On the recursion operators for integrable equations

TL;DR

This paper discusses an alternative method for constructing recursion operators for integrable equations, which relies only on initial generalized symmetries, and demonstrates its effectiveness with classical examples.

Contribution

It introduces a new approach to find recursion operators using only initial symmetries, differing from traditional Hamiltonian or Lax-based methods.

Findings

Method successfully applied to KdV, Krichever-Novikov, and Kaup-Kupershmidt equations

Efficient for discrete models and continuous integrable systems

Provides a simpler alternative to existing construction techniques

Abstract

It is widely known that the recursion operator is a very important component of integrability. It allows one to describe in a compact form both hierarchies of the generalized symmetries and infinite series of the local conservation laws. In the literature, we can find several methods for constructing recursion operators, some of them use the Hamiltonian approach and the others are based on the Lax representation of the equation. In the present article we discuss on an alternative method, suggested in \cite{HabKhaTMP18}, which is connected only with the first several generalized symmetries of the given equation. Efficiency of the method is illustrated with the examples of KdV, Krichever-Novikov and Kaup-Kupershmidt equations and discrete models.