Absence of wave packet diffusion in disordered nonlinear systems

Numerical methods for generation and characterization of disordered aperiodic photonic lattices

We introduce numerical modeling of two different methods for the deterministic randomization of two-dimensional aperiodic photonic lattices based on Mathieu beams, optically induced in a photorefractive media. For both methods we compare light transport and localization in such lattices along the propagation, for various disorder strengths. A disorder-enhanced light transport is observed for all disorder strengths. With increasing disorder strength light transport becomes diffusive-like and with further increase of disorder strength the Anderson localization is observed. This trend is more noticeable for longer propagation distances. The influence of input lattice intensity on the localization effects is studied. The difference in light transport between two randomization methods is attributed to various levels of input lattice intensity. We observe more pronounced localization for one of the methods. Localization lengths differ along different directions, due to the crystal and lattice anisotropy. We analyze localization effects comparing uniform and on-site probe beam excitation positions and different probe beam widths.

Nonlinear Schrödinger waves in a disordered potential: Branched flow, spectrum diffusion, and rogue waves

The problem of nonlinear Schrödinger (NLS) waves in a disordered potential arises in many physical occasions, such as hydrodynamics, optics, and cold atoms. It provides a paradigm for studying the interaction between nonlinearity and random effect, but the current results are far from perfect. In this paper, we systematically simulate the turbulent waves for the focusing NLS equation with dynamical (time-dependent) random potentials, where the enhanced branching structures evolve into branched soliton flows as the nonlinearity increases. In this process, the occurrence of rogue waves for short times results from the interplay of linear random focusing and modulation instability. While the nonlinear spectral analysis reveals that for longer times, it is due to a self-organization of larger solitons competing with breakup of intermediate solitons. On the other hand, we found that the strong nonlinearity can significantly increase the width of the linear (Fourier) spectrum for several time scales, but its spreading rate becomes suppressed, which has a dependence on the correlation length of the potential. We hope that our findings will facilitate a deeper understanding of the nonlinear waves interacting with disordered media.

Deconfinement of classical Yang-Mills color fields in a disorder potential

We study numerically and analytically the behavior of classical Yang-Mills color fields in a random one-dimensional potential described by the Anderson model with disorder. Above a certain threshold, the nonlinear interactions of Yang-Mills fields lead to chaos and deconfinement of color wavepackets with their subdiffusive spreading in space. The algebraic exponent of the second moment growth in time is found to be in the range of 0.3-0.4. Below the threshold, color wavepackets remain confined even if a very slow spreading at very long times is not excluded due to subtle nonlinear effects and the Arnold diffusion for the case when initially color packets are located in close vicinity. In the case of large initial separation of color wavepackets, they remain well confined and localized in space. We also present the comparison with the behavior of the one-component field model of discrete Anderson nonlinear Schrödinger equation with disorder.

Effects of interplay between disorder and anharmonicity on heat conduction

Heat conduction through a disordered Fermi-Pasta-Ulam-β (DFPU-β) chain is studied. The presence of disorder makes the heat current behave significantly different from that of the ordered Fermi-Pasta-Ulam-β (FPU-β) chain. Thanks to the interplay between disorder and anharmonicity, a nonmonotonic-monotonic transition occurs when the disorder strength increases. That is, a peak for the heat current emerges for weak disorder; however, monotonic increasing of the heat current shows up for strong disorder. This can be understood based on the competition between two effects of anharmonicity on phonons, namely, delocalization and phonon-phonon scattering, which is shown by the spectral decomposition of heat current.

Transport, correlations, and chaos in a classical disordered anharmonic chain

We explore transport properties in a disordered nonlinear chain of classical harmonic oscillators, and thereby identify a regime exhibiting behavior analogous to that seen in quantum many-body-localized systems. Through extensive numerical simulations of this system connected at its ends to heat baths at different temperatures, we computed the heat current and the temperature profile in the nonequilibrium steady state as a function of system size N, disorder strength Δ, and temperature T. The conductivity κ_{N}, obtained for finite length (N), saturates to a value κ_{∞}>0 in the large N limit, for all values of disorder strength Δ and temperature T>0. We show evidence that for any Δ>0 the conductivity goes to zero faster than any power of T in the (T/Δ)→0 limit, and find that the form κ_{∞}∼e^{-B|ln(CΔ/T)|^{3}} fits our data. This form has earlier been suggested by a theory based on the dynamics of multioscillator chaotic islands. The finite-size effect can be κ_{N}<κ_{∞} due to boundary resistance when the bulk conductivity is high (the weak disorder case), or κ_{N}>κ_{∞} due to direct bath-to-bath coupling through bulk localized modes when the bulk is weakly conducting (the strong disorder case). We also present results on equilibrium dynamical correlation functions and on the role of chaos on transport properties. Finally, we explore the differences in the growth and propagation of chaos in the weak and strong chaos regimes by studying the classical version of the out-of-time-ordered commutator.

