The Benjamin-Ono Hierarchy with Asymptotically Reflectionless Initial Data in the Zero-Dispersion Limit

TL;DR

This paper investigates the zero-dispersion limit of the Benjamin-Ono hierarchy with general positive initial data, deriving formulas for conserved densities in terms of solutions to the Burgers hierarchy.

Contribution

It provides explicit formulas for the limits of conserved densities in the Benjamin-Ono hierarchy using solutions of the inviscid Burgers hierarchy, extending understanding of zero-dispersion asymptotics.

Findings

Limits of conserved densities expressed as alternating sums of Burgers solutions

Established weak/distributional convergence in the zero-dispersion limit

Connected Benjamin-Ono hierarchy behavior to Burgers hierarchy solutions

Abstract

We study the Benjamin-Ono hierarchy with positive initial data of a general type, in the limit when the dispersion parameter tends to zero. We establish simple formulae for the limits (in appropriate weak or distributional senses) of an infinite family of simultaneously conserved densities in terms of alternating sums of branches of solutions of the inviscid Burgers hierarchy.