We construct small-amplitude surface water waves travelling over a large but finite depth in a fluid with concentrated vorticity. The proof is based on a bifurcation-like argument, but where the bifurcation is rather done from something that is not a water wave solution per se, but a radial and exponentially decaying ground state of a semi-linear elliptic problem in the whole plane.

The proof is based on several different parts, where some precise reflection estimates at the surface and flat bed play a major role, as does the fact that everything is exponentially decaying away from the concentrated vorticity. Finally, the vertical replacement of the centre of vorticity with respect to the surface and flat bed is necessary to guarantee a travelling solitary wave whose motion is governed by the very concentrated, but everywhere present, vorticity.

This is joint work with C. Zeng (Georgia Tech) and S. Walsh (Missouri).