map studied by Feigenbaum. However, because our map has a q u a r t i c ( r a t h e r than q u a d r a t i c ) maxim m, it is t o be expected t h a t t h e r e w i l l be d i f f e r e n t critical exponents,' as we now discuss. This l i m i t i n g point has s e v e r a l i n t e r e s t i n g p r o p e r t i e s which we s h a l l d i s c u s s below. (B)
C r i t i c a l exponents and the u n i v e r s a l f u n c t i o n
There a r e two relevant eigen-directions at the critical point (0 , -1.59490). Along a relevant eigen-direction and f o r l a r g e N, we have
where ST i s the c r i t i l a l eigenvalue g t h i s eigen-direction. One eigend i r e c t i o n is along the i s where 6T = 7.2851. The o t h e r eigen-direction is ( 1 , -1.2347) -where ' ; 6 2.8571. Note t h a t the kigenbaum f i x e d point has only one relevant d i r e c t i o n with 6 F = 4.6692. In phase-transition language, a c r i t i c a l point with two relevant d i r e c t i o n s which a l s o serves as the end-point of a c r i t i c a l l i n e is known as a " t r i c r i t i c a l " point. As we s h a l l see, our fixed point is indeed a t r i c r i t i c a l point. See Discussion Section below.
P
atPP
A t the c r i t i c a l point ( 0 , -1.59490),
t h e function is s e l f - s i m i l a r near x = 0 t o within a proper rescaling. The s c a l e f a c t o r here is aT = -1.6903. J u s t as 1%the Feigenbaum case, the t r i c r i t i c a l point has i t s own u n i v e r s a l f u n c t i o n f T ( x ) obeying the same f u n c t i o n a l equation pf:(2)(x/p)
= f%(x),
p
In the neighborhood of x = 0, f;(x) fT(x) * 1
-
.
-1.6903
can be represented approximately by
+ 0.01301 x8 x12 - 0.062035 x16 +
1.83413 x4
+ 0.31188 (C)
-
-***
.
(3.3)
An i d e n t i t y
F are
The second exponent 6 i 2 ) and t h e s c a l e f a c t o r
.
6$2) = a; This r e l a t i o n is exact2)as can be shown as follows: If vector belonging t o 6,
.
f ( x ) = f;(x> + Eh(x),
0
2.28 h r t h e r e a c t i o n e x h i b i t s simple p e r i o d i c o s c i l l a t i o n s . When T i s increased w i t h i n t h e range between these l i m i t s , t h e r e a c t i o n passes through an a l t e r n a t i n g sequence o f p e r i o d i c and c h a o t i c regimes; we c a l l these regimes P1 ( T < 0.87 h r ) , C1, P2, Ct, bCI , P ' , (T > 2.28 h r ) . The time r o p o r t i o n a j t o -log[Br-1) and t h e dependence o f t h e bromide i o n PO en 1 corresponding power spectra and 2D phase p o r t r a i t s are shown f o r t h e f i r s t s i x regimes i n F i g u r e s 1-6. (The bromide p o t e n t i a l i s p l o t t e d i n r e l a t i v e u n i t s w i t h a zero o f f s e t ; t h e p o t e n t i a l o s c i l l a t e s i n t h e range 155-200 mV.)
tpjdl
...
As Figures 1, 3, and 5 i l l u s t r a t e , each regime Pk i s c h a r a c t e r i z e d by: ( 1 ) a p e r i o d i c time s e r i e s w i t h each p e r i o d c o n t a i n i n g one l a r g e amplitude r e l a x a t i o n o s c i l l a t i o n ( o f t h e t y p e i n regime P1) and k - 1 small amplitude n e a r l y s i n u s o i d a l o s c i l l a t i o n s ( o f t h e type i n regime P I ) , ( 2 ) a power spectrum c o n s i s t i n g o f a s i n g l e i n s t r u m e n t a l l y sharp fundamental \requency component and i t s harmonics, and (3) a phase p o r t r a i t t h a t i s a l i m i t c y c l e . As Figures 2, 4, and 6 i l l u s t r a t e , each regime c k i s c h a r a c t e r i z e d by: ( 1 ) a nonperiodic o s c i l l a t i n g time series, ( 2 ) a broadband power spectrum, and (3) a phase p o r t r a i t t h a t i s a strange a t t r a c t o r ( a s w i l l be discussed i n t h e f o l l o w i n g s e c t i o n ) . One way i n which we have d i s t i n g u i s h e d t h e p e r i o d i c and c h a o t i c regimes i s by comparing zero-frequency i n t e r c e p t s o f t h e background n o i s e l e v e l i n the power spectra f o r t h e d i f f e r e n t regimes. I n t h e p e r i o d i c regimes t h e broadband noise l e v e l , which i s presumably due t o i n s t r u m e n t a l n o i s e ( i n c l u d i n g f l u c t u a t i o n s i n f l o w r a t e , s t i r r i n g r a t e , e t c . ) , i s t y p i c a l l y two t o t h r e e orders o f magnitude lower than t h e broadband s p e c t r a l i n t e n s i t y a t low frequencies f o r t h e c h a o t i c regimes; t h i s i s i l l u s t r a t e d f o r P2 and C2 i n F i g u r e 7. The sequence o f dynamical regimes we have observed i s summarized i n F i g u r e 8, which shows t h e broadband s p e c t r a l noise l e v e l as a f u n c t i o n o f residence time. Each p e r i o d i c and c h a o t i c regime c l e a r l y e x i s t s over some range o f T , although t h e e x t e n t of t h e regimes, drawn as equal i n F i g u r e 8, has n o t been determined.
COMPLEX DYNAMICS IN A CHEMICAL REACTION
41 1
h
I-
+ ., 4J
v
m
F i g u r e 1 . The t i m e dependence o f t h e b r a n i d e i o n p o t e n t i a l and t h e c o r r e s p o n d i n q power spectrum and phase p o r t r a i t ( w i t h T = 8.8 s) for d a t a o b t a i n e d i n regime PI w i t h T = 0.49 h r .
J.C. ROUX er al.
412
I
I
I
I
121I IB
I
-7
-$ -2 1-3
e
-4
-5
Y
-62m
I
I
I
0.0f
0.02
0.a3
fwquency
I
1
0.04
0.05
(HI)
I
F i g u r e 2. The time dependence o f t h e bromide i o n p o t e n t i a l and t h e corresponding power spectrum and phase p o r t r a i t ( w i t h T = 8.8 s ) f o r d a t a obtained i n regime C w i t h T = 0.90 h r . The v e r t i c a l dashed l i n e shows t h e l o c a t i o n o f t h e Poincare an!l r e t u r n maps discussed i n t h e section e n t i t l e d D e t e r m i n i s t i c Chaos.
':E/
COMPLEX DYNAMICS IN A CHEMICAL REACTION
413
ftl
1 W
I
I
I
I
1
I I?#IF
-1
3 -* c
f -3 L-4
B
-5
h
I-
+.,
c1
Y
m
B(ti1 Figure 3. The time dependence of t h e branide i o n p o t e n t i a l and t h e corresponding power spectrum and phase p o r t r a i t ( w i t h T = 8.8 s) f o r data obtained i n regime P2 w i t h T = 1.03 h r .
J. C. ROUX et al.
414
Time (seconds)
-I
2-2 E
g
c
g
e
-3 -4 -5
-%.a.W
-
I
I
1
I
0.01
0.02
0.a3
0.04
Fnquency (Hz)
0.05
Figure 4. The time dependence of t h e bromide i o n p o t e n t i a l and t h e corresponding power spectrum and phase p o r t r a i t ( w i t h T = 8.8 s ) f o r data obtained i n regime C p with T = 1 . 1 1 hr.
COMPLEX D YNAMICS IN A CHEMICAL REACTION
-5
t
415
' l l ' t l
Figure 5. The time dependence o f the branide ion potential and the corresponding power spectrum and phase portrait (with T = 8.8 s ) for data obtained in regime P 3 with T = 1.25 hr.
J. C. ROUX et al.
416
D 0
.r
Plrm 735
387
180
360
540
720
900
1060
1260
1440
1620
le00
Time (seconds)
-I
3 -2 E
!i
-3
L
g
-4
Q -5
Fmquency fHz)
Figure 6. The time dependence o f the bromide ion potential and the corresponding power spectrum and phase portrait (with T = 8.8 s ) for data obtained in regime C 3 with T 1.38 hr.
COMPLEX DYNAMICS IN A CHEMICAL REACTION
417
DETERMINISTIC CHAOS We have c a l l e d t h e a t t r a c t o r s f o r t h e regimes ck " s t r a n g e a t t r a c t o r s , " w h i c h i m p l i e s t h a t t h e system i s d e t e r m i n i s t i c ( s e e e.g., L a n f o r d [18]). As S h o w a l t e r .__ et . a-l-. . [19] r i g h t l y emphasize, a l l e x p e r i m e n t a l systems a r e i n e v i t a b l y s u b j e c t t o e x t e r n a l p e r t u r b a t i o n s which r e s u l t i n n o n p e r i o d i c b e h a v i o r ; " t h e r e f o r e , f a i l u r e t o observe e x a c t r e p e t i t i o n i n a chemical s t i r r e d t a n k r e a c t o r cannot prove t h e system would be t r u l y c h a o t i c i n t h e complete absence o f p e r t u r b a t i o n " . We concur completely. No experiment w i l l ever be a b l e t o make an a b s o l u t e d i s t i n c t i o n between s t o c h a s t i c and d e t e r m i n i s t i c processes, and, i n f a c t , such a d i s t i n c t i o n cannot even be made i n p r i n c i p l e : as Kac has p o i n t e d o u t , a s t o c h a s t i c process can always i n p r i n c i p l e be d e v i s e d t o d e s c r i b e any n o i s y d a t a even when t h e d a t a can be simply and e l e g a n t l y d e s c r i b e d by a d e t e r m i n i s t i c model [20].
