A general uncertainty principle for partial differential equations

Abstract

We consider the coupled equations(rt−qt)+2A0(L+)(rq)=0, where L+ is the integro-differential operatorL+=12i(∂x−2r∫−∞xdyq2r∫−∞xdyr−2q∫−∞xdyq−∂x+2q∫−∞xdyr,) and A0(z) is an arbitrary ratio of entire functions. We study two main cases: the first one when the potentials |q|,|r|→0 as |x|→∞ and the second one when r=−1 and |q|→0 as |x|→∞. 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⁡(−x1+δ), x≥0 and δ>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.