The course is centered on the Hamiltonian aspects of integrable systems of Ordinary and, especially, Partial Differential Equations, with a focus on the geometrical side. Integrability will mean existence of a “sufficient number” of conservation laws.
Contents & schedule.
1) Symplectic and Poisson geometry: a reminder. Extensions to PDEs.
2) The Marsden-Weinstein, Dirac and Marsden-Ratiu reduction schemes.
3) Lie-Poisson structures on (duals of) Lie algebras. Drinfel’d-Sokolov reduction on loop
algebras and equations of Korteweg - de Vries (KdV) - type
4) Bihamiltonian structures and integrability.
5) From the Hamiltonian structure of the Euler incompressible equations to the
Hamiltonian structure of water-waves and, finally, to the bihamiltonian structure of the
KdV equation.