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This paper has been accepted for publication in Physica D. To download a preprint of this paper just click here.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1206131. Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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The Scattering Transform for the Benjamin-Ono Equation in the Small-Dispersion Limit

Peter D. Miller and Alfredo N. Wetzel

Department of Mathematics, University of Michigan, Ann Arbor


Using exact formulae for the scattering data of the Benjamin-Ono equation valid for general rational potentials recently obtained in [S2], 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.