Integrable and Superintegrable Systems

bitegrable and Supenntegrable Systems

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Integrable and Superintegrable Systems Edited by

Boris A. Kupershmidt The University of Tennessee Space

World Scientific Singapo9e°mmWrMm^ndon

Institute

• Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

INTEGRABLE AND SUPERINTEGRABLE SYSTEMS Copyright © 1990 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 photo­ copying, recording or any information storage and retrieval system now known or to be invented, without written perimission from the Publisher.

ISBN 981-02-0316-0

Printed in Singapore bjCt^/?gWteGfe)M^©/feA Pte- Ltd.

V

PREFACE When the modern era in the thoroughly classical field of integrable sys­ tems burst forth in the mid 1960's with the discovery of curious objects dubbed 'solitons', few could have dreamt that the field would develop with such aston­ ishing speed, depth and breadth as what took place in the ensuing quarter of a century, and it is safe to say that fewer still can imagine what surprises will emerge in the next quarter. In fact, the field has split up in so many sub fields that it is quite difficult to keep pace with what's going on right now. Is there any way to ameliorate the plight of those who, for whatever rea­ son, are interested in the subject, be they middle-aged and upper-middle-aged practitioners, specialists in other fields, and especially beginners (if there are any left) and graduate students? This collection offers an attempt to assist the reader in forming a good (though not all-encompassing impression of the current state of the area of integrable systems; the authors are active develop­ ers of the subject, most of them have had a hand or two in enlivening it, and their papers describe what each one of them is working on nowadays or has been working on lately, — their individual perspectives on some of the most interesting recent advances will provide the reader with a global picture of in­ tegrable systems as it emerges from these proceedings of the best conference that has ever been.

B. A. Kupershmidt

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VII

CONTENTS Preface

v

The Main Soliton Theorem I. Cherednik

1

Functional Bethe Ansatz E. K. Sklyanin

8

Integrabihty in Models of Two-Dimensional Turbulence Y. Murometz and S. Razboynick

34

Solitons, Numerical Chaos and Cellular Automata M. J. Ablowitz, B. M. Herbst, and J. M. Reiser

46

The Unstable Nonlinear Schrodinger Equation T. Yajima and M. Wadati

80

Classification of Integrable Equations R. K. Dodd

102

List of All Integrable Hamiltonian Systems of General Type With Two Degrees of Freedom. "Physical Zone'' in This Table A. T. Fomenko

134

Finite-Dimensional Soliton Systems S. N. M. Ruijsenaars

165

Relativistic Analogs of Basic Integrable Systems J. Gibbons and B. A. Kupershmidt

207

Liouville Generating Functions for Isospectral Flow in Loop Algebras M. R. Adams, J. Hamad, and J. Huriubise

232

viii

A Loop Algebra Decomposition for Korteweg-de Vries Equations R. J. Schilling

257

Energy Dependent Spectral Problems: Their Hamiltonian Structures and Miura Maps A. P. Fordy

280

Commuting Differential Operators over Integrable Hierarchies F. Guil

307

Lie Superalgebra Structure on Eigenfunctions, and Jets of the Resolvent's Kernel Near the Diagonal of an n-th Order Ordinary Differential Operator T. Khovanova

321

Superstring Schwartz Derivative and the Bott Cocycle A. 0. Radul

336

Super Miura Transformations, Sup»r Schwarzian Derviatives and Super Hill Operators P. Mathieu

352

1

THE MAIN SOLITON THEOREM IVAN CHEREDNIK A. N. Belozevsky Laboratory of Molecular Biology and Bioorganic Chemistry, Moscow State University, Moscow 119899, USSR

