Existence and stability of interfacial capillary-gravity solitary waves with constant vorticity

TL;DR

This paper proves the existence of small-amplitude capillary-gravity solitary waves with constant vorticity and analyzes their stability, revealing conditions under which these waves are orbitally stable or unstable.

Contribution

It introduces a new existence proof for small-amplitude solitary waves in a two-fluid system with vorticity and characterizes their stability using a novel parameter-dependent criterion.

Findings

Existence of small-amplitude solitary waves in the system.

Explicit stability criterion based on system parameters.

Identification of conditions for orbital stability and instability.

Abstract

In this paper, we consider capillary-gravity waves propagating on the interface separating two fluids of finite depth and constant density. The flow in each layer is assumed to be incompressible and of constant vorticity. We prove the existence of small-amplitude solitary wave solutions to this system in the strong surface tension regime via a spatial dynamics approach. We then use a variant of the classical Grillakis--Shatah--Strauss (GSS) method to study the orbital stability/instability of these waves. We find an explicit function of the parameters (Froude number, Bond number, and the depth and density ratios) that characterizes the stability properties. In particular, conditionally orbitally stable and unstable waves are shown to be possible.