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

## TL;DR

This paper investigates the parametric instabilities of vortex ring oscillations in a periodic Bose-Einstein condensate, revealing conditions for instability, numerical growth of modes, and recurrence to initial states, with comparisons to other density profiles.

## Contribution

It introduces a detailed analysis of vortex ring instabilities in a $z$-periodic BEC, including the derivation of an integrable Hamiltonian model and numerical validation.

## Key findings

- Parametric instabilities occur near specific energy values $E_m^{(p)}$.
- Unstable modes grow rapidly at small $
ho(z)$ modulation amplitude $\\sim 0.03$.
- Vortex rings exhibit recurrence to initial states after instability growth.

## 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 $H\propto \sigma(E_2^{(1)}-E)(|b_+|^2 + |b_-|^2) -\epsilon(b_+ b_- + b_+^* b_-^*) +u(|b_+|^4 +|b_-|^4) + w |b_+|^2|b_-|^2$ for two complex envelopes $b_\pm(t)$. The results have been compared to parametric instabilities of 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 $\alpha\approx 16/7$.