It turns out that after years of writing papers on soliton theory I have not expressed anywhere my personal subjective attitude towards this theory. Maybe this paper is the right place. I'll try here to look at soliton theory from the mathematical point of view as if forgettting its physical origin and wide applications to concrete equations. It does not mean t h a t I underestimate the latter. A large part of my book Algebraic Methods in Soliton Theory (to be published soon by D. Reidel) is devoted just to this (classic) soliton theory. Moreover, for several years I have been taking part in practical activities on the realization of inverse scattering technique by computers and other devices. Many have heard of course about one-solitons (~ sech) in optical fibers. But maybe it is not common knowledge that there are experimental devices making it possible to investigate nonlinear eflfects in optical fibers more delicate than the stablility of one-solitons. The results are in perfect harmony with soliton theory. I believe that the 1990's will be the years of the "soliton boom" in engineering, and hope that some of my and my colleagues' proposals can be useful in future "solitonization". Nevertheless, when trying to visualize the possible future development of soliton theory, I prefer to pay the main attention to mathematical arguments. Let (£ be a simple Lie algebra, r(u) — a function of u 6 C taking values in (£ ® (£. We assume that r[u) = t/u + r ( u ) for some analytical r in a neighbourhood of u = 0, where t = J2 Ia ® Ia, {! see [1-3,6], one can rewrite (1.1) and (1.2) into a more compact form R(u-v)T(1)(u)T(2)(v) = T(y)=(P0 TT (y _v m ) •

(1-11)

m=l

Substituting the polynomial *>(y) (1.11) into Baxter's equation (1.9), dividing both parts by " n

following

satisfy

relations

[y.yJ = [X",Y-] = 0, m

n

m

(2.6a)

P

Y*y =(y±T)5 JY*, m n

Proof-

n

the

n 1

mn

(2.6b)

m

Y V = A(y ±2). n n n Z The commutativity of y

(2.6c) was established in

p

subsection 2.2. In order to prove the relation (2.6b) for Y consider the relation n

(U-V+T})B(u)A(v)

= (U-V)A(V)B(U)

+ 7)B(V)A(U)

which can be derived from (1.1'), (1.5) and (1.6). Substitute y for v and recall the operator ordering (2.5). n

Using B(y )=0, see (2.4), one obtains n

(u-y )Y~B(u) = (u-y+TJ)B(U)Y~. n n n Using the formula (2.4) for B(u) none obtains the eguality Y~cr(y) = cr(E~y)Y~ n n for every symmetric polynomial cr(y) .n Lemma 1. Let ne{l,...,N) and ee{ + ,-) be fixed. Then for every polynomial p(y) there exists such symmetric polynomial cr(y) that for all yeV

°"(y) = P(y)

and

IT(EJ) = p(E^y) . n n

n n

The proof is simple. Note that it uses the part c) of Condition 5. Using the Lemma 1 we can replace symmetric polynomial cr(y) in (2.7) with arbitrary polynomial p(y) and, in particular, with p(y)=y which proves the relation (2.6b) for Y . The relation (2.6b) for Y is derived guite analogously from the relation (u-v-7))B(u)D(v) = (u-v)D(v)B(u) - T?B(V)D(U) . The commutativity (2.6a) of Y follows

from

n

commutativity of the families A(u) and D(u). The relations (2.6c) follow from the definition (1.7) of the guantum determinant A(u), see [18,19]. + Applying the operators Y to the function weFun(V),

19 u ( y ) = l and

putting A ; ( y ) == [Y±w](y)

(2 8)

±

one obtains the following realization of the operators Yn . Theorem 2. The action of the operators Y± on the functions feFun(V)sW is given by the formula [Y"f](y)

.

Functions

A"(y)

n

satisfy

= A"(y)f(E-y). n

the

following

tf~ = {yeV 1 iTy Then

n

: 3 )

A;(y)

b)

A

c)

= 0 A

> >

A

n

'

(2.9)

n

conditions. Let

t V).

n

VyeV*. n

VyeV.

< *y> = < ( y ) A * ( E ; y ) E

>K (E±y)

= A(yn±|)

VyeV.

