The (2+1)-dimensional nonisospectral relativistic Toda hierarchy related to the generalized discrete Painlevé hierarchy

Abstract

In this paper, we will concentrate on the topic of integrable discrete hierarchies in 2+1 dimensions, and their connection with discrete Painlevé hierarchies. By considering a (2+1)-dimensional nonisospectral discrete linear problem, two new (2+1)-dimensional nonisospectral integrable lattice hierarchies—the 2+1 nonisospectral relativistic Toda lattice hierarchy and the 2+1 nonisospectral negative relativistic Toda lattice hierarchy—are constructed. It is shown that the reductions of the two new 2+1 nonisospectral lattice hierarchies lead to the (2+1)-dimensional nonisospectral Volterra lattice hierarchy and the (2+1)-dimensional nonisospectral negative Volterra lattice hierarchy. We also obtain two new (1+1)-dimensional nonisospectral integrable lattice hierarchies and two new ordinary difference hierarchies which are direct reductions of the two 2+1 nonisospectral integrable lattice hierarchies. One of the two difference hierarchies yields our previously obtained generalized discrete first Painlevé (dPI) hierarchy and another one yields a generalized alternative discrete second Painlevé (alt-dPII) hierarchy.