Narrow Results

The unsmooth boundary has a great influence on the solitary wave form of a nonlinear wave equation. It this work, we for the first time ever propose the fractal regularized long-wave equation which can describe the shallow water waves under the unsmooth boundary (such as the fractal seabed). The fractal variational principle is established and is proved to have a strong minimum condition by the He–Weierstrass theorem. Then, the solitary wave solution is obtained by the fractal variational method which can reduce the order of differential equation and obtain the optimal solution by the stationary condition. Finally, the impact of the unsmooth boundary on the solitary wave is presented. It shows that the fractal order can affect the wave morphology, but cannot affect its peak value. 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.

In this paper, for the first time in pass records, we create the fractal fourth-order nonlinear Ablowitz–Kaup–Newell–Segur (FFONAKNS) shoal water wave mold under an unsmooth boundary or in microgravity of space. With the aid of fractal traveling wave variation (FTWV) and fractal semi-inverse technology (FSIT), the fractal variational principle (FVP) is achieved, and then, using He–Weierstrass function, the strong minimum necessary condition is proved. Afterwards, the solitary wave solution is attained by FVP and minimum stationary conditions. Finally, the effect of a non-smooth border on solitary wave is deliberated and demeanors of solutions are displayed via 3D isohypse. The fractal dimension can impact the waveform, but not its apex value. The solitary wave solution (SWS) is given, and the exhibition of the technology used is not only creditable but also significant.