Quasiperiodic perturbations of Stokes waves: Secondary bifurcations and stability

TL;DR

This paper introduces a novel numerical approach using canonical conformal variables and Fourier-Floquet-Hill methods to analyze the spectrum and stability of Stokes waves, revealing new bifurcation points and extending stability results.

Contribution

The authors develop a spectral, matrix-free numerical method for eigenvalue problems in Stokes wave stability, identifying new bifurcation points and extending stability analysis to quasiperiodic eigenfunctions.

Findings

Identified new bifurcation points near the limiting Stokes wave.

Extended stability analysis to quasiperiodic eigenfunctions.

Confirmed and extended existing results on wave instabilities.

Abstract

We develop a numerical method based on canonical conformal variables to study two eigenvalue problems for operators fundamental to finding a Stokes wave and its stability in a 2D ideal fluid with a free surface in infinite depth. We determine the spectrum of the linearization operator of the quasiperiodic Babenko equation, and provide new results for eigenvalues and eigenvectors near the limiting Stokes wave identifying new bifurcation points via the Fourier-Floquet-Hill (FFH) method. We conjecture that infinitely many secondary bifurcation points exist as the limiting Stokes wave is approached. The eigenvalue problem for stability of Stokes waves is also considered. The new technique is extended to allow finding of quasiperiodic eigenfunctions by introduction of FFH approach to the canonical conformal variables based method. Our findings agree and extend existing results for the Benjamin-Feir, high-frequency and localized instabilities. For both problems the numerical methods are based on Krylov subspaces and do not require forming of operator matrices. Application of each operator is pseudospectral employing the fast Fourier transform (FFT), thus enjoying the benefits of spectral accuracy and $O(N \log N)$ numerical complexity. Extension to nonuniform grid spacing is possible via introducing auxiliary conformal maps.