Canonical conformal variables based method for stability of Stokes waves

We study the stability of Stokes waves in an ideal fluid of infinite depth. The perturbations that are either coperiodic with a Stokes wave (superharmonics) or integer multiples of its period (subharmonics) are considered. The eigenvalue problem is formulated using the conformal canonical Hamiltonian variables and admits numerical solution in a matrix-free manner. We find that the operator matrix of the eigenvalue problem can be factored into a product of two operators: a self-adjoint operator and an operator inverted analytically. Moreover, the self-adjoint operator matrix is efficiently inverted by a Krylov-space-based method and enjoys spectral accuracy. Application of the operator matrix associated with the eigenvalue problem requires only (Formula presented.) flops, where N is the number of Fourier modes needed to resolve a Stokes wave. Additionally, due to the matrix-free approach, (Formula presented.) storage for the matrix of coefficients is no longer required. The new method is based on the shift-invert technique, and its application is illustrated in the classic examples of the Benjamin–Feir and the superharmonic instabilities. Simulations confirm numerical results of preceding works and recent theoretical work for the Benjamin–Feir instability (for small amplitude waves), and new results for large amplitude waves are shown.