Introduction to classical integrable systems

Introduction to Classical Integrable Systems OLIVIER BABELON Laboratoire de Physique Théorique et Hautes Energies, Universités Paris VI–VII DENIS BERNARD Service de Physique Théorique de Saclay, Gif-sur-Yvette MICHEL TALON Laboratoire de Physique Théorique et Hautes Energies, Universités Paris VI–VII published by the press syndicate of the university of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom cambridge university press The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org c O. Babelon, D. Bernard & M. Talon 2003
This book 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 2003 A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Babelon, Olivier, 1951– Introduction to classical integrable systems / Olivier Babelon, Denis Bernard, Michel Talon. p. cm. – (Cambridge monographs on mathematical physics) Includes bibliographical references and index. ISBN 0 521 82267 X 1. Dynamics. 2. Hamiltonian systems. I. Bernard, Denis, 1961– II. Talon, Michel, 1952– III. Title. IV. Series. QA845 .B32 2003 531′ .163–dc21 2002034955 ISBN 0 521 82267 X hardback Contents 1 Introduction 1 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 Integrable dynamical systems Introduction The Liouville theorem Action–angle variables Lax pairs Existence of an r-matrix Commuting flows The Kepler problem The Euler top The Lagrange top The Kowalevski top The Neumann model Geodesics on an ellipsoid Separation of variables in the Neumann model 5 5 7 10 11 13 17 17 19 20 22 23 25 27 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 Synopsis of integrable systems Examples of Lax pairs with spectral parameter The Zakharov–Shabat construction Coadjoint orbits and Hamiltonian formalism Elementary flows and wave function Factorization problem Tau-functions Integrable field theories and monodromy matrix Abelianization Poisson brackets of the monodromy matrix The group of dressing transformations 32 33 35 41 49 54 59 62 65 72 74 vii viii Contents 3.11 Soliton solutions 79 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 Algebraic methods The classical and modified Yang–Baxter equations Algebraic meaning of the classical Yang–Baxter equations Adler–Kostant–Symes scheme Construction of integrable systems Solving by factorization The open Toda chain The r-matrix of the Toda models Solution of the open Toda chain Toda system and Hamiltonian reduction The Lax pair of the Kowalevski top 86 86 89 92 94 96 97 100 105 109 115 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 Analytical methods The spectral curve The eigenvector bundle The adjoint linear system Time evolution Theta-functions formulae Baker–Akhiezer functions Linearization and the factorization problem Tau-functions Symplectic form Separation of variables and the spectral curve Action–angle variables Riemann surfaces and integrability The Kowalevski top Infinite-dimensional systems 124 125 130 138 142 145 149 153 154 156 162 164 167 169 175 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 The closed Toda chain The model The spectral curve The eigenvectors Reconstruction formula Symplectic structure The Sklyanin approach The Poisson brackets Reality conditions 178 178 181 182 184 191 193 196 200 7 7.1 The Calogero–Moser model The spin Calogero–Moser model 206 206 Contents ix 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 Lax pair The r-matrix The scalar Calogero–Moser model The spectral curve The eigenvector bundle Time evolution Reconstruction formulae Symplectic structure Poles systems and double-Bloch condition Hitchin systems Examples of Hitchin systems The trigonometric Calogero–Moser model 208 210 214 216 218 220 221 223 226 232 239 244 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 Isomonodromic deformations Introduction Monodromy data Isomonodromy and the Riemann–Hilbert problem Isomonodromic deformations Schlesinger transformations Tau-functions Ricatti equation Sato’s formula The Hirota equations Tau-functions and theta-functions The Painlevé equations 249 249 251 262 264 270 272 277 278 280 282 290 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Grassmannian and integrable hierarchies Introduction Fermions and GL(∞) Boson–fermion correspondence Tau-functions and Hirota bilinear identities The KP hierarchy and its soliton solutions Fermions and Grassmannians Schur polynomials From fermions to pseudo-differential operators The Segal–Wilson approach 299 299 303 308 311 314 316 322 328 331 10 10.1 10.2 10.3 10.4 The KP hierarchy The algebra of pseudo-differential operators The KP hierarchy The Baker–Akhiezer function of KP Algebro-geometric solutions of KP 338 338 341 344 348 x Contents 10.5 10.6 10.7 10.8 10.9 10.10 10.11 The tau-function of KP The generalized KdV equations KdV Hamiltonian structures Bihamiltonian structure The Drinfeld–Sokolov reduction Whitham equations Solution of the Whitham equations 352 355 359 363 364 370 379 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 The KdV hierarchy The KdV equation The KdV hierarchy Hamiltonian structures and Virasoro algebra Soliton solutions Algebro-geometric solutions Finite-zone solutions Action-angle variables Analytical description of solitons Local fields Whitham’s equations 382 382 386 392 394 398 408 414 419 425 433 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 The Toda field theories The Liouville equation The Toda systems and their zero-curvature representations Solution of the Toda field equations Hamiltonian formalism Conformal structure Dressing transformations The affine sinh-Gordon model Dressing transformations and soliton solutions N -soliton dynamics Finite-zone solutions 443 443 445 447 454 456 463 467 471 474 481 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 Classical inverse scattering method The sine-Gordon equation The Jost solutions Inverse