On a direct algorithm for constructing recursion operators and Lax pairs for integrable models

TL;DR

This paper introduces an algorithm for constructing recursion operators and Lax pairs for integrable models, utilizing invariant manifolds and Darboux transformations to systematically derive these operators.

Contribution

The paper presents a novel algorithm that expresses recursion operators as ratios of differential operators, connecting invariant manifolds and Darboux transformations for integrable equations.

Findings

Recursion operators can be represented as ratios of differential operators.

The method links invariant manifolds with Lax pairs and Dubrovin-Weierstrass equations.

The approach provides a systematic way to construct recursion operators for integrable models.

Abstract

We suggested an algorithm for searching the recursion operators for nonlinear integrable equations. It was observed that the recursion operator $R$ can be represented as a ratio of the form $R=L_1^{-1}L_2$ where the linear differential operators $L_1$ and $L_2$ are chosen in such a way that the ordinary differential equation $(L_2-\lambda L_1)U=0$ is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter $\lambda\in \textbf{C}$. For constructing the operator $L_1$ we use the concept of the invariant manifold which is a generalization of the symmetry. Then for searching $L_2$ we take an auxiliary linear equation connected with the linearized equation by the Darboux transformation. Connection of the invariant manifold with the Lax pairs and the Dubrovin-Weierstrass equations is discussed.