Cosymmetries and Nijenhuis recursion operators for difference equations

TL;DR

This paper explores cosymmetries and co-recursion operators for difference equations, introduces a new factorisation method, and proves the Nijenhuis property of a key recursion operator for the Viallet equation, with applications to related discretisations.

Contribution

It presents a new factorisation of recursion operators, proves Nijenhuis property for a specific operator, and extends these concepts to Yamilov's discretisation of the Krichever-Novikov equation.

Findings

Introduces a new factorisation for difference equation recursion operators.

Proves the Nijenhuis property of the recursion operator for the Viallet equation.

Applies the framework to Yamilov's discretisation of the Krichever-Novikov equation.

Abstract

In this paper we discuss the concept of cosymmetries and co--recursion operators for difference equations and present a co--recursion operator for the Viallet equation. We also discover a new type of factorisation for the recursion operators of difference equations. This factorisation enables us to give an elegant proof that the recursion operator given in arXiv:1004.5346 is indeed a recursion operator for the Viallet equation. Moreover, we show that this operator is Nijenhuis and thus generates infinitely many commuting local symmetries. This recursion operator and its factorisation into Hamiltonian and symplectic operators can be applied to Yamilov's discretisation of the Krichever-Novikov equation.