Suppressed-to-enhanced thermal transport in a Fermi-Pasta-Ulam superlattice: Mediation roles of solitons and phonons

Managing thermal transport in nanostructured materials possesses both theoretical and application value in thermoelectric and microelectronics design. Though a suppressed thermal conductivity could be easily achieved through disorder-induced phonon scattering in a superlattice, it is challenging to enhance thermal transport in a periodically designed lattice. In this paper, we show the possibility of mediating thermal conductivity from a suppressed to an enhanced value in a Fermi-Pasta-Ulam β superlattice with periodic cells of arithmetically increased nonlinearity. When the cell length is increased, thermal conductivity in the superlattice crosses over a suppressed region into an enhanced one and it is even higher than in a homogeneous lattice with the same nonlinearity strength. The mediation originates from the long-lived nonlinear wave packets as solitons across the disorder-induced interface between cells of the superlattice, while at the same time the normal vibrational modes as phonons are suppressed. Our result shows a promising strategy to manipulate thermal transport over a wide range in a superlattice with strong nonlinearity.

Chaotic wave-packet spreading in two-dimensional disordered nonlinear lattices

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 Schrödinger equation. We find that in both models (a) the wave packet's second moment asymptotically evolves as t^{a_{m}} with a_{m}≈1/5 (1/3) for the weak (strong) chaos dynamical regime, in agreement with previous theoretical predictions [S. Flach, Chem. Phys. 375, 548 (2010)CMPHC20301-010410.1016/j.chemphys.2010.02.022]; (b) chaos persists, but its strength decreases in time t since the finite-time maximum Lyapunov exponent Λ decays as Λ∝t^{α_{Λ}}, with α_{Λ}≈-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 α_{Λ} values.

Composition Methods for Dynamical Systems Separable into Three Parts

New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a way that each part is explicitly solvable. The methods are obtained by applying different optimization criteria and preserve geometric properties of the continuous problem by construction. Different numerical examples exhibit their improved performance with respect to previous splitting methods in the literature.

Seltrapping in flat band lattices with nonlinear disorder

We study the transport properties of an initially localized excitation in several flat band lattices, in the presence of nonlinear (Kerr) disorder. In the weak nonlinearity regime, the dynamics is controlled by the degeneracy of the bands leading to a linear form of selftrapping. In the strong nonlinearity regime, the dynamics of the excitations depends strongly on the local environment around the initial excitation site that leads to a highly fluctuating selfrapping profile. For a binary nonlinear disorder, it is shown that the spreading of the flat band fundamental mode, is completely inhibited for a finite fraction of all cases. This fraction corresponds to the fraction of times the same value of (random) nonlinearity is assigned to all sites of the fundamental mode.

Transient diffusion and two-regime localization of discrete breatherlike excitations in nonlinear Schrödinger lattice with disorder

We systematically simulate and analyze the motion of discrete breatherlike excitations (DBEs) in the nonlinear Schrödinger lattice with random potentials. A universal transient diffusion of the DBEs is observed for short timescales (t≲10^{3}). For longer timescales (t up to 10^{5}), the DBEs become localized. Such localization, depending on the DBE powers, has two different regimes: Regime I is the Anderson-like localization induced by the disorder, while Regime II is an enhanced localization attributed to both the disorder and discreteness. Our study is expected to shed light on understanding the interplay between disorder and strong nonlinearity, from the diffusive transport and localization properties of nonlinear localized excitations in random media.

