Symmetries and integrability of difference equations

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Geometric group theory II, G.A. NIBLO & M.A. ROLLER (eds) Shintani zeta functions, A. YUKIE Arithmetical functions, W. SCHWARZ & J. SPILKER Representations of solvable groups, O. MANZ & T.R. WOLF Complexity: knots, colourings and counting, D.J.A. WELSH Surveys in combinatorics, 1993, K. WALKER (ed) Local analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY Polynomial invariants of finite groups, D.J. BENSON Finite geometry and combinatorics, F. DE CLERCK et al Symplectic geometry, D. SALAMON (ed) Independent random variables and rearrangement invariant spaces, M. BRAVERMAN Arithmetic of blowup algebras, WOLMER VASCONCELOS Microlocal analysis for differential operators, A. GRIGIS & J. SJOSTRAND Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI, W. METZLER & A.J. SIERADSKI (eds) The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN Invariant potential theory in the unit ball of Cn, MANFRED STOLL The Grothendieck theory of dessins d'enfant, L. SCHNEPS (ed) Singularities, JEAN-PAUL BRASSELET (ed) The technique of pseudodifferential operators, H.O. CORDES Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMITH Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT & J. HOWIE (eds) Ergodic theory and its connections with harmonic analysis, K. PETERSEN & I. SALAMA (eds) An introduction to noncommutative differential geometry and its physical applications, J. MADORE Groups of Lie type and their geometries, W.M. KANTOR & L. DIMARTINO (eds) Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD & W.M. OXBURY (eds) Arithmetic of diagonal hypersurfaces over finite fields, F.Q. GOUVEA & N. YUI Hilbert C*-modules, E.C. LANCE Groups 93 Galway / St Andrews I, CM. CAMPBELL et al (eds) Groups 93 Galway / St Andrews II, CM. CAMPBELL et al (eds) Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, V. KOWALENKO, N.E. FRANKEL, M.L. GLASSER & T. TAUCHER Number theory 1992-93, S. DAVID (ed) Stochastic partial differential equations, A. ETHERIDGE (ed) Quadratic forms with applications to algebraic geometry and topology, A. PFISTER Surveys in combinatorics, 1995, PETER ROWLINSON (ed) Algebraic set theory, A. JOYAL & I. MOERDIJK Harmonic approximation, S.J. GARDINER Advances in linear logic, J.-Y. GIRARD, Y. LAFONT & L. REGNIER (eds) Analytic semigroups and semilinear initial boundary value problems, KAZUAKITAIRA Computability, enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN & S.S. WAINER (eds) A mathematical introduction to string theory, S. ALBEVERIO, J. JOST, S. PAYCHA, S. SCARLATTI Novikov conjectures, index theorems and rigidity I, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Novikov conjectures, index theorems and rigidity II, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Ergodic theory of Zd actions, M. POLLICOTT & K. SCHMIDT (eds) Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E.V. FLYNN Semigroup theory and its applications, K.H. HOFMANN & M.W. MISLOVE (eds) The descriptive set theory of Polish group actions, H. BECKER & A.S. KECHRIS Finite fields and applications, S. COHEN & H. NIEDERREITER (eds) Introduction to subfactors, V. JONES & V.S. SUNDER Number theory 1993-94, S. DAVID (ed) The James forest, H. FETTER & B. GAMBOA DE BUEN Sieve methods, exponential sums, and their applications in number theory, G.R.H. GREAVES, G. HARMAN & M.N. HUXLEY (eds) Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TODOROV (eds) Clifford algebras and spinors, P. LOUNESTO Stable groups, FRANK O. WAGNER Surveys in combinatorics, 1997, R.A. BAILEY (ed) Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds) Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds) Model theory of groups and automorphism groups, D. EVANS (ed) Geometry, combinatorial designs and related structures, J.W.P. HIRSCHFELD, S.S. MAGLIVERAS & M.J. DE RESMINI (eds) p-Automorphisms of finite p-groups, E.I. KHUKHRO Analytic number theory, Y. MOTOHASHI (ed) Tame topology and o-minimal structures, LOU VAN DEN DRIES The atlas of finite groups: ten years on, ROBERT CURTIS & ROBERT WILSON (eds) Characters and blocks of finite groups, G. NAVARRO Grobner bases and applications, B. BUCHBERGER & F. WINKLER (eds) Geometry and cohomology in group theory, P. KROPHOLLER, G. NIBLO, R. STOHR (eds) The ^-Schur algebra, S.DONKIN Galois representations in arithmetic algebraic geometry, A.J. SCHOLL & R.L. TAYLOR (eds) Symmetries and integrability of difference equations, P.A. CLARKSON & F.W. NIJHOFF (eds)

London Mathematical Society Lecture Note Series. 255

Symmetries and Integrability of Difference Equations

Edited by Peter A. Clarkson University of Kent at Canterbury Frank W. Nijhoff University of Leeds

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521596992 © Cambridge University Press 1999 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 A catalogue recordfor this publication is available from the British Library ISBN-13 978-0-521-59699-2 paperback ISBN-10 0-521-59699-8 paperback Transferred to digital printing 2006 Cambridge University Press has no responsibility for the persistence or accuracy of email addresses referred to in this publication.

Contents Preface

ix

List of Participants

xv

1 Partial Difference Equations Discrete linearisable Gambier equations A.K. Common, E. Hesameddini and M. Musette

3

Generalized Backlund transformation a n d new explicit solutions of t h e two-dimensional Toda equation X.-B. Hu and P.A. Clarkson 15 Different aspects of relativistic Toda chain S. Kharchev, A. Mironov and A. Zhedanov

23

2 Integrable Mappings Integrable symplectic maps A.P. Fordy

43

An iterative process on quartics a n d integrable symplectic maps J.P. Frangoise and 0. Ragnisco 56 Integrable mappings of K d V t y p e a n d hyperelliptic addition formulae F. W. Nijhoff and V.Z. Enolskii 64 i?-matrix hierarchies, integrable lattice systems a n d their integrable discretizations Yu.B. Suris 79

3 Discrete Geometry Discrete conformal maps and surfaces A.I. Bobenko

97

T h e Backlund transformation for discrete isothermic surfaces J. Cieslinski 109

vi

Contents

Integrable discrete geometry with ruler and compass A. Doliwa

122

Self-dual Einstein spaces and a discrete Tzitzeica equation, a permutability theorem link W. Schief

137

4 Asymptotic Analysis New solutions of nonstationary Schrodinger and KadomtsevPetviashvili equations M.J. Ablowitz and J. Villaroel

151

On asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert method P. Bleher and A. Its

165

A new spectral transform for solving the continuous and spatially discrete heat equations on simple trees P. Bressloff and A.S. Fokas

178

5 Discrete Painleve Equations The discrete Painleve I hierarchy C. Cresswell and N. Joshi

197

Rational solutions to dPIV J. Hietarinta and K. Kajiwara

206

The discrete Painleve II equation and the classical special functions K. Kajiwara

217

Freud's equations for orthogonal polynomials as discrete Painleve equations A. Magnus

228

Contents

vii

6 Symmetries of Difference Equations An approach to master symmetries of lattice equations B. Fuchssteiner and W.-X. Ma

247

Symmetries and generalized symmetries for discrete dynamical systems D. Levi 261 Nonlinear difference equations with superposition formulae P. Winternitz 275

7 Numerical Methods and Miscellaneous Generalised Painleve truncation: expansion in Riccati pseudopotentials A. P. Fordy and A. Pickering 287 Symplectic Runge-Kutta schemes W. Oevel

299

8 Cellular Automata Soliton Cellular Automata C. Gilson

313

Painleve equations and cellular automata B. Grammaticos and A. Ramani

325

Soliton cellular automaton S. Moriwaki, A. Nagai, J. Satsuma, T. Tokihiro, M. Torii, D. Takahashi and J. Matsukidaira

334

viii

Contents

9 q-Special Functions and g-Difference Equations Fourier-Gauss transform of some ^-special functions N.M. Atakishiyev

345

The Wilson bispectral involution: some elementary examples F.A. Grunbaum and L. Haine

353

Factorisation of Macdonald polynomials V.B. Kuznetsov and E.K. Sklyanin

370

10 Quantum Aspects and Yang-Baxter Equations Schrodinger equation on quantum homogeneous spaces F. Bonechi, R. Giachetti, E. Sorace and M. Tarlini

