Symmetry and Lie-Frobenius reduction of differential equations

TL;DR

This paper explains how twisted symmetries facilitate the reduction of nonlinear differential equations through a Lie-Frobenius framework, emphasizing distributions of vector fields in involution rather than Lie algebras.

Contribution

It introduces a Lie-Frobenius reduction approach that broadens the understanding of twisted symmetries beyond traditional Lie algebra methods.

Findings

Twisted symmetries are as effective as standard symmetries in reducing nonlinear equations.

The effectiveness is explained via distributions of vector fields in involution, not just Lie algebras.

The approach generalizes symmetry analysis by focusing on Frobenius involution.

Abstract

Twisted symmetries, widely studied in the last decade, proved to be as effective as standard ones in the analysis and reduction of nonlinear equations. We explain this effectiveness in terms of a Lie-Frobenius reduction; this requires to focus not just on the prolonged (symmetry) vector fields but on the distributions spanned by these and on systems of vector fields in involution in Frobenius sense, not necessarily spanning a Lie algebra.