Soliton Equations and their Algebro-Geometric Solutions: Volume 1, (1+1)-Dimensional Continuous Models

This book offers a detailed treatment of a class of algebro-geometric solutions and their representations.

Fritz Gesztesy (Author), Helge Holden (Author)

9780521753074, Cambridge University Press

Hardback, published 5 June 2003

518 pages
22.9 x 15.2 x 3.3 cm, 0.784 kg

'The book is very well organized and carefully written. It could be particularly useful for analysts wanting to learn new methods coming from algebraic geometry.' EMS Newsletter

The focus of this book is on 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 as well as 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). The presentation is rigorous, detailed, and self-contained, with ample background material provided in various appendices. Detailed notes for each chapter together with an exhaustive bibliography enhance the presentation offered in the main text.

Introduction
1. The KdV hierarchy
2. The sGmKdV hierarchy
3. The AKNS hierarchy
4. The classical massive Thirring system
5. The Camassa–Holm hierarchy
Appendix A. Algebraic curves and their theta functions
Appendix B. KdV-type curves
Appendix C. AKNS-type curves
Appendix D. Asymptotic spectral parameter expansions
Appendix E. Lagrange interpolation
Appendix F. Symmetric functions
Appendix G. KdV and AKNS Darboux-type transformations
Appendix H. Elliptic functions
Appendix I. Herglotz functions
Appendix J. Weyl-Titchmarsh theory
List of symbols
Bibliography
Index.

Subject Areas: Nonlinear science [PBWR], Geometry [PBM]