Stat. Field Theory Course
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**Integrable Systems**

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Gregorio Falqui

**CONTENTS**

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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