Vortex stability in nearly two-dimensional Bose-Einstein condensates   with attraction
  We perform accurate investigation of stability of localized vortices in an effectively two-dimensional ("pancake-shaped") trapped BEC with negative scattering length. The analysis combines computation of the stability eigenvalues and direct simulations. The states with vorticity S=1 are stable in a third of their existence region, $0<N<(1/3)N_{\max}^{(S=1)}$, where $N$ is the number of atoms, and $N_{\max}^{(S=1)}$ is the corresponding collapse threshold. Stable vortices easily self-trap from arbitrary initial configurations with embedded vorticity. In an adjacent interval, $(1/3)N_{\max }^{(S=1)}<N<$ $\allowbreak 0.43N_{\max}^{(S=1)}$, the unstable vortex periodically splits in two fragments and recombines. At $N>$ $\allowbreak 0.43N_{\max}^{(S=1)}$, the fragments do not recombine, as each one collapses by itself. The results are compared with those in the full 3D Gross-Pitaevskii equation. In a moderately anisotropic 3D configuration, with the aspect ratio $\sqrt{10}$, the stability interval of the S=1 vortices occupies $\approx 40%$ of their existence region, hence the 2D limit provides for a reasonable approximation in this case. For the isotropic 3D configuration, the stability interval expands to 65% of the existence domain. Overall, the vorticity heightens the actual collapse threshold by a factor of up to 2. All vortices with $S\geq 2$ are unstable. 
Dumitru Mihalache, Dumitru Mazilu, Boris A. Malomed, Falk Lederer