We introduce a simplified model for wave turbulence theory—the Wick nonlinear Schrödinger equation, of which the main feature is the absence of all self‐interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture.

中文翻译：

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Wick 非线性薛定谔方程的非均匀湍流

我们引入波浪湍流理论的简化模型——Wick非线性薛定谔方程，其主要特征是其解的相关展开中不存在所有自相互作用。对于该模型，我们推导了几个波动动力学方程，这些方程控制其解在不同状态下的有效统计行为。在均匀设置中，初始相关性是平移不变的，我们得到了一个类似于形式理论预测的波动动力学方程。在非均匀设置中，我们获得了一个波动动力学方程，该方程描述了解的波包的统计行为，同时考虑了波包的传输和波包之间的碰撞。另一个波动力学方程在文献中似乎是新的，也出现在这种设置的一定比例范围内，并提供了更精细的碰撞图。