Hamiltonian Theory of Soliton Equations
**Hamiltonian Theory of Soliton Equations **

Gregorio Falqui (SISSA)

The aim of the course is to provide the students with
some basics of the theory of soliton equations of KdV-type,
with some emphasis to their Hamiltonian aspects.

Schedule: November - December 1997

**Program**

The KdV equation and its Lax representation.
The algebra of microdifferential operators in one variable.
Lax representation for the Gel'fand-Dickey hierarchies.
Soliton solutions to the Gel'fand-Dickey hierarchies.

Poisson Manifolds.
Poisson structures on (duals of) Lie Algebras.
Poisson actions of Lie algebras.
The Marsden-Weinstein and Marsden-Ratiu reduction
theorems.

The KdV theory as a reduction from **SL(2)**.
The bihamiltonian theory of the KdV hierarchy.
Towards the KP theory: the Sato approach.
The KP theory as a system of conservation laws.

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