Proof of modulational instability of Stokes waves in deep water

TL;DR

This paper rigorously proves the spectral modulational instability of small-amplitude steady periodic water waves in both finite and infinite depth, confirming long-standing observations about wave instability in deep water.

Contribution

It introduces a new, self-contained method to prove modulational instability for water waves in both finite and infinite depth, extending previous finite-depth results.

Findings

Spectral modulational instability is proven for deep water waves.

The approach is self-contained and applicable to both finite and infinite depth cases.

Confirms long-standing observations of wave instability in deep water.

Abstract

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.