Internal solitary waves with trapped cores in multilayer shallow water

We consider a nonlinear system of equations that in the Boussinesq approximation describes near-bottom and near-surface large-amplitude internal waves propagating under a cover in multilayer stratified shallow water. We obtain smooth steady-state soliton-like solutions of the equations of motion in the form of symmetric and nonsymmetric mode-2 waves adjoining a given constant flow. We show that the construction of a smooth solution in which one of the layers has a finite length (trapped core) can lead to the formation of a singularity. In the class of functions with piecewise smooth first derivatives, a method for constructing solutions with trapped cores is proposed. For multilayer shallow water equations, we give examples of steady-state solutions describing soliton-like structures and flows with trapped cores.