Self-similarity and recurrence in stability spectra of near-extreme Stokes waves

TL;DR

This paper investigates the stability spectra of near-extreme Stokes waves in deep fluid, revealing self-similar recurrence patterns and providing new numerical insights into their instability mechanisms.

Contribution

It introduces matrix-free numerical methods to analyze the stability of high-amplitude waves near the steepest wave, uncovering self-similar spectral recurrence phenomena.

Findings

Spectral self-similarity observed near the origin

Numerical evidence supports conjectures on instability patterns

Recurrent spectral features as wave steepness increases

Abstract

We consider steady surface waves in an infinitely deep two--dimensional ideal fluid with potential flow, focusing on high-amplitude waves near the steepest wave with a 120 degree corner at the crest. The stability of these solutions with respect to coperiodic and subharmonic perturbations is studied, using new matrix-free numerical methods. We provide evidence for a plethora of conjectures on the nature of the instabilities as the steepest wave is approached, especially with regards to the self-similar recurrence of the stability spectrum near the origin of the spectral plane.