Solitary water waves with discontinuous vorticity

Abstract

We investigate the existence of solitary gravity waves traversing a two-dimensional body of water that is bounded below by a flat impenetrable ocean bed and above by a free surface of constant pressure. Our main interest is constructing waves of this form that exhibit a discontinuous distribution of vorticity. More precisely, this means that the velocity limits both upstream and downstream to a laminar flow that is merely Lipschitz continuous. We prove that, for any choice of background velocity with this regularity, there exists a global curve of solutions bifurcating from a critical laminar flow and including waves arbitrarily close to having stagnation points. Each of these waves has an axis of even symmetry, and the height of their streamlines above the bed decreases monotonically as one moves to the right of the crest.