Chaos and Anderson localization in disordered classical chains: Hertzian versus Fermi-Pasta-Ulam-Tsingou models

We numerically investigate the dynamics of strongly disordered 1D lattices under single-particle displacements, using both the Hertzian model, describing a granular chain, and the α+β Fermi-Pasta-Ulam-Tsingou model (FPUT). The most profound difference between the two systems is the discontinuous nonlinearity of the granular chain appearing whenever neighboring particles are detached. We therefore sought to unravel the role of these discontinuities in the destruction of Anderson localization and their influence on the system's chaotic dynamics. Our results show that the dynamics of both models can be characterized by: (i) localization with no chaos; (ii) localization and chaos; (iii) spreading of energy, chaos, and equipartition. The discontinuous nonlinearity of the Hertzian model is found to trigger energy spreading at lower energies. More importantly, a transition from Anderson localization to energy equipartition is found for the Hertzian chain and is associated with the "propagation" of the discontinuous nonlinearity in the chain. On the contrary, the FPUT chain exhibits an alternate behavior between localized and delocalized chaotic behavior which is strongly dependent on the initial energy excitation.

Parity-Induced Thermalization Gap in Disordered Ring Lattices

The gaps separating two different states widely exist in various physical systems: from the electrons in periodic lattices to the analogs in photonic, phononic, plasmonic systems, and even quasicrystals. Recently, a thermalization gap, an inaccessible range of photon statistics, was proposed for light in disordered structures [Nat. Phys. 11, 930 (2015)NPAHAX1745-247310.1038/nphys3482], which is intrinsically induced by the disorder-immune chiral symmetry and can be reflected by the photon statistics. The lattice topology was further identified as a decisive role in determining the photon statistics when the chiral symmetry is satisfied. Being very distinct from one-dimensional lattices, the photon statistics in ring lattices are dictated by its parity, i.e., odd or even sited. Here, we for the first time experimentally observe a parity-induced thermalization gap in strongly disordered ring photonic structures. In a limited scale, though the light tends to be localized, we are still able to find clear evidence of the parity-dependent disorder-immune chiral symmetry and the resulting thermalization gap by measuring photon statistics, while strong disorder-induced Anderson localization overwhelms such a phenomenon in larger-scale structures. Our results shed new light on the relation among symmetry, disorder, and localization, and may inspire new resources and artificial devices for information processing and quantum control on a photonic chip.

Chaos and Anderson-like localization in polydisperse granular chains

We investigate the dynamics of highly polydisperse finite granular chains. From the spatiospectral properties of small vibrations, we identify which particular single-particle displacements lead to energy localization. Then, we address a fundamental question: Do granular nonlinearities and the resulting chaotic dynamics destroy this energy localization? Our numerical simulations show that for moderate nonlinearities, the overall system behaves chaotically, and spreading of energy occurs. However, long-lasting chaotic energy localization is observed for particular single-particle excitations in the presence of the nonsmooth nonlinearities. On the other hand, for sufficiently strong nonlinearities, the granular chain reaches energy equipartition. In this case, an equilibrium chaotic state is reached independent of the initial position excitation.

Direct measurement of superdiffusive energy transport in disordered granular chains

Energy transport properties in heterogeneous materials have attracted scientific interest for more than half of a century, and they continue to offer fundamental and rich questions. One of the outstanding challenges is to extend Anderson theory for uncorrelated and fully disordered lattices in condensed-matter systems to physical settings in which additional effects compete with disorder. Here we present the first systematic experimental study of energy transport and localization properties in simultaneously disordered and nonlinear granular crystals. In line with prior theoretical studies, we observe in our experiments that disorder and nonlinearity—which individually favor energy localization—can effectively cancel each other out, resulting in the destruction of wave localization. We also show that the combined effect of disorder and nonlinearity can enable manipulation of energy transport speed in granular crystals. Specifically, we experimentally demonstrate superdiffusive transport. Furthermore, our numerical computations suggest that subdiffusive transport should be attainable by controlling the strength of the system’s external precompression force.

Wave propagation is often nonlinear in character, yet the interplay between disorder and nonlinearity remains elusive. Kim et al. use experiments and corroborating numerical simulations to investigate this phenomenon and demonstrate superdiffusive energy transport in disordered granular chains.

