Global Bifurcation of Capillary-Gravity Water Waves With Overhanging Profiles and Arbitrary Vorticity

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 reformulation of Bernoulli's equation, the problem is equivalently cast into the form "identity plus compact", which is amenable to Rabinowitz's 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 curves and global families of solutions, bifurcating from laminar flows with a flat surface, are constructed.