Theory of intrinsic localized modes in diatomic chains: beyond the rotating wave approximation

Intrinsic localized modes in the gap of a diatomic chain with free ends are discussed in detail by going beyond the rotating wave approximation. We include in the time dependence of the displacements terms up to cos (2ωt). We consider a finite chain of particles interacting with nearest-neighbour interactions. We study amplitudes of the intrinsic localized modes smaller than 0.25 A. In this range of amplitudes the full potential can be well represented by an expansion in powers of the displacements up to fourth-order terms. The use of a force constant model allows us to simplify the problem. As a test case we consider a chain of LiI atoms. We found intrinsic localized modes in the gap. The amplitudes of the first harmonic term (cos (ωt)) are of even or odd parity, whereas we prove that the amplitudes of the static part and those of the second harmonic can have only even symmetry. The main result of the paper is that the amplitudes associated with the second harmonic are two or three orders of magnitude less than those of the first harmonic. Furthermore, the frequency of the localized modes are modified by less than 1% by the inclusion of the second harmonic.