Proof of the transverse instability of Stokes waves

TL;DR

This paper rigorously proves that Stokes waves are transversely unstable to three-dimensional perturbations, confirming numerical predictions and extending previous two-dimensional analyses to three dimensions.

Contribution

It provides the first rigorous proof of three-dimensional transverse instability of Stokes waves, including detailed calculations up to third order in wave amplitude.

Findings

Transverse perturbations cause exponential growth in Stokes waves.

First rigorous proof of 3D instabilities in Stokes waves.

Extends previous 2D instability results to 3D case.

Abstract

A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. In 1981 McLean discovered via numerical methods that Stokes waves at infinite depth are unstable with respect to transverse perturbations of the initial data. Even for a Stokes wave that has very small amplitude $\varepsilon$, we prove rigorously that transverse perturbations, after linearization, will lead to exponential growth in time. To observe this instability, extensive calculations are required all the way up to order $O(\varepsilon^3)$. All previous rigorous results of this type were merely two-dimensional, in the sense that they only treated long-wave perturbations in the longitudinal direction. This is the first rigorous proof of three-dimensional instabilities of Stokes waves.