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Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies
, Graz Technical University, Austria
, F. Gesztesy
, University of Missouri, Columbia, MO
, H. Holden
, Norwegian University of Science and Technology, Trondheim, Norway
, and G. Teschl
, Institut für Reine und Angewandte Mathematik, Aachen, Germany
| | Memoirs of the American Mathematical Society
1998; 79 pp; softcover
ISBN-13: 978-0-8218-0808-5 List Price: US$47
Individual Members: US$28.20
Institutional Members: US$37.60
Order Code: MEMO/135/641
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.
Graduate students, research mathematicians and theoretical physicists working in completely integrable systems.
Table of Contents
- The Toda hierarchy, recursion relations, and hyperelliptic curves
- The stationary Baker-Akhiezer function
- Spectral theory for finite-gap Jacobi operators
- Quasi-periodic finite-gap solutions of the stationary Toda hierarchy
- Quasi-periodic finite-gap solutions of the Toda hierarchy and the time-dependent Baker-Akhiezer function
- The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy
- Spectral theory for finite-gap Dirac-type difference operators
- Quasi-periodic finite-gap solutions of the Kac-van Moerbeke hierarchy
- Hyperelliptic curves of the Toda-type and theta functions
- Periodic Jacobi operators
- Examples, \(g-0,1\)