Nonlinear Evolution Equations and Painlevé Test

The following sections are included:

• Contents

• Preface

Much attention has been focused on the classification of dynamical systems as integrable and nonintegrable ones. When we consider classical mechanics, the periodic Toda lattice, described by a Hamiltonian function with N degrees of freedom, is a well-known example of an integrable system. In the 2N-dimensional phase space there are N first integrals and there is a Lax representation. In field theory the Korteweg-de Vries equation is an integrable system. The initial value problem of this equation can be solved with the help of the inverse scattering transform. Moreover, there are an infinite number of conservation laws. In quantum field theory the best known example of an integrable system is the quantum nonlinear Schrödinger equation. This system can be solved with the help of the Bet he ansatz or quantum inverse scattering transform. However, most dynamical systems are nonintegrable. In classical mechanics we find among these nonintegrable systems those with chaotic behaviour…

The following sections are included:

• Method of Sonya Kovalevskaya

• Painlevé Property

• Variational equation

• Painlevé Test

• Integrability

• Painlevé Test and Integrability

• Quasi- and Weak Painlevè Test

• The Painlevé Transcendents

• The Schwarzian Derivative

• Painlevé Test and Chaotic Behaviour

• Self-Similar Natural Boundaries

• Computer Algebra and Painlevé Test

The following sections are included:

• Conservative Systems

• Nambu Mechanics

• Dissipative Systems

• Construction of Solutions

The following sections are included:

• Introduction

• Ziglin's Theorems

• Applications

The following sections are included:

• Introduction

• Examples
  • Completely Integrable Systems
  • Nonintegrable Systems

The following sections are included:

• Definitions and Technique

The following sections are included:

• Introduction

• Lax Pairs and Bäcklund Transformation

• Symmetries

• Recursion Operator

• Linearizable Equations

• Kowalewski Exponents and Conservation Laws

• Resonances and Bäcklund Transformations

The following sections are included:

• Hirota Technique

• Connection with Painlevé Test

Here we study the deformation of Painlevé-series under a similarity reduction. It is shown that, under certain conditions, the Painlevé expansion of the reduced equation is formally equal to that of the given equation (Strampp and Renner (1988))…

The following sections are included:

• Introduction

• Burgers' Equation and a Hierarchy of Burgers' Equations

• Korteweg de Vries Equation

• Modified Korteweg de Vries Equation

• Boussinesq Equation

• One-Dimensional Nonlinear Schrödinger Equation

• Integrable Nonlinear Klein-Gordon Equations

• Calogero-Degasperis-Fokas Equation

• (1 + 1) Dimensional SO(2) Invariant σ Model

• Reduced Maxwell Bloch Equations

• Self-Dual Yang-Mills Equations

The following sections are included:

• Introduction

• Nonintegrable Nonlinear Klein-Gordon Equations
  • u⁴-Field Equation
  • Double Sine-Gordon Equation

• Discrete Boltzmann Equations

• Modified Burgers Equations

• Nonlinear Field Equation Describing the Relaxation to a Maxwell Distribution

The Painlevé transcendents described in section 1.8 play an important rôle in exactly solvable models in statistical mechanics. Here we discuss the appearence of the third Painlevé transcendent in the expression for the spin-spin correlation functions of the rectangular two-dimensional Ising model (Barouch et al (1973), Wu et al (1976)) and the appearence of the fifth Painlevé transcendent in the expression for the n-particle reduced density matrix of the one-dimensional impenetrable Bose gas (Jimbo et al (1980))…

The following sections are included:

• REFERENCES

• Index of Symbols

• Index