Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies

Abstract

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 finitegap 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.

Jacobi Operators and Completely Integrable Nonlinear Lattices

“…Suppose H satisfies (H.10.1) and let H σ1 be constructed as in Theorem 11.4 with n). Then the transmission T σ1 (k) and reflection coefficients R ±,σ1 (k) of H σ1 in terms of the corresponding scattering data T (k), R ± (k) of H are given by 35) where 0). Moreover, the norming constants γ ±,σ1,j associated with λ j ∈ σ pp (H σ1 ) in terms of γ ±,j , corresponding to H, read…”

“…The approach is inspired by ideas of McKean [169]. Some parts are taken from [35] (see also [173]), but again the material has been extended to a larger class of operators and augmented.…”

“…The first results seem due to Krichever (see, e.g., [156] - [160]). The main reference for this chapter is again [35]. Additional references are [74], [169], [173], [230], and [175].…”

“…III a). It follows essentially the monograph [35] (with some extensions). An existence and uniqueness theorem for the semi infinite case seems to appear first in [65], the form stated here is taken from [220].…”

“…Chapter 13. Section 1 and 2 closely follow [35] with several additions made. The BakerAkhiezer function, the fundamental object of our approach, goes back to the work of Baker [23], Burchnall and Chaundy [36], [37], and Akhiezer [8].…”