F. Gesztesy, H. Holden, J. Michor, and G. Teschl:

# Soliton Equations and Their Algebro-Geometric Solutions

Volume II: (1+1)-Dimensional Discrete Models

Cambridge studies in advanced mathematics, Vol. 114

Cambridge University Press, Cambridge, 2008

## About

The focus of this book is on algebro-geometric solutions of completely integrable, nonlinear, difference-differential equations in (1+1)-dimensions, also known as soliton equations. Explicitly treated integrable models include the Toda, Kac-van Moerbeke, and Ablowitz-Ladik hierarchies. 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 I

Here you find more information about Volume I
## Review

The book is described in a review by Emma Previato in MathSciNet (Mathematical Reviews) [MR2446594] and by Yuri V. Rogovchenko in Zentralblatt
[Zbl 1151.37056].

## Errata and Addenda

[pdf] (Last updated: June 27, 2011).
Any comments and corrections are most welcome.

Please send email to
Fritz Gesztesy,
Helge Holden, Johanna Michor or Gerald Teschl.
## How to get it

- Cambridge University Press: US,
UK

## Link to the authors

- Fritz Gesztesy, University of Missouri, Columbia, USA [homepage]
- Helge Holden, Norwegian University of Science and Technology, Trondheim, Norway [homepage]
- Johanna Michor, University of Vienna, Austria [homepage]
- Gerald Teschl, University of Vienna, Austria [homepage]

## Table of contents

Introduction

1. The Toda hierarchy

2. The Kac-van Moerbeke hierarchy

3. The Ablowitz-Ladik hierarchy

Appendix A. Algebraic curves and their theta functions in a nutshell

Appendix B. Hyperelliptic curves of the Toda-type

Appendix C. Asymptotic spectral parameter expansions

Appendix D. Lagrange interpolation

List of symbols

Bibliography

Index

Errata and Addenda for Volume I

## Sample chapter