  We consider the coupled equations \begin{equation*} \begin{pmatrix}r_t\\
-q_t\end{pmatrix}+2A_0(L^+)\begin{pmatrix}r\\ q\end{pmatrix}=0, \end{equation*}
where $L^+$ is the integro-differential operator \begin{equation*}
L^+=\frac{1}{2\I}\begin{pmatrix}\partial_x-2r\int_{-\infty}^xdyq&
2r\int_{-\infty}^xdyr\\ -2q\int_{-\infty}^xdyq&
-\partial_x+2q\int_{-\infty}^xdyr.\end{pmatrix} \end{equation*} and $A_0(z)$ is
an arbitratry ratio of entire functions. We study two main cases: the first one
when the potentials $|q|,|r|\to 0$ as $|x|\to\infty$ and the second one when
$r=-1$ and $|q|\to0$ as $|x|\to\infty$. In such conditions we prove that there
cannot exist a solution different from zero if at two different times the
potentials have a strong decay. This decay is of exponential rate:
$\exp(-x^{1+\delta})$, $x\geq 0$ and $\delta>0$ is a constant. As particular
cases we will cover the Korteweg-de Vries equation, the modified Korteweg-de
Vries equation and the nonlinear Schr\"odinger equation.
