Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data

TL;DR

This paper derives explicit formulas for the scattering data of the Benjamin-Ono equation with rational initial data, enabling precise spectral analysis for arbitrary simple pole conditions.

Contribution

It provides new explicit formulas for Jost solutions, eigenfunctions, and scattering data for the Benjamin-Ono equation with rational initial conditions.

Findings

Explicit formulas for Jost solutions and eigenfunctions.

Derivation of Evans function for eigenvalues.

Formulas for phase constants and reflection coefficient.

Abstract

We compute the scattering data of the Benjamin-Ono equation for arbitrary rational initial conditions with simple poles. Specifically, we obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient.