Adequate dynamical perspective of traveling wave solutions to the perturbed Boussinesq equation appearing in ocean engineering

In the current research, the modified extended Fan sub-equation approach is employed for creating the precise traveling wave solution for the perturbed Boussinesq equation. The perturbed Boussinesq equation proved essential to comprehending and investigating shallow water wave dynamics that describes an extended amplitude, slightly nonlinear estimation which is deployed in marine and coastal infrastructure construction to simulate wave action in shallow oceans and ports. Certain novel solutions for complex trigonometric, complex hyperbolic trigonometric, single and dual non-degenerate Jacobi elliptic patterns are generated by assigning appropriate constraints for the limiting factors. There is a wide range of soliton solutions which incorporate the Z-pattern, simple and singular bell pattern or simple and singular bright, anti-kink pattern, simple and singular anti-bell pattern or simple and singular dark, W-pattern, V-pattern and M-pattern solitary wave solutions. These create several visualizations, including surface, 2D and contour illustrations that demonstrate the computational investigation of the equation. To solve the given system of algebraic equations and generate visual depictions, Mathematica 14 program is utilized. This paper also provides a physical overview of the PBE's solutions and applications. Ultimately, the imposed technique is considered to be more powerful and successful than other methods, and the solutions found can improve our knowledge of soliton frameworks in ocean engineering. The discoveries are intriguing that enhance our knowledge of shallow water wave model and prove potential in addressing the stochastic features of several complex physical concepts originating from ocean engineering.