颗粒链中的周期行波和双光子
Periodic travelling waves and compactons in granular chains
摘要：
我们研究了一种不寻常的类型的周期性行波的传播链相同的珠子通过赫兹的接触力相互作用。由于赫兹相互作用的特殊性质（在压缩下完全非线性，在没有接触的情况下消失），每个珠子周期性地经历压缩阶段，然后自由飞行。我们证明了这种波的存在接近二进制振荡，并数值继续这些解决方案时，他们的波长增加。在长波极限，我们观察到他们的收敛冲击配置文件组成的小压缩区域接近孤立波，交替与大域的自由飞行珠速度小。我们给出正式的参数来证明这种渐近行为，使用匹配技术和以前的孤立波解的结果。这种波的数值发现意味着存在的压缩波，即压缩支撑的压缩波传播速度恒定，取决于振幅和宽度的波。珠子是静止的，在波的外部被相等的间隙分开，并且波到达的每个珠子在有限的时间间隔内移动有限的距离。低于临界波数，我们观察到快速不稳定的周期性行波导致一个无序的制度。
英文摘要：
We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz's contact forces. Each bead periodically undergoes a compression phase followed by a free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behaviour, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wavenumber, we observe fast instabilities of the periodic travelling waves leading to a disordered regime.
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