Cartan's Equivalence Method and it's Applications in Differential Equations

Abstract

The fundamental equivalence problems are to determine whether two exterior differential systems can be transformed into each other by a suitable change of variables. The symmetry group of a geometric object can be regarded as a special case of the general equivalence problems, where symmetry is interpreted merely as self-equivalence of the exterior differential systems. Applications of Cartan's equivalence to symmetries of differential equations, are considered. The examples include interrelations between the nonlinear acoustics equations, the solutions of equivalence problems for the classes of linear parabolic equations and nonlinear wave equations.