387

Local Yang-Baxter relations associated with Hirota's discrete equation R.M. Kashaev

402

Some algebraic solutions of discrete equations from anticommuting variables C. Viallet

410

^-combinatorics and quantum integrability A.Yu. Volkov

420

Preface The second international conference on "Symmetries and Integrability of Difference Equations" (SIDE II) was held at the University of Kent at Canterbury, July 1-5, 1996. It was the successor of a first meeting on the same topic held in Esterel (Quebec, Canada), under the auspices of the Centre de Recherches Mathematiques (CRM) of the University of Montreal in 1994, cf. the Proceedings of that meeting, [1]. Like in Esterel, this SIDEII meeting aimed at bringing together researchers working in the general field of discrete systems and difference equations with emphasis on symmetries and integrability. The subject area of the meeting is relatively young: in the last decade, and particularly during the last few years, a great deal of progress has been made on the mathematical aspects of discrete integrable systems, including integrable dynamical mappings, ordinary and partial difference equations, lattice solitons, discrete versions of the Painleve equations, symmetry approaches and singularity analysis, and applications to numerical analysis, computer science and Physics. The two SIDE meetings have brought together many leading experts in the various aspects of this field, coming from quite different backgrounds. As such the interdisciplinary nature of the meeting is reflected in the present volume, marking contributions in a large variety of fields. It is important to give some explanation as to why we believe that combining the efforts in the fields of discrete systems is significant, especially in this day and age. From a general perspective, research into discrete systems has greatly lagged behind some other fields, notably the far more developed theories of ordinary and partial differential equations. One should not forget, of course, the ambitious programme of the Birkhoff school at the beginning of the century, to develop the analytic aspects of linear difference equations (cf. [8]). The great tradition of the early 1900's, with the works of Painleve and his school [19, 20, 21, 22], dealing with the classification of ordinary differential equations, was in line with the spirit of developing analogous theories for difference equations. Furthermore, a vast amount of knowledge, dating back to the classic works of the 19th century and earlier, has been accumulated, mostly relating to finite-element methods and the classic techniques of numerical analysis (cf. [9]). It is somehow tragic that this tradition doesn't seem to have been continued in recent years, with the notable exceptions of the fields of ^-special functions and orthogonal polynomials (cf. [23, 24, 25, 26]). From the point of view of algebra and geometry, symmetry approaches

x

Preface

have been at the heart of the development of effective methods for integrating differential equations (cf. [12, 13, 14]). However, no comprehensive theory of symmetry methods exists to date for difference equations, although new impetus has been given to such a quest in recent years (cf. [15]). It is only now that symmetry reduction techniques, which have nowadays become so very much a standard tool in the analysis of partial differential equations, are being developed for difference and differential-difference equations. One may ask what is the reason for this difference in development between the comparable theories for differential equations on the one hand and difference equations on the other hand. One explanation is probably the lack of motivation. Whereas in the "classical era", mathematicians such as Birkhoff naturally considered difference equations alongside the "allied" theories of (continuous) differential equations, after the second world war mathematical theories were dominated by continuum models. Thus there was no longer as much interest in studying the admittedly more difficult theory of difference equations. Also in physics, there has been an overall domination of continuum models to describe the processes of nature. It is understandable from one perspective: the continuum is the state of matter as it appears to us in our daily observation of nature, and in order to capture the smoothness of all movement and the coherence of all material we need the mathematical tools of differential calculus. This point of view reached its cumulation in theoretical high energy physics, especially in the fifties and sixties, with the significant developments of relativistic quantum field theory, a continuum theory in extremis, which was expected to provide the basis for the description for the fundamental forces in nature. However, it is known now that this theory has severe limitations exactly because of the difficulties of uniting on the one hand quantum mechanics, which is inherently an algebraic and discrete theory, with the notions of continuum functions on the other. One step has been the development of string theory, and more recently the theory of membranes, where in a partial fashion the pointwise structure of quantum fields has been generalised by allowing extended objects as our fundamental building blocks of nature. But even this may not be enough and we should contemplate taking the bold step of acknowledging the assumption that at the quantum mechanical level the continuum description of space-time actually is no longer valid and that an inherently discrete description should take precedence. This point of view, revolutionary as it might seem, has actually been proposed on many occasions (cf. [2, 3]), and there have been recent attempts to put these ideas into practice (cf. [7]). From a general perspective,

Preface

XI

one needs a much more developed theory of difference equations to be able to classify ordinary and partial difference equations, to develop the analytical tools to study their solutions and to get control over the intrinsically nonlocal features of their behaviour. For this one needs examples that can be rigorously treated. Thus we are led to study exactly integrable discrete systems in all their appearances. The SIDE meetings, the second of which the present volume gives an account, try to bring together and unite the various developments in the different areas of research where discrete systems play a role. In such a way, we have seen in recent years on the discrete level the unification of the theory of ordinary differential equations, notably the Painleve equations, with the theory of orthogonal polynomials, [24, 25], the unification of numerical analysis [26] and convergence algorithms with the theory of integrable discrete lattices, the unification of difference geometry [16], with graph theory and combinatorics, unification of quantum field theory with the theory of g-special functions and the unification of soliton equations and cellular automata viz. neural networks. This is the aim of the present volume: to demonstrate the linking elements between the various disciplines dealing with discrete systems. The contributions are written in such a way that they give a brief overview of the state of the art whilst reporting some original research in the subject area. In this way, we hope that the volume can on the one hand assist to familiarise young researcher with this relatively new field of research, and on the other hand serve as a benchmark for the present-day understanding in the various fields. Hopefully, the volume forms an inspiration for further research and so help to establish the links between the various communities working on discrete systems. The contributions to the present volume cover roughly the following topics: 1. Special Functions and Difference Equations. Discrete Painleve equations, difference- and q-difference orthogonal polynomials, separation of variables. 2. Algebraic aspects. Quantum algebras and representation theory, associated special functions. 3. Computational and Numerical Aspects. Formal theory of orthogonal polynomials, soliton cellular automata, symplectic and volume-preserving integrators, integrable versus nonintegrable discretisations in computation.

Preface

Xll

4. Symmetry aspects. Symmetries of difference equations, similarity reductions, integration techniques through symmetries. 5. Analytic aspects. Analysis of difference equations, isomonodromic deformation theory for discrete systems, asymptotics of orthogonal polynomials. 6. Geometry. Discrete curves and surfaces, connections with discrete soliton systems, visualisation techniques. 7. Applications. Neural networks, coding theory and cryptology, data compression, mathematical biology and economics, integrable algorithms. All these subjects are highly interconnected, and it is the objective of the present volume to demonstrate clearly to the readers how much all these issues are intertwined. We would like to thank Dr Elizabeth Mansfield for her considerable assistance in the organisation of the meeting and doing most of the "running around", which was especially appreciated as one of us couldonly get around in a wheelchair at the time of the meeting. We also thank Pilar Gordoa, Koryn Grant, Andrew Hicks, Michael Ody, Andrew Pickering, Thomas Priestley, Barry Vowden and Helen Webster for their help with the meeting. The meeting was supported by grants from the London Mathematical Society and the University of Kent at Canterbury Research Fund, which are both gratefully acknowledged.

Peter A Clarkson (University of Kent at Canterbury) Frank W Nijhoff (University of Leeds)

References [1] D. Levi, L. Vinet and P. Winternitz (Editors), "Symmetries and Integrability of Difference Equations", Proceedings of the first SIDE meeting, Esterel, Quebec, Canada, May 1995, CRM Proceedings and Lecture Notes Series, 9 American Mathematical Society, Providence, RI (1996).

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[2] A. Einstein, Physics and Reality. Printed in: Essays in Physics, Philosophical Library, New York, (1950). [3] R.P. Feynman, Simulating Physics with Computers, Int. J. Theor. Phys. 21 (1982) 467. [4] Y. Nambu, Field Theory and Galois' Fields, Int. J. Theor. Phys. 21 (1982) 625-636. [5] T.D. Lee, Can time be a discrete variable? Phys. Lett. 122B (1983) 217-220. [6] G. 't Hooft, Quantization of Discrete Deterministic Theories by Hilbert Space Extension, Nucl. Phys. B342 (1990) 471-485. [7] G. 't Hooft, K. Isler and S. Kalitzin, Quantum Field theoretic behaviour of a deterministic cellular automaton, Nucl. Phys. B386 (1992) 495-519. [8] G.D. Birkhoff, General theory of linear difference equations, Trans. Amer. Math. Soc. 12 (1911) 243-284; The generalized Riemann Problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. Am. Acad. Arts Sci. 49 (1913) 521-568. [9] N.E. Norlund, (1923).

