Spreading for the generalized nonlinear Schrödinger equation with disorder

Veksler, Hagar; Krivolapov, Yevgeny; Fishman, Shmuel

doi:10.1103/physreve.80.037201

Cited by 59 publications

(54 citation statements)

References 20 publications

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“…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 !…”

Section: A the Disordered Quartic Klein-gordon Modelmentioning

confidence: 99%

“…In particular, for single-site excitations the wave packet's spreading leads to an increase of the second moment according to m 2 $ t 1=3 , both in the diffusive as well as the self-trapping case. 9,11,19,21 Currently, a greatly debatable problem is the long time behavior of wave packet spreading in disordered nonlinear lattices. 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.…”

Section: A the Disordered Quartic Klein-gordon Modelmentioning

confidence: 99%

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Complex statistics and diffusion in nonlinear disordered particle chains

Antonopoulos

Bountis

Skokos

et al. 2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We investigate dynamically and statistically diffusive motion in a Klein-Gordon particle chain in the presence of disorder. In particular, we examine a low energy (subdiffusive) and a higher energy (self-trapping) case and verify that subdiffusive spreading is always observed. We then carry out a statistical analysis of the motion in both cases in the sense of the Central Limit Theorem and present evidence of different chaos behaviors, for various groups of particles. Integrating the equations of motion for times as long as 10 9 , our probability distribution functions always tend to Gaussians and show that the dynamics does not relax onto a quasi-periodic KAM torus and that diffusion continues to spread chaotically for arbitrarily long times.

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Section: A the Disordered Quartic Klein-gordon Modelmentioning

confidence: 99%

Section: A the Disordered Quartic Klein-gordon Modelmentioning

confidence: 99%

Complex statistics and diffusion in nonlinear disordered particle chains

Antonopoulos

Bountis

Skokos

et al. 2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

show abstract

“…in different lattice models [8]. The interplay of these two localization mechanisms, nonlinearity and disorder, has been studied extensively in the recent years [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. In most of these studies, an initially localized wavepacket was shown to lead to delocalization and a sub-diffusive spreading of the energy, for sufficiently large nonlinearities.…”

Section: Introductionmentioning

confidence: 99%

Energy transport in one-dimensional disordered granular solids

2016

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We investigate the energy transport in one-dimensional disordered granular solids by extensive numerical simulations. In particular, we consider the case of a polydisperse granular chain composed of spherical beads of the same material and with radii taken from a random distribution. We start by examining the linear case, in which it is known that the energy transport strongly depends on the type of initial conditions. Thus, we consider two sets of initial conditions: i) an initial displacement and ii) an initial momentum excitation of a single bead. After establishing the regime of sufficiently strong disorder, we focus our studies on the role of nonlinearity for both sets of initial conditions. By increasing the initial excitation amplitudes we are able to identify three distinct dynamical regimes with different energy transport properties: a near linear, a weakly nonlinear and a highly nonlinear regime. Although energy spreading is found to be increasing for higher nonlinearities, in the weakly nonlinear regime no clear asymptotic behavior of the spreading is found. In this regime, we additionally find that energy, initially trapped in a localized region, can be eventually detrapped and this has a direct influence on the fluctuations of the energy spreading. We also demonstrate that in the highly nonlinear regime, the differences in energy transport between the two sets of initial conditions vanish. Actually, in this regime the energy is almost ballistically transported through shock-like excitations.

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“…An important aspect of our system is the interplay between disorder and nonlinearity, which has been studied extensively in recent years [15][16][17][18][19][20][21][22][23][24][25][26][27][28], [29,Sec. 7.4].…”

mentioning

confidence: 99%

Chaoticity without thermalisation in disordered lattices

Tieleman

Skokos

Lazarides

2014

EPL

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We study chaoticity and thermalization in Bose-Einstein condensates in disordered lattices, described by the discrete nonlinear Schrödinger equation (DNLS). A symplectic integration method allows us to accurately obtain both the full phase space trajectories and their maximum Lyapunov exponents (mLEs), which characterize their chaoticity. We find that disorder destroys ergodicity by breaking up phase space into subsystems that are effectively disjoint on experimentally relevant timescales, even though energetically, classical localisation cannot occur. This leads us to conclude that the mLE is a very poor ergodicity indicator, since it is not sensitive to the trajectory being confined to a subregion of phase space. The eventual thermalization of a BEC in a disordered lattice cannot be predicted based only on the chaoticity of its phase space trajectory.Introduction In this Letter, we bring together the topics of disorder, nonlinearity, chaos, ergodicity, and Bose-Einstein condensation (BEC) in optical lattices. Inspired by the realisation of disordered optical potentials in experiments with ultracold atomic gases [1, 2], we explore the question of thermalisation in such systems. Previous theoretical works have, amongst others, addressed the topics of Bose and Anderson glasses [3], Anderson localisation [4,5], and Lifshits glasses [6]. In [6], various regimes of interaction strengths were investigated, and it was found that for sufficiently strong interactions, a disordered BEC is expected. We will focus on this regime, and study the Bose-Hubbard model [7,8], which describes a bosonic gas in a lattice, in the mean-field approximation. The resulting model can be obtained by discretising the Gross-Pitaevskii equation, and is also known as the discrete nonlinear Schrödinger equation (DNLS) [9]. In this Letter, we pose and answer the following question: what is the effect of disorder on thermalisation and ergodicity in the mean-field Bose-Hubbard model / DNLS?The connection between chaoticity and thermalisation was recently discussed in the disorder-free case [10]. Intuitively, one would expect chaotic trajectories to thermalise, since unlike regular ones, they are not confined to the neighborhood of stable periodic orbits. Not being confined, the expectation is that they are able to cover the available phase space, and that the system is therefore well-described by the microcanonical ensemble, since only the energy and particle number are conserved. We show that this expectation is not correct, by explicitly demonstrating the absence of equiprobability of states on the microcanonical shell. Since the energies involved are higher than the disorder potential, classical localisation cannot be responsible for this effect.The most commonly employed method of chaos detection, which quantifies the sensitive dependence on initial conditions, is the evaluation of the maximum Lyapunov exponent (mLE) [11][12][13][14]. In [10], a positive mLE, which

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Spreading for the generalized nonlinear Schrödinger equation with disorder

Cited by 59 publications

References 20 publications

Complex statistics and diffusion in nonlinear disordered particle chains

Complex statistics and diffusion in nonlinear disordered particle chains

Energy transport in one-dimensional disordered granular solids

Chaoticity without thermalisation in disordered lattices

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