Absence of Wave Packet Diffusion in Disordered Nonlinear Systems

We study the spreading of an initially localized wave packet in two nonlinear chains (discrete nonlinear Schrödinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverges with time. We find that the participation number of a wave packet does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times we find a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wave packet, we rule out the possibility of slow energy diffusion. The dynamical state could approach a quasiperiodic solution (Kolmogorov-Arnold-Moser torus) in the long time limit.

Figure 1

KG: energy distribution at $t=6\times {10}^7$ (black solid line) and $t=1.2\times {10}^8$ (red dashed line) in AS. Initial single site excitation with energy $E=1$, $V=0.25$. Insets: profiles of the strongest excited AMs in real space.

Figure 2

Same as in Fig. 1 but on a logarithmic scale. Top panel: KG, $t=1.2\times {10}^8$, AS. Bottom panel: DNLS, $W=4$, $t=1.2\times {10}^8$, real space.

Figure 3

$P$ and $m_2$ vs time, on logarithmic scale. Parameters as in Fig. 2. Top panel: KG, AS. Bottom panel: DNLS, real space.

Figure 4

$I(\omega )$ for KG and DNLS. Parameters as in Fig. 3. Inset: magnification of the main peak.