Integrable and Superintegrable Systems

396 Pages·1991·134.614 MB·English

by  Kupershmidt B.A. (ed.)

bitegrable and
Supenntegrable Systems
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Integrable and
Superintegrable Systems
Edited by
Boris A. Kupershmidt
The University of Tennessee Space Institute
World Scientific
Singapo9e°mmWrMm^ndon • 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 that 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 I ® I, {!<*} C (E is an orthonormal
a a
a
base with respect to the Killing form (,)K- Put 1a = a®l®l®...,2a =
2
1 ® a ® 1 ® ... ,... ,'J(a ® fc) = 'aJfc,((a ® fc)c)i<- = a(6, <:)#- for a, fc,c being
from the universal enveloping algebra J7((£) of (E (or from any of its quotient
algebras). We impose the condition 12r(u) + 21r(—u) = 0.
The main soliton theorem. The following 4 assertions are equivalent:
(a)
[13r(u)+23r(v),12r(u-v)} = {l3r(u),23r(v)\ for any u, v E C ; (1)
(b) the relation
CL(u),2L(v)} = [lL(u) + 2L(v),l2r(u-v)} (2)
defines a Poisson bracket on the indeterminate coefficients (and functionals
of them) of the generic element L(u) for the Lie algebra (£ of meromorphic
functions with values in (£;
(c) all the functions
L(u) = £ Res(12r(u - v)2 L(v)) dv) , (3)
r K
where the sum is over the set of the poles of L, L belongs to (£ above, form the
Lie subalgebra (£ C (£;
r
(d) the equations
dG/dui = K^2 ijr(u< - Uj)G (4)
i&
for a function G of u = (u,-) taking values in £/((£) ® ... ® !/((£) are pairwise
compatible (satisfy the cross-derivative integrability conditions). ■
You can prove the theorem with ease. But do not hurry to question its
title. I'll try to convince you that this theorem is a powerful tool.
(a) Yang-Baxter equations. Relation (1) is very interesting by itself. But
I prefer to discuss here only its quantum analog — the so-called Yang-Baxter
identity
12R(u)13R(u + v)23R(v) = 23R(v)13R( + v)12R(u) . (5)
u
This relation results in (1) if R depends on h and R(u;h) = 1 + hr(u) + o(h).
Identity (5) plays the key role for two-dimensional integrable quantum models.
But it does not exhaust its significance. To illustrate it I'll tell a little story.