Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity

TL;DR

This paper develops a new mathematical framework to analyze two-dimensional periodic capillary-gravity water waves with arbitrary vorticity, allowing for overhanging profiles and stagnation points, and constructs global solution curves bifurcating from laminar flows.

Contribution

It introduces a novel Babenko-type reformulation and applies Rabinowitz's bifurcation theorem to study complex water wave profiles without geometric restrictions.

Findings

Constructed local and global solution curves bifurcating from laminar flows.

Allowed for overhanging wave profiles and stagnation points.

Provided a comprehensive mathematical framework for arbitrary vorticity flows.

Abstract

We study two-dimensional periodic capillary-gravity water waves propagating at the free surface of water in a flow with arbitrary, prescribed vorticity over a flat bed. Using conformal mappings and a new Babenko-type reformulation of Bernoulli's equation, the problem is equivalently cast into the form "identity plus compact", which is amenable to Rabinowitz' global bifurcation theorem, while no restrictions on the geometry of the surface profile and no assumptions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local and global solution curves, bifurcating from laminar flows with a flat surface, are constructed.