  The dynamical instabilities and ensuing dynamics of singly- and
doubly-quantized vortex states of Bose-Einstein condensates with attractive
interactions are investigated using full 3D numerical simulations of the
Gross-Pitaevskii equation. With increasing the strength of attractive
interactions, a series of dynamical instabilities such as quadrupole, dipole,
octupole, and monopole instabilities emerge. The most prominent instability
depends on the strength of interactions, the geometry of the trapping
potential, and deviations from the axisymmetry due to external perturbations.
Singly-quantized vortices split into two clusters and subsequently undergo
split-merge cycles in a pancake-shaped trap, whereas the split fragments
immediately collapse in a spherical trap. Doubly-quantized vortices are always
unstable to disintegration of the vortex core. If we suddenly change the
strength of interaction to within a certain range, the vortex splits into three
clusters, and one of the clusters collapses after a few split-merge cycles. The
vortex split can be observed using a current experimental setup of the MIT
group.
