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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