The scattering transform for the Benjamin-Ono equation in the small-dispersion limit

TL;DR

This paper rigorously analyzes the scattering data of the Benjamin-Ono equation in the small-dispersion limit, providing precise asymptotic formulas and confirming existing conjectures with new detailed insights.

Contribution

It offers the first rigorous asymptotic analysis of the scattering data for the Benjamin-Ono equation in the small-dispersion regime, validating and extending previous conjectures.

Findings

Asymptotic formulae for the reflection coefficient

Precise eigenvalue location and density estimates

Dependence of phase constants on eigenvalues

Abstract

Using exact formulae for the scattering data of the Benjamin-Ono equation valid for general rational potentials recently obtained by Miller and Wetzel (2015), we rigorously analyze the scattering data in the small-dispersion limit. In particular, we deduce precise asymptotic formulae for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant (associated with each eigenvalue) on the eigenvalue itself. Our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering, and they also provide new details not previously reported in the literature.