Group-Invariant Solutions

Abstract

When one is confronted with a complicated system of partial differential equations arising from some physically important problem, the discovery of any explicit solutions whatsoever is of great interest. Explicit solutions can be used as models for physical experiments, as benchmarks for testing numerical methods, etc., and often reflect the asymptotic or dominant behaviour of more general types of solutions. The methods used to find group-invariant solutions, generalizing the well-known techniques for finding similarity solutions, provide a systematic computational method for determining large classes of special solutions. These group-invariant solutions are characterized by their invariance under some symmetry group of the system of partial differential equations; the more symmetrical the solution, the easier it is to construct. The fundamental theorem on group-invariant solutions roughly states that the solutions which are invariant under a given r-parameter symmetry group of the system can all be found by solving a system of differential equations involving r fewer independent variables than the original system. In particular, if the number of parameters is one less than the number of independent variables in the physical system: r = p - 1, then all the corresponding group-invariant solutions can be found by solving a system of ordinary differential equations.