We prove unique continuation principles for solutions of evolution Schrodinger equations with time dependent potentials. These correspond to uncertainly principles of Paley-Wiener type for the Fourier transform. Our results extends to a large class of semi-linear Schrodinger equation.
In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schrodinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.
In this work we continue our study initiated in cite{GFGP} on the uniqueness properties of real solutions to the IVP associated to the Benjamin-Ono (BO) equation. In particular, we shall show that the uniqueness results established in cite{GFGP} do not extend to any pair of non-vanishing solutions of the BO equation. Also, we shall prove that the uniqueness result established in cite{GFGP} under a hypothesis involving information of the solution at three different times can not be relaxed to two different times.
We study the existence and stability of the standing waves for the periodic cubic nonlinear Schrodinger equation with a point defect determined by a periodic Dirac distribution at the origin. This equation admits a smooth curve of positive periodic solutions in the form of standing waves with a profile given by the Jacobi elliptic function of dnoidal type. Via a perturbation method and continuation argument, we obtain that in the case of an attractive defect the standing wave solutions are stable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself. In the case of a repulsive defect, the standing wave solutions are stable in the subspace of even functions of $H^1_{per}$ and unstable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself.