The equality a) is obvious, b) follows from (2.6a) and c) follows from (2.6c). Note that the isomorphism W=Fun(V) is not defined uniquely but the gauge transformation f i—>f is possible f(y)

=p(y) f(y)

where p(y) is arbitrary function on V having no zeroes. The + functions A (y) are then transformed as follows n

, A" (y) n

_"

p(y) ± + — A"(y) P ( E "n y ) "

f o r y e YT

0

for y e V

+

(2 11)

n

±

One can consider the definition

(2.5) of Y" as the n

interpolation problem for the polynomials A(u) and D(u) . Combining this observation with the formulas (2.4) for B(u) and (1.7) for A(u) one obtains the following expressions + for the matrix elements of T(u) in terms of Y~y. Theorem 3. The operator polynomials A(u), B(u), C(u) and D(u) are given by the formulas: B(u)

— BNN

ft (u- -yj> ■ ■

"

A(U)

= 1

(2 12a)

n

n=l

u-y

N

= AH ft (u-y p ) + I

n=l

,

17 m =1



y -y n

m

(2 12b)

20 D(u) = DN ft ( u - y j + I n=l

r

u-y

n

n=l

y

(y)l %1

=

The m a t r i x T(U) i s t h e p r o d u c t (1.4) of L r (u) =

u-p

n

expfgj

-exp(-q) 0

the

canonical

- ^ n ^ L-operators

1

(3.1)

n

The R-matrix is R(u)=u-iT)?>. The quantum determinant is A(u)=l. The operator B(u) coincides with the generating function for hamiltonians of the open Toda chain whose spectrum is well known. The resulting spectrum V=Rx...xR is continuous. It presents a serious obstacle for rigorous mathematical derivation of the algebra (2.6) and its representation (2.9) which is not overcome at present. The ± . . . problem is that the shift En in (2.9) should be imaginary: y ±ii). On the physical level of rigor one can simply n

+

postulate the necessary Y y algebra by analogy with (2.6). The algebra possesses the involution y - y ±iT) + + * n ID (IT) = Y" y n - *«> TT y _ y n

which follows from the involution T a g ( u ) = T a g ( u ) • T n e representations of the algebra can be constructed by analogy with the Propositions 1 and 2. The unsolved problem is to choose the factorization (1.16) of the quantum

28 determinant which could produce via (2.20a), (2.9) and (2.12) the representation T(u) isomorphic to the original one descibed in terms of the L-operators (3.1). In the paper [19] the conjecture was made that the +u

right choice equation

is

A+=i~

which

leads

to

the

separated

i"(P(Y+iTJ) + i~V(y-i-n) = r(y))) ■

( 2 - 22e )

It remains to notice that formulae (2.1)-(2.7) and (1.8) imply the inequali­ ties: As?0,

BS^Q,

Ds^0.

(2.23)

39

Remark 2.24. Our system (2.22) has An + 1 complex variables consisting of 2n (complex) triads. It has the following invariant subsystem consisting of n triads: 6(s) = W c ( a ) = 0 ,

V

1 < s < n,

(2.26a)

(2. 22a ,7

«d(»

& M = BSDJ

1

1

dH/dpd{s), dH/dad{s).

(4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.10)

d k =--dH/d(3k,

(4.11)

A=

(4.12)

■-

-dH/dak. —

42

Proof: by direct calculation. Remark 4.13. We do not use the relations (2.19) in Sections 3-5. Remark 4.14. If we remove formulae (4.5)-(4.8) from the set of equa­ tions (4.3)-(4.12), the remaining formulae provide a canonical Hamiltonian structure for the n-triads system (2.26). For the case n = 1 this canonical structure is well known (see, e.g., {12]).

,,

§ 5. Integrability

Set ws:=Us Xs

+ Vs,

(5-1)

:= Us-Vs.

Theorem 5.3.

