Accurate numerical determination of a self-preserving quantum vortex ring

We compute simultaneously the translational speed, the magnitude and the phase of a quantum vortex ring for a wide range of radii, within the Gross-Pitaevskii model, by imposing its self preservation in a co-moving reference frame. By providing such a solution as the initial condition for the time-dependent Gross-Pitaevskii equation, we verify a posteriori that the ring’s radius and speed are well maintained in the reference frame moving at the computed speed. Convergence to the numerical solution is fast for large values of the radius, as the wavefunction tends to that of a straight vortex, whereas a continuation technique and interpolation of rough solutions are needed to reach convergence as the ring tends to a disk. Comparison with other strategies for generating a quantum ring reveals that all of them seem to capture quite well the translational speed, whereas none of them seems to preserve the radius with the accuracy reached in the present work.