The crossover from strong to weak chaos for nonlinear waves in disordered systems

Abstract

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.

Destruction of Anderson localization by nonlinearity in kicked rotator at different effective dimensions

“…The later studies indeed confirmed that in DANSE the exponent α is approximately the same as in the KNR [4,7,9]. The examples of probability spreading in the FMKNR at d = 1, 2, 3, 4 in the model M 1 are shown in Fig.…”

“…It is interesting to note that the exponent α = 1/2 corresponds to a so-called regime of "strong chaos" [9,10,33]. Indeed, the numerical simulations performed in [9,10] introduced a randomization of phases of linear eigenmodes after a fixed time scale τ ∼ 1 showing numerically that in such a case α = 1/2.…”

“…Indeed, the numerical simulations performed in [9,10] introduced a randomization of phases of linear eigenmodes after a fixed time scale τ ∼ 1 showing numerically that in such a case α = 1/2. This relation can be understood on a basis of simple estimates in a following way: the equations of amplitudes of linear modes C m in the interaction representation have a form i∂C/∂t ∼ βC 3 [2,4,7].…”

Complex statistics and diffusion in nonlinear disordered particle chains

“…Understanding the effect of nonlinearity on the localization properties of wave packets in disordered systems has attracted the attention of many researchers to date. 9,11,13,14,[19][20][21][22][23][24][25][26][27][28] Most of these studies consider the evolution of an initially localized wave packet and show that it spreads subdiffusively for moderate nonlinearities, while for strong enough nonlinearities a substantial part of it is self-trapped. In such works, one typically analyzes normalized norm or energy distributions z l E l = P N i¼1 E i !…”

“…Recently, it was conjectured 15,16 that chaotically spreading wave packets will asymptotically approach KAM torus-like structures in phase-space, while numerical simulations typically do not show any sign of slowing down of the spreading behavior. 13,14,29 Nevertheless, for particular disordered nonlinear models some numerical indications of a possible slowing down of spreading have been reported. 30,31 Thus, we decided to implement the ideas of Tsallis statistics to shed new light on this problem.…”