the PAINLEVE HANDBOOK.

Preface to the Second Edition
Preface
Outline
	References
Contents
List of Figures
List of Tables
Acronyms
1 Introduction
	1.1 Singularities in the Complex Plane
		1.1.1 Perturbative Method
		1.1.2 Nonperturbative Method
	1.2 Painlevé Property and the Six Transcendents
	References
2 Singularity Analysis: Painlevé Test
	2.1 Kowalevski-Gambier Method
		2.1.1 Lorenz Model
		2.1.2 Kuramoto-Sivashinsky (KS) Equation
		2.1.3 Cubic Complex Ginzburg-Landau (CGL3) Equation
		2.1.4 Duffing-van der Pol Oscillator
		2.1.5 Hénon-Heiles System
	2.2  The α-Method
		2.2.1 Theoretical Background
		2.2.2 The Practical Method
		2.2.3 Example
		2.2.4 General Necessary Conditions (Order m and Degree One)
	2.3 Fuchsian Perturbative Method
	2.4 NonFuchsian Perturbative Method
	2.5 Method for First Order Equations
	References
3 Integrating Ordinary Differential Equations
	3.1 Integrable Situation
		3.1.1 First Integrals and Integration of the Lorenz Model
			3.1.1.1 Case (1,1/2,0)
			3.1.1.2 Case (2,1,1/9)
			3.1.1.3 Case (0,1/3,r)
			3.1.1.4 Case (1,0,r)
		3.1.2 General Traveling Wave of KdV Equation
		3.1.3 General Traveling Wave of NLS Equation
		3.1.4 Integration of First Order Autonomous ODEs
	3.2 Partially Integrable Situation
		3.2.1 The Three ODEs KS, CGL3, CGL5
		3.2.2 First Order Equations and Elliptic Functions
		3.2.3 Second Order Equations and Meromorphic Functions
		3.2.4 The Two Classes of Algebraic ODEs
	3.3 First Class of Partially Integrable Equations
		3.3.1 Criterium of Residues
		3.3.2 Hermite Decomposition
		3.3.3 Computation of the Entire Part of a Hermite Decomposition
		3.3.4 Handling Cases with Several Laurent Series
		3.3.5 Subequation Method
		3.3.6 Example Jacobi
		3.3.7 All Meromorphic Traveling Waves of KS Equation
		3.3.8 All Meromorphic Solutions of the Duffing-van der Pol Oscillator
		3.3.9 All Meromorphic Traveling Waves of CGL3 and CGL5
			3.3.9.1 Laurent Series M(ξ) of CGL3/5
			3.3.9.2 Criterium of Residues for Elliptic Solutions
			3.3.9.3 Solutions M
			3.3.9.4 All Meromorphic Solutions of CGL3
			3.3.9.5 Selected Meromorphic Solutions of CGL5
		3.3.10 Particular Class of Polynomials in tanh
		3.3.11 Particular Class of Polynomials in tanh and sech
	3.4 Second Class of Partially Integrable Equations
		3.4.1 Infinite Number of Laurent Series, Elliptic Subequation
		3.4.2 Infinite Number of Laurent Series, One-Family Truncation with S(x)=0
		3.4.3 Single-valued Solutions of the Bianchi IX Cosmological Model
	References
4 Partial Differential Equations: Painlevé Test
	4.1 On Reductions
	4.2 Soliton Equations
	4.3 Painlevé Property for PDEs
	4.4 Painlevé Test
		4.4.1 Optimal Expansion Variable
		4.4.2 Integrable Situation, Example of KdV
		4.4.3 Partially Integrable Situation, Example of KPP
	References
5 From the Test to Explicit Solutions of PDEs
	5.1 Global Information from the Test
	5.2 Building N-Soliton Solutions
	5.3 Tools of Integrability
		5.3.1 Lax Pair
		5.3.2 Darboux Transformation
		5.3.3 Crum Transformation
		5.3.4 Singular Part Transformation
		5.3.5 Nonlinear Superposition Formula
	5.4 Choosing the Order of Lax Pairs
		5.4.1 Second-Order Lax Pairs and Their Privilege
		5.4.2 Third-Order Lax Pairs
	5.5 Singular Manifold Method
		5.5.1 Algorithm
		5.5.2 Level of Truncation and Choice of Variable
	5.6 Application to Integrable Equations
		5.6.1 One-Family Cases: KdV and Boussinesq
			5.6.1.1 The Case of KdV
			5.6.1.2 The Case of Boussinesq
			5.6.1.3 Resonant Triad of One-Soliton Solutions
		5.6.2 Two-Family Cases: Sine-Gordon and Modified KdV
			5.6.2.1 The Sine-Gordon Equation
			5.6.2.2 The Modified Korteweg-de Vries Equation
			5.6.2.3 The Breather of Nonlinear Schrödinger
		5.6.3 Third Order: Sawada–Kotera and Kaup–Kupershmidt
			5.6.3.1 The Help from the Gambier Classification
			5.6.3.2 Singularity Structure of SK and KK Equations
			5.6.3.3 Truncation with a Second Order Lax Pair
			5.6.3.4 Truncation with a Third Order Lax Pair and G5
			5.6.3.5 Truncation with a Third Order Lax Pair and G25
			5.6.3.6 Bäcklund Transformation
			5.6.3.7 Nonlinear Superposition Formula
	5.7 Application to Partially Integrable Equations
		5.7.1 One-Family Case: Fisher Equation
		5.7.2 Two-Family Case: KPP Equation
	5.8 Reduction of the Singular Manifold Method to the ODE Case