Solitons in a modified discrete nonlinear Schrödinger equation

We study the bulk and surface nonlinear modes of a modified one-dimensional discrete nonlinear Schrödinger (mDNLS) equation. A linear and a modulational stability analysis of the lowest-order modes is carried out. While for the fundamental bulk mode there is no power threshold, the fundamental surface mode needs a minimum power level to exist. Examination of the time evolution of discrete solitons in the limit of strongly localized modes, suggests ways to manage the Peierls-Nabarro barrier, facilitating in this way a degree of soliton steering. The long-time propagation of an initially localized excitation shows that, at long evolution times, nonlinear effects become negligible and as a result, the propagation becomes ballistic. The qualitative similarity of the results for the mDNLS to the ones obtained for the standard DNLS, suggests that this kind of discrete soliton is an robust entity capable of transporting an excitation across a generic discrete medium that models several systems of interest.

Dynamics of interacting electrons under effect of a Morse potential

We consider interacting electrons moving in a nonlinear Morse lattice. We set the initial conditions as follows: electrons were initially localized at the center of the chain and a solitonic deformation was produced by an impulse excitation on the center of the chain. By solving quantum and classical equations for this system numerically, we found that a fraction of electronic wave function was trapped by the solitonic excitation, and trapping specificities depend on the degree of interaction among electrons. Also, there is evidence that the effective electron velocity depends on Coulomb interaction and electron-phonon coupling in a nontrivial way. This association is explained in detail along this work. In addition, we briefly discuss the dependence of our results with the type of initial condition we choose for the electrons and lattice.

Phase transitions at high energy vindicate negative microcanonical temperature

The notion of negative absolute temperature emerges naturally from Boltzmann's definition of "surface" microcanonical entropy in isolated systems with a bounded energy density. Recently, the well-posedness of such construct has been challenged, on account that only the Gibbs "volume" entropy-and the strictly positive temperature thereof-would give rise to a consistent thermodynamics. Here we present analytical and numerical evidence that Boltzmann microcanonical entropy provides a consistent thermometry for both signs of the temperature. In particular, we show that Boltzmann (negative) temperature allows the description of phase transitions occurring at high energy densities, at variance with Gibbs temperature. Our results apply to nonlinear lattice models standardly employed to describe the propagation of light in arrays of coupled wave guides and the dynamics of ultracold gases trapped in optical lattices. Optically induced photonic lattices, characterized by saturable nonlinearity, are particularly appealing because they offer the possibility of observing states and phase transitions at both signs of the temperature.

A self-consistent theory of localization in nonlinear random media

The self-consistent theory of localization is generalized to account for a weak quadratic nonlinear potential in the wave equation. For spreading wave packets, the theory predicts the destruction of Anderson localization by the nonlinearity and its replacement by algebraic subdiffusion, while classical diffusion remains unaffected. In 3D, this leads to the emergence of a subdiffusion-diffusion transition in place of the Anderson transition. The accuracy and the limitations of the theory are discussed.

Anisotropy Enhancement of Thermal Energy Transport in Supported Black Phosphorene

Thermal anisotropy along the basal plane of materials possesses both theoretical importance and application value in thermal transport and thermoelectricity. Though common two-dimensional materials may exhibit in-plane thermal anisotropy when suspended, thermal anisotropy would often disappear when supported on a substrate. In this Letter, we find a strong anisotropy enhancement of thermal energy transport in supported black phosphorene. The chiral preference of energy transport in the zigzag rather than the armchair direction is greatly enhanced by coupling to the substrate, up to a factor of approximately 2-fold compared to the suspended one. The enhancement originates from its puckered lattice structure, where the nonplanar armchair energy transport relies on the out-of-plane corrugation and thus would be hindered by the flexural suppression due to the substrate, while the planar zigzag energy transport is not. As a result, thermal conductivity of supported black phosphorene shows a consistent anisotropy enhancement under different temperatures and substrate coupling strengths.

Energy transport in one-dimensional disordered granular solids

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: an initial displacement and an initial momentum excitation of a single bead. After establishing the regime of sufficiently strong disorder, we focus our study 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 shocklike excitations.

Superdiffusive transport and energy localization in disordered granular crystals

We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder-an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)-and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment m̃(2) of the energy has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and which is particularly evident for an Anderson-like disorder. By contrast, for velocity excitations, we find that a standard scaling m̃(2)∼t(γ) (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m̃(2) and the inverse participation ratio P(-1) satisfy scaling relations m̃(2)∼t(γ) and P(-1)∼t(-η), and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η<0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly γ≈1.7 and η≈1) for all three types of disorder and for both types of initial conditions.

