A mapping function approach applied to some classes of nonlinear equations

TL;DR

This paper introduces a mapping function approach to derive solutions for various nonlinear equations, extending existing methods and uncovering new classes of solutions for models like the nonlinear Klein-Gordon, Camassa-Holm, and Benjamin-Bona-Mahony equations.

Contribution

It presents a systematic mapping technique that connects known nonlinear equations to new models, broadening the scope of solvable nonlinear equations in scalar field theories.

Findings

Solutions for nonlinear Klein-Gordon, Camassa-Holm, and Benjamin-Bona-Mahony equations are obtained.

The approach reveals oscillating solutions via the Weierstrass equation.

A larger class of nonlinear equations is systematically explored and characterized.

Abstract

In this work, we study some models of scalar fields in 1+1 dimensions with non-linear self-interactions. Here, we show how it is possible to extend the solutions recently reported in the literature for some classes of nonlinear equations like the nonlinear Klein-Gordon equation, the generalized Camassa-Holm and the Benjamin-Bona-Mahony equations. It is shown that the solutions obtained by Yomba [1], when using the so-called auxiliary equation method, can be reached by mapping them into some known nonlinear equations. This is achieved through a suitable sequence of translation and power-like transformations. Particularly, the parent-like equations used here are the ones for the $\lambda \phi^4$ model and the Weierstrass equation. This last one, allow us to get oscillating solutions for the models under analysis. We also systematize the approach in order to show how to get a larger class of nonlinear equations which, as far as we know, were not taken into account in the literature up to now.