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.

Readership

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

Table of Contents

• 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