Universality in the Small-Dispersion Limit of the Benjamin-Ono Equation

TL;DR

This paper investigates the small-dispersion limit of the Benjamin-Ono equation for rational initial data, revealing universal profiles near critical points and comparing these with similar results for the Korteweg-de Vries equation.

Contribution

It introduces a detailed analysis of universal limiting profiles in the Benjamin-Ono equation's small-dispersion limit, extending understanding of dispersive wave behavior.

Findings

Universal limiting profiles near caustic and critical points

Comparison with KdV equation results

Method involving contour integrals and saddle point analysis

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.