We numerically investigate the characteristics of chaos evolution during wave packet spreading in two typical one-dimensional nonlinear disordered lattices: the Klein-Gordon system and the discrete nonlinear Schrödinger equation model. Completing previous investigations [38] we verify that chaotic dynamics is slowing down both for the so-called 'weak' and 'strong chaos' dynamical regimes encountered in these systems, without showing any signs of a crossover to regular dynamics. The value of the finite-time maximum Lyapunov exponent Λ decays in time t as Λ ∝ t α Λ , with αΛ being different from the αΛ = −1 value observed in cases of regular motion. In particular, αΛ ≈ −0.25 (weak chaos) and αΛ ≈ −0.3 (strong chaos) for both models, indicating the dynamical differences of the two regimes and the generality of the underlying chaotic mechanisms. The spatiotemporal evolution of the deviation vector associated with Λ reveals the meandering of chaotic seeds inside the wave packet, which is needed for obtaining the chaotization of the lattice's excited part.
We implement several symplectic integrators, which are based on two part splitting, for studying the chaotic behavior of one-and two-dimensional disordered Klein-Gordon lattices with many degrees of freedom and investigate their numerical performance. For this purpose, we perform extensive numerical simulations by considering many different initial energy excitations and following the evolution of the created wave packets in the various dynamical regimes exhibited by these models. We compare the efficiency of the considered integrators by checking their ability to correctly reproduce several features of the wave packets propagation, like the characteristics of the created energy distribution and the time evolution of the systems' maximum Lyapunov exponent estimator. Among the tested integrators the fourth order ABA864 scheme [59] showed the best performance as it needed the least CPU time for capturing the correct dynamical behavior of all considered cases when a moderate accuracy in conserving the systems' total energy value was required. Among the higher order schemes used to achieve a better accuracy in the energy conservation, the sixth order scheme s11ABA82 6 exhibited the best performance.
We have identified, in the original paper, confusing use of the symbol ξ l denoting the normalized energy [disordered Klein-Gordon (DKG) system] and norm density [disordered discrete nonlinear Schrödinger equation (DDNLS)], which only affects the presentation of some of the provided information about the setup of our numerical simulations. Thus, the following changes should be made in the first paragraph of Sec. III: (a) p l = ± √ 2ξ l should become p l = ± √ 2ξ l H K (line 4), (b) ξ l = 1 should become ξ l S = 1 (line 7), and (c) H K = Lξ l should become H K (line 11). Furthermore, in the last entry of the presentation of the studied cases in Sec. III A, ξ l should become ξ l H K for the DKG system and ξ l S for the DDNLS model
We introduce a modified version of the disordered Klein-Gordon lattice model, having two parameters for controlling the disorder strength: D, which determines the range of the coefficients of the on-site potentials, and W, which defines the strength of the nearestneighbor interactions. We fix W = 4 and investigate how the properties of the system's normal modes change as we approach its ordered version, i.e. D → 0. We show that the probability density distribution of the normal mode's frequencies takes a 'U'-shaped profile as D decreases. Furthermore, we use two quantities for estimating the mode's spatial extent, the so-called localization volume V (which is related to the mode's second moment) and the mode's participation number P. We show that both quantities scale as ∝ D−2 when D approaches zero and we numerically verify a proportionality relation between them as V/P ≈ 2.6.
We investigate the performance of various methods of symplectic integration, which are based on two part splitting of the integration operator, for the numerical integration of multidimensional Hamiltonian systems.We implement these schemes to study the behaviour of the one-dimensional quartic Klein Gordon disordered lattice with many degrees of freedom (of the order of a few hundreds) and compare their efficiency for the weak chaos regime of the system's dynamics. For this reason we perform extensive numerical simulations for each considered integration scheme. In this process, the second moment and the participation number of propagating wave packets, along with the system's relative energy error and the required CPU time are registered and compared.
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