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

TL;DR

This paper develops and justifies asymptotic models for internal and surface wave propagation in a two-fluid system, introducing symmetric hyperbolic systems and rigorously deriving the KdV approximation, with analysis of different physical configurations.

Contribution

It introduces a new family of symmetric hyperbolic systems for internal and surface waves, and provides a rigorous justification of the KdV approximation in this context.

Findings

Four KdV equations effectively describe wave propagation.

The models' validity depends on depth and density ratios.

Numerical comparisons confirm the theoretical approximations.

Abstract

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.