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In this paper, we derive asymptotic models for the propagation of two andthree-dimensional gravity waves at the free surface and the interface betweentwo layers of immiscible fluids of different densities, over an uneven bottom.We assume the thickness of the upper and lower fluids to be of comparable size,and small compared to the characteristic wavelength of the system (shallowwater regimes). Following a method introduced by Bona, Lannes and Saut based onthe expansion of the involved Dirichlet-to-Neumann operators, we are able togive a rigorous justification of classical models for weakly and stronglynonlinear waves, as well as interesting new ones. In particular, we derivelinearly well-posed systems in the so called Boussinesq/Boussinesq regime.Furthermore, we establish the consistency of the full Euler system with thesemodels, and deduce the convergence of the solutions.

Asymptotic shallow water models for internal waves in a two-fluid system with a free surface