As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton 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. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results.
- 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
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Reviews & endorsements
Reviews of Volume 1: '… this is a book that I would recommend to any student of mine, for clarity and completeness of exposition … Any expert as well would enjoy the book and learn something stimulating from the sidenotes that point to alternative developments. We look forward to volumes two and three!' Mathematical Reviews
'The book is very well organized and carefully written. It could be particularly useful for analysts wanting to learn new methods coming from algebraic geometry.' EMS Newsletter
Review of Volume 2: 'As with the first part, the book is very well written and carefully organised and it is a pleasure to read it.' EMS Newsletter
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Product details
- Date Published: September 2008
- format: Hardback
- isbn: 9780521753081
- length: 448 pages
- dimensions: 234 x 152 x 27 mm
- weight: 0.74kg
- availability: 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.
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Authors
Fritz Gesztesy, University of Missouri, Columbia
Fritz Gesztesy is Professor of Mathematics at the University of Missouri, Columbia.
Helge Holden, Norwegian University of Science and Technology, Trondheim
Helge Holden is Professor of Mathematics at the Norwegian University of Science and Technology.
Johanna Michor, Universität Wien, Austria
Johanna Michor is a Postdoctoral Fellow in the Faculty of Mathematics at the University of Vienna.
Gerald Teschl, Universität Wien, Austria
Gerald Teschl is Associate Professor of Mathematics at the University of Vienna.