Quasiperiodic perturbations of Stokes waves: Secondary bifurcations and stability

We develop a numerical method to study eigenvalue problems for operators fundamental to Stokes wave and its stability in a 2D ideal fluid with a free surface and infinite depth. The method allows to determine the spectrum of the linearization operator of the quasiperiodic Babenko equation. We illustrate by providing new results for eigenvalues and eigenvectors near the limiting Stokes wave and identify new bifurcation point to double-period waves. An infinite number of such points is conjectured as the limiting Stokes wave is approached. The eigenvalue problem for stability is also considered. The method in [1] is extended to allow finding of quasiperiodic eigenfunctions by introducing the Fourier-Floquet-Hill (FFH) approach in canonical conformal variables. Our findings agree and extend existing results for the Benjamin-Feir, high-frequency and localized instabilities, see also Refs. [2]. The numerical method is matrix-free and is based on Krylov subspaces. All operators appearing in the problems are pseudospectral and employ the fast Fourier transform (FFT), thus enjoying the benefits of spectral accuracy and O(Nlog⁡N) numerical complexity.