Global bifurcation theory for periodic traveling interfacial gravity-capillary waves

TL;DR

This paper establishes the existence of global bifurcation branches for periodic traveling gravity-capillary waves with arbitrary densities and parameters, allowing for complex interface geometries and singularities.

Contribution

It extends global bifurcation theory to a broad class of interfacial waves, including cases with non-graph interfaces and potential singularities.

Findings

Existence of either closed solution loops or unbounded solution sets.

Solutions can include waves with arbitrarily high speed or length.

Interfaces may self-intersect or develop singularities.

Abstract

We consider the global bifurcation problem for spatially periodic traveling waves for two-dimensional gravity-capillary vortex sheets. The two fluids have arbitrary constant, non-negative densities (not both zero), the gravity parameter can be positive, negative, or zero, and the surface tension parameter is positive. Thus, included in the parameter set are the cases of pure capillary water waves and gravity-capillary water waves. Our choice of coordinates allows for the possibility that the fluid interface is not a graph over the horizontal. We use a technical reformulation which converts the traveling wave equations into a system of the form "identity plus compact." Rabinowitz' global bifurcation theorem is applied and the final conclusion is the existence of either a closed loop of solutions, or an unbounded set of nontrivial traveling wave solutions which contains waves which may move arbitrarily fast, become arbitrarily long, form singularities in the vorticity or curvature, or whose interfaces self-intersect.