# Unique continuation

A question arising by analogy from the theory of unique continuation in elliptic equations, and also in control theory, is the following: if ${\displaystyle u\,}$ is a solution to a nonlinear Schrodinger equation, and ${\displaystyle u(t_{0})\,}$ and ${\displaystyle u(t_{1})\,}$ is specified on a domain ${\displaystyle D\,}$ at two different times ${\displaystyle t_{0},t_{1}\,}$, does this uniquely specify the solution everywhere at all other intermediate times?

• For the 1D cubic NLS, with ${\displaystyle D\,}$ equal to a half-line, and u assumed to vanish on ${\displaystyle D\,}$, this is in Zg1997.
• For general NLS with analytic non-linearity, and with u assumed compactly supported, this is in Bo1997b.
• For ${\displaystyle D\,}$ the complement of a convex cone, and arbitrary NLS of polynomial growth with a bounded potential term, this is in KnPoVe2003
• For ${\displaystyle D\,}$ in a half-plane, and allowing potentials in various Lebesgue spaces, this is in IonKn-p
• A local unique continuation theorem (asserting that a non-zero solution cannot vanish on an open set) is in Isk1993