This paper presents an existence theory for small-amplitude
solitary-wave solutions to the classical water-wave problem in the absence
of surface tension and with an arbitrary distribution of vorticity. The hydrodynamic
problem is formulated as an in nite-dimensional Hamiltonian
system in which the horizontal spatial direction is the time-like variable.
A centre-manifold reduction technique is employed to reduce the system
to a locally equivalent Hamiltonian system with one degree of freedom.
The phase portrait of the reduced system contains a homoclinic orbit, and
the corresponding solution of the water-wave problem is a solitary wave