Energy and magnetization transport in nonequilibrium macrospin systems

We investigate numerically the magnetization dynamics of an array of nanodisks interacting through the magnetodipolar coupling. In the presence of a temperature gradient, the chain reaches a nonequilibrium steady state where energy and magnetization currents propagate. This effect can be described as the flow of energy and particle currents in an off-equilibrium discrete nonlinear Schrödinger (DNLS) equation. This model makes transparent the transport properties of the system and allows for a precise definition of temperature and chemical potential for a precessing spin. The present study proposes a setup for the spin-Seebeck effect, and shows that its qualitative features can be captured by a general oscillator-chain model.

Electron-soliton dynamics in chains with cubic nonlinearity

In our work, we consider the problem of electronic transport mediated by coupling with solitonic elastic waves. We study the electronic transport in a 1D unharmonic lattice with a cubic interaction between nearest neighboring sites. The electron-lattice interaction was considered as a linear function of the distance between neighboring atoms in our study. We numerically solve the dynamics equations for the electron and lattice and compute the dynamics of an initially localized electronic wave-packet. Our results suggest that the solitonic waves that exist within this nonlinear lattice can control the electron dynamics along the chain. Moreover, we demonstrate that the existence of a mobile electron-soliton pair exhibits a counter-intuitive dependence with the value of the electron-lattice coupling.

Complex statistics and diffusion in nonlinear disordered particle chains

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 Kolmogorov-Arnold-Moser torus and that diffusion continues to spread chaotically for arbitrarily long times.

How nonlinear interactions challenge the three-dimensional Anderson transition

In disordered systems, our present understanding of the Anderson transition is hampered by the possible presence of interactions between particles. We demonstrate that in boson gases, even weak interactions deeply alter the very nature of the Anderson transition. While there still exists a critical point in the system, below that point a novel phase appears, displaying a new critical exponent, subdiffusive transport, and a breakdown of the one-parameter scaling description of Anderson localization.

Kinetic theory of nonlinear diffusion in a weakly disordered nonlinear Schrödinger chain in the regime of homogeneous chaos

We study the discrete nonlinear Schröinger equation with weak disorder, focusing on the regime when the nonlinearity is, on the one hand, weak enough for the normal modes of the linear problem to remain well resolved but, on the other, strong enough for the dynamics of the normal mode amplitudes to be chaotic for almost all modes. We show that in this regime and in the limit of high temperature, the macroscopic density ρ satisfies the nonlinear diffusion equation with a density-dependent diffusion coefficient, D(ρ) = D(0)ρ(2). An explicit expression for D(0) is obtained in terms of the eigenfunctions and eigenvalues of the linear problem, which is then evaluated numerically. The role of the second conserved quantity (energy) in the transport is also quantitatively discussed.

Nonequilibrium chaos of disordered nonlinear waves

Do nonlinear waves destroy Anderson localization? Computational and experimental studies yield subdiffusive nonequilibrium wave packet spreading. Chaotic dynamics and phase decoherence assumptions are used for explaining the data. We perform a quantitative analysis of the nonequilibrium chaos assumption and compute the time dependence of main chaos indicators--Lyapunov exponents and deviation vector distributions. We find a slowing down of chaotic dynamics, which does not cross over into regular dynamics up to the largest observed time scales, still being fast enough to allow for a thermalization of the spreading wave packet. Strongly localized chaotic spots meander through the system as time evolves. Our findings confirm for the first time that nonequilibrium chaos and phase decoherence persist, fueling the prediction of a complete delocalization.

Decohering localized waves

In the absence of confinement, localization of waves takes place due to randomness or nonlinearity and relies on their phase coherence. We quantitatively probe the sensitivity of localized wave packets to random phase fluctuations and confirm the necessity of phase coherence for localization. Decoherence resulting from a dynamical random environment leads to diffusive spreading and destroys linear and nonlinear localization. We find that maximal spreading is achieved for optimal phase fluctuation characteristics, which is a consequence of the competition between diffusion due to decoherence and ballistic transport within the mean free path distance.

