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Research Reports in Physics

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Ecodynamics Contributions to Theoretical Ecology Editors: w. Wolff, C.-J. Soeder, and F. R. Drepper

Nonlinear Waves 1 Dynamics and Evolution Editors: A. V. Gaponov-Grekhov, M.1. Rabinovich, and J. Engelbrecht

Nonlinear Waves 2 Dynamics and Evolution Editors: A. V. Gaponov-Grekhov, M.1. Rabinovich, and J. Engelbrecht

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Nonlinear Evolution Equations and Dynamical Systems Editors: S. Carillo and O. Ragnisco

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s. Carillo O. Ragnisco (Eds.)

Nonlinear Evolution Equations and Dynamical Systems

With 15 Figures

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Sandra Carillo Dipartimento di Metodi e Modelli Matematici per Ie Scienze Applicate, Universita di Roma "La Sapienza", via A. Scarpa 10,1-00161 Roma, Italy

Orlando Ragnisco Dipartimento di Fisica, Universita di Roma "La Sapienza", P. Ie A. Moro 2, 1-00185 Roma, Italy

ISBN-13:978-3-540-51983-6 e-ISBN-13:978-3-642-84039-5 DO I: 10.1007/978-3-642-84039-5

Libary of Congress Cataloging-in-Publication Data. Nonlinear evolution equations and dynamical sy­stems 1 S. Carillo O. Ragnisco, eds. p. cm.--(Research reports in physics) "Vth Workshop on Nonlinear Evolution Equations and Dynamical Systems, took place July 2-16, 1989 in Crete atthe Orthodox Aca­demy" --Pref. Includes bibliographical references.lSBN-13:978-3-540-51983-6(U.S.:alk.paper)1.Evolution equations, Nonlinear--Congresses. 2. Differentiable dynamical systems--Congresses. I. Carillo, S. (Sandra). 1955 -II. Ragnisco, O. (Orlando), 1946 -III. Workshop on Nonlinear Evolution Equations and Dynamical Systems (5th: 1989: Orthodox Academy) IV. Series. QA377.N658 1990 515'.353--dc20 90-9714

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Preface*

The Fifth Workshop on Nonlinear Evolution Equations and Dynamical Systems took place July 2-16, 1989 in Crete at the Orthodox Academy, a modem building in a splendid environment by the sea near the village of Kolymbari, not far from Chania.

The Workshop was carried out in the same spirit as the previous ones, held in Crete (1980, 1983), Baia Verde near Gallipoli (Italy, 1985), and Balaruc near Montpellier (1987). Its main purpose was to bring together, from all over the world, scientists engaged in research on nonlinear systems, either interested in their underlying mathematical properties or in their physical applications.

Accordingly, many talks were devoted to present methods of solution (like the inverse scattering transform) and to the investigation of structural (geometri­cal and/or algebraic) properties of (continuous and discrete) nonlinear evolution equations. Peculiar nonlinear systems, such as mappings and cellular automata, have also been discussed.

Applications to various fields of physics, namely, quantum field theory, fluid dynamics, general relativity and condensed matter physics have been considered.

A special effort has been made to ensure a large attendance by researchers coming from countries with nonconvertible currency. There were 89 participants from 22 countries: USSR (18), Italy (16), USA (9), Greece (7), Germany (6), UK (6), the Netherlands (5), France (4), Turkey (3), Australia and the Republic of China (2), Bulgaria, Canada, Finland, Japan, Mexico, Poland, People's Republic of China, Spain, Sweden, Switzerland, Yugoslavia (1). Remarkably, almost all participants gave a lecture or presented a poster: indeed, there were 17 long talks (1 hour), 60 short talks (25 minutes) and 11 posters. In addition to the scheduled program, many informal exchanges of ideas and free discussions characterized the workshop which, thus, was rich in opportunities for a fruitful scientific cooperation.

