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

TL;DR

This paper rigorously analyzes the direct scattering problem for the Benjamin--Ono equation, establishing properties of Jost solutions and scattering coefficients, which are essential for solving the equation via inverse scattering transform.

Contribution

It provides a rigorous foundation for the existence, uniqueness, and asymptotic behavior of Jost solutions in the BO equation's scattering theory, advancing the inverse scattering method.

Findings

Proved existence and uniqueness of Jost solutions.

Derived formulas relating scattering coefficients.

Analyzed asymptotic behavior of scattering data.

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.