Generalization of the double reduction theory

Abstract

In a recent work [1, 2] Sjoberg remarked that generalization of the double reduction theory to partial differential equations of higher dimensions is still an open problem. In this note we have attempted to provide this generalization to find invariant solution for a non linear system of qth order partial differential equations with n independent and m dependent variables provided that the non linear system of partial differential equations admits a nontrivial conserved form which has at least one associated symmetry in every reduction. In order to give an application of the procedure we apply it to the nonlinear (2 + 1) wave equation for arbitrary function f (u) and g(u).