Potential Symmetries

As defined previously, a symmetry group of a differential equation is a group which maps any solution of the differential equation to another solution of the differential equation. In previous chapters we considered symmetries defined by infinitesimal transformations whose infinitesimals depend on independent variables, dependent variables, and derivatives of dependent variables. Such symmetries are local symmetries since at any point x the infinitesimals are determined if u(x) is sufficiently smooth in some neighborhood of x. In Chapters 5 and 6, by enlarging the classes of local symmetries admitted by given differential equations from point symmetries to contact symmetries, and, still more generally, to Lie-Bäklund symmetries, we could find more conservation laws, construct mappings to related differential equations, and determine more invariant solutions.