(i) The non-zero Poisson brackets between the integrals

H,ws,xs,fs,gs,^>s, {fs,9s}

(5.2)

and V>s are given by the formulae:

= -ws/Ds,

{fs,ws}

= 2\sgs,

{ws,gs}

= 2\sfs,

A, := -A.B.DJ1, {s} = -KXs,

{fs,Xs}

(5.4s) (5.5)

= ^K^s,

{Xs,ips} = 2Xs oo) could be linearized by inverse scattering methods.

In fact, Ablowitz, -Kaup, Newell and Segur [11]

demonstrated that these ideas apply to a class of nonlinear evolution equations, including the physically interesting sine-Gordon,

49 «K

— u « + sin « = 0.

(3)

Relying on the integrability properties of the solition equations, it is possible to develop the geometry of their infinite dimensional phase space, see for example, [12] and the references therein. In particular, Ercolani, Forest and McLaughlin [12] show how the homoclinic orbits of the sine-Gordon equation, (3), derive from its breather solutions. In the case of the NLS equation, (2), the homoclinic orbits derive from the dark-hole soliton solutions of the defocusing NLS equation,

iut + uxx - 2 u V = 0.

(4)

Homoclinic orbits are generically related to soliton solutions and other examples where one can establish a correspondence between soliton solutions and homoclinic orbits, include the modified KdV equation, u, + 6u2ux + uxxx = 0,

(5)

and the complex modified KdV equation,

tt( + 6uu'ux + uxxx = 0.

(6)

As in the case of nonlinear dynamical systems, the homoclinic orbits of the soliton equations are also structurally unstable and are 'broken' under small perturbations. In fact, it is found that small perturbations of some of the soliton equations, most notably the NLS and sine-Gordon equations, may also lead to temporal chaos, see for example, [13,12,14] and the references therein. In these studies, the homoclinic structure of the underlying unperturbed partial differential equation has proven to be the key in understanding the mechanism resonsible for the chaos. It has been observed that certain discretizations of the NLS equation such as the Fourier spectral scheme, may induce" irregular (even with a positive Lyapunov exponent) temporal behavior at intermediate levels of grid refinement, see, for ex­ ample, [15,16]. In sections 2 and 3 we argue, concentrating on a finite difference scheme, that the underlying homoclinic structure of the NLS equation provides the

50 mechanism responsible for the destabilization of the standard discretizations of the NLS equation. For this reason, the instabihty is refered to as Numerical Homoclinic Instabihty (NHI), see [17,18,19,20]. An important ingredient in our line of argu­ ment is the availability of the integrable discretization of the NLS equation, due to Ablowitz and Ladik [22],

iUn + ^ ( t f » + i + Vn-i - 2Vn) + UnUZ(Un+1 + Un-i) = 0,

(7)

which tends to the cubic NLS in the continuous limit, h —» 0. Due to its integrability, it has its own homoclinic orbits and we find quasi-periodic solutions at all levels of discretization. Theoretical studies of this system may be found in, [22,23], among others. The aforementioned discussion centered around problems in 1+1 dimension. However there are natural extensions to problems in higher dimensions, e.g., the Kadomtsev-Petviashvili (KP) equation (a 2+1 dimensional generalization of KdV), = -3a2Uyy,

(ut + 6uux + uxxx)x

(8)

and the so-called Davey-Stewartson (DS) equations (a 2+1 dimensional generaliza­ tion of the NLS equation),

iut + ! 3 + £«!+; 3=1

(22)

j=0

on the interval - c o < i < oo where a{ denotes values at site i and time t with a' taking on values 0,1 only. The new state at level t + 1 is given by t+1 1

_ \ 0 1 1

if S(ati+1) is odd or zero if S(a* +1 ) is even, nonzero

(23)

The computation is carried out by sweeping from left to right, assuming that at the initial time there is a finite number of nonzero sites, a', and that to the left we always have an infinite number of zeros. We shall refer to r as the radius. Figure 10 shows two particles, initially completely separated by a large number of zeros, emerging intact after an interaction. Note that a ' 1 ' is represented by

66

Figure 11: Solitonic interaction, represented by its sum profile.

a black box and a '0' by a blank space. The resemblence with the KdV soliton interaction of Figure 1 is striking. Note, for instance, a similar behavior in the phase shift. Changing the representation of the CA to plot the 'sum profile' allows the resemblence to be even more pronounced, as shown in Figure 11 (note that the CA rule is not changed). The sum profile at level t is simply given by S{at),

-co
= \Q

S{k

if k is a nonnegative integer otherwise.

