Dynamics of vortices in two-dimensional Bose-Einstein condensates

We derive the asymptotic motion equations of vortices for the time-dependent Gross-Pitaevskii equation with a harmonic trap potential. The asymptotic motion equations form a system of ordinary differential equations which can be regarded as a perturbation of the standard Kirchhoff problem. From the numerical simulation on the asymptotic motion equations, we observe that the bounded and collisionless trajectories of three vortices form chaotic, quasi 2-or quasi 3-periodic orbits. Furthermore, a new phenomenon of 1:1-topological synchronization is observed in the chaotic trajectories of two vortices.