Second order integrability conditions for difference equations. An integrable equation

TL;DR

This paper establishes second order integrability conditions for difference equations, introduces a new integrable equation, and demonstrates its integrability through symmetries, conservation laws, and a Lax representation.

Contribution

It presents novel second order integrability conditions, constructs a new integrable difference equation, and links its symmetries to known integrable lattices.

Findings

New second order integrability conditions for difference equations

A newly identified integrable equation satisfying these conditions

Construction of symmetries, conservation laws, and a Lax pair for the new equation

Abstract

Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws is discussed. In the generic case, nonlocal conservation laws are also generated. A new integrable equation satisfying the second order integrability conditions is presented and its integrability is established by the construction of symmetries, conservation laws and a 3x3 Lax representation. Finally, the relation of the symmetries of this equation to a generalized Bogoyavlensky lattice and a new integrable lattice are derived.