T. V. Laptyeva

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

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. Copyright c EPLA, 2010

Nonlinear waves in disordered chains: Probing the limits of chaos and spreading

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.

Anderson Localization or Nonlinear Waves: A Matter of Probability

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems (localization versus propagation) is under intense theoretical debate and experimental study. We resolve this dispute showing that, unlike in the common hypotheses, the answer is probabilistic rather than exclusive. 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 generalize to higher dimensions as well.

WAVE INTERACTIONS IN LOCALIZING MEDIA — A COIN WITH MANY FACES

A variety of heterogeneous potentials are capable of localizing linear noninteracting waves. In this work, we review different examples of heterogeneous localizing potentials which were realized in experiments. We then discuss the impact of nonlinearity induced by wave interactions, in particular, its destructive effect on the localizing properties of the heterogeneous potentials.

Subdiffusion of nonlinear waves in quasiperiodic potentials

We study the time evolution of wave packets in one-dimensional quasiperiodic lattices which localize linear waves. Nonlinearity (related to two- body interactions) has a destructive effect on localization, as observed recently for interacting atomic condensates (Lucioni et al 2011 Phys. Rev. Lett. 106 230403). We extend the analysis of the characteristics of the subdiffusive dynamics to large temporal and spatial scales. Our results for the second moment m2 consistently reveal an asymptotic m2 t 1/3 and an intermediate m2 t 1/2 law. At variance with purely random systems (Laptyeva et al 2010 Europhys. Lett. 91 30001), the fractal gap structure of the linear wave spectrum strongly favours intermediate self-trapping events. Our findings give a new dimension to the theory of wave packet spreading in localizing environments.

Subdiffusion of nonlinear waves in two-dimensional disordered lattices

We perform high-precision computational experiments on nonlinear waves in two-dimensional disordered lattices with tunable nonlinearity. While linear wave packets are trapped due to Anderson localization, nonlinear wave packets spread subdiffusively. Various speculations on the growth of the second moment as tα are tested. Using fine statistical averaging we find agreement with predictions from Flach S., Chem. Phys., 375 (2010) 548, which supports the concepts of strong and weak chaos for nonlinear wave propagation in disordered media. We extend our approach and find potentially long-lasting intermediate deviations due to a growing number of surface resonances of the wave packet.

The weak-password problem: Chaos, criticality, and encrypted p-CAPTCHAs

Vulnerabilities related to weak passwords are a pressing global economic and security issue. We report a novel, simple, and effective approach to address the weak-password problem. Building upon chaotic dynamics, criticality at phase transitions, CAPTCHA recognition, and computational round-off errors, we design an algorithm that strengthens the security of passwords. The core idea of our simple method is to split a long and secure password into two components. The first component is memorized by the user. The second component is transformed into a CAPTCHA image and then protected using the evolution of a two-dimensional dynamical system close to a phase transition, in such a way that standard brute-force attacks become ineffective. We expect our approach to have wide applications for authentication and encryption technologies.

Do nonlinear waves in random media follow nonlinear diffusion equations

Abstract Probably yes, since we find a striking similarity in the spatio-temporal evolution of nonlinear diffusion equations and wave packet spreading in generic nonlinear disordered lattices, including self-similarity and scaling. We discuss, analyze and compare nonlinear diffusion equations with compact or exponentially decaying interactions, and generalized dependences of the diffusion coefficient on the density. Our results strongly support applicability to wave packet spreading in disordered nonlinear lattices.

Quantum subdiffusion with two- and three-body interactions

Abstract We study the dynamics of a few-quantum-particle cloud in the presence of two- and three-body interactions in weakly disordered one-dimensional lattices. The interaction is dramatically enhancing the Anderson localization length ξ1 of noninteracting particles. We launch compact wave packets and show that few-body interactions lead to transient subdiffusion of wave packets, m2 ~ tα, α< 1, on length scales beyond ξ1. The subdiffusion exponent is independent of the number of particles. Two-body interactions yield α ≈ 0.5 for two and three particles, while three-body interactions decrease it to α ≈ 0.2. The tails of expanding wave packets exhibit exponential localization with a slowly decreasing exponent. We relate our results to subdiffusion in nonlinear random lattices, and to results on restricted diffusion in high-dimensional spaces like e.g. on comb lattices.

FEW PARTICLE DIFFUSION IN LOCALIZING POTENTIALS: CHAOS AND REGULARITY

В данной работе изучается динамика распространения волновых пакетов в моделях нескольких взаимодействующих квантовых частиц с разными видами пространственной модуляции. Для одной частицы или, что эквивалентно, многих невзаимодействующих частиц, известно, что в случае пространственного беспорядка все собственные состояния становятся локализованными, а в случае квазипериодической неоднородности существует порог перехода к локализации по силе неоднородности. В другом предельном случае – многих взаимодействующих частиц – задача решалась в среднеполевом приближении, в рамках нелинейного дискретного уравнения Шредингера. Здесь наблюдалось разрушение локализации за счет нелинейности, возникающего динамического хаоса. Основными наблюдаемыми свойствами были субдиффузия волновых пакетов, их самоподобие в асимптотическом пределе, зависимость показателя субдиффузии от порядка нелинейности. В настоящей работе показано, что эти свойства обнаруживаются и для нескольких квантовых частиц в решетке с беспорядком, при том, что условия среднеполевого приближения не выполнены. Тем не менее квантовый хаос обеспечивает подобную динамику. При этом показатель субдиффузии уменьшается при увеличении порядка взаимодействия, так же как и в нелинейных уравнениях. В случае квазипериодического потенциала в модели нескольких взаимодействующих частиц наблюдается квантовая регулярная динамика и почти баллистическое распространение волновых пакетов. При этом малая добавка беспорядка разрушает квантовую регулярную динамику.