Finite-amplitude internal solitary waves in a shear flow of a two-layer fluid

We consider a nonlinear system of equations that describes the propagation of finite-amplitude internal waves in a two-layer stratified shallow water under a cover in the Boussinesq approximation. Within the framework of this model, we study solitary waves taking into account the non-hydrostatic pressure distribution in one or both layers, and also consider the case of weakly nonlinear waves. In the class of traveling waves, this model is reduced to an ordinary differential equation for determining the profile of a solitary wave in a shear flow. This equation admits an analytical study. The conditions for the existence of solitary waves adjacent to a given constant flow are determined by two dimensionless parameters related to the wave speed and the velocity shear of the undisturbed flow. Examples of solitary waves are given. These examples demonstrate the effects of the velocity shear on the waveform.