Internal Doubly Periodic Gravity-Capillary Waves with Vorticity

We consider a multifluid system with several free interfaces. For this system we prove existence of three-dimensional steady gravity-capillary waves with nonzero vorticity. We obtain nonzero vorticity by prescribing the relative velocity fields to be Beltrami fields, for which the vorticity and velocity are parallel. The main result is a multiparameter 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 multiparameter bifurcation. This theorem is presented in an appendix as an abstract result that can be applied directly to other problems.