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

Abstract

We prove that if $u_1,u_2$ are solutions of the Benjamin-Ono equation defined in $(x,t)\in \mathbb{R}\times [0,T]$ which agree in an open set $\hat{I}\text{c◯}\subset \mathbb{R}\times [0,T]$, 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.