An alternative proof of modulation instability of Stokes waves in deep water

TL;DR

This paper provides an alternative proof of the modulational instability of Stokes waves in deep water, extending the spectral stability analysis using the Evans function approach to infinite depth scenarios.

Contribution

It generalizes the Evans function method to infinite depth water, offering a new proof of Benjamin-Feir instability and analyzing spectral stability at resonant frequencies.

Findings

Small amplitude waves are always low-frequency unstable in deep water.

No additional instability at non-zero resonant frequencies.

Provides an alternative proof to existing results for infinite depth cases.

Abstract

We generalize the periodic Evans function approach recently used to study the spectral stability of Stokes wave and gravity-capillary (including Wilton ripples) in water of finite depth to study spectral stability of Stokes waves in water of infinite depth. We prove waves of sufficiently small amplitude are always low-frequency unstable regardless of the wave number and gravity, giving an alternative proof for the Benjamin-Feir modulational instability in the infinite depth case. Here, the first proof for the infinite depth case is recently obtained by Nguyen and Strauss. We also study the spectral stability at non-zero resonant frequencies and find no additional instability.