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This study deals with asymptotic models for the propagation ofone-dimensional internal waves at the interface between two layers ofimmiscible fluids of different densities, under the rigid lid assumption andwith a flat bottom. We present a new Green-Naghdi type model in theCamassa-Holm (or medium amplitude) regime. This model is fully justified, inthe sense that it is consistent, well-posed, and that its solutions remainclose to exact solutions of the full Euler system with corresponding initialdata. Moreover, our system allows to fully justify any well-posed andconsistent lower order model; and in particular the so-called Constantin-Lannesapproximation, which extends the classical Korteweg-de Vries equation in theCamassa-Holm regime.

A new fully justified asymptotic model for the propagation of internal waves in the Camassa-Holm regime