Figure 1

(Color online) Evolution of the atomic density according to (a) the exact solution (2), (b) TF approximation with ${\nu }_i=0.02$, (c) “Mexican hat,” and (d) uniform ${\rho }_i^2=1$ initial conditions. In (b), (c), and (d), the initial phase distribution ${\theta }_i$ is taken from Eq. (3). The initial time is $t_i=-3$. The inset in (c) shows the initial densities obtained from Eq. (2) (dashed line) and from the “Mexican hat” approximation ${\psi }_{MH}$ (solid line). In (b), (c), and (d), we used $g=0.05$.Reuse & Permissions

Figure 2

(Color online) Evolution of the atomic density starting at the lower (left column, $\sigma =-1$, $E_0=-0.9368$) and upper (right column, $\sigma =1$, $E_1=-0.7332$) edges of the first lowest band of the OL with $V=3$. As the initial condition, we used Eq. (8) with $t_i=-600$ and $\epsilon =0.05$ (panels a,b), $\epsilon =0.1$ (panels c–f). The rogue wave is observed without dissipative losses (panels a–d) and with inelastic interactions (panels e,f, where $g=0.5$). The upper insets show the respective initial shapes at $t_i$ (dashed lines) and ones at $t=0$, when rogue waves reach their maxima (solid lines) [gray and white strips correspond to half-periods of OL, when $-V\text{cos}\left(2x\right)<0$ and $-V\text{cos}\left(2x\right)>0$, respectively].Reuse & Permissions