Nonlinear Evolution Equations and Dynamical Systems: Proceedings of the Icm2002 Satellite Conference, Yellow Mountains, China 15-18 August 2002

Proceedings of the I C M 2 0 0 2 Satellite Conference

I 1

1 II 1 m_M_—

7m

iHifillili^ Editors

Cheng Yi

• Hu Sen • Li Yishen • Peng Chiakuei

total Evolution Eptions

and

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P r o c e e d i n g s of t h e I C M 2 0 0 2

Satellite

Conference

Nonlidr Evolution Eqodtioos and

Dpdiidl Systpms Yellow Mountains, China

1 5 - 1 8 August 2002

Editors

Cheng Yi, Hu Sen, Li Yishen University of Science & Technology of China &

Peng Chiakuei Chinese Academy of Sciences

\IJP World Scientific NEW JERSEY

• LONDON

• SINGAPORE • SHANGHAI • H O N G K O N G

• TAIPEI • BANGALORE

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

NONLINEAR EVOLUTION EQUATIONS AND DYNAMCAL SYSTEMS Proceedings of the ICM2002 Satellite Conference Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-276-3

This book is printed on acid-free paper. Printed in Singapore by Mainland Press

PREFACE

"Nonlinear Evolution Equations and Dynamical Systems" was held at the Yellow Mountains of Anhui province from August 15-18, 2002 as one of the ICM satellite conferences. Professor Gu Chaohao served as the Chairman of the Scientific Committee and Professor Cheng Yi served as the Chairman of the Organizing Committee. The current proceedings recorded some of the materials presented there. There were about 50 scientists participating the conference. Talks covers several fields such as nonlinear evolution equations and integrable systems, infinite dimensional algebra, conformal field theory and geometry. Some of them were invited to give talks at the Beijing ICM 2002 conference and not all participants are able to contribute their papers here. Much of the works were done by Dr. He Jingsong and we wish to express our thanks to him. Dr. Zuo Dafeng helps a great deal in editing the proceedings. The conference was unable to be held without the support from staffs of the math department. And the last but not the least we wish to acknowledge of financial support from the University of Science and Technology of China and the China 973 program. The editors Cheng Yi, Hu Sen, Li Yishen and Peng Chiakuei

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CONTENTS

Preface

v

Soliton Excitations and Periodic Waves without Dispersion Relation in (2 + l)-Dimensional Dispersive Long Wave Equation

1

Chun-Li Chen and Sen-Yue Lou Global Bifurcation from the Eigenvalues of Some Semi-linear Biharmonic Equations

13

Zu-Chi Chen and Kun Zhao The Multisoliton Solutions of the KP Equation with Self-consistent Sources

25

Shu-Fang Deng, Deng-Yuan Chen and Da-Jun Zhang A Direct Extension of Lie Algebra An-\

33

Fukui Guo, Yufeng Zhang and Qingyou Yan On Gauge-gravity Correspondence and Open-closed String Duality

39

Sen Hu and Xiao-Jun Wang Integrable Semi-discretizations of the AKNS Equation and the Hirota-Satsuma Equation

49

Xing-Biao Hu and Hon- Wah Tarn. Differential Equations and Conformal Field Theories

61

Yi-Zhi Huang Lie Symmetries for Lattice Equations D. Levi

73

viii

Contents

Multi-scale Reduction for Differential Difference Equations and Integrability

79

D. Levi p-Harmonic Maps with Applications

91

Shihshu Walter Wei The Infinitely Many Conservation Laws of the Lax Integrable Differential-difference Systems

109

Da-Jun Zhang and Deng-Yuan Chen Quivers and Hopf Algebras

121

Fred van Oystaeyen and Pu Zhang Theory of Bidirectional Solitons on Water

135

Jin E. Zhang and Yishen Li A Finite Dimensional Integrable System Associated with BKK Soliton Equation

149

Jinshun Zhang The Novel N-Soliton Solutions of Equation for Shallow Water Waves

157

Yi Zhang, Shu-Fang Deng and Deng- Yuan Chen A New Integrable Hierarchy and its Expansive Integrable Model

165

Yufeng Zhang and Qingyou Yan Extremal Functions of Sobolev-Poincare Inequality

173

Meijun Zhu

List of the Participants

183

SOLITON EXCITATIONS A N D P E R I O D I C WAVES W I T H O U T D I S P E R S I O N RELATION I N ( 2 + l ) - D I M E N S I O N A L D I S P E R S I V E LONG WAVE EQUATION* CHUN-LI CHEN Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, P.R. China clchen@sjtu. edu. en SEN-YUE LOU Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, P.R. China Department of Physics, Ningbo University, Ningbo, 315211, P.R. China sylou@sjtu. edu. en

Using the nonstandard and standard truncations of a modified Conte's invariant Painleve expansion for the dispersive long wave equation system, two types of soliton excitations without any dispersive relations are found. Periodic waves expressed by Jacobi elliptic functions are found by the truncations of a special extended Painleve expansion. The soliton solutions are special cases of the corresponding two of the given periodic solutions. The dispersion relations of the solutions are crucially dependent on the boundary conditions.

