Generalized invariant manifolds for integrable equations and their applications

TL;DR

This paper introduces generalized invariant manifolds that are compatible with the linearization of integrable equations, enabling the construction of solutions, Lax pairs, and recursion operators more effectively.

Contribution

It extends the method of differential constraints by focusing on compatibility with linearized equations, facilitating the derivation of integrability attributes.

Findings

Constructs particular solutions using generalized invariant manifolds.

Provides a method to derive Lax pairs and recursion operators.

Enhances the understanding of integrability in nonlinear equations.

Abstract

In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the nonlinear partial differential equations. Its essence is in adding to the nonlinear PDE, a much simpler, as a rule ordinary, differential equation, compatible with the given one. Then any solution of the ODE is a particular solution of the PDE as well. However the main problem is to find this compatible ODE. Our generalization is that we look for an ordinary differential equation that is compatible not with the nonlinear partial differential equation itself, but with its linearization. Such a generalized invariant manifold is effectively sought. Moreover, it allows one to construct such important attributes of integrability theory as Lax pairs and recursion operators for integrable nonlinear equations. In this paper, we show that they provide a way to construct particular solutions to the equation as well.