Unstable Stokes waves

TL;DR

This paper analyzes the spectral instability of small-amplitude periodic Stokes waves in water, developing a new analytical approach and providing evidence that all such waves are likely spectrally unstable.

Contribution

It introduces a periodic Evans function method to prove spectral instability of Stokes waves beyond the Benjamin--Feir instability and offers numerical validation for the conjecture of universal instability.

Findings

Spectral instability occurs near the origin for certain wave numbers.

Resonance can induce instability even outside the Benjamin--Feir regime.

Numerical evidence supports the conjecture that all small-amplitude Stokes waves are unstable.

Abstract

We investigate the spectral instability of a $2\pi/\kappa$ periodic Stokes wave of sufficiently small amplitude, traveling in water of unit depth, under gravity. Numerical evidence suggests instability whenever the unperturbed wave is resonant with its infinitesimal perturbations. This has not been analytically studied except for the Benjamin--Feir instability in the vicinity of the origin of the complex plane. Here we develop a periodic Evans function approach to give an alternative proof of the Benjamin--Feir instability and, also, a first proof of spectral instability away from the origin. Specifically, we prove instability near the origin for $\kappa>\kappa_1:=1.3627827\dots$ and instability due to resonance of order two so long as an index function is positive. Validated numerics establishes that the index function is indeed positive for some $\kappa<\kappa_1$, whereby there exists a Stokes wave that is spectrally unstable even though it is insusceptible to the Benjamin--Feir instability. The proofs involve center manifold reduction, Floquet theory, and methods of ordinary and partial differential equations. Numerical evaluation reveals that the index function remains positive unless $\kappa=1.8494040\dots$. Therefore, we conjecture that all Stokes waves of sufficiently small amplitude are spectrally unstable. For the proof of the conjecture, one has to verify that the index function is positive for $\kappa$ sufficiently small.