As a partner to Volume 1: Dimensional Continuous Models, this book provides a self-contained introduction to solition equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices.
Author(s): Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl
Series: Cambridge Studies in Advanced Mathematics 114
Edition: 1
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 450
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 9
Acknowledgments......Page 11
Introduction......Page 13
1.1 Contents......Page 37
1.2 The Toda Hierarchy, Recursion Relations, Lax Pairs, and Hyperelliptic Curves......Page 38
1.3 The Stationary Toda Formalism......Page 53
1.4 The Stationary Toda Algebro-Geometric Initial Value Problem......Page 84
1.5 The Time-Dependent Toda Formalism......Page 96
1.6 The Time-Dependent Toda Algebro-Geometric Initial Value Problem......Page 115
1.7 Toda Conservation Laws and the Hamiltonian Formalism......Page 129
1.8 Notes......Page 157
2.1 Contents......Page 173
2.2 The KM Hierarchy and its Relation to the Toda Hierarchy......Page 174
2.3 The Stationary KM Formalism......Page 184
2.4 The Time-Dependent KM Formalism......Page 190
2.5 Notes......Page 193
3.1 Contents......Page 198
3.2 The Ablowitz–Ladik Hierarchy, Recursion Relations, Zero-Curvature Pairs, and Hyperelliptic Curves......Page 199
3.3 Lax Pairs for the Ablowitz–Ladik Hierarchy......Page 214
3.4 The Stationary Ablowitz–Ladik Formalism......Page 232
3.5 The Stationary Ablowitz–Ladik Algebro-Geometric Initial Value Problem......Page 248
3.6 The Time-Dependent Ablowitz–Ladik Formalism......Page 261
3.7 The Time-Dependent Ablowitz–Ladik Algebro-Geometric Initial Value Problem......Page 279
3.8 Ablowitz–Ladik Conservation Laws and the Hamiltonian Formalism......Page 293
3.9 Notes......Page 326
Appendix A: Algebraic Curves and Their Theta Functions in a Nutshell......Page 336
Appendix B: Hyperelliptic Curves of the Toda-Type......Page 365
Appendix C: Asymptotic Spectral Parameter Expansions and Nonlinear Recursion Relations......Page 377
Appendix D: Lagrange Interpolation......Page 397
List of Symbols......Page 407
Bibliography......Page 410
Index......Page 435
Errata and Addenda for Volume......Page 438
