Soliton Equations and Their Algebro-Geometric Solutions
Soliton Equations and Their Algebro-Geometric Solutions
Helge Holden, Norwegian University of Science and Technology, Trondheim
Johanna Michor, Universität Wien, Austria
Gerald Teschl, Universität Wien, Austria
$160.00
USDAs 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.
- A detailed treatment of the class of algebro-geometric solutions and their representations in terms of Riemann theta functions
- Rigorous and self-contained presentation at graduate level
- Four appendices and an exhaustive bibliography provide extensive background material
Reviews & endorsements
"The authors do a superb job at clarity and detail; very subtle features of existence, almost periodicity and reality conditions are also developed. Paraphrasing journalist John Gunther, "tell your audience what you are going to say, say it, then tell them what you said". I called the theory beautiful because it combines algebra and analysis, to provide satisfying, exact solutions to important nonlinear problems that arise in the real world. It does so by giving a geometric explanation of the persistence of symmetry, and allows the modeling of (multi-) solition solutions. This book, in my opinion, does the theory full justice."
Emma Previato, Mathematical Reviews
Product details
September 2008Hardback
9780521753081
448 pages
234 × 152 × 27 mm
0.74kg
Available
Table of Contents
- Acknowledgments
- Introduction
- 1. The Toda hierarchy
- 2. The Kac–van Moerbeke hierarchy
- 3. The Ablowitz–Ladik hierarchy
- A. Algebraic Curves and Their Theta Functions in a Nutshell
- B. Hyperelliptic Curves of the Toda-Type
- C. Asymptotic Spectral Parameter Expansions
- D. Lagrange Interpolation
- List of Symbols
- Bibliography
- Index
- Errata and Addenda for Volume I.
