Solitary waves of the fractal Whitham-Broer-Kaup equation in shallow water

In the current work, we propose the fractal Whitham-Broer-Kaup equation which can well describe the propagation of shallow water travelling along unsmooth boundary (such as the fractal seabed). By the Semi-inverse method, we establish its fractal variational principle, which is proved to have a strong minimum condition by He-Weierstrass theorem. Then the fractal variational method is used to seek its solitary wave solution. The impact of the fractal order on the behaviors of the solitary wave is presented through the 3-D plots and 2-D curves. The finding in this paper is important for the coast protection and expected to bring a light to the study of the fractal theoretical basis in the geosciences.