Fermi-Pasta-Ulam格点上的椭圆余弦波
Cnoidal Waves on Fermi-Pasta-Ulam Lattices
摘要：
本文研究了无限多粒子链在非线性弹簧作用下的运动方程[ \ddot{q}_n = V '（q_{n+1} -q_n）-V'（q_n- q_{n-1}）]，其最近邻势为一般的$V$ .我们表明，这个链进行精确的空间周期行波的轮廓是渐近的，在一个小振幅长波制度，KdV椭圆余弦波。离散波有三个有趣的特征：（1）作为精确的行波，它们在无限时间内保持其形状，而不是仅仅达到形式渐近分析所建议的波长$^{-3}$量级的时间尺度，（2）与孤立波不同，它们携带每个粒子的非零能量，（3）它们的KdV连续对应物的类似行为表明它们在彼此非线性相互作用下的长时间稳定性。与费米-帕斯塔-乌拉姆复发现象的连接表示。证明涉及Friesecke和Pego（1999）的重整化方法对周期设置的适应，以及具有KdV椭圆余弦波势的周期薛定谔算子的谱理论。
英文摘要：
We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion [\ddot{q}_n = V'(q_{n+1}-q_n) - V'(q_n-q_{n-1})] with generic nearest-neighbour potential $V$. We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength$^{-3}$ suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of Friesecke and Pego (1999) to a periodic setting and the spectral theory of the periodic Schrödinger operator with KdV cnoidal wave potential.
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