Vorlesungen u'ber Differenzenrechnung,

Kopenhagen

[10] E.L. Ince, "Ordinary Differential Equations", Dover, New York (1956). [11] R.L. Devaney, "An Introduction to Chaotic Dynamical Addison-Wesley, New York (1989).

Systems",

[12] S. Lie, "Theorie der Transformationengruppen", B.G. Teubner Verlag, Leipzig (1888, 1890, 1893). [13] P.J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts Math., vol. 107, Springer-Verlag, New York (1993). [14] G.W. Bluman and S. Kumei, "Symmetries and Differential Equations", Appl. Math. Sci., vol. 81, Springer-Verlag, New York (1989). [15] S. Maeda, The similarity method for J. Appl. Math. 38 (1987) 129-134.

difference equations IMA

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[16] R. Sauer, " Differenzengeometrie", Springer-Verlag, Berlin (1970). [17] L. Bianchi, "Lezioni di geometria differenziale", Zanechelli, Bologna (1929). [18] G. Darboux, "Legons sur la theorie generate des surfaces et les applications geometriques du calcul infinitesimaF, vol. 1—4, Gauthier-Villars, Paris (1887-1896). [19] P. Painleve, Memoire sur les equations differentielles dont Vintegrale generate est uniforme, Bull. Soc. Math. France 28 (1900) 201-261; Sur les equations differentielles du second ordre et d1ordre superieur dont Vintegrale generate est uniforme, Acta Math. 25 (1902) 1-85. [20] B. Gambier, Sur les equations differentielles du second ordre et du premier degre dont Vintegrale generate est a points critiques fixes, Acta Math. 33 (1909) 1-55. [21] J. Chazy, Sur les equations differentielles du troisieme ordre et d'ordre superieur dont Vintegrale generate a ses points critiques fixes, Acta Math. 34 (1911) 317-385. [22] M.R. Gamier, Sur des equations differentielles du troisieme ordre dont Vintegrale est uniforme et sur une classe d'equations nouvelles d'ordere superieur dont Vintegrale a ses points critiques fixes, Ann. Sci. de l'ENS vol. X X I X , # 3, (1912), 1-126. [23] G. Gasper and M. Rahman, "Basic Hypergeometric Series", Cambridge University Press, Cambridge (1990). [24] G. Szego, "Orthogonal Polynomials", Amer. Math. Soc. Colloq. Publ., vol. 23, Providence, RI (1982). [25] G. Freud, "Orthogonal Polynomials", Pergamon Press, New York (1971). [26] C. Brezinski, "Pade-Type Approximation and General Orthogonal Polynomials", (Birkhauser, Boston, 1980).

Participants Name Ablowitz, Mark Atakishiyev, Natig Bobenko, Alexander Bullough, Robin Capel, Hans Cieslinski, Jan Clarkson, Peter Common, Alan Doliwa, Adam Dorodnitsyn, Vladimir Estevez, Pilar Fokas, Athanassios Fordy, Allan Gandarias, Maria Gibbons, John Gilson, Claire Gordoa, Pilar Grammaticos, Basil Grant, Koryn Griinbaum Alberto Harnad, John Hesameddini, Esmaeil Hicks, Andrew Hietarinta, Jar mo Hu, Xing-Biao Hydon, Peter Its, Alexander Kajiwara, Kenji Kashaev, Rinat Kruskal, Martin Kuznetsov, Vadim Lambert, Franklin Levi, Decio Luther, Gregory

Affiliation University of Colorado, Boulder, USA Universidad Nacional Autonoma de Mexico, Mexico Technische Universitat Berlin, Germany UMIST, Manchester, UK University of Amsterdam, Netherlands Warsaw University Division in Bialystok, Poland University of Kent, Canterbury, UK University of Kent, Canterbury, UK Warsaw University, Poland Keldysh Institute of Applied Mathematics, Moscow, Russia Universidad de Salamanca, Spain Imperial College, London, UK University of Leeds, UK Universidad de Cadiz, Spain Imperial College, London, UK University of Glasgow, UK University of Kent, Canterbury, UK Universite Paris VII, France University of Kent, Canterbury, UK University of California, Berkeley, USA Concordia University, Montreal, Canada University of Kent, Canterbury, UK University of Kent, Canterbury, UK University of Turku, Finland Chinese Academy of Sciences, Bejing, China University of Surrey, Guildford, UK Indiana University-Purdue University at Indianapolis, USA Doshisha University, Kyoto, Japan Steklov Mathematical Institute, St. Petersburg, Russia Rutgers University, New Brunswick, USA University of Leeds, UK Vrije Universiteit Brussel, Belgium Universita di Roma III, Italy University of Notre Dame, USA

XVI

Lutzewisch, Shanna Ma, Wen-Xiu Magnus, Alphonse Mansfield, Elizabeth Mikhailov, Alexandre Mironov, Andrei Mugan, Ugurhan Musette, Micheline Nijhoff, Frank Ody, Michael Oevel, Walter Papageorgieu, Vassilis Pickering, Andrew Priestley, Thomas Ragnisco, Orlando Satsuma, Jumkichi Schief, Wolfgang Smart, Nigel Sorace, Emanuele Springael, John Suris, Yuri Takhtajan, Leon Tamizhmani, Kilkothur Umeno, Ken Veselov, Alexander Viallet, Claude Volkov, Alexandre Vowden, Barry Vu, Khai Webster, Helen Willox, Ralph Winternitz, Pavel

Participants Universitat Paderborn, Germany Universitat Paderborn, Germany Universite Catholique de Louvain, Belgium University of Kent, Canterbury, UK University of Leeds, UK Lebedev Physics Institute, Moscow, Russia Bilkent University, Turkey Vrije Iniversiteit Brussel, Belgium University of Leeds, UK University of Kent, Canterbury, UK Universitat Paderborn, Germany Technical University of Crete, Chania, Greece University of Kent, Canterbury, UK University of Kent, Canterbury, UK Universita di Roma III, Italy University of Tokyo, Japan University of New South Wales, Sydney, Australia University of Kent, Canterbury, UK Universita di Firenze, Italy Vrije Universiteit Brussel, Belgium Universitat Bremen, Germany State University of New York at Stony Brook, USA Pondicherry University, India Institute of Physical and Chemical Research, Saitama, Japan Loughborough University, UK Universite Paris VI, France Steklov Mathematical Institute, St. Petersburg, Russia University of Kent, Canterbury, UK Monash University, Melbourne, Australia University of Kent, Canterbury, UK Vrije Universiteit Brussel, Belgium Universite de Montreal, Canada

Chapter 1

Partial Difference Equations

Discrete linearisable Gambier equations A K Common*, E Hesameddini* and M Musette* * Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, UK. * Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium Abstract We propose two candidates for discrete analogues to the nonlinear Ermakov equation. The first discretisation of degree two possesses the two features which characterise its linearisation in the continuum while the second form of degree one is directly linearisable into a third order equation. We extend our procedure of discretisation, based on an association with a two dimensional conformal mapping, to a nonlinear equation of the Gambier classification linearisable into a fourth order equation.

1

Introduction

In 1906, Gambier [8] reported that the second order nonlinear differential equation PZZX

I**

where c ^ 0 is a constant and p, q two arbitrary functions of x, is linearisable into a third order equation. Making the transformation z = py, one obtains for y the following equation \

-c = o,

(2)

where

/(„-«„ + !(& -§(&)" ,

,3,

4

1. Partial Difference Equations

which by derivation, is simply related to the third order linear equation y*x* + 4/jfc + 2fxy = 0 .

