Sine-Gordon expansion method to construct the solitary wave solutions of a family of 3D fractional WBBM equations

The Sine-Gordon expansion (SGE) method is used to develop exact traveling wave solutions of a family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations. The governing equations are reduced to standard ordinary differential equations via companionable wave transformation. The order of the projected polynomial-type solution is figured out using the homogeneous stability approach based on the renowned Sine-Gordon equation. This solution's replacement corresponds to the earlier stage. A system of algebraic equations (SAE) is formed by comparing the coefficients of the powers of the projected solution. A robust coefficient scheme supplies the necessary relationships between parameters and coefficients to develop solutions. Some solutions are modeled for various parameter combinations. Several hyperbolic function solutions are created using the mentioned method. The MATLAB program displays the answers' graphical exemplifications in three-dimensional (3D) surface plots, contour plots, and 2D line plots. The resulting solutions to the equation, with the appropriate parameters, are used to depict the absolute wave configurations in all displays. Additionally, it can conclude that the discovered solutions and their physical characteristics could aid in understanding how shallow water waves propagate in nonlinear dynamics.