Existence and modulation of traveling waves in particles chains

We consider an infinite particle chain whose dynamics are governed by the following system of differential equations:$$\ddot q_n = V'(q_{n-1} - q_n) - V'(q_n - q_{n+1}), n=1,2,\ldots, $$where q n ( t ) is the displacement of the n th particle at time t along the chain axis and denotes differentiation with respect to time. We assume that all particles have unit mass and that the interaction potential V between adjacent particles is a convex C ∞ function. For this system, we prove the existence of C ∞, time‐periodic, traveling‐wave solutions of the formq n ( t ) = q ( wt kn + where q is a periodic function q ( z ) = q ( z +1) (the period is normalized to equal 1), ω and k are, respectively, the frequency and the wave number, is the mean particle spacing, and can be chosen to be an arbitrary parameter. We present two proofs, one based on a variational principle and the other on topological methods, in particular degree theory. For small‐amplitude waves, based on perturbation techniques, we describe the form of the traveling waves, and we derive the weakly nonlinear dispersion relation. For the fully nonlinear case, when the amplitude of the waves is high, we use numerical methods to compute the traveling‐wave solution and the non‐linear dispersion relation. We finally apply Whitham's method of averaged Lagrangian to derive the modulation equations for the wave parameters α, β, k , and ω. © 1999 John Wiley & Sons, Inc.