The Alber equation is a moment equation for the nonlinear Schrödinger equation, formally used in ocean engineering to investigate the stability of stationary and homogeneous sea states in terms of their power spectra. In this work we present the first well-posedness theory for the Alber equation with the help of an appropriate equivalent reformulation. Moreover, we show linear Landau damping in the sense that, under a stability condition on the homogeneous background, any inhomogeneities disperse and decay in time. The proof exploits novel $ L^2 $ space-time estimates to control the inhomogeneity and our result applies to any regular initial data (without a mean-zero restriction). Finally, the sufficient condition for stability is resolved, and the physical implications for ocean waves are discussed. Using a standard reference dataset (the "North Atlantic Scatter Diagram") it is found that the vast majority of sea states are stable, but modulationally unstable sea states do appear, with likelihood $ O(1/1000); $ these would be the prime breeding ground for rogue waves.

中文翻译：

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Alber方程的强解和单向波谱的稳定性

Alber方程是非线性Schrödinger方程的矩方程，正式用于海洋工程，以研究静态和均质海态的功率谱稳定性。在这项工作中，我们借助于适当的等效公式，提出了Alber方程的第一个适定性理论。此外，我们表现出线性Landau阻尼的意义是，在均匀背景下的稳定条件下，任何不均匀性都会随时间散布和衰减。该证明利用新颖的$ L ^ 2 $时空估计来控制不均匀性，我们的结果适用于任何常规初始数据（无均值零约束）。最后，解决了稳定的充分条件，并讨论了对海浪的物理影响。使用标准参考数据集（“