An existence theory for three-dimensional periodic travelling gravity–capillary water waves with bounded transverse profiles

Abstract

This article presents a rigorous existence theory for three-dimensional gravity–capillary water waves which are uniformly translating and periodic in one horizontal spatial direction x and have a nontrivial transverse profile in the other z. The hydrodynamic equations are formulated as an infinite-dimensional Hamiltonian system in which z is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system of ordinary differential equations. A family of straight lines C 1, C 2,… in an appropriate two-dimensional parameter space is identified at which the number of purely imaginary eigenvalues of the linear problem changes: at each point on one of these lines two real eigenvalues become purely imaginary by passing through zero. There are also codimension-two points: the line C k intersects each of the lines C k+1 , C k+2 ,… in precisely one point. General statements concerning the existence of waves which are periodic or quasiperiodic in z are made by applying standard tools in Hamiltonian-systems theory to the reduced equations. Moreover, a critical curve in parameter space is found at which a two-dimensional Stokes wave and a three-dimensional wave with a spatially localised and exponentially decaying transverse profile simultaneously bifurcate from the uniform flow. This curve is piecewise linear: it contains one line segment from each of C 1, C 2,… .