1. Introduction It is well known that the Painleve analysis developed by WTC (WeissTobor-Canvela) : not only is one of the most powerful methods to prove the integrability of a model, but also can be used to find some exact solutions no matter whether the model is integrable or not. In 1989, Conte 2 proposed an invariant version of the WTC approach which is corresponding to the re-summation of the usual WTC approach. In Ref.3, Pickering proposed a nonstandard truncation approach basing on the Conte's invariant Painleve analysis. The closed truncation expansion in the Conte's analysis is related *The work was supported by the National Outstanding Youth Foundation of China (No. 19925522), and the National Natural Science Foundation of Zhejiang Province of China. 1

2

Chun-Li Chen & Sen- Yue Lou

to a special type of nontruncated summation in the usual WTC approach. Similar to the Conte's consideration, Lou4 obtained some other types of expansions to study the Painleve properties, which can be used to find some new solutions. Usually the soliton solution of a nonlinear system is a special case of the periodic wave solution that is expressed by a Jacobi elliptic function. However, using the usual truncation forms of the WTC's and/or Conte's Painleve expansion and the Pickering's modification, one can not find such types of periodic solutions. In this paper, we will try to find the periodic solutions which are the generalizations of the soliton solutions by the standard and nonstandard truncations of a special extended Painleve expansion given in Ref.4. In Ref.5, it was proved that the 2+1-dimensional integrable dispersive long wave equations Uyt + Vxx+ -(U2)xy

=0

vt + (uv + u + uxy)x

.^

=0

possesses a weak Lair pair and Kac-Moody-Virasoro type algebra, have a single-valued expansion and still fail the Painleve test in the ARS algorithm and in the WTC approach because multiple-value expansions about some movable characteristic singularity manifolds are allowed for this model. In this paper, two differential types of soliton solutions without any dispersion relation are obtained for DLWE by the standard and nonstandard truncation of a modified Conte's invariant Painlev'e expansion. Two types of Periodic wave solutions without any dispersive relation is found. In section 2 of this paper, we outline the extended Painleve expansion for the DLWE (1). Using the nonstandard and standard truncations of a special extended Painleve expansion, a slightly modification of the Conte's expansion (a special form of the Pickering's modification), we obtain soliton solutions without dispersion relation. Section 3 is devoted to find periodic solutions of (1) by using another special type of extended Painleve expansion. The last section is a simple summary and discussions. 2. Extended Painleve Expansions It has been proved that for the DLWE (1) there is a single valued expansion about the noncharacteristic singular manifold5 which has the form oo

« =Eu/"1. j=0

oo

« = 5>^'- 2 j=0

(2)

(2+l)-Dimensional

Dispersive Long Wave Equation

3

with ,„,

(y

(3)

and for arbitrary functions i>2, 113, V3 and V4. Using the standard truncated Painleve expansion of (2), solution of the DLWE(l) can be given. From (2) and (3), we know that Eq.(l) possesses two branches in the standard Painleve analysis. So from the Conte's expansion one can obtain some new exact solutions. In Ref.4, it is pointed out that the Painleve expansion form (2) can be modified as 00

00

j=0

j=0

with £ being determined by N

N

N

i=o

j=o

j=o

When we take JV = 2, the general expansion (4) with (5) is just the Pickering's modification6. Now using the nonstandard truncation of (4), one can find some types of new exact solutions. More specifically, we fix the expansion function as

fo^gsEA-x,

=(^-|^)

x

(6)

with A being an arbitrary constant and (f> being an arbitrary function of space-time variables. When we take A = 0, the modified expansion (4) with (6) will be reduced back to the usual Conte's expansion. As in the usual Conte's expansion, the coefficients Uj and Vj are all invariant under the Mobius transformation. From the special selection (6), (5) becomes 9* = - 1 + - ( A - gf gy = C-Cx(\-g) gt = K-

Kx(\

+ ±(Cxx-CS)(\-g)2

(7)

-g) + ±(KXX - KS)(X -

g)\

where s=3fxx\

<j>xxx

c

_ _(py_

K

=

_!tl.

(a)

4

Chun-Li Chen & Sen-Yue

Lou

which are the Mobius transformation invariants. It is straightforward to prove that all the compatibility conditions gxy — gyx, gxt = gtx, 9ty = 9yt are satisfied automatically because of (8). When the expansion function of (4) is selected as g shown by (6), the corresponding nonstandard truncation form reads u = — + f/i + U2g,

v = -^ + ^-+V2 92

9

+ V3g + Vi92.

(9)

9

Substituting the nonstandard truncated expansion (9) with (7) into (1) and vanishing all the coefficients of different powers of g, we can find a set of complicated over-determined equations to fix the functions Uj, (j = 0,1,2), Vfe, (k = 0 . . . 4), S, C and K. However, if we only want to find the single soliton solution, we can take them as constants simply. Omitting the detailed calculations to solve these over-determined equations, we only list the final result:

U0 = ±(SX2 - 2), Vi = SCX(2 - SX), U2 = ±S,

V0 = j(SX2-2)2, V2 = - 1 ,

U1=K-±SX, 2

V3 = -S CX,

(10)

Vi = ]-S2C.

And general solution of (7) reads /2

V2Stenh(^y/Srj) g = X+ j ^ ,

r] = x + Cy + Kt.

