LAUSR

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 $$E_m^{\left( p \right)} = 2m\sqrt {{m^2} - 1} /p$$Em(p)=2mm2−1/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.