Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity

Abstract

This paper presents an existence theory for small-amplitude Stokes and 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 infinite-dimensional Hamiltonian system in which the horizontal spatial coordinate is the time-like variable. A centre-manifold technique is used to reduce the system to a locally equivalent Hamiltonian system with one degree of freedom for values of a dimensionless parameter α near its critical value α ⋆ . The phase portrait of the reduced system contains a homoclinic orbit for α < α ⋆ and a family of periodic orbits for α > α ⋆ ; the corresponding solutions of the water-wave problem are respectively a solitary wave of elevation and a family of Stokes waves.