(11)

By the way, using the standard truncated form of the extended Painleve expansion (4) with (6), i.e, (9) with U2 = V3 = V4 = 0, one can obtain another type of soliton solution that has the form _ Q X 2 _|_ 9

u = ± — — — ±SX + CK,

, = W-»)'_c«(w-») + cav_cg..1, 2#2

g

(12)

2

However, the soliton solutions (9) with (10) and (12) are obviously not equivalent because u in (9) is bell or ring type soliton while u in (12) is kink or anti-kink soliton. There is no also dispersion relation also for the both types of soliton solution (12) because both K and C are arbitrary.

(2+1)-Dimensional

Dispersive Long Wave Equation

5

3. Periodic Solutions of the DLWE Usually, the single soliton solutions of integrable models are special limited cases of the elliptic function solutions. In our knowledge, the periodic solutions expressed by Jacobi or Weierstrass elliptic functions cannot be obtained by the WTC's standard truncated expansion, Conte's standard and nonstandard truncated expansion, the Pickering's modification and the two-singular manifold approach. If we take N —> oo in (5) and select the functions Sj, Cj and Kj such that the summations become some closed forms, then it is also possible to obtain some types of new exact solutions of the DLWE by using the standard and nonstandard truncations of the extended Painleve expansion (4). For instance, a special type of summation form of (5) may have the forms M

M

M

X>-(-D/^-. (13) \

\

j=0

3=0

Using the expansion functions expressed by (13) in (4), we may use both the standard and nonstandard truncated expansions to find new exact solutions. To find the solutions with elliptic functions, we taKe ST, Cj and Kj as constants and M = 4. The compatibility conditions of (13) for Sj, Cj and kj being constants reads

/vj — K

(14)

Sj •

When £ is determined by (13) with M = 4 and (14), the nonstandard truncation form can be taken as

U =

^

+ U l

+ U2t

v=^

+ ^-+V2

+ V^ + V4e-

(15)

while the standard truncation has the same form as (15) but with U2 = V3 = V4= 0. Substituting (15), (13), (14) and M = 4 into Eq.(l) and vanishing all the coefficients of different powers of £, we can solve out the equation system

6

Chun-Li Chen & Sen-Yue Lou

yields three nontrivial cases. The first case,

V0 = -C-Ul

Vl

-2CS3U°

=

3U2 1 V2 = Y^J2 (9cU0U$ + 8cs 2 - 18cUis2 - 18C/|)

v

2cs

_

U0s3

3

c 2

v

3kU2 - 2s3

(16)

U$

U*

corresponds to the nonstandard truncation. The second case, Vo = -\ul, 5o = ^ ,

Vx = -U0c(k + Ui), s ^ - ^ k + Ut),

V2 = 2Ui.kc + dJ\ + k2c - 1 - cs2

U2 = V3 = V4 = 0

(17) is related to the standard truncation while the last case is equivalent to the second case. Now the remained problem is to solve out the equation system (13) and give out the explicit expressions of the expansion functions. In these cases, the expansion functions can be expressed by the usual Jacobi elliptic functions with help of the mapping deformation approach proposed in Ref.7. Using the results of 7 , we may obtain many kinds of periodic solutions of Eq.(l). More explicitly, In 7 , It is proved that sn(m) is a solution of the equation /x2 = A / 2 + / x / 4 + C,

A = - ( l + m 2 ),

M

= 2m 2 ,

C = 1.

So, we can use this kind of / as the expansion functions. Moreover, it is easy to check that the equation (13) is form invariant under the Mobious transformation. In other words, if we make the transformation S^Pr^rAa^-azb^O),

(18)

then the function g satisfies the same equation as (13) 4

9y _ 9t

9x =

\ We know that the expansion function £ can be expressed explicitly with help of the standard Jacobi elliptic functions by the appropriate selections

(2+1)-Dimensional

Dispersive Long Wave Equation

7

of the constants ai, b\, a2 and b2 such that s[ = s'3 = 0.

(20)

In Ref.7, various solutions of (19) had been listed in a table. Using the results of Ref.7, we may obtain many kinds of periodic solutions of the DLWE (1). We write down here some final results with two special cases, which are the generalizations of the soliton solutions listed in the last section. So if we rewrite the arbitrary constants in the first case (16) as P{qa\rn2 + qa\ + 3m 2 + 3q2a24

a\U2

2 3b2p{q2a\ - m2) «3 =

z—5

y 2 b2 ,

£21)

01 =

2qa\ a2 where p, m and q are new arbitrary constants. Then the general solution of (13) with (16) and (21) has the form £

=

ff

b +b a'

=

V^ sn ( 7 ?)'

V = VP(X + CV + kt)

(22)

where sn(rj) is the usual Jacobi elliptic sine function and m is the modular of the function sn^. The corresponding periodic solution of Eq.(l) reads a2Al92

+ A2

_4Blgi-2a2B2g2

2

JpCx(alg -\y

2

+ Bz 2 2

2

C (a g -1)

(23)

where A\ = 3pm2 + a\pq2 — 2a\pq — 2a\pqrr? — kC\ A2 = —2qa2pm2 +pm2 + 3q2a\p — 2qa\p + kCi C\ = a2a^q{qa\

- l)(qa?, - m 2 )

Bi = cpq4a2 - a2qz + q2{l + m2 -

6cpm2)a2+

qm2(4cpm2 + Acp —l)a\ — Scpm4 B2 = q3al[3cpqa2 - 6cp - 6cpm2 - 1] + g 2 a|[m 2 (4cpm 2 + lOcp + 1) + 1 + 4cp] - m2q(l + 6cpm2 + &cp)a\ + 3cpm4 £3 = -3cpqAa\ + q3(4cp - 1 + 4cpm2)a2i + q2(l+ m2 — 6cpm2)a,2 — a^qm2 + cpmA. In Fig. 1(a), the periodic waves are plotted for small modular (m = 1/2) for the fields u, v which are expressed by (23) with m = 1/2, a2 = 1, c = 1, q = 0.04, p = 0.1, a = - 1

(24)

8

Chun-Li Chen & Sen-Yue

Lou

Fig. 1(b) shows the soliton lattice structure of (23) with m = 0.5, a2 = 1, c = 1, q = 4/25, p = 0.1, a = - 1 .

fig. 1a

0.25-

A

/ \ / \ -0.3

(25)

j

\

\

/-

0.35

/

^

\

-0.4

\ \

V

-6

/

/^ \

0.45-

\

w

•

.

-2

A\ \

/

.

/

\ > /

u

\/

-4

/

9

2

4

~ 6

fig.lb -0.7 -0.8 -0.9-1-1.1-1.2-

/

'

\

\

\ /

\J -4

/

-2

2

/

\j 2

4

6

Fig. 1. (a)The plot of the periodic solution (23) with (24) for the DLWE equation for small m (m = 0.5).(b)The structures of the soliton lattice solution (23) with (29) and m = 0.5 for the DLWE equation.

(2+l)-Dimensional

Dispersive Long Wave Equation

9

Similarly, for the second case (17), if we rewrite the parameters as 3p

«3 = —, f—^[qa -2q + m2) + ( - 2 m 2 + q + 2(l q(a2 -ax)2 2p(q-l)(m2-q) s4 q(a2-al)2 2 S2 = -,—-—T~[q4(6q-m q{a2 - ai) 2 + a 2 (6m 2 -qm2 - q)]

m2q)ai],

- 1) - 4 q a l a 2 { m 2 + 1) (26)

- qal){d{m2 - gap q(a2 - ax) 2a2ai(l+m2)p a2k + U0 4pm2 0% Ui Uo(a2-ai) a2 Uoa2{a2~a\)q where m, q, a i , a2 and p are all arbitrary constants. Then the general solution of (13) with (17) and (26) becomes U0

±y/qp(al2

aiWftnq

^

( x + cy +

fet)>

(27)

1 + y/q siiT]

In this case, the final periodic solution of Eq.(l) has a simple form A6g + A7 (ai + a2g)C2 '

V

=P

F

B5g2 - 2qa2a1B6g2 2- qB7 (ai a2g) C / , . +, „_„s*r*

(28)

with AQ =qpaia2(m2

+ 1) — a2kC2 — 2pm2a\

A-j =2q pa\ — pa\qa2{vr? + 1) — C2ka\ B5 =q3a%(cpm2 - 1 + cp) + Q 2 a 2 a2(l + m2 - 6cpm2) + afa^qm2 (icpm

+ 2>cp — 1) — 2pcafm4

i?6 =o-2q + q(2cpm — m — cpm — 1 — pc)a\a2 + a^m BY =2a\cpq3 + q2(l — Scpm2 — Zpc)a\a\ -f q(6cpm2 — m2 — l)aia 2 + m (1 — pc — pcm2)a\ C2=±

\Jqp{a\ - qa22){alm2 - qa%)

In Fig. 2(a), the periodic waves are plotted for small modular (m = 1/2) for the fields u, v which are expressed by (28) with m = 1/2, a2 = 1, al = 1/2, c = 1, q = 1/25, p = 0.1, a = - 1

(29)

Fig.2(b) shows the soliton lattice structure of (28) with m = 1/2, a2 = 1, ai = - 1 , c = 2, q = 0.02, p = 0.1, o = 1

(30)

10

Chun-Li Chen & Sen-Yue

Lou

fig.2a

A

III

-0.6-

i \

'J -15

-10

-5

§

f^

/~\

pi r ^/' |

|V_/'1

M

-0.9-

' i

1

i

;

V 5

fig 2b

-0.6- ,^,

-0.8-

II

';

HH H A/\ A A .

1

-0.8-

-0.7-

/-N

r\

) M /' M

-0.5-

-0.7

r\

, /if --•

\J

"~\

"•— 15

10

f~\

"i ^ i

\-/ l\

/i

j^l

-1-

\

-1.1-1.2-

'^'

-15

-10

-5

v

v'

V

g

5

10

V 15

Fig. 2. (a)The plot of the periodic solution (28) with (29) for the DLWE equation for small m (m = 0.5).(b)The structures of the soliton lattice solution (28) with (30) and m = 0.5 for DLWE.

