An analysis of a new approximate model for two-dimensional water waves with constant vorticity

In this thesis, we investigate the water-wave problem for two-dimensional steady waves with constant vorticity. We introduce a new approximate model that, in its simplest form, reduces to a planar dynamical system. We then analyze this system for various energy levels and vorticity constants, studying its phase portraits to determine the locations and types of solutions that arise. In the unidirectional regime, where the flow preserves its direction, we recover classical periodic and solitary wave solutions, including both smooth profiles and extreme waves with sharp crests. Additionally, we identify new classes of waves propagating over flows that contain a critical layer. These solutions exhibit a discontinuity in the derivative of the surface profile at a point along the side of the wave. We therefore regard them as potential prototypes of overhanging waves for the full Euler equations.