,,_» (35)

Since (34) is a pth order scheme we need p initial conditions which we choose as Xj = 0, t = — 1 , . . . , —p. Notice that this choice of initial conditions imply that XQ = 1. This difference formula yields a spatially periodic sequence of the form,

•

POPOPO

•

(36)

where each O is a string of at least r +1 zeros separating the last 1 in one P from the first 1 in the next P . The procedure is then to compute Xi from the linear difference equation, (34), stopping after the first P has been computed; the details may be found in [41]. The formula for period two particles is obtained in a similar spirit and is given by,

,-*. + (1 -*,-,) [(,(;£) + ,(i = £ = ii)

(37)

where p = 2r—d, d\ and d2 denote the displacements during'the first and second time steps respectively, d = d\ + d2 and 6(k) is defined as before. The inital conditions are given as for the period one particle, again leading to a spatially periodic particle that can be truncated to the desired period two particle. For example, if we choose the radius r = 3, with intermediate displacements, di = 1 and d2 = 4, then Jthe period two particle is constructed, using

Xi

= Xi_5 + (1 - 2xj_5) [S ( i ) + S ( i z i ) ] ,

(38)

where the initial conditions are Xi = 0, — 5 < i < — 1. The resulting particle is given by 1000111001011010011100010 where members of the set of box sites are indicated by a bar. Particles with higher periodicity can be constructed in a similar way. However, for these higher periodic particles one has to deal with a phenomenon called splitting.

73 This happens when the particle splits into two or more separate particles, loosing its periodicity. We have recently obtained a characterizaton of situations where this will- happen, in terms of solutions of a diophantine equation [42].

4.4

Soliton Interactions in Higher Dimensions

Many of the ideas of the preceeding subsections can be extended to higher dimensions and here we demonstrate soliton interactions in 2+1 dimensions. Consider a two-dimensional grid and let o{- denotes the value at site (i, j) at time level t. In order to advance to the next time level the one-dimensional PRFA is applied to each of the successive horizontal levels, sweeping from left to right. Next the PRFA is successively applied to each of the vertical levels, sweeping from top to bottom (these sweeping directions are clearly not important). The two-dimensional automaton is represented by its sum profile by adding the horizontal and vertical sums. The main observation is that we find that in multi-dimensions it is easy to construct localized solitary waves. Indeed, interactions of these waves can be solitonic. Figures 12 and 13 show the interaction of two different pairs of particles in 2+1 dimensions (r — 3). We start initially with two localized particles. The time evolutions of the two individual particles, without any interaction, are shown in Figures 12a,b and 13a,b. As the time evolves, the images of the particles at the old positions are frozen. In Figures 12c and 13c the particles are allowed to interact. Again the particles emerge virtually intact from the interaction, apart from a phase shift - the particles behave like solitons. These ideas are generalized in a straightforward manner to three spatial dimen­ sions, or, for that matter, n + 1 dimensions, for any n > 1. We apply the one dimensional PRFA successively in each of the spatial directions in order to advance to the next time level. Again the order in which the rule is applied is not impor­ tant. It is hard to illustrate soliton interactions in dimensions higher than 2+1 in a visually attractive way and the interested reader is referred to [21] for illustrations of soliton interactions in 3+1 dimensions. Finally we note that the way in which the PRFA is generalized here, increases dissipation. By this we mean that the 'energy' (see, [43] for a discussion of the

74

(a)

(b)

(c)

Figure 12: Soliton interaction in 2+1 dimension.

