Existence and modulation of traveling waves in particle chains
We consider an infinite particle chain whose dynamics are governed by the following system of differential equations: q̈n = V′ (qn-1 - qn) - V′ (qn - qn+1), n = 1,2, . . . , where qn(t) is the displacement of the nth 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 form qn(t) = q(wt - kn) + βt - αn, 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, α is the mean particle spacing, and β 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 nonlinear dispersion relation. We finally apply Whitham's method of averaged Lagrangian to derive the modulation equations for the wave parameters α, β, k, and w. © 1999 John Wiley & Sons, Inc.
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