The book contains many worked examples and is suitable for use as a textbook on graduate courses.It also provides a comprehensive reference for researchers already working in the field.

This book provides a thorough introduction to the theory of classical integrable systems，discussing the various approaches to the subject and explaining their interrelations.The book begins by introducing the central ideas of the theory of integrable systems，based on Lax representations，loop groups and Riemann surfaces.These ideas are then illustrated with detailed studies of model systems.The connection between isomon- odromic deformation and integrability is discussed，and integrable field theories are covered in detail.The KP，KdV and 'lbda hierarchies are explained using the notion of Grassmannian，vertex operators and pseudo-differential operators.A chapter is devoted to the inverse scattering method and three complementary chapters cover the necessary mathematical tools from symplectic geometry，Riemann surfaces and Lie algebras.

The book contains many worked examples and is suitable for use as a textbook on graduate courses.It also provides a comprehensive reference for researchers already working in the field.

OLIVIER BABELON has been a member of the Centre National de la Recherche Sci- entifique （CNRS） since 1978.He works at the Laboratoire de Physique Theorique et Hautes Energies （LPTRE） at the University of Paris VI-Paris Ⅷ.He is main fields of interest are particle physics，gauge theories and integrables systems.

MICHEL TALON has been a member of the CNRS since 1977.He works at the LPTHE at the University of Paris VI-Paris VII.He is involved in the computation of radiative corrections and anomalies in gauge theories and integrable systems.

DENIS BERNARD has been a member of the CNRS since 1988.He currently works at the Service de Physique Theorique de Saclay.His main fields of interest are conformal field theories and integrable systems，and other aspects of statistical field theories，including statistical turbulence.