Rome January 1990

F. Calogero D. Levi

A. Verganelakis

*The complete manuscript was received by Springer-Verlag on February 14, 1990

v

Introduction

Research on Nonlinear Dynamical Systems has been widely pursued during the last few decades. The "nonlinear world" has revealed a rich and fascinating phenomenology, whose description either requires novel mathematical tools or entails a revision of concepts and theories developed in the nineteenth century mathematics. Surprisingly, results which, during about fifty years, were thought of as old and out-of-date have recently turned out to be a source for further fruitful investigations.

This book is devoted to current research on the "narrow" but nevertheless il­luminating window on the nonlinear world provided by integrable systems. Even though a worldwide accepted rigorous definition of integrability for nonlinear evolution equations is not yet available, those systems generally termed integrable share many common remarkable mathematical properties: for instance, they are linearizable through the Inverse Scattering Transform, possess a Hamiltonian struc­ture and a Lax representation.

Basic achievements in this field have been the celebrated paper by Zabuski and Kruskal [1] on the "recurrence" phenomenon in the Fermi-Pasta-Ulam nonlinear lattice, and the Lax formulation [2] of the Korteweg-de Vries equation, together with the fundamental results obtained by Gardner, Greene, Kruskal and Miura [3] and Zakharov and Faddeev [4] on the linearization of this equation and its Hamiltonian nature. The existence of stable purely nonlinear modes, the so­called solitons, for integrable Nonlinear Evolution Equations, and their particle­like behavior motivated a large interest among applied scientists; a number of remarkable monographs on the subject is now available (e.g. ref. [5-10]).

As long as the tree was growing, more and more branches originated from its trunk. One of the main ones was the line of research aimed at discovering new integrable systems, both finite- and infinite-dimensional, and at finding solutions by appropriate techniques, such as the Inverse Scattering (Spectral) Transform, the Direct Linearization Method, the Dressing Method, and the Hirota Bilinear Approach. On the other hand, people tried to understand, by means of alge­braic and geometrical tools, the mathematical structure underlying integrability: the role played by bi-Hamiltonian structures, hereditary recursion operators, mas­ter symmetries and infinite-dimensional Lie algebras of Kac-Moody-Virasoro type has been pointed out. Moreover, the investigation of periodic problems in the framework of algebraic geometry allowed to establish a deep connection between integrable systems and string theory, while the search for integrable quantum sys­tems and the related r-matrix formalism provided cornerstones for the construction of the theory of quantum groups.

VII

In our opinion, for the large variety of topics that have been covered, and for the quality of the contributions, these proceedings give a good and up-to-date picture of the state of the art in the field. They are not intended to provide an exhaustive self-contained description of the whole subject, but rather to give an outline of the most recent and relevant results in a way that, hopefully, should stimulate the interested reader (not necessarily a specialist) to get further acquainted with this exciting domain. Thus, most of the contributions are rather short, but each of them is followed by a long list of references.

The entire material has been rearranged and divided into sections in order to provide a guideline for the reader. However, not all the papers, and not all the lectures in Kolymbari, fit exactly into a single section. Indeed, in some cases the subject covered by the author(s) is related to two or even three different sections. Thus, we tried to identify the most characteristic features in each contribution.

In the first section all the papers referring to multidimensional integrable sys­tems are collected: a major role is played by the recent discovery of coherent structures in two space dimensions, and by their mathematical description. In particular, special solutions of the Davey-Stewartson (DS) equation have been presented and discussed by A.S. Fokas, P.M. Santini and by M. Boiti et al.; the origin of boundary conditions for DS systems is discussed by M.J. Ablowitz, S.V. Manakov and C.L. Shultz. Further interesting results have been obtained by c.R. Gilson et aI., who studied rational solutions to the Kadomtsev-Petviashvili (KP) equation, and by Z. Jiang, who proposed a systematic construction of scattering data in two space dimensions.

The second section comprises the contributions concerned with the problem of establishing criteria and tests of integrability. Accordingly, Painleve analy­sis, Lie-point symmetries approach and the Hirota method are applied and dis­cussed. Specifically, J. Hietarinta introduces the notion of "Hirota integrability"; P.A. Clarkson discusses new similarity reductions for the Boussinesq and KP equation, and P. Broadbridge shows how a computer algebra assisted search for Lie-Backlund symmetries may help in finding exact solutions.

