Coupled wave method in the theory of a diatomic anharmonic lattice

We use the coupled wave method to investigate a one-dimensional nonlinear diatomic lattice. We consider the case where the two different lattice atoms have similar masses. This narrows the forbidden frequency gap between the longitudinal acoustic and optic phonon branches. As a result, the nonlinear shift of the gap's borders becomes comparable with the width of the gap itself. We find the exact solution, in the form of only two waves propagating towards each other. This is an important first step towards a full three-dimensional solution. We calculate the ratio of the two waves' amplitudes, and analyse the dispersive properties of the diatomic chain. Of special interest is how stable localized modes emerge in the forbidden frequency gap, both stationary and slowly propagating along the chain. We show that the envelope of their amplitudes has the form of solitary waves. What is crucial for such localized modes to appear is the spatial phase modulation of the particles' vibrations. This is because it makes the chain transparent at the margins of the forbidden gap, due to the self-action of the wave (optical switching).