This book is about algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions; also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary and time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces).
Author(s): Fritz Gesztesy, Helge Holden
Series: Cambridge Studies in Advanced Mathematics 79
Publisher: Cambridge University Press
Year: 2003
Language: English
Pages: 519
Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
Acknowledgments......Page 13
Introduction......Page 15
1.1 Contents......Page 33
1.2 The KdV Hierarchy, Recursion Relations, and Hyperelliptic Curves......Page 34
1.3 The Stationary KdV Formalism......Page 43
1.4 The Time-Dependent KdV Formalism......Page 78
1.5 General Trace Formulas......Page 102
1.6 Notes......Page 119
2.1 Contents......Page 137
2.2 The sGmKdV Hierarchy, Recursion Relations, and Hyperelliptic Curves......Page 138
2.3 The Stationary sGmKdV Formalism......Page 148
2.4 The Time-Dependent sGmKdV Formalism......Page 163
2.5 Notes......Page 187
3.1 Contents......Page 191
3.2 The AKNS Hierarchy, Recursion Relations, and Hyperelliptic Curves......Page 192
3.3 The Stationary AKNS Formalism......Page 204
3.4 The Time-Dependent AKNS Formalism......Page 226
3.5 The Classical Boussinesq Hierarchy......Page 242
3.6 Notes......Page 251
4.1 Contents......Page 256
4.2 The Classical Massive Thirring System, Recursion Relations, and Hyperelliptic Curves......Page 257
4.3 The Basic Algebro-Geometric Formalism......Page 263
4.4 Theta Function Representations of u, v, u ,v......Page 281
4.5 Notes......Page 297
5.1 Contents......Page 300
5.2 The CH Hierarchy, Recursion Relations, and Hyperelliptic Curves......Page 301
5.3 The Stationary CH Formalism......Page 307
5.4 The Time-Dependent CH Formalism......Page 322
5.5 Notes......Page 336
Appendix A Algebraic Curves and Their Theta Functions in a Nutshell......Page 341
Notes......Page 367
Appendix B Hyperelliptic Curves of the KdV-Type......Page 369
Appendix C Hyperelliptic Curves of the AKNS-Type......Page 381
Appendix D Asymptotic Spectral Parameter Expansions and Nonlinear Recursion Relations......Page 394
Notes......Page 408
Appendix E Lagrange Interpolation......Page 410
Notes......Page 414
Appendix F Symmetric Functions, Trace Formulas, and Dubrovin-Type Equations......Page 415
Notes......Page 437
Appendix G KdV and AKNS Darboux-Type Transformations......Page 439
KdV Darboux-Type Transformations......Page 440
AKNS Darboux-Type Transformations......Page 448
Notes......Page 457
Appendix H Elliptic Functions......Page 459
Notes......Page 463
Appendix I Herglotz Functions......Page 464
Notes......Page 468
Appendix J Spectral Measures and Weyl–Titchmarsh m-Functions for Schrödinger Operators......Page 469
Notes......Page 478
List of Symbols......Page 480
Bibliography......Page 483
Index......Page 515
