Differential-difference equations associated with the fractional Lax operators

TL;DR

This paper explores integrable hierarchies linked to spectral problems involving difference operators, leading to discrete analogs of well-known equations like Sawada--Kotera and Kaup--Kupershmidt, with explicit solutions and an $r$-matrix framework.

Contribution

It introduces new nonlinear differential-difference equations as inhomogeneous generalizations of Bogoyavlensky lattices, connecting discrete models to classical integrable equations.

Findings

Provides explicit solutions for the new lattices

Establishes an $r$-matrix formulation for the hierarchies

Shows these lattices as discrete analogs of known continuous equations

Abstract

We study integrable hierarchies associated with spectral problems of the form $P\psi=\lambda Q\psi$ where $P,Q$ are difference operators. The corresponding nonlinear differential-difference equations can be viewed as inhomogeneous generalizations of the Bogoyavlensky type lattices. While the latter turn into the Korteweg--de Vries equation under the continuous limit, the lattices under consideration provide discrete analogs of the Sawada--Kotera and Kaup--Kupershmidt equations. The $r$-matrix formulation and several simplest explicit solutions are presented.