A method for the dynamics of vortices in a Bose-Einstein condensate: analytical equations of the trajectories of phase singularities

We present a method to study the dynamics of a quasi-two dimensional Bose-Einstein condensate which contains initially many vortices at arbitrary locations. We present first the analytical solution of the dynamics in a homogeneous medium and in a parabolic trap for the ideal non-interacting case. For the homogeneous case this was introduced in the context of photonics. Here we discuss this case in the context of Bose-Einstein condensates and extend the analytical solution to the trapped case, for the first time. This linear case allows one to obtain the trajectories of the position of phase singularities present in the initial condensate along with time. Also, it allows one to predict some quantities of interest, such as the time at which a vortex and an antivortex contained in the initial condensate will merge. Secondly, the method is complemented with numerical simulations of the non-linear case. We use a numerical split-step simulation of the non-linear Gross-Pitaevskii equation to determine how these trajectories and quantities of interest are changed by the presence of interactions. We illustrate the method with several simple cases of interest both in the homogeneous and parabolically trapped systems.