Propagation of modulated waves in narrow-bandpass one-dimensional lattices.

Abstract

We consider the problem of the propagation of modulated waves in one-dimensional discrete lattices with linear bandpass-type dispersion relation. We are interested specifically in the cases where the gap frequency f_{0} and the cutoff frequency f_{max} verify 0<f_{0}<f_{max}<2f_{0}. Analytical investigations of such models commonly use the rotating wave approximation in which only the fundamental harmonic is taken into consideration. This approach is routinely justified by the physical argument that the zeroth and higher harmonic frequency terms cannot propagate, being out of band. Using a simple model of electrical lattice as a case study, we show analytically that accounting for the generation of those terms is indispensable for the accurate prediction of modulated solitary waves supported by the model. Moreover, a carefully designed numerical investigation of the propagation of an exact and consistent second-order approximation of such waves reveals that the second-order harmonic included is transmitted throughout the network the same as the fundamental harmonic is. Thus, we unveil the pitfall of the rotating wave approximation that is detrimental to its reliability for most of the models to which it has commonly been employed. We also suggest how to avoid this pitfall.