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In the present work, we explore the existence, stability and dynamics ofsingle and multiple vortex ring states that can arise in Bose-Einsteincondensates. Earlier works have illustrated the bifurcation of such states, inthe vicinity of the linear limit, for isotropic or anisotropicthree-dimensional harmonic traps. Here, we extend these states to the regime oflarge chemical potentials, the so-called Thomas-Fermi limit, and explore theirproperties such as equilibrium radii and inter-ring distance, for multi-ringstates, as well as their vibrational spectra and possible instabilities. Inthis limit, both the existence and stability characteristics can be partiallytraced to a particle picture that considers the rings as individual particlesoscillating within the trap and interacting pairwise with one another. Finally,we examine some representative instability scenarios of the multi-ring dynamicsincluding breakup and reconnections, as well as the transient formation ofvortex lines.

Single and Multiple Vortex Rings in Three-Dimensional Bose-Einstein Condensates: Existence, Stability and Dynamics