Construction of Mappings Relating Differential Equations

Abstract

A symmetry of a PDE is a transformation (mapping) of its solution manifold into itself, i.e., it is a transformation that maps any solution of the PDE into another solution of the same PDE. Invariant solutions (similarity solutions) are solutions that map into themselves. If a symmetry of a given PDE is a point symmetry, then invariant solutions arise constructively from a reduced differential equation with fewer independent variables [Ovsiannikov [(1962), (1982)]; Bluman & Cole (1974); Olver (1986); Bluman & Kumei (1989); Stephani (1989); Bluman & Anco (2002); Cantwell (2002)].

In this chapter, we consider the problem of determining whether there exists a mapping of a given PDE into a target PDE of interest and to construct such a mapping when it exists. A target PDE is either a specific PDE or a member of a class of PDEs. The target PDE is locally equivalent to the given PDE if the mapping is invertible. The invertible mapping is not necessarily unique if a target PDE is a member of a class of PDEs. It is shown that the situation for showing existence and then finding such a mapping is especially fruitful when the target PDE (or target class of PDEs) is completely characterized by a class of contact symmetries (which only exist as point symmetries in the case of a system of PDEs).