Three-dimensional stability of leapfrogging quantum vortex rings

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 $\xi$ 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 ($\Delta W\sim$ 0.01--0.05) between $m$-pairs $(1,2)$ and $(2,3)$ at $\Lambda \approx$ 4--12 thus corresponding to micro/mesoscopic sizes of vortex rings in the case of superfluid ${}^4$He. With four and more rings, at least for $W>0.1$, resonances overlap for all $\Lambda$ and no stable domains exist.

7 pages, 9 figures, the published version plus an additional remark

References in corpus (3)

• Anomalous translational velocity of vortex ring with finite-amplitude Kelvin waves
• Scattering and leapfrogging of vortex rings in a superfluid
• Long-lived quantum vortex knots

Cited by in corpus (4)

• Quasi-stable quantum vortex knots and links in anisotropic harmonically trapped Bose-Einstein condensates
• Symmetry-breaking instability of leapfrogging vortex rings in a Bose-Einstein condensate
• Theoretical framework bridging classical and quantum mechanics for the dynamics of cryogenic liquid helium-4 using smoothed-particle hydrodynamics
• Vortex knots on three-dimensional lattices of nonlinear oscillators coupled by space-varying links