A forced KdV equation including the special topography effect is derived to describe nonlinear long wave and solitary eddy based on the quasi-geostrophic potential vorticity model. We obtain the theoretical solution of the equation and the concrete form of stream function through perturbation theory and multi-scale analysis methods. It is found that the joint effect of weak shear basic flow and topography can change the cyclone and anticyclone structure of eddy, and in the meantime topographic structure affects the East-West propagation direction of solitary wave. Finally, according to the interaction between nonlinear long wave and topography by pseudo spectral numerical method, the topographic height is related to the amplitude, wavelength and wave velocity of the excited wave train, and the topography affects not only the spatial structure of wave, but also the amplitude of wave.

The paper deals with a non-linear barotropic vorticity equation in a shear flow, the model is applied to get the non-linear Rossby solitary wave with the effects of topography and other external forcing. Basing on the perturbation method, the variable coefficient KdV equation is derived for Rossby waves. The periodic-like solution for the equation is obtained with the help of Jacobi elliptic functions, the solitary solutions can also be obtained in the limit case. The analysis indicates that the wave amplitude and velocity will be related to the topography effect. It is also shown that the external forcing plays an important role in evolution of the waves.

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.

In this article, using multi-scale and perturbation methods, a partial differential model of Rossby solitary waves with complete Coriolis force effect is obtained, which is called the combined ZK-mZK equation. This equation can reflect the prop­agation characteristics of Rossby waves in space, and is more suitable for real oceans and atmosphere than the (1+1)-dimensional model. According to the pa­rameter composition of the new model, it can be seen that the effect of the complete Coriolis force affects not only the longitudinal structure of the model, but also the amplitude structure of the non-linear long wave.