Article
Memoirs of the Amer. Math. Soc. 135/641, 1998 [DOI: 10.1090/memo/0641]

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

Wolfgang Bulla, Fritz Gesztesy, Helge Holden, and Gerald Teschl

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.

MSC91: Primary 39A70, 35Q58; Secondary 39A12, 35Q51
Keywords: Jacobi operators, Toda hierarchy, Kac-van Moerbeke hierarchy

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