Estevez–Mansfield–Clarkson equation: Investigation of Breathers, two waves, and solitary wave solutions in shallow water phenomena and engineering fluid dynamics

This article focuses on investigating the Estevez-Mansfield-Clarkson equation describing the complex dynamics of shallow water waves and the physics of fluids. For such equations, analytically wave solutions are of the utmost importance in numerical and theoretical studies. Firstly, we introduce a traveling wave transformation, and then the recently developed methods known as the modified generalized Riccati equation mapping approach and the modified generalized exponential rational function technique are applied. A variety of soliton solutions, like dark, bright, singular, bright-dark, dark-singular, as well as the hyperbolic, periodic, and exponential solutions, are secured. Secondly, the Hirota method is applied to the studied equation, and we get the bilinear form, and a variety of breathers and two-wave solutions are extracted. Breather waves are solitary waves that demonstrate periodic structure in either space or time, as well as partial localization. In a variety of physical domains, such as ocean engineering, fluid mechanics, optics, hydrodynamics, and quantized superfluidity, breathers have been observed to perform essential functions in nonlinear physics. The variety of graphs in three dimensional with projection and two dimensional with variation of time have been plotted to observe the dynamics of the waves. The outcomes of this work could help to confirm the efficiency of the used techniques and improve understanding of the nonlinear dynamics exhibited by the system at issue. The gained results significantly help to clarify nonlinear science and the nonlinear wave disciplines in higher dimensions.