Recurrent bifurcations of stability spectra for steep Stokes waves in a deep fluid

TL;DR

This paper analyzes the stability spectra bifurcations of steep Stokes waves in deep fluids, revealing recurrent patterns and deriving normal forms using advanced pseudo-differential operator theory.

Contribution

It extends the analytic stability analysis of Stokes waves to singular pseudo-differential operators, identifying four key bifurcation types and validating normal form predictions numerically.

Findings

Four types of bifurcations are identified and characterized.

Normal forms accurately predict spectral band behaviors.

Numerical computations confirm theoretical predictions for wave steepness thresholds.

Abstract

We study the modulational stability problem for the traveling periodic waves (called Stokes waves) in an infinitely deep fluid by using pseudo-differential operators in conformal variables. We derive the criteria and the normal forms for four bifurcations which are repeated recurrently when the steepness of the Stokes wave is increased towards the highest wave with the peaked profile. The four bifurcations are observed in the following order: (a) new figure-8 bands appearing at each extremal point of speed, (b) degeneration of figure-8 bands resulting in vertical slopes, (c) new circular bands around the origin appearing at each period-doubling bifurcation, and (d) reconnection of figure-$\infty$ bands at each extremal point of energy. Our work uses the analytic theory of Stokes waves developed previously for Babenko's equation. The novelty of our work is the analytic extension of the modulational stability problem for singular pseudo-differential operators in terms of the Floquet parameter. The derivation of the normal form uses some structural assumptions which are known to be true for the Stokes waves. For the first and second bifurcation cycles, we compute numerically with a higher-order accuracy the actual values of wave steepness for which the structural assumptions are satisfied and the numerical coefficients of the normal forms to show the excellent agreement between the normal form theory and the numerical approximations of the spectral bands.