Modulation instability in helicoidal spin-orbit coupled open Bose-Einstein condensates

We introduce a vector form of the cubic complex Ginzburg-Landau equation describing the dynamics of dissipative solitons in the two-component helicoidal spin-orbit coupled open Bose-Einstein condensates (BECs), where the addition of dissipative interactions is done through coupled rate equations. Furthermore, the standard linear stability analysis is used to investigate theoretically the stability of continuous-wave (cw) solutions and to obtain an expression for the modulational instability gain spectrum. Using direct simulations of the Fourier space, we numerically investigate the dynamics of the modulational instability in the presence of helicoidal spin-orbit coupling. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the threshold for amplitude perturbations.

Figure 1

Panels ${\left(\mathrm{a}j\right)}_{j=1,2,3,4}$ show the contour plots of the MI gain in the $(K,C_1)$ plane for (a1)-(a2) $C_2=C_3=2$, (a3)-(a4) $C_2=C_3=-2$. Panels ${\left(\mathrm{b}j\right)}_{j=1,2,3,4}$ display the contour plot of the MI gain in the $(K,C_3)$ plane for (b1)-(b2) $C_1=C_2=2$ and (b3)-(b4) $C_1=C_2=-2$, with $n_0=1$, ${\alpha }_0=1$ and $C_4=0.5$.

Figure 2

MI gain in the $({\alpha }_0,{\beta }_0)$ plane for ${\left(\mathrm{a}j\right)}_{j=1,2,3,4}$ $C_3>0$ and ${\left(\mathrm{b}j\right)}_{j=1,2,3,4}$ $C_3<0$. The coefficients $C_1$ and $C_2$ are taken such that in panels (a1)-(b1) $C_1>0$, $C_2<0$, (a2)-(b2) $C_1=C_2>0$, (a3)-(b3) $C_1=C_2<0$, and (a4)-(b4) $C_1<0$, $C_2>0$, with $n_0=1$ and $K=1.5$.

Figure 3

The contour plots show the gain of MI in the $(K,{\alpha }_0)$ plane for ${\left(\mathrm{a}j\right)}_{j=1,2,3,4}$ $C_3>0$ and ${\left(\mathrm{b}j\right)}_{j=1,2,3,4}$ $C_3<0$. The coefficients $C_1$ and $C_2$ are taken such that in panels (a1)-(b1) $C_1>0$, $C_2<0$, (a2)-(b2) $C_1=C_2>0$, (a3)-(b3) $C_1=C_2<0$, and (a4)-(b4) $C_1<0$, $C_2>0$, with $n_0=1$, ${\beta }_0=1$, and $C_4=0.5$, with $n_0=1$.

Figure 4

Spatiotemporal development of MI for ${\alpha }_0=0$, and ${\beta }_0$ taking the values ${\left(\mathrm{a}j\right)}_{j=1,2}$ ${\beta }_0=0$, ${\left(\mathrm{b}j\right)}_{j=1,2}$ ${\beta }_0=1.5$, and ${\left(\mathrm{c}j\right)}_{j=1,2}$ ${\beta }_0=2.5$, with the other parameter values being $C_1=C_2=5$, $C_3=2$, $C_4=0.5$, $K=1.5$, $n_0=1$, ${\mathrm{\Delta }}_0=0.01$.

Figure 5

Spatiotemporal development of MI for ${\alpha }_0=0.5$, and ${\beta }_0$ taking the values ${\left(\mathrm{a}j\right)}_{j=1,2}$ ${\beta }_0=0$, ${\left(\mathrm{b}j\right)}_{j=1,2}$ ${\beta }_0=1.5$, and ${\left(\mathrm{c}j\right)}_{j=1,2}$ ${\beta }_0=2.5$, with the other parameter values being $C_1=C_2=5$, $C_3=2$, $C_4=0.5$, $K=1.5$, $n_0=1$, ${\mathrm{\Delta }}_0=0.01$.

Figure 6

Spatiotemporal development of MI for ${\beta }_0=-1.5$, and $C_3=2$, with ${\alpha }_0$ taking the values ${\left(\mathrm{a}j\right)}_{j=1,2}$ ${\alpha }_0=0$, ${\left(\mathrm{b}j\right)}_{j=1,2}$ ${\alpha }_0=0.25$, and ${\left(\mathrm{c}j\right)}_{j=1,2}$ ${\alpha }_0=0.5$, with the other parameter values being $C_1=C_2=5$, $C_3=2$, $C_4=0.5$, $K=1.5$, $n_0=1$, ${\mathrm{\Delta }}_0=0.01$.

Figure 7

Spatiotemporal development of MI for ${\beta }_0=-1.5$, and $C_3=-2$, with ${\alpha }_0$ taking the values ${\left(\mathrm{a}j\right)}_{j=1,2}$ ${\alpha }_0=0$, ${\left(\mathrm{b}j\right)}_{j=1,2}$ ${\alpha }_0=0.25$, and ${\left(\mathrm{c}j\right)}_{j=1,2}$ ${\alpha }_0=0.5$, with the other parameter values being $C_1=C_2=5$, $C_3=2$, $C_4=0.5$, $K=1.5$, $n_0=1$, ${\mathrm{\Delta }}_0=0.01$.