Dynamics and pattern formation of ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable interactions

We investigate the dynamics and pattern formation of two ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable intercomponent interaction via numerical simulation of the time-dependent Gross-Pitaevskii equation. Both the black and gray ring dark solitons are considered for cases where the modulation frequency of the intercomponent interaction is resonant or nonresonant with the one of the trapping potential. Our results show that in the presence of periodic modulation of the intercomponent interaction not only are the lifetimes of the ring dark solitons largely extended but also their decaying dynamics are dramatically affected. Before snaking instability sets in, new ring dark solitons are formed, and both the numbers and depths of the ring dark solitons exhibit collective oscillations. With the development of instability, the system exhibits different decaying processes, and a variety of decay profiles, such as vortex necklace, distorted octagon, vortex-antivortex ring, and cross, are formed, showing a strong dependence on the modulation frequency of the intercomponent interaction and the initial depth of the ring dark soliton.

Figure 1

The time evolutions of two black RDSs with resonant modulation frequency of the intercomponent interaction $\omega =0.028$ for $t=$ (a) 0, (b) 30, (c) 110, (d) 180, (e) 210, and (f) 240, where the top row is for component 1 and the bottom one is for component 2. The initial depths of two such black RDSs $cos{\varphi }_1\left(0\right)=cos{\varphi }_2\left(0\right)=1$, the initial locations $R_{10}=27.9$ and $R_{20}=28.9$, eccentricity $e_c=0$, and the harmonic trap frequency $\mathrm{\Omega }=0.028$.

Figure 2

The 1D density distributions at $t=30$ (a, b) and at $t=110$ (c, d) for the two-component system, corresponding to Figs. 1 and 1, respectively. Parameters are chosen as the same as those in Fig. 1.

Figure 3

The time evolutions of the system following Fig. 1 for $t=$ (g) 270, (h) 300, (i) 320, (j) 380, (k) 510, and (l) 630. Parameters are chosen as the same as those in Fig. 1.

Figure 4

The time evolutions of two black RDSs for $t=$ (a) 170, (b) 300, (c) 380, and (d) 530, where the top row is for component 1 and the bottom one is for component 2. The modulation frequency of the intercomponent interaction $\omega =0.01$. Other parameters are the same as those in Fig. 1.

Figure 5

The time evolutions of two black RDSs for $t=$ (a) 270, (b) 300, (c) 380, and (d) 530, where the top row is for component 1 and the bottom one is for component 2. The modulation frequency of the intercomponent interaction $\omega =0.05$. Other parameters are the same as those in Fig. 1.

Figure 6

The time evolutions of two deep gray RDSs for $t=$ (a) 410, (b) 430, (c) 520, and (d) 650, where the top row is for component 1 and the bottom one is for component 2. The initial depths of two such RDSs $cos{\varphi }_1\left(0\right)=cos{\varphi }_2\left(0\right)=0.67$, and the modulation frequency of the intercomponent interaction $\omega =0.028$. Other parameters are the same as those in Fig. 1.

Figure 7

The dependence of the lifetime of both black (blue line) and deep gray RDSs (red line) on the modulation frequency.

Figure 8

The time evolutions of two black RDSs with resonant modulation frequency of the intercomponent interaction $\omega =0.028$ for $t=$ (a) 110, (b) 210, (c) 300, (d) 380, and (e) 510 and for a larger computational box (typically, twice the size of Fig. 1). Other parameters are the same as those in Fig. 1.

Figure 9

The time evolutions of two black RDSs with modulation frequency $\omega =0.01$ for $t=$ (a) 170, (b) 300, (c) 380, and (d) 530 and for a larger computational box (typically, twice the size of Fig. 1). Other parameters are the same as those in Fig. 1.

Figure 10

The time evolutions of two black RDSs with modulation frequency $\omega =0.05$ for $t=$ (a) 270, (b) 300, (c) 380, and (d) 530 and for a larger computational box (typically, twice the size of Fig. 1). Other parameters are the same as those in Fig. 1.

Figure 11

The time evolutions of two black RDSs in the presence of a small (5%) random perturbation to both the amplitude and phase at every point, and for $t=$ (a) 40, (b) 130, (c) 170, and (d) 190. The modulation frequency of the intercomponent interaction $\omega =0.028$. Other parameters are the same as those in Fig. 1.

Figure 12

The same as in Fig. 11 but for modulation frequency $\omega =0.01$.

Figure 13

The same as in Fig. 11 but for modulation frequency $\omega =0.05$.