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In this paper, we study solitary waves propagating along the surface of aninfinitely deep body of water in two or three dimensions. The waves are actedupon by gravity and capillary effects are allowed --- but not required --- onthe interface. We assume that the vorticity is localized in the sense that itsatisfies certain moment conditions, and we permit there to be finitely manypoint vortices in the bulk of the fluid in two dimensions. We also consider atwo-fluid model with a vortex sheet.  Under mild decay assumptions, we obtain precise asymptotics for the velocityfield and free surface, and relate this to global properties of the wave. Forinstance, we rule out the existence of waves whose free surface elevations havea single sign and of vortex sheets with finite angular momentum. Building onthe work of Shatah, Walsh, and Zeng, we also prove the existence of families oftwo-dimensional capillary-gravity waves with compactly supported vorticitysatisfying the above assumptions. For these waves, we further show that thefree surface is positive in a neighborhood of infinity, and that theasymptotics at infinity are linked to the net vorticity.

Existence, nonexistence, and asymptotics of deep water solitary waves with localized vorticity