The decomposition method and Maple procedure for finding first integrals of nonlinear PDEs of any order with any number of independent variables

TL;DR

This paper introduces a new decomposition method for solving nonlinear PDEs of any order with multiple variables, utilizing iterative auxiliary PDEs, and provides a Maple implementation for automatic solution finding.

Contribution

The paper presents a novel decomposition approach for nonlinear PDEs, including an explicit Maple procedure, enabling automatic derivation of solutions for complex equations.

Findings

Method successfully finds closed-form solutions for various nonlinear PDEs.

The decomposition process reduces complex PDEs to simpler auxiliary systems.

The Maple implementation demonstrates practical applicability and efficiency.

Abstract

In present paper we propose seemingly new method for finding solutions of some types of nonlinear PDEs in closed form. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. It is shown that decomposition process can be done by iterative procedure(s), each step of which is reduced to solution of some auxiliary PDEs system(s) for one dependent variable. Moreover, we find on this way the explicit expression of the first-order PDE(s) for first integral of decomposable initial PDE. Remarkably that this first-order PDE is linear if initial PDE is linear in its highest derivatives. The developed method is implemented in Maple procedure, which can really solve many of different order PDEs with different number of independent variables. Examples of PDEs with calculated their general solutions demonstrate a potential of the method for automatic solving of nonlinear PDEs.