On intense energy exchange and localization in periodic FPU dimer chains

Abstract

Dynamics of periodically heterogeneous, nonlinear lattices is a subject of intense research in great variety of aspects of theoretical and applied physics. In fact almost all the studies pursued to date in the field of lattice dynamics have been mostly concerned with the analysis of stationary and near stationary regimes. However, not much has been done with respect to highly non-stationary processes existing in periodically heterogeneous nonlinear lattices such as FPU dimers etc. The system under consideration in the present paper is the FPU dimer chain given to periodic boundary conditions and composed of two identical cells of particles. Each cell of the chain comprises exactly one heavy particle succeeded by a group of N light particles of identical masses (i.e. 1:N dimer chain). In the current work we present an extensive analytical study of the mechanism of the formation of highly non-stationary regimes excited in FPU dimer chains subject to periodic boundary conditions. In particular we show that the initial excitation provided to a single group of light particles (belonging to a single cell), may lead to the two opposing regimes, namely energy localization on the same (initially excited) group of light particles as well as the recurrent (near complete) energy exchanges between the two groups of light particles of both cells. It is important to emphasize that for both the regimes heavy particles remain near completely stationary. An analytical study of the dynamics of the system under consideration reveals the threshold value of the parameter of nonlinearity (stiffness nonlinearity) above which the initial energy supplied to a specific group of light particles of the chain gets permanently localized on it. The developed analytical procedure correctly predicts the threshold value for the transition from the regime of strong energy exchange between the two groups of light particles to the permanent energy localization on a single group. It should be emphasized that to date the analytical procedure developed in the paper is the only possible tool for addressing the highly non-stationary regimes which by no means can be replaced with the well-known modal analysis. Results of analytical predictions are in a spectacular correspondence with those of numerical simulations.