Three-dimensional stability of leapfrogging quantum vortex rings

It is shown by numerical simulations within a regularized Biot-Savart law that dynamical systems of two or three leapfrogging coaxial quantum vortex rings having a core width xi and initially placed near a torus of radii R-0 and r(0) can be three-dimensionally (quasi-) stable in some regions of parameters Lambda = ln(R-0/xi) and W = r(0)/R-0. At fixed Lambda, stable bands on W are intervals between non-overlapping main parametric resonances for different (integer) azimuthal wave numbers m. The stable intervals are most wide (Delta W similar to 0.01-0.05) between m-pairs (1, 2) and (2, 3) at Lambda approximate to 4-12, thus corresponding to micro/mesoscopic sizes of vortex rings in the case of superfluid He-4. With four and more rings, at least for W > 0.1, resonances overlap for all Lambda and no stable domains exist. Published by AIP Publishing.