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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