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 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 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).

We consider a mechanical lattice inspired by the Su-Schrieffer-Heeger model along with cubic Klein-Gordontype nonlinearity. We investigate the long-time dynamics of the nonlinear edge states, which are obtained by nonlinear continuation of topological edge states of the linearized model. Linearly unstable edge states delocalize and lead to chaos and thermalization of the lattice. Linearly stable edge states also reach the same fate, but after a critical strength of perturbation is added to the initial edge state. We show that the thermalized lattice in all these cases shows an effective renormalization of the dispersion relation. Intriguingly, this renormalized dispersion relation displays a unique symmetry, i.e., its square is symmetric about a finite squared frequency, akin to the chiral symmetry of the linearized model.