Stat. Field Theory Course

Integrable Systems

Gregorio Falqui


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.

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