On the Toda and Kac–van Moerbeke hierarchies

Abstract

Abstract. We provide a comprehensive treatment of the single and double commutation method as a tool for constructing soliton solutions of the Toda and Kac-van Moerbeke hierarchy on arbitrary background. In addition, we present a novel construction based on the single commutation method. As an illustration we compute the N -soliton solution of the Toda and Kac-van Moerbeke hierarchy.

Jacobi Operators and Completely Integrable Nonlinear Lattices

“…The first rigorous results were given by Guseinov [124], but unfortunately without proof. Additional results are given in [125], [126], and [224]. This chapter extends the results of [124] and gives (to the best of my knowledge) the first proof in print.…”

“…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]. The Burchnall-Chaundy polynomial in the case of commuting differential expressions was first obtained in [36], [37] (cf.…”

“…His formulas clearly apply to the entire Toda hierarchy upon using the t dependence of the norming constants given in (13.72) and coincide with those obtained using the double commutation method (i.e., the last example in Section 14.5). All results from the fourth section are taken from [219] and [224].…”

“…The Bäcklund transformation connecting the Toda and the Kac-van Moerbeke equations has first been considered in [234] and [239] (see also [56], [151], [152]). The results presented here are mainly taken from [35], [117] and [220]. An alternative approach to the modified Toda hierarchy, using the discrete analog of the formal pseudo differential calculus, can be found in [161], Ch.…”

Inverse scattering transform for the Toda hierarchy with quasi-periodic background

“…First of all we recall the basic existence and uniqueness theorem for the Toda hierarchy (see, e.g., [12], [13], or [14, Section 12.2]).…”