front-matter
Chapter 1
1 Analytic Methods
2 Painlev´e Analysis
4 Lie-Algebraic and Group-Theoretical Methods
5 Bihamiltonian Structures
Chapter 2
1 Fundamentals of Waves
2 IST for Nonlinear Equations in 1+1 Dimensions
3 Scattering and the Inverse Scattering Transform
4 IST for 2+1 Equations
Chapter 3
1 General Introduction: Who Cares about Integrability?
2 Historical Presentation: From Newton to Kruskal
3 Towards a Working De.nition of Integrability
3.1 Complete Integrability
3.2 Partial and Constrained Integrability
4 Integrability and How to Detect It
4.1 Fixed and Movable Singularities
4.2 The Ablowitz-Ramani-Segur Algorithm
5 Implementing Singularity Analysis: From Painlev´e to ARS and Beyond
6 Applications to Finite and In.nite Dimensional Systems
6.1 Integrable Di.erential Systems
6.2 Integrable Two-Dimensional Hamiltonian Systems
6.3 In.nite-Dimensional Systems
7 Integrable Discrete Systems Do Exist!
8 Singularity Con.nement: The Discrete Painlev´e Property
9 Applying the Con.nement Method: Discrete Painlev´e Equations and Other Systems
9.1 The Discrete Painlev´e Equations
9.2 Multidimensional Lattices and Their Similarity Reductions
9.3 Linearizable Mappings
10 Discrete/Continuous Systems: Blending Con.nement with Singularity Analysis
10.1 Integrodi.erential Equations of the Benjamin-Ono Type
10.2 Multidimensional Discrete/Continuous Systems
10.3 Delay-Di.erential Equations
11 Conclusion
Chapter 4
1 Why the Bilinear Form?
2 From Nonlinear to Bilinear
2.1 Bilinearization of the KdV Equation
2.2 Another Example: The Sasa-Satsuma Equation
2.3 Comments
3 Constructing Multi-soliton Solutions
3.1 The Vacuum, and the One-Soliton Solution
3.2 The Two-Soliton Solution
3.3 Multi-soliton Solutions
4 Searching for Integrable Evolution Equations
4.1 KdV
4.2 mKdV and sG
4.3 nlS
Chapter 5
Introduction
1 Lie Bialgebras
1.1 An Example: sl(2, C)
1.2 Lie-Algebra Cohomology
1.3 De.nition of Lie Bialgebras
1.4 The Coadjoint Representation
1.5 The Dual of a Lie Bialgebra
1.6 The Double of a Lie Bialgebra. Manin Triples
1.7 Examples
1.8 Bibliographical Note
2 Classical Yang-Baxter Equation and r-Matrices
2.2 The Classical Yang-Baxter Equation
2.3 Tensor Notation
2.4 R-Matrices and Double Lie Algebras
2.5 The Double of a Lie Bialgebra Is a Factorizable Lie Bialgebra
2.6 Bibliographical Note
3 Poisson Manifolds. The Dual of a Lie Algebra. Lax Equations
3.1 Poisson Manifolds
3.2 The Dual of a Lie Algebra
3.3 The First Russian Formula
3.4 The Traces of Powers of Lax Matrices Are in Involution
3.5 Symplectic Leaves and Coadjoint Orbits
3.6 Double Lie Algebras and Lax Equations
3.7 Solution by Factorization
3.8 Bibliographical Note
4 Poisson Lie Groups
4.1 Multiplicative Tensor Fields on Lie Groups
4.2 Poisson Lie Groups and Lie Bialgebras
4.3 The Second Russian Formula (Quadratic Brackets)
4.5 The Dual of a Poisson Lie Group
4.6 The Double of a Poisson Lie Group
4.7 Poisson Actions
4.8 Momentum Mapping
4.9 Dressing Transformations
4.10 Bibliographical Note
Appendix 1 The ‘Big Bracket’ and Its Applications
Appendix 2 The Poisson Calculus and Its Applications
Background on Manifolds, Lie Algebras and Lie Groups, and Hamiltonian Systems
A. Fundamental Articles on Poisson Lie Groups
B. Books and Lectures on Poisson Manifolds, Lie Bialgebras, r-Matrices, and Poisson Lie Groups
C. Further Developments on Lie Bialgebras, r-Matrices and Poisson Lie Groups
D. Bibliographical Note Added in the Second Edition
Chapter 6
1 Introduction
2 Nonlinear-Regular-Singular Analysis
2.1 The Painlev´e Property
2.3 The Painlev´e Test
2.4 Necessary versus Su.cient Conditions for the Painlev´e Property
2.5 A Direct Proof of the Painlev´e Property for ODEs
2.6 Rigorous Results for PDEs
3 Nonlinear-Irregular-Singular Point Analysis
3.1 The Chazy Equation
3.2 The Bureau Equation
4 Coalescence Limits
Chapter 7
1st Lecture: Bihamiltonian Manifolds
2nd Lecture: Marsden–Ratiu Reduction
3rd Lecture: Generalized Casimir Functions
4th Lecture: Gel’fand–Dickey Manifolds
5th Lecture: Gel’fand–Dickey Equations
5.2. The Auxiliary Eigenvalue Problem
5.3 Dressing Transformations
6th Lecture: KP Equations
7th Lecture: Poisson–Nijenhuis Manifolds
8th Lecture: The Calogero System
Chapter 8
1 Introduction
2 Hirota’s Method
3 Algebraic Inentities
4 Extensions
4.1 q-Discrete Toda Equation
4.2 Trilinear Formalism
4.3 Ultra-discrete Systems
5 Concluding Remarks
Chapter 9
1 Introduction
2 Generalities
2.1 Basic Theorem: Linear Case
2.2 Two Examples
3 Quadratic Case
3.1 Abstract Case: Poisson Lie Groups and Factorizable Lie Bialgebras
3.2 Duality Theory for Poisson Lie Groups and Twisted Spectral Invariants
3.3 Sklyanin Bracket on G(z)
4 Quantization
4.1 Linear Case
4.2 Quadratic Case. Quasitriangular Hopf Algebras