A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models

Abstract

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L₁⁻¹ L₂, where the linear differential operators L₁ and L₂ are chosen such that the ordinary differential equation (L₂ −λL₁)U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L₂, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L₁\(\tilde U\) = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution \(\tilde U\) of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.