Influence of disorder on generation and probability of extreme events in Salerno lattices

Extreme events (EEs) in nonlinear and/or disordered one-dimensional photonic lattice systems described by the Salerno model with on-site disorder are studied. The goal is to explain particular properties of these phenomena, essentially related to localization of light in the presence of nonlinear and/or nonlocal couplings in the considered systems. Combining statistical and nonlinear dynamical methods and measures developed in the framework of the theory of localization phenomena in disordered and nonlinear systems, particularities of EEs are qualitatively clarified. Findings presented here indicate that the best environment for EEs' creation are disordered near-integrable Salerno lattices. In addition, it is been shown that the leading role in the generation and dynamical properties of EEs in the considered model is played by modulation instability, i.e., by nonlinearities in the system, although EEs can be induced in linear lattices with on-site disorder too.

Figure 1

The curve $\zeta \left(d\right)$ in the log-log presentation: black curve with squares: $\mathrm{\Gamma }=0.001$; red curve with circles: $\mathrm{\Gamma }=0.07$; blue line with upper triangles: $\mathrm{\Gamma }=0.1$; dark cyan line with down triangles: $\mathrm{\Gamma }=1$; solid magenta line with diamonds: $\mathrm{\Gamma }=2$; solid orange line with left triangles: $\mathrm{\Gamma }=5$; dark yellow line with right triangles: $\gamma =0,\mu =1$; solid olive line with stars: linear case ($\gamma =0,\mu =0$); dashed blue line with empty upper triangles: $\gamma =1,\mu =0$; dashed magenta line with empty diamonds: $\gamma =2,\mu =0$; dashed orange line with empty left triangles: $\gamma =5,\mu =0$. The curves with slopes $\alpha =-1/2$ and $\alpha =-2/3$ are added as guiding lines.

Figure 2

The normalized probability $P_{ee}=P_h(h>h_{th})$ as a function of parameter $\mathrm{\Gamma }$ for two values of disorder parameter $d=0$ and $d=4$ (a). The rectangular area marked in (a) corresponding to small $\mathrm{\Gamma }$ is separately shown (b). Let us note that $\mathrm{\Gamma }=0$ corresponds to the case with only nonlocal nonlinearity (AL limit, $\gamma =0$).

Figure 3

(a) and (b) Evolution of scaled amplitudes in the linear lattice for (a) $d=0.1$ and (b) $d=2$. On both graphs, the maximum amplitude value is noted ($A_{max}$); (c) the effective average width $w_{\mathrm{eff}}$ of the excitations formed in the linear lattice as a function of time for different disorder levels.

Figure 4

Comparison of normalized height probability density (a) $P_h\left(h\right)$ and return time probability $P_r$ (normalized so that the surface below each curve is equal to 1) as a function of $r/R$ (b) for different disorder levels in the linear case ($\gamma =0,\mu =0$). The fitting curve for cases with $d>1$ is approximately $y=a{x}^b$. An example of a fitting curve is plotted for (b) $d=6$ with parameters $a=0.029,b=-2.06$, the adjusted $R^2$ [28] is 0.99989. Here $y=log\left(P_r\right),x=log(r/R)$, and the threshold value of the amplitude for the return time calculation is $q=q_2$ (Table 1).

Figure 5

(a) The evolution of scaled amplitudes with $\mathrm{\Gamma }=1$ and $d=4$, and the maximum amplitude ($A_{max}$) is marked on the graph. (b) The effective average width $w_{\mathrm{eff}}$ of the excitations formed in the lattice as a function of time for different disorder levels. (c) The normalized height probability density $P_h\left(h\right)$ on the log-log scale for several disorder values (the arrow indicates the position of the plateau).

Figure 6

Return time probability $P_r$ (normalized so that the surface below each curve is equal to 1) as a function of $r/R$ for $\mathrm{\Gamma }=1$ and threshold $q=h_{th}=2.2h_s$ for different disorder levels. An example of fitting the $P_r(r/R)$ curve for $d=6$ is given in the inset of the figure: The fitting function is $y=a{x}^b$ with $a=0.0048$ and $b=-1.36$, and the adjusted $R^2$ is 0.99228, $y=log\left(P_r\right)$, and $x=log(r/R)$.

Figure 7

(a) Evolution of scaled amplitudes with $\mathrm{\Gamma }=0.1$ for $d=0$. (b) Effective average width $w_{\mathrm{eff}}$ of excitations formed in the lattice as a function of time for different disorder levels. (c) The normalized height probability density $P_h\left(h\right)$ for several disorder values.

Figure 8

Return time probability $P_r$ (normalized so that the surface below each curve is equal to 1) as a function of $r/R$ for $\mathrm{\Gamma }=0.1$ and threshold $q=h_{th}$ for different disorder levels with a corresponding fitting curve for $d=6$. The value of fitting parameter $b$ is $-1.45$, whereas the adjusted $R^2$ is 0.99374.

Figure 9

Evolution of scaled amplitudes with $\mathrm{\Gamma }=0.0001$ for (a) $d=0$ and (b) $d=4$. (c) Effective average width $w_{\mathrm{eff}}$ of the excitations formed in the lattice as a function of time for different disorder levels. (d) The normalized height probability density $P_h\left(h\right)$ for several disorder values.

Figure 10

Return time probability (normalized so that the surface below each curve is equal to 1) as a function of $r/R$ for $\mathrm{\Gamma }=0.0001$ and threshold $q=h_{th}=2.2h_s$ for $d=0,2,4,6$. Whereas the curve for $d=0$ can be fitted by two power law curves (two allometric functions), those with $d>0$ possess the humps and cannot be fitted. The dashed lines on the graphs indicate the slope of $P_r$ curves for $d=0$ and $d=6$. The positions of the humps are denoted by arrows.