经典可积系统导论 内容简介

This book provides a thorough introduction to the theory ofclassical integrable systems,discussing the various approachesto the subject and explaining their interrelatio.The bookbegi by introducing the central ideas of the theory ofintegrable systems，based on Lax representatio，loop groups andRiemann surfaces.These ideas are then illustrated with detailedstudies of model systems.The connection between isomon- odromicdeformation and integrability is discussed，and integrable fieldtheories are covered in detail.The KP，KdV and 'lbda hierarchiesare explained using the notion of Grassmannian，vertex operatoand pseudo-differential operato.A chapter is devoted to theinvee scattering method and three complementary chapte coverthe necessary mathematical tools from symplectic geometry，Riemann surfaces and Lie algebras.The book contai many worked examples and is suitable foruse as a textbook on graduate coues.It also provides acompreheive reference for researche already working in thefield.OLIVIER BABELON has been a member of the Centre National dela Recherche Sci- entifique （CNRS） since 1978.He works at theLaboratoire de Physique Theorique et Hautes Energies （LPTRE） atthe Univeity of Paris VI-Paris Ⅷ.He is main fields of interestare particle physics，gauge theories and integrables systems.MICHEL TALON has been a member of the CNRS since 1977.Heworks at the LPTHE at the Univeity of Paris VI-Paris VII.He isinvolved in the computation of radiative correctio andanomalies in gauge theories and integrable systems.DENIS BERNARD has been a member of the CNRS since 1988.Hecurrently works at the Service de Physique Theorique deSaclay.His main fields of interest are conformal field theoriesand integrable systems，and other aspects of statistical fieldtheories，including statistical turbulence.

经典可积系统导论 本书特色

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.

经典可积系统导论 目录

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 modifiedⅥln9-Baxter equations
4.2 Algebraic meaning of the classical Yan9-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 T0da 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 Caloger0-Moser model
……
8 Isomonodromic deformations
9 Grassmannian and integrable hierarchies
10 The KP hierarchy
11 The KdV hierarchy
12 The Toda field theories
13 Classical inverse scattering method
14 Symplectic geometry
15 Riemann surfaces
16 Lie algebras
Index