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Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies
W. Bulla, 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
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Memoirs of the American Mathematical Society
1998; 79 pp; softcover
Volume: 135
ISBN-10: 0-8218-0808-7
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.

Features:

• 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.

• Introduction
• 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, $$g-0,1$$
• Acknowledgments
• Bibliography