Consider the cubic nonlinear Schrödinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

中文翻译：

---

非线性Schrödinger方程长时间行为的波动湍流描述

考虑在d维环面上设置的三次非线性Schrödinger方程，其傅里叶系数具有相位均匀且独立的相位的数据。我们表明，平均而言，傅立叶系数模的演化是由在非平凡的时间尺度上由所谓的波动力学方程控制的，该波动方程是在波湍流理论中预测的。