Linear determining equations, differential constraints and invariant solutions

TL;DR

This paper introduces a method using linear determining equations to find differential constraints compatible with PDEs, generalizing classical Lie operator methods, and discusses invariant solutions for specific equations like the nonlinear heat equation.

Contribution

It presents a new approach with linear determining equations for constructing differential constraints and invariant solutions, extending classical methods.

Findings

Generalizes classical determining equations for PDEs

Provides conditions for invariant solutions under involutive distributions

Applies method to nonlinear heat and Gibbons-Tsarev's equations

Abstract

A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations used in the search for admissible Lie operators. As applications of this approach non-linear heat equations and Gibbons-Tsarev's equation are discussed. We introduce the notion of an invariant solution under an involutive distribution and give sufficient conditions for existence of such a solution.