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In the present work, we explore analytically and numerically the co-existenceand interactions of ring dark solitons (RDSs) with other RDSs, as well as withvortices. The azimuthal instabilities of the rings are explored via theso-called filament method. As a result of their nonlinear interaction, thevortices are found to play a stabilizing role on the rings, yet their effect isnot sufficient to offer complete stabilization of RDSs. Nevertheless, completestabilization of the relevant configuration can be achieved by the presence ofexternal ring-shaped barrier potentials. Interactions of multiple rings arealso explored, and their equilibrium positions (as a result of their owncurvature and their tail-tail interactions) are identified. In this case too,stabilization is achieved via multi-ring external barrier potentials.

Pairwise Interactions of Ring Dark Solitons with Vortices and other Rings: Stationary States, Stability Features and Nonlinear Dynamics