An asymptotic model for internal capillary-gravity waves in deep water

TL;DR

This paper develops an asymptotic bi-directional model for internal capillary-gravity waves in deep water, incorporating surface tension effects, and introduces related unidirectional models with analysis of well-posedness and solitary wave solutions.

Contribution

It derives a new asymptotic bi-directional model for internal waves with surface tension and analyzes its mathematical properties and solutions.

Findings

Well-posedness of the new models established

Existence of solitary wave solutions demonstrated

Generalization to a regularized Benjamin equation provided

Abstract

Considered in this paper is a bi-directional model for the propagation of interfacial capillary-gravity waves in a two-layer system of fluids with rigid lid condition for the upper layer and lower layer with a much larger or infinite depth. The system is derived from a reformulation of the Euler equations for internal waves with nonnegligible surface tension effects in the interface and the corresponding asymptotic model under the Benjamin-Ono regime. Another unidirectional model, so-called regularized Benjamin equation, generalizing the Benjamin equation, is also introduced. Well-posedness of the new equations and existence of solitary wave solutions are discussed.