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

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation