Book

**Cambridge Studies in Advanced Mathematics, Volume 114, Cambridge University Press, Cambridge, 2008**

## Soliton Equations and Their Algebro-Geometric Solutions. Volume II: (1+1)-Dimensional Discrete Models

### Fritz Gesztesy, Helge Holden, Johanna Michor, and Gerald Teschl

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.

** MSC2000:** Primary 37K15, 37K10; Secondary 39A12, 35Q55

**Keywords:** *Toda hierarchy, Kac-van Moerbeke hierarchy, Ablowitz-Ladik hierarchy, discrete NLS, algebro-geometric solutions.*

Errata, Addenda, and More

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Ordering Details

Publisher: | Cambridge University Press | |

Series: | Cambridge Studies in Advanced Mathematics | |

Volume: | 114 | |

Publication Year: | 2008 | |

ISBN: | 978-0-5217-5308-1 | |

Paging: | 450 pp; hardcover | |

List Price: | £75 | |

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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.

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.

Volume I

Here is some further information on Soliton Equations and Their Algebro-Geometric Solutions. Volume I:
(1+1)-Dimensional Continuous Models by Fritz Gesztesy and Helge Holden.