Suppression of Spatial Chaos via Noise-Induced Growth of Arrays of Spatial Solitons

Domain walls with oscillatory tails are commonplace in models of spatially extended nonlinear optical devices. Their interaction and locking at discrete distances lead to asymptotically stable spatial disorder. We show that noise in the presence of domain walls with oscillatory tails can suppress spatial disorder by privileging highly correlated dynamical states consisting of arrays of spatial solitons.

Figure 1

Real part of a generic stationary solution $A_1^s$ showing a disordered distribution of defects. The parameters are ${\gamma }_0=0.2$, $E=4$, and $a=0.5$.

Figure 2

(a) Depiction of a typical $u(x,t)$ for the noise intensity $D=0.251$ after a transient has been eliminated. The two colors denote the sign of $u$ and dark lines are the zero-crossing points $x_k(t)$. (b) Power spectra of $u(x,t)$ for $D=0.282$ averaged over $8\times {10}^4$ time steps: ${\gamma }_0=0.2$ (solid line) and ${\gamma }_0=0.4$ (bold dashed line). The peak frequency shifts to the right on increasing ${\gamma }_0$. Shaded plot corresponds to the spectrum of $u(x,t)$ in the deterministic case and is averaged over ${10}^3$ random initial conditions: ${\gamma }_0=0.2$; other parameters are as in Fig. 1.

Figure 3

(a) Stationary solutions of (1): unstable soliton (dotted line), stable soliton (dashed line), stable double soliton (bold solid line). The inset shows a plot of the thresholds ${\mathcal{E}}_a$ (dashed line) and ${\mathcal{E}}_d$ (solid line) against ${\gamma }_0$. (b) Defect density: ${\gamma }_0=0.2$ (circles) and ${\gamma }_0=0.6$ (squares). Dashed line corresponds to ${\rho }_p$. Other parameters are $E=4$ and $a=0.5$.

Figure 4

(a) Histograms of defect zero-crossing distances. Peaks correspond to equilibrium characteristic distances $s_j$: $D=0.282$ (solid line) and $D=0.708$ (dashed line). The actual first three characteristic distances are indicated by arrows. (b) Strength of the peaks $P_0$, $P_1$, and $P_2$ on varying $D$. Other parameters are as in Fig. 1.