Abstract. Dynamics of large amplitude internal waves in two-layers of shallow water is considered.It is demonstrated that in laboratory experiments the subsurface waves of depression over a shelf may be simulated by internal symmetric solitary waves of the mode 2 ("lump-like" waves). The mathematical model describing the propagation and decaying of large internal waves in two-layer fluid is introduced. It is a variant of Choi-Camassa equations with hydrostatic pressure distribution in one of the layers. It is shown that the numerical scheme developed for the GreenNaghdi equations in open channel flows may be applied for the description of large amplitude internal waves over a shelf.

Abstract. The evolution of large amplitude internal waves propagating towards the shore and more specifically the run up phase over the "swash" zone is considered. The mathematical model describing the generation, interaction, and decaying of solitary internal waves of the second mode in the interlayer is proposed. The exact solution specifying the shape of solitary waves symmetric with respect to the unperturbed interface is constructed. It is shown that, taking into account the friction on interfaces in the mathematical model, it is possible to describe adequately the change in the phase and amplitude characteristics of two solitary waves moving towards each other before and after their interaction. It is demonstrated that propagation of large amplitude solitary internal waves of depression over a shelf could be simulated in laboratory experiments by internal symmetric solitary waves of the second mode.

The problem of a homogeneous heavy liquid flow over a local obstacle is considered in the longwave approximation. The steady and unsteady waves in the vicinity of the obstacle are described by second-order models of the shallow-water theory and their hyperbolic approximations. The flow in the vicinity of the leading and trailing edges of bluff bodies (sills and steps) is studied. The solution of the problem of the blocked zone upstream of the step is constructed.Key words: homogeneous liquid, equations of the shallow-water theory, dispersion effects, flow over a sill.