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This paper has been submitted for publication in the Annales de l'Institut Henri Poincaré C, Analyse Non Linéare. 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-2204896 (Miller). 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 Benjamin-Ono initial-value problem for rational data with application to long-time asymptotics and scattering

Elliot Blackstone, Louise Gassot, Patrick Gérard, and Peter D. Miller

EB and PDM: Department of Mathematics, University of Michigan, Ann Arbor
LG: CNRS and Department of Mathematics, University of Rennes, Rennes, France
PG: Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay, Orsay, France

Abstract:

We show that the initial-value problem for the Benjamin-Ono equation on R with L2(R) rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The dimension of the determinant depends on the number of simple poles of the rational initial data only and the matrix elements depend explicitly on the independent variables (t,x) and the dispersion coefficient ϵ. This allows for various interesting asymptotic limits to be resolved quite efficiently. As an example, and as a first step towards establishing the soliton resolution conjecture, we prove that the solution with initial datum equal to minus a soliton exhibits scattering.