  The three-dimensional study of the ring vortex solitons is conducted for both
attractive and repulsive BECs subject to harmonic potential confinement. A
family of stationary ring vortex solitons, which is defined by the radial
excitation number and the winding number of the intrinsic vorticity, are
obtained numerically for a given atomic interaction strength. We find that
stabilities of the ground and radially excited states of the ring vortex
soliton are dependent on the winding number differently. The ground state of
the ring vortex soliton with the large winding number is unstable dynamically
against random perturbation. The radially excited state of the ring vortex
soliton with large winding number corresponds to the increased collapse
threshold, and therefore can be made stable for sufficiently small atomic
interaction strengths. The ground and radially excited states also demonstrate
different dynamical evolutions under large atomic interaction strengths. The
former exhibits simultaneously symmetrical splitting in the transverse plane,
while the latter displays periodically expand-merge cycles in the longitudinal
direction.
