Universal Spreading of Wave Packets in Disordered Nonlinear Systems

In the absence of nonlinearity all eigenmodes of a chain with disorder are spatially localized (Anderson localization). The width of the eigenvalue spectrum and the average eigenvalue spacing inside the localization volume set two frequency scales. An initially localized wave packet spreads in the presence of nonlinearity. Nonlinearity introduces frequency shifts, which define three different evolution outcomes: (i) localization as a transient, with subsequent subdiffusion; (ii) the absence of the transient and immediate subdiffusion; (iii) self-trapping of a part of the packet and subdiffusion of the remainder. The subdiffusive spreading is due to a finite number of packet modes being resonant. This number does not change on average and depends only on the disorder strength. Spreading is due to corresponding weak chaos inside the packet, which slowly heats the cold exterior. The second moment of the packet grows as $t^{\alpha }$. We find $\alpha =\frac{1}{3}$.

Figure 1

$m_2$ and $P$ vs time in log-log plots. Left plots: The DNLS model with $W=4$, $\beta =0.1$, 1, 4.5 [(b), blue; (g), green; (r) red]. Right plots: The KG chain with $W=4$ and initial energy $E=0.05$, 0.4, 1.5 [(b), blue; (g), green; (r) red]. The disorder realization is kept unchanged for each of the models. Dashed straight lines are guides to the eye for exponents $1/3$ ($m_2$) and $1/6$ ($P$), respectively.

Figure 2

$m_2$ (in arbitrary units) vs time in log-log plots in regime (ii) and different disorder strengths. Lower set of curves: plain integration (without dephasing). Upper set of curves: integration with dephasing of NMs (see text). Dashed straight lines with exponents $1/3$ (no dephasing) and $1/2$ (dephasing) are guides to the eye. Left panel: The DNLS model, $W=4$, $\beta =3$ (blue); $W=7$, $\beta =4$ (green); $W=10$, $\beta =6$ (red). Right panel: The KG chain, $W=10$, $E=0.25$ (blue), $W=7$, $E=0.3$ (red), $W=4$, $E=0.4$ (green). The curves are shifted vertically in order to give maximum overlap within each group.

Figure 3

Left plot: Probability densities $\mathcal{W}(x)$ of NMs being resonant (see text for details). $W=4$, 7, 10 (from top to bottom). Right plot: Probability densities $\mathcal{P}(d)$ of peak distances between resonant NM pairs for $W=4$; (o) orange, $x_c=0.1$; (b) black, $x_c=0.01$ (see text for details). Inset: The eigenvectors of a resonant pair of double-peaked states (solid and dashed lines).