It is proven that small-amplitude steady periodic water waves with infinite depth are unstable with respect to long-wave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. It has been proven rigorously only in the case of finite depth. We provide a completely different and self-contained approach to prove the spectral modulational instability for water waves in both the finite and infinite depth cases. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.

中文翻译：

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深水中斯托克斯波调制不稳定性的证明

已证明具有无限深度的小振幅稳定周期性水波对于长波扰动是不稳定的。半个多世纪前，Benjamin 和 Feir 首次观察到这种调制不稳定性。仅在有限深度的情况下得到严格证明。我们提供了一种完全不同且独立的方法来证明在有限和无限深度情况下水波的光谱调制不稳定性。© 2022 Courant Institute of Mathematics 和 Wiley Periodicals LLC。