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.
We study scaling properties of energy spreading in disordered strongly nonlinear Hamiltonian lattices. Such lattices consist of nonlinearly coupled local linear or nonlinear oscillators, and demonstrate a rather slow, subdiffusive spreading of initially localized wave p ackets. We use a fractional nonlinear diffusion equation as a heuristic model of this process, and confirm that the scaling predictions resulting from a self-similar solution of this equation are indeed applicable to all studied cases. We s how that the spreading in nonlinearly coupled linear oscillators slows down compared to …
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 [EPL {\bf 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 observa…
We perform novel energy and norm density resolved wave packet spreading studies in the disordered Gross-Pitaevskii (GP) lattice to confine energy density fluctuations. We map the locations of GP regimes of weak and strong chaos subdiffusive spreading in the 2D density control parameter space and observe strong chaos spreading over several decades. We obtain a renormalization of the ground state due to disorder, which allows for a new disorder-induced phase of disconnected insulating puddles of matter due to Lifshits tails. Inside this Lifshits phase, the wave packet spreading is substantially …
We study effects of weak nonlineary on localization of waves in disordered Stark ladder corresponding to propagation in presence of disorder and a static field. Our numerical results show that nonlinearity leads to delocalization with subdiffusive spreading along the ladder. The exponent of spreading remains close to its value in absence of the static field.
We study the dynamical and chaotic behavior of a disordered one-dimensional elastic mechanical lattice which supports translational and rotational waves. The model used in this work is motivated by the recent experimental results of B. Deng et al., Nat. Commun. 9, 3410 (2018). This lattice is characterized by strong geometrical nonlinearities and the coupling of two degrees-of-freedom (DoFs) per site. Although the linear limit of the structure consists of a linear Fermi-Pasta-Ulam-Tsingou lattice and a linear Klein-Gordon (KG) lattice whose DoFs are uncoupled, by using single site initial exci…
In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find a nice agreement of the scaling predicted from the NDE with the spreading results from extensive numerical studies. Moreover, we show that the scaling works also for regular lattices with strongly nonli…
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 $a_m \approx 1/5$ ($1/3$) for the weak (strong) chaos dynamical regime, in agreement with previous theoretical predictions [S.~Flach, Chem.~Phys.~{\bf 375}, 548 (2010)], (b) chaos persists, but its strength decreases in time $t…
Implementing the Generalized Alignment Index (GALI) method of chaos detection we investigate the dynamical behavior of the nonlinear disordered Klein-Gordon lattice chain in one spatial dimension. By performing extensive numerical simulations of single site and single mode initial excitations, for several disordered realizations and different disorder strengths, we determine the probability to observe chaotic behavior as the system is approaching its linear limit, i.e. when its total energy, which plays the role of the system's nonlinearity strength, decreases. We find that the percentage of c…
We study energy propagation in locally time-periodically driven disordered nonlinear chains. For frequencies inside the band of linear Anderson modes, three different regimes are observed with increasing driver amplitude: 1) Below threshold, localized quasiperiodic oscillations and no spreading; 2) Three different regimes in time close to threshold, with almost regular oscillations initially, weak chaos and slow spreading for intermediate times, and finally strong diffusion; 3) Immediate spreading for strong driving. The thresholds are due to simple bifurcations, obtained analytically for a si…
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 $m_2$ 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 large…
Localization of waves by disorder is a fundamental physical problem encompassing a diverse spectrum of theoretical, experimental and numerical studies in the context of metal-insulator transitions, the quantum Hall effect, light propagation in photonic crystals, and dynamics of ultra-cold atoms in optical arrays, to name just a few examples. Large intensity light can induce nonlinear response, ultracold atomic gases can be tuned into an interacting regime, which leads again to nonlinear wave equations on a mean field level. The interplay between disorder and nonlinearity, their localizing and …
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 sufficient…
We study spreading wave packets in a disordered nonlinear ladder with broken time-reversal symmetry induced by synthetic gauge fields. The model describes the dynamics of interacting bosons in a disordered and driven optical ladder within a mean-field approximation. The second moment of the wave packet $m_{2} = g t^α$ grows subdiffusively with the universal exponent $α\simeq 1/3$ similar to the time-reversal case. However the prefactor $g$ is strongly modified by the field strength and shows a non-monotonic dependence. For a weak field, the prefactor increases since time-reversal enhanced back…
We study properties of energy spreading in a lattice of elastically colliding harmonic oscillators (Ding-Dong model). We demonstrate that in the regular lattice the spreading from a localized initial state is mediated by compactons and chaotic breathers. In a disordered lattice the compactons do not exist, and the spreading eventually stops, resulting in a finite configuration with a few chaotic spots.
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 thermaliz…
The dynamics of a disordered nonlinear chain can be either regular or chaotic with a certain probability. The chaotic behavior is often associated with the destruction of Anderson localization by the nonlinearity. In the presentwork it is argued that at weak nonlinearity chaos is nucleated locally on rare resonant segments of the chain. Based on this picture, the probability of chaos is evaluated analytically. The same probability is also evaluated by direct numerical sampling of disorder realizations and quantitative agreement between the two results is found.
The Discrete Nonlinear Schroedinger Equation with a random potential in one dimension is studied as a dynamical system. It is characterized by the length, the strength of the random potential and by the field density that determines the effect of nonlinearity. The probability of the system to be regular is established numerically and found to be a scaling function. This property is used to calculate the asymptotic properties of the system in regimes beyond our computational power.
This thesis focuses on the mechanisms of energy transport in multidimensional heterogeneous lattice models, studying in particular the case of the Klein-Gordon model of coupled anharmonic oscillators in one and two spatial dimensions. We report the effects of the type of the impurity (heterogeneity) parameter on the systems' transport properties and classifies the transport mechanisms of the nonlinear versions of the models into various dynamical regimes. We also identify computationally efficient numerical integration techniques, including the so-called symplectic integrators, and implement t…
We investigate dynamically and statistically diffusive motion in a chain of linearly coupled 2-dimensional symplectic McMillan maps and find evidence of subdiffusion in weakly and strongly chaotic regimes when all maps of the chain possess a saddle point at the origin and the central map is initially excited. In the case of weak coupling, there is either absence of diffusion or subdiffusion with $q>1$-Gaussian probability distributions, characterizing weak chaos. However, for large enough coupling and already moderate number of maps, the system exhibits strongly chaotic ($q\approx 1$) subdiffu…
While symplectic integration methods based on operator splitting are well established in many branches of science, high order methods for Hamiltonian systems that split in more than two parts have not been studied in great detail. Here, we present several high order symplectic integrators for Hamiltonian systems that can be split in exactly three integrable parts. We apply these techniques, as a practical case, for the integration of the disordered, discrete nonlinear Schroedinger equation (DDNLS) and compare their efficiencies. Three part split algorithms provide effective means to numericall…
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 Gaussian…
The analysis of dissipation and dephasing in driven mesoscopic devices requires a distinction between two notions of quantum irreversibility. One ("Loschmidt echo") is related to "time reversal", while the other is related to "driving reversal". In the latter context the time of maximum return (compensation) should substitute the inappropriate notion of "echo" time. Non-perturbative features manifest themselves in the energy spreading process. This is demonstrated for the prototype random-matrix Wigner model, where the compensation time and the system response exhibit a non-universal scaling b…
