## Integrable systems: the KdV equation, direct and inverse scattering on the line

### Prof. Tamara Grava

Number of cycles: 2

First semester: Tue 14-15:30, Thu 14-15:30 (starting from October 11)

Contents:

- Hamiltonian systems: preliminaries;
- Lie-Poisson bracket, symplectic manifolds;
- Liouville integrable systems;
- Integrable systems in infinite dimensions: the Korteweg de Vries equation;
- Lax Pair;
- direct scattering on the line for the Schrodinger equation;
- reflectionless potentials and n-soliton solutions;
- inverse scattering on the line and Riemann-Hilbert problem;
- Bloch spectrum of the Schrodinger operator with a periodic potential, monodromy matrix;
- Theory of the KdV hierarchy, bi-Hamiltonian structure and recursion operator. Schouten brackets and bihamiltonian manifolds.