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 M1 and M2 and interact in accordance to a general nonlinear force F acting between the nearest neighbors. Their motion is described by the system of equations qqn = 1/M1(F(yn-1-yn)-F(yn-yn+1)), qqn+1 = 1/M2(F(yn-yn+1)-F(yn+1-yn+2)), where {yn}n = -∞^∞ 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-ωt. We determine the nonlinear dispersion relation. We also show that the system sustains binary oscillations with arbitrarily large amplitude.

Duke Scholars

Author Stephanos Venakides Mathematics