Determination of maximal symmetry groups of classes of differential equations

Abstract

A symmetry of a differential equation is a transformation which leaves invariant its family of solutions. As the functional form of a member of a class of differential equations changes, its symmetry group can also change. We give an algorithm for determining the structure and dimension of the symmetry group(s) of maximal dimension for classes of partial differential equations. It is based on the application of differential elimination algorithms to the linearized equations for the unknown symmetries. Existence and Uniqueness theorems are applied to the output of these algorithms to give the dimension of the maximal symmetry group. Classes of differential equations considered include ODE of form uxx = f(x, u, ux), Reaction-Diffusion Systems of form ut-uxx = f(u, v), vt-vxx = g(u, v), and Nonlinear Telegraph Systems of form vt = ux, vx = C(u, x)ux+B(u, x).