10 March 2005
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.
- Differential Equations and Applications Seminar