Dynamics of interacting dark soliton stripes

In the present work we examine the statics and dynamics of multiple parallel dark soliton stripes in a two-dimensional Bose-Einstein condensate. Our principal goal is to study the effect of the interaction between the stripes on the transverse instability of the individual stripes. The cases of two-, three-, and four-stripe states are studied in detail. We use a recently developed adiabatic invariant formulation to derive a quasianalytical prediction for the stripe equilibrium position and for the Bogoliubov–de Gennes spectrum of excitations of stationary stripes. We subsequently test our predictions against numerical simulations of the full two-dimensional Gross-Pitaevskii equation. We find that the number of unstable eigenmodes increases as the number of stripes increases due to (unstable) relative motions between the stripes. Their corresponding growth rates do not significantly change, although for large chemical potentials, the larger the stripe number, the larger the maximal instability growth rate. The instability induced dynamics of multiple stripe states and their decay into vortices are also investigated.

Figure 1

Cross sections ($y=\mathrm{const}$) along the $x$ direction of typical wave functions $\mathrm{\Psi }$ [which are found from the GPE (1) using $u(x,y,t)=\mathrm{\Psi }\left(x\right)exp(-i\mu t)$] corresponding from top to bottom to one, two, three, and four dark solitons, respectively. In all cases $\mathrm{\Omega }=1$ and $\mu =5$ (left) and $\mu =40$ (right) correspond to chemical potential values close to the linear (small density) and Thomas-Fermi (large density) limits, respectively. Note that these 2D stationary states are homogeneous in the $y$ direction, as the potential (2) is only $x$ dependent. All quantities in this and subsequent figures are dimensionless; see text.

Figure 2

Top: Equilibrium position $x_0=x_0^{\left(\mathrm{eq}\right)}$ for the two-stripe case, as obtained from the GPE (1) (red line) and the AI PDE model (6) (blue circles). Bottom: Difference $\mathrm{\Delta }{x}_0$ between the equilibrium position from the GPE (1) and the one predicted by the AI PDE model (6). Notice the remarkably good agreement even near the linear limit.

Figure 3

Comparison between the two dark soliton stripes' stability spectra for the full GPE and the analytical prediction based on the AI PDE. Depicted are the real (${\lambda }_r$; top subpanels) and imaginary (${\lambda }_i$; bottom subpanels) parts of the stability eigenvalues, $\lambda ={\lambda }_r+i{\lambda }_i$, vs the chemical potential $\mu$ (note that ${\lambda }_r$ is scaled by $\sqrt{\mu }$). For the numerical domain we use $L_x=16$ as well as $L_y=2$ (top set of panels) and $L_y=4$ (bottom set of panels). In the top subpanels (displaying ${\lambda }_r$) the real part of the eigenvalues from the full GPE is depicted by the small red dots, while the thick pink (light gray) curves depict the real part of the eigenvalues for the effective AI PDE model and the thick violet (dark gray) curves depict the (unstable) $n\ge 1$ in-phase modes; see Eq. (9). In the bottom subpanels (displaying ${\lambda }_i$) the imaginary part of the eigenvalues from the full GPE is depicted by the small blue dots, while the thick orange (light gray) curves depict the real part of the eigenvalues for the effective AI PDE model, the thick green (dark gray) horizontal line depicts the single-stripe $n=0$ (stable) in-phase mode $\text{Im}\left(\lambda \right)=\mathrm{\Omega }/\sqrt{2}$, and the thick gray horizontal lines correspond to the 1D TF spectrum; see Eq. (8).

Figure 4

Dynamics for in-phase modes corresponding, from top to bottom, to the one-, two-, and three-stripe states for $\mu =40$. The color bars on the right-hand side correspond to the density ${\left|u(x,y,t)\right|}^2$. The system was initialized at $t=0$ with an $N$-stripe state perturbed by the mode $Acos\left(k_{n}y\right)$ in the transverse direction, where $k_n=n\pi /L_y$ with $A={10}^{-3}$, $n=5$, and $L_y=8$ (only the portion corresponding to $-5\le y\le 5$ at the times indicated is shown). Note that the dynamics, including the growth rate, seem to be closely analogous in the different cases.