1
(b)
I
I
h
I'
3
I
0.01
I
I
I
0.02
0.03
0.04
FR€QU€NCY (Hz)
G 5
F i g u r e 7. The i n t e r c e p t s a t z e r o frequency o f l i n e s drawn t h r o u g h t h e broadband background n o i s e a r e i l l u s t r a t e d f o r ( a ) a p e r i o d i c regime ( P 2 ) and ( b ) a c h a o t i c regime (C,)
.
J.C. ROUX et al.
410
Recognizing t h e s e d i f f i c u l t i e s , we now p r e s e n t t h r e e d a t a analyses t h a t p r o v i d e s u c c e s s i v e l y s t r o n g e r s u p p o r t f o r t h e i n t e r p r e t a t i o n o f regime C1 as c o r r e s p o n d i n g t o d e t e r m i n i s t i c chaos. F i r s t c o n s i d e r t h e 2D P o i n c a r e map shown i n F i g u r e 9. T h i s map was c o n s t r u c t e d f r o m a 3D R u e l l e - t y p e p o r t r a i t t h a t has, i n a d d i t i o n t o t h e two axes shown i n t h e phase p o r t r a i t i n F i g u r e 2 , a t h i r d a x i s ( p e r p e n d i c u l a r t o t h e f i r s t two axes) o b t a i n e d w i t h T -17.6 s. The P o i n c a r e map i n F i g u r e 9 was formed by t h e i n t e r s e c t i o n o f t h e 3D p&ise space o r b i t s w i t h a p l a n e t h a t passes (normal t o t h e paper) t h r o u g h t h e dashed l i n e i n F i g u r e 2. The o b s e r v a t i o n t h a t t h e P o i n c a r e map i s a smooth c u r v e ( F i g u r e 9) r a t h e r t h a n a s c a t t e r o f p o i n t s suggests t h a t t h e process i s d e t e r m i n i s t i c ; however, a s t o c h a s t i c process c o u l d be d e v i s e d t o g i v e t h e c u r v e i n F i g u r e 9. As a second t e s t o f t h e c h a r a c t e r of t h e dynamics, c o n s i d e r a r e t u r n ma d c o n s t r u c t e d f r o m t h e a t t r a c t o r i n F i g u r e 2. L e t X(n) b e t h e v a l u e o f t when an o r b i t i n t h e 20 p o r t r a i t crosses t h e dashed l i n e i n F i g u r e 2, and l e t X(nt1) b e t h e o r d i n a t e t h e n e x t t i m e t h e o r b i t c r o s s e s t h e dashed l i n e . A r e t u r n m p o b t a i n e d by p l o t t i n g t h e o r d e r e d p a i r s [ X ( n ) , X ( n t l ) ] f o r a l l n i s shown i n F i g u r e 10. Again t h e d a t a d e s c r i b e a smooth c u r v e t h a t suggests t h a t t h e system i s d e t e r m i n i s t i c r a t h e r t h a n s t o c h a s t i c : g i v e n any i n i t i a l X(n), a l l subsequent X ( n t k ) a r e determined by t h e map. The s t r o n g e s t evidence we have i n d i c a t i n g t h a t C1 corresponds t o d e t e r m i n i s t i c chaos i s g i v e n by t h e v a l u e of t h e Lyapunov exponent. Lyapunov exponents p r o v i d e a q u a n t i t a t i v e measure o f t h e degree o f chaos o r “ s e n s i t i v e dependence on i n i t i a l c o n d i t i o n s ” f o r n o n l i n e a r systems. A p o s i t i v e exponent means t h a t nearby o r b i t s w i l l e x p o n e n t i a l l y separate; t h e system i s t h e n h i g h l y s e n s i t i v e t o t h e i n i t i a l conditions. On t h e o t h e r hand, i f t h e exponent i s n e g a t i v e , nearby o r b i t s w i l l e x p o n e n t i a l l y converge; t h e system i s t h e n i n s e n s i t i v e t o i n i t i a l C o n d i t i o n s .
-1
-1
1
I
1
I
1
I
1
I
1 I
I
1 I
I
1 I
1 I
IAI
I
I
-
1 I
I
I P
I
I
I \
I
RESIDENCE TIME (hr) F i g u r e 8. The a l t e r n a t i n g sequence o f p e r i o d i c and c h a o t i c regimes i s shown by t h i s graph o f t h e broadband s p e c t r a l n o i s e ( z e r o - f r e q u e n c y i n t e r c e p t s ) measured as a f u n c t i o n o f r e s i d e n c e time. I n t h e p e r i o d i c regimes Pk t h e n o i s e i s i n s t r u m e n t a l (see t h e a t t r a c t o r s i n F i g u r e s 1, 3, and 5 ) , w h i l e i n t h e c h a o t i c regimes c k t h e n o i s e i s s e v e r a l o r d e r s o f magnitude l a r g e r . The c u r v e i s drawn t o g u i d e t h e eye.
COMPLEX DYNAMICS IN A CHEMICAL REACTION
419
For a 1D map t h e Lyapunov exponent has a maximum v a l u e o f I n 2 = 0.69. Robert Shaw has c a l c u l a t e d t h e Lyapunov exponent f o r t h e map shown i n F i g u r e 10 and has o b t a i n e d 0.520.1. The p o s i t i v e Lyapunov exponent g i v e s t h e r a t e o f d i v e r g e n c e o f o r b i t s i n t h e a t t r a c t i n g s h e e t shown i n c r o s s s e c t i o n i n t h e P o i n c a r e map ( F i g u r e 9). I n c o n t r a s t , o r b i t s o f f o f t h e sheet converge e x p o n e n t i a l l y f a s t t o t h e sheet ( t h e Lyapunov exponent i n d i r e c t i o n s normal t o t h e s h e e t i s n e g a t i v e ) . T h e r e f o r e , t h e i n e v i t a b l e s c a t t e r i n o u r e x p e r i m e n t a l d a t a due t o f l u c t u a t i o n s i n t h e environment i s much l e s s n o t i c e a b l e i n t h e P o i n c a r e map (where t h e s c a t t e r i s a l o n g t h e c u r v e i n F i g u r e 9) t h a n i n t h e r e t u r n map ( F i g u r e 10).
++
n
c
cy
+
c U
m
t*
*+ r)
*+'
+*
d
i
/" B ( t +T) F i g u r e 9. A P o i n c a r e map ( w i t h T = 8.8 s) f o r t h e s t r a n g e a t t r a c t o r i n regime C shown i n F i g u r e 2. The map was formed by t h e i n t e r s e c t i o n o f t h e 3D o r b i t s wit!! t h e p l a n e (normal t o t h e paper) p a s s i n g t h r o u g h t h e dashed l i n e i n F i g u r e 2.
J.C. ROUX et al.
420
MODEL
STUDIES
Numerical s t u d i e s o f a mathematical model o f t h e BZ r e a c t i o n have p l a y e d an i m p o r t a n t r o l e i n t h e d e s i g n and guidance o f o u r e x p e r i m e n t a l program. I n these s t u d i e s a f o u r - v a r i a b l e model based on t h e Oregonator o f F i e l d and Noyes [21,22] has been found t o e x h i b i t a v a r i e t y o f phenomena r e l e v a n t t o e x p e r i m e n t a l s t u d i e s i n a s t i r r e d f l o w reactor. These i n c l u d e m u l t i p l e s t a t i o n a r y and o s c i l l a t o r y s t a t e s , h y s t e r e s i s , b u r s t s o f o s c i l l a t i o n , complex p e r i o d i c and n o n p e r i o d i c s t a t e s , and two d i s t i n c t sequences o f t r a n s i t i o n s i n v o l v i n g mixed p e r i o d i c and c h a o t i c s t a t e s . D e t a i l s of t h e model and i t s p r e d i c t i o n s a r e p r e s e n t e d elsewhere [12,14],
F i g u r e 10. A r e t u r n map f o r t h e s t r a n g e a t t r a c t o r i n regime C shown i n F i g u r e 2. The map was formed by p l o t t i n g as o r d e r e d p a i r s [X(n),X(n+l)] t h e successive values X(n) o f t h e o r d i n a t e o f phase space o r b i t s when t h e y c r o s s t h e dashed l i n e i n F i g u r e 2.
