Abstract
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.
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