On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data

TL;DR

This paper investigates the behavior of solutions to the Benjamin-Ono equation as dispersion tends to zero, establishing a weak limit analogous to methods used for the Korteweg-de Vries equation.

Contribution

It develops a new approach to analyze the zero-dispersion limit for the Benjamin-Ono equation, extending techniques from KdV analysis.

Findings

Established existence of the zero-dispersion limit in a weak sense

Developed an analogue of Lax-Levermore method for Benjamin-Ono

Provided insights into the asymptotic behavior of solutions

Abstract

We study the Cauchy initial-value problem for the Benjamin-Ono equation in the zero-disperion 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.