(4)

The expression of the general solution of (2) in terms of two solutions (^15^2) of the second order linear equation Vw + f(x)tl> = 0

(5)

was previously given by Ermakov [6, 4] in 1880 as

y(x) = atf(x) with a^y — (I2 = —cW~2/2, the constant W representing the Wronskian of the two solutions of (5). At this time, Appell [2] also found the relation (6) in searching for the link between the solution of the third order linear equation (4) and the square of the general solution of (5). Moreover, the transformation y(x) = x = (3 + EUJ + au + w(7, w) - \-y{u, u>)

(13)

with N x N matrix E satisfying £J + J£T = 0 where / is such that

(14)

1. Partial Difference Equations

5, a ) = 6TIoc = Sicti + S2a2 + • • • 6pap — and 1 < p < N. One can show [11] that specific transformations on the dependent and independent variables of the set of equations (13) lead to a standard form which in case N = 2, / = I ux

n

2

I with e2 — ± 1 , is

= Ai + D2uv

(16)

vx = A2(x)-jD2(u2-e2v2)

(17)

where Ai, D2 are constant and A2(x) a function of x. Eliminating v from the two coupled first order differential equations (16-17), we obtain for u(x) uuxx - t£ + 2AlUx - l-A\ - A2D2u2 + jD22u4

= 0,

(18)

or in the variable y = u" 1 Wx* ~ \vl + 1My2yx + i ^ y 4 + A2D2y2 - jD22 = 0.

(19)

We see that this is equivalent to (2) when

AlS0,

Mx) = 2-M, c=^.

(20)

Let us note that in the variables w = — (u + v)/2, ?/ = u" 1 and for e2 = — 1 the system (16-17) becomes

^ W

o

(21)

= 0

(22)

=

l)

which corresponds to the system given by Gambier [9] in association with the equation of class XXVII in the particular case n = 2. If A\ = 0, instead of having two Riccati equations in "cascade", the second equation of the set (21-22) is linear and possesses the particular solution y — i\)\i\)2 where (^1^2) are two solutions of the linear equation

^ +^ W

(23)

COMMON, HESAMEDDINI & MUSETTE with w = (Log^1)?a,/D2 and the Wronskian W(ipi,ip2) = D2. Hence performing a linear transformation on (^1,^2), one recovers the solution (6) with ai/^4, v = i>2/ij>4, w =

In the particular case Ax = 0, the linear system degenerates in three coupled equations (^1 = K = constant) and the variable ^ 4 with the identification D2A2(x) = 2f(x) satisfies the third order linear equation (4).

3

Discrete forms of the Ermakov equation

There are of course infinitely many ways of constructing nonlinear difference equations which tend to the equation (2) in the continuum. We will discuss two approaches based on the connection with the conformal Riccati system and Schwarz derivative respectively. Firstly, we will obtain a discrete form whose general solution is connected to a linear second order difference equation in the same way as in the continuum. The conformal Riccati equations in (13) are the infinitesimal counterpart [1] of the discrete conformal transform of the vector a? in R^ given by epA

[a; + 7a;2]

(29)

8

1. Partial Difference Equations

where A is a general Lorentz transformation and 7, a are in RN and p is a scalar. The most obvious and natural way to discretise (13) is then to consider (29) as a mapping from w(n) to w(n + 1). To obtain the discrete analogy of the equation (2) we take TV = 2 as in §2 and the standard form of (29) with A = /,/? = 0, 7 = ( °J, OL = r^J. The mapping (29) may then be written in the component form: u(n)

u(n v(n

2

2e j2(n)v(n) + [u2 — a2(n) =

e2v2(n)]e2^{n)}

,(30)

2e2l2(n)v(n) + [u2(n) + e2v2(n))s272(n)}> (31)

and one recovers the standard form (16-17) by setting x = nh,

u(n) —> u(x), v(n) —* v(x) he2D2 , a2in) =

(32)

and taking the continuum limit h —> 0. To linearise the set (30-31) we make the definition

w(n) = u2{n) + e2v2(n) u(n) = j y - r ,

(33)

v(n) —

w(n) =

,

(34)

and obtain the four linear recurrence relations:

)

= A(n)

(35)

+ 1) / with

( 1 A=

0

2e2all2 2

1 + 2£ a 2 7 2

£2ai7j 72(l

2

+ 2£ a 272 )

2a! 1fe2(a2 + 72af2) l + e 2 7 2 ( 2 a 2 + 72^22)

k c

2e 2 7 2

F2^2 £

72

OL\ \ Oi2 2

(36)

I2 /

and a^2 = a\ + e 2 ^ . If ai(n) = 0 , 6 - e2A\(, = 0.

(63)

The nonlinear equation for y(x) = ^(a:)"1 -] 2

E2h2—\ (

£2h2=\

yy - — F j \vy-—Y\

, with Y = 1 - hAlV (64)

is of second degree in f like (41) and is linearisable by the transformation y(x) = (d(n + 1) — £i(n))/(hAi£i(n)) into a fourth order linear difference equation. Let us remark that in the continuum limit h —+ 0 equations (63) and (64) become respectively 2A2£lfXX + A2^hx

- e 2 A'6 = 0,

1 1 e2 2 2 4 2 - ^yl + 2AlV yx + -A lV + A2(x)y - - = 0,

(65) (66)

with y = £i,a;/(Ai£i) as it was given in the thesis of Gambier [9] .

5

Conclusion

This paper is an update of a previous contribution on the same subject [3]. The new results reported here are the explicit linearisation of the second order nonlinear difference equation (41) into a second order difference equation and the expression (64) of a discrete equation linearisable into a fourth order difference equation. A group theory approach similar to ours is also applied by Winternitz (see these Proceedings) to derive a discretisation of the Pinney equation equivalent to (41). Equations (2) and (19) represent particular cases of the Gambier equation XXVII which can be explicitly linearised. Grammaticos and Ramani [10] have proposed a discrete analogue of this last equation in the form of two discrete Riccati equations in " cascade". They use the singularity confinement criterion to generate a two-dimensional mapping but never write the explicit form of its general solution in terms of linear difference equations.

14

I. Partial Difference Equations

Acknowledgements During this conference we have benefitted from an enlightening discussion with Dr W. Schief concerning the discretisation and linearisation of the Ermakov equation (2). We thank him for his advice. M.M. also thanks R. Conte for his interest in a discretisation of first degree. A.K.C. and M.M. would like to thank the British Council and the National Fonds voor Wetenschappelijk Onderzoek for financial support for exchange visits during which much of this work was carried out. E.H. would like to thank the government of the Islamic Republic of Iran for their sponsorship of his study at the University of Kent. M.M. acknowledges financial support from the project IUAP III funded by the Belgian government.

References [1] Anderson R.L., Harnad J. and Winternitz P., Physica D 4 (1982) 164-182. [2] Appell P., Comptes Rendus 91 (1880) 211-214. [3] Common A.K., Hesameddini E. and Musette M., J. Phys. A. 29 (1996) 6343-6352. [4] Common A.K., Hesameddini E. and Musette M., Theory of nonlinear special functions u the Painleve transcendents", eds L.Vinet and P.Winternitz ( Springer, Berlin, 1997). [5] Conte R. and Musette M., Phys.Lett. 224A 101-110. [6] Ermakov V.P., Univ.Izv.Kiev Series III, 9 (1880) 1-25. [7] Faddeev L.D. and Takhtajan L.A., Springer Lect.Notes Phys. 246(1986) 166-179. [8] Gambier B., Comptes Rendus 142 (1906) 1403-1406. [9] Gambier B., Acta Mathematica 33 (1910) 1-55. [10] Grammaticos B. and Ramani A., Physica A 223 (1996) 125-136. [11] Hesameddini E., Thesis (1996), University of Kent. [12] Schief W.K., A discrete Pinney equation, to be published in Applied Mathematics Letters.

Generalized Backlund transformation and new explicit solutions of the two-dimensional Toda equation Xing-Biao Hu^* and Peter A Clarkson* t State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific Engineering Computing, Academia Sinca, P.O. Box 2719, Beijing 100080, P.R. China * Institute of Mathematics and Statistics University of Kent at Canterbury Canterbury, Kent CT2 7NF, UK Abstract In this paper, we obtain a generalized Backlund transformation for the bilinear representation of the two-dimensional Toda lattice equation, in which Backlund parameters are functions of x, y, n. Furthermore, corresponding nonlinear superposition formula are derived and using these results, we obtain some new explicit solutions of the two-dimensional Toda lattice equation.

