The IVP for the Benjamin-Ono equation in weighted Sobolev spaces II

TL;DR

This paper investigates the uniqueness properties of solutions to the Benjamin-Ono equation in weighted Sobolev spaces, showing limitations of previous results and the necessity of multiple time points for certain uniqueness theorems.

Contribution

It demonstrates that previous uniqueness results do not extend to non-vanishing solutions and that information at three times cannot be reduced to two for uniqueness.

Findings

Uniqueness results do not hold for non-vanishing solutions.

Uniqueness at three times cannot be replaced by two.

Previous conditions for uniqueness are shown to be sharp.

Abstract

In this work we continue our study initiated in \cite{GFGP} on the uniqueness properties of real solutions to the IVP associated to the Benjamin-Ono (BO) equation. In particular, we shall show that the uniqueness results established in \cite{GFGP} do not extend to any pair of non-vanishing solutions of the BO equation. Also, we shall prove that the uniqueness result established in \cite{GFGP} under a hypothesis involving information of the solution at three different times can not be relaxed to two different times.