Abundant analytical closed-form solutions and various solitonic wave forms to the ZK-BBM and GZK-BBM equations in fluids and plasma physics

Abstract

Solitary wave profiles of various forms can be found in many disciplines, including oceanography, marine engineering, plasma physics, optical fibers, fluid dynamics, and mathematical sciences. In this work, we use the generalized exponential rational function (GERF) method to find the closed-form wave solutions to two nonlinear evolution equations, the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZK-BBM) equation, and the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony (GZK-BBM) equation. The GERF method under the wave transformation is a very simple, efficient, robust, and straightforward technique for solving various nonlinear partial differential equations. By utilizing this technique, we obtain thirty different kinds of exact solitary wave solutions for governing equations. These newly solitary wave solutions are generated in terms of the rational function, trigonometric function, hyperbolic function, and exponential function forms, all of which play crucial roles in many areas of nonlinear sciences and engineering. Furthermore, we present the two-dimensional and three-dimensional wave profiles of some achieved solutions under the best suitable choices of involved arbitrary constants with dependable range space to understand the dynamical wave behaviors of the solutions, which makes this research more commendable. The GZK-BBM equation demonstrates how the model behaves when long waves propagate, which has a nonlinear and dissipative impact.