Séminaire Analyse Numérique et EDP
A Nekhoroshev theorem for some (smoothing) perturbations of the Benjamin-Ono equation with initial data close to finite gap tori
17
oct. 2024
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Intervenant : Dario Bambusi
Institution : Università degli studi di Milano
Heure : 14h00 - 15h00
Lieu : 3L8
 
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
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