It is always of considerable interest in obtaining exact analytical solutions of nonlinear equations which have many important applications in several aspects of physics as well as other natural and applied sciences. In this paper, we discuss the following nonlinear differential-difference equation d2 hn+l + ftn-i = 2hn + — — (ln/i n ) (1)

oxoy

which was first introduced by Darboux in 1915 [1]. By introducing a new variable Qn related to hn given by

16

I. Partial Difference Equations

we can represent (1) in the form - = exp(Q n -i - Qn) - exp(Q n - Qn+1). dxdy

(2)

We shall refer to this equation as the two-dimensional Toda lattice equation. There have been several studies of equations (1) or (2) (see, e.g. [2-14]). Now, we consider corresponding bilinear equation of (1) [DxDy - 4sinh 2 (!A0 + 2H(x,y,n)]fn.fn

=0

(3)

which is connected with (1) by the transformation d2 oxoy where H(xyy,n) = H0(x,y)n + Hi(x,y) with H0(x,y), Hi(x,y) functions of x and y. In the following, we shall generalize the results in [9] and present a generalized Backlund transformation (BT) for the bilinear equation (3). Furthermore the corresponding nonlinear superposition formula is derived and some particular exact solutions of (3) are given. Let fn be a solution of (3) and f'n be a solution of [DxDy - 4sinh2(f A,) + 2H{x,y,n))f'n.fn

=0

(4)

where H(x,y,n) = HQ(x,y)n + H\(x,y) with JHQ(x,y)1Hi(x,y) functions of x, y. If we can find two equations which relate fn and f'n and satisfy P = fn2[DxDy -fl[DxDy

- 4sinh 2 (|A l ) + 2H]fn.fn - 4sinh 2 (!A0 + 2H]fn.fn = 0

then this is a BT. Here we show that [Dx + A"1 exp(-D n ) + / x ] / n . / ; = 0 ±Dn) - Aexp(|Dn) + 7 e x p ( - | D n ) ] / n - / ; = 0

(5) (6)

indeed provides a BT connecting (3) and (4), where A = X(x, y) is an arbitrary function of x, y, fj, = /x(x, y, n) satisfies [iv — H(x, y, n) — i7(x, y, n) and 7 = n(ln X)y + 7(3;, y) with 7(3:, y) an arbitrary function. In fact, making use of (A.1)-(A.3),(5) and (6), P can be rewritten as

P = 2Dy(Dxfn.f'n).fnti

ftff

HU&CLAKKSON

17

= 4sinh(|A l ) { [^Z?y = 0 Therefore, we have the following result: Proposition 1. A BT connecting (3) and (4) is given by (5) and (6) with A = \{x,y) being an arbitrary function of 2, y, \i = n{x,y,n) satisfies \iy = H{x,y,n) — H{x,y,n) and 7 = n(lnA)y + 7(2,2/), with 7(2,2/) an arbitrary function. Remark: When A, \x and 7 are arbitrary constants and H = H = 0, Proposition 1 becomes a result in [9]. As an application of Proposition 1, we give some examples. Example 1. If we choose fn = 1, A = —1, a =

, 7 = —; x-y 2(2 - y) and H = 0, then it is easily verified that f'n = Jn{x — y) satisfies (5) and (6), where H = -, rr and Jn(z) is the standard Bessel function. Thus {x-y)2 Jn(x — y) satisfies DxDy - 4sinh 2 (|A l ) + , and K =

™ , 2 Jn{x - y)*Jn{x - y) = 0

n d2 rr2 + 1 + •£-£- In Jn{x - y) {x - y) dxdy

is a solution of (1). Example 2. If we choose fn = 1, A = — (2/y)1/2, fi = —, 7 = ZJX

*y

and H = 0, then it is easily verified that f'n = Jn{2y/xy) satisfies (5) and (6), where H = 0 and Jn is the Bessel function. Example 3. If we choose fn = 1, A is a constant, // = —A"1, 7 = A and H = 0, then it is easily verified that f'n = n+\~lx + \y + c (with c a constant) satisfies (5) and (6), and H = 0.

18

J. Partial Difference Equations In the following, we just consider the case \xv = 0, i.e. fj(T

o, n) = TJ(r

?/

rj\

In this case, Proposition 1 can be modified to be Proposition 1'. A BT for (3) is [Dx + A"1 exp(-Dn) + ,x]/ n ./; = 0 [Dyexp(—\Dn) — Aexp(|J5n) -f 7exp(—\Dn))fn* f'n = 0

(7) (8)

where A = A(x, y), \x = /i(x, n) are arbitrary functions and 7 = n(lnA)y + 7(3;, y), with 7 an arbitrary function. We shall represent the transformation (7,8) symbolically by fn -^? f'n. Thus from Examples 2 and 3, we have '2a:'

4t/

'

T

^n

^

Henceforth, we denote fn{x,y) = f{n,x,y) = f(n) = f without confusion. Proposition 2. Let / 0 be a solution of (3) and suppose that /», i = 1,2, is a solution of (3), which is related by / 0 under BT (7,8) with (Aj,/ij,7j), i.e. / 0 ( A ^ 7 i ) / n z = 1,2, AiA2 7^ 0, /j ^ 0, j = 0,1,2. Then / 12 defined by exp(-|D n )/ 0 ./i2 = k(x,y)[\iexp(-\Dn)

- A 2 exp(|D n )]/i»/ 2

(9)

where k(x,y) is some function of x, y, is a new solution which is related by /1 and / 2 under BT (7,8) with parameters (A2, \xi + A, 72 + [ln(fcAiA2)]y) and (Ai,/xi + A,7i + [ln(A;AiA2)]2/), respectively, provided that (JfcAi)* v

JL

,

/ jj

x /

=

\

,

x /

-j \

A 1

A

fi2{x,n) — /42(x, n — 1) + A

hold for some function A = A(x, y, n). Proof. First, from [{Dx + A + A^1 exp(-D n ) + /X2)/0./2]/i = 0

^~in\

(11)

HU k CLARKSON

19

we have

-Dxfi(n).f2(n)

+ (MI

- ^ ^ e x p ( - D n ) / 0 ( n ) . / 1 2 ( n ) = 0.

(12)

Second, from

f

/

d\

}

< exp I — 5 - I [Ai^Da; 4- exp(-£) n ) -h \i/J>i]fo»fi} / 2 W

{

\ dnj

J

- jexp ( - — j [A2^x + exp(-J9n) + A2/i2]/o*/2^/i(n) = 0 we obtain, after some detailed calculations,

-/iW/2(n-A2/x2(n - l)A(n)/ 2 (n - 1) = 0.

(13)

Thus, using (10,11) and (13), we obtain {[A2L»X + exp(-U B ) + A2(/z2 = A2/i(n)[§I>x/o(n - l)./i 2 (n) - ^ A i D . / ^ n - l)./ 2 (n) - ^ A a D . A C n ) . ^ ^ - 1) - KfcAOx/^n - l)/ 2 (n) Ak\2f1(n)f2(n +|(*A 2 ),/ 1 (n)/ 2 (n - 1) + A*Ai/x(n - l)/ 2 (n) /i(n - l)/ 2 (n) - fcA2/i2(n)/i(n)/2(n - 1)]

H . / , ^ - 1) + KkAiUIn - l)/2(n) - l)/i(n - l)/ 2 (n) (14) which implies that [Dx + X2l exp(-D B ) + (ju2(n) + A)]/i./« = 0. Similarly we have [£>, + Ar1 exp(-D n ) + (Ml(n) + A)]/ 2 ./ 1 2 = 0.

20

I. Partial Difference Equations

Finally, since /i and / 2 are two solutions of (3), then we have 0 =

fi[DxDy -fl[DxDy - 4sinh2(i£>n) + 2H]f2.f2 = 2Dy(DJ1.f2).f1f2 - 4 sinh(|Z) n )[exp(iD n )/i •/ 2 ] • [exp(-i£>n)

= -2A, [ 4_ . +-sinh(±A l ){[Aiexp(-±A l ) exp(-iD n )/ 1 (

ex

P (l|^

7i(« + |)/2(n)/i2(» + 1) - Ai/2(n + l)/i 2

Mn-

l)/i(n)exp f i j M

{Dyexp(-iZ?n)/2(n)./12(n)

7i(n)exp(-|D n )/ 2 (n)./i 2 (n) - Aiexp(iD n )/ 2 (n)./i 2 (n) fcA1A2exp(-iZ)n)/2(n)./12(n)} (15) which implies that

[IVxpHi^-Ai exp^M Similarly, we can show that ) { ( ) [ ] } | D

n

) ] /

1

. /

1 2

= 0.

