Part A:
Lie Groups and Hamilton Structures of Fluid Dynamics.
a1) Lie Groups and Lie Algebras: basic notions.
a2) The Euler equation of the (generalized) rigid body and invariant metrics
on a Lie group.
a3) Hamiltonian systems with symmetry, moment maps and hamiltonian reductions:
some examples.
a4) Euler equations of fluid dynamics and the diffeomorphism group
a5) The Korteweg - de Vries (KdV) equation as an Euler equation.
Part B:
Integrable PDEs in 1+1 dimensions
b1) The KdV equation: elemntary solutions and the Lax representation
b2) The Inverse Scattering Method and the KdV Soliton solutions.
b3) Hamiltonian structures of KdV: complete integrability.
b4) Finite gap solutions and their geometrical properties.
Some bibliography:
V. Arnol'd, B. Khesin, Topological methods in Hydrodynamics, Springer 1998;
S. Novikov, S. Manakov, L. Pitaevskii, V. Zakharov, Theory of Solitons,
Consultants Bureau, 1984.
P. Drazin, R. Johnson, Solitons: an Introduction, Cambridge 1989.