Transverse Dynamics of Two-Dimensional Gravity–Capillary Periodic Water Waves

Abstract

We study the transverse dynamics of two-dimensional gravity–capillary periodic water waves in the case of large surface tension. In this parameter regime, predictions based on model equations suggest that periodic traveling waves are stable with respect to two-dimensional perturbations, and unstable with respect to three-dimensional perturbations which are periodic in the direction transverse to the direction of propagation. In this paper, we confirm the second prediction. We show that, as solutions of the full water-wave equations, the periodic traveling waves are linearly unstable under such three-dimensional perturbations. In addition, we study the nonlinear bifurcation problem near these transversely unstable two-dimensional periodic waves. We show that a one-parameter family of three-dimensional doubly periodic waves is generated in a dimension-breaking bifurcation. The key step of this approach is the analysis of the purely imaginary spectrum of the linear operator obtained by linearizing the water-wave equations at a periodic traveling wave. Transverse linear instability is then obtained by a perturbation argument for linear operators, and the nonlinear bifurcation problem is studied with the help of a center-manifold reduction and the classical Lyapunov center theorem.