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posted on 25.08.2005by Mark D. Groves, M. Haragus
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.