Self-trapping threshold in disordered nonlinear photonic lattices

We investigate numerically and experimentally the influence of coupling disorder on the self-trapping dynamics in nonlinear one-dimensional optical waveguide arrays. The existence of a lower and upper bound of the effective average propagation constant allows for a generalized definition of the threshold power for the onset of soliton localization. When compared to perfectly ordered systems, this threshold is found to decrease in the presence of coupling disorder.

Modeling the transport of interacting matter waves in a disordered system by a nonlinear diffusion equation

We model the expansion of an interacting atomic Bose-Einstein condensate in a disordered lattice with a nonlinear diffusion equation normally used for a variety of classical systems. We find approximate solutions of the diffusion equation that well reproduce the experimental observations for both short and asymptotic expansion times. Our study establishes a connection between the peculiar shape of the expanding density profiles and the microscopic nonlinear diffusion coefficients.

Enhanced distribution of a wave-packet in lattices with disorder and nonlinearity

We show, numerically and experimentally, that the presence of weak disorder results in an enhanced energy distribution of an initially localized wave-packet, in one- and two-dimensional finite lattices. The addition of a focusing nonlinearity facilitates the spreading effect even further by increasing the wave-packet effective size. We find a clear transition between the regions of enhanced spreading (weak disorder) and localization (strong disorder).

Nonequilibrium discrete nonlinear Schrödinger equation

We study nonequilibrium steady states of the one-dimensional discrete nonlinear Schrödinger equation. This system can be regarded as a minimal model for the stationary transport of bosonic particles such as photons in layered media or cold atoms in deep optical traps. Due to the presence of two conserved quantities, namely, energy and norm (or number of particles), the model displays coupled transport in the sense of linear irreversible thermodynamics. Monte Carlo thermostats are implemented to impose a given temperature and chemical potential at the chain ends. As a result, we find that the Onsager coefficients are finite in the thermodynamic limit, i.e., transport is normal. Depending on the position in the parameter space, the "Seebeck coefficient" may be either positive or negative. For large differences between the thermostat parameters, density and temperature profiles may display an unusual nonmonotonic shape. This is due to the strong dependence of the Onsager coefficients on the state variables.

Wave-packet spreading dynamics under a noninstantaneous nonlinearity: self-trapping, defocusing, and focusing

Special localized wave modes show up in several physical scenarios including BEC in optical lattices, nonlinear photonic crystals, and systems with strong electron-phonon interaction. These result from an underlying nonlinear contribution to the wave equation that is usually assumed to be instantaneous. Here we demonstrate that the relaxation process of the nonlinearity has a profound impact in the wave-packet dynamics and in the formation of localized modes. We illustrate this phenomenology by considering the one-electron wave packet spreading in a C60 buckball structure whose dynamics is governed by a discrete nonlinear Schrödinger equation with a Debye relaxation of the nonlinearity. We report the full phase diagram related to the spacial extension of the asymptotic wave packet and unveil a complex wave-packet dynamical behavior.

Effective noise theory for the nonlinear Schrödinger equation with disorder

For the nonlinear Shrödinger equation with disorder it was found numerically that in some regime of the parameters Anderson localization is destroyed and subdiffusion takes place for a long time interval. It was argued that the nonlinear term acts as random noise. In the present work, the properties of this effective noise are studied numerically. Some assumptions made in earlier work were verified, and fine details were obtained. The dependence of various quantities on the localization length of the linear problem were computed. A scenario for the possible breakdown of the theory for a very long time is outlined.

Interacting ultracold bosons in disordered lattices: sensitivity of the dynamics to the initial state

We study the dynamics of a nonlinear one-dimensional disordered system obtained by coupling the Anderson model with the Gross-Pitaevskii equation. We introduce a single quantity globally characterizing the localization of the system. This quantity obeys a scaling law with respect to the width of the initial state, which can be used to characterize the dynamics independently of the initial state.

Optical surface modes in the presence of nonlinearity and disorder

We investigate numerically the effect of the competition of disorder, nonlinearity, and boundaries on the Anderson localization of light waves in finite-size, one-dimensional waveguide arrays. Using the discrete Anderson-nonlinear Schrödinger equation, the propagation of the mode amplitudes up to some finite distance is monitored. The analysis is based on the calculated localization length and the participation number, two standard measures for the statistical description of Anderson localization. For relatively weak disorder and nonlinearity, a higher disorder strength is required to achieve the same degree of localization at the edge than in the interior of the array, in agreement with recent experimental observations in the linear regime. However, for relatively strong disorder and/or nonlinearity, this behavior is reversed and it is now easier to localize an excitation at the edge than in the interior.

Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation

We elucidate the case in which the Ablowitz-Ladik (AL)-type discrete nonlinear Schrödinger equation (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple one-dimensional (1D) discrete chain. The strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The present work is a nontrivial extension of our preceding one [Sobirov et al., Phys. Rev. E 81, 066602 (2010)] on the continuum NLSE to the discrete case. We find (1) the solution on each bond is a part of the universal (bond-independent) AL soliton solution on the 1D discrete chain, but it is multiplied by the inverse of the square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule; and (3) under findings 1 and 2, there exist an infinite number of constants of motion. As a practical issue, with the use of an AL soliton injected through the incoming bond, we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds.

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.

Observation of subdiffusion in a disordered interacting system

We study the transport dynamics of matter-waves in the presence of disorder and nonlinearity. An atomic Bose-Einstein condensate that is localized in a quasiperiodic lattice in the absence of atom-atom interaction shows instead a slow expansion with a subdiffusive behavior when a controlled repulsive interaction is added. The measured features of the subdiffusion are compared to numerical simulations and a heuristic model. The observations confirm the nature of subdiffusion as interaction-assisted hopping between localized states and highlight a role of the spatial correlation of the disorder.

Localization-delocalization transition and subdiffusion of discrete nonlinear Schrödinger equation in three dimensions

Localization-delocalization transition in a three-dimensional discrete nonlinear Schrödinger equation (DNLSE) with random potential is investigated, and the effect of nonlinearity is clarified numerically. By Thouless-number analysis, it is shown that the nonlinearity tends to delocalize the stationary states in a three-dimensional DNLSE. The spreading of a wave packet in a localized regime is also clarified for this system, and the subdiffusive behavior is observed in the three-dimensional system when the nonlinearity of the system is sufficiently strong. In addition, a one-dimensional DNLSE with random potential having a finite correlation length is studied, and it is suggested that the power-law exponent of the subdiffusion in DNLSE is not a universal quantity.

Wave packet dynamics in one-dimensional linear and nonlinear generalized Fibonacci lattices

The spreading of an initially localized wave packet in one-dimensional linear and nonlinear generalized Fibonacci (GF) lattices is studied numerically. The GF lattices can be classified into two classes depending on whether or not the lattice possesses the Pisot-Vijayaraghavan property. For linear GF lattices of the first class, both the second moment and the participation number grow with time. For linear GF lattices of the second class, in the regime of a weak on-site potential, wave packet spreading is close to ballistic diffusion, whereas in the regime of a strong on-site potential, it displays stairlike growth in both the second moment and the participation number. Nonlinear GF lattices are then investigated in parallel. For the first class of nonlinear GF lattices, the second moment of the wave packet still grows with time, but the corresponding participation number does not grow simultaneously. For the second class of nonlinear GF lattices, an analogous phenomenon is observed for the weak on-site potential only. For a strong on-site potential that leads to an enhanced nonlinear self-trapping effect, neither the second moment nor the participation number grows with time. The results can be useful in guiding experiments on the expansion of noninteracting or interacting cold atoms in quasiperiodic optical lattices.

Nonlinear relativistic and quantum equations with a common type of solution

Generalizations of the three main equations of quantum physics, namely, the Schrödinger, Klein-Gordon, and Dirac equations, are proposed. Nonlinear terms, characterized by exponents depending on an index q, are considered in such a way that the standard linear equations are recovered in the limit q→1. Interestingly, these equations present a common, solitonlike, traveling solution, which is written in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics. In all cases, the well-known Einstein energy-momentum relation is preserved for arbitrary values of q.

Resonant states and wavepacket super-diffusion in intra-chain correlated ladders with diluted disorder

In this work, we study a tight-binding Hamiltonian model system of a binary correlated ladder with diluted disorder. We introduce intra-chain correlations between the on-site potentials by imposing that ϵ(i, s) = - ϵ(i, - s) where s = ± 1 indexes the two ladder chains. Further, we consider each ladder chain as composed of inter-penetrating ordered and random sub-chains. We show that the presence of a random on-site distribution in one of the inter-penetrating chains leads to Anderson localization except at a specific symmetric pair of energy eigenmodes. Further, by integrating the time-dependent Schroedinger equation, we follow the time-evolution of an initially localized one-electron wavepacket. We report that the remaining delocalized resonant modes are responsible for a super-diffusive spread of the wavepacket dispersion while the wavepacket participation function remains finite. A scaling analysis of the wavepacket distribution shows that it obeys a universal scaling form with the development of a power-law tail followed by a super-diffusively evolving cutoff. We obtain three exponents characterizing this super-diffusive dynamics and show that they satisfy a simple scaling relation.

