The Cartesian coordinate system is not sufficient to study wave propagation on the coastline or in the sea where the terrain is extremely complicated, so it is necessary to study it in an unconventional coordinate system, fractals. In this paper, from the governing equations of fluid, the fractional nonlinear Schrödinger equation is derived to describe the evolution of Rossby waves in fractal by using multi-scale analysis and perturbation method. Based on the equation, the rogue-wave solution is obtained by the integral preserving transformation to explain some serious threats at sea.

In this study, the propagation of internal solitary waves in oceans at great depths was analyzed. Using multi-scale analysis and perturbation expansion, the basic equation is simplified to the classical Benjamin–Ono equation with variable coefficients. To better describe the propagation characteristics of solitary waves, we derived a higher-order variable-coefficient integral differential (Benjamin–Ono) equation. Subsequently, the bilinear form of the model was derived using Hirota’s bilinear method, and a multi-soliton solution was obtained. Based on the multi-soliton solution of the model, we further studied the interaction of the soliton, which led to the discovery of Mach reflection. Some conclusions were drawn, which are of potential value for further study of solitary waves in the ocean.