Dynamical behavior of water wave phenomena for the 3D fractional WBBM equations using rational sine-Gordon expansion method.

To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.