On a method for constructing the Lax pairs for nonlinear integrable equations

TL;DR

The paper introduces a direct, effective algorithm for constructing Lax pairs for nonlinear integrable equations, applicable to both continuous and discrete models, facilitating the derivation of conservation laws and symmetry hierarchies.

Contribution

It presents a novel, straightforward method to find Lax pairs and recursion operators directly from nonlinear equations, differing from classical approaches.

Findings

The method successfully constructs Lax pairs for previously unsolved equations.

The Lax pairs obtained enable the generation of infinite conservation laws.

The approach is effective for hyperbolic PDEs using invariant manifolds and Laplace sequences.

Abstract

We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.