Structure of Symmetry Groups via Cartan's Method: Survey of Four Approaches

TL;DR

This survey reviews four recent methods for computing Maurer-Cartan structure equations of symmetry groups in differential equations, illustrating their applications through examples like contact equivalence and transformations between notable equations.

Contribution

It provides a comprehensive comparison of four approaches for analyzing symmetry groups using Cartan's method, highlighting their effectiveness and applications.

Findings

Four methods effectively compute Maurer-Cartan equations.

Applications include contact equivalence of hyperbolic equations.

Transformations between complex differential equations are achieved.

Abstract

In this review article we discuss four recent methods for computing Maurer-Cartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations and finding a contact transformation between the generalized Hunter-Saxton equation and the Euler-Poisson equation.