Traveling wave solutions for wave equations with two exponential nonlinearities

TL;DR

This paper introduces a straightforward method for deriving traveling wave solutions of wave equations with exponential nonlinearities, encompassing classical equations like Liouville and sine-Gordon, with solutions expressed via hypergeometric functions or elliptic functions.

Contribution

It presents a simple approach to obtain explicit traveling wave solutions for equations with exponential nonlinearities, including new solutions for Liouville, Tzitzeica, and sine-Gordon variants.

Findings

Explicit solutions in terms of hypergeometric functions when the integrand's constant term is zero.

Traveling wave solutions involving elliptic functions for nonzero constant term cases.

Application of solutions to nonlinear physics and dynamical systems models.

Abstract

We use a simple method that leads to the integrals involved in obtaining the traveling wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in terms of hypergeometric functions are obtained while when that term is nonzero all the basic traveling wave solutions of Liouville, Tzitzeica and their variants, as well as sine/sinh-Gordon equations with important applications in the phenomenology of nonlinear physics and dynamical systems are found through a detailed study of the corresponding elliptic equations