Chaotic wave packet spreading in two-dimensional disordered nonlinear lattices

TL;DR

This study investigates wave packet spreading in two-dimensional disordered nonlinear lattices, revealing power-law growth, persistent but weakening chaos, and localized chaotic seeds that lead to thermalization, supported by extensive numerical simulations.

Contribution

It provides the first detailed numerical analysis of wave packet spreading and chaos decay in 2D disordered nonlinear lattices, confirming theoretical predictions and proposing a universal scaling law.

Findings

Wave packet's second moment grows as t^{1/5} (weak chaos) and t^{1/3} (strong chaos).

Finite time Lyapunov exponent decays as t^{-0.37} (weak chaos) and t^{-0.46} (strong chaos).

Localized chaotic seeds wander, inducing thermalization of the wave packet.

Abstract

We reveal the generic characteristics of wave packet delocalization in two-dimensional nonlinear disordered lattices by performing extensive numerical simulations in two basic disordered models: the Klein-Gordon system and the discrete nonlinear Schr\"{o}dinger equation. We find that in both models (a) the wave packet's second moment asymptotically evolves as $t^{a_m}$ with $a_m \approx 1/5$ ($1/3$) for the weak (strong) chaos dynamical regime, in agreement with previous theoretical predictions [S.~Flach, Chem.~Phys.~{\bf 375}, 548 (2010)], (b) chaos persists, but its strength decreases in time $t$ since the finite time maximum Lyapunov exponent $\Lambda$ decays as $\Lambda \propto t^{\alpha_{\Lambda}}$, with $\alpha_{\Lambda} \approx -0.37$ ($-0.46$) for the weak (strong) chaos case, and (c) the deviation vector distributions show the wandering of localized chaotic seeds in the lattice's excited part, which induces the wave packet's thermalization. We also propose a dimension-independent scaling between the wave packet's spreading and chaoticity, which allows the prediction of the obtained $\alpha_{\Lambda}$ values.