On uniqueness results for the Benjamin equation

TL;DR

This paper investigates the limits of existing uniqueness results for the Benjamin equation, demonstrating their non-extendability for non-vanishing solutions and establishing new local uniqueness conditions based on initial data agreement.

Contribution

It proves the non-extendability of previous uniqueness results for non-vanishing solutions and introduces new local uniqueness theorems based on initial data agreement.

Findings

Previous uniqueness results cannot be extended to non-vanishing solutions.

Local uniqueness holds if solutions and their time derivatives agree on an open set at initial time.

A refined version of the local uniqueness result is also provided.

Abstract

We prove that the uniqueness results obtained in \cite{urrea} for the Benjamin equation, cannot be extended for any pair of non-vanishing solutions. On the other hand, we study uniqueness results of solutions of the Benjamin equation. With this purpose, we showed that for any solutions $u$ and $v$ defined in $\R\times [0,T]$, if there exists an open set $I\subset \R$ such that $u(\cdot,0)$ and $v(\cdot,0)$ agree in $I$, $\p_t u(\cdot,0)$ and $\p_t v(\cdot,0)$ agree in $I$, then $u\equiv v$. To finish, a better version of this uniqueness result is also established.