Thus we have completed the proof of Proposition 2. In what follows, we give some examples as an illustrative application of the above results. Example 4. Setting / 0 = 1, H — 0 and Ai5 % = 1,2, are constants. Then as shown in Example 3, we have

Furthermore, by use of (9), we obtain the following solution /12 = Ai(n - 1 + X[lx + \iy + ci)(n + A^x + \2y + c2) -A 2 (n + X[lx + Xxy + Ci)(n - 1 + A^1^ + X2y + c2). (16)

HU & CLARKSON

21

In general, continuing along this line, we can obtain a hierarchy of rational solutions. Example 5. Setting / 0 = 1, H = 0, Ai is a constant and A2 = — (x/y)1/2. Then we have

>

Jn(2y/xy)

Furthermore if we set k = 1 and A = 0. Then it is easily verified that (10) and (11) hold. By use of (9), we have a solution of (3) with H = 0: /12 = Ai(n - 1 + X{lx + Aiy + ci) Jn{2y/xy)

Example 6. Setting / 0 = 1, H = 0, Ai is a constant and A2 = —(x then we have

/

L »2x'

± Ay

I

T

/

o

/

\

where A and B are constants. Furthermore, if we set k = 1 and A = 0, then it is easily verified that (10) and (11) hold. Thus, using (9), we have the following solution of (3) with H = 0:

i(l - e~A)y}) J n (2v^) + {x/y)1'2 (1 + Bexp{An rV

" 1)* + Ax(l - e~A)y}) Jn_1(2v^y)

(17)

In this paper, we have given a generalized BT of (3), in which Backlund parameters are functions of re, y, n. Furthermore, corresponding nonlinear superposition formula has been shown. Using these results, we have obtained some new explicit solutions of the two-dimensional Toda lattice equation (1).

Acknowledgement This work of XBH was supported by National Natural Science Foundation of China.

22

1. Partial Difference Equations

Appendix The following bilinear operator identities hold for arbitrary functions a, 6, c and d: (DxDya.a)b2 - a2DxDyb.b = 2Dy(Dxa.b).ab (A.I) [sinh2(eL>n)a.a]&2 - a2[sinh2(eDn)6.6] = sinh(cDB)[exp(cJDn)a.6].[exp(-eDn)o.6]

^

'

Dy[exp(2eDn)a*b] mdb = smh.(eDn)[Dy exp(eDn)a»b] • [exp(eZ)n)a»6] (A.3)

References 1. Darboux G 1915 Lecons sur la theorie des surfaces vol 2, 2nd edn (Paris:Gauthier-Villars) 2. Mikhailov A V 1978 Pisma Zh. Eksp. Teor. Fiz. 77 24 3. Mikhailov A V 1979 JETP Lett. 30 443 4. Fordy A P and Gibbons J 1980 Commun. Math. Phys. 77 2 5. Nakamura A 1983 J. Phys. Soc. Japan 52 380 6. Saitoh N, Takeno S and Takizawa E 11985 J. Phys. Soc. Japan 54 3701 7. Levi D and Winternitz P 1993 J. Math. Phys. 34 3713 8. Matveev V B and Salle M A 1991 Darboux Transformations and Solitons (BerlimSpringer) 9. Hu X B 1994 J. Phys. A: Math. Gen. 27 201 10. Kajiwara K and Satsuma J 1991 J. Math. Phys. 32 506 11. Mcintosh I 1994 Nonlinearity 7 85 12. Villarroel J and Ablowitz M J 1992 Phys. Lett. A 163 293 13. Villarroel J and Ablowitz M J 1993 Physica D 65 48 14. Villarroel J and Ablowitz M J 1994 J. Phys. A: Math. Gen 27 931

Different Aspects of Relativistic Toda Chain

2

S. Kharchev 1 , A. Mironov2 and A. Zhedanov3 1 ITEP, Bol. Cheremushkinskaya, 25, Moscow, 117 259, Russia Theory Department, P.N. Lebedev Physics Institute, Leninsky prospect, 53, Moscow, 117924, Russia 3 Physics Department, Donetsk State University, Donetsk, 340 055, Ukraine

Abstract We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). We also show that the RTC is gauge equivalent to the discrete AKNS hierarchy and the unitary matrix model. Relativistic Toda molecule hierarchy is also considered, along with the forced RTC. The simple approach to the discrete RTC hierarchy based on Darboux-Backlund transformation is proposed.

1

Introduction

Since the paper of Ruijsenaars [1], where has been proposed, the relativistic Toda chain (RTC) system was investigated in many papers [2]-[4]. This system can be defined by the equation: *.=

i - qn)

eqn)--—r 1 + e2exp{qn - qn-i) which transforms to the ordinary (non-relativistic) Toda chain (TC) in the evident limit e —> 0. The RTC is integrable, which was discussed in different

24

1. Partial Difference Equations

frameworks (see, for example, [2]-[4] and references therein). The RTC can be obtained as a limit of the general Ruijsenaars system [1]. In this paper we are going to review different Lax representations of the RTC, and to establish numerous relations of it with many well-known integrable systems like AKNS, unitary matrix model etc. It is also shown that the RTC hierarchy can be embedded to the 2DTL hierarchy [5]. Besides, we discuss the forced RTC hierarchy and its finite analog, the relativistic Toda molecule. At the end of this short paper we describe the simple approach to discrete evolutions of the RTC which is based on the notion of the Darboux-Backlund transformations and can be considered as a natural generalization of the corresponding notion in the usual Toda chain theory. For more details we refer the reader to our lengthy paper [6].

2

Lax representation for RTC

Let us describe the Lax representation for the standard RTC equation. The usual procedure to obtain integrable non-linear equations consists of the two essential steps: i) To find appropriate spectral problem for the Baker-Akhiezer function(s). ii) To define the proper evolution of this function with respect to isospectral deformations. Lax representation by three-term recurrent relation. In the theory of the usual Toda chain the first step implies the discretized version of the Schrodinger equation (see [7], for example). In order to get the relativistic extension of the Toda equations, one should consider the following "unusual" spectral problem *n+l(*) + «n*»(*) = z{$n(z) + bn9n^(z)}

,

UGZ

(2)

representing a particular discrete Lax operator acting on the Baker-Akhiezer function $n(z)> This is a simple three-term recurrent relation (similar to those for the Toda and Volterra chains) but with "unusual" spectral dependence. As for the second step, one should note that there exist two distinct integrable flows leading to the same equation (1). As we shall see below, the spectral problem (2) can be naturally incorporated into the theory of twodimensional Toda lattice (2DTL) which describes the evolution with respect to two (infinite) sets of times (tfi,£2,...) > (^_i, t_2,...) (positive and

KHARCHEV, MIRONOV & ZHEDANOV

25

negative times, in accordance with [5]). Here we describe the two particular flows (at the moment, we deal with them "by hands", i.e. introducing the corresponding Lax pairs by a guess) which lead to the RTC equations (1). The most simple evolution equation is that with respect to the first negative time and has the form

| £ = *•*-!

(3)

with some (yet unknown) Rn. The compatibility condition determines Rn in terms of an and bn Rn = ^2L and leads to the following equations of motion: dan

bn+1

bn

dbn

= In order to get (1), we should identify an = exp(-ep n );

bn = -e2 exp(qn - qn_i - epn)

(5)

Note that in this parameterization the " Hamiltonian" Rn depends only on coordinates # n 's: Rn = — e2 exp(qn — qn-i). Performing the proper rescaling of time in (4) we reach the RTC equation As we noted already, the evolution (3), which leads to the RTC equations is not the unique one. The other possible choice leading to the same equations is ^

= -U*n-z9n-i)

(6)

The compatibility condition of (2) and (6) gives the equations —

= -an(bn+1

-bn)

— = -bn(bn+1

- 6n_i + a n _i - an)

(7)

This leads to the same RTC equation (1). 2 x 2 matrix Lax representation. The same RTC equation can be obtained from the matrix Lax operator depending on the spectral parameter [3] (generalizing the Lax operator for the TC [7]). Then the RTC arises as the compatibility condition for the following 2 x 2 matrix equations:

where

26

1. Partial Difference Equations 2

e exp(qn -tfn_!+ epn_i)

eC1 expi-q^

+ ep^)

-eC" 1 exp(^ n ) \

1 - (" 2 + e2 )

[

'

One can easily reduce these equations to the system (2) and (3). To conclude this section, we remark that L-operator (9), which determines the RTC is not unique; moreover, it is not the simplest one. Indeed, we shall see that there exists the whole family of the gauge equivalent operators, which contains more "natural" ones and includes, in particular, the well known operator generating the AKNS hierarchy. From general point of view, the whole RTC hierarchy is nothing but AKNS and vice versa.

