  The dynamics of vortex ring pairs in the homogeneous nonlinear Schr\"odinger
equation is studied. The generation of numerically-exact solutions of traveling
vortex rings is described and their translational velocity compared to revised
analytic approximations. The scattering behavior of co-axial vortex rings with
opposite charge undergoing collision is numerically investigated for different
scattering angles yielding a surprisingly simple result for its dependence as a
function of the initial vortex ring parameters. We also study the leapfrogging
behavior of co-axial rings with equal charge and compare it with the dynamics
stemming from a modified version of the reduced equations of motion from a
classical fluid model derived using the Biot-Savart law.
