Figure 1

Snapshots of real-time propagation of the GPE (2) in a polar grid at (a) $t=0.0$, (b) $t=1.5$, (c) $t=3.0$, and (d) $t=4.4/{\omega }_x$ with the harmonic torus (3) with $k=96$, $R_S=8.5$, and $C_{2\text{D}}=400$, but with a slight added anisotropy at $\theta =\pi /2$ [34] (we replace $R_S=8.3$ at the $\theta =\pi /2$ lattice points, but the exact form of the anisotropy was not observed to matter). The transverse oscillations of the snake instability are seen to start immediately originating at the trap anisotropy and propagate along the ring with a finite speed, in accordance with the model discussed in the text. Without the anisotropy the RDS is stable.Reuse & Permissions

Figure 2

Snapshots of real-time propagation of the GPE (1) in a rectangular grid at (a) $t=0.0$ and (b) $t=2.8/{\omega }_x$ with the same parameters as in Fig. 1, excluding the anisotropy. The transverse oscillations of the snake instability are seen to originate over the whole ring, resulting in the collapse of the ring dark soliton (producing eventually a necklace of vortex-antivortex pairs).Reuse & Permissions

Figure 3

Hydrodynamical force (6) (solid line) around the notch of a planar dark soliton at $x=0$ (dashed line). Note that the force is symmetrical about the notch $x=0$ of the (planar) soliton $n={\tanh }^2(x/\sqrt{2})$, so there is no net force. Note also that this solution and therefore the force corresponds to a homogeneous background with no external potential.Reuse & Permissions

Figure 4

Numerical propagation of the GPE in real time over a time period of $t=42.6/{\omega }_x$ with $C_{2\text{D}}=400$ and $V_{\mathrm{trap}}$ given by Eq. (3) with $k=1$ and $R_S=7.5$. The soliton is printed at the radius given by Eq. (7). The initial ring dark soliton decays into a vortex-antivortex necklace, but is revived several times later on. Between the ring dark soliton states the vortices oscillate radially and azimuthally in a circular fashion changing pairs (the sense of the circulation depends on whether we have a vortex or an antivortex). (a) The upper row shows the density while the lower row shows the phase. From left to right, the snapshots correspond to the times $t=0.0/{\omega }_x$, $13.5/{\omega }_x$, $15.4/{\omega }_x$, $16.2/{\omega }_x$, $18.6/{\omega }_x$, and $23.4/{\omega }_x$. (b) Same as (a), but the times are $t=25.0/{\omega }_x$, $27.0/{\omega }_x$, $32.4/{\omega }_x$, $33.6/{\omega }_x$, $36.0/{\omega }_x$, and $42.6/{\omega }_x$ respectively.Reuse & Permissions

Figure 5

Correspondence between the predicted [dashed line; see Eq. (8)] and measured (solid line) number of vortex-antivortex pairs for the harmonic toroidal potential (3) with $k=1$ and $C_{2\mathrm{D}}=400$. The healing length was measured by finding a best fit to $\frac{\mu -\frac{1}{4}{(r-R_S)}^2}{C_{2\mathrm{D}}}{\tanh }^2\left(\frac{r-R_S}{\sqrt{2}\xi }\right)$. The shaded areas show where the complex Bogoliubov modes such that $q$ is a multiple of 4 are active.Reuse & Permissions

Figure 6

Results of the Bogoliubov analysis. In all of the panels we have used the harmonic toroidal potential (3) with $k=1$ and $C_{2\mathrm{D}}=400$. (a) Typical spectrum of Eq. (4) in the complex eigenvalue plane with $q=8$ for a toroidal condensate corresponding to $R_S=8.5$ with no soliton or excitations. Note that all the eigenvalues are real, signifying dynamical stability of the condensate. (b) Same as (a), but with the soliton printed at the radius given by Eq. (7). Note the appearance of an imaginary mode ${\Omega }_8=0.613i$ related to the snake instability. The imaginary mode is shown in more detail in Fig. 7. (c) All of the imaginary eigenvalues ${\Omega }_q$ of Eq. (4) with the ring dark soliton (up to $q=12$), corresponding to modes with a dynamical instability, labeled by the value of $q$. There is only at most one imaginary eigenvalue per $q$. The inset shows the reciprocal numbers giving the decay times. The solid black line without markers is given by $\Omega =i/(0.135R_S+0.444)$. The shading is the same as in Fig. 5.Reuse & Permissions

Figure 7

Imaginary Bogoliubov mode ${\Omega }_8=0.613i$ of Fig. 6b visualized in more detail. The main figure shows the wave functions $|{\varphi }_0{|}^2$ (solid line) and $|{\varphi }_0+u+\overline{v}{|}^2$ (dashed line) at $\theta =2\pi /8$. Note that we have chosen our normalization of $u$ and $v$ for the sake of clarity. The inset shows how the location of the soliton's notch is affected by the Bogoliubov mode over the whole ring. The mode mimics the snake instability and coincides exactly with the locations of the vortices (exaggerated for clarity). For this radius ($R_S=8.5$), the $q=8$ mode is the fastest [see Fig. 6c], and it produces $q=N_v=8$ pairs, in accordance with observations (see Fig. 5).Reuse & Permissions