Algebro-geometric Quasi-periodic Finite-gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

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.

It features a simple and unified treatment of the topic.

It has self-contained development. There are novel results for the Kac-van Moerbeke hierarchy and its algebro-geometric solutions.