Wave propagation and resonance in a one-dimensional nonlinear discrete periodic medium

We consider wave propagation in a nonlinear infinite diatomic chain of particles as a discrete model of propagation in a medium whose properties vary periodically in space. The particles have alternating masses M-1 and M-2 and interact in accordance to a general nonlinear force F acting between the nearest neighbors. Their motion is described by the system of equations y(n) = 1/M-1 (F(y(n,1) + y(n)) + F(y(n) + y(n+1))); y(n+1) = 1/M-2 (F(y(n) + y(n+1)) + F(y(n+1) + y(n+2))); where {y(n)}(n=-infinity)(infinity) is the position of the nth particle. Using Fourier series methods and tools from bifurcation theory, we show that, for nonresonant wave-numbers k; this system admits nontrivial small-amplitude traveling wave solutions g and h; depending only on the linear combination z = kn + omega t. We determine the nonlinear dispersion relation. We also show that the system sustains binary oscillations with arbitrarily large amplitude.