The Benjamin-Feir instability in the infinite depth

TL;DR

This paper rigorously proves that small amplitude Stokes' waves in infinite-depth water are spectrally unstable to slow modulations, confirming a long-standing theoretical prediction by Benjamin and Feir.

Contribution

It provides a rigorous mathematical proof of the Benjamin-Feir instability for small amplitude waves in infinite-depth water.

Findings

Small amplitude Stokes' waves are spectrally unstable to slow modulation.

The instability is rigorously justified, confirming Benjamin and Feir's formal argument.

The result applies to two-dimensional, irrotational flows with infinite depth.

Abstract

We prove that a Stokes' periodic wave of sufficiently small amplitude, traveling under gravity at the free surface of a two dimensional, infinitely deep, and irrotational flow, is spectrally unstable to slow modulation, rigorously justifying Benjamin and Feir's formal argument.