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 t
hen 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. 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. 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 systems tangent dynamics. In particular, we consider the disordered discrete nonlinear Schrodinger (DDNLS) equation of one (1D) and two (2D) spatial dimensions. We present detailed computations of the time evolution of the systems maximum Lyapunov exponent (MLE--$Lambda$), and the related deviation vector distribution (DVD). We find that although the systems MLE decreases in time following a power law $t^{alpha_Lambda}$ with $alpha_Lambda <0$ for both the weak and strong chaos regimes, no crossover to the behavior $Lambda propto t^{-1}$ (which is indicative of regular motion) is observed. 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.
We study the chaotic dynamics of graphene structures, considering both a periodic, defect free, graphene sheet and graphene nanoribbons (GNRs) of various widths. By numerically calculating the maximum Lyapunov exponent, we quantify the chaoticity for
a spectrum of energies in both systems. We find that for all cases, the chaotic strength increases with the energy density, and that the onset of chaos in graphene is slow, becoming evident after more than $10^4$ natural oscillations of the system. For the GNRs, we also investigate the impact of the width and chirality (armchair or zigzag edges) on their chaotic behavior. Our results suggest that due to the free edges the chaoticity of GNRs is stronger than the periodic graphene sheet, and decreases by increasing width, tending asymptotically to the bulk value. In addition, the chaotic strength of armchair GNRs is higher than a zigzag ribbon of the same width. Further, we show that the composition of ${}^{12}C$ and ${}^{13}C$ carbon isotopes in graphene has a minor impact on its chaotic strength.
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 Schr
{o}dinger equation. We find that in both models (a) the wave packets 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 $t$ since the finite time maximum Lyapunov exponent $Lambda$ decays as $Lambda propto t^{alpha_{Lambda}}$, with $alpha_{Lambda} approx -0.37$ ($-0.46$) for the weak (strong) chaos case, and (c) the deviation vector distributions show the wandering of localized chaotic seeds in the lattices excited part, which induces the wave packets thermalization. We also propose a dimension-independent scaling between the wave packets spreading and chaoticity, which allows the prediction of the obtained $alpha_{Lambda}$ values.
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-Tsing
ou (FPUT) chain and the one- and two-dimensional disordered, discrete nonlinear Schrodinger 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 systems 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 $SRKN^a_{14}$ exhibit the best performance, respectively for moderate and high accuracy levels in the case of the FPUT chain, while for the DDNLS models $s9mathcal{ABC}6$ and $s11mathcal{ABC}6$ (moderate accuracy), along with $s17mathcal{ABC}8$ and $s19mathcal{ABC}8$ (high accuracy) proved to be the most efficient schemes.
