Jost Solutions and the Direct Scattering Problem of the Benjamin--Ono Equation

Abstract: In this paper, we present a rigorous study of the direct scattering problem that arises from the complete integrability of the Benjamin--Ono (BO) equation. In particular, we establish existence, uniqueness, and asymptotic properties of the Jost solutions to the scattering operator in the Fokas--Ablowitz inverse scattering transform (IST). Formulas relating different scattering coefficients are proven, together with their asymptotic behavior with respect to the spectral parameter. This work is an initial step toward the construction of general solutions to the BO equation by IST.