3

RTC and unitary matrix model, AKNS, etc.

Now we are going to describe the generalized RTC hierarchy as well as its connection with some other integrable systems. We start our investigation from the framework of orthogonal polynomials Unitary matrix model. It is well-known that the partition function rn of the unitary one-matrix model can be presented as a product of norms of the biorthogonal polynomial system [8]. Namely, let us introduce a scalar product of the form * = / ^ e x p J

ZmZ

Y.^rnzm-t-mz-m) A(z)B(z-1)

(11)

m>0

Let us define the system of polynomials biorthogonal with respect to this scalar product

=hJnk

(12)

Then, the partition function r n of the unitary matrix model is equal to the product of /i n 's: k=0

The polynomials are normalized as follows:

*n(z) = zn + ... + Sn.1,

l(z) = zn + ... + S*^,

S-^^sl (14)

These polynomials satisfy the following recurrent relations:

${z)

= z${z) +

Sz^{z-') *(z)

{

'

x The signs of positive and negative times are defined in this way to get the exact correspondence with the times introduced in [5].

KHARCHEV, MIRONOV k ZHEDANOV

27

and "•n+1

(16)

K

The above relations can be written in several equivalent forms. First, it can be presented in the form analogous to (2): (17)

} (is)

J

V-l J From the first relation and using (2) and (5), one can immediately read off n

i

? = - eexp(-ep x p ( e pn));; *->n-l

—?? = - ee2 exp(^fn - qn.x)

(19)

^n-1

Thus, the orthogonality conditions (12) lead exactly to the spectral problem for the RTC. We should stress that equations (17), (18) can be derived from the unitary matrix model. Using the orthogonal conditions, it is also possible to obtain the equations which describe the time dependence of $ n , $£ . Differentiating (12) with respect to times t\ , t_i gives the evolution equations: , (21) (see general evolution equations with respect to higher flows below). The compatibility conditions give the following nonlinear evolution equations: 9Sn __

dS

n _

hn+i

n*

h

n+l .

oT " " ^ X " '

dSn _

dS* _

».i " "

hn+i

(

.

* ^n+1

n+1

K

As a consequence, in the polynomial case, (24)

28

I. Partial Difference Equations

These are exactly relativistic Toda equations written in somewhat different form. Indeed, from (24), (22), (23) and (16) one gets2

log K=

h

log

~{k

hn

) {K

hn+i

log K+

)

K

(25) On the other hand, the RTC is a particular case of the 2DTL hierarchy. Indeed, let us introduce the key objects in the theory of integrable systems the r-functions, which are defined through the relation hn = r n + i/r n . Then, with the help of (22)-(24), one can show that the functions r n satisfy the first equation of the 2DTL: dtA-iWrn = -T-^r1

(26)

Therefore, it is natural to assume that the higher flows generate the whole set of non-linear equations of the 2DTL in spirit of [5]. This is indeed the case. This completes the derivation of the RTC from the unitary matrix model. RTC versus AKNS and "novel" hierarchies. Now let us demonstrate the correspondence between RTC and AKNS system. We have already seen that the orthogonality conditions naturally lead to the 2x2 formulation of the problem generated by the unitary matrix model:

d_

dh -71

Q-k

q

C*

/ 1

(ft*

(Equations (28), (29) follow from (20)-(21) and the original spectral problem (15)). Put $ n = W 2 ~ 1 / 4 F n , $* = ^" n / 2 + 1 / 4 F*. Then the spectral problem (27) can be rewritten in the matrix form /

z?

\

(30) 2

The same equation holds for t_i-flow.

KHARCHEV,

MIRONOV k ZHEDANOV

29

where Sn

^(AKNS) =fC

\

( = z i/a

(3i)

This is the Lax operator for the discrete AKNS [9]. Obviously the evolution equations (28), (29) can be written in terms of Fni F* as

,-iZ ~ l-mZ ) ( J 1 ZmZ lm>0 J Let us note that D (D*T) is the upper (lower) triangle matrix with the units on the diagonal (because of (14)). This representation is nothing but the Riemann-Hilbert problem for the forced hierarchy. Now taking the determinant of the both sides of (57), one gets n-l

due to formula (13). The remaining last step is to observe that

f

exp ( £ (tmzm - t_mz-

K (60)

i.e.

rB(t) = det [ ^ ( - ^ y jf ^

exp | Y.^zm - N (62) for some N. This system should be called iV — 1-particle relativistic Toda molecule, by analogy with the non-relativistic case and is nothing but RTCreduction of the two-dimensional Toda molecule [13,14]3. 3

Sometimes the Toda molecule is called non-periodic Toda [15]. It is an immediate generalization of the Liouville system.

KHARCHEV, MIRONOV k ZHEDANOV

35

sl(N) Toda can be described by the kernel A(z, w) )

(63)

where f^k\z) and g^k\z) are arbitrary functions. From this description, one can immediately read off the corresponding determinant representation (55). Indeed, equation (26) and condition (62) implies that log To and logr^ satisfy the free wave equation d^dt^ logr0 = d^dt^ logr/v = 0. Since the relative normalization of r n 's is not fixed, we are free to choose TQ = 1. Then, ro(t) = 1, rN(t) = x(*i)x(*-i)> where \{h) and x(*-i) are arbitrary functions. 2DTL with boundary conditions (5) was considered in [13]. The solution to (26) in this case is given by [14]:

rn(t) = det d^i-d^y-'nit)

(64)

with

where functions a^(t) and a'^(t_i) satisfy det d^a^t)

= x(t),

This result coincides with that obtained by substituting into (55) the kernel A(z,w) of the form (63). Lax representation. In all our previous considerations, we dealt with infinite-dimensional matrices. Let us note that the Toda molecule can be effectively treated in terms of N x N matrices like the forced case could be described by the quarter-infinite matrices. This allows one to deal with the both identities (53) since all the products of finite matrices are well-defined. To see this, one can just look ai> the recurrent relation (43) and observe that there exists the finite-dimensional subsystem of (N) polynomials which is decoupled from the whole system. The recurrent relation for these polynomials can be considered as the finite-matrix Lax operator (which still does not depend on the spectral parameter, in contrast to (9)). Indeed, from (43) and condition (62), i.e. hn/hn-i = 0 as n > N (the Toda molecule conditions in terms of S-variables read as Sn = 5* = 1 for n > N — 2 or n > 0), one can see that zVN(z) = VN+1(z) - VN(z),

zVN+1{z) = VN+2(z) - VN+I(Z)

etc. (66)

36

1. Partial Difference Equations

i.e. all the polynomials Vn with n > N are trivially expressed through VNTherefore, the system can be effectively described by the dynamics of only some first polynomials (i.e. has really finite number of degrees of freedom). Certainly, all the same is correct for the star-polynomials V* although, in this case, it would be better to use the original non-singularly normalized polynomials $*. Now let us look at the corresponding Lax operators (49)-(50). They are getting quite trivial everywhere but in the left upper corner of the size N x N [6]. Therefore, one can restrict himself to the system of N polynomials "Pn, n = 0 , l , . . . , i V — 1 and the finite matrix Lax operators (of the size N X N). Now one needs only to check that this finite system still has the same evolution equations (51). It turns out to be the case only for the first N — 1 times. This is not so surprising, since, in the finite system with N — 1 degrees of freedom, only first N — 1 time flows are independent. Therefore, if looking at the finite matrix Lax operators, one gets the dependent higher flows. On the other hand, if one embeds this finite system into the infinite 2DTL, one observes that the higher flows can be no longer described inside this finite system. Just the finite system is often called relativistic Toda molecule. To simplify further notations, we introduce, instead of 5 n , *S*, the new variables sn = ( - ) n + 1 S n , s* = (~)n+1S*. Then, one can realize that the Lax operator can be constructed as the product of simpler ones C = CN^N-I • • • £\-> where Ck is the unit matrix wherever but a 2 x 2-block: 1

Gk

1

Gk =

3*4-

4

(67)

/ Analogously, C =

with

C\..

