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 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 (∆W ∼ 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 4 He. With four and more rings, at least for W > 0.1, resonances overlap for all Λ and no stable domains exist.

Symmetry-breaking instability of leapfrogging vortex rings in a Bose-Einstein condensate

“…The rings in this arrangement also exhibit periodic leapfrogging dynamics such that the triangles of the vortex cores rotate around their centers. The axial symmetry is retained for a long time in the leapfrogging dynamics for the triangular configuration, as reported for the vortex-filament model [16,23]. We have numerically confirmed that the axially-symmetric leapfrogging dynamics continue at least until t ≃ 500 for Figs.…”

“…They oscillate and never grow for m = 6. The growth of the m = 6 mode is slow and only affects the long-time dynamics [23]. The time evolution of C m for the rectilinear initial arrangement in Fig.…”

“…In Bose-Einstein condensates (BECs) of ultracold atomic gases, quantized vortex rings have been created through the decay of solitons and directly observed by imaging the density distribution [4][5][6][7]. The dynamics of quantized vortex rings in superfluids have been theoretically studied by many researchers [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Quasistable quantum vortex knots and links in anisotropic harmonically trapped Bose-Einstein condensates

“…A focal theme of interest within this nexus of topological charge, nonlinearity and spatial confinement has been the study of vortex rings and simple filaments [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] whose interaction dynamics and even leapfrogging [32][33][34] have been considered. An even more demanding 3D territory that has been less explored (especially so experimentally) has been that of vortex knot structures.…”