Determine the stability of localized wave-borne vortices

Determine the stability, including linear stability, of localized water waves carrying submerged vortices in regimes beyond the cases currently established, with particular emphasis on capillary-gravity wave-borne vortices.

Background

The paper studies traveling capillary-gravity waves in finite-depth water that carry either a point vortex or a hollow vortex. It establishes existence of small-amplitude, fast-moving point-vortex waves and constructs hollow-vortex waves, and it proves conditional orbital stability for the point-vortex family constructed in the paper.

The authors situate these results within a broader literature in which stability results for wave-borne vortices are scarce. Thus, the unresolved issue concerns the stability of other localized wave-borne vortex configurations and parameter regimes, including stability at the linear level.