# Uniqueness Properties of Solutions to the Benjamin-Ono equation and related models

## TL;DR

This paper establishes strong uniqueness properties for solutions to the Benjamin-Ono equation and related non-local models, showing solutions that agree on an open set must be identical, extending to various boundary conditions and models.

## Contribution

It proves a new strong uniqueness theorem for solutions of the Benjamin-Ono equation and extends it to a broader class of non-local equations, including the intermediate long wave equation.

## Key findings

- Solutions agreeing on an open set are identical.
- Uniqueness results extend to initial value and periodic boundary problems.
- Stronger uniqueness versions are presented for the Benjamin-Ono equation.

## Abstract

We prove that if $u_1,\,u_2$ are solutions of the Benjamin-Ono equation defined in $ (x,t)\in\R \times [0,T]$ which agree in an open set $\Omega\subset \R \times [0,T]$, then $u_1\equiv u_2$. We extend this uniqueness result to a general class of equations of Benjamin-Ono type in both the initial value problem and the initial periodic boundary value problem. This class of 1-dimensional non-local models includes the intermediate long wave equation. Finally, we present a slightly stronger version of our uniqueness results for the Benjamin-Ono equation.