Sub-diffusive spreading and anomalous localization in a 2D Anderson model with off-diagonal nonlinearity

Sales, M.O.; Dias, W. S.; Neto, Adhemar Ranciaro; Lyra, Marcelo L.; Moura, F. A. B. F. de

doi:10.1016/j.ssc.2017.11.001

Cited by 8 publications

(6 citation statements)

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“…It is worth noting that due to the computational difficulty of the problem very few numerical results for the 2D DDNLS system exist in the literature (e.g. [39,65]). We consider again two sets of initial conditions:…”

Section: The 2d Disordered Discrete Nonlinear Schrödinger Equation Symentioning

confidence: 99%

“…In this case the corrector Hamiltonian K of Eq. (A.7) becomes 38) and the operator e τL KZ is given by the following set of equations e τL KZ : 39) with γ i = E 2 J − 4 cos(2(q i+1 − q i )) − 4 cos(q i−1 − 2q i + q i+1 ) − 4 cos(2(q i−1 − q i )) + 2 sin(q i+2 − q i+1 ) sin(q i − q i+1 ) + 2 sin(q i−2 − q i−1 ) sin(q i − q i−1 ) γ i+1 = E 2 J 2 cos(q i−1 − 2q i + q i+1 ) + 4 cos(2(q i+1 − q i )) + 2 cos(q i+2 − 2q i+1 + q i ) γ i−1 = E 2 J 2 cos(q i−1 − 2q i + q i+1 ) + 4 cos(2(q i − q i−1 )) + 2 cos(q i−2 − 2q i−1 + q i ) γ i+2 = E 2 J 2 cos(q i+2 − q i+1 ) cos(q i − q i+1 ) γ i−2 = E 2 J 2 cos(q i−2 − q i−1 ) cos(q i − q i−1 )…”

Section: The Xy Model Of a Josephson Junctions Arraymentioning

confidence: 99%

“…The numerical integration of these sets of ODEs can be performed by any general purpose integrator. For example Runge-Kutta family schemes are still used [38,39]. Another category of integrators is the so-called symplectic integrators (SIs), which were particularly designed for Hamiltonian systems (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

Danieli¹,

Manda²,

Mithun³

et al. 2019

Mathematics in Engineering

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We investigate the computational performance of various numerical methods for the integration of the equations of motion and the variational equations for some typical classical many-body models of condensed matter physics: the Fermi-Pasta-Ulam-Tsingou (FPUT) chain and the one-and two-dimensional disordered, discrete nonlinear Schrödinger equations (DDNLS). In our analysis we consider methods based on Taylor series expansion, Runge-Kutta discretization and symplectic transformations. The latter have the ability to exactly preserve the symplectic structure of Hamiltonian systems, which results in keeping bounded the error of the system's computed total energy. We perform extensive numerical simulations for several initial conditions of the studied models and compare the numerical efficiency of the used integrators by testing their ability to accurately reproduce characteristics of the systems' dynamics and quantify their chaoticity through the computation of the maximum Lyapunov exponent. We also report the expressions of the implemented symplectic schemes and provide the explicit forms of the used differential operators. Among the tested numerical schemes the symplectic integrators ABA864 and S RKN a 14 exhibit the best performance, respectively for moderate and high accuracy levels in the case of the FPUT chain, while for the DDNLS models s9ABC6 and s11ABC6 (moderate accuracy), along with s17ABC8 and s19ABC8 (high accuracy) proved to be the most efficient schemes.are the 2N × 2N elements of the Hessian matrix D 2 H (x(t)) of the Hamiltonian function H computed on the phase space trajectory x(t). Eq. (3.5) is linear in w(t), with coefficients depending on the system's trajectory x(t). Therefore, one has to integrate the variational equations (3.5) along with the equations of motion (3.2), which means to evolve in time the general vector X(t) = (x(t), δx(t)) by solving the systemẊH (x(t)) · δx(t).( 3.7) In what follows we will briefly describe several numerical schemes for integrating the set of equations (3.7).

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Section: The 2d Disordered Discrete Nonlinear Schrödinger Equation Symentioning

confidence: 99%

Section: The Xy Model Of a Josephson Junctions Arraymentioning

confidence: 99%

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Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

Danieli¹,

Manda²,

Mithun³

et al. 2019

Mathematics in Engineering

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“…In the presence of nonlinearity the dynamics becomes more complicated as the system's normal modes (NMs) couple and chaos appears. Thus, the interplay of disorder and nonlinearity has attracted extensive attention in theory [12,13,14,15,16,17,18,19,20,21,22,23,24,25], numerical simulations [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,…”

Section: Introductionmentioning

confidence: 99%

Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method

Senyange,

Skokos

2021

Preprint

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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 chaotic cases diminishes as the energy decreases leading to exclusively regular motion on multidimensional tori. We also discriminate between localized and spreading chaos, with the former dominating the dynamics for lower energy values. In addition, our results show that single mode excitations lead to more chaotic behaviors for larger energies compared to single site excitations. Furthermore, we demonstrate how the GALI method can be efficiently used to determine a characteristic chaoticity time scale for the system when strong enough nonlinearites lead to energy delocalization in both the so-called 'weak' and 'strong chaos' spreading regimes.

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“…One of the main obstacles in such studies is the very large computational effort required for the long time simulation of these models and especially of the 2D DDNLS system. In [4] the wave packet spreading in the 2D DDNLS model for the (what was later called) weak chaos regime was studied up to t = 10 6 (dimensionless) time units, while in [16] a similar model, including also non-diagonal nonlinear terms, was considered. In both cases statistical analyses over a few disorder realizations were performed.…”

Section: Introductionmentioning

confidence: 99%

Chaotic wave packet spreading in two-dimensional disordered nonlinear lattices

Manda,

Senyange,

Skokos

2019

Preprint

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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 am with a m ≈ 1/5 (1/3) for the weak (strong) chaos dynamical regime, in agreement with previous theoretical predictions [10], (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 dimensionindependent scaling between the wave packet's spreading and chaoticity, which allows the prediction of the obtained α Λ values.

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Sub-diffusive spreading and anomalous localization in a 2D Anderson model with off-diagonal nonlinearity

Cited by 8 publications

References 50 publications

Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method

Chaotic wave packet spreading in two-dimensional disordered nonlinear lattices

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