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 Bose-Einstein condensate 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) Nmax (S=1), where N is the number of atoms, and Nmax (S=1) is the corresponding collapse threshold. Stable vortices easily self-trap from arbitrary initial configurations with embedded vorticity. In an adjacent interval, (13) Nmax (S=1) <N<0.43 Nmax (S=1), the unstable vortex periodically splits in twofragments and recombines. At N>0.43 Nmax (S=1), the fragments do not recombine, as each one collapses by itself. The results are compared with those in the full three-dimensional (3D) Gross-Pitaevskii equation. In a moderately anisotropic 3D configuration, with the aspect ratio 10, the stability interval of the S=1 vortices occupies ≈40% of their existence region, hence the two-dimensional (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≥2 are unstable.