Destruction of Anderson Localization by a Weak Nonlinearity

We study numerically the spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schrödinger lattice with disorder. We demonstrate that above a certain critical strength of nonlinearity the Anderson localization is destroyed and an unlimited subdiffusive spreading of the field along the lattice occurs. The second moment grows with time $\propto t^{\alpha }$, with the exponent $\alpha$ being in the range 0.3–0.4. For small nonlinearities the distribution remains localized in a way similar to the linear case.

Figure 1

The dependence of the second moment $\sigma$ of the probability distribution $w_n$ on time $t$ for two disorder strengths $W=2$ (top full red/gray curve) and $W=4$ (bottom full black/blue curve) for $\beta =1$; dotted curves of same color show data at $\beta =0$ (for large $t$ only the average value is shown by a horizontal line). The values of ${log}_{10}\sigma$ are averaged over 8 disorder realizations and over time intervals $\Delta ({log}_{10}t)\approx 0.1$. Dashed lines show the numerical fits ${log}_{10}\sigma =\alpha {log}_{10}t+\eta$ with $\alpha =0.344\pm 0.003$, $\eta =1.76\pm 0.02$ (done for $3\le {log}_{10}t\le 8$ for $W=2$) and $\alpha =0.306\pm 0.002$, $\eta =0.94\pm 0.01$ (done for $2\le {log}_{10}t\le 8$ for $W=4$). The full straight line shows the slope $\alpha =0.4$. Here and below the logarithms are decimal.

Figure 2

Probability distribution $w_n$ over lattice sites $n$ at $W=4$ for $\beta =1$, $t={10}^8$ (top blue/solid curve) and $t={10}^5$ (middle red/gray curve); $\beta =0$, $t={10}^5$ (bottom black curve; the order of the curves is given at $n=500$). At $\beta =0$ a fit $\ln w_n=-(\gamma |n|+\chi$) gives $\gamma \approx 0.3$, $\chi \approx 4$. The values of ${log}_{10}{w}_n$ are averaged over the same disorder realizations as in Fig. 1.

Figure 3

Same as in Fig. 2 but with $W=2$. At $\beta =0$ a fit $\ln w_n=-(\gamma |n|+\chi )$ gives $\gamma \approx 0.06$, $\chi \approx -3$. The values of $\ln w_n$ are averaged over the same disorder realizations as in Fig. 1.

Figure 4

Dependence of the second moment $\sigma$ of probability distribution $w_n$ on time $t$ for different strengths of nonlinearity $\beta =1$, 0.1, 0.03, 0 at $W=4$ (curves from top to bottom at ${log}_{10}t=4.5$, respectively). Data are shown for one particular disorder realization.

Figure 5

Probability distribution $w_n$ over lattice sites $n$ for the same disorder realization as in Fig. 4, for $W=4$, $t={10}^8$ and for different nonlinearities $\beta =1$, 0.1, 0.03, 0 (from top to bottom; for clarity 3 bottom curves have additional vertical shifts $-10$, $-20$, and $-30$ compared to the top one).