A spatial dynamics approach to three-dimensional gravity-capillary steady water waves

A new approach to the question of the existence of small-amplitude, uniformly translating, two-dimensional capillary-gravity water waves was proposed by Kirchgässner (1988), who suggested writing the governing equations as an infinite-dimensional, quasilinear dynamical system in which the horizontal coordinate is the time-like variable. The centre-manifold reduction theorem of Mielke (1988) may then be used to show that this dynamical system is locally equivalent to a system of ordinary differential equations whose solution set can, in theory, be analysed. This method has become known as 'spatial dynamics' and is the basis for several existence theories for two-dimensional capillary-gravity water waves (see Iooss (1995) for a review). It was later noticed by Mielke (1991, ch. 9) and Baesens & MacKay (1992) that the two-dimensional steady water-wave problem can be written as a Hamiltonian system, and their observations were placed upon a secure functional-analytic foundation by Groves & Toland (1997). Mielke (1991) demonstrated that the reduced system of ordinary differential equations inherits the Hamilto-nian structure, and this additional feature of the reduction process has since been exploited in recent existence theories for infinite families of multi-humped solitary waves (Buffoni, Groves & Toland 1996; Buffoni & Groves 1999). This article is concerned with the application of spatial dynamics methods to a three-dimensional steady water-wave problem in which the waves are uniformly translating in one horizontal direction and periodic in the other. The issue was raised recently by Haragus-Courcelle & Illichev (1998), who studied a model equation for three-dimensional water waves of this type. The main difficulty in the full three-dimensional steady water-wave problem is the appropriate choice of variables for a correct formulation as a dynamical system. This matter is settled in the present article. Physically motivated arguments based upon a well-known formal variational formulation of the problem are used to select variables whose mathematical suitability is confirmed a posteriori. The dynamical system obtained in this fashion is Hamiltonian and amenable to the centre-manifold reduction procedure described above. The first contribution to the study of variational formulations of the water-wave problem was made by Luke (1967), who published a formal variational principle that recovers the water-wave equations. A version of Luke's variational principle for the three-dimensional steady water-wave problem under consideration here is given in Section 2.2. It depends upon the velocity potential φ(x, y, z) and the free-surface elevation η(x, z) and involves integrals over the unknown fluid domain D η = {(x, …