A Spatial Dynamics Theory for Doubly Periodic Travelling Gravity-Capillary Surface Waves on Water of Infinite Depth

Abstract

In this article we show how Kirchgassner’s spatial dynamics approach can be used to construct doubly periodic travelling gravity-capillary surface waves on water of infinite depth. The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable and the infinite-dimensional function space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. The imaginary part of the spectrum of the linearised Hamiltonian vector field consists of essential spectrum at the origin and a finite number of eigenvalues whose distribution is described geometrically. Periodic solutions to the spatial Hamiltonian system are detected using Iooss’s generalisation of the reversible Lyapunov centre theorem; these solutions correspond to doubly periodic solutions to the travelling water-wave problem. For a generic choice of the periodic domain there exist values of the physical parameters at which a doubly periodic wave with this domain exists.