Vortex stability in nearly two-dimensional Bose-Einstein condensates with attraction

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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⁽ˢ⁼¹⁾, where N is the number of atoms, and N⁽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⁽ˢ⁼¹⁾<N< 0.43N⁽ˢ⁼¹⁾, the unstable vortex periodically splits in two fragments and recombines. At N> 0.43N⁽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 √10, the stability interval of the S=1 vortices occupies ≈ 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≥ 2 are unstable.