COMPLEX DYNAMICS IN A CHEMICAL REACTION
42 1
O f p a r t i c u l a r i n t e r e s t h e r e a r e t h e sequences o f t r a n s i t i o n s which occur as t h e i n v e r s e r e s i d e n c e t i m e i s v a r i e d . The o s c i l l a t i o n s which appear f i r s t i n t h e model as ~ - 1 i n c r e a s e s a r e n e a r l y s i n u s o i d a l . Further increase i n T - ~ reveals a p e r i o d - d o u b l i n g sequence o f t r a n s i t i o n s o f t h e t y p e discussed i n t h i s volume by Feigenbaum [see a l s o Ref. 231. For s t i l l l a r g e r 1 - 1 . a second t y p e o f o s c i l l a t i o n appears i n t h e model s t u d i e s , h a v i n g l a r g e r a m p l i t u d e and l o w e r frequency t h a n t h o s e p r e v i o u s l y found. It i s i n t h i s r e g i o n o f T-1 t h a t t h e p e r i o d i c - c h a o t i c sequence observed i n o u r experiments i s found i n t h e model. The model p r e d i c t i o n s o f t h e p e r i o d i c - c h a o t i c sequence, which were o b t a i n e d p r i o r t o t h e conduct o f t h e experiments, a r e i n e x c e l l e n t q u a l i t a t i v e agreement w i t h t h e experiments [12]. We expect t h e q u a n t i t a t i v e agreement t o improve w i t h parameter v a r i a t i o n and model r e f i n e m e n t now i n progress.
DISCUSS I O N
A s i m i l a r sequence o f a l t e r n a t i n g p e r i o d i c and c h a o t i c regimes has been observed i n experiments by Hudson e t al.[7] and by t h e Bordeaux group [10,24]. However. t h e sequences o f regimes observed i n our experiments [12] and i n p r e v i o u s experiments [7,10,24] have t h e o o s i t e dependence on r e s i d e n c e t i m e T . The e x p e r i m e n t a l c o n d i t i o n s a r e s i m i k c e p t f o r t h e range i n r e s i d e n c e t i m e s s t u d i e d : 0.08 < T < 0.2 h r f o r p r e v i o u s experiments and 0.5 < T > 1 and t h e c o n d i t i o n f o r s t o c h a s t i c i t y expected f o r an imposed f i e l d i s exceeded. s i m u l a t i o n t h e c o n d i t i o n f o r q u a s i l i n e a r theory h - l = s a t is f ied.
Also, throughout t h e
1) and i f ) ( h - l )are s a t i s f i e d d u r i n g t h e s i m u l a t i o n we must conclude t h a t iii) t h e quasigaussian f l u c t u a t i o n assumption i s n o t v a l i d i n t h e sel f - c o n s i s t e n t f i e l d case. This statement i s n o t s u r p r i s i n g i n i t s e l f since we know t h a t t h e s e l f - c o n s i s t e n t Poisson equation i s nonlinear. As a r e s u l t nongaussian e l e c t r i c f i e l d f l u c t u a t i o n s w i l l evolve from i n i t i a l l y gaussian e l e c t r i c f i e l d f l u c t u a t i o n s . SELF-CONSISTENT STATISTICAL THEORIES OF PLASMA TURBULENCE Introduction The understanding o f t h e s e l f - c o n s i s t e n t s t a t i s t i c a l d e s c r i p t i o n o f Vlasov t u r bulence i s i n a new stage o f development today and t h e study o f t h i s problem w i l l c e r t a i n l y be t h e s u b j e c t o f f u t u r e research i n t h e n o n l i n e a r physics o f plasmas. I n t h e p r e s c r i b e d e l e c t r i c f i e l d problem we were concerned w i t h t h e motion o f p a r t i c l e s i n random forces. The p r o p e r t y f o r t h e e l e c t r i c f i e l d o f b e i n g p r e s c r i b e d d i d n o t appear e x p l i c i t l y i n t h e d e r i v a t i o n o f t h e equations o f e v o l u t i o n f o r < f > and . These equations (eq. 3.6 and 3.7) a c t u a l l y reduce t o two Fokker-Planck equations and t h e y can be e a s i l y d e r i v e d by computing t h e f r i c t i o n and d i f f u s i o n c o e f f i c i e n t s corresponding t o t h e motion o f a p a r t i c l e i n t h e random f i e l d . T h i s i s t h e reason why i t has been thought f o r a long t i m e t h a t t h e s t a t i s t i c a l d e s c r i p t i o n o f t h e p a r t i c l e motion i n s e l f - c o n s i s t e n t Vlasov t u r b u lence was t h e same as i n the p r e s c r i b e d e l e c t r i c f i e l d case. We have, however, t o keep i n mind t h a t a basic v a l i d i t y c o n d i t i o n f o r q u a s i l i n e a r t h e o r y as w e l l as f o r t h e Fokker-Planck equation i s t h e quasigaussian p r o p e r t y o f t h e e l e c t r i c f i e l d . I n t h e s t o c h a s t i c a c c e l e r a t i o n problem t h i s p r o p e r t y was always imposed, e.g., through random phases, b u t i n t h e Vlasov-Poisson case i t i s necessary t o check t h e v a l i d i t y o f t h i s hypothesis.
-
-
We consider e x p l i c i t l y a s e l f - c o n s i s t e n t Vlasov t u r b u l e n c e s i t u a t i o n f o r which t h e c o n d i t i o n s 3.31 and 3 . 3 i i a r e s a t i s f i e d . We assume a l s o t h e usual weak turbulence hypothesis t h a t assigns N, we1 1-defined frequencies wn(k) t o a wave number k. These frequencies have t o be computed s e l f - c o n s i s t e n t l y through a n o n l i n e a r d i s p e r s i o n r e l a t i o n . For s i m p l i c i t y we w i l l t a k e N, = 1. A l l t h e q u a n t i t i e s such as k,
wb, t c are now computed through t h e instantaneous values
o f t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d . I n e q u a l i t y 3.3i) i s e a s i l y s a t i s f i e d i n unstable s i t u a t i o n s . I n e q u a l i t y 3 . 3 i i ) i s u s u a l l y considered as t h e v a l i d i t y c o n d i t i o n o f q u a s i l i n e a r t h e o r y and i s f u l f i l l e d , f o r instance, i n t h e warm beam-plasma i n s t a b i l i t y . We d e f i n e as quasigaussian t h e o r i e s t h e s t a t i s t i c a l d e s c r i p t i o n s o f Vlasov t u r bulence which assume a p r i o r i t h a t t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d has quasigaussian p r o p e r t i e s . I n t h i s s e c t i o n we w i l l f i r s t p r e s e n t t h e usual argument given f o r j u s t i f y i n g t h i s assumption. We w i l l then show t h a t t h e quasigaussian t h e o r i e s are s e l f - c o n t r a d i c t o r y i n t h e regimes where t h e n o n l i n e a r i t y p l a y s a r o l e . The reason f o r discrepancy w i l l be e x h i b i t e d through a p e r t u r b a t i v e c a l culation: the s t a t i s t i c a l properties o f the self-consistent e l e c t r i c f i e l d
TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS
are not characterized by a unique c o r r e l a t i o n l e n g t h stochastic d e c o r r e l a t i o n l e n g t h Rh = vth.
449
= l / A k b u t a l s o by the
We close t h i s Section by describing some o f the features o f the d i r e c t i n t e r a c t i o n approximation ( D I A ) which i s a renormalized theory which takes i n t o account c e r t a i n nongaussian e f f e c t s . The quasi gaussi an hypothesis I n the e a r l i e r d e r i v a t i o n s o f quasi1 i n e a r theory the random-phase assumption was usually i m p l i c i t l y made. z z 9 z 3 The underlying reason has been made e x p l i c i t by OlNei124 as follows: Let us assume on physical grounds t h a t w e l l above t h e stochastic threshold the e l e c t r i c f i e l d i s i t s e l f t u r b u l e n t and i s characterized by an a u t o c o r r e l a t i o n I n a f i n i t e box o f length L, the F o u r i e r components o f the e l e c t r i c length Qeff. f i e l d ar& defined as:
1 L
Ek(t) =
dx E(x,t) exp-ikx
I f we now d i v i d e t h e i n t e r v a l [O,L]
w i t h k = n2n
eff i n t o segments o f l e n g t h go where gc
, namely t h e q u a s i l i n e a r equation 3.6 and t h e Dupree equation 3.7. The f r i c t i o n and d i f f u s i o n c o e f f i c i e n t s i n v o l v e t h e s p e c t r a l d e n s i t y < l E k l 2>. 2 2 ) The spectrum < I E k l > i s then computed by w r i t i n g Poisson's equation as: k2 = (471)2q2 Jdvdv' c 6 f 6 f ' > k
(5.1)
We thus o b t a i n a closed s e t o f t h r e e equations f o r t h e t h r e e unknowns < f > , < 6 f
2
6 f ' > , and .
