  We derived here in a systematic way, and for a large class of scaling
regimes, asymptotic models for the propagation of internal waves at the
interface between two layers of immiscible fluids of different densities, under
the rigid lid assumption and with a flat bottom. The full (Euler) model for
this situation is reduced to a system of evolution equations posed spatially on
$\R^d$, $d=1,2$, which involve two nonlocal operators. The different asymptotic
models are obtained by expanding the nonlocal operators with respect to
suitable small parameters that depend variously on the amplitude, wave-lengths
and depth ratio of the two layers. We rigorously derive classical models and
also some model systems that appear to be new. Furthermore, the consistency of
these asymptotic systems with the full Euler equations is established.
