Optical solitary waves and soliton solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation

In this investigation, an appropriate traveling wave transformation has been employed to analyze the fourth-order nonlinear $(3+1)$-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) equation for an offshore structure. In addition to being integrable and having constant coefficients, the examined model represents fluid flow in the context of an offshore structure. The aforesaid nonlinear system is subjected to the initial application of two trustworthy and dependable approaches, namely the improved Bernoulli sub-equation function method and the modified extended $\operatorname{tanh}$-function method. Investigating and obtaining certain explicitly exact traveling waves, periodic waves, and soliton solutions is the major objective. The generated solutions take the form of trigonometric hyperbolic functions, exponential functions, rational functions, and multiple forms of trigonometric functions. The proposed solutions are both novel and important in that they provide light on the relevant aspects of the physical phenomena. The characteristics of the solutions have been displayed in a variety of figures, including two- and three-dimensional ones, for the best visual optical discernment.