\ Gk

Gk =

4 - sks*k

-i sk

(68)

1 One can trivially see that £k£k — £k£>k — 1> a nd, therefore, one obtains CC = CC = l (cf. (53)). From these formulas, one obtains that d e t £ = d e t £ = 1 which reminds once more of the sl(N) algebra. More generally, the factorization property of the Lax operators opens the wide road to the group theory interpretation of the RTC molecule - see [6].

KHARCHEV, MIRONOV & ZHEDANOV

6

37

Discrete evolutions and limit to Toda chain

Darboux-Backlund transformations. Now we are going to discuss some discrete evolutions of the RTC given by the Darboux-Backlund transformations and their limit to the usual Toda chain. One can easily take the continuum limit of the formulas of this section to reproduce the TC as the limit of the RTC, both with the standard continuous evolutions. The discrete evolution equations in the RTC framework were recently introduced by [3] in a little bit sophisticated way. Here we outline the simple approach based on the notion of the Darboux-Backlund transformation (DBT). More details will be presented in the separate publication [16]. Let discrete index i denote the successive DBT's. The spectral problem now can be written as follows:

(69)

= z

Let us define the first forward DBT (treating it as a discrete analog of (3)): $ n (j -f. \\z\ - 0 , one can rewrite all the equations describing the first forward DBT as follows: an(t + A) = a n _i(t)

t x

bn{t + A) = &n_i(t)

""+*

x

jay—7TT"

(76)

(77)

After the rescaling Z{ —»- g A, one gets from (72) a ^ ^ —gAbn/an and, in the continuum limit, A —+ 0 the last two equations lead directly to (4). The analogous equations can be written for the second forward DBT but with Z{ rescaled as z\; —» - ~ . It is clear that, in the limit A —»• 0, one reproduces the continuum equations (7). Limit to Toda chain. Now let us make the following expansion (compare with (5)) a n ( 0 ^ 1 - e p n ( t ) ; bn{i) -e2Rn(i)

z ~ 1 + e\ ; Introduce also functions

Zi ~ 1 + e\i

$n(i) *»(t) - enVn(i)

(79)

It is easy to see that (70) leads to the forward DBT for the non-relativistic Toda chain if one identifies «W(i) ~ tAn{i)

(80)

Indeed, in the limit e —» 0, one gets the standard Toda spectral and evolution equations. There also exist some other interesting limits [6] leading to the modified discrete Toda equations [3].

KHARCHEV, MIRONOV k ZHEDANOV

7

39

Concluding remarks

From the point of view of studying the RTC hierarchy itself, the most promising representation is that describing the relativistic Toda hierarchy as a particular reduction of the two-dimensional Toda lattice hierarchy. However, even this quite large enveloping hierarchy is still insufficient. Loosely speaking, the Toda lattice is too "rigid" to reproduce both the continuous and discrete flows of the RTC. Therefore, one should embed the RTC into a more general system which admits more natural reductions. This is done in the forthcoming publication [16], where we show that the RTC has a nice interpretation if considering it as a simple reduction of the two-component KP (Toda) hierarchy. It turns out that, in the framework of the 2-component hierarchy, the continuous AKNS system, Toda chain hierarchy and the discrete AKNS (which is equivalent to the RTC how we proved in this paper) can be treated on equal footing. We acknowledge V. Fock, A. Marshakov and A. Zabrodin for the discussions. A.Z. is also grateful to V.Spiridonov, S.Suslov and L.Vinet for stimulating discussions. The work of S.K. is partially supported by grants RFFI96-02-19085, INTAS-93-1038 and by Volkswagen Stiftung, that of A.M. - by grants RFFI-96-02-16210(a), INTAS-96-2058 and Volkswagen Stiftung.

References 1. S.N. Ruijsenaars, Comm. Math. Phys., 133 (1990) 217-247 2. M. Bruschi and 0 . Ragnisco, A129 (1988) 21-25; Phys. Lett, A134 (1989) 365-370; Inverse Problems, 5 (1989) 389-405 3. Yu. Suris, Phys. Lett, A145 (1990) 113-119; Phys. Lett, A156 (1991) 467-474; Phys. Lett, A180 (1993) 419-433; solv-int/9510007 4. Y. Ohta, K. Kajiwara, J. Matsukidaira, J. Satsuma, solv-int/9304002 5. K. Ueno and K. Takasaki, Adv.Stud. in Pure Math., 4 (1984) 1-95 6. S. Kharchev, A. Mironov and A. Zhedanov, hep-th/9606144 7. L. Faddeev and L. Takhtadjan, Hamiltonian methods in the theory of solitons, Springer, Berlin, 1987 8. V. Periwal and D.i Shevits, Phys. Rev. Lett, 64 (1990) 1326-1335 M.J. Bowick, A. Morozov and D. Shevits, Nucl Phys., B354 (1991) 496-530

40

J. Partial Difference Equations 9. M.J. Ablowitz and J.E. Ladik, J. Math. Phys., 17 (1976) 1011-1018

10. I. Merola, 0 . Ragnisco and Tu Gui-Zhang, solv-int/9401005 A. Kundu and 0 . Ragnisco, hep-th/9401066 11. S. Kharchev and A. Mironov, Int. J. Mod. Phys., A17 (1992) 4803-4824 12. S. Kharchev, A. Marshakov, A. Mironov and A. Morozov, Nucl. Phys., B397 (1993) 339-378; hep-th/9203043 13. R. Hirota, J. Phys. Soc. Japan, 57 (1987) 4285-4288 R. Hirota, Y. Ohta and J. Satsuma, Progr. Theor. Phys. SuppL, 94 (1988) 59-72 14. A. Leznov and M. Saveliev, Physica, 3D (1981) 62 15. M. Olshanetsky and A. Perelomov, Phys. Rep., 71 (1981) 71-313 16. S. Kharchev, A. Mironov and A. Zhedanov, Discrete and Continuous Relativistic Toda in the Framework of 2-component Toda hierarchy, to

appear

Chapter 2

Integrable Mappings

Integrable Symplectic Maps A.P. Fordy Department of Applied Mathematical Studies and Centre for Nonlinear Studies, University of Leeds, Leeds LS2 9JT, UK Email: allan@amsta. leeds .ac.uk Abstract We consider the construction of canonical transformations which preserve some given Hamiltonian function and thus the corresponding Hamiltonian flow. The methods employed depend upon whether or not we already know a Lax representation for the continuous system. The paper thus falls naturally into two parts: 1. In the case that we know a Lax representation for the continuous system we use a special similarity transformation. Our examples are finite reductions of integrable PDEs with the canonical transformation being a finite reduction of Darboux-Backlund transformation. 2. In the case that we are just given a Hamiltonian function, without any knowledge of other structures, then we use a novel form of the Hamilton-Jacobi equation to directly construct a canonical transformation. Examples include (a special case of) the McMillan map and Backlund transformations for some of the Painleve equations.

1

Introduction

An important ingredient in the theory of continuous Hamiltonian dynamics is the canonical transformation. The interpretation is that we change to a new set of canonical co-ordinates, with the coefficients of the symplectic form being unchanged. An alternative interpretation is to actively change points in our symplectic manifold M 2n , {q^Pi) *-¥ &(q, p)5Pi(q5 p))> in such a way that n

n

A dpi = Y, dqi A dp{.

44

2. Integrable Mappings

In such a case we call this a symplectic map (p : M2n —> M 2n , which can then be iterated. A given function / : M2n -» % on phase space will change value under such a map and this defines a new function / : M2n —>-7£, given by: 7(q,p) = /(5(q,p),p(q,p)). If /(q, p) = /(q, p), meaning that the functional form of / is unchanged by the transformation, then / is said to be an invariant of the symplectic map, analogous to a first integral in the continuous case. In [9] Veselov proved a discrete version of the Arnol'd-Liouville theorem, which led him to the definition: Definition 1.1 (Integrable Symplectic Map) A symplectic map