4. Summary and Discussions In summary, for the DLWE (1) there are two types of soliton solutions. For the first type of soliton solutions, both the fields u and v are bell or ring type soliton solutions. For the second type of soliton solution, the field u is kink or anti-kink soliton solution while the field v is bell or ring type

(2+1)-Dimensional

Dispersive Long Wave Equation

11

soliton solution. Though these types of similar soliton solutions had also been obtained by many other authors using different authors, some further arbitrary constant had been included in the soliton solutions such that there is no dispersion relations needed. The usual dispersion relation can be recovered when we put the suitable boundary conditions on the soliton solutions that means the dispersion relations of the soliton solutions are crucially related the boundary conditions. Usually, both the standard and nonstandard truncations of the both the WTC's and the Conte's expansions cannot be used to find the exact elliptic function solutions. In this paper, we use standard and nonstandard truncations of a special limited case of the extended Painleve expansion to obtain some types of periodic solutions which are expressed by the Jacobi elliptic functions. When the modular of the elliptic function tends to one, two types of periodic solutions tends to the equivalent two types of soliton solutions respectively while other types of periodic solutions tends to constant or zero solutions. According to the results of this paper, many interesting open questions are worthy of studying further. For instance, whether the soliton (or solitary wave) excitations without any dispersion relation can be found in other nonlinear systems and/or real physical systems? Can the non-dispersion soliton solutions be preserved for multi-soliton solutions? Whether the soliton solutions without dispersion relations can be obtained by other types of well known approaches like the Darboux transformation and the inverse scattering transformation? References 1. 2. 3. 4. 5. 6. 7.

J. Weiss, M. Tabor, G. Carnevale, J. Math. Phys., 24, 522 (1983). R. Conte, Phys. Lett. A, 140, 383 (1989). A. Pickering, J. Phys. A. Math. Gen. , 26, 4395(1993). S. Y. Lou, Z. Naturforsch., 53a, 251 (1998). S.Y.Lou, Phys.Lett.A, 176, 96 (1993). A. Pickering, J. Math. Phys., 37, 1894 (1996). S. Y. Lou and G. J. Ni, J. Math. Phys., 30, 1614 (1989); S. Y. Lou, G. X. Huang and G. J. Ni, Phys. Lett. A, 146, 45 (1991); S. Y. Lou, G. X. Huang and G. J. Ni, Commun. Theor. Phys., 17, 67 (1992).

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GLOBAL BIFURCATION F R O M T H E EIGENVALUES OF SOME SEMILINEAR BIHARMONIC EQUATIONS* ZU-CHI CHEN f AND KUN ZHAO Department of Mathematics University of Science and Technology of China Hefei, Anhui 230026, China * chenzcQustc. edit, en

This paper is devoted to the global bifurcation phenomena from the eigenvalues of some semilinear biharmonic equations in an open bounded domain that contains the origin with homogeneous Dirichlet boundary condition. Our main tools are based on the operator theory, the Leray-Schauder degree theory and the Rabinowitz's global bifurcation theory.

1.

Introduction

This paper is concerned with global bifurcation for semilinear biharmonic operator {-A)2u

= Xg(x)f(u),

x£tt

(1)

with homogeneous Dirichlet boundary condition du « = o, - = o, ov

ieffi

(2)

where A G R, and v is the unit outside normal of dfl. This b o u n d a r y value problem has the following weak formulation / AuAvdx

in

= f \g(x)f(u)vdx,Vv

Jo.

€ D2'2{fl),

u e D2'2(Q)

(3)

in the s t a n d a r d space D2,2{Q), i.e, the closure of the Co°(fi) functions with respect to the norm ||W|||,2,2(Q) = fn \Au\2dx. Here Q is an open bounded domain in RN(N > 4) t h a t contains the origin and dflis sufficient smooth. Here we state the general hypothesis which will be assumed throughout the paper The project supported by NSF of China (No. 10071080 and No. 10101024.) 13

14 Zu-Chi Chen & Kun Zhao

(G) g is a smooth function, at least C 1,Q (fi) for some a G (0,1), such that g G L Af / 4 (n) n L°°(n) and g(x) > 0 , Vz G ft. (F) / : R -> [0,oo) is a smooth function such that /(0) = 0, /'(0) > 0 and f(u) > 0 for all u £ 0. Also / ' , / " G L°°(ft) and there is K* > 0 such that |/(s)| < K*\s\ for all s e f l . It's well known that bifurccation phenomena has been studied for decades of years by many authors. For example, M. A. Del Pino and R. F. Manasevich studied some bifurcation phenomena associated with the p - Laplacian operator 3 . And V. K. Le in Ref. 2 generalized this result, he discussed the global bifurcation result for equations asociated with the p — Laplacian like ( — Laplacian) operators -div(<j)(\ Vu\) Vu) = F{x,

u,X),x€Q.

In the case of biharmonic operator, in Ref.l N. M. Stavrakakis studied it's bifurcation phenomena in the standard energy space D2>2(RN), but his work is concerned only with bifurcation from single point. And the single point bifurcation results for the biharmonic operator in bounded domain case have been studied in many papers 8,9,10 . In this paper we shall discuss the global bifurcation result for the biharmonic operator in a bounded domain in RN. The result in this paper has the global property that embraces the single point bifurcation result. So our paper is different from those papers 8 , 9 , 1 °. In section 2 we will first study the linearized problem for (1) and (2). Besause we'll apply the standard global bifurcation theory, the compactness and the topological degree properties of the solution operator of the linearized problem will be studied in this section. In section 3 we shall discuss the global bifurcation for (1) and (2) by an added hypothesis on the function / . The result is achieved by the standard global bifurcation theory introduced by P. Rabinowitz in Ref. 2. Notation: For simplicity, we use the symbol ||.|| p for the norm ||.||iP(fi) and Lp for the space Lp{9.) and £>2,2 for the space D 2 , 2 (fi). 2. The Linearized Problem In this section we shall discuss the linearized problem of (1) and (2) (-A)2u u

= Xg(x)u,

i6(l,

= o, ^ = o, x G an.

