  In this work we present a systematic study of the three-dimensional extension
of the ring dark soliton examining its existence, stability, and dynamics in
isotropic harmonically trapped Bose-Einstein condensates. Detuning the chemical
potential from the linear limit, the ring dark soliton becomes unstable
immediately, but can be fully stabilized by an external cylindrical potential.
The ring has a large number of unstable modes which are analyzed through
spectral stability analysis. Furthermore, a few typical destabilization
dynamical scenarios are revealed with a number of interesting vortical
structures emerging such as the two or four coaxial parallel vortex rings. In
the process of considering the stability of the structure, we also develop a
modified version of the degenerate perturbation theory method for
characterizing the spectra of the coherent structure. This semi-analytical
method can be reliably applied to any soliton with a linear limit to explore
its spectral properties near this limit. The good agreement of the resulting
spectrum is illustrated via a comparison with the full numerical Bogolyubov-de
Gennes spectrum. The application of the method to the two-component ring
dark-bright soliton is also discussed.
