Anderson Localization or Nonlinear Waves: A Matter of Probability

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems (localization versus propagation) is under intense theoretical debate and experimental study. We resolve this dispute showing that, unlike in the common hypotheses, the answer is probabilistic rather than exclusive. At any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results generalize to higher dimensions as well.

Figure 1

Schematic dependence of the probability ${\mathcal{P}}_V$ for wave packets to stay localized (dark area) together with the complementary light area of spreading wave packets versus the wave packet volume $V$ (either initial or attained at some time $t$) for three different orders of nonlinearity $\gamma <4$, $\gamma =4$, and $\gamma >4$.

Figure 2

Second moment (see text for definition) of a spreading single-site excitation for two different disorder realizations versus time, $d=1$, $\gamma =4$, $E=1$. Dashed and dash-dotted lines with the slopes $1/3$ and $1/2$, respectively, guide the eye. Inset: Second moment of a spreading and a nonspreading single-site excitation for two different disorder realizations versus time, $d=1$, $\gamma =4$, $E=0.05$.

Figure 3

Sorted participation numbers $P_i$ (see text for definition) for $N_r=1000$ different disorder realizations $1\le i\le N_r$ at $t={10}^8,5\times {10}^8,{10}^9$ (bottom to top) for $E=0.08$ ($bl$), $E=0.05$ ($v$), $E=0.03$ ($b$), $E=0.02$ ($g$), $E=0.002$ ($o$). Inset: Single-site localized fractions (${\mathcal{P}}_{L=1}$): numerics by symbols, linear fits for FSW and for KG by solid lines.