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This paper has been submitted for publication in Communications in Mathematical Physics. 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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Universality in the small-dispersion limit of the Benjamin-Ono equation

Elliot Blackstone, Peter D. Miller, and Matthew D. Mitchell

EB and PDM: Department of Mathematics, University of Michigan, Ann Arbor
MDM: Department of Physics, University of Michigan, Ann Arbor

Abstract:

We examine the solution of the Benjamin-Ono Cauchy problem for rational initial data in three types of double-scaling limits in which the dispersion tends to zero while simultaneously the independent variables either approach a point on one of the two branches of the caustic curve of the inviscid Burgers equation, or approach the critical point where the branches meet. The results reveal universal limiting profiles in each case that are independent of details of the initial data. We compare the results obtained with corresponding results for the Korteweg-de Vries equation found by Claeys-Grava in three papers. Our method is to analyze contour integrals appearing in an explicit representation of the solution of the Cauchy problem, in various limits involving coalescing saddle points.

Left: the solution of Benjamin-Ono and its approximation near the soliton edge for \(\epsilon=2^{-9}\). Center: the solution of Benjamin-Ono and its approxmation near the harmonic edge for \(\epsilon=2^{-9}\). Right: the universal profile \(U(T,X)\) near the catastrophe point for various values of \(T\).