On uniqueness and decay of solution to the Hirota equation

Abstract

We address the question of the uniqueness of solution to the initial value problem associated to the equation \begin{equation*} \partial_{t}u+i\alpha \partial^{2}_{x}u+\beta \partial^{3}_{x}u+i\gamma|u|^{2}u+\delta |u|^{2}\partial_{x}u+\epsilon u^{2}\partial_{x}\overline{u} = 0, \quad x,t \in \mathbb{R}, \end{equation*} and prove that a certain decay property of the difference $u_1-u_2$ of two solutions $u_1$ and $u_2$ at two different instants of times $t=0$ and $t=1$, is sufficient to ensure that $u_1=u_2$ for all the time.