A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori. (English) Zbl 07873282
Summary: 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 from the 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 \(\epsilon^{-\frac{1}{2(N+1)}} \).
MSC:
| 37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35Q51 | Soliton equations |
| 35B20 | Perturbations in context of PDEs |
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