  We have studied collective modes of quasi-2D Bose-Einstein condensates with
multiply-charged vortices using a variational approach. Two of the four
collective modes considered exhibit coupling between the vortex dynamics and
the large-scale motion of the cloud. The vortex presence causes a shift in all
frequencies of collective modes even for the ones that do not couple
dynamically with the vortex-core. The coupling between vortex and large-scale
collective excitations can induce the multi-charged vortex to decay into
singly-charged vortices with the quadrupole mode being one possible channel for
such a decay. Therefore a thorough study was done about the possibility to
prevent the vortex decay by applying a Gaussian potential with its width
proportional to the vortex-core radius and varying its height. In such way, we
created a stability diagram of height versus interaction strength which has
stable regions due the static Gaussian potential. Furthermore, by using a
sinusoidal time-modulation around the average height of the Gaussian potential,
we have obtained a diagram for the parametric resonance which can prevent the
vortex decay in regions where static potential can not.
