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

Gregorio Falqui (SISSA)

The aim of the course is to introduce to the students
some basic algebraic and geometrical aspects of the theory of
soliton equations of KdV-type.

Special attention will be paid to their Hamiltonian aspects.

Schedule: November 1998 -- January 1999

First Lecture: Monday 16/11/1998, 4. pm, room F, SISSA Main building

**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.
KP theory: the Sato approach.

Poisson Manifolds.
Bihamiltonian manifolds. Lenard recursion and generalized Casimir
functions.
Compatible Poisson structures on (duals of) Lie Algebras
and cohomology.
Poisson structures on the dual of the Virasoro algebra.
The bihamiltonian theory of the KdV hierarchy.
The KP theory as a system of conservation laws.

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