Ring Dark Solitons and Vortex Necklaces in Bose-Einstein Condensates

We introduce the concept of ring dark solitons in Bose-Einstein condensates. We show that relatively shallow rings are not subject to the snake instability, but a deeper ring splits into a robust ringlike cluster of vortex pairs, which performs oscillations in the radial and azimuthal directions, following the dynamics of the original ring soliton.

Figure 1

Evolution of a ring dark soliton (RDS) with $R(0)=28.9$ and $\cos \phi (0)=0.6$. (a),(b) The initial profile shown by cross-section and gray-scale density plots, where RDS corresponds to a gray ring. (c) RDS shrinks to the minimum radius (at $t=40$). (d) Beginning of the emission of dark concentric rings (at $t=160$).

Figure 2

Evolution of RDS with $R(0)=28.9$ and $\cos \phi (0)=0.76$. (a) Spontaneous undulations developed on the ring at $t=60$. (b) Four vortex pairs are formed, at $t=70$.

Figure 3

Evolution of four vortex pairs created by the instability of RDS from Fig. 2 for (a) $t=120$, (b) $t=240$, (c) $t=420$, and (d) $t=540$. The vortex pairs remain on the ring and move along it, so that the configuration oscillates between the x- and $+$-like configurations, while the ring itself periodically shrinks and expands between $R_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $R_{\mathrm{m}\mathrm{a}\mathrm{x}}$.

Figure 4

Evolution of an original stationary ring soliton with $\cos \phi (0)=1$ and $R(0)=R_0=28.9$. The initial configuration remains undistorted up to $t=40$. (a) Undulations of RDS at $t=40$. (b) Sixteen vortex pairs, formed by $t=60$ as a result of the snake instability. Further snapshots are shown for (c) $t=240$ and (d) $t=760$. The initial set of $16$ vortex pairs reduces first to $12$, then to $8$, and eventually to $4$ pairs, through successive annihilations. Four surviving vortex pairs stay at $r=R_0$, oscillating between the x- and $+$-like patterns.