75

(a)

(b)

(c)

Figure 13: Soliton interaction in 2+1 dimension.

76 'energy'associated with the PRFA) decrease is greater in higher dimensions and that a larger number of particles will loose their character under propagation by this rule. Nevertheless, the fact that localized solitary waves exist and the fact that some particles maintain their character, even under interactions as we have seen, merit our interest in these straightforward generalizations. Also, these generalizations are by no means the only possibilities. In fact, 2+1 dimensional soli ton interactions have been observed for rules based on the FRT mentioned above. Although not quite as straightforward as the ones described above, preliminary studies show that they may be less dissipative. Acknowledgements

This work (MJA) is partially supported by the TCSF, Grants

No. DMS-8803471, the Office of Naval Research, Grant No. N00014-88-K-0447 and the Air Force Office of Scientific Research, Grant No. AFOSR-88-0073. One of us (BMH) would like to express his appreciation for the hospitality of the University of Colorado and the support of the University of the Orange Free State and colleagues in the department of Applied Mathematics. fPermanent address: Department of Applied Mathematics, University of the Orange Free State, Bloemfontein 9300, South Africa.

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79 [32] J.T. Stuart and R.C. DiPrima. Proc. R. Soc. London, A362, p27 (1978). [33] C.R. Doering, J.D. Gibbon, D.D. Holm and B.Nicolaenko. Phys. Rev. Lett., 59, p2911 (1987) and Nonlinearity, 1, p279 (1988). [34] H.C. Yuen and B.M. Lake. Phys. Fluids, 18, p956 (1975). [35] H.C. Yuen and W.E. Ferguson. Phys, Fluids, 2 1 , pl275 (1978). [36] R. Hirota. Direct methods of finding exact solutions of nonlinear evolution equations, in Backlund Transformations, R.M. Muira, ed., Lecture Notes in Mathematics 515, Springer -Verlag, New York (1976). [37] N. Ercolani, M.G. Forest and D.W. McLaughlin. Notes on Melnikov integrals for models of the driven pendulum chain. Preprint (1989). [38] T.S. Papatheodorou, M.J. Ablowitz and Y.G. Saridakis. Stud. Appl. Math., 79, pl73 (1988). [39] M.F. Maritz. Soliton behavior in cellular automata and difference equations related to cellular automata. Technical Report 2/88, Department of Applied Mathematics, University of the Orange Free State (1988). [40] A.S. Fokas, E. Papadopoulou, Y. G. Saridakis and M.J. Ablowitz. Interaction of simple particles in soliton cellular automata. Stud. Appl. Math., 8 1 , pl53180 (1989). [41] J. M. Keiser. On the computation of periodic particles for cellular automata. Masters Thesis, Clarkson University (1989). [42] J.M. Keiser and M.J. Ablowitz. Private Communication. [43] C.H. Goldberg. Parity filter automata. Preprint, Department of Computer Science, Princeton University (1987).

80

The Unstable Nonlinear Schrodinger Equation TetsuYAJIMA and Miki WADATI Institute of Physics, College of Arts and Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153, JAPAN.

ABSTRACT A nonlinear evolution equation iqx + qit + 2 | q \2q = 0 , which we term unstable nonlinear Schrodinger equation, is introduced to investigate soliton phenomena in unstable systems. The equation describes a competition between instability and nonlinearity. Initial value problem is solved by the inverse scattering method. Based on the exact results, properties and roles of soli tons in unstable nonlin­ ear systems are discussed.

1. INTRODUCTION In this paper we study a nonlinear evolution equation iqx + qu + 2\q\2q

= 0.

(1.1)

We refer to eq. (1.1) as unstable nonlinear Schrodinger (UNS) equa­ tion [1]. The UNS equation may be considered as a prototype am­ plitude equation for the study of soliton phenomena in unstable sys­ tems. It is easy to notice that interchange of space i and time t in eq. (1.1) leads to the conventional nonlinear Schrodinger equation ift + qXx + 2 | q | 2 q=0.

