Past

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.

When studying invariant quantities and stability of discretization schemes for time-dependent differential equations(ODEs), Backward error analysis (BEA) has proven itself an invaluable tool. Although the established results give very accurate estimates, the known results are generally given for "worst case" scenarios. By taking into account the structure of the differential equations themselves further improvements on the estimates can be established, and sharper estimates on invariant quantities and stability can be established. In the talk I will give an overview of BEA, and its applications as it stands emphasizing the shortcoming in the estimates. An alternative strategy is then proposed overcoming these shortcomings, resulting in a tool which when used in connection with results from dynamical systems theory gives a very good insight into the dynamics of discretized differential equations.

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