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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
- Fritz Gesztesy, University of Missouri, Columbia, USA [homepage]
- Helge Holden, Norwegian University of Science and Technology, Trondheim, Norway [homepage]
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
- Introduction [pdf]
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