Disorder-induced soliton transmission in nonlinear photonic lattices

We address soliton transmission and reflection in nonlinear photonic lattices embedded into uniform Kerr nonlinear media. We show that by introducing disorder into the guiding lattice channels, one may achieve soliton transmission even under conditions where regular lattices reflect the input beam completely. In contrast, in the parameter range in which the lattice is almost transparent for incoming solitons, disorder may induce a significant reflection.

Scaling of energy spreading in strongly nonlinear disordered lattices

To characterize a destruction of Anderson localization by nonlinearity, we study the spreading behavior of initially localized states in disordered, strongly nonlinear lattices. Due to chaotic nonlinear interaction of localized linear or nonlinear modes, energy spreads nearly subdiffusively. Based on a phenomenological description by virtue of a nonlinear diffusion equation, we establish a one-parameter scaling relation between the velocity of spreading and the density, which is confirmed numerically. From this scaling it follows that for very low densities the spreading slows down compared to the pure power law.

Heat conduction in disordered harmonic lattices with energy-conserving noise

We study heat conduction in a harmonic crystal whose bulk dynamics is supplemented by random reversals (flips) of the velocity of each particle at a rate λ. The system is maintained in a nonequilibrium stationary state (NESS) by contacts with white-noise Langevin reservoirs at different temperatures. We show that the one-body and pair correlations in this system are the same (after an appropriate mapping of parameters) as those obtained for a model with self-consistent reservoirs. This is true both for the case of equal and random (quenched) masses. While the heat conductivity in the NESS of the ordered system is known explicitly, much less is known about the random mass case. Here we investigate the random system with velocity flips. We improve the bounds on the Green-Kubo conductivity obtained by Bernardin [J. Stat. Phys. 133, 417 (2008)]. The conductivity of the one-dimensional system is then studied both numerically and analytically. This sheds some light on the effect of noise on the transport properties of systems with localized states caused by quenched disorder.

Asymptotic energy profile of a wave packet in disordered chains

We investigate the long-time behavior of a wave packet initially localized at a single site n₀ in translationally invariant harmonic and anharmonic chains with random interactions. In the harmonic case, the energy profile <e(n)(t)> averaged on time and disorder decays for large |n-n₀| as a power law <e(n)(t)> ≈ C|n-n₀|(⁻η), where η=⁵/₂ and ³/₂ for initial displacement and momentum excitations, respectively. The prefactor C depends on the probability distribution of the harmonic coupling constants and diverges in the limit of weak disorder. As a consequence, the moments <m(ν)(t)> of the energy distribution averaged with respect to disorder diverge in time as t(β(ν)) for ν ≥ 2, where β=ν+1-η for ν>η-1 . Molecular-dynamics simulations yield good agreement with these theoretical predictions. Therefore, in this system, the second moment of the wave packet diverges as a function of time despite the wave packet is not spreading. Thus, this only criterion, often considered earlier as proving the spreading of a wave packet, cannot be considered as sufficient in any model. The anharmonic case is investigated numerically. It is found for intermediate disorder that the tail of the energy profile becomes very close to those of the harmonic case. For weak and strong disorders, our results suggest that the crossover to the harmonic behavior occurs at much larger |n-n₀| and larger time.

Statistics of wave interactions in nonlinear disordered systems

We study the properties of mode-mode interactions for waves propagating in nonlinear disordered one-dimensional systems. We focus on (i) the localization volume of a mode which defines the number of interacting partner modes, (ii) the overlap integrals which determine the interaction strength, (iii) the average spacing between eigenvalues of interacting modes, which sets a scale for the nonlinearity strength, and (iv) resonance probabilities of interacting modes. Our results are discussed in the light of recent studies on spreading of wave packets in disordered nonlinear systems and are related to the quantum many-body problem in a random chain.