However, i t i s necessary t o go beyond t h i s l e v e l i n o r d e r t o
compute t h e i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n s o f t h e e l e c t r i c f i e l d o f h i g h e r order than 2. This scheme would be c o n s i s t e n t o n l y i f these c o r r e l a t i o n funct i o n s s a t i s f y t h e d e f i n i t i o n o f a quasigaussian process given i n 52. An i n d i r e c t demonstration o f t h e i n c o n s i s t e n c y o f t h i s scheme i n t h e regime (k2D)1/3 >> y f a L * has been given by Adam, Lava1 and P e ~ m e . They ~ ~ solved t h e Dupree equation 3.7 and by i n s e r t i n g i t s s o l u t i o n i n t o Poisson's equation 5.1; they found a growth r a t e d i f f e r e n t from t h e q u a s i l i n e a r p r e d i c t i o n . This r e s u l t , combined w i t h q u a s i l i n e a r equation 3.6 does n o t a l l o w energy conservation. These authors conclude t h a t t h e q u a s i l i n e a r theory i s i n c o n s i s t e n t i n one dimension and i n t h e regime ( k DQ'L')1/3
>> y f e L * and showed t h a t t h e c o n t r i b u t i o n o f t h e 4-
point irreducible correlation function i s not negligible, i n contradiction t o t h e quasigaussian hypothesis. Even though t h e quasigaussian scheme appears t o be i n v a l i d , t h e s o l u t i o n o f t h e Dupree equation 3.7 e x h i b i t s c e r t a i n f e a t u r e s worth b e i n g mentioned. I n t h e
l i m i t o f i n e q u a l i t i e s 3 . 3 i ) and ii).and f o r Ik(v+) v-I (k2DQ*L.)1/3),
and Ix-/v+I
]
The l a s t term o f t h e R.H.S. o f Eq. 5.2 i s c a l l e d t h e mode-coupling term s i n c e i t couples t o g e t h e r t h e k and k-k(v+), k + k(v+) F o u r i e r components o f < 6 f 6 f ' > . The r a t i o R = (k2DQ*L')1/3/yf*L* t u r n s o u t t o be a measure o f t h e n o n l i n e a r i t y and can be considered as a Reynolds number. For R > 1 t h e mode c o u p l i n g terms can be shown t o be o f t h e same order o f magnitude as t h e o t h e r and generate terms. The mode c o u p l i n g terms g i v e a c o n t r i b u t i o n f o r v - v ' = w k / k
TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS
451
harmonics coupled together through t h e resonant p a r t i c l e s . As a r e s u l t , t h e c o r r e l a t i o n f u n c t i o n c 6 f 6 f ' > i s found t o have t h e f o l l o w i n g convenient harmonic expansion
= cgn(v-)
exp
n k(v+)(x-x')
] [-inw exp
Ik(v+)l (t-t')]
(5.3)
I n t h e regime R >> 1, where t h e n o n l i n e a r i t y p l a y s a r o l e , t h e c o r r e l a t i o n funct i o n i s thus the sum o f an i n f i n i t e number o f harmonics coupled together. When i n s e r t e d i n t o Poisson equations t h e c o n t r i b u t i o n coming from t h i s harmonic coupl i n g appears as a source term r a d i a t i n g coherently. We can, t h e r e f o r e , say t h a t t h e n o n l i n e a r i t y r e i n f o r c e s t h e coherence o f t h e s o l u t i o n . As a consequence, Adam, Laval, and Pesme indeed have found an increase o f t h e growth r a t e compared t o the q u a s i l i n e a r r e s u l t . This f e a t u r e i s generalized i n a s e l f - c o n s i s t e n t way i n the D I A . The Dupree "clump theory" l i k e w i s e p r e d i c t s an enhanced r a d i a t i o n due t o macroscopic g r a n u l a t i o n o f t h e phase space d e n s i t y a s o r t o f enhanced discreteness noise due o n l y t o n o n l i n e a r processes i n t h e continous Vlasov approximation. The Dupree t h e o r y a p p l i e s t o a "strong" turbulence l i m i t i n which no d e f i n i t e normal mode frequency e x i s t s f o r a given k. Therefore k(v) = wk(,,)/v i s not defined i n t h i s l i m i t .
-
The r e s u l t s o f Adam, Laval and Pesme apply i n t h e l i m i t o f q u a s i l i n e a r o r d e r i n g (eq. 3 . 3 i and 3 . 3 i i ) when yk/wk 1 2 3 1 On s u b s t i t u t i o n o f t h e expansion (5.10) f o r each Ek we f i n d f i r s t o f a l l t h e usual terms <E19 E-kl (l)
E-$ ( l ) E-k3 ( l ) > which, because o f t h e gaussian assumption
f o r E ( l ) can be c a s t i n t o a sum o f <E ( l ) ~ ( l ) > < E ( ~ ) E ( ~ ) .> L e t us now compute t h e nongaussian c o n t r i b u t i o n s .
A t t h e lowest order we o b t a i n t h e sum o f f o u r
terms l i k e <E(3)E(1)E(1) E ( l ) > w i t h t k m = 0. The terms E(2) a r e indeed s m a l l e r kj k kR s i n c e E(u,k) i n Eq. 5.11a has t o be taken a t frequency w = 0 by a f a c t o r y /w k Pe o r f 20 and cannot be made small. Pe From Eq. 5.11b we f i n d
ki
453
TURBULENCE AND SELF-CONSISTENT FIELDS IN PLASMAS
where Qk = wk + i y k and 2Q = R k l + Rkll + Rk-kl-kll expression i n t o <E(3)E(1)E(1)E(1)>
.
After substitution o f t h i s
and use o f t h e gaussian assumption f o r E (1) ,
we o b t a i n t h e f o l l o w i n g expression:
exp- i w
( tl- t2) exp- i w k 2 ( tl- t 3) exp- iwk3( tl- t4) kl
W
W
(2kl+k2+k3)
3 -2iu
+ permutations (kl,k2,
k3)
, with u = y
(5.13)
kl The Here t h e m a t r i x element V has been e x p l i c i t l y evaluated u s i n g 5.8b. d i e l e c t r i c f u n c t i o n has been taken t o be on resonance, i , e . , wkl + wk2 + wk3 which i m p l i e s t h a t frequencies and wave numbers have n o t a l l t h e same +k +k kl2 3 sign. We have a l s o assumed t h a t t h e d i s p e r s i o n o f t h e r o o t s wk i s weak so
2:w
t h a t e(m)
N
(E)
.
2 i ~ t . ~ .
The expression 5.13 enables us t o compute t h e
contribution o f the irreducible 4-point correlation function i n the evaluation o f the mode amplitudes. A t the lowest order we have
which leads t o t h e r e s u l t : (5.15)
D. PESME, D.F. DUBOlS
454
The m o d i f i c a t i o n o f t h e spectrum as g i v e n by t h e expression i n s i d e t h e b r a c k e t of Eq. 5.15 shows again t h a t t h e mode c o u p l i n g terms l e a d t o an increase o f t h e energy t r a n s f e r from resonant p a r t i c l e s towards waves. T h i s r e s u l t i s s t r i c t l y v a l i d o n l y f o r y!'L'>>
( k 2 D ) , l l 3 b u t i t c l e a r l y shows t h a t t h e i r r e d u c i b l e
4 - p o i n t c o r r e l a t i o n f u n c t i o n g i v e s a n o n - n e g l i g i b l e c o n t r i b u t i o n as soon as (k2D)1'3
2
y i ' L. which i n v a l i d a t e s t h e quasigaussian assumption.
By F o u r i e r transform o f Eq. 5.13 we o b t a i n t h e expression i n r e a l space o f t h e 4-point correlation function <E(3)(x,,tl)
E(l)(x2,t2)
E(')(x3,t3)
E(')(x4,t4)>
=
(5.16) expi kl(xl-x2)
expi k2(x1-x3)
expi k3(x1-x4)
I f the expression i n t h e c u r l y brackets o f t h e R.H.S. o f Eq. 5.13 was a smooth f u n c t i o n o f kl, kz and k3, then t h i s c o r r e l a t i o n f u n c t i o n would be e s s e n t i a l l y a product o f t h r e e two-point c o r r e l a t i o n f u n c t i o n s o f t h e form
$([,r) = 2 exp-i[wkr-k[l For c o r r e l a t i o n s along t h e unperturbed o r b i t s l a t i o n time
T~
5
= v t , t h i s has t h e usual c o r r e -
= I A k ( v - V g ) l - l discussed p r e v i o u s l y .
On t h e c o n t r a r y , however, the expression i n c u r l y brackets o f t h e R.H.S. o f Eq. 5.13 i s a peaked f u n c t i o n f o r vlkl+kzl
5 u.
2
I n t h e regime R = ( k 0)
1/3 /yk
>> 1, Eq. 5.15 shows t h a t t h g s e r i e s expansion i n power o f E ( l ) i s n o t convergFnt. 9
The f r e e propagator g
may then be replaced by t h e renormalized propagator [(k21f*L')~]3
m
gw,k
= (-q/m$
drexp
i(w-kv)r.
exp
-
3 av
(5.17)
0
I n the e s t i m a t i o n o f order o f magnitudes i t i s s u f f i c i e n t t o r e p l a c e u = y i m L . by v = ( k DQ'L*)1/3.
The nongaussian c o r r e l a t i o n f u n c t i o n i s , t h e r e f o r e , charac-
t e r i z e d by t h e time (k2DQ*L*)-1'3
which i s o f t h e o r d e r o f t h e i n v e r s e o f Kolmo-
gorov entropy hQ. L' = 0.36( k2DQ*L'
)ll3 encountered e a r l i e r .
Since ( k2D)-1'3
>>
tC
we see t h a t s e l f - c o n s i stency i n t r o d u c e s a tendency toward s e l f - o r g a n i z a t i o n o f t h e t u r b u l e n t sys tem.