(1.2)

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We refer to eq. (1.2) as stable nonlinear Schrodinger (SNS) equa­ tion. Since the SNS equation has been proved to be a soliton system (completely integrable system) [2,3], readers might wonder that the solvability of the UNS equation is trivial. However, eq. (1.1) is a second order partial differential equation in time and initial value problem is very different from that of eq. (1.2). To have some insight into the physical significance of the UNS equation (1.1), we proceed from simple analysis on the evolution of a small disturbance. Linearized equation of (1.1) is iqo,x + qo,u = 0 . Substitution of qo(x,t) = Aexp(ikx

(1.3)

— iuit) into (1.3) yields a dis­

persion relation u2 = -k.

(1.4)

Therefore, a small disturbance with positive k components, however small it is, might exponentially grow. Because of this instability, numerical analysis of eq. (1.1) is rather delicate. We set k = 4ri2, q0(x,t)

u = 2irj,

T) > 0 ,

= Ae^2x+2r>i.

(1.5) (1.6)

As time goes on, we have to consider a nonlinear effect due to 2 | q \2 q. To see this, we expand q(x,t) as q{x,t) = e^'iAe2"* + eB^t)

+ e2B2(t) + •••)•

(1-7)

The constant e is a "smallness" parameter for perturbative calcula­ tion and we finally take e = 1. Substituting (1.7) into eq. (1.1), we

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have n-l

- A W +^ r

+ 2 E 5;(t)5m(t)5B_,_m_1(*) = o, 7,m=0

n = l,2,3,---, ,

(1.8) BQ(t) = Ae2*.

(1.9)

We solve (1.8) with (1.9) iteratively and the solution is

Bn(t) = (-l-0r)nAe^^^.

(1.10)

Thus, summing up contributions of all orders, we arrive at «(x,t) = e 4i " 2 * e 2 ' 1 A ( l + E

A?

(-if^r

e4

"'")

n=l

1ir\ e*«V*+»>

(1.11)

cosh(2?7i + p) where with real constants, p and , we have set A = -title*4*

.

(1.12)

This illustrates that the instability in eq.(l.l) will not continue but be suppressed by the nonlinearity. The outline of this paper is the following. In §2, we derive the UNS equation (1.1) from a set of equations describing electron beam plasma. In §3, we solve initial value problem of the UNS equation through the inverse scattering method[4]. In §4, using the results of §3, we investigate properties of solitons in unstable systems. The last section is devoted to concluding remarks.

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2.

D E R I V A T I O N OF T H E U N S E Q U A T I O N

The UNS equation (1.1) was derived in plasma physics[5,6]. We con­ sider a plasma system where an electron beam is injected under high frequency electric field. By the continuity relation and the Bernoulli equation, we have the relations among the density n and the velocity u of electrons in plasma and beam. Using subscripts p and b meaning plasma and beam, respectively, we have ^L

+ V-(npup)

= 0,

(2.1a)

2 £ + (iv V K = - - E W - - ^ Vn, , r

at drib

at

^

*

m

mnp

+ V-(n 6 u 6 ) = 0 ,

(2.1c)

+ (u,V)u l = - i # ) .

at

(2.1b)

*

(2.1d)

m

Here J&W is the electric field, Tp the plasma temperature, m the electron mass and — e the electron charge. In (2.1d), the term in­ cluding the beam temperature is neglected, because we assume that the beam velocity is sufficient large and we take only high frequency part of the field. The densities and velocities are devided into three parts: the average , the high frequency and the low frequency parts. These are distinguished by the superscripts 0, h, and I. We investigate the high frequency part of (2.1). The higher order terms, such as npup

\ of the high frequency part are considered to

be small. Eliminating up and u&, we have



=

aw?

=

v

A

47re

- ^V 2 )4jren£*\ m

(2.4a)

)E 2(1 + K), the condition for k which gives complex u> is

k