scattering as a Riemann--Hilbert problem Time evolution of the scattering data The Gelfand--Levitan--Marchenko equation Soliton solutions Poisson brackets of the scattering data Action--angle variables 486 486 487 496 497 498 502 505 510 Contents xi 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Symplectic geometry Poisson manifolds and symplectic manifolds Coadjoint orbits Symmetries and Hamiltonian reduction The case M = T ∗ G Poisson–Lie groups Action of a Poisson–Lie group on a symplectic manifold The groups G and G∗ The group of dressing transformations 516 516 522 525 532 534 538 540 542 15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11 15.12 15.13 15.14 15.15 Riemann surfaces Smooth algebraic curves Hyperelliptic curves The Riemann–Hurwitz formula The field of meromorphic functions of a Riemann surface Line bundles on a Riemann surface Divisors Chern class Serre duality The Riemann–Roch theorem Abelian differentials Riemann bilinear identities Jacobi variety Theta-functions The genus 1 case The Riemann–Hilbert factorization problem 545 545 547 549 549 551 553 554 554 556 559 560 562 563 567 568 16 16.1 16.2 16.3 16.4 16.5 16.6 Lie algebras Lie groups and Lie algebras Semi-simple Lie algebras Linear representations Real Lie algebras Affine Kac–Moody algebras Vertex operator representations 571 571 574 580 583 587 592 Index 599 1 Introduction The aim of this book is to introduce the reader to classical integrable systems. Because the subject has been developed by several schools having different perspectives, it may appear fragmented at first sight. We develop here the thesis that it has a profound unity and that the various approaches are simply changes of point of view on the same underlying reality. The more one understands each approach, the more one sees their unity. At the end one gets a very small set of interconnected methods. This fundamental fact sets the tone of the book. We hope in this way to convey to the reader the extraordinary beauty of the structures emerging in this field, which have illuminated many other branches of theoretical physics. The field of integrable systems is born together with Classical Mechanics, with a quest for exact solutions to Newton’s equations of motion. It turned out that apart from the Kepler problem which was solved by Newton himself, after two centuries of hard investigations, only a handful of other cases were found. In the nineteenth century, Liouville finally provided a general framework characterizing the cases where the equations of motion are “solvable by quadratures”. All examples previously found indeed pertained to this setting. The subject stayed dormant until the second half of the twentieth century when Gardner, Greene, Kruskal and Miura invented the Classical Inverse Scattering Method for the Korteweg– de Vries equation, which had been introduced in fluid mechanics. Soon afterwards, the Lax formulation was discovered, and the connection with integrability was unveiled by Faddeev, Zakharov and Gardner. This was the signal for a revival of the domain leading to an enormous amount of results, and truly general structures emerged which organized the subject. More recently, the extension of these results to Quantum Mechanics already led to remarkable results and is still a very active field of research. 1 2 1 Introduction Let us give a general overview of the ideas we present in this book. They all find their roots in the notion of Lax pairs. It consists of presenting the equations of motion of the system in the form L̇(λ) = [M (λ), L(λ)], where the matrices L(λ) and M (λ) depend on the dynamical variables and on a parameter λ called the spectral parameter, and [ , ] denotes the commutator of matrices. The importance of Lax pairs stems from the following simple remark: the Lax equation is an isospectral evolution equation for the Lax matrix L(λ). It follows that the curve defined by the equation det (L(λ) − µI) = 0 is time-independent. This curve, called the spectral curve, can be seen as a Riemann surface. Its moduli contain the conserved quantities. This immediately introduces the two main structures into the theory: groups enter through the Lie algebra involved in the commutator [M, L], while complex analysis enters through the spectral curve. As integrable systems are rather rare, one naturally expects strong constraints on the matrices L(λ) and M (λ). Constructing consistent Lax matrices may be achieved by appealing to factorization problems in appropriate groups. Taking into account the spectral parameter promotes this group to a loop group. The factorization problem may then be viewed as a Riemann–Hilbert problem, a central tool of this subject. In the group theoretical setting, solving the equations of motion amounts to solving the factorization problem. In the analytical setting, solutions are obtained by considering the eigenvectors of the Lax matrix. At any point of the spectral curve there exists an eigenvector of L(λ) with eigenvalue µ. This defines an analytic line bundle L on the spectral curve with prescribed Chern class. The time evolution is described as follows: if L(t) is the line bundle at time t then L(t)L−1 (0) is of Chern class 0, i.e. is a point on the Jacobian of the spectral curve. It is a beautiful result that this point evolves linearly on the Jacobian. As a consequence, one can express the dynamical variables in terms of theta-functions defined on the Jacobian of the spectral curve. The two methods are related as follows: the factorization problem in the loop group defines transition functions for the line bundle L. The framework can be generalized by replacing the Lax matrix by the first order differential equation (∂λ − Mλ (λ))Ψ = 0, where Mλ (λ) depends rationally on λ. The solution Ψ acquires non-trivial monodromy when λ describes a loop around a pole of Mλ . The isomonodromy problem consists of finding all Mλ with prescribed monodromy data. The solutions depend, in general, on a number of continuous parameters. The deformation equations with respect to these parameters form an integrable system. The theta-functions of the isospectral approach are then promoted to more general objects called the tau-functions. 