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.

- Dorfman, I.Y. and Athorne, C., On the nonsymmetric Novikov-Veselov hierarchy, Phys. Lett. A 182 (1993) 369-372MathSciNet Review.
- Athorne, C. and Dorfman, I.Y., The Hamiltonian structure of the Ablowitz-Kaup-Newell-Segur hierarchy, J. Math. Phys. 34 (1993) 3507--3517, MathSciNet Review.
- Athorne, C., Local Hamiltonian Structures of multicomponent KdV equations, J. Phys. A 21 (1988) 4549-4556.MathSciNet Review.
- Athorne, C. and Fordy, A.P., Integrable equations in (2+1)-dimensions associated with symmetric and homogeneous spaces, J. Math. Phys. 28 (1987) 2018-2024, MathSciNet Review.
- Athorne, C. and Fordy, A.P., Generalized KdV and MKdV equations associated with symmetric spaces, J. Phys. A: 20 (1987) 1377-1386 MathSciNet Review.