A d e t a i l e d diagrammatic expansion leads t o t h e conclusion t h a t i n t h e regime (k2D)1'3
>> y k a l l t h e i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n s c o n t r i b u t e a t t h e same
order o f magnitude as the q u a s i l i n e a r term. T h i s r e s u l t i n v a l i d a t e s t h e t h e o r i e s which c o n s i s t o f a simple r e n o r m a l i z a t i o n o f t h e propagator such as developed by
TURBULENCE AND SELF-CONSISTENT FIELDS IN PLASMAS
Rudakov and Tsytovich, Choi and H o r t ~ n . ~In~ the regime defined by ii) they lead to the quasilinear result since they don't contain the ling term of Eq. 5.2. It can be shownz8 that a partial summation of irreducible correlation functions leads to the Dupree equation 3.7. we may summarize the different levels of renormalization: 1.
455
3.3i) and mode coupthe neglected Therefore,
WEAK TURBULENCE THEORY - This makes the usual quasilinear prediction but suffers from the defect of secular divergences arising in the neglected irreducible correlation functions. These secularities are removed by renormalization of the propagator which leads to:
2.
SIMPLY RENORMALIZED THEORIES - These again make the usual quasilinear prediction but still suffer from non-negligible contributions from neglected irreducible correlation functions. A partial summation of the irreducible correlation functions leads to:
3.
DUPREE'S THEORY - This theory predicts a zero order modification of the quasilinear result coming from mode coupling terms. However, this theory considers only part of t h e irreducible correlation functions and is not properly self-consistent.
SELF-CONSISTENT STATISTICAL DESCRIPTIONS AND THE DIA It appears from the considerations above that most familiar statistical theories of Vlasov turbulence until recently, have been quasigaussian theories in the sense that they explicitly or implicitly make the assumption of quasigaussian electric field correlations. A self-consistent statistical description of Vlasov turbulence must take into account the relationship between perturbations in the averaged phase space distribution = and perturbations in the mean electric field <El> = <E(xltl)> as well as the relationship between the These fluctuations in these quantities: 6fl = fl- and 6E1 = EI-<E1>. relationships are, of course, imposed by Poisson's eqn. (4.1). For the discussion here we follow closely the results of reference 29. From the Vlasov eqn. (32) the well-known equations for the mean distribution function is easily found: r
(5.18)
This couples to the fluctuation correlation function . The averaged Poisson equation is (5.19) V1 * <E1> = 4nq J dvl - 4nqN We have added an artificial phase space source density q1 = q(xlxltl) which we can consider as a probe of the system. Suppose for r) = 0 the system of equations has a well-defined solution , <El>; we then add a local phase space source perturbation 6r) which changes to + 6 and <E1> to <E1> + &<El>. The increments 6 are functional differentials such as those used in the calculus of variations of continuous fields. A relationship between the increments 6, 6<E> and 6<rp is readily found from eq. (5.18) to be of the form
0. PESME, 0.F. DUBOIS
456
G;:
6
a
+
r126<E2> = 6ql v1
(5.20)
By t a k i n g t h e f u n c t i o n a l d i f f e r e n t i a l o f eqn. (5.18) c o e f f i c i e n t s as
i t i s easy t o i d e n t i f y t h e
(5.21)
(5.22) To s i m p l i f y w r i t i n g t h e equations we a r e u s i n g a summation convention where, f o r example, :;G
6
Jdv2Jdx2Jdt2 G;:
6, etc. and 6(1-2)
6(vl-v2)
6(x1-x2)6 (tl-t2). The l a s t terms on t h e r i g h t hand s i d e o f eqns. (5.21) and (5.22) c o n t a i n f u n c t i o n a l d e r i v a t i v e s o f t h e c o r r e l a t i o n f u n c t i o n w i t h respect t o changes i n t h e mean values and <E>, r e s p e c t i v e l y . These funct i o n a l d e r i v a t i v e s completely describe t h e renormal i z a t i o n o f t h e response funct i o n s G12 and rI2. The response f u n c t i o n G12
i s c a l l e d t h e " q u a s i p a r t i c l e propagator"; w i t h o u t
r e n o r m a l i z a t i o n i t i s c l e a r from eqn. (5.21)
t h a t G12
simply describes p a r t i c l e
propagation along unperturbed o r b i t s ( i n t h e mean f i e l d
<El> which i s zero i n
a homogeneous system.) The r e n o r m a l i z a t i o n term takes i n t o account t h e s e l f c o n s i s t e n t response o f the f l u c t u a t i n g t u r b u l e n t "background" through which t h e p a r t i c l e propagates. I n an imposed f l u c t u a t i o n l i m i t t h i s propagator reduces t o t h e f a m i l i a r Dupree renormalized propagator g i v e n i n eqn. 3.9. The response f u n c t i o n
aVlrl2
i s c a l l e d t h e " q u a s i p a r t i c l e p o t e n t i a l source."
I t takes i n t o account t h e e f f e c t i v e source d e n s i t y 6q which must a r i s e when t h e r e i s a change i n t h e mean e l e c t r o s t a t i c f i e l d 6<E> i n o r d e r t o keep < f > constant. Without r e n o r m a l i z a t i o n , t h i s term i s simply p r o p o r t i o n a l t o a . v1 The r e n o r m a l i z a t i o n o f t h i s q u a n t i t y i s n o t f a m i l i a r b u t was a c t u a l l y considered
by K a d o m ~ t e vi ~n ~Section I11 o f h i s 1965 book (equation 21a). (Kadomstevls renormalized theory has some elements i n common w i t h t h e D I A , b u t d i f f e r s i n several important respects r e l a t e d t o s e l f - c o n s i s t e n c y . ) We next consider t h e f l u c t u a t i o n s . By s u b t r a c t i n g eq. (5.18) from eqn. (3.2) and n o t i n g t h e d e f i n i t i o n s (5.21) and (5.22) we can d e r i v e t h e exact equation f o r bf,
6fl
=
6f;nCA
+ ;q/m)
G12 8,
[6f26E2-
2
-
(5.23) 6 6f3
6
-
6
D. PESME. D.F. DUBOIS
We have i n d i c a t e d t h a t 6fl
457
can be separated i n t o a coherent p a r t p r o p o r t i o n a l t o
t h e e l e c t r o s t a t i c f l u c t u a t i o n 6E and an incoherent p a r t .
I n a s p a t i a l l y homo-
geneous plasma i t i s easy t o see t h a t t h e F o u r i e r component 6 f p h i s p r o p o r t i o n a l t o 6Ek.
For t h e incoherent p a r t terms l i k e 6f16E1
such as
i n v o l v e F o u r i e r convolutions
6fk-k16Ekl ( w i t h 6Ekz0=0) and thus i n v o l v e s t h e c o u p l i n g o f d i f f e r e n t
F o u r i e r modes. I f t h e expression (5.23) i s i n s e r t e d i n t o Poisson's equation Vl* GE1=4nqJdvl
[ 6 f i o h + 6 f i n c ] one can rearrange t h e coherent c o n t r i b u t i o n t o w r i t e
V1 where
-
E~~
cI2
6L2 = 4nqJdv16fl
inc
(5.24)
i s t h e d i e l e c t r i c operator 29
The coherent p a r t has produced t h e s u s c e p t i b i l i t y f u n c t i o n i n t h e d i e l e c t r i c operator. For a homogeneous s t a t i o n a r y system, t h e F o u r i e r c o e f f i c i e n t o f E can be w r i t t e n
2
2
E(k,w) = 1 + (4nq /mk ) Jdv3 Jdv
i k * a r (k,w) v4 v4
(k,w)
Gv
4
3 4
We see from eqn. (5.24) t h a t t h e charge d e n s i t y associated w i t h 6finC a c t s as an incoherent source f o r t h e e l e c t r o s t a t i c f l u c t u a t i o n . For example, t h e twop o i n t e l e c t r o s t a t i c c o r r e l a t i o n f u n c t i o n can be w r i t t e n as
i n terms o f t h e inverse d i e l e c t r i c operator.
Or
n
(5.26b) f o r t h e s p e c t r a l i n t e n s i t y i n a homogeneous, s t a t i o n a r y case. The two-point c o r r e l a t i o n f u n c t i o n f o r t h e phase space d i s t r i b u t i o n can l i k e w i s e be w r i t t e n as c6f16fll
> = (q 2 /m 2 1 G 1 2 ~ v 2 ~ 2 3 G 1 1 2 1 ~ v ; ~ 2 ~ 3 ~ ~ ~ E 3 ~ E 3 ~ >
where P12
6(1-2)P1,2,-P126~l'-21)
1
which can be seen f r o m t h e d e f i n i t i o n o f 6finc, eq. 5.23, t o i n v o l v e 4 - p o i n t and higher c o r r e l a t i o n f u n c t i o n s . The vanishing o f t h e cross c o r r e l a t i o n terms i s made e x p l i c i t l y i n t h e theory o f Dupree ( r e f . 2 eqn. (21)). The equations w r i t t e n i n t h i s s e c t i o n up t o t h i s p o i n t a r e f o r m a l l y exact. We inc6f inc, see t h a t the system i s n o t closed since t h e c o r r e l a t i o n f u n c t i o n > > 1f o r propagator i n eq. 5.1. a one-dimensional one-component plasma we have been a b l e t o show t h a t t h e s e l f c o n s i s t e n t f l u c t u a t i o n terms cannot be neglected. A s i m i l a r equation r e s u l t s f o r the renormalized Tv(k,u) which we w i l l n o t discuss here. F u r t h e r r e s u l t s a r e given i n reference 29. I f the D I A approximation o f eqn. (5.30) f o r i s i n s e r t e d i n t o eq. (5.27) we o b t a i n t h e f o l l o w i n g equation f o r t h e two-point c o r r e l a t i o n f u n c t i o n
460
D. PESME, D.F. DUBOIS
r
s . c. A
= (q2/m2) G12~v;23G1121~vh~2131
(5.34)
When the terms labelled S.C. are dropped in this expression and only the imposed field renormalization terms in G are retained, this equation can be shown to reduce to the Dupree equation for >lwhere n i s the plasma d e n s i t y and AD t h e Debye length.
