Generalization of the double reduction theory

TL;DR

This paper extends the double reduction theory to higher-dimensional nonlinear PDEs, providing a method to find invariant solutions for systems with conserved forms and symmetries, demonstrated on a (2+1) wave equation.

Contribution

It generalizes the double reduction theory to higher dimensions for nonlinear PDEs with conserved forms and symmetries, enabling invariant solution derivation.

Findings

Successfully generalized the double reduction theory to higher-dimensional PDEs.

Applied the method to a nonlinear (2+1) wave equation with arbitrary functions.

Provided a systematic approach for finding invariant solutions in complex PDE systems.

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).