Integrability / A. V. Mikhailov (ed.).

Format:

Book

Contributor:

Mikhailov, Alexander V., 1953-

Series:

Lecture notes in physics ; 767.

Lecture notes in physics, 0075-8450 ; 767

Language:

English

Subjects (All):

Mathematical physics.

Integral equations.

Physical Description:

xiii, 339 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : Springer, [2009]

Summary:

This is a unique collection of lectures on integrability, intended for graduate students or anyone who would like to master the subject from scratch, and written by leading experts in the field including Fields Medallist Serge Novikov. Since integrable systems have found a wide range of applications in modern theoretical and mathematical physics, it is important to recognise integrable models and, ideally, to obtain a global picture of the integrable world. The main aims of the book are to present a variety of views on the definition of integrable systems; to develop methods and tests for integrability based on these definitions; and to uncover beautiful hidden structures associated with integrable equations.

Contents:

Introduction / A.V. Mikhailov 1

1 Symmetries of Differential Equations and the Problem of Integrability / A.V. Mikhailov, V.V. Sokolov 19

1.1 Introduction 19

1.2 Symmetries and First Integrals of Finite-Dimensional Dynamical Systems 20

1.3 Basic Concepts of the Symmetry Approach 32

1.4 Modifications and Generalizations 52

1.5 Short Description of Solved Classification Problems and References 73

2 Number Theory and the Symmetry Classification of Integrable Systems / J.A. Sanders, J.P. Wang 89

2.1 Introduction 89

2.2 The Symbolic Method 91

2.3 An Implicit Function Theorem 96

2.4 Symmetry-Integrable Evolution Equations 98

2.5 Evolution Systems with k Components 105

2.6 One Symmetry Does not Imply Integrability 108

2.7 Concluding Remarks, Open Problems and Further Development 113

2.8 Some Irreducibility Results by F. Beukers 114

2.9 The Filtered Lie Algebra Version of the Implicit Function Theorem 115

3 Four Lectures: Discretization and Integrability. Discrete Spectral Symmetries / S.P. Novikov 119

3.1 Introduction 119

3.2 Continuous and Discrete Spectral Symmetries of 1D Systems and Spectral Theory of Operators. 1D Continuous Schrödinger Operator and Its Discrete Analogue 120

3.3 2D Schrödinger Operator. Discrete Spectral Symmetries, Spectral Theory of the Selected Energy Level and Space/Lattice Discretization 124

3.4 Discretization of the 2D Schrödinger Operators and Laplace Transformations on the Square and Equilateral Lattices 128

3.5 2D Manifolds with the Colored Black-White Triangulation. Integrable Systems on a Trivalent Tree 133

4 Symmetries of Spectral Problems / A. Shabat 139

4.1 Lie-Type Symmetries 139

4.2 Discrete Symmetries 153

5 Normal Form and Solitons / Y. Hiraoka, Y. Kodama 175

5.1 Introduction 175

5.2 Perturbed KdV Equation 178

5.3 Conserved Quantities and N-Soliton Solutions 180

5.4 Symmetry and the Perturbed Equation 184

5.5 Normal Form Theory 187

5.6 Interactions of Solitary Waves 195

5.7 Examples 201

6 Multiscale Expansion and Integrability of Dispersive Wave Equations / A. Degasperis 215

6.1 Introduction 215

6.2 Nonlinear Schrödinger-Type Model Equations and Integrability 225

6.3 Higher Order Terms and Integrability 234

7 Painlevé Tests, Singularity Structure and Integrability / A.N.W. Hone 245

7.1 Introduction 245

7.2 Painlevé Analysis for ODEs 249

7.3 The Ablowitz-Ramani-Segur Conjecture 254

7.4 The Weiss-Tabor-Carnevale Painlevé Test 257

7.5 Truncation Techniques 261

7.6 Weak Painlevé Tests 267

7.7 Outlook 273

8 Hirota's Bilinear Method and Its Connection with Integrability / J. Hietarinta 279

8.1 Why the Bilinear Form? 279

8.2 From Nonlinear to Bilinear (KdV) 280

8.3 Multisoliton Solutions for the KdV Class 283

8.4 Soliton Solution for the mKdV and sG Class 290

8.5 The Nonlinear Schrödinger Equation 294

8.6 Hierarchies 303

8.7 Bilinear Bäcklund Transformation 305

8.8 The Three-Soliton Condition as an Integrability Test 306

8.9 From Bilinear to Nonlinear 310

8.10 Conclusions 312

9 Integrability of the Quantum XXZ Hamiltonian / T. Miwa 315

9.1 Integrability 315

9.2 Symmetry 319.

Notes:

Includes bibliographical references and index.

ISBN:

9783540881100

3540881107

3540881115

9783540881117

OCLC:

248993496