**Memoirs of the American Mathematical Society**

1998;
79 pp;
Softcover

MSC: Primary 39; 35;

Print ISBN: 978-0-8218-0808-5

Product Code: MEMO/135/641

List Price: $47.00

Individual Member Price: $28.20

**Electronic ISBN: 978-1-4704-0230-3
Product Code: MEMO/135/641.E**

List Price: $47.00

Individual Member Price: $28.20

# Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

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*W. Bulla; F. Gesztesy; H. Holden; G. Teschl*

In this work, the authors provide a self-contained discussion of all real-valued quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies of completely integrable evolution equations. The approach utilizes algebro-geometric methods, factorization techniques for finite difference expressions, as well as Miura-type transformations. Detailed spectral theoretic properties of Lax pairs and theta function representations of the solutions are derived.

Simple and unified treatment of the topic.

Self-contained development.

Novel results for the Kac-van Moerbeke hierarchy and its algebro-geometric solutions.

#### Table of Contents

# Table of Contents

## Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

- Contents vii8 free
- Chapter 1. Introduction 112 free
- Chapter 2. The Toda Hierarchy, Recursion Relations, and Hyperelliptic Curves 718 free
- Chapter 3. The Stationary Baker-Akhiezer Function 1425
- Chapter 4. Spectral Theory for Finite-Gap Jacobi Operators 2334
- Chapter 5. Quasi-Periodic Finite-Gap Solutions of the Stationary Toda Hierarchy 2839
- Chapter 6. Quasi-Periodic Finite-Gap Solutions of the Toda Hierarchy and the Time-Dependent Baker-Akhiezer Function 3243
- Chapter 7. The Kac-van Moerbeke Hierarchy and its Relation to the Toda Hierarchy 4152
- Chapter 8. Spectral Theory for Finite-Gap Dirac-Type Difference Operators 4758
- Chapter 9. Quasi-Periodic Finite-Gap Solutions of the Kac-van Moerbeke Hierarchy 5061
- Appendix A. Hyperelliptic Curves of the Toda-Type and Theta Functions 5869
- Appendix B. Periodic Jacobi Operators 6475
- Appendix C. Examples, g = 0,1 7182
- Acknowledgments 7586
- Bibliography 7687

#### Readership

Graduate students, research mathematicians and theoretical physicists working in completely integrable systems.