Hamiltonian and recursion operators for a discrete analogue of the Kaup-Kupershmidt equation

TL;DR

This paper investigates a new integrable differential-difference equation related to the Kaup-Kupershmidt equation, constructing its recursion operator and bi-Hamiltonian structures to reveal its algebraic properties.

Contribution

It introduces a novel integrable discrete equation, derives its recursion operator, and establishes its bi-Hamiltonian structures, extending the understanding of discrete integrable systems.

Findings

Constructed a recursion operator for the discrete equation

Proved the recursion operator is Nijenhuis

Presented bi-Hamiltonian structures for the equation

Abstract

In this paper we study the algebraic properties of a new integrable differential-difference equation. This equation can be seen as a deformation of the modified Narita-Itoh-Bogoyavlensky equation and has the Kaup-Kupershmidt equation in its continuous limit. Using its Lax representation we explicitly construct a recursion operator for this equation and prove that it is a Nijenhuis operator. Moreover, we present the bi-Hamiltonian structures for this new equation.