Spectral Stability of Deep Two-Dimensional Gravity-Capillary Water Waves

Abstract

In this contribution we study the spectral stability problem for periodic traveling gravity-capillary waves on a two-dimensional fluid of infinite depth. We use a perturbative approach that computes the spectrum of the linearized water wave operator as an analytic function of the wave amplitude/slope. We extend the highly accurate method of Transformed Field Expansions to address surface tension in the presence of both simple and repeated eigenvalues, then numerically simulate the evolution of the spectrum as the wave amplitude is increased. We also calculate explicitly the first nonzero correction to the flat-water spectrum, which we observe to accurately predict the stability (or instability) for all amplitudes within the disk of analyticity of the spectrum. With this observation in mind, the disk of analyticity of the flat state spectrum is numerically estimated as a function of the Bond number and the Bloch parameter, and compared to the value of the wave slope at the first finite amplitude eigenvalue collision. © 2012 by the Massachusetts Institute of Technology.