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This paper was originally published in Comm. Pure Appl. Math. 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-0807653. 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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On the Zero-Dispersion Limit of the Benjamin-Ono Cauchy Problem for Positive Initial Data

Peter D. Miller and Zhengjie Xu
Department of Mathematics, University of Michigan, Ann Arbor


We study the Cauchy initial-value problem for the Benjamin-Ono equation in the zero-dispersion limit, and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by Lax and Levermore to analyze the corresponding limit for the Korteweg-de Vries equation.

Here is a physical experiment by Roberto Camassa and Richard McLaughlin of the University of North Carolina, Chapel Hill, demonstrating the trapping of particles of a contaminant by a density interface whose motion can under some assumptions be modeled by the Benjamin-Ono equation: