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

TL;DR

This paper introduces a new fully justified Green-Naghdi type asymptotic model for internal wave propagation in the Camassa-Holm regime, extending classical models and ensuring solutions closely match the full Euler system.

Contribution

It presents a novel Green-Naghdi type model for internal waves in the Camassa-Holm regime, with rigorous justification and the ability to validate lower order models like the Constantin-Lannes approximation.

Findings

Model is consistent and well-posed

Solutions remain close to full Euler system solutions

Extends classical KdV to 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, well-posed, and that its solutions remain close to exact solutions of the full Euler system with corresponding initial data. Moreover, our system allows 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 equation in the Camassa-Holm regime.