KdV ’95: Proceedings of the International Symposium held in Amsterdam, The Netherlands, April 23–26, 1995, to commemorate the centennial of the publication of the equation by and named after Korteweg and de Vries

506 Pages·1995·31.82 MB·English

by  M. J. Ablowitz

KdV '95
PROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM
HELD IN AMSTERDAM, THE NETHERLANDS, 1995
KdV '95
Proceedings of the International Symposium
held in Amsterdam, The Netherlands, April 23-26,1995,
to commemorate the centennial of the publication
of the equation by and named after
Korteweg and de Vries
Edited by Michiel Hazewinkel, Hans W. Capel
and Eduard M. de Jager
Reprinted from Acta Applicandae Mathematicae, Volume 39, 1995
Springer Science+Business Media, B.V.
Library of Congress Cataloging-in-Publication Data
A CLP. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-4011-2 ISBN 978-94-011-0017-5 (eBook)
DOI 10.1007/978-94-011-0017-5
Printed on acid-free paper
All Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover 1st edition 1995
No part of the material protected by this copyright notice may be reproduced or utilized
in any form or by any means, electronic or mechanical, including photocopying, record
ing, or by any information storage and retrieval system, without written permission from
the copyright owner.
Table of Contents
E. M. DE JAGER / Preface
PART I: INVITED PLENARY LECTURES
M. J. ABLOWI1Z, S. CHAKRAVARTY and B. M. HERBST / Integra-
bility, Computation and Applications 5
D. G. CRIGHTON / Applications of KdV 39
L. FADDEEV / Instructive History of the Quantum Inverse Scattering
Method 69
AKIRA HASEGAWA / Optical Solitons in Communications: From Inte-
grability To Controllability 85
A. J. KOX / Korteweg, de Vries, and Dutch Science at the Tum of the
Century 91
I. KRICHEVER / Algebraic-Geometrical Methods in the Theory of In-
tegrable Equations and Their Perturbations 93
MARTIN D. KRUSKAL / An ODE to a PDE: Glories of the KdV Equa-
tion. An Appreciation of the Equation on Its lOOth Birthday! 127
FRANK NUHOFF and HANS CAPEL / The Discrete Korteweg-de Vries
Equation 133
NORMAN J. ZABUSKY / Coherent Structure Visiometrics: From the
Soliton to HEC 159
PART II: INVITED CONTRIBUTIONS
M. BOrn, F. PEMPINELLI, and A. POGREBKOV / The KPI Equation
with Unconstrained Initial Data 175
R. K. BULLOUGH and P. J. CAUDREY / Solitons and the Korteweg-
de Vries Equation: Integrable Systems in 1834-1995 193
FRANCESCO CALOGERO / Integrable Nonlinear Evolution Equations
and Dynamical Systems in Multi dimensions 229
vi TABLE OF CONTENTS
PETIER A. CLARKSON and ELIZABETII L. MANSFIELD I Symme
try Reductions and Exact Solutions of Shallow Water Wave
Equations 245
P. G. ESTEVEZ and S. B. LEBLE I A KdV Equation in 2+ 1 Dimensions:
Painleve Analysis, Solutions and Similarity Reductions 277
A. S. FOKAS I The Korteweg-de Vries Equation and Beyond 295
V. A. GALKIN and V. V. RUSSKIKH I On the Background of Limit Pass
for Korteweg-de Vries Equation as the Dispersion Vanishes 307
F. GESZTESY and H. HOLDEN I On New Trace Formulae for Schrodinger
Operators 315
B. GRAMMATICOS, V. PAPAGEORGIOU, and A. RAMANI I KdV
Equations and Integrability Detectors 335
CHAOHAO GU I Generalized Self-Dual Yang-Mills Flows, Explicit
Solutions and Reductions 349
W. HEREMAN and W. ZHUANG I Symbolic Software for Soliton
Theory 361
B. G. KONOPELCHENKO I Solitons of Curvature 379
R. A. KRAENKEL, 1. G. PEREIRA and M. A. MANNA I The Reductive
Perturbation Method and the Korteweg-de Vries Hierarchy 389
GEOFF A. LATIIAM and EMMA PREVIAT O / Darboux Transforma
tions for Higher-Rank Kadomtsev-Petviashvili and Krichever-No-
vikov Equations 405
YOSHIMASA NAKAMURA and YUH KODAMA I Moment Problem
of Hamburger, Hierarchies of Integrable Systems, and the Positivity
of Tau-Functions 435
F. PEMPINELLI I New Features of Soliton Dynamics in
2 + 1 Dimensions 445
A. Yeo REDNIKOV, M. G. VELARDE, Yu. S. RYAZANTSEV, A. A. NE
POMNYASHCHY and V. N. KURDYUMOV I Cnoidal Wave Trains
and Solitary Waves in a Dissipation-Modified Korteweg-de Vries
Equation 457
JEAN-CLAUDE SAUT I Recent Results on the Generalized Kadomtsev-
Petviashvili Equations 477
RAINER SCHIMMING I An Explicit Expression for the Korteweg-
de Vries Hierarchy 489
LEEN VAN WIJNGAARDEN I Evolving Solitons in Bubbly Flows 507
Acta Applicandae Mathematicae 39: 1-2, 1995. 1
© 1995 Kluwer Academic Publishers.