The third section is mainly devoted to the Inverse Scattering (or Spectral) approach: it was, however, our choice not to include the already mentioned con­tributions on the solution of the Davey-Stewartson equation, to emphasize the physical (two space dimensions) context with respect to the used mathematical techniques. Hence, this section contains essentially 1+ I-dimensional problems.

Some papers are centered on the spectral transform itself, while in other ones special solutions of certain nonlinear PDEs are constructed.

To the former class pertain the studies on the semi-infinite Toda-Iattice by Y.M. Berezanski, on the elliptic sinh-Gordon equation by M. Jaworski and D. Kaup, on the Marchenko equation and its approximate solutions by D. Atkinson and on a suitably perturbed Korteweg-de Vries equation by V.K. Mel'nikov. The papers by J.H. Lee on a linear system of Zakharov-Shabat type, by U. Mugan and A.S. Fokas on a Riemann-Hilbert boundary value problem related to the third Painleve equation, and by B.G. Konopelchenko on nonlinear evolution equations for eigenfunctions of the Lax operator can also be grouped here.

VIII

To the latter class can be ascribed the results on periodic solutions for the Nonlinear SchrOdinger Equation by J.J. Lee and on the N double pole solution for the modified Korteweg-de Vries equation by K. Konno.

Here the contributions by F. Calogero and by S. De Lillo on C-integrable systems (Le., systems solvable by an explicit change of variables) are also included. Although they do not rely upon the solution of an underlying spectral problem, they are indeed based on a linearization procedure!

The fourth section, by far the largest of these proceedings, concerns the al­gebraic and geometrical aspects of integrability. Integrability is understood to be an extension to nonlinear partial differential (or differential-difference) equations of the notion of Liouville integrability in Hamiltonian Classical Mechanics. New developments under this perspective are collected here.

The symplectic structure of the multisoliton manifold related to completely in­tegrable systems is investigated by B. Fuchssteiner. Then, the role of the so-called master-symmetries and, in particular, their relevance to construction of the canon­ical action-angle variables for integrable systems is studied in the contributions by S. Carillo and B. Fuchssteiner and by G. Oevel, B. Fuchssteiner and M. Blaszak:. In this context a computer algebra algorithm to compute master-symmetries is described by W. Wiwianka and B. Fuchssteiner.

A unifying approach to integrable systems is pursued, by different techniques (geometric reduction theory, classical Yang-Baxter algebra), in the papers by W. Oevel and O. Ragnisco, O. Ragnisco and P. Santini and C. Morosi and G. Tondo. Symplectic operators and associated representations of the triangle group are dis­cussed by I. Dorfman.

The Hamiltonian Structure related to polynomial spectral problems is consid­ered by A.P. Fordy, while M. Antonowicz and M. Blaszak report on a non-standard Hamiltonian description of integrable systems. Super-Hamiltonian operators are investigated by E.D. Van der Lende and H.G.J. Pijls. Grinewich and Orlov present results on the action of Virasoro group on Riemann surfaces that are also relevant for string theories.

An unusual application of combinatorial results to Hamiltonian systems is presented by G.Z. Th. The section closes with the contribution by G. Gorni and G. Zampieri on "cone-potentials"; this is the sole paper in these proceedings devoted to Classical Hamiltonian Systems with a finite number of degrees of freedom.

In the fifth section some "unconventional" papers, mainly concerned with map­pings, are collected. F.W. Nijhoff et al. report on a class of integrable systems in a three dimensional lattice regarded as an example of a class of integrable non­linear mappings. Some exciting results describing the soliton-like behavior of a special kind of cellular automata (the so-called filter automata) are presented by E. Papadopoulou et al. A cubic mapping is studied by P. Petek, and, last, the stability of soliton-like solutions is investigated by F.V. Kusmartsev.

