均相动力学wave equation的推导：更长的时间尺度
Derivation of the homogeneous kinetic wave equation: longer time scales
摘要：
我们考虑的非线性薛定谔方程设置在一个平坦的环面上，在该制度，这是紧张导致动力学波动方程，特别是，数据是随机的，并蔓延到高频在弱非线性制度。我们追求我们以前的论文的调查，并表明，在环面是标准的情况下，只有考虑缩放有允许收敛的戴森系列的动力学时间尺度。我们还表明，对于一般的二次色散关系（非矩形环面），戴森系列收敛显着更长的时间尺度，我们能够达到一个任意小的多项式误差为一个更大的缩放的动力学时间。这些结果表明色散关系的精确结构的重要性，更具体地说，一些双线性量的equidistribution属性类似于对相关性来自它。
英文摘要：
We consider the nonlinear Schrödinger equation set on a flat torus, in the regime which is conjectured to lead to the kinetic wave equation; in particular, the data are random, and spread up to high frequency in a weakly nonlinear regime. We pursue the investigations of our previous paper, and show that, in the case where the torus is the standard one, only the scaling considered there allows convergence of the Dyson series up to the kinetic time scale. We also show that, for generic quadratic dispersion relations (non rectangular tori), the Dyson series converges on significantly longer time scales; we are able to reach the kinetic time up to an arbitrarily small polynomial error for a larger set of scalings. These results show the importance of the exact structure of the dispersion relation, more specifically of equidistribution properties of some bilinear quantities akin to pair correlations derived from it.
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