Symmetry Groups of Differential Equations

Abstract

The symmetry group of a system of differential equations is the largest local group of transformations acting on the independent and dependent variables of the system with the property that it transform solutions of the system to other solutions. The main goal of this chapter is to determine a useful, systematic, computational method that will explicitly determine the symmetry group of any given system of differential equations. We restrict our attention to connected local Lie groups of symmetries, leaving aside problems involving discrete symmetries such as reflections, in order to take full advantage of the infinitesimal techniques developed in the preceding chapter. Before pressing on to the case of differential equations, it is vital that we deal adequately with the simpler situation presented by symmetry groups of systems of algebraic equations, and this is done in the first section. Section 2.2 investigates the precise definition of a symmetry group of a system of differential equations, which requires knowledge of how the group elements actually transform the solutions. The corresponding infinitesimal method rests on the important concept of "prolonging" a group action to the spaces of derivatives of the dependent variables represented in the system. The key "prolongation formula" for an infinitesimal generator of a group of transformations, given in Theorem 2~'36, then provides the basis for the systematic determination of symmetry groups of differential equations. Applications to physically important partial differential equations, including the heat equation, Burgers' equation, the Korteweg-de Vries equation and Euler's equations for ideal fluid flow are presented in Section 2.4. In the case of ordinary differential equations, Lie showed how knowledge of a one-parameter symmetry group allows us to reduce the order of the equation by one. In particular, a first order equation with a known one-P. J. Olver, Applications of Lie Groups to Differential Equations