Dynamics of Two Vortex Rings in a Bose - Einstein Condensate

In this paper, we consider the dynamics of two interacting point vortex rings in a Bose - Einstein condensate. The existence of an invariant manifold corresponding to vortex rings is proved. Equations of motion on this invariant manifold are obtained for an arbitrary number of rings from an arbitrary number of vortices. A detailed analysis is made of the case of two vortex rings each of which consists of two point vortices where all vortices have same topological charge. For this case, partial solutions are found and a complete bifurcation analysis is carried out. It is shown that, depending on the parameters of the Bose - Einstein condensate, there are three different types of bifurcation diagrams. For each type, typical phase portraits are presented.