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 $O(\epsilon^{\frac{1}{2(N+1)}})$ 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-Kapeller's formulae for the Hamiltonian of the BO equation in Birkhoff variables.

 

Joint work with Patrick Gerard