Potential Symmetries of Partial Differential Equations

Abstract

The last two chapters discussed point symmetries and non-classical symmetries. These types of symmetry are local symmetries because the coordinates are involved in the local transformations in a direct way. This chapter discusses a completely different type of symmetry. We not only consider the original PDEs A = 0 but also derived systems of PDEs whose solutions are solutions of the original equations. The new associated system of PDEs is treated by the methods discussed in the previous sections. The result of this treatment are symmetries not only depending on the local variables of the original equation but also on variables of the affiliated system of PDEs. Thus, we get a new type of symmetry depending on an extended set of variables. Such symmetries are generally called non-local symmetries. A special type of non-local symmetry is a potential symmetry. Our interest in this chapter are potential symmetries of PDEs.