Involutive reductions and solutions of differential equations

Abstract

This work introduces the so-called involutive reduction procedure to simplify and solve differential equations. The method is based on symmetry analysis, which was developed by the Norwegian mathematician Sophus Lie. The involutive reduction procedure uses the given differential equation itself and the invariant surface condition of this differential equation. The invariant surface condition incorporates the symmetries of the problem. This coupled system of partial differential equations, which is in general nonlinear, is solved by an involutive reduction procedure. This procedure couples for the first time an involutive simplification algorithm with a heuristic solution algorithm. The combination of these two procedures helps to find new solutions of nonlinear differential equations, as is shown in various examples.