Parametric instability of oscillations of a vortex ring in a $z$-periodic Bose-Einstein condensate and the recurrence to starting state

Abstract

The dynamics of deformations of a quantum vortex ring in a Bose-Einstein condensate with periodic equilibrium density $\rho (z)=1-\epsilon \cos z$ has been considered within the local induction approximation. Parametric instabilities of the normal modes with azimuthal numbers $\pm m$ have been revealed at the energy integral $E$ near values $E_m^{(p)}=2m\sqrt{m^2-1}/p$, where $p$ is the resonance order. Numerical simulations have shown that already at $\epsilon \sim 0.03$ a rapid growth of unstable modes with $m=2$, $p=1$ to magnitudes of order of unity is typical, which is then followed, after a few large oscillations, by fast return to a weakly excited state. Such behavior corresponds to an integrable Hamiltonian of the form $H\propto \sigma (E_2^{(1)}-E)(|b_{+}{|}^2+|b_{-}{|}^2)-\epsilon (b_{+}{b}_{-}+b_{+}^{\ast }{b}_{-}^{\ast })+u(|b_{+}{|}^4+|b_{-}{|}^4)+w|b_{+}{|}^2|b_{-}{|}^2$ for two complex envelopes $b_{\pm }(t)$. The results have been compared to parametric instabilities of vortex ring in condensate with density $\rho (z,r)=1-r^2-\alpha z^2$, which take place at $\alpha \approx 8/5$ and at $\alpha \approx 16/7$.

arXiv comment5 pages, 5 figures, important new material added

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• Quasi-stable quantum vortex knots and links in anisotropic harmonically trapped Bose-Einstein condensates
• Dynamics of interacting dark soliton stripes
• Stable and unstable vortex knots in a trapped Bose-Einstein condensate
• Discrete vortices on spatially nonuniform two-dimensional electric networks
• Vortex knots on three-dimensional lattices of nonlinear oscillators coupled by space-varying links