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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
H. Holden Norwegian University of Science & Technology, Trondheim
G. Teschl Institut für Reine und Angewandte Mathematik, Aachen, Germany
Available Formats:
Electronic ISBN: 978-1-4704-0230-3
Product Code: MEMO/135/641.E
79 pp
List Price: $47.00 MAA Member Price:$42.30
AMS Member Price: $28.20 Click above image for expanded view 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 H. Holden Norwegian University of Science & Technology, Trondheim G. Teschl Institut für Reine und Angewandte Mathematik, Aachen, Germany Available Formats:  Electronic ISBN: 978-1-4704-0230-3 Product Code: MEMO/135/641.E 79 pp  List Price:$47.00 MAA Member Price: $42.30 AMS Member Price:$28.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1351998
MSC: Primary 39; 35;

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.

• Chapters
• 1. Introduction
• 2. The Toda hierarchy, recursion relations, and hyperelliptic curves
• 3. The stationary Baker-Akhiezer function
• 4. Spectral theory for finite-gap Jacobi operators
• 5. Quasi-periodic finite-gap solutions of the stationary Toda hierarchy
• 6. Quasi-periodic finite-gap solutions of the Toda hierarchy and the time-dependent Baker-Akhiezer function
• 7. The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy
• 8. Spectral theory for finite-gap Dirac-type difference operators
• 9. Quasi-periodic finite-gap solutions of the Kac-van Moerbeke hierarchy
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Volume: 1351998
MSC: Primary 39; 35;

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.

• Chapters
• 1. Introduction
• 2. The Toda hierarchy, recursion relations, and hyperelliptic curves
• 3. The stationary Baker-Akhiezer function
• 4. Spectral theory for finite-gap Jacobi operators
• 5. Quasi-periodic finite-gap solutions of the stationary Toda hierarchy
• 6. Quasi-periodic finite-gap solutions of the Toda hierarchy and the time-dependent Baker-Akhiezer function
• 7. The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy
• 8. Spectral theory for finite-gap Dirac-type difference operators
• 9. Quasi-periodic finite-gap solutions of the Kac-van Moerbeke hierarchy
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