15. Sturrock, P. A.,
Phys. Rev.
141, 186
(1966).
16. Thomson, J. J., and Benford, G . , J. Math. Phys. 17. Dupree, T. H., Phys. F l u i d s
2 , 334
14,531 (1973).
(1972).
18. Pesme, D., and B r i s s e t , A., t o be published. 19. Adam, J-C., Laval, G., and Pesme, D., i n Proceedings o f t h e I n t e r n a t i o n a l Conference on Plasma Physics, Nagoya, Japan, 1980. 20. O ' N e i l , T. M., and Malmberg, J. H., Phys. F l u i d s
11, 1754 (1968).
21. Bezzerides, B., Gitomer, S. J., and Forslund, D. J., Phys. Rev. L e t t . 44, 651 (1980). 22. Drummond, W. E . , and Pines, D., Nucl. Fusion, Suppl. P a r t 3, 1049 (1962). Vedenov, A. A., Velikhov, E . P. and Sagdeev, R. Z . , Nucl. Fusion, Suppl. P a r t 2, 465 (1962).
z,
1816 (1964). Zaslavsky, 23. Aamodt, R. E . , and Drummond, W. E . , Phys. F l u i d s G. M., and C h i r i k o v , B. V., Usp. F i z . Nauk 2,195 (1972) [Sov. Phys. Usp. 2 , 549 (1972)l. 24. O ' N e i l , T. M., Phys. Rev. L e t t . 25. Adam, J. C . , Laval, G.,
33, 73
and Pesme, D.,
26. Laval, G., and Pesme, D., Phys. L e t t e r s
(1974).
43,
Phys. Rev. L e t t .
E, 266
1671 (1979).
(1980).
27. Rudakov, L. I., and Tsytovich, V. N., Plasma Phys. 13, 213 (1971). D., and Horton, W. J r . , Phys. F l u i d s 11, 2048 (1974r 28. Adam, J. C . , Laval, G., (1980).
Choi,
and Pesme, D . , Suppl. J. de Physique 4 l , C3-383,
29. DuBois, D. F., Phys. Rev. A
2,865
(1981).
30. Kadomtsev, B. B., Plasma Turbulence (Academic Press, New York, 1965). 31. Kraichnan, R. H., J. Math. Phys.
2, 124 (1961).
32. Orszag, S. A., and Kraichnan, R. H., Phys. F l u i d s 33. DuBois, D. F . , and Espedal, M., 34. Krommes, J. A.,
and Kleva, R. G.,
Plasma Phys.
20, 1209
Phys. F l u i d s
35. Krommes, J. A., and Similon, P., Phys. F l u i d s 36. M a r t i n , P. C.,
lo, 1720
Siggia, E. D., and Rose, H. A.,
(1967).
(1978).
2,2168 (1979).
23, 1553
(1980).
Phys. Rev. A
8,
423 (1973).
464
0. PESME. D.F. DUBOIS
appears to be only a restriction on the level 37. The condition of electric field fluctuations whereas the condition for smallness of the vertex corrections to the D I A must also involve some restriction on . The fluctuation level is asymptotically in time proportional to and both are bounded. Convergence would seem possible for sufficiently small and smooth initial fluctuations < 6 f2>.
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (edc.) @ North-Holland Publishing Company, 1982
465
SEW-FOCUSMG TENDENCIES OF 'RIE NaJLMEAR SCHF~DINGERAND ZAKHAW~VEQOATIONS HARVEY A. RC6E AND D. F. WBOIS
IDS A l m s National Laboratory
P. 0. Box 1663 LOS AlamoS, NM
07545
The Zakharov equations and an associated nonlinear SchrUdinger (NLS) equation are models of long wavelength, interacting Langmuir waves. Although there seems to be a parameter regime i n which these models consist of weakly interacting waves, t h i s regime is pre-empted by self-focusing; which, i n the case of the NLS equation, can generate a singularity i n a f i n i t e time. 'Ihe implications for a s t a t i s t i c a l theory are discussed.
The problem of Langmuir turbulence i n plasnas is one of the conceptually s h p l e s t and most studied problems in nonlinear plasma physics.1,2,3 'Ihe physical situation is that of a c-letely ionized, urmagnetized plasm w i t h electron temperature mch higher than the ion temperature where the linear collective modes of the system are longitudinal Langmuir modes and ion-sound modes and transverse electromagnetic modes. A useful &el which we w i l l use here and which describes many of the interesting non-linear features of the system is the model of Zakharov4 which explicitly separates the f a s t and slow time scales of the system. Weak turbulence theory is built on the linear collective modes of the system. It has been studied since the early 1 9 6 0 ' ~ .It~ is well-known1r2 that t h i s theory predicts a transfer of Langmuir wave intensity to low wave nunbers by a cascade of three wave processes of the form 8 8' + s, i.e., the three wave process which converts a Langrmir wave ( 8 ) w i t h wave nunber into a Langmuir k' and vice wave ( 8 ' ) w i t h nunber k' and an ion-sound wave w i t h wave nunber versa. 'Ihe frequency matching condition for these processes is the well-knm one
*
-
H.A. ROSE, D.F. DUBOIS
(1.1)
112
Langmuir frequency and cs = (Tdmj is the ion-sound speed where we express temperature in energy u n i t s . If Langmuir energy is injected a t some f i n i t e k >> (1/3) ( m d m i ) 1/2km = k* then Langmuir i n t e n s i t y cascades t o lower k ' < k u n t i l i.) the intensity is low enough for some f i n i t e $in > k* so that the mode hin is stabilized by linear collisional damping or ii.) unstable modes for k < k* are excited for which further decay by t h e process (1.1) are not allowed. "his piling up of Langmuir i n t e n s i t y a t low k has been called the "Langmuir condensate." The question then arises as t o how t h e energy i n t h i s condensate is dissipated. A t t e m p t s have been made to invoke weak turbulence processes such as wave-wave scattering, i.e., Ll + L2 = a3 + i4. However, as we w i l l show here explicitly the weak turbulence scheme breaks down for low wavenumbers k < k*. is that coherent or incoherent Langmuir oscillations a t long wavelengths are unstable t o modulational instability i n which regions of excess Langmuir oscillations deplete the local electron and ion density by action of t h e ponderomtive force, and the density depletion through its effect on t h e Langmuir wave phase speed, a t t r a c t s the trajectories of nearby waves. *is instability involves the competition of linear dispersive and dissipative terms w i t h the nonlinear "self-focusing" effect of the ponderomotive force; t h i s is a case of effective "Reynold's number" of order unity. The linear stage of the modulational instability does not directly transfer energy t o the high k region where Landau damping can be an effective dissipation process. It
The conceptual
picture of the nonlinear evolution of the modulational instability derives from a combination of computer experiments and special analytic solutions.' In the case of one dimension, it a w a r s 5 t 6 that the modulational instability can develop into a set of spiky structures which may be loosely related t o the known soliton i n i t i a l value solutions of the undriven, When the system is undamped Zakharov or nonlinear SchrUdinger equations. driven, e.g., by a uniform pump, there is conflicting evidence5 that the& spiky I n two or three structures may, i n fact, undergo a singular collapse. dimensions it is'well-known from t h e pioneering work of Zakharov4 and m r e recent work by Goldman and Nicholson6 that a localized Langmuir wavepacket w i l l tend t o collapse i f the electrostatic energy i n the packet W s a t i s f i e s
SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EOUATION
W >
nTe
467
dk
f(-)2
kD
where Ak is the width of the spectrum in wave nunber and f is a constant of order unity. The ultimate evolution of the collapsing wave packet presumably involves strong wave particle interaction i n which the energy of the packet is given up t o very energetic electrons. In two or three dimensions it appears that direct collapse can be initiated without t h e intermediate stage of modulat ional instability
.
is conjectured that the nonlinear stage of the modulational instability results i n snall scale structures of the order of several Debye lengths which It
dissipate energy by wave particle interaction. Galeev, e t al.' argue that snall scale density fluctuations are created by sound radiation from supersonically collapsing wave packets; these snall scale density fluctuations provide an enhanced absorption (analogous to inverse Bremsstrahlung) of longer wavelength Langmuir waves which terminates the collapse. In our research we have attempted t o formulate a systematic s t a t i s t i c a l theory of Langmuir turbulence using the "strong" turbulence or renormalization methods of Kraichnansrg which were originally applied to incompressible fluid turbulence. The simplest renormalized theory, the so-called Direct Interaction Approximation (DIA) has been successful i n describing moderate Reynold's number fluid turbulence so it seemed reasonable t o apply it t o the Langmuir turbulence problem w i t h "Reynold's nunber" of order unity.
