The generalized Kupershmidt deformation for constructing new discrete integrable systems

TL;DR

This paper introduces a generalized Kupershmidt deformation method to generate new discrete integrable systems from existing hierarchies like Toda, Kac-van Moerbeke, and Ablowitz-Ladik, expanding the tools for integrable system construction.

Contribution

It proposes a novel generalized Kupershmidt deformation approach based on bi-Hamiltonian structures to create new discrete integrable systems with explicit Lax representations.

Findings

Constructed new discrete integrable systems from known hierarchies.

Provided Lax representations for the deformed systems.

Demonstrated the effectiveness of the generalized deformation method.

Abstract

KdV6 equation can be described as the Kupershmidt deformation of the KdV equation (see 2008, Phys. Lett. A 372: 263). In this paper, starting from the bi-Hamiltonian structure of the discrete integrable system, we propose a generalized Kupershmidt deformation to construct new discrete integrable systems. Toda hierarchy, Kac-van Moerbeke hierarchy and Ablowitz-Ladik hierarchy are considered. The Lax representations for these new deformed systems are presented. The generalized Kupershmidt deformation for the discrete integrable systems provides a new way to construct new discrete integrable systems.