Spreading of wave packets in disordered systems with tunable nonlinearity

We study the spreading of single-site excitations in one-dimensional disordered Klein-Gordon chains with tunable nonlinearity $|u_{l}|^{\sigma} u_{l}$ for different values of $\sigma$. We perform extensive numerical simulations where wave packets are evolved a) without and, b) with dephasing in normal mode space. Subdiffusive spreading is observed with the second moment of wave packets growing as $t^{\alpha}$. The dependence of the numerically computed exponent $\alpha$ on $\sigma$ is in very good agreement with our theoretical predictions both for the evolution of the wave packet with and without dephasing (for $\sigma \geq 2$ in the latter case). We discuss evidence of the existence of a regime of strong chaos, and observe destruction of Anderson localization in the packet tails for small values of $\sigma$.Comment: 9 pages, 7 figure

Chaoticity without thermalisation in disordered lattices

We study chaoticity and thermalization in Bose-Einstein condensates in disordered lattices, described by the discrete nonlinear Schr\"odinger equation (DNLS). A symplectic integration method allows us to accurately obtain both the full phase space trajectories and their maximum Lyapunov exponents (mLEs), which characterize their chaoticity. We find that disorder destroys ergodicity by breaking up phase space into subsystems that are effectively disjoint on experimentally relevant timescales, even though energetically, classical localisation cannot occur. This leads us to conclude that the mLE is a very poor ergodicity indicator, since it is not sensitive to the trajectory being confined to a subregion of phase space. The eventual thermalization of a BEC in a disordered lattice cannot be predicted based only on the chaoticity of its phase space trajectory

Characteristics of chaos evolution in one-dimensional disordered nonlinear lattices

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 Schr\"{o}dinger equation model. Completing previous investigations \cite{SGF13} 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 $\Lambda$ decays in time $t$ as $\Lambda \propto t^{\alpha_{\Lambda}}$, with $\alpha_{\Lambda}$ being different from the $\alpha_{\Lambda}=-1$ value observed in cases of regular motion. In particular, $\alpha_{\Lambda}\approx -0.25$ (weak chaos) and $\alpha_{\Lambda}\approx -0.3$ (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 $\Lambda$ reveals the meandering of chaotic seeds inside the wave packet, which is needed for obtaining the chaotization of the lattice's excited part.Comment: 11 pages, 10 figure

How does the Smaller Alignment Index (SALI) distinguish order from chaos?

The ability of the Smaller Alignment Index (SALI) to distinguish chaotic from ordered motion, has been demonstrated recently in several publications.\cite{Sk01,GRACM} Basically it is observed that in chaotic regions the SALI goes to zero very rapidly, while it fluctuates around a nonzero value in ordered regions. In this paper, we make a first step forward explaining these results by studying in detail the evolution of small deviations from regular orbits lying on the invariant tori of an {\bf integrable} 2D Hamiltonian system. We show that, in general, any two initial deviation vectors will eventually fall on the ``tangent space'' of the torus, pointing in different directions due to the different dynamics of the 2 integrals of motion, which means that the SALI (or the smaller angle between these vectors) will oscillate away from zero for all time.Comment: To appear in Progress of Theoretical Physics Supplemen