Spreading of energy in the Ding-Dong model

Abstract

We study the properties of energy spreading in a lattice of elastically colliding harmonic oscillators (Ding-Dong model). We demonstrate that in the regular lattice the spreading from a localized initial state is mediated by compactons and chaotic breathers. In a disordered lattice, the compactons do not exist, and the spreading eventually stops, resulting in a finite configuration with a few chaotic spots.

Energy spreading in strongly nonlinear disordered lattices

“…We have found that with the FNDE it is possible to explain in a consistent way the subdiffusive spreading behavior and the energy scaling of spreading states. Analysis of self-similar solutions of the FNDE not only predicts a subdiffusive spreading, but also induces a scaling of time and energy of the spreading process according to relations (21) and (22), which depend on parameters γ and a, responsible for the index of the fractional time derivative and of the nonlinearity, respectively. We tested these scaling laws on a class of nonlinearly coupled oscillators with different values of the nonlinear indices κ (local nonlinearity) and λ (coupling nonlinearity).…”

“…In contrast, the strongly nonlinear lattices studied here have only a nearestneighbor coupling without a random coupling coefficient. Similarly to the studies before, we analyze the excitation times T (L) as a function of excitation length L for different energies E to check the predictions of the FNDE scaling (22). At first, we report the results for κ = 2 and λ = 4.…”

“…From (22) one finds that for γ = 1 the slope of the rescaled curves is simply given by a(w) + 1. Thus we evaluate this slope by means of a polynomial fit and plot the resulting numerical value for a(w) in the insets in figures 9(b) and 10(b).…”

“…A complete theoretical understanding of the subdiffusive behavior has not been presented, but it is mostly agreed that the spreading in these models is induced by weak chaos. However, the true asymptotic behavior is still discussed in some of these models with recent claims that spreading might stop due to an extinction of chaos [17,21,22].…”

Introduction to Focus Issue: Statistical mechanics and billiard-type dynamical systems

“…This is done by Gaspard and Gilbert, 21 who analyze the energy transfer processes of interacting hard spheres. Energy spreading is discussed in the work of Roy and Pikovsky 22 in the regular lattice of the Ding-Dong model. It is natural to ask if billiard particles can acquire unlimited energy when colliding with a moving boundary.…”