Applications of Symmetry Methods to Partial Differential Equations

A continuous symmetry of a system of partial differential equations (PDEs) is a transformation that leaves invariant the solution manifold of the system, i.e, it maps (deforms) any solution of the system into a solution of the same system. This definition is topological in nature. However, in practice, the direct calculation of the continuous symmetries of a given system of PDEs restricts one to consider symmetries that are local transformations acting on its space of independent variables, dependent variables and their derivatives. Lie’s algorithm to determine Lie groups of point transformations (point symmetries) of differential equations was presented in Bluman & Anco (2002) [see also Ovsiannikov [(1962), (1982)]; Bluman & Cole (1974); Olver (1986); Bluman & Kumei (1989); Stephani (1989); Hydon (2000); Cantwell (2002)]. Point symmetries arise from solutions of linear systems of determining equations for components of infinitesimal generators for the independent and dependent variables of a given PDE system, where these components themselves depend only on the given PDE system’s independent and dependent variables. Point transformations acting on the space of the given independent and dependent variables of a given PDE system can be extended (prolonged) to point transformations acting on the space of the given independent variables, dependent variables, and their derivatives to any finite order.

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).

Up to now, for a given PDE system, we have considered the calculation and application of its local symmetries (point, contact or higher-order) as well as the calculation of its local conservation laws. In particular, it has been shown how to use local symmetries to map solutions to other solutions; how to use local symmetries of given and target PDEs as an aid in relating them; how to use point or contact symmetries to determine whether a given PDE system can be mapped invertibly to some PDE system belonging to a target class of PDE systems that is completely characterized by its point symmetries as well as determine an explicit mapping when one exists; how to use multipliers yielding local conservation laws to determine whether a given nonlinear PDE system can be mapped invertibly to some linear PDE system as well as determine a specific mapping when one exists. Moreover, as it is well known, local symmetries can be used to find specific solutions (invariant solutions) of PDEs; this application is considered and extended in Chapter 5.

In Chapter 3, it was shown how one can systematically construct a set (tree) of PDE systems nonlocally related to a given PDE system. In particular, local conservation laws of a PDE system lead to augmented nonlocally related (potential) systems that explicitly include nonlocal (potential) variables. Moreover, further nonlocally related PDE systems (nonlocally related subsystems) arise when one or more dependent variables (including dependent variable(s) arising after a point transformation that involves an interchange of dependent and independent variable(s)) are excluded from a PDE system or its potential systems, through differential relations. In Section 3.5, an algorithm for the construction of an extended tree of nonlocally related systems was outlined. In particular, n local conservation laws of a given PDE system lead to a tree of up to 2ⁿ-1 nonlocally related potential systems. A tree is further extended by considering subsystems of both the given PDE system and its nonlocally related potential systems as well as by considering potential systems arising from conservation laws (whose multipliers have an essential dependence on potential variables) of its nonlocally related potential systems.

In this chapter, we consider three further topics on symmetry methods for PDEs. In particular, it is shown how symmetry methods can be used and further adapted to systematically construct particular solutions of a PDE system, how to find nonlocally-related systems that could yield nonlocal symmetries and/or nonlocal conservation laws in the case of a PDE system with three or more independent variables, and how to find local symmetries and local conservation laws through symbolic manipulation software.