1 Introduction 3 One can study the behaviour around each singularity of the differential operator quite independently. In the group theoretical version, the above extension of the framework corresponds to centrally extending the loop groups. Around a singularity the most general extended group is the group GL(∞) which corresponds to the KP hierarchy. It can be represented in a fermionic Fock space. Fermionic monomials acting on the vacuum yield decomposed vectors, which describe an infinite Grassmannian introduced by Sato. In this setting, the time flows are induced by the action of commuting one-parameter subgroups, and the tau-function is defined on the Grassmannian, i.e. the orbit of the vacuum, and characterizes it. Finally the Plücker equations of the Grassmannian are identified with the equations of motion, written in the bilinear Hirota form. We have tried, as much as possible, to make the book self-contained, and to achieve that each chapter can be studied quite independently. Generally, we first explain methods and then show how they can be applied to particular examples, even though this does not correspond to the historical development of the subject. In Chapter 2 we introduce the classical definition of integrable systems through the Liouville theorem. We present the Lax pair formulation, and describe the symplectic structure which is encoded into the so-called rmatrix form. In Chapter 3 we explain how to construct Lax pairs with spectral parameter, for finite and infinite-dimensional systems. The Lax matrix may be viewed as an element of a coadjoint orbit of a loop group. This introduces immediately a natural symplectic structure and a factorization problem in the loop group. We also introduce, at this early stage, the notion of tau-functions. In Chapter 4 we discuss the abstract group theoretical formulation of the theory. We then describe the analytical aspects of the theory in Chapter 5. In this setting, the action variables are g moduli of the spectral curve, a Riemann surface of genus g, and the angle variables are g points on it. We illustrate the general constructions by the examples of the closed Toda chain in Chapter 6, and the Calogero model in Chapter 7. The following two Chapters, 8 and 9, describe respectively the isomonodromic deformation problem and the infinite Grassmannian. Soliton solutions are obtained using vertex operators. Chapters 10 and 11 are devoted to the classical study of the KP and KdV hierarchies. We develop and use the formalism of pseudo-differential operators which allows us to give simple proofs of the main formal properties. Finite-zone solutions of KdV allow us to make contact with integrable systems of finite dimensionality and soliton solutions. In the next Chapter, 12, we study the class of Toda and sine-Gordon field theories. We use this opportunity to exhibit the relations between 4 1 Introduction their conformal and integrable properties. The sine-Gordon model is presented in the framework of the Classical Inverse Scattering Method in Chapter 13. This very ingenious method is exploited to solve the sineGordon equation. The last three chapters may be viewed as mathematical appendices, provided to help the reader. First we present the basic facts of symplectic geometry, which is the natural language to speak about Classical Mechanics and integrable systems. Since mathematical tools from Riemann surfaces and Lie groups are used almost everywhere, we have written two chapters presenting them in a concise way. We hope that they will be useful at least as an introduction and to fix notations. Let us say briefly how we have limited our discussion. First we choose to remain consistently at a relatively elementary mathematical level, and have been obliged to exclude some important developments which require more advanced mathematics. We put the emphasis on methods and we have not tried to make an exhaustive list of integrable systems. Another aspect of the theory we have touched only very briefly, through the Whitham equations, is the study of perturbations of integrable systems. All these subjects are very interesting by themselves, but the present book is big enough! A most active field of recent research is concerned with quantum integrable systems or the closely related field of exactly soluble models in statistical mechanics. When writing this book we always had the quantum theory present in mind, and have introduced all classical objects which have a well-known quantum counterpart, or are semi-classical limits of quantum objects. This explains our emphasis on Hamiltonians methods, Poisson brackets, classical r-matrices, Lie–Poisson properties of dressing transformations and the method of separation of variables. Although there is nothing quantum in this book, a large part of the apparatus necessary to understand the literature on quantum integrable systems is in fact present. The bibliography for integrable systems would fill a book by itself. We have made no attempt to provide one. Instead, we give, at the end of each chapter, a short list of references, which complements and enhances the material presented in the chapter, and we highly encourage the reader to consult them. Of course these references are far from complete, and we apologize to the numerous authors having contributed to the domain, and whose due credit is not acknowledged. Finally we want to thank our many colleagues from whom we learned so much and with whom we have discussed many parts of this book.