Elastic scattering of traveling phonons by discrete breathers as a time-periodic scatterer in the one-dimensional discrete nonlinear Schrödinger model: Exact analysis

Abstract

Discrete nonlinear Schrödinger model is a nonlinear lattice model used to investigate different nonlinear phenomena arising in many physical contexts. In this work we used this model to observe linear traveling phonon scattering by time-periodic discrete breathers in the piecewise smooth (PWS) version of nonlinearity. In one-dimensional system the single-channel scattering is found to be elastic (with the incoming and outgoing fluxes of energy being equal to each other). Considering one-site symmetric breather solution we are able to calculate the exact expression for the transmission coefficient (T) or the ratio of the transmitted to incident flux of energy in such a system using transfer-matrix formalism. From the condition of perfect transmission ( T=1 , and hence reflection coefficient R=0 ) or the ratio of reflected to incident flux of energy for elastic scattering is zero, our observation shows that perfect transmission happens to appear at the threshold of a localized mode, which occurs at the band edge of the extended eigenmodes (plane waves). We have also presented the results obtained from the condition of perfect reflection. The advantage of using the PWS version of nonlinearity in the model is that all the results derived are exact. Numerical simulations complement our results.