A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves

This article presents a rigorous existence theory for small-amplitude three-dimensional travelling water waves. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable. Wave motions which are periodic in a second, different horizontal direction are detected using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. A catalogue of bifurcation scenarios is compiled by means of a geometric argument based upon the classical dispersion relation for travelling water waves. Taking all parameters into account, one finds that this catalogue includes virtually any bifurcation or resonance known in Hamiltonian systems theory. Nonlinear bifurcation theory is carried out for a representative selection of bifurcation scenarios; solutions of the reduced Hamiltonian system are found by applying results from the well-developed theory of finite-dimensional Hamiltonian systems such as the Lyapunov centre theorem and the Birkhoff normal form. We find oblique line waves which depend only upon one spatial direction which is not aligned with the direction of wave propagation; the waves have periodic, solitary-wave or generalised solitary-wave profiles in this distinguished direction. Truly three-dimensional waves are also found which have periodic, solitary-wave or generalised solitary-wave profiles in one direction and are periodic in another. In particular, we recover doubly periodic waves with arbitrary fundamental domains and oblique versions of the results on threedimensional travelling waves already in the literature.