Preface
The now famous paper "On the Change of Form of Long Waves Advancing
in a Rectangular Canal and on a New Type of Long Stationary Waves" by
D. J. Korteweg and G. de Vries appeared in the Philosophical Magazine a cen
tury ago. For the board of the Dutch Association for Mathematical Physics, this
centennial is a welcome opportunity to commemorate the event with an interna
tional symposium on those developments in mathematics and physics that have
their roots in what is now known as the Korteweg-de Vries equation. The cir
cumstances that Korteweg held the chair of mathematics and mechanics at the
University of Amsterdam from 1881 until 1918 and that de Vries wrote his the
sis on the above-mentioned long waves under the supervision of Korteweg are
motives to organize this symposium in Amsterdam.
The new type of long stationary waves were called cnoidal waves (in analogy
with sinusoidal waves) by Korteweg and de Vries. In the particular case that the
modulus of the Jacobian elliptic function equals one, the wave takes the form
of the "rounded, smooth and well defined heap of water", discovered by Scott
Russell in 1834 and coined much later in 1965 by Zabusky and Kruskal as the
soliton.
Meanwhile, it has become commonplace to state that the rediscovery of the
soliton by Zabusky and Kruskal precursed a breakthrough in nonlinear analysis
and mechanics. Besides a better understanding of nonlinear wave phenomena this
also brought new important developments in the theory of completely integrable
systems.
The board of the Dutch Association of Mathematical Physics and the organiz
ing scientific committee are pleased with the response of many mathematicians
and physicists to the invitation to participate in the symposium; in particular we
thank those participants who have contributed to the symposium by giving a
lecture or presenting a poster.
A very fascinating feature of the theory of solitons lies in the application of
many branches of mathematics, such as analysis, differential geometry, algebraic
geometry, topology and infinite-dimensional algebraic structures. Apart from this,
one should also mention a multitude of applications in physics, for instance in
statistical mechanics, point mechanics, hydrodynamics, optics and field theory.
These proceedings reflect all these aspects and are in line with the aims and
scope that the scientific committee had in mind. Besides this, attention has also
been paid to the interesting history of solitons.
2 PREFACE
The symposium has been held under the auspices of the Royal Netherlands
Academy of Arts and Sciences, KNAW.
Nowadays, a symposium is not possible without external help and funding;
we express our gratitude to the following organizations and institutions for their
generous support, financial or otherwise: the Royal Netherlands Academy of
Arts and Sciences; University of Amsterdam; Municipality of the City of Ams
terdam; Foundation for Fundamental Research of Matter, FOM; Foundation for
Mathematics "Mathematical Centre", SMC; Foundation "Physics"; Centre for
Mathematics and Informatics, CWI; Committee "Nonlinear Systems"; Nether
lands Organization for Scientific Research; Johan Enschede BV, Haarlem; IBM
Nederland BV, Amsterdam; Kluwer Academic Publishers, Dordrecht; OCE van
der Grinten NV, Venlo; PIT Research, Leidschendam; Shell Nederland BV, Rot
terdam.
To give PhD students the opportunity to get the optimal profit from the sym
posium activities, it was considered appropriate to organize during the week
preceding the symposium, an introductory course on integrable systems connect
ed with the KdV equation and its relatives. The lectures of this course will be
published by the Centre for Mathematics and Informatics in the CWI Syllabus
senes.
We are indebted to the following organizations who made this course pos
sible by granting financial support: Erasmus Inter University Cooperation Pro
gramme, ICP 94; Research School JM Burgers Centre, Delft; Research School
MRI, Utrecht; Research School Thomas Stieltjes, Leiden; Dutch Association for
Mathematical Physics, FOMISMC.
Further, it is our duty to acknowledge the secretarial support of Mrs M. 1. van
der Kooij, University of Twente, who conducted the secretariat of the symposium,
Mrs R. Koopmans, Eindhoven University, who assisted the treasurer, and Mrs
Ph. Zijlstra, Universty of Amsterdam, who was engaged with the administration
of the PhD course.
Finally, we highly appreciate that these proceedings have been published as
a special issue of Acta Applicandae Mathematica. It is our pleasure to thank
Kluwer Academic Publishers for this, and to thanks them for their generous
financial support.
E. M. DE JAGER
President Scientific Committee KdV'95
Part I: Invited Plenary Lectures
Acta Applicandae Mathematicae 39: 5-37, 1995. 5
© 1995 Kluwer Academic Publishers.
