关于格点的孤子分解
On Soliton Resolution for a Lattice
摘要：
色散型演化偏微分方程的孤子分辨猜想（模糊地）是有限能量的一般初始数据渐近地产生一组后退孤子和一个衰减的背景辐射。在这封信中，我们调查了一个可能的扩展这一猜想的费米-Pasta-Ulam-Tsingou型离散格（而不是偏微分方程）在两种情况下：有限能量的初始数据的情况下，更一般的情况下，初始数据是一个周期函数的短程扰动。在第二种情况下，严格的结果户田晶格的启发，我们建议，孤子分辨率现象被取代的东西有点更复杂：一个周期函数的短程扰动实际上会引起不同的现象在不同的地区。除了区域的（渐近）纯周期性和区域的孤子在周期性的背景下，我们还观察到“调制”区域的快速振荡与缓慢变化的参数，如振幅和相位。我们已经进行了一些数值计算，以调查如果这种分割（纯周期性+孤子+调制振荡）持续任何离散格的Fermi-Pasta-Ulam-Tsingou型。对于小扰动的可积格，如线性调和格，朗缪尔链和Toda格，这是真的。但一般来说，甚至混沌现象也会发生。
英文摘要：
The soliton resolution conjecture for evolution PDEs of dispersive type states (vaguely) that generic initial data of finite energy give rise asymptotically to a set of receding solitons and a decaying background radiation. In this letter, we investigate a possible extension of this conjecture to discrete lattices of the Fermi-Pasta-Ulam-Tsingou type (rather than PDEs) in two cases: the case of finite energy initial data and a more general case where the initial data are a short range perturbation of a periodic function. In the second case, inspired by rigorous results on the Toda lattice, we suggest that the soliton resolution phenomenon is replaced by something somewhat more complicated: a short range perturbation of a periodic function actually gives rise to different phenomena in different regions. Apart from regions of (asymptotically) pure periodicity and regions of solitons in a periodic background, we also observe "modulated" regions of fast oscillations with slowly varying parameters like amplitude and phase. We have conducted some numerical calculations to investigate if this trichotomy (pure periodicity + solitons + modulated oscillations) persists for any discrete lattices of the Fermi-Pasta-Ulam-Tsingou type. For small perturbations of integrable lattices like the linear harmonic lattice, the Langmuir chain and the Toda lattice, this is true. But in general even chaotic phenomena can occur.
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