Parametric instability of oscillations of a vortex ring in a $z$-periodic Bose condensate and return to the initial state

The dynamics of deformations of a quantum vortex ring in a Bose condensate with the periodic equilibrium density $\rho(z)= 1-\epsilon\cos z$ has been considered in the local induction approximation. Parametric instabilities of normal modes with the azimuthal numbers $\pm m$ at the energy integral $E$ near the values $E_m^{(p)}=2m\sqrt{m^2-1}/p$, where $p$ is the order of resonance, have been revealed. Numerical experiments have shown that the amplitude of unstable modes with $m=2$ and $p=1$ can sharply increase already at $\epsilon\sim 0.3$ to values about unity. Then, after several fast oscillations, fast return to a weakly perturbed state occurs. Such a behavior corresponds to the integrable Hamiltonian $H\propto\sigma(E_2^{(1)}-E)(|b_+|^2+|b_-|^2)- \epsilon(b_+ b_- + b_+^*b_-^*)+u(|b_+|^4+|b_-|^4)+w |b_+|^2|b_-|^2$ for two complex envelopes $b_{\pm}(t)$. The results have been compared to parametric instabilities of the vortex ring in the condensate with the density $\rho(z, r)=1-r^2-\alpha z^2$, which occur at $\alpha\approx 8/5$ and $16/7$.