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 [EPL {\bf 91}, 30001 (2010)] and consider strong disorder, which favors Anderson localization. We probe the limit of infinite disorder strength and study Fr\"ohlich-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.Comment: 11 pages, 7 figure

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 $m_2$ 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 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.Comment: 5 pages, 3 figures. Submitted Europhysics Letter

High order three part split symplectic integrators: Efficient techniques for the long time simulation of the disordered discrete nonlinear Schroedinger equation

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 numerically study the asymptotic behavior of wave packet spreading in the DDNLS - a hotly debated subject in current scientific literature.Comment: 5 Figures, Physics Letters A (accepted

Efficient integration of the variational equations of multi-dimensional Hamiltonian systems: Application to the Fermi-Pasta-Ulam lattice

We study the problem of efficient integration of variational equations in multi-dimensional Hamiltonian systems. For this purpose, we consider a Runge-Kutta-type integrator, a Taylor series expansion method and the so-called `Tangent Map' (TM) technique based on symplectic integration schemes, and apply them to the Fermi-Pasta-Ulam $\beta$ (FPU-$\beta$) lattice of $N$ nonlinearly coupled oscillators, with $N$ ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique--which shows the best performance among the tested algorithms--and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.Comment: 14 pages, 6 figure

Wave interactions in localizing media - a coin with many faces

A variety of heterogeneous potentials are capable of localizing linear non-interacting 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.Comment: Review submitted to Intl. Journal of Bifurcation and Chaos Special Issue edited by G. Nicolis, M. Robnik, V. Rothos and Ch. Skokos 21 Pages, 8 Figure

Application of new dynamical spectra of orbits in Hamiltonian systems

In the present article, we investigate the properties of motion in Hamiltonian systems of two and three degrees of freedom, using the distribution of the values of two new dynamical parameters. The distribution functions of the new parameters, define the S(g) and the S(w) dynamical spectra. The first spectrum definition, that is the S(g) spectrum, will be applied in a Hamiltonian system of two degrees of freedom (2D), while the S(w) dynamical spectrum will be deployed in a Hamiltonian system of three degrees of freedom (3D). Both Hamiltonian systems, describe a very interesting dynamical system which displays a large variety of resonant orbits, different chaotic components and also several sticky regions. We test and prove the efficiency and the reliability of these new dynamical spectra, in detecting tiny ordered domains embedded in the chaotic sea, corresponding to complicated resonant orbits of higher multiplicity. The results of our extensive numerical calculations, suggest that both dynamical spectra are fast and reliable discriminants between different types of orbits in Hamiltonian systems, while requiring very short computation time in order to provide solid and conclusive evidence regarding the nature of an orbit. Furthermore, we establish numerical criteria in order to quantify the results obtained from our new dynamical spectra. A comparison to other previously used dynamical indicators, reveals the leading role of the new spectra.Comment: Published in Nonlinear Dynamics (NODY) journal. arXiv admin note: text overlap with arXiv:1009.1993 by other author

Universal spreading of wavepackets in disordered nonlinear systems

In the absence of nonlinearity all eigenmodes of a chain with disorder are spatially localized (Anderson localization). The width of the eigenvalue spectrum, and the average eigenvalue spacing inside the localization volume, set two frequency scales. An initially localized wavepacket spreads in the presence of nonlinearity. Nonlinearity introduces frequency shifts, which define three different evolution outcomes: i) localization as a transient, with subsequent subdiffusion; ii) the absence of the transient, and immediate subdiffusion; iii) selftrapping of a part of the packet, and subdiffusion of the remainder. The subdiffusive spreading is due to a finite number of packet modes being resonant. This number does not change on average, and depends only on the disorder strength. Spreading is due to corresponding weak chaos inside the packet, which slowly heats the cold exterior. The second moment of the packet is increasing as $t^{\alpha}$. We find $\alpha=1/3$.Comment: 4 pages, 3 figure

Interplay Between Chaotic and Regular Motion in a Time-Dependent Barred Galaxy Model

We study the distinction and quantification of chaotic and regular motion in a time-dependent Hamiltonian barred galaxy model. Recently, a strong correlation was found between the strength of the bar and the presence of chaotic motion in this system, as models with relatively strong bars were shown to exhibit stronger chaotic behavior compared to those having a weaker bar component. Here, we attempt to further explore this connection by studying the interplay between chaotic and regular behavior of star orbits when the parameters of the model evolve in time. This happens for example when one introduces linear time dependence in the mass parameters of the model to mimic, in some general sense, the effect of self-consistent interactions of the actual N-body problem. We thus observe, in this simple time-dependent model also, that the increase of the bar's mass leads to an increase of the system's chaoticity. We propose a new way of using the Generalized Alignment Index (GALI) method as a reliable criterion to estimate the relative fraction of chaotic vs. regular orbits in such time-dependent potentials, which proves to be much more efficient than the computation of Lyapunov exponents. In particular, GALI is able to capture subtle changes in the nature of an orbit (or ensemble of orbits) even for relatively small time intervals, which makes it ideal for detecting dynamical transitions in time-dependent systems.Comment: 21 pages, 9 figures (minor typos fixed) to appear in J. Phys. A: Math. Theo

Delocalization of wave packets in disordered nonlinear chains

We consider the spatiotemporal evolution of a wave packet in disordered nonlinear Schr\"odinger and anharmonic oscillator chains. In the absence of nonlinearity all eigenstates are spatially localized with an upper bound on the localization length (Anderson localization). Nonlinear terms in the equations of motion destroy Anderson localization due to nonintegrability and deterministic chaos. At least a finite part of an initially localized wave packet will subdiffusively spread without limits. We analyze the details of this spreading process. We compare the evolution of single site, single mode and general finite size excitations, and study the statistics of detrapping times. We investigate the properties of mode-mode resonances, which are responsible for the incoherent delocalization process.Comment: 14 pages, 9 figures, accepted for publication in Phys. Rev.

Numerical integration of variational equations

We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom, and investigate their efficiency in accurately reproducing well-known properties of chaos indicators like the Lyapunov Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs). We find that the best numerical performance is exhibited by the \textit{`tangent map (TM) method'}, a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton's equations of motion by the repeated action of a symplectic map $S$, while the corresponding tangent map $TS$, is used for the integration of the variational equations. A simple and systematic technique to construct $TS$ is also presented.Comment: 27 pages, 11 figures, to appear in Phys. Rev.