(4)

(5)

Global Bifurcation from the Eigenvalues

15

Let the space D2'2 be the closure of the Co°(fi) functions with respect to the norm ||u||D2,2 = ( / \Au\2dx)1/2, Jn 5 It can be shown that

D2* =

u e D2'2.

{ueL2N'W-Q\AueL2}

and the embedding jr>2,2 ^

L2N/(N-i)

is continuous and D2'2 embeds into L 2 compactly. Thus, the space £>2'2 is a reflexive Banach space. Define the inner product in £>2,2 given by (u,v) = / AuAvdx,Vu,v e D2'2. Jn By the maximum principle for the Laplacian equation, it is easy to check that (u, v) satisfies the three properties of the standard inner product. And since £>2'2 is a reflexive Banach space, thus D 2 ' 2 is a Hilbert space. Define the bilinear form by 0(u,v) = / g(x)uvdx Jn for a l l u . u e £>2,2. Then, \P(u,v)\ = | / g{x)uvdx\ Jn < I \g\\u\\v\dx Jn < IMI./V/4||w||2;V/(JV-4)IMl2;V/(iV-4)Since D2'2 embeds into L2N^N~4\ that

so there exists a constant K > 0 such

IMl2JV/(JV-4) < K\\u\\Di.i. 22

(6)

Thus (3 is bounded in D ' . Hence by the Riesz representation theorem we can define a bounded linear operator B such that (3(u,v) = (Bu,v) for all u,v e D2'2. So the weak formulation of the problem (2.1)(2.2) can be given by

{(I-XB)u,v)

=0

16

Zu-Chi Chen & Kun Zhao

for all v G D2'2. We call XB the solution operator of (4) and (5). The following lemma shows the compactness of B. Lemma 2.1: Suppose g satisfies the hypothesis (G). Then B is a selfadjoint linear compact operator. Proof. 1. Prom the definition of B it is obvious that B is linear. Let u,v € D2'2, then (Bu,v) = / guvdx — j gvudx = (Bv,u) = JQ Jn

(u,Bv),

so B is self-adjoint. 2. Let {un} be a bounded sequence in D2'2, thus for all m, n > 0 and 4> G D2)\ = | Jng(um - un)4>dx\ ^ /nlsllwm -un\\4>\dx u

< HsIloollUm -

n||2iV/(Af+4)||2.2.

Since dfl is sufficient smooth thus D2,2 embeds into L2 compactly. Because {un} is bounded in D2'2 thus there is a subsequence of {un}, still denoted by {un}, is convergent in L2. And since L2 embeds into L2N/(N+4) thus {un} is convergent in L 2W/(iV+4)

H e n c e

for

a n y

£ >

Q)

w h e n

m, n is sufficiently large | | w m - Un\\2N/(N+4)

< yn

I,

,

•"• H a l l o o

and thus ||J5UTO - B U „ | |

D

2,2
1 and u ^ 0 such that Hu={\*

-K){I-KB)~lBu,

then u-(K

+ (A* - X,)/n)Bu

=0

Global Bifurcation from the Eigenvalues

19

and A, + (A* - X,)/IM G C(B)

n (A*, A*).

Because C(B) D (A„, A* = {Ai}, whence fi = (A* - A.)/(Ai - A,). is the unique eigenvalue of (A* — A*)(7 - A*B) _ 1 B that is greater than 1. By the Leray — Schauder formula degD2,2(I - (A* - A,)(7 - A . B ^ B , b, 0) =

(-if0,

where L3Q is the algebraic multiplicity of /x as an eigenvalue of the compact operator (A* - A,)(7 - A . B ) - ^ . Now suppose that n is the smallest positive integer such that [Hi - (A* - A.)(I - A*B) _ 1 B] n u = 0, i.e. ( 7 - A , B ) - " ( / - A i B ) n u = 0. If n = 1, then clearly liu = [(A* - A.)(J - A . B ) - 1 ^ , i.e., ^ - ^ n = [(A* - A,)(7 - A . B J ^ B K Ai — A*

( J - A „ B ) u = (Ai -A»)Bu, u = X\Bu. If n > 2, put itj = [/i7 - (A* - A»)(7 - A . B ^ B ] " - ^ , i = 1,2, then obviously «i ^ 0. Since (7 - A,J5) _ n (7 - A I B ) " M = 0 and (7 - A*B) and (7 — AiS) are commutative and B is linear, then (/ - AiB)«i = (7 - A i B K " " ^ / - A i B ) " " 1 ^ - A . B ) - * " " 1 ^ = ^~1{I - A,B)(J - A,B)-"(7 - AiB) n u = 0 and ((/ - A.B)ui,ui) = 1 r](TK)- Hence by (7) we can see that (3Q — 7, this completes the proof of theorem 2.1. 3. Global Bifurcation Results for the Biharmonic Operator In this section we will discuss the Global Bifurccation of the solution of (1) and (2) by the standard bifurcation theory introduced by P.Rabinowitz. First we define the solution operator of (1) and (2) P : R x £>2,2 -* D2 € D 2 ' 2 , where (,} denotes the inner product in D2,2. Lemma 3.1: The operator P is well defined by (8). Proof. For fixed u € D2'2 we define the following functional F{4) = A / gf(u)cf>dx Jn

