  In the present work, we explore analytically and numerically the co-existence
and interactions of ring dark solitons (RDSs) with other RDSs, as well as with
vortices. The azimuthal instabilities of the rings are explored via the
so-called filament method. As a result of their nonlinear interaction, the
vortices are found to play a stabilizing role on the rings, yet their effect is
not sufficient to offer complete stabilization of RDSs. Nevertheless, complete
stabilization of the relevant configuration can be achieved by the presence of
external ring-shaped barrier potentials. Interactions of multiple rings are
also explored, and their equilibrium positions (as a result of their own
curvature and their tail-tail interactions) are identified. In this case too,
stabilization is achieved via multi-ring external barrier potentials.
