Polygons for finding exact solutions of nonlinear differential equations

TL;DR

This paper introduces a novel geometric method using polygons and power geometry algorithms to find exact solutions of nonlinear differential equations, successfully applied to several complex equations.

Contribution

It presents a new polygon-based approach for solving nonlinear differential equations, expanding the toolkit for exact solution methods.

Findings

Found new exact solitary wave solutions for several nonlinear equations.

Applied the method to generalized Korteveg--de Vries--Burgers and Kuramoto--Sivashinsky equations.

Demonstrated the effectiveness of the polygon method in solving fifth-order nonlinear evolution equations.

Abstract

New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for finding one -- parameter exact solutions of the generalized Korteveg -- de Vries -- Burgers equation, the generalized Kuramoto - Sivashinsky equation, and the fifth -- order nonlinear evolution equation. All these nonlinear equations contain the term $u^mu_x$. New exact solitary waves are found.