Perturbation theory for the Benjamin-Ono equation

We develop a perturbation theory for the Benjamin-Ono (BO) equation. This p erturbation theory is based on the inverse scattering transform for the BO equation, which was originally developed by Fokas and Ablowitz and recently refined by Kaup and Matsuno. We find the expressions for the variations of the scattering data with respect to the potential, as well as the dual exp ression for the variation of the potential in terms of the variations; of t he scattering data. This allows us to introduce the squared eigenfunctions for the BO equation, whose completeness and orthogonality in both x- and la mbda-spaces we also establish. We consider the two most important applicati ons of the developed machinery. First, we present an explicit first-order s olution of the BO equation driven by a small perturbation. Second, we intro duce the Poisson bracket and a set of the canonical action-angle variables for the BO equation, and thus demonstrate its complete integrability as a H amiltonian dynamical system.