We observe a crossover from strong to weak chaos in the spatiotemporal evolution of multiple site excitations within disordered chains with cubic nonlinearity. Recent studies have shown that Anderson localization is destroyed, and the wave packet spreading is characterized by an asymptotic divergence of the second moment m2 in time (as t 1/3 ), due to weak chaos. In the present paper, we observe the existence of a qualitatively new dynamical regime of strong chaos, in which the second moment spreads even faster (as t 1/2 ), with a crossover to the asymptotic law of weak chaos at larger times. We analyze the pecularities of these spreading regimes and perform extensive numerical simulations over large times with ensemble averaging. A technique of local derivatives on logarithmic scales is developed in order to quantitatively visualize the slow crossover processes.

We probe the limits of nonlinear wave spreading in disordered chains which are known to localize linear waves. We particularly extend recent studies on the regimes of strong and weak chaos during subdiffusive spreading of wave packets [Europhys. Lett. 91, 30001 (2010)] and consider strong disorder, which favors Anderson localization. We probe the limit of infinite disorder strength and study Fröhlich-Spencer-Wayne models. We find that the assumption of chaotic wave packet dynamics and its impact on spreading is in accord with all studied cases. Spreading appears to be asymptotic, without any observable slowing down. We also consider chains with spatially inhomogeneous nonlinearity, which give further support to our findings and conclusions.

To understand typical dynamics of an open quantum system in continuous time, we introduce an ensemble of random Lindblad operators, which generate Markovian completely positive evolution in the space of density matrices. Spectral properties of these operators, including the shape of the spectrum in the complex plane, are evaluated by using methods of free probabilities and explained with non-Hermitian random matrix models. We also demonstrate universality of the spectral features. The notion of ensemble of random generators of Markovian qauntum evolution constitutes a step towards categorization of dissipative quantum chaos.

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems is under intense theoretical debate and experimental study. We resolve this dispute showing that 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 are generalized to higher dimensions as well. [6,7].The pioneering experiments [8] demonstrate both the principal possibility to study this effect, but also the severe length and time scale limitations which restrict a quantitative analysis. Theoretical studies become therefore central here. Two faces of Anderson localization are usually probed: the absence of the wavepacket spreading [11][12][13][14][15][16][17][18][19][20] (the focus of this paper) and vanishing conductivity [9,10]. Wave packets, for not too weak nonlinearities, have been found to spread subdiffusively, disobeying AL at least over many time scales [11][12][13], and remarkable progress in studying the regimes and characteristics of wave packet spreading has been achieved [16][17][18][19][20].Still, the original questions are open. It remains debated, whether the observed spreading will continue infinitely or slow down and even stop to restore AL, once the wave packet densities become substantially small, as conjectured in Ref. [14,15,24]. It is unknown, whether there exists a lower bound on the nonlinearity strength, beyond which wave packets do obey AL, and therefore do not spread at all; or are some time scales diverging in this limit? Unavoidable finite size, time, energy, and precision limitations will make the most advanced numerical results not entirely convincing. To achieve a breakthrough, a novel approach is in need.The probabilistic description of AL in terms of the measure of localized regular trajectories in phase space (periodic orbits and tori) appears a promising solution for nonlinear disordered systems. Indeed, the mere assumption that a wave packet is launched on a chaotic trajectory leads to the conclusion that chaos remains forever: Arnol'd conjecture, unproved but widely accepted, states the uniqueness of the chaotic region in phase space [21]. If initial conditions are chaotic, then the trajectory will be unbounded in phase space, corresponding to unlimited spreading of a wavepacket. If they belong to a localized periodic orbit or torus, then the spreading is absent.The progress has been quite limited up to now due to the difficulty of the task. Persistence of tori has been proved for a special class of infinite systems [22] and for finite tori dimensionality [23] only. The non-zero probability of existence of infinite-dimensional tori in generic case (for small enough ener...

Abstract. We review recent progress in the dynamics of nonlinear lattice waves in heterogeneous media, which enforce complete wave localization in the linear wave equation limit, especially Anderson localization for random potentials, and AubryAndre localization for quasiperiodic potentials. Additional nonlinear terms in the wave equations can either preserve the phase-coherent localization of waves, or destroy it through nonintegrability and deterministic chaos. Spreading wave packets are observed to show universal features in their dynamics which are related to properties of nonlinear diffusion equations.