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 $|q|\to 0$ 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: $\exp (-x^{1+\delta })$, $x\ge 0$ 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 Schrödinger equation.

arXiv comment20 pages, some misprints have been corrected and some examples have been added. To appear in Journal of Mathematical Analysis and Applications

References in corpus (2)

• Uniqueness for discrete Schrodinger evolutions
• A Dynamic Uncertainty Principle for Jacobi Operators