(8)

Global Bifurcation from the Eigenvalues

21

for all G D2'2. Since / satisfies (F) and g satisfies (G) then \F()\ D2'2 by A(X, u) = P(X, u) - XBu . Lemma 3.2: A is a complete continuous operator. Proof. 1. For any u,v, G D2'2 and A, fj, G R by (F) and (G) we have \(A(\,u)-A(»,v),)\, = \(A(X,u)-A(X,v),4>) + (A(X,v) - A(n,v),)\ A u u = l /n 3 [ / W - - / ( ) + * # d a ; + (A - M) /n 9[f(u) - u)dx\ < |A| / n | 5 |[|/(«) " /(v)l + l« - v\]Wx + | A - H L |s|[|/(«)| + M]M2,2 and P is compact, then we need only to show that P is compact. In fact, let {un} be a bounded sequence in D2'2 then |(P(A,um)-P(A,un),0)| =

\Xfng[f(um)-f(un)]ci>dx\

< |A|sup r e ( o,i)||/ (um +ru n )||jr,oo/ n |gi|| U-m -un\\ 1 uniformly for A in bounded intervals. Since it has been shown1 that the algebraic

22

Zu-Chi Chen & Kun Zhao

multiplicity of Ai is 1, then from theorem 2.1 we can see that the Leray — Schauder degree degD2,2(/- X*B,s,0) = — degD2,2(I - X*B,s,0). Let 0 = {(A, u) G R x D2'2\u ^ 0, u - XBu - A(X, u) = 0} U {[A„ A*] x {0}} and K be the connected component of 0 that contains [A,, A*] x {0}. Then we have Theorem 3.1: K is described as above, then either (a) K is unbounded in R x D2'2 or (6)Kn{(.R\[A»,A*])x{0}}^$. Proposition 3.1: K contains no points of the form (Ao,0) in {(R \ [A*, A*]) x {0}, where A0 ^ XiProof. Suppose that n contains a point (Ao,0) where Ao ^ Ai. So we can construct a sequence {(Xn,un)} C K such that un > 0, for all n G N and x G fi, un —> 0 in D 2 , 2 , and A„ —» Ao, as n —» oo. Put vn = un/\\un\\£,2,2. Since un G K, then wn = P ( A n , u n ) , i.e. un = XnBun + R(Xn,un). Since B is a linear operator, there results that vn = XnBv„ + .R(An,iin)/||i(n||£>2,2. Because ||u n ||D 2 . 2 = 1 and B is compact and R(Xn,un) = 0(||un||£>2,2), then Vn —¥ vo in -D2'2, as n —> oo and VQ = XoBvo. Since ||UO||D2.2 = 1 and un > 0 then VQ > 0. Thus vo is a positive solution of (2.1)(2.2) and Ao is an eigenvalue of (2.1)(2.2) corresponding to a positive eigenfunction. But it has been shown in [1] that Ai is the unique positive eigenvalue of (2.1)(2.2) corresponding to a positive eigenfunction. Thus Ai = Ao, a contradiction. Hence K contains no points of the form (A, 0), where A ^ X\. Proposition 3.1 shows that K is unbounded in R x D2'2. Remark 3.1: By the similar method, the global bifurcation phenomena of the biharmonic operator can be generalized to the polyharmonic operator A m ( m > 2) in the spaces Dm'2 (m = 3, • • •) with the norm given by \\u\\%m,2 = f \km/2u\2dx,m

Jn

= 2k,

f \A(m-V/2u\2dx,m

Jo.

= 2k + 1. (9)

The key point is the algebraic multiplicity of the first eigenvalue of (9). References 1. N. Stavrakakis, Semilinear Biharmonic Problems on R , Reaction and Diffusion System (Trieste 1995), 365-376.

Global Bifurcation from the Eigenvalues 23 2. Vy. khoi. Le, Global Bifurcation in some degenerate quasilinear elliptic equations by a variational inequality approach, Nonlinear Analysis, 46, 567-589 (2001). 3. Manuel. A. Del Pino and Raul. F. Manasevich, Global Bifurcation from the eigenvalues of the p—Laplacian, J. Diff. Equations, 92 no. 2, 226-251 (1991). 4. K. J. Brown and N. Stavrakakis, Global Bifurcation results for a Semilinear Elliptic Equation on all of RN,Duke-Math-Journal, 85 no. 1, 75-94 (1996) .

