Introduction ................................................... 1
A.V. Mikhailov
References ................................................. 15
1 Symmetries of Differential Equations and the Problem of
Integrability .............................................. 19
A.V. Mikhailov and V.V. Sokolov
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
References ................................................. 85
2 Number Theory and the Symmetry Classification of
Integrable Systems ......................................... 89
J.A. Sanders and J.P. Wang
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 к 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
References ................................................ 116
3 Four Lectures: Discretization and Integrability.
Discrete Spectral Symmetries .............................. 119
S.P. Novikov
3.1 Introduction ......................................... 119
3.2 Continuous and Discrete Spectral Symmetries of ID
Systems and Spectral Theory of Operators. ID
Continuous Schrodinger Operator and Its Discrete
Analogue ............................................. 120
3.3 2D Schrodinger Operator. Discrete Spectral
Symmetries, Spectral Theory of the Selected Energy
Level and Space/Lattice Discretization ............... 124
3.4 Discretization of the 2D Schrodinger 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
References ................................................ 137
4 Symmetries of Spectral Problems ........................... 139
A. Shabat
4.1 Lie-Type Symmetries .................................. 139
4.2 Discrete Symmetries .................................. 153
References ................................................ 172
5 Normal Form and Solitons .................................. 175
Y. Hiraoka and Y. Kodama
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
References ................................................ 212
6 Multiscale Expansion and Integrability of Dispersive
Wave Equations ............................................ 215
A. Degasperis
6.1 Introduction ......................................... 215
6.2 Nonlinear Schrodinger-Type Model Equations and
Integrability ........................................ 225
6.3 Higher Order Terms and Integrability ................. 234
References ................................................ 243
7 Painleve Tests, Singularity Structure and Integrability ... 245
A.N.W. Hone
7.1 Introduction ......................................... 245
7.2 Painleve Analysis for ODEs ........................... 249
7.3 The Ablowitz-Ramani-Segur Conjecture ................. 254
7.4 The Weiss-Tabor-Carnevale Painleve Test .............. 257
7.5 Truncation Techniques ................................ 261
7.6 Weak Painleve Tests .................................. 267
7.7 Outlook .............................................. 273
References ................................................ 275
8 Hirota's Bilinear Method and Its Connection with
Integrability ............................................. 279
J. Hietarinta
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 Schrodinger Equation ................... 294
8.6 Hierarchies .......................................... 303
8.7 Bilinear Backlund Transformation ..................... 305
8.8 The Three-Soliton Condition as an Integrability
Test ................................................. 306
8.9 From Bilinear to Nonlinear ........................... 310
8.10 Conclusions .......................................... 312
References ................................................ 312
9 Integrability of the Quantum XXZ Hamiltonian .............. 315
T. Miwa
9.1 Integrability ........................................ 315
9.2 Symmetry ............................................. 319
References ................................................ 323
Index ........................................................ 325
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