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.

翻译：我们开发了一种基于正则共形变量的数值方法，用于研究两个算子特征值问题，这两个算子对于求解二维无限深理想流体自由表面上斯托克斯波及其稳定性至关重要。通过确定准周期Babenko方程线性化算子的谱，并利用Fourier-Floquet-Hill方法识别极限斯托克斯波附近的新分岔点，我们提供了特征值与特征向量的新结果。我们推测，随着极限斯托克斯波的逼近，存在无穷多个二次分岔点。此外，本文还考虑了斯托克斯波稳定性的特征值问题。通过将FFH方法引入基于正则共形变量的方法，该新技术得以扩展，能够求解准周期本征函数。我们的结果与现有的Benjamin-Feir不稳定性、高频不稳定性及局部不稳定性研究一致，并拓展了相关结论。针对这两个问题，数值方法均基于Krylov子空间，无需显式形成算子矩阵。每个算子的应用采用赝谱方法结合快速傅里叶变换，因此兼具谱精度与O(N log N)数值复杂度的优势。通过引入辅助共形映射，该方法可推广至非均匀网格间距的情形。