Figure 1

(Color) Results of simulation of Eq. (1). (a) and (b) show stationary patterns, (c)–(e) show a turbulent one. In (a)–(c) we show profiles of $\left|A\right|$ (dotted red line), $\left|\overline{A}\right|$ (solid green line), and $\left|B\right|$ (long-dashed blue line). The system size $L$ is specified in each frame. In (b), thin pink solid line (largely occluded by the green one) shows a snapshot from simulation with specially prepared initial state, see text. In panels (d) and (e), showing space-time plots of $\left|\overline{A}\right|$ and $\left|B\right|$, respectively, red color (darker in b/w print) encodes values close to 1, blueish color (brighter in b/w print) encodes values close to 0. In (d) and (e), the time span of simulations is 5000 dimensionless time units and $L=7.5$.Reuse & Permissions

Figure 2

The real part of the stability eigenvalues $\lambda$ of $z_{\text{soq}}$ in dependence on $k$ (the wave number of perturbation). $k_c\approx 2\pi /5.10$ denotes the critical wave number.Reuse & Permissions

Figure 3

(Color online) Stationary profiles of $r$ for different system sizes $L$. On the left, $r=\left|z\right|$ (solid red line) and $b$ (dashed blue line) in comparison to $\Delta$ (horizontal short-dashed line) are depicted, the corresponding stability eigenvalues of the Jacobian matrix are on the right.Reuse & Permissions

Figure 4

(Color) Space-time plots of the coarse-grained order parameter $\left|\overline{z}\right|$ (a) and of $b$ (b) obtained by numerical integration of Eq. (5) with $L=7.5$. Red color (darker in b/w print) represents larger values close to one, bluish (brighter in b/w print) represents smaller values close to zero. We show simulation results from 1500 to 3000 dimensionless time units. The discretization in space $x$ was done on 1024 sites. (c) Final snapshot of integration from (a) and (b): Solid red line shows non-coarse-grained $r=\left|z\right|$, green short-dashed line shows the magnitude of the coarse-grained $\overline{z}$, and long-dashed blue line represents profile of $b$.Reuse & Permissions