In applying these methods t o the Langrmir turbulence problem one is innnediately faced with problems not encountered i n the u s u a l applications t o isotropic, homogeneous Navier-Stokes turbulence. Among these problems is the existence of weakly damped, oscillatory modes corresponding i n the l i m i t of large k t o the unperturbed (Langmuir and ion sound) modes of the system and the existence of nonlinear normal modes due to self focusing, i.e., modulational instability which goes over t o transient self trapping i n the steady state. These problems corrplicate both the nunerical solution of the DIA equations and further analytic approximations such as the "pole approximation" discussed below.
H.A. ROSE, D.F. DUBOIS
468
2.
THE
ZAKH Aw ) v
EQUATIONS
L e t $ ( x , t ) be the (complex valued) low frequency envelope of potential. %en the Zakharov equations a r e 4 1 1 r 2
the e l e c t r o s t a t i c
where $*(x,t) is the c q l e x conjugate of $ ( x , t ) . are using a dimensionless set of u n i t s here’ i n which distances a r e measured is the electron Debye length, t h e is i n u n i t s of 32(mi/nQ 1/2X where X 3De De measured in units Of T(mi/%)upe-l where is the electron plasma frequency, We
WP,
e l e c t r i c f i e l d strengths a r e measured i n u n i t s of (me/mi) 1/2(16nno~J3) where no is the average ion or electron density and Te is t h e electron temperature and the density fluctuation n is measured i n u n i t s of %(4m$3mi). Phenommological Langmuir wave damping ve and ion wave damping u i ( i n the above units) have been added to the equations; we can be chosen to be nonlocal i n real space to mimic Landau damping i f desired. We w i l l not discuss the conditions for v a l i d i t y of these equations here since they have been given elsewhere.’ Cur goal is to study the s t a t i s t i c a l theory of these model nonlinear equations; our hope is t h a t further physical refinements can be added l a t e r in a straightforward way. They can be c a s t i n t o Hamiltonian form’ when the d i s s i p a t i v e coefficients, ve and wi, vanish by formally regarding $ and $* as independent f i e l d s arid by the hydrodynamic flux p o t e n t i a l u:
introducing
(2.3a)
(2.3b)
-an =
at
0%
(2.3~)
SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EQUATIONS
469
(2.3d)
Define the Hamiltonian H a s the volume integral of t h e density h.
+ '[2
h = V2$V2$*
(_OU)~
+
n2]
+ n_O$*_O$* .
Then equations ( 2 ) a r e equivalent, when ue = u i = 0, t o the canonical system.
-an = -
u -a =
6H/6u
at
at
6H/6n
In addition t o H , equations (2.5) conserve: the plasna nunber N,
N = I_O$*_O$*dz= IE*E*dz
(2.6)
where E is the corrplex e l e c t r i c f i e l d envelope,
P, =
J{+
i(q,,aE;Z/ax,,,
- Debye radius) the plasna could evolve t o a state which is strongly collapsed relative t o the i n i t i a l conditions, but still be i n a physical regime where the NLS equation is still valid.
Of
As collapse continues i n time t h e density w i l l not keep up w i t h the rapid variations of E, and the adiabatic relation (3.3) m u s t be replaced by the Zakharov equations. These too w i l l become invalid as collapse intensifies, and t o properly account for the interaction between fields and particles the Vlasov equation is needed.
SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EOUATION
475
i n the s t r i c t sense that (6rI2 + 0 , then the singularity would be a t a p i n t , i.e., a set w i t h dimension zero. Even though t h i s many not be realized, it still m u s t be true that whatever the nature of the singularity, infinite field strengths are present because otherwise the dispersive term i n (3.1) would dominate t h e nonlinear term, and the resulting linear SchrUdinger equation is w e l l behaved. Since N is conserved, t h e regions where IEl is becoming stronger and stronger must f i l l a snaller and snaller fraction of t h e spacial volume, and t h i s is what we mean by intermittency. If collapse did occur for solutions of the NLS equation, -
i n the n e x t section can accurately describe a strongly intermittent solution of the N L S equation. However, there are physically relevant parameters r e g h e s i n which collapse is cut off by dissipative effects so that the intermittency of the solutions to the highly idealized NLS equation is not actually realized. S t i l l , there is a tendency to self-focus, so that a theory that is more comprehensive than weak turbulence theory is called for. I t is unlikely that the s t a t i s t i c a l theories of the k i n d
6.
discussed
STATISTICAL THEORY OF THE ZAKHAKN AND NLS EQUATIorUS
Since the NLS equation is a long wavelength l i m i t of the Zakharov equations, it is possible t o develop a s t a t i s t i c a l theory of the former by taking the long wavelength limit of the l a t t e r . Conversely since, by solving (2.2) for n in terms of IVJ1I2, and substituting i n (2.1) , the Zakharov equation for J, h a s the form of a generalized NLS equation, a s t a t i s t i c a l theory of a NLS equation could be used for the Zakharov equations. has been found that t h e DIA12 as applied t o the Zakharov equations is essentially equivalent13 t o the Screened Potential Approximation for the NLS equation ( i - e . , the classical field theory analog of the quantun mechanical plasna approximation that is needed t o understand electrodynamic screening of charged particles). The propagation of infinitessimal disturbances i n t h e density field n, d,of (2.2) (disturbances of I E I 2 i n 3.1) is modified by polarization-like effects, which for the Zakharov equation DIA, amounts to replacing k 2 , t h e fourier transform of (-a2/ax2) i n the inverse of the sound wave propagator, It
H.A. ROSE, D.F. DUBOlS
476
by k2
+
k20(k,w)
where o(k,w) is t h e transform of o ( x , t ) ,
-
o ( x , t ) = i[R$J(xt)C*(xt) c.c.]
and where R$J is the mean propagator of infinitessimal disturbances i n
$J,
and
c(xt) = - y. = x, T - T * = t. Hombgeneous turbulence has been results presented for d = 1, w i t h similar results for d > 1.
with y
While
the
traditional
polarization of
assmed
charged particles
and
the
involves a
Debye rearrangement of charge density over an infinite distance (i.e., screening) when a local charge inhomogeneity l a s t s a long time, i n the Zakharov and NLS equations, there can be transient but long lived (or even unstable) trapping of Langmuir waves by density inhomogeneities i f the wave nunber bandwidth of t h e spectrum of fluctuations of E is mall compared t o (W)ll2. These are the self-focusing and modulational instability mechanisms mentioned earlier. For the special case of a narrow bandwidth spectrum of Langmuir wave fluctuations centered a t k = 0, we have compared the linear response functions a s predicted by the DIA with a mnte Carlo simulation of a few fourier modes extracted from t h e Zakharov equations and found qualitative agreement. l 2
As an aid i n understanding t h e structure of the DIA, and also for the purpose of finding a method of solution that is more efficient than nunerical integration of the DIA, we12 have developed a consolidated continued fraction technique that is applied to the space and time fourier transformed correlation and linear response functions C(kw) and R ( k w ) . L e t u s recall that the form of the DIA is
477
SELF-FOCUSING OF THE NONLINEAR SCHRODINGER EQUATION
where G is a generalized propagator that includes both C and R; and quadratic functional. Equation (5.1) is replaced by
1
is a
Under some mild restrictions it is easy t o show that each iterate, Ga(kw) is a rational function of w and hence is characterized by a nmber, n ( t ) , of poles and residues. Since n ( a ) is a rapidly growing function of a , a nunerical investigation of the possible convergence of (5.2) must involve some kind of consolidation of the poles a t each iteration. For have examimed, t h i s technique (using rough, ad hoc appears to succeed i n the sense that convergence the fixed point solution t o (5.2) i n good agreement DIA, determined by nunerical integration.
the particular cases that we methods of consolidation) is frequently obtained, w i t h w i t h the solution t o the
I t is natural t o interpret the collection of poles of G(kw) for fixed k , a s nonlinear normal modes of the turbulent system because they depend on the strength and spectral distribution of fluctuations. I n addition t o modes t h a t correspond t o turbulently modified continuations of the linear normal modes (these are the only ones allowed by the various weak turbulence theories) there are new modes that correspond t o the dynamics of self-focusing and modulational instability.
Although t h i s technique is still nunerically ccinplex, it should serve as an intermediary i n the construction of a simpler phenanenological theory a t the level of the Eddy damped quasi normal approximations"+ that have proved t o be of great u t i l i t y i n the study of fully developed fluid turbulence.
REFERENCES 1.
w
useful review a r t i c l e s on the subject of Langmuir turbulence are: and D. ter Haar, "Langmuir Turbulence and Modulational Instability", Physics mports 43c (1978) 43 and L. I. Fudakov and V. N. Bytovich, "Strong Langmuir Turbulence", Physics &ports 40 (1978) 1. S. G. Thornhell
H.A. ROSE, D.F. DUBOIS
478
2.
B. B. Kadomstev, "Plasma V. N. Tsytovich,
Turbulence",
"Theory
Academic
Press,
of Turbulent Plasma",
New York,
1965.
Consultants Bureau, New
York I 1977. 3.
A. A. Vedenov and L. I. Rudakov, Sov. Phys. Doklady 9 (1965) 1073.
4.
V. E. Zakharov, Sov. Phys. JEW 35 (1972) 908.
5.
J. J. Thomson, F. J. Faehl,
6.
M. V. Goldman
and W. L. Kruer, Phys. Rev. Lett. 3 1 (1973) 918. G. J. MDrales, Y. C. Lee and R. B. white, Phys. Rev. Lett. 32 (1974) 457.
Astrophysical
J.
