A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori

We consider a perturbation of the Benjamin Ono equation with periodic boundary conditions on a segment. We consider the case where the perturbation is Hamiltonian and 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 $\text{cO}({\epsilon }^{\frac{1}{2(N+1)}})$ close to their initial value for times exponentially long with ${\epsilon }^{-\frac{1}{2(N+1)}}$.