Symmetry approaches for reductions of PDEs, differential constraints and Lagrange-Charpit method

TL;DR

This paper unifies various symmetry-based methods for reducing PDEs by connecting classical and modern approaches through rigorous compatibility and homological techniques.

Contribution

It introduces a unifying framework for PDE reduction methods, linking classical symmetry approaches with recent advances in compatibility theory.

Findings

Established rigorous connections between symmetry methods and differential constraints.

Applied homological methods to analyze compatibility of overdetermined PDE systems.

Provided a comprehensive unifying approach for PDE reduction techniques.

Abstract

Many methods for reducing and simplifying differential equations are known. They provide various generalizations of the original symmetry approach of Sophus Lie. Plenty of relations between them have been noticed and in this note a unifying approach will be discussed. It is rather close to the classical differential constraint method, but we provide certain rigorous results basing on recent advances in compatibility theory of non-linear overdetermined systems and homological methods for PDEs.