Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

Abstract

Combining algebro-geometric methods and factorization techniques for finite difference expressions we provide a complete and self-contained treatment of all real-valued quasi-periodic finite-gap solutions of both the Toda and Kac-van Moerbeke hierarchies. In order to obtain our principal new result, the algebro-geometric finite-gap solutions of the Kac-van Moerbeke hierarchy, we employ particular commutation methods in connection with Miura-type transformations which enable us to transfer whole classes of solutions (such as finite-gap solutions) from the Toda hierarchy to its modified counterpart, the Kac-van Moerbeke hierarchy, and vice versa.

LaTeX, to appear in Memoirs of the Amer. Math. Soc

Cited by in corpus (10)

• Algebro-Geometric Solutions of the Boussinesq Hierarchy
• Scattering Theory for Jacobi Operators with Quasi-Periodic Background
• Stability of the periodic Toda lattice under short range perturbations
• Soliton Solutions of the Toda Hierarchy on Quasi-Periodic Backgrounds Revisited
• Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds
• Trace Formulas in Connection with Scattering Theory for Quasi-Periodic Background
• On the Equivalence of Different Lax Pairs for the Kac-van Moerbeke Hierarchy
• Lieb-Robinson Bounds for the Toda Lattice
• KdV-Volterra chain
• On uniqueness properties of solutions of the Toda and Kac-van Moerbeke hierarchies