Integrability, Computation and Applications
M. J. ABLOWITZ, S. CHAKRAVARTY
Program in Applied Mathematics, University of Colorado, Boulder; CO 80309, U.S.A.
and
B. M. HERBST
Department of Applied Mathematics, The University of the Orange Free State,
Bloemfontein 9300, South Africa
(Received: 13 January 1995)
Abstract. The study of integrable systems and the notion of integrability has been re-energized
with the discovery that infinite-dimensional systems such as the Korteweg-<le Vries equation are
integrable. In this paper, the following novel aspects of integrability are described: (i) solutions
of Darboux, Brioschi, Halphen-type systems and their relationships to monodromy problems and
automorphic functions, (ii) computational chaos in integrable systems, (iii) we explain why we
believe that homodinic structures and homoclinic chaos associated with nonlinear integrable wave
problems, will be observed in appropriate laboratory experiments.
Mathematics Subject Classification (1991): 58F07.
Key words: integrability, KdV equation, nonlinear integrable wave functions, monodromy prob
lems, automorphic functions.
1. Introduction
The study of 'integrable' or exactly solvable systems has a long history. Classical
mathematicians such as Euler, Lagrange, Liouville, Riemann, Poincare, Painleve
amongst many others, investigated nonlinear systems which could either be inte
grated more or less explicitly or possessed special analytic structures in the
complex plane. Perhaps surprisingly, there is still no single adequate definition
of 'integrability'. Certainly explicit integration of nonlinear systems in the real
domain should be considered as integrable, as should cases where suitable trans
formations exist to allow an elementary solution, such as occurs in the Hamilto
nian case with action-angle variables (often called integrability in the Liouville
sense). Less clear is the still developing notion of integrability in the complex
plane. For example, if the general solution of an ODE has appropriate analyt
ic properties, such as (i) having poles as the only movable singular points (a
movable singular point is one whose location depends on the initial conditions; a
fixed singular point is fixed by the coefficients in the equation) or more generally
(ii) being everywhere single-valued in its domain of existence, then we consider
the equation to be integrable in the complex plane. Many examples of nonlinear
6
M. I. ABLOWITZ ET AL.
ODEs have shown that when (i) or (ii) are satisfied then the equation falls into
a class in which the general solution can be either obtained by explicit integra
tion or can be linearized via an associated Riemann-Hilbert (RH) factorization
problem.
The investigation of integrable systems has been an interesting and active
field in recent years due to the fact that numerous physically interesting infinite
dimensional systems have been linearized via the method of the Inverse Scattering
Transform (1ST) and large classes of explicit solutions, such as soliton solutions,
have been obtained. The best known example is the Korteweg--de Vries (KdV)
equation,
Ut + 6uux + Uxxx = 0, (1)
which these proceedings have been named after. The KdV equation arises in
many physical problems. It is the canonical equation describing weakly disper
sive and weakly nonlinear wave phenomena arising in shallow water waves (this
was the original application of Korteweg and de Vries), internal waves in fluids,
plasma waves, lattice dynamics, etc.
The 1ST method yields the general solution to the Cauchy problem on the
infinite line corresponding to rapidly decaying initial conditions (cf. [1-3]), a
special case of this is the general N -soliton solution. The 1ST method uses direct
and inverse scattering analysis to obtain a matrix RH factorization problem which
leads to a Gel'fand-Levitan-Marchenko integral equation as the linearization of
KdV. The KdV equation has been investigated in a variety of contexts and we
note that there is one other case in which the general initial-value solution is
known. This is the periodic boundary-value problem where the solution of the
KdV equation can be expressed in terms of Riemann theta functions of arbitrary
genus (cf. [1, 2]).
The 1ST is an effective tool to obtain linearizations and solutions to many
other well known nonlinear wave equations in (1 + 1) dimensions, e.g., the
nonlinear SchrOdinger equation (NLS), sine-Gordon equation, three wave inter
action (1 + 1) equations, etc. Moreover, it is significant that there are many
(2 + 1) -dimensional equations which have solutions via the 1ST method. The
best known are the Kadomtsev-Petviashvili (KP), Davey-Stewartson (DS) and
three wave interaction (2 + 1) equations. We note that the 1ST method needs
to be generalized appropriately in order to handle (2 + 1) problems; it was
found that nonlocal RH problems and DBAR problems played essential roles
(cf. [3]).
In this paper, we shall discuss two issues related to integrability. In the first
part, we shall focus on specific differential equations which have novel proper
ties. These equations are obtained from a well known integrable system in four
dimensions, the so-called self-dual Yang-Mills (SDYM) system. The other topic
we shall discuss involves the computation of integrable systems in the vicinity of
what are called homoclinic manifolds. In these regions, the solutions of the non-