A note on the uniqueness properties of solutions for the Schr\"odinger-Korteweg de Vries system

In this work we prove that if $(u_i,v_i)$, $i=1,2$, are smooth enough solutions of the coupled Schr\"odinger-Korteweg-de Vries system \begin{align*} \left. \begin{array}{rl} i u_t+\partial_x^2 u &\hspace{-2mm}=\beta uv - |u|^2 u,\\ \partial_t v + \partial_x^3 v &\hspace{-2mm}=\gamma \partial_x |u|^2-\frac12\partial_x (v^2) \end{array} \right\} \end{align*} with appropriate decay at infinity such that at two different times $t_0=0$ and $t_1=1$ satisfy that $$u_1(0)-u_2(0),u_1(1)-u_2(1),v_1(0)-v_2(0),v_1(1)-v_2(1)\in H^1(e^{ax^{2}}dx),$$ for $a>0$ big enough, then $u_1=u_2$ and $v_1=v_2$. (Let us recall that $f\in H^1(e^{ax^{2}} dx)$ iff $f\in L^2(e^{ax^{2}}dx)$ and $\partial_x f\in L^2(e^{ax^{2}}dx)$).