F. Gesztesy and H. Holden:

Soliton Equations and Their Algebro-Geometric Solutions

Volume I: (1+1)-Dimensional Continuous Models

Cambridge studies in advanced mathematics, Vol. 79
Cambridge University Press, Cambridge, 2003
First announcement

About

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. Detailed notes for each chapter together with an extensive bibliography enhance the presentation offered in the main text.

Volume II

Here you find more information about Volume II

Review

The book is described in a featured review by Emma Previato on MathSciNet (Mathematical Reviews) [MR1992536]
A review in the Newsletter of the European Mathematical Society (Dec 2004) written by Vladimir Soucek [EMS Review]
A review in Zentralblatt by Ma Wen-Xiu [Zbl pre01971279]
Book review by Emma Previato in Bulletin of the American Mathematical Society [Bull. AMS Review]

Errata and Addenda

[pdf] (Last updated: April 5, 2018) Any additional comments and corrections are most welcome.
Please send email to Fritz Gesztesy or Helge Holden.

How to get it

Link to the authors

Table of contents

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 in a nutshell
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

Sample chapter