1 Introduction

2 Integrable dynamical systems

 2.1 Introduction

 2.2 The Liouville theorem

 2.3 Action-angle variables

 2.4 Lax pairs

 2.5 Existence of an r-matrix

 2.6 Commuting flows

 2.7 The Kepler problem

 2.8 The Euler top

 2.9 The Lagrange top

 2.10 The Kowalevski top

 2.11 The Neumann model

 2.12 Geodesics on an ellipsoid

 2.13 Separation of variables in the Neumann model

3 Synopsis of integrable systems

 3.1 Examples of Lax pairs with spectral parameter

 3.2 The Zakharov-Shabat construction

 3.3 Coadjoint orbits and Hamiltonian formalism

 3.4 Elementary flows and wave function

 3.5 Factorization problem

 3.6 Tau-functions

 3.7 Integrable field theories and monodromy matrix

 3.8 Abelianization

 3.9 Poisson brackets of the monodromy matrix

 3.10 The group of dressing transformations

 3.11 Soliton solutions

4 Algebraic methods

 4.1 The classical and modified Yang-Baxter equations

 4.2 Algebraic meaning of the classical Yang-Baxter equations

 4.3 Adler-Kostant-Symes scheme

 4.4 Construction of integrable systems

 4.5 Solving by factorization

 4.6 The open Toda chain

 4.7 The r-matrix of the Toda models

 4.8 Solution of the open Toda chain

 4.9 Toda system and Hamiltonian reduction

 4.10 The Lax pair of the Kowalevski top

5 Analytical methods

 5.1 The spectral curve

 5.2 The eigenvector bundle

 5.3 The adjoint linear system

 5.4 Time evolution

 5.5 Theta-functions formulae

 5.6 Baker-Akhiezer functions

 5.7 Linearization and the factorization problem

 5.8 Tau-functions

 5.9 Symplectic form

 5.10 Separation of variables and the spectral curve

 5.11 Action-angle variables

 5.12 Riemann surfaces and integrability

 5.13 The Kowalevski top

 5.14 Infinite-dimensional systems

6 The closed Toda chain

 6.1 The model

 6.2 The spectral curve

 6.3 The eigenvectors

 6.4 Reconstruction formula

 6.5 Symplectic structure

 6.6 The Sklyanin approach

 6.7 The Poisson brackets

 6.8 Reality conditions

7 The Calogero-Moser model

 7.1 The spin Calogero-Moser model

 7.2 Lax pair

 7.3 The r-matrix

 7.4 The scalar Calogero-Moser model

 7.5 The spectral curve

 7.6 The eigenvector bundle

 7.7 Time evolution

 7.8 Reconstruction formulae

 7.9 Symplectic structure

 7.10 Poles systems and double-Bloch condition

 7.11 Hitchin systems

 7.12 Examples of Hitchin systems

 7.13 The trigonometric Calogero-Moser model

8 Isomonodromic deformations

 8.1 Introduction

 8.2 Monodromy data

 8.3 Isomonodromy and the Riemann-Hilbert problem

 8.4 Isomonodromic deformations

 8.5 Schlesinger transformations

 8.6 Tau-functions

 8.7 Ricatti equation

 8.8 Sato's formula

 8.9 The Hirota equations

 8.10 Tau-functions and theta-functions

 8.11 The Painleve equations

9 Grassmannian and integrable hierarchies

 9.1 Introduction

 9.2 Fermions and GL

 9.3 Boson-fermion correspondence

 9.4 Tau-functions and Hirota bilinear identities

 9.5 The KP hierarchy and its soliton solutions

 9.6 Fermions and Grassmannians

 9.7 Schur polynomials

 9.8 From fermions to pseudo-differential operators

 9.9 The Segal-Wilson approach

10 The KP hierarchy

 10.1 The algebra of pseudo-differential operators

 10.2 The KP hierarchy

 10.3 The Baker-Akhiezer function of KP

 10.4 Algebro-geometric solutions of KP

 10.5 The tau-function of KP

 10.6 The generalized KdV equations

 10.7 KdV Hamiltonian structures

 10.8 Bihamiltonian structure

 10.9 The Drinfeld-Sokolov reduction

 10.10 Whitham equations

 10.11 Solution of the Whitham equations

11 The KdV hierarchy

 11.1 The KdV equation

 11.2 The KdV hierarchy

 11.3 Hamiltonian structures and Virasoro algebra

 11.4 Soliton solutions

 11.5 Algebro-geometric solutions

 11.6 Finite-zone solutions

 11.7 Action-angle variables

 11.8 Analytical description of solitons

 11.9 Local fields

 11.10 Whitham's equations

12 The Toda field theories

 12.1 The Liouville equation

 12.2 The Toda systems and their zero-curvature represe

 12.3 Solution of the Toda field equations

 12.4 Hamiltonian formalism

 12.5 Conformal structure

 12.6 Dressing transformations

 12.7 The affine sinh-Gordon model

 12.8 Dressing transformations and soliton solutions

 12.9 N-soliton dynamics

 12.10 Finite-zone solutions

13 Classical inverse scattering method

 13.1 The sine-Gordon equation

 13.2 The Jost solutions

 13.3 Inverse scattering as a Riemann-Hilbert problem

 13.4 Time evolution of the scattering data

 13.5 The Gelfand-Levitan-Marchenko equation

 13.6 Soliton solutions

 13.7 Poisson brackets of the scattering data

 13.8 Action-angle variables

14 Symplectic geometry

 14.1 Poisson manifolds and symplectic manifolds

 14.2 Coadjoint orbits

 14.3 Symmetries and Hamiltonian reduction

 14.4 The case M=T*G

 14.5 Poisson-Lie groups

 14.6 Action of a Poisson-Lie group on a symplectic manifold

 14.7 The groups G and G*

 14.8 The group of dressing transformations

15 Riemann surfaces

 15.1 Smooth algebraic curves

 15.2 Hyperelliptic curves

 15.3 The Riemann-Hurwitz formula

 15.4 The field of meromorphic functions of a Riemann surface

 15.5 Line bundles on a Riemann surface

 15.6 Divisors

 15.7 Chern class

 15.8 Serre duality

 15.9 The Riemann-Roch theorem

 15.10 Abelian differentials

 15.11 Riemann bilinear identities

 15.12 Jacobi variety

 15.13 Theta-functions

 15.14 The genus 1 case

 15.15 The Riemann-Hilbert factorization problem

16 Lie algebras

 16.1 Lie groups and Lie algebras

 16.2 Semi-simple Lie algebras

 16.3 Linear representations

 16.4 Real Lie algebras

 16.5 Affine Kac-Moody algebras

 16.6 Vertex operator representations

Index