**Speaker**

Dario Bambusi, University of Milan

**Abstract**

We consider the Benjamin Ono equation with periodic boundary conditions on a segment. We add a small Hamiltonian perturbation and consider the case where the corresponding Hamiltonian vector field is analytic as a map form energy space to itself. Let \(\epsilon\) be the size of the perturbation. We prove that for initial data close in energy norm to an \(N\)-gap state of the unperturbed equation all the actions of the Benjamin Ono equation remain \(\mathcal{O}(\epsilon^{\frac{1}{2(N+1)}})\) close to their initial value for times exponentially long with \(\epsilon^{-\frac{1}{2(N+1)}}\).

The result is made possible by the use of Gerard-Kappeler’s formulae for the Hamiltonian of the BO equation in Birkhoff variables.

Joint work with Patrick Gerard.