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

摘要

The dynamics of deformations of a quantum vortex ring in a Bose condensate with the periodic equilibrium density rho(z) = 1 - I mu 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 , 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 I mu 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 ae sigma(E (2) ((1)) - E)(|b (+)|(2) + |b (-)|(2))-I mu(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 rho(z, r) = 1 - r (2) - alpha z (2), which occur at alpha ae 8/5 and 16/7.