Existence and modulation of traveling waves in particle chains

We consider an infinite particle chain whose dynamics are governed by the f ollowing system of differential equations: q(n) = V'(q(n-1) - q(n)) - V'(q(n) - g(n+1)), n = 1, 2,..., 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 tha t all particles have unit mass and that the interaction potential V between adjacent particles is a convex C-infinity function. For this system, we pr ove the existence of C-infinity, time-periodic, traveling-wave solutions of the form q(n)(t) = q(wt - kn) + beta t - alpha n, where q is a periodic function q(z) = q(z + 1) (the period is normalized to equal 1), w and k are, respectively, the frequency and the wave number, al pha is the mean particle spacing, and beta can be chosen to be an arbitrary parameter. We present two proofs, one based on a variational principle and the other o n topological methods, in particular degree theory. For small-amplitude waves, based on perturbation techniques, we describe th e form of the traveling waves, and we derive the weakly nonlinear dispersio n relation. For the fully nonlinear case, when the amplitude of the waves i s high, we use numerical methods to compute the traveling-wave solution and the nonlinear dispersion relation. We finally apply Whitham's method of averaged Lagrangian to derive the modu lation equations for the wave parameters alpha, beta, k, and w. (C) 1999 Jo hn Wiley & Sons, Inc.