Vortex-ring solutions of the Gross-Pitaevskii equation for an axisymmetrically trapped Bose-Einstein condensate

We propose a numerical scheme for obtaining the stationary vortex-ring solutions of the Gross-Pitaevskii (GP) equation for an axisymmetrically trapped Bose-Einstein condensate (BEC). The effective energy functional and the associated GP equation are derived by assuming a trial phase profile for the wavefunction that is subject to the condition of circulation quantization on the rz plane. The wavefunction of the vortex ring is determined by solving the ground state of the effective GP equation numerically. Application of our method to the formation of a three-dimensional Skyrmion in a trapped two-component BEC is demonstrated