The nonlinear Schrödinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that there exists a kinetic time scale depending on the frequency localization of the data and on the strength of the nonlinearity, on which the expectation of the squares of moduli of Fourier modes evolve according to an effective equation: the so‐called kinetic wave equation. When the kinetic time for our setup is 1, we prove this conjecture up to an arbitrarily small polynomial loss. When the kinetic time is larger than 1, we obtain its validity on a more restricted time scale. The key idea of the proof is the use of Feynman interaction diagrams both in the construction of an approximate solution and in the study of its nonlinear stability. We perform a truncated series expansion in the initial data, and obtain bounds in average in various function spaces for its elements. The linearized dynamics then involves a linear Schrödinger equation with a corresponding random potential whose operator norm in Bourgain spaces we are able to estimate on average. This gives a new approach for the analysis of nonlinear wave equations out of equilibrium, and gives hope that refinements of the method could help settle the conjecture.

中文翻译：

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关于齐次动波方程的推导


考虑了以随机高斯场作为初始数据的弱非线性状态下的非线性薛定谔方程。问题设置在圆环上任何大于 2 的维度上。统计物理学中的一个猜想是，存在一个动力学时间尺度，具体取决于数据的频率定位和非线性的强度，在这个尺度上，傅里叶模数平方的期望根据一个有效的方程演变：即所谓的动力学波动方程。当我们设置的动力学时间为 1 时，我们将这个猜想证明为任意小的多项式损失。当动力学时间大于 1 时，我们在更有限的时间尺度上获得其有效性。该证明的关键思想是在构建近似解和研究其非线性稳定性时使用费曼交互图。我们在初始数据中执行截断级数展开，并获取其元素在各种函数空间中的 average 边界。然后，线性动力学涉及一个线性薛定谔方程，该方程具有相应的随机势，我们能够平均估计其在布尔干空间中的算子范数。这为分析失衡的非线性波动方程提供了一种新方法，并有望改进该方法有助于解决猜想。