Figure 5

Eigenfunctions corresponding to the in-phase (IP; left) and out-of-phase (OP; right) unstable modes of the stationary two-stripe state $\mathrm{\Psi }\left(x\right)$ for $L_y=2$ and $\mu =40$. Shown are the $k_1$ modes with transverse dependence given by $cos\left(k_{n}y\right)$ with $k_n=n\pi /L_y$. The eigenmodes corresponding to other values of $n$ are very similar (results not shown here). The thick solid (blue) and dotted (red) lines correspond, respectively, to the real and imaginary parts of the eigenfunction $w\left(x\right)$. The corresponding normalized steady state, $\mathrm{\Psi }\left(x\right)/\sqrt{\mu }$, is depicted by the thin black line. Note that the real part of the eigenmodes has localized “humps” that, when out of phase, produce in-phase motion of the dark solitons and vice versa. The reason for this apparent contradiction is that the dark solitons are themselves of opposite phase: in this figure the left soliton corresponds to $-tanh$ while the right one corresponds to $+tanh$.

Figure 6

Dynamical destabilization and collision of two dark soliton stripes. The background color map corresponds to full GPE numerics while the green overlaid curves correspond to the AI reduction The corresponding systems are initialized symmetrically with the right dark soliton stripe at $x_0\left(y\right)={\xi }_0+{\sum }_{n=1}^{\nu }{\varepsilon }_{n}sin(2\pi ny/L_y+{\phi }_n)$ with ${\varepsilon }_n=0.01$, ${\phi }_n=(n-1)L_y\pi /10$, ${\xi }_0=0.45$ (i.e., close to the steady state equilibrium $x_0^{\left(\mathrm{eq}\right)}=0.368$), and $\mu =40$. The top two rows of panels correspond to a single-mode perturbation ($\nu =1$) with $L_y=2$ while the bottom two rows of panels correspond to a perturbation containing five modes ($\nu =5$) and $L_y=8$. Within each set of panels the top and bottom rows (and their color bars) correspond to the density $\left[{\left|u(x,y,t)\right|}^2\right]$ and phase ($-\pi$ to $\pi$) of the field at the indicated times.

Figure 7

Similar to Fig. 3 (for two stripes), but now for the three- and four-stripe cases, for $L_x=16$ and $L_y=2$.

Figure 8

Same as Fig. 5 but for the three-stripe (top) and four-stripe (bottom) states. Parameters correspond to $L_y=2$ and $\mu =40$ for the transverse modes involving the motion of the dark soliton stripes. The notation $\pm 1:\pm 1:\pm 1$ is used to denote a normal mode that has a displacement from the steady state of the first, second, and third dark solitons in the positive ($+$) or negative ($-$) direction. The same notation is used for the four-stripe case.

Figure 9

Top panel: Comparison of the fully in-phase oscillation mode $(n=0)$ frequency for different numbers of dark soliton stripes using the same $L_y=2$. $\mathrm{DS}N$ denotes the $N$-stripe state. Note that they all converge to the 1D single dark soliton results $\mathrm{\Omega }/\sqrt{2}$ (see horizontal dashed line) corresponding to the IP oscillation of the $N$ stripes. Bottom panel: The most unstable mode growth rate for different numbers of dark soliton stripes. Note there is an interesting crossover behavior. The growth rate is larger for the one-stripe state near the linear limit, but in the TF limit more dark solitons are more unstable. Moreover, in the latter limit there is a (weak, yet) monotonic increase of the growth rate with $N$. The DS1, DS2, DS3, and DS4 cases are depicted, respectively, with the red (gray), blue (dark gray), black, and yellow (light gray) curves.

Figure 10

Dynamical destabilization of two-stripe states using different small random perturbations. Each set of two rows of panels corresponds, from top to bottom, to two different realizations for a random perturbation of amplitude ${10}^{-1}$, ${10}^{-2}$, ${10}^{-4}$, and ${10}^{-12}$, respectively.

Figure 11

Dynamical destabilization of the three- and four-stripe states as obtained from the full GPE. In these cases, we have added a small (${10}^{-4}$ amplitude) random noise. Both states, following the spectral picture, break with the most unstable mode, and give rise to numerous vortex pairs. Note that, despite the existence of more unstable modes, the maximum decay rates are about the same for different numbers of stripes.