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

TL;DR

This paper proves a Nekhoroshev-type stability result for perturbed Benjamin-Ono equations with initial data near finite gap tori, showing actions remain close over exponentially long times.

Contribution

It establishes a Nekhoroshev theorem for certain Hamiltonian perturbations of the Benjamin-Ono equation with initial data close to finite gap solutions.

Findings

Actions remain close to initial values for exponentially long times.

Stability estimate depends on the size of the perturbation and the number of gaps.

Results extend Nekhoroshev theory to a nonlocal dispersive PDE.

Abstract

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 $\cO(\epsilon^{\frac{1}{2(N+1)}})$ close to their initial value for times exponentially long with $\epsilon^{-\frac{1}{2(N+1)}}$.