Symmetry, compatibility and exact solutions of PDEs

TL;DR

This paper explores compatibility conditions for overdetermined PDE systems, linking symmetries to invariant solutions, and identifies cases where equations with large symmetry algebras can be integrated explicitly.

Contribution

It generalizes compatibility criteria for PDEs using brackets and establishes conditions for invariant solutions related to Lie symmetries, including models with integrable large symmetry algebras.

Findings

Compatibility criteria for overdetermined PDEs are extended.

Conditions for invariant solutions under Lie symmetries are provided.

Models with large symmetry algebras can be integrated explicitly.

Abstract

We discuss various compatibility criteria for overdetermined systems of PDEs generalizing the approach to formal integrability via brackets of differential operators. Then we give sufficient conditions that guarantee that a PDE possessing a Lie algebra of symmetries has invariant solutions with respect to this Lie algebra. These conditions hold for generic=nondegenerate pairs (equation,symmetry) in certain dimension range for the Lie algebra. Finally we discuss models of equations with large symmetry algebras, which eventually lead to integration in closed form.