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: 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 + 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.

我们考虑一个无限粒子链，其动力学由以下微分方程组控制：其中\(q_n(t)\)是第\(n\)个粒子在时间\(t\)沿链轴的位移，并且表示对时间的求导。我们假设所有粒子具有单位质量，并且相邻粒子之间的相互作用势\(V\)是一个凸的\(C^{\infty}\)函数。对于这个系统，我们证明了形如\(q_n(t)=q(\omega t - kn + \cdots)\)的\(C^{\infty}\)、时间周期的行波解的存在性，其中\(q\)是一个周期函数\(q(z)=q(z + 1)\)（周期归一化为等于\(1\)），\(\omega\)和\(k\)分别是频率和波数，\(\cdots\)是平均粒子间距，并且\(\cdots\)可以被选为一个任意参数。我们给出两种证明，一种基于变分原理，另一种基于拓扑方法，特别是度理论。对于小振幅波，基于微扰技术，我们描述行波的形式，并且我们推导出弱非线性色散关系。对于完全非线性的情况，当波的振幅较高时，我们使用数值方法来计算行波解和非线性色散关系。我们最后应用惠瑟姆平均拉格朗日方法来推导波参数\(\alpha\)、\(\beta\)、\(k\)和\(\omega\)的调制方程。©1999约翰威立父子公司