Phys. Rev. Lett.
D. W. Nicholson,
and
D. R. Nicholson,
M. V. Goldman,
223 (1978)
P. Hoyne,
and
605. D. R. Nicholson
41
(1979)
406.
J. C. Weatherall,
and
M. V. Goldman,
Phys. Fluids 2L (1978) 1766. 7.
A. A. Galeev,
R. 2. Sagdeev,
V. D. Shapiro,
and V. I. Schevchenko, JEW
Letters 24 (1976) 21. 8.
R. H. Kraichnan, J. Fluid Mech. 3 (1959) 32.
9.
R. H. Kraichnan, Phys. Fluids 7 (1964) 1163.
10.
V. E. Zakharov
and E. A. KuznetSov, JEW
11. R. H. Kraichnan, Phys. Fluids 12.
(1978) 458.
(1967) 1417.
D. F. Dueois and H. A. Mse, " S t a t i s i t i c a l Theories of Langmuir Turbulence:
The Direct I n t e r a c t i o n Approximation Responses, Los A l m s report LA-UR-80-3603 (accepted f o r publication i n Phys. Rev. A ) . I.
13.
D. F. Dubois, D. R. Nicholson, and
H. A. Mse,
"Statistical
Theories of
Langmuir Turbulence 11 ( i n preparation). 14.
For a review of these kinds of turbulence approximations see: H. A. Mse and P. L. S u l m , Journal d e Physique 2 (1978) 441.
Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (4s.) @ North-Holland Publishing Company, 1982
479
CHAOTIC OSCILLATIONS I N A SIMPLIFIED MODEL FOR LANGMUIR WAVES P. K. C . Wang
School o f Engineering and Applied Science University o f California Los Angeles, C a l i f o r n i a U.S.A.
The r e s u l t s o f a f u r t h e r study of the chaotic s o l u t i o n s o f t h e single-mode equations corresponding t o Zakharov's model f o r Langmuir waves i n a plasma are described b r i e f l y .
Recently, i t was found [ l l t h a t the following single-mode equations corresponding t o Zakharov's model f o r Langmuir waves i n a plasma have nonperiodic chaotic solutions
where
Here, a - b denotes he usual scalar product o f two vectors and b i n the r e a l Eucl idFan space IRt; $, i s the orthonormal ized eigenfunction o f the Laplacian operator corresponding t o the eigenvalue < 0; Ym and m r are t h e phenomenol o g i c a l damping c o e f f i c i e n t s ; E and nm are r e s p e c t i v e l y t h e c o e f f i c i e n t s o f density n i n terms o f $,, defined on expansions f o r the e l e c t r i c f i d d E and a bounded s p a t i a l domain Q C R N . E EIR represents the normalized amplitude o f an external s p a t i a l l y homogeneous3lectric f i e l d (pump) o s c i l l a t i n g a t the electron plasma frequency. I n 111, chaotic s o l u t i o n s were discovered through numerical experimentation and b i f u r c a t i o n analysis w i t h Il&lp as a v a r i a b l e parameter. The s t r u c t u r e and the onset o f these s o l u t i o n s were n o t explored i n detail. Here, we summarize the r e s u l t s o f a f u r t h e r study o f t h e chaotic solutions o f ( 1 ) - ( 3 ) .
icp
1.
B i f u r c a t i o n o f E q u i l i b r i u m States
For fixed 6,,, > 0 and v a r i a b l e parameter
Il&11',
the t r i v i a l s o l u t i o n 2 = (nmm.~m,E+,)
(O,O,Q+ iQ) i s the o n l y e q u i l i b r i u m s t a t e when 0
ri
11&,11'
< E&,
where
480
P. K.C. WANG
A t 11&112 = E&, a n o n t r i v i a l e q u i l i b r i u m s t a t e z emerges and i t b i f u r c a t e s i n t o two d i s t i n z t e q u i l i b r i u m states and 3 when *I&,IIZ > E&. For t h e case where 1 1 h 1 1 i s f i x e d and ym i s a v a r i a b l e parameter, t h e t r i v i a l s o l u t i o n i s the For 0 < y, < y,$, and y, # where o n l y e q u i l i b r i u m s t a t e when ym = 0.
%,
we have, i n a d d i t i o n t o the t r i v i a l e q u i l i b r i u m state, two d i s t i n c t n o n t r i v i a l e q u i l i b r i u m s t a t e s and they merge i n t o one when y = yc. When ym increases beyond ,y; the n o n t r i v i a l e q u i l i b r i u m s t a t e s abruptyy digappear. A t y, = o n l y a t r i v i a l and a n o n t r i v i a l e q u i l i b r i u m s t a t e s e x i s t .
%,
2. Chaotic O s c i l l a t i o n s
I n the case where ym and rmcorrespond t o Landau damping, Y v a r i e s r a p i d l y f o r small p m X ~> 0, where AD i s the Debye length. It i s interest t o determine the threshold value o f Ym f o r the quenching o r onset o f c h a o t i c o s c i l l a t i o n s . A d e t a i l e d numerical study was made f o r the one dimensiona w i t h fi = ]O,L[, , !-11= n/L, L = n/m, r l = 2, a1 = ( 2 /L)/(3n) and 91(x) = JB = -8nm!?L!0r the t y p i c a l case where :E = 2.669, we have =,.5.3358. I i can be v e r i f i e d t h a t Hopf b i f u r c a t i o n ( w i t h respect t o the parameter Y = l / v l ) takes place about the e q u i l i b r i u m s t a t e z+ when Y YH = 4.35. As Y i s decreased from YH, i t was found t h a t the soeutions a b i u t zg change from b e r i o d i c t o almost p e r i o d i c and become c h a o t i c when Y Y =1.55. The power spectra o f the e l e c t r i c f i e l d evolve from e s s e n t i a l l y d l s z r e i e spectra t o broadband spectra as Y crosses YT. The phenomena appear t o be more complex than simple p e r i o d We a l s o obtained numerically the p o i n t s doubling as observed i n many systems. generated by Poincar6 maps corresponding t o various hyperplanes i n z-space which Typical r e s u l t s are sFown i n are transverse t o the c h a o t i c t r a j e c t o r i e s . Figures 1 and 2. Evidently, t h e p o i n t s appear t o l i e i n the neighborhood o f some curve i n R 3 , implying t h a t the Poincar6 map i s e s s e n t i a l l y one-dimensional However, t h e graphs o f t h e Poincar6 maps i n terms o f some curve i n nature. A more d e t a i l e d d e s c r i p t i o n o f t h e aboveparameter are n o t r e a d i l y obtainable. mentioned r e s u l t s i s given i n reference [Z].
o'P
$-F"
YT
REFERENCES
[l]Wan ,P.K.C. ,"Nonlinear O s c i l l a t i o n s o f Langmuir Waves", J. Math. Physics, 21 81980) 398-407. [2] Wang,P.K.C. ,"Further Study o f Chaotic O s c i l l a t i o n s o f S i m p l i f i e d Zakharov's Model f o r Langmuir Turbulence", Univ. o f C a l i f . Engr. Rprt. No. UCLA-ENG8013 (March,1980).
0 0
+
0
0
0
0
N 0' 0 0
N
0 0
W
N 0
0
N 0
W 0
0
0
0
0' 0
0
m
rn'
0
0
0
m0
m0
3 '"'
-c
0
0
0'
c
0
0
-
y. 0
m
0
-=ln -I
+ 0 0
-20.00
0
-60.00
ION DENSITY
0- 00.00
P. K. C. WANG
I00
40
-20
n -80 ..
.:.
.. .
.
."
-1 40
-200
-100
-80
-60
n
-40
-20
Fig. 2: Isometric representation of points generated by Poincar; map (for the solution shown in Fig.1) onto the hyperplane: Im E 0 in z-space.
-
483
INDEX OF CONTRIBUTORS
Andrei, N., 169 A r i s , R., 247 Bardos, C., 289 Bishop, A.R., 195 Brauner, C.M. , 335 Campbell , D. K. , 195 Chang, S.-J., 395 Crandall, M.G., 117 Dubois,
D.F., 425, 465
Emery, V.J., 159 Etemad, S . , 209 Feigenbaum, M.J. , 379 F i f e , P.C., 267 Flaschka, H., 65 Foias, C., 317 Fremond, M., 109 FriaZ, A., 109 Gollub, J.P., Greene, J.M.,
403 423
Heeger, A.J., Holland, R.R. Howes, F.A. , Hyman, J.M.,
209 229 329 91
Jeandel, D.,
335
Kumar, P.,
229
Leach, Lewis, Lions, Lions,
P.G.L., 133 H.R., 133 J.-L., 3 P.L., 17
Mathieu, J., 335 McCormick, W.D., 409 Newell, A.C., Newman, W . I . ,
65 147
Orszag, S.A.,
367
Palmore, J.I., 127 Patera, A.T., 367 Pesme, D. , 435 Raupp, M. , 109 Rice, M.J., 189 Rose, H.A., 465 ROUX, J.C. , 409 Ruck, H.M., 237 S a t t i n g e r , D.H., 51 S c h r i e f f e r , J.R. , 179 Singer, I.M., 35 Swinney, H.L., 409 Temam, R., 317 Turner, J . S . , 409 Wang, P.K.C., 479 Wortis, M. , 395 Wright, J.A., 395
This Page Intentionally Left Blank