On stability of vortices in three-dimensional self-attractive Bose–Einstein condensates

Results of accurate analysis of stability are reported for localized vortices in the Bose–Einstein condensate (BEC) with the negative scattering length, trapped in an anisotropic potential with the aspect ratio Ω. The cases of Ω≫1Ω≫1 and Ω≪1Ω≪1 correspond to the “pancake” (nearly-2D) and “cigar-shaped” (nearly-1D) configurations, respectively (in the latter limit, the vortices become “tubular” solitons). The analysis is based on the 3D Gross–Pitaevskii equation. The family of solutions with vorticity S=1S=1 is accurately predicted by the variational approximation. The relative size of the stability area for the vortices with S=1S=1 (which was studied, in a part, before) increases with the decrease of Ω   in terms of the number of atoms, but decreases in terms of the chemical potential. All states with S⩾2S⩾2 are unstable, while the stability of the ordinary solitons (S=0S=0) obeys the Vakhitov–Kolokolov criterion. The stability predictions are verified by direct simulations of the full 3D equation.