In the last section some applications of the concepts and tools of integrable systems in various physical settings are collected. In the context of Quantum Field Theory, M. Olshanetsky proposes a Lie-algebraic approach to the Wess-Zumino­Witten model, while H.J. Munczek and D.W. McKay consider a composite-field

IX

model in Quantum Chromodymamics. An application to ocean dynamics is pre­sented by P. Ripa; also in relation with fluid dynamics, a forced KdV type of equation is considered by R. Grimshaw. An application to solid state physics is studied by V.G. Bar'yakhtar et al.. The volume closes with an application to gen­eral relativity: B. Gaffet discusses the completely integrable Einstein equations in the stationary case (Ernst equation).

Rome January 1990

References

Sandra Carillo Orlando Ragnisco

[1) NJ. Zabusky, MD. Kruskal: Interaction of solitons in a collisionless pJasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240-243 (1984)

[2) PD. Lax: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467-490 (1968)

[3) C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura: Method for solving the Koneweg-de Vries equations. Phys. Rev. Lett., 19, 1095-1097 (1967)

[4) V.E. Zakharov, LD. Fadeev: Koneweg-de Vries equation, a completely integrable Hamiltonian system. Funct. Anal. Appl. 5, 280-287 (1971)

[5) F. calogero, A. Degasperis: "Spectral transform and solitons I", Studies in Mathematics and Its Application, Vol. 13 (North-Holland, Amsterdam 1980)

[6) V.E. Zak:ltarov, S.V. Manakov, S.P. Novikov, L.P. Pitayevsky: Theory of Solitons. The Method of the Inverse Scattering Problem (Nauka, Moscow 1980)

[7) MJ. Ablowitz, H. Segur: Solitons and the Inverse Scattering Transform (Siam, Philadelphia 1981) [8) R.K. Dodd, J.C. Eilheck, J.D. Gibbon, H.C. Morris: Solitons and Nonlinear Wave Equations (Academic, London

1982) [9) A.C. Newell: Solitons in Mathematics and Physics (Siam, Philadelphia 1985) [10) LD. Fadeev, L.A. Takhtajan: Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, Heidelberg. New

York 1987)

x

Contents

Part I Integrable Systems in (2 + 1)-Dimensions

Solitons and Dromions, Coherent Structures in a Nonlinear World By P.M. Santini and A.S. Fokas (With 2 Figures) ................ 2

Boundary Value Problems in 1 + 1 and in 2 + 1, the Dressing Method, and Cellular Automata By A.S. Fokas (With 2 Figures) ............................ 14

Exponentially Localized Solitons in 2 + 1 Dimensions By M. Boiti, J. Leon, L. Martina, and F. Pempinelli .............. 26

On the Boundary Conditions of the Davey-Stewartson Equation By M.J. Ablowitz, C.L. Shultz, and S.V. Manakov ............... 29

Rational Solutions to the 1\vo-Component K-P Hierarchies By C.R. Gilson, J.J.C. Nimmo, and N.C. Freeman . . . . . . . . . . . . . . . . 32

Construction of Inverse Data in Multidimensions By Zhuhan Jiang ...................................... 36

Part IT Criteria and Tests of Integrability: Painleve Property, Hirota Method, Lie-Backlund Symmetries

Examples of Nonclassical Similarity Reductions By P.A. Clarkson ...................................... 42

Equations That Pass Hirota's Three-Soliton Condition and Other Tests of Integrability By J. Hietarinta ....................................... 46

Selection of Solvable Nonlinear Evolution Equations by Systematic Searches for Lie Backlund Symmetries By P. Broadbridge ..................................... 51

Part 1lI Spectral Methods and Related Topics, C-Integrable Systems

Inverse Problems of Spectral Analysis and the Integration of Nonlinear Equations By Yu.M. Berezansky ................................... 56

XI

The Inverse Scattering Transform for the Elliptic Sinh-Gordon Equation By M. Jaworski and D. Kaup .............................. 64

Reflection Coefficients and Poles By D. Atkinson ....................................... 68

A N x N Zakharov-Shabat System with a Quadratic Spectral Parameter By Jyh-Hao Lee ....................................... 73

On Integration of the Korteweg-de Vries Equation with a Self-consistent Source By V.K. Mel'nikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

On the Initial Value Problem of the Third Painleve Equation By U. Mugan and A.S. Fokas (With 1 Figure) .................. 82

