On form-preserving point transformations of partial differential equations

Abstract

New identities are presented relating arbitrary order partial derivatives of u(x, t) and u′(x′, t′) for the general point transformation x′ = P(x, t, u), t′ = Q(x, t, u), u′ = R(x, t, u). These identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 + 1 partial differential equations. These results facilitate the search for point symmetries, both discrete and continuous (Lie), and assist the search for point transformations which reduce equations to canonical, but similar, form. A simple test for the existence of hodograph-type transformations between equations of similar form is given.