On the support of solutions to the NLS-KdV system

Abstract

It is shown that if $(u,v)$ is a sufficiently smooth solution of the initial value problem associated with the Schrödinger-Korteweg-de Vries system such that there exist $a, b\in\mathbb{R}$ with $\operatorname{supp}u(t_j)\subseteq(a,\infty)$ (or$(-\infty,a)$) and $\operatorname{supp}v(t_j)\subseteq(b,\infty)$ (or $(-\infty,b)$), for $j=1,2 \ (t_1\neq t_2)$, then $u\equiv v\equiv0$.