		5.8.1 From Lax Pair to Isomonodromic Deformation
		5.8.2 From BT to Birational Transformation
		5.8.3 From NLSF to Contiguity Relation
		5.8.4 Reformulation of the Singular Manifold Method: An Additional Homography
	References
6 Integration of Hamiltonian Systems
	6.1 Various Integrations
	6.2 Cubic Hénon-Heiles Hamiltonians
		6.2.1 Second Invariants
		6.2.2 Separation of Variables
			6.2.2.1 Case β/ α=-6 (KdV5)
			6.2.2.2 Cases β/ α=-1 (SK) and -16 (KK)
		6.2.3 Direct Integration
	6.3 Quartic Hénon-Heiles Hamiltonians
		6.3.1 Second Invariants
		6.3.2 Separation of Variables
			6.3.2.1 Case 1:2:1 (Manakov System)
			6.3.2.2 Cases 1:6:1 and 1:6:8
			6.3.2.3 Case 1:12:16
		6.3.3 Painlevé Property
	6.4 Final Picture for HH3 and HH4
	References
7 Discrete Nonlinear Equations
	7.1 Generalities
	7.2 Discrete Painlevé Property
	7.3 Discrete Painlevé Test
		7.3.1 Method of Polynomial Growth
		7.3.2 Method of Singularity Confinement, Simplified Version
		7.3.3 Method of Singularity Confinement, Complete Version
		7.3.4 Method of Perturbation of the Continuum Limit
	7.4 Discrete Riccati Equation
	7.5 Discrete Lax Pairs
	7.6 Exact Discretizations
		7.6.1 Ermakov-Pinney Equation
		7.6.2 Elliptic Equation
	7.7 Discrete Versions of NLS
	7.8 A Sketch of Discrete Painlevé Equations
		7.8.1 Analytic Approach
		7.8.2 Geometric Approach
		7.8.3 Summary of Discrete Painlevé Equations
	References
8 Selected Problems Integrated by Painlevé Functions
	8.1 Bonnet Surfaces
	8.2 Solution of a Real Boundary Value Problem
	8.3 Random Matrices and Integrable Systems of PDEs
	8.4 Persistence
	8.5 Probability Law of the Airy Kernel
	8.6 Three-Wave Resonant Interaction
	8.7 Bianchi IX System with Self-Dual Weyl Tensor
	8.8 Conformal Blocks
	8.9 Superintegrable Potentials
	8.10 Benjamin-Feir Instability and PI
	References
9 FAQ (Frequently Asked Questions)
	References
A The Classical Results of Painlevé and Followers
	A.1 Groups of Invariance of the Painlevé Property
	A.2 Irreducibility. Classical Solutions
	A.3 Classifications
		A.3.1 First Order First Degree ODEs
		A.3.2 First Order Higher Degree ODEs
		A.3.3 Second Order First Degree ODEs
		A.3.4 Second Order Higher Degree ODEs
		A.3.5 Third Order First Degree ODEs
		A.3.6 Fourth Order First Degree ODEs
		A.3.7 Higher Order First Degree ODEs
		A.3.8 Second Order First Degree PDEs
B More on the Painlevé Transcendents
	B.1 Chronology of the Discovery of PVI
	B.2 The Two Representations of PVI
	B.3 Coalescence Cascade
	B.4 Painlevé Equations in Elliptic or Degenerate Elliptic Coordinates
	B.5 Tau-Functions of PVI
	B.6 Invariances of PVI
	B.7 The Two Points of View on Painlevé Equations: Analytic, Geometric
	B.8 How Many Painlevé Equations: Six, Eight or Five?
	B.9 The Eight Continuous Painlevé Equations
	B.10 Geometric Characterizations of all Discrete and Continuous Painlevé Equations
	B.11 Affine Weyl Groups
	B.12 Invariance Under Homographies
	B.13 Invariance Under Birational Transformations
		B.13.1 Normal Sequence
		B.13.2 Biased Sequence
	B.14 Invariance Under Nonbirational Transformations
	B.15 Hamiltonian Representations
	B.16 Lax Pairs
		B.16.1 Matrix Lax Pairs in Rational Coordinates
			B.16.1.1 Matrix Lax Pairs Holomorphic in the Four Parameters
			B.16.1.2 Matrix Lax Pairs Diagonally Symmetric
		B.16.2 Spectral Curves
		B.16.3 Matrix and Scalar Lax Pairs in Elliptic Coordinates
		B.16.4 Scalar Lax Pairs in Rational Coordinates
		B.16.5 Scalar Lax Pairs and Quantum Correspondence
	B.17 Classical Solutions
		B.17.1 One-Parameter Classical Solutions
		B.17.2 Zero-Parameter Classical Solutions
	B.18 Expansions Near a Critical Singularity
	B.19 Connection Problem
	B.20 Order of Growth of the Solutions of Pn
	B.21 Elliptic Functions of Three Arguments
		B.21.1 From Rational to Elliptic Coordinates
		B.21.2 From Elliptic to Rational Coordinates
C Brief Presentation of the Elliptic Functions
	C.1 The Notations of Jacobi and Weierstrass
	C.2 The Symmetric Notation of Halphen
	C.3 Successive Degeneracies
	C.4 Three Representations of Elliptic Functions
D Basic Introduction to the Nevanlinna Theory
E The Bilinear Formalism
	E.1 Bilinear Representation of a PDE
	E.2 Bilinear Representation of a Bäcklund Transformation
F Algorithm for Computing Laurent Series
	References
Index