A New Fully Justified Asymptotic Model for the Propagation of Internal Waves in the Camassa--Holm Regime

Abstract

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