  We study here some asymptotic models for the propagation of internal and
surface waves in a two-fluid system. We focus on the so-called long wave regime
for one dimensional waves, and consider the case of a flat bottom. Starting
from the classical Boussinesq/Boussinesq system, we introduce a new family of
equivalent symmetric hyperbolic systems. We study the well-posedness of such
systems, and the asymptotic convergence of their solutions towards solutions of
the full Euler system. Then, we provide a rigorous justification of the
so-called KdV approximation, stating that any bounded solution of the full
Euler system can be decomposed into four propagating waves, each of them being
well approximated by the solutions of uncoupled Korteweg-de Vries equations.
Our method also applies for models with the rigid lid assumption, and the
precise behavior of the KdV approximations depending on the depth and density
ratios is discussed for both rigid lid and free surface configurations. The
fact that we obtain {\it simultaneously} the four KdV equations allows us to
study extensively the influence of the rigid lid assumption on the evolution of
the interface, and therefore its domain of validity. Finally, solutions of the
Boussinesq/Boussinesq systems and the KdV approximation are rigorously compared
and numerically computed.
