Exploring shallow water wave phenomena: A fractional approach to the Whitham-Broer-Kaup-Boussinesq-Kupershmidt system

This study generalizes the Whitham-Broer-Kaup-Boussinesq-Kupershmidt (WBKBK) system to a fractional-order model using conformable derivatives, aiming to better capture wave dissipation, anomalous dispersion, and memory effects in shallow water waves. The Modified Extended Direct Algebraic Method (MEDAM) is employed to derive a series of solitary wave solutions, including kink waves, periodic waves, and solitons. These solutions are validated through numerical simulations using the quartic B-spline collocation method, demonstrating excellent agreement between analytical and numerical results. A linear stability analysis of a representative kink wave solution illustrates the robustness of the derived solutions under specific parameter conditions. This research enriches the theoretical understanding of fractional WBKBK systems and provides valuable tools for modeling complex wave dynamics in fluid dynamics, plasma physics, and nonlinear optics, with potential applications in coastal engineering and marine resource development.