Perturbation theory for the Benjamin–Ono equation

We develop a perturbation theory for the Benjamin–Ono (BO) equation. This perturbation 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 expression for the variation of the potential in terms of the variations of the scattering data. This allows us to introduce the squared eigenfunctions for the BO equation, whose completeness and orthogonality in both xand λ-spaces we also establish. We consider the two most important applications of the developed machinery. First, we present an explicit first-order solution of the BO equation driven by a small perturbation. Second, we introduce the Poisson bracket and a set of the canonical action-angle variables for the BO equation, and thus demonstrate its complete integrability as a Hamiltonian dynamical system.