Three dimensional travelling gravity-capillary water waves

The classical gravity-capillary water-wave problem is the

study of the irrotational flow of a three-dimensional perfect

fluid bounded below by a flat, rigid bottom and above by a

free surface subject to the forces of gravity and surface

tension. In this lecture I will present a survey of currently

available existence theories for travelling-wave solutions of

this problem, that is, waves which move in a specific

direction with constant speed and without change of shape.

The talk will focus upon wave motions which are truly

three-dimensional, so that the free surface of the water

exhibits a two-dimensional pattern, and upon solutions of the

complete hydrodynamic equations for water waves rather than

model equations. Specific examples include (a) doubly

periodic surface waves; (b) wave patterns which have a

single- or multi-pulse profile in one distinguished

horizontal direction and are periodic in another; (c)

so-called 'fully-localised solitary waves' consisting of a

localised trough-like disturbance of the free surface which

decays to zero in all horizontal directions.

I will also sketch the mathematical techniques required to

prove the existence of the above waves. The key is a

formulation of the problem as a Hamiltonian system with

infinitely many degrees of freedom together with an

associated variational principle.