8 research outputs found
Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems
In this thesis we study the chaotic behavior of multidimensional Hamiltonian systems in the presence of nonlinearity and disorder. It is known that any localized initial excitation in a large enough linear disordered system spreads for a finite amount of time and then halts forever. This phenomenon is called Anderson localization (AL). What happens to AL when nonlinearity is introduced is an interesting question which has been considered in several studies over the past decades. Recent works focussing on two widely–applicable systems, namely the disordered Klein-Gordon (DKG) lattice of anharmonic oscillators and the disordered discrete nonlinear Schr¨odinger (DDNLS) equation, mainly in one spatial dimension suggest that nonlinearity eventually destroys AL. This leads to an infinite diffusive spreading of initially localized wave packets whose extent (measured for instance through the wave packet's second moment m2) grows in time t as t αm with 0 < αm < 1. However, the characteristics and the asymptotic fate of such evolutions still remain an issue of intense debate due to their computational difficulty, especially in systems of more than one spatial dimension. Two different spreading regimes, the so-called weak and strong chaos regimes, have been theoretically predicted and numerically identified. As the spreading of initially localized wave packets is a non-equilibrium thermalization process related to the ergodic and chaotic properties of the system, in our work we investigate the properties of chaos studying the behavior of observables related to the system's tangent dynamics. In particular, we consider the DDNLS model of one (1D) and two (2D) spatial dimensions and develop robust, efficient and fast numerical integration schemes for the long-time evolution of the phase space and tangent dynamics of these systems. Implementing these integrators, we perform extensive numerical simulations for various sets of parameter values. We present, to the best of our knowledge for the first time, detailed computations of the time evolution of the system's maximum Lyapunov exponent (MLE–Λ) i.e. the most commonly used chaos indicator, and the related deviation vector distribution (DVD). We find that although the systems' MLE decreases in time following a power law t αΛ with αΛ < 0 for both the weak and strong chaos cases, no crossover to the behavior Λ ∝ t −1 (which is indicative of regular motion) is observed. By investigating a large number of weak and strong chaos cases, we determine the different αΛ values for the 1D and 2D systems. In addition, the analysis of the DVDs reveals the existence of random fluctuations of chaotic hotspots with increasing amplitudes inside the excited part of the wave packet, which assist in homogenizing chaos and contribute to the thermalization of more lattice sites. Furthermore, we show the existence of a dimension-free relation between the wave packet spreading and its degree of chaoticity between the 1D and 2D DDNLS systems. The generality of our findings is confirmed, as similar behaviors to the ones observed for the DDNLS systems are also present in the case of DKG models
Stability Properties of 1-Dimensional Hamiltonian Lattices with Non-analytic Potentials
We investigate the local and global dynamics of two 1-Dimensional (1D)
Hamiltonian lattices whose inter-particle forces are derived from non-analytic
potentials. In particular, we study the dynamics of a model governed by a
"graphene-type" force law and one inspired by Hollomon's law describing
"work-hardening" effects in certain elastic materials. Our main aim is to show
that, although similarities with the analytic case exist, some of the local and
global stability properties of non-analytic potentials are very different than
those encountered in systems with polynomial interactions, as in the case of 1D
Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Our approach is to study the motion
in the neighborhood of simple periodic orbits representing continuations of
normal modes of the corresponding linear system, as the number of particles
and the total energy are increased. We find that the graphene-type model is
remarkably stable up to escape energy levels where breakdown is expected, while
the Hollomon lattice never breaks, yet is unstable at low energies and only
attains stability at energies where the harmonic force becomes dominant. We
suggest that, since our results hold for large , it would be interesting to
study analogous phenomena in the continuum limit where 1D lattices become
strings.Comment: Accepted for publication in the International Journal of Bifurcation
and Chao
Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions
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 Schr\"odinger 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 and
exhibit the best performance, respectively for moderate and high accuracy
levels in the case of the FPUT chain, while for the DDNLS models
and (moderate accuracy), along with
and (high accuracy) proved to be the
most efficient schemes.Comment: Accepted for publication in Mathematics in Engineerin
Wave-packet spreading in the disordered and nonlinear Su-Schrieffer-Heeger chain
We numerically investigate the characteristics of the long-time dynamics of a
single-site wave-packet excitation in a disordered and nonlinear
Su-Schrieffer-Heeger model. In the linear regime, as the parameters controlling
the topology of the system are varied, we show that the transition between two
different topological phases is preceded by an anomalous diffusion, in contrast
to Anderson localization within these topological phases. In the presence of
Kerr nonlinearity this feature is lost due to mode-mode interactions. Direct
numerical simulations reveal that the characteristics of the asymptotic
nonlinear wave-packet spreading are the same across the whole studied parameter
space. Our findings underline the importance of mode-mode interactions in
nonlinear topological systems, which must be studied in order to define
reliable nonlinear topological markers.Comment: 14 pages, 11 figure
Skin modes in a nonlinear Hatano-Nelson model
Non-Hermitian lattices with non-reciprocal couplings under open boundary
conditions are known to possess linear modes exponentially localized on one
edge of the chain. This phenomenon, dubbed non-Hermitian skin effect, induces
all input waves in the linearized limit of the system to unidirectionally
propagate toward the system's preferred boundary. Here we investigate the fate
of the non-Hermitian skin effect in the presence of Kerr-type nonlinearity
within the well-established Hatano-Nelson lattice model. Our method is to probe
the presence of nonlinear stationary modes which are localized at the favored
edge, when the Hatano-Nelson model deviates from the linear regime. Based on
perturbation theory, we show that families of nonlinear skin modes emerge from
the linear ones at any non-reciprocal strength. Our findings reveal that, in
the case of focusing nonlinearity, these families of nonlinear skin modes tend
to exhibit enhanced localization, bridging the gap between weakly nonlinear
modes and the highly nonlinear states (discrete solitons) when approaching the
anti-continuum limit with vanishing couplings. Conversely, for defocusing
nonlinearity, these nonlinear skin modes tend to become more extended than
their linear counterpart. To assess the stability of these solutions, we
conduct a linear stability analysis across the entire spectrum of obtained
nonlinear modes and also explore representative examples of their evolution
dynamics.Comment: 12 pages, 8 figure
Thermalization in the one-dimensional Salerno model lattice
The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes. © 2021 American Physical Society.Peer-reviewed manuscript: [https://vinar.vin.bg.ac.rs/handle/123456789/9660