Nonlinear Equations for Soliton Eigenfunctions Are the 1ST Integrable Equations By B.G. Konopelchenko ................................. 87

The Geometry and Completeness of the Two-Phase Solutions of the Nonlinear SchrOdinger Equations By J.E. Lee and M.P. Tsui ............. . . . . . . . . . . . . . . . . . . . 94

N Double Pole Solution and Its Initial Value Problem for the Modified Korteweg-de Vries Equation By K. Konno ......................................... 98

C-Integrable Generalization of a System of Nonlinear PDE's Describing Nonresonant N-Wave Interactions By F. Calogero ....................................... 102

The Burgers Equation: Initial/Boundary Value Problems on the Semiline By S. De Lillo ........................................ 105

PartN Algebraic Approach to Integrability and Hamiltonian Theory

The Tangent Bundle for Multisolitons: Ideal Structure for Completely Integrable Systems By B. Fuchssteiner ..................................... 114

Action-Angle Variables and Asymptotic Data By G. Oevel, B. Fuchssteiner, and M. Bl'aszak .................. 123

The Action-Angle Transformation for the Korteweg-de Vries Equation By S. Carillo and B. Fuchssteiner ........................... 127

Algorithms to Detect Complete Integrability in 1 + 1 Dimension By W. Wiwianka and B. Fuchssteiner ........................ 131

GN Manifolds, Yang-Baxter Equations and ILW Hierarchies By C. Morosi and G. Tondo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 136

XII

Integral and Discrete Evolution Equations: A Unified Approach By O. Ragnisco and P.M. Santini ........................... 140

An Abstract Tri-Hamiltonian Lax Hierarchy By W. Oevel and o. Ragnisco ............................. 144

On Symplectic and Hamiltonian Differential Operators By I.Ya. Dorfman ...................................... 148

On a Non-Standard Hamiltonian Description of NLEE By M. Antonowicz and M. Bfaszak . . . . . . . . . . . . . . . . . . . . . . . . .. 152

Energy Dependent Spectral Problems: Their Hamiltonian Structures, Miura Maps and Master Symmetries By A.P. Fordy (With 1 Figure) ............................. 157

Super Hamiltonian Operators and Lie Superalgebras By E.D. van der Lende and H.G.I. Pijls ....................... 161

Higher (Non-isospectral) Symmetries of the Kadomtsev-Petviashvili Equations and the Virasoro Action on Riemann Surfaces By P.G. Grinevich and A.Yu. Orlov ......................... 165

A Combinatorial Rule to Hirota's Bilinear Equations By Th Guizhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 170

Liouville-Arnold Integrability for Scattering Under Cone Potentials By G. Gorni and G. Zampieri . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 173

Part V Mappings, Cellular Automata and Solitons

Lattice Equations and Integrable Mappings By V.G. Papageorgiou, F.W. Nijhoff, and H.W. Capel (With 1 Figure) 182

Recent Developments in Soliton Cellular Automata By E.P. Papadopoulou (With 3 Figures) ....................... 186

Cubic Equation, Newton's Method and Analytic Functions By P. Petek .......................................... 190

Singularity of Differential Mappings and Stability of Solitons By F.V. Kusmartsev (With 3 Figures) ........................ 195

Part VI Physical Applications

Action-Angle Variables in the Quantum Wess-Zumino-Witten Model By M. Olshanetsky ..................................... 202

On the Derivation of Propagator and Bound State Equations and S-Matrix Elements for Composite States By H.J. Munczek and D.W. McKay ......................... 205

XIII

Resonant Flow over Topography By R. Grimshaw (With 2 Figures)

Taxonomy of Ocean Stability Conditions

209

By P. Ripa .......................................... 212

Kinetic Equations and Soliton Diffusion in Low-Dimensional Magnets By V.G. Bar'yakhtar, B.A. Ivanov, A.K. Kolezhuk, and E.V. Tartakovskaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 216

On Einstein's Equations with Two Commuting Killing Vectors By B. Gaffet ......................................... 219

SUbject Index ........................................ 225

List of Participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 227

Index of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 233

XIV