Dynamic nature of analytical soliton solutions of the nonlinear ZKBBM and GZKBBM equations

Abstract

A solitary wave, characterized as a localized perturbation in a medium, emerges as a result of a delicate equilibrium between nonlinear and dispersive phenomena. Solitons, a subtype of solitary waves, exhibit persistent shape and velocity during propagation, representing a fundamental phenomenon observed widely in natural systems and possessing various applications in nonlinear dynamics. This investigation focuses on two nonlinear evolution equations (NEEs), specifically the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation and the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony (GZKBBM) equation, which find relevance in domains such as fluid dynamics and ocean engineering. Utilizing an ansatz-based methodology, soliton solutions of both bright and dark characteristics are derived, alongside exploration of rogue wave-type solutions. Notably, the manifestation of dark, bright, and rogue waves aligns with the physical interpretation of the generated solitons. Computational simulations conducted using Wolfram Mathematica aim to provide a comprehensive description of the physical phenomena. The novelty of this study lies in its unreported investigation, contributing new insights into the solitonic dynamics within the considered models.