论文标题
均匀动力学方程的推导：更长的时间尺度
Derivation of the homogeneous kinetic wave equation: longer time scales
论文作者
论文摘要
我们考虑在扁平的圆环上设置的非线性schrödinger方程，该方程在猜想中导致动力波方程。特别是，数据是随机的，并在弱非线性方向上扩散到高频。我们对以前的论文进行了调查，并表明，在圆环是标准的情况下，只有在那里考虑的缩放率才能使戴森系列的收敛到动力学时间尺度。我们还表明，对于通用二次分散关系（非矩形托里），戴森级数会在明显更长的时间尺度上收敛。对于较大的量表，我们能够到达动力学时间至任意小的多项式误差。这些结果表明了分散关系的确切结构的重要性，更具体地说明了某些双线性量的等均分配属性，类似于从中得出的配对相关性。
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.
