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We develop a numerical method based on canonical conformal variables to studytwo eigenvalue problems for operators fundamental to finding a Stokes wave andits stability in a 2D ideal fluid with a free surface in infinite depth. Wedetermine the spectrum of the linearization operator of the quasiperiodicBabenko equation, and provide new results for eigenvalues and eigenvectors nearthe limiting Stokes wave identifying new bifurcation points via theFourier-Floquet-Hill (FFH) method. We conjecture that infinitely many secondarybifurcation points exist as the limiting Stokes wave is approached. Theeigenvalue problem for stability of Stokes waves is also considered. The newtechnique is extended to allow finding of quasiperiodic eigenfunctions byintroduction 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 numericalmethods are based on Krylov subspaces and do not require forming of operatormatrices. Application of each operator is pseudospectral employing the fastFourier transform (FFT), thus enjoying the benefits of spectral accuracy and$O(N \log N)$ numerical complexity. Extension to nonuniform grid spacing ispossible via introducing auxiliary conformal maps.

Quasiperiodic perturbations of Stokes waves: Secondary bifurcations and stability