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 ξ and initially placed near a torus of radii R₀ and r₀ can be three-dimensionally (quasi-)stable in some regions of parameters Λ = ln(R₀/ξ) and W = r₀/R₀. At fixed Λ, 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 (ΔW ∼ 0.01-0.05) between m-pairs (1, 2) and (2, 3) at Λ ≈ 4-12, thus corresponding to micro/mesoscopic sizes of vortex rings in the case of superfluid ⁴He. With four and more rings, at least for W > 0.1, resonances overlap for all Λ and no stable domains exist.