Dynamical generation of interwoven soliton trains by nonlinear emission in binary Bose-Einstein condensates

We propose a method for the generation of trains of alternating bright solitons in two-component Bose-Einstein condensates, using controlled emission of nonlinear matter waves in the uncoupled regime with spatially varying intraspecies interaction and out-of-phase oscillations of the ground states in the trap. Under this scheme, solitons are sequentially launched from the different components and interact with each other through phase-independent cross coupling. We obtain an analytical estimation of the critical condition for soliton emission using a geometric guiding model, in analogy with integrated optical systems. In addition, we show how strong initial perturbations in the system can trigger the spontaneous generation of supersolitons, i.e., localized phononlike excitations of the soliton trains. Finally, we demonstrate the controllable generation of slow and fast supersolitons by adding external localized potentials in the nonlinear region.

Figure 1

Upper panel: Spatial profile $U\left(x\right)$ of the eigenstate featuring $N=N_{\max }$ with $x_0=6$. Lower panel: Shape of the trapping potential with $V_0=-0.3$, $d=2$, and $l=4$. In the upper panel, shaded domains indicate the regions where just self-interactions (gray) or both self- and cross interactions (blue) are active.

Figure 2

(a) Existence curves of the eigenmodes $N\left(\right|\mu \left|\right)$ in the scalar case ($c_{jk}=0$, $j\ne k$) for different positions of the nonlinear boundary: $x_0=2,3,4$, and 5. Solid and dashed lines correspond to the stable and unstable domains, respectively. (b) Dependence of the amplitude of the solution at the boundary $U(x=x_0)$ as a function of $N$. The black points correspond to $N=N_{\max }$. Other parameters are $c=1$, $V_0=-0.3$, $d=2$, $l=4$. (c)–(e) Profiles of the stationary states for different chemical potentials $\mu$, in the case $x_0=4$.

Figure 3

(a) Dependence of the critical amplitude $U_{\mathrm{cr}}={\left.\left(U\right)\right|}_{x=x_0}$ on $x_0$ at the point $N=N_{\max }$, calculated numerically solving Eq. (3) (black dashed line) and analytical prediction (red solid line) taken from the geometric nonguiding condition (4). Other parameters are $V_0=-0.3$, $l=4$. (b) Dependence $U_{\mathrm{cr}}\left(x_0\right)$ estimated numerically for different widths of the trap $d$. The value of $d$ is indicated for each of the curves. Dashed line corresponds to the condition $x_0=d$.

Figure 4

(a) Temporal dynamics of the density ${\left|u(x,t)\right|}^2$ for the eigenstate featuring $N=N_{\max }=19.84$ under small perturbations. The initial condition considered reads $N=(1+\delta )N_{\max }$, with $\delta =0.02$. (b) Evolution of the amplitude of the solution at the point $x=x_0$. The dashed red line indicates the outcome of ideal unperturbed dynamics, whereas the solid black line corresponds to the perturbed dynamics shown in (a). (c)–(f) Profiles of the density of the solution at the points indicated in (b). The final number of particles localized in the trap is $N_f\approx 11.8$ and the norm of each soliton is $\approx$2.8. Other parameters are $c=1$, $V_0=-0.3$, $d=2$, $l=4$.

Figure 5

(a) Density plot of $|u_1{|}^2-{\left|u_2\right|}^2$ with $c_{jk}=0$. The boundary for intraspecies interaction $x_0$ is shown as a dashed horizontal line. (b) Density profiles of the components $|u_1{|}^2$ (red solid line) and $|u_2{|}^2$ (black dashed line) taken at $t=300$. Other parameters are $c_{jj}=1$, $x_0=6$, $V_0=-0.3$, $d=2$, $l=4$, and $v=0.003$.

Figure 6

Propagation dynamics for (a) weak ($\left|v\right|=0.003$) and (b) strong ($\left|v\right|=0.03$) initial perturbations. Other parameters are the same as in Fig. 5 with $c_{jk}=-1$ and $x_{00}=15$.

Figure 7

Generation of (a) slow and (b) fast SSs in the weakly perturbed configuration shown in Fig. 6a by switching narrow repulsive potentials at (a) $t_i=196$ with $\tau =6$ and (b) $t_i=206$ with $\tau =3$. Parameters of the localized potentials are the following: $\zeta =3$, $x_d=15$, and $\xi =0.25$. The black dots show the time-space position of the external potential $V_g\left(x\right)$. The arrows show the directions of the SS trajectories.

Figure 8

Head-on collision of slow and fast SSs generated by switching on narrow repulsive potentials at $t_i=99$ with duration $\tau =2.3$, and $t_i=306$ with duration $\tau =2$. Other parameters are the same as in Fig. 7.