Spectral Stability of Deep Two-Dimensional Gravity Water Waves: Repeated Eigenvalues

The spectral stability problem for periodic traveling waves on a two-dimensional fluid of infinite depth is investigated via a perturbative approach, computing the spectrum as a function of the wave amplitude beginning with a flat surface. We generalize our previous results by considering the crucially important situation of eigenvalues with multiplicity greater than one (focusing on the generic case of multiplicity two) in the flat water configuration. We use this extended method of transformed field expansions (which now accounts for the resonant spectrum) to numerically simulate the evolution of the eigenvalues as the wave amplitude is increased. We observe that there are no instabilities that are analytically connected to the flat state: The spectrum loses its analyticity at the Benjamin–Feir threshold. We complement the numerical results with an explicit calculation of the first nonzero correction to the linear spectrum of resonant deep water waves. Two countably infinite families of collisions of eigenvalues with opposite Krein signature which do not lead to instability are presented.