Explicit formula for the Benjamin--Ono equation with square integrable and real valued initial data and applications to the zero dispersion limit

TL;DR

This paper extends Gérard's explicit solution formula for the Benjamin--Ono equation to include square integrable, real-valued initial data, and applies it to analyze the zero dispersion limit with more singular initial conditions.

Contribution

It provides a new explicit formula for solutions with less regular initial data and extends the zero dispersion limit analysis to more singular cases.

Findings

Extended Gérard's formula to square integrable initial data

Derived an explicit solution formula for the zero dispersion limit

Discovered an integral equality potentially useful in other contexts

Abstract

In this paper, we extend G{\'e}rard's formula for the solution of the Benjamin--Ono equation on the line to square integrable and real valued initial data. Combined with this formula, we also extend the G{\'e}rard's formula for the zero dispersion limit of the Benjamin--Ono equation on the line to more singular initial data. In the derivation of the extension of the formula for the zero dispersion limit, we also find an interesting integral equality, which might be useful in other contexts.