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

摘要

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