论文标题
Korteweg-de Vries和Fermi-Pasta-ulam-tsingou：准单向波的渐近可合转性
Korteweg-de Vries and Fermi-Pasta-Ulam-Tsingou: asymptotic integrability of quasi unidirectional waves
论文作者
论文摘要
在本文中，我们构建了在费米 - 帕斯塔 - 乌拉姆（FPU）链中准单向波的歧管的高阶扩展。我们还近似于该流形的动力学。作为扰动参数，我们使用$ h^2 = 1/n^2 $，其中$ n $是链的粒子数。众所周知，准单向波的动力学被Korteweg-de Vries（KDV）方程描述为一阶。在这里，我们证明了二阶的动力学由KDV层次结构中前两个非平地方程的组合控制 - 对于FPU电位中的任何参数选择。另一方面，我们发现只有当FPU电位的参数满足条件时，那么KDV层次结构中前三个非平凡方程的组合将准单向波的动力学确定为三阶的动力学。 TODA链满足了所需的条件。我们的结果表明了为什么FPU链（FPU悖论）在时间尺度上比KDV近似所解释的时间更长的时间尺度上的可接近综合行为，以及如何对系统的最终热量化负责的集成性（与KDV层次结构的分解）负责。
In this paper we construct a higher order expansion of the manifold of quasi unidirectional waves in the Fermi-Pasta-Ulam (FPU) chain. We also approximate the dynamics on this manifold. As perturbation parameter we use $h^2=1/n^2$, where $n$ is the number of particles of the chain. It is well known that the dynamics of quasi unidirectional waves is described to first order by the Korteweg-de Vries (KdV) equation. Here we show that the dynamics to second order is governed by a combination of the first two nontrivial equations in the KdV hierarchy -- for any choice of parameters in the FPU potential. On the other hand, we find that only if the parameters of the FPU potential satisfy a condition, then a combination of the first three nontrivial equations in the KdV hierarchy determines the dynamics of quasi unidirectional waves to third order. The required condition is satisfied by the Toda chain. Our results suggest why the close-to-integrable behavior of the FPU chain (the FPU paradox) persists on a time scale longer than explained by the KdV approximation, and also how a breakdown of integrability (detachment from the KdV hierarchy) may responsible for the eventual thermalization of the system.
