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

Abstract

The dynamics of deformations of a quantum vortex ring in a Bose condensate with the periodic equilibrium density ρ(z) = 1 − ϵ cos z has been considered in the local induction approximation. Parametric instabilities of normal modes with the azimuthal numbers ±m at the energy integral E near the values \(E_m^{\left( p \right)} = 2m\sqrt {{m^2} - 1} /p\), 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 ϵ ~ 0.03 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 ∝ σ(E₂⁽¹⁾ − E)(|b₊|² + |b_-|²)-ϵ(b₊b_- + b₊*b_-*)+u(|b₊|⁴ + |b_-|⁴)+w|b₊|²|b_-|² for two complex envelopes b_±(t). The results have been compared to parametric instabilities of the vortex ring in the condensate with the density ρ(z, r) = 1 − r² − αz², which occur at α ≈ 8/5 and 16/7.