An analytical study of the time-fractional extended shallow-water wave equation in (3 + 1)-dimension with two different derivatives and their comparison

Abstract

In ocean physics, an essential mathematical framework for examining the dynamic behavior of waves is the (3 + 1)-dimensional generalized shallow-water wave equation. This approach is driven by the growing need to incorporate nonlinear and anomalous behaviors in shallow-water wave propagation into more realistic mathematical models. This motivation is a key consideration for improving coastal hazard prediction, mitigating tsunami impacts, optimizing renewable energy extraction, and deepening our understanding of complex coastal processes. In this paper, exact solutions of the fractional generalized shallow-water wave equation are constructed using two alternative methods: the extended modified auxiliary equation mapping method and the F-expansion approach. The extended modified auxiliary equation mapping method yielded nineteen exact solutions across two main sets, while the F-expansion method produced solutions for seventeen different cases. To visualize these, 2D and 3D graphical representations have been generated for several solutions using fractional parameter values α ∈ (0; 1], including sample values such as 0.2, 0.5, 0.7, and 0.78, to illustrate how the order of the derivative affects the soliton profile. The results show that decreasing α leads to broader and smoother soliton structures. Finally, the modulation instability of the governing model is also investigated, confirming that the established results are stable.