A stability of nearly limiting Stokes waves to superharmonic perturbations is considered numerically in approximation of an infinite depth. Investigation of the stability properties can give one an insight into the evolution of the Stokes wave. The new, previously inaccessible branches of superharmonic instability were investigated. Our numerical simulations suggest that eigenvalues of linearized dynamical equations, corresponding to the unstable modes, appear as a result of a collision of a pair of purely imaginary eigenvalues at the origin, and a subsequent appearance of a pair of purely real eigenvalues: a positive and a negative one that are symmetric with respect to zero. Complex conjugate pairs of purely imaginary eigenvalues correspond to stable modes, and as the steepness of the underlying Stokes wave grows, the pairs move toward the origin along the imaginary axis. Moreover, when studying the eigenvalues of linearized dynamical equations we find that as the steepness of the Stokes wave grows, the real eigenvalues follow a universal scaling law, that can be approximated by a power law. The asymptotic power law behavior of this dependence for instability of Stokes waves close to the limiting one is proposed. Surface elevation profiles for several unstable eigenmodes are made available through http://stokeswave.org website.

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 Ofalse(NlogNfalse)$O(N\log N)$ flops, where N is the number of Fourier modes needed to resolve a Stokes wave. Additionally, due to the matrix‐free approach, Ofalse(N2false)$O(N^2)$ 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.