A General Uncertainty Principle for Partial Differential Equations

Abstract

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.