Controlling the transverse instability of dark solitons and nucleation of vortices by a potential barrier

We study possibilities to suppress the transverse modulational instability (MI) of dark-soliton stripes in two-dimensional Bose-Einstein condensates (BEC’s) and self-defocusing bulk optical waveguides by means of quasi-one-dimensional structures. Adding an external repulsive barrier potential (which can be induced in BEC by a laser sheet, or by an embedded plate in optics), we demonstrate that it is possible to reduce the MI wave number band, and even render the dark-soliton stripe completely stable. Using this method, we demonstrate the control of the number of vortex pairs nucleated by each spatial period of the modulational perturbation. By means of the perturbation theory, we predict the number of the nucleated vortices per unit length. The analytical results are corroborated by the numerical computation of eigenmodes of small perturbations, as well as by direct simulations of the underlying Gross-Pitaevskii/nonlinear Schrödinger equation.

Figure 1

Integral (12) as a function of the rescaled width $\overline{\sigma }=\beta \sigma$.

Figure 2

(a) Inverse width $\beta$ of the dark soliton as a function of width $\sigma$ of the potential barrier, with height $A$ ranging from $A=-1$ (top curves) to $A=+1$ (bottom curves) in steps of $0.25$. The thick (blue) curves depict the width of the numerical solution, obtained by fitting its profile to $\tanh (\beta x)$. The thin (red) lines represent the results predicted by the variational approximation, see Eq. (13). Panels (b) and (c) depict, respectively, two solutions for $(A,\sigma )=(1,1)$ and $(A,\sigma )=(-1,1.5)$, the latter case corresponding to the potential trough, rather than a barrier. The thick (blue) line depicts the numerically found steady state, while the thin (red) line represents the fitted $\tanh (\beta x)$ profile. The transverse profile of the potential barrier is depicted by the dashed (green) line. The chemical potential is fixed to be $\mu =1$.

Figure 3

The dynamics induced by the transverse modulational instability of a dark-soliton stripe in the absence of the external harmonic trap. Shown are different density snapshots at indicated moments of time. The initial condition corresponds to a stationary stripe with an added random perturbation of relative size ${10}^{-5}$. The top panel: the case of the attractive potential trough, with $A=-1$ and $\sigma =1.5$. The middle panel: no potential barrier or trough ($A=0$). The bottom panel: the case of the repulsive potential barrier, with $A=1$ and $\sigma =1.5$. Note the faster destabilization and nucleation of a larger number of vortices (see dark circular spots for later times) in the case of the attractive potential trough (the top panel), in comparison to the case when the trough is absent (the middle panel), and the complete stabilization of the dark-soliton stripe by the repulsive potential barrier in the bottom panel. The simulations in this figure were carried out in the domain of $(x,y)\in [-15,15]\times [-15,15]$.

Figure 4

The growth rate of the transverse modulational instability of dark-soliton stripes at different values of the height and width of the potential barrier $A$ and $\sigma$. The panels correspond to the increasing width (a) $\sigma =0.5$, (b) $\sigma =1$, and (c) $\sigma =1.5$, as indicated by the dashed lines in Fig. 5. Each panel displays the real part of the eigenvalue, for the potential trough or barrier with heights $A=-1$ [(blue) circles], $A=0$ [(green) squares], and $A=1$ [(red) triangles]. The computations corresponding to this figure were carried out in the domain of $(x,y)\in [-50,50]\times [-50,50]$.

Figure 5

Regions of the transverse modulational instability of the dark-soliton stripe for different parameters of the potential barrier. Depicted is the real part of the eigenvalue (the lighter color represents the stronger instability, while black corresponds to the stability) as a function of wave number $k$ and the barrier’s height $A$. Different panels correspond to increasing width of the barrier: (a) $\sigma =0.5$, (b) $\sigma =1$, and (c) $\sigma =1.5$. The black and white solid lines depict the prediction of the critical wave number as per Eq. (15). Vertical dashed lines represent the cuts depicted in Fig. 4.

Figure 6

Dynamics of the transverse modulational instability of a dark-soliton stripe in the BEC loaded into the harmonic trap of strength $\omega =0.05$. Shown are different snapshots of the density at the times indicated in the figure. The initial condition corresponds to a stationary stripe to which a random perturbation of relative size ${10}^{-5}$ was added. The top panel shows the case with the attractive potential trough of depth $A=-0.8$ and width $\sigma =1.5$. The middle panel: the situation in the absence of the channel potential ($A=0$). The bottom panel: the case of the repulsive potential barrier of height $A=0.4$ and width $\sigma =1.5$. Note that the number of nucleated vortices can be controllably varied from about $16$ for $A=-0.8$ (in the top panel) to none for $A\gtrsim 1$ (not shown here).

Figure 7

The number of nucleated vortices ($N_v$) resulting from the snaking instability of the dark-soliton stripe in the BEC loaded into the harmonic trap versus the strength ($A$) of the channel potential. The number of vortices is measured by direct count of the number of empty cores present in the density profile after the snaking instability is fully developed. The solid line depicts half of the theoretical prediction of Eq. (17) (see text). Circles: the trap’s strength corresponds to $\omega =0.05$ and depicted is the half number of the vortices ($N_v/2$). Squares: the trap’s strength corresponds to $\omega =0.1$. The width of potential channel is $\sigma =1.5$ for both data sets.