Dynamical exploration of novel soliton solutions of the modified Benjamin–Bona–Mahoni and Eckhaus equations based on the extended hyperbolic function method

Abstract

In this article, through the utilization of the extended hyperbolic function method (EHFM), we derive the exact solutions for the modified Benjamin–Bona–Mahoni (mBBM) and Eckhaus equations. The EHFM introduces various novel solutions, including dark, bright, singular, singular-periodic, kink solitons, and some rational function solutions. Furthermore, due to a deeper understanding of the physical structure of these solutions, we used Maple and MATLAB software to generate 3D and contour representations of some solutions. The obtained outcomes are innovative regarding the considered equations, and the findings demonstrate that the approach is practical, concise, and direct, which can be applied to describe other physical phenomena.