5. Swanson C. A., The Best Sobolev Constant.Applic.Analysis, 47, 227-239 (1992). 6. Toland J. F., A Leray-Schauder degree calculation leading to non-standard Global Bifurcation results, Bull. London. Math. Soc,15 ,149-154 (1983). 7. Hile.G.N. and Yeh.R.Z., Inequalities gor eigenvalues of the Biharmonic Operator, Pacific Journal of Math, 112 no. 1, 115-133 (1984). 8. Dalmasso R., Elliptic Equations of order 2m in Annular Domains, Trans. Amer. Math. Soc, 347 no.9, 3573-3585 (1995). 9. Edmunds D. E., Fortunato D. and Jannelli E., Critical Exponents, Critical Dimensions and the Biharmonic Operator, Arch. Rat. Mech. An,112,269-289 (1992). 10. Fleckinger J. and Lapidus M. L., Eigenvalues for Elliptic Boundary Value Problems with an indefinite Weight Function, Trans. Amer. Math. Soc, 295, 305-324 (1986). 11. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of second order, 2nd ed., Grundlehren. Math. Wiss. 224, Springer-Verlag, Berlin, 1983. 12. P. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3, 161-202 (1973).

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T H E MULTISOLITON SOLUTIONS OF T H E K P EQUATION W I T H SELF-CONSISTENT SOURCES*

SHU-FANG DENG, DENG-YUAN CHEN AND DA-JUN ZHANG Department

of Mathematics, Shanghai 200436,

Shanghai University, P.R. China

The KP equation with self-consistent sources is derived through the linear problem of the KP system. The Multisoliton solutions for the KP equation with self-consistent sources are presented by using Hirota method and Wronskian technique. The coincidence of these solutions is shown by direct computation. The novel multisoliton solutions of the KP equation with a self-consistent source are also obtained by Hirota method.

1. Introduction In recent years the study of the soliton equations with self-consistent sources (SESCS) has received considerable attention. The SESCS are usually used to describe interactions of long and short solitary waves in different physical field. Some ways to solve SESCS are provided. Refs.l, 2 construct the integration of the KdV equation with a self-constraint source and the nonlinear Schrodinger equation with a source was constructed through the inverse scattering transform without use of explicit evolution equations of eigenfunction. Refs.3, 4 use the d- method and guage transformation to find soliton solution for the modified Manakov system with self-consistent source. Recently, Zeng et.al. developed a simple treatment of the singularity in the evolution of eigenfunctions and obtained the explicit soliton solutions of some SESCS such as the KdV, AKNS, modified KdV, etc. hierarchies with self-consistent sources through the inverse scattering method 5 - 8 Also, the Darboux transformation for the Kaup-Newll and AKNS hierarchy with self-consistent sources were presented in Refs.9, 10. Since the evolution of eigenfunction for the SESCS possesses singularity in spectral parameter, which required special skill to treat the evolution of scattering data, so finding the soliton solutions for SESCS by above methods are quite complicated. In this paper, we first derive the KP equation "This project is supported by the National Science Foundation of China. 25

26

Shu-Fang Deng, Deng-Yuan

Chen & Da-Jun

Zhang

with self-consistent sources(KPESCS) by use of compatible condition of linear problems. Then we present a set of dependent variable transformations to write out the bilinear form of the KPESCS and by which we can construct multisoliton solutions through the Hirota's approach. 11 On the basis of this, we use Wronskian technique to give Wronski determinant 12 of soliton solution. Also we prove coincidence of the N-soliton solutions obtained by Hirota method and Wronskian technique. Last we state simply the main results for the novel soliton solutions of the KP equation with a self-consistent source. 2. The K P Equation with Self-consistent Sources Consider the spectral problem and its adjoint associated with the KP equation $y = $xx + « $ ,

(1)

-tfj, = Vxx + utf.

(2)

Suppose time evolution of the eigenfunction $ is given by St = A*,

(3)

where A is aoperator function of d and d~l (d = g | and d~1d = dd~l = 1). The compatibility of (1) and (3) requires that A satisfy + ld2 + u,A} = 0,

(4)

2Axd - Axx - [u, A].

(5)

A = aQd3 + aid2 + a2d + a3+a^d-1^,

(6)

ut-Ay or ut = AyNow we take

where a,- (j = 0,1,2,3) are undetermined functions of u and its derivatives, and a is an arbitrary constant. Substituting (6) into (5) and equating coefficients powers of d, we obtain ut = az,y - a-3,xx + a-oUxxx + o,\uxx + a-2Ux ~ 2a($*)x, 02,y — 2az,x — d2,xx + Saouxx + 2a\ux = 0,

( 14 )

®j,V = $j,xx + U$j,

(15)

-•yjty

=

tyjtXX+uVj,

(16)

while the operator A becomes A = - 4 d 3 - 6ud - 3ux - 3{d~luy)

1 N + - ^ Qjd^Vj.

(17)

j=l

3. Hirota Form's Solution In this section, we shall give the soliton solution of the KPESCS by use of Hirota method. With the help of the dependent variable transformations

u = 2(ln/) IX> *j = J,

^ = J.

(J = 1,2,---,7V),

(18)

the KPESCS (14)-(16) can be transformed into the bilinear forms 1

N

(DxDt + Dx + 3D2y)f . / = --J^gj.hj, 1

3=1

(19)

28

Shu-Fang Deng, Deng-Yuan

Chen & Da-Jun

Dvgj.f

Zhang

= D2xgJ.f,

(20)

-Dyhj.f = D2xhj.f, where D is the well-known Hirota bilinear operator DlxD™D?a.b=

(dx-

dx,)l(dy-dv,)m(dt-

(21)

dt,ra(x,y,t)b(x',y',t')\x,=XiV,=ytt,=t.

Expanding / ,