A class of solitary and periodic wave structures in (3+1)-dimensional space-time fractional KdV-BBM equation with Jacobi elliptic functions

This paper considers the (3+1)-dimensional space-time fractional variable Korteweg-de Vries-Benjamin-Bona-Mahoney (KdV-BBM) equation. This equation is of critical importance in studying long-wave phenomena in nonlinear dispersive media. It is commonly accepted that classical equations fail to capture memory and fractal properties of physical phenomena. In fluid dynamics and plasma physics, fractional equations accurately capture the memory effect in nonlinear one-way wave propagation. Thus, we employ a fractional complex transformation that reduces the controlling nonlinear partial differential equations (NPDE) to a nonlinear ordinary differential equation (NODE). The newly introduced Jacobi elliptic function expansion method is systematically implemented to capture various types of exact or analytical wave solutions. Our findings explicitly reveal that by tuning the $\beta$ parameter in the fractional variable KdV-BBM equation, one can capture solitary and double periodic nonlinear structures in terms of Jacobi sine and cosine functions. The major outcome is that it enormously regulates the topology of various solitary and double periodic nonlinear waves. Therefore, our investigation explicitly reveals that conformable fractional derivatives serve to be highly efficient operators in exploring higher-dimensional nonlinear or solitary and double periodic waves. The anticipated research could be considered a foundation that could be used in future experiments in nonlinear optics and water flows.