Hamiltonian aspects of integrable systems.
(See also: resources and, in particular, research topics.)
Soliton equations, such as the Korteweg-de Vries (KdV) and Nonlinear Schrodinger (NLS) equations and their higher dimensional extensions, the Kadomstev-Petviashvili (KP) and Davey-Stewartson (DS) equations, share with most finite mechanical systems the property of being Hamiltonian. The property has to be suitably defined for these effectively ``infinite dimensional'' systems by applying to matrix and integro-differential rings the exterior differential algebra of Lie algebra complexes. A nice introduction to the fundamental aspects of this work is to be found in the book Dirac Structures by Irene Ya. Dorfman.
Of major concern over recent years has been the inclusion within this general picture of all the known examples of integrable partial differential equations.