# Internal doubly periodic gravity-capillary waves with vorticity

## TL;DR

This paper proves the existence of three-dimensional steady gravity-capillary waves with vorticity in a multi-fluid system, using bifurcation theory and explicit conditions for in-phase and off-phase motions across interfaces.

## Contribution

It introduces a multi-parameter bifurcation framework for steady gravity-capillary waves with vorticity, including explicit conditions for different interface motions.

## Key findings

- Existence of 3D steady gravity-capillary waves with vorticity.
- Explicit conditions for in-phase and off-phase motions.
- A general bifurcation theorem applicable to similar problems.

## Abstract

We consider a multi-fluid system with several free interfaces. For this system we prove existence of three-dimensional steady gravity-capillary waves with non-zero vorticity. We obtain non-zero vorticity by prescribing the relative velocity fields to be Beltrami fields, for which the vorticity and velocity are parallel. The main result is a multi-parameter bifurcation result for small amplitude waves given in two variants: a first theorem guaranteeing existence under some general parameter assumptions; and a second specific but less exhaustive theorem, for which the assumptions may be explicitly verified, yielding the existence of both in-phase and off-phase motions in the different layers. The proof relies on an implicit function theorem corresponding to multi-parameter bifurcation. This theorem is presented in an appendix as an abstract result that can be applied directly to other problems.