Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii   equation
  Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation. 
Robert L. Jerrard and Didier Smets
math.AP