  We demonstrate that the evolution of superflows in interacting persistent
currents of ultracold gases is strongly affected by symmetry breaking of the
quantum vortex dynamics. We study counter-propagating superflows in a system of
two parallel rings in regimes of weak (a Josephson junction with tunneling
through the barrier) and strong (rings merging across a reduced barrier)
interactions. For the weakly interacting toroidal Bose-Einstein condensates,
formation of rotational fluxons (Josephson vortices) is associated with
spontaneous breaking of the rotational symmetry of the tunneling superflows.
The influence of a controllable symmetry breaking on the final state of the
merging counter-propagating superflows is investigated in the framework of a
weakly dissipative mean-field model. It is demonstrated that the population
imbalance between the merging flows and the breaking of the underlying
rotational symmetry can drive the double-ring system to final states with
different angular momenta.
