Traveling wave solutions of nonlinear partial differential equations

TL;DR

This paper introduces a straightforward algebraic method to generate traveling wave solutions for various nonlinear partial differential equations, including models with compactons and peakons, expanding the known solution classes.

Contribution

The paper presents a novel algebraic approach applicable to both integrable and non-integrable PDEs, producing new classes of solutions such as compactons and peakons.

Findings

Derived new classes of compacton solutions.

Generated novel peakon solutions.

Applicable to a broad range of nonlinear PDEs.

Abstract

We propose a simple algebraic method for generating classes of traveling wave solutions for a variety of partial differential equations of current interest in nonlinear science. This procedure applies equally well to equations which may or may not be integrable. We illustrate the method with two distinct classes of models, one with solutions including compactons in a class of models inspired by the Rosenau-Hyman, Rosenau-Pikovsky and Rosenau-Hyman-Staley equations, and the other with solutions including peakons in a system which generalizes the Camassa-Holm, Degasperis-Procesi and Dullin-Gotwald-Holm equations. In both cases, we obtain new classes of solutions not studied before.