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 $u\,$ is a solution to a nonlinear Schrodinger equation, and $u(t_{0})\,$ and $u(t_{1})\,$ is specified on a domain $D\,$ at two different times $t_{0},t_{1}\,$, does this uniquely specify the solution everywhere at all other intermediate times?

• For the 1D cubic NLS, with $D\,$ equal to a half-line, and u assumed to vanish on $D\,$, this is in Zg1997.
• For general NLS with analytic non-linearity, and with u assumed compactly supported, this is in Bo1997b.
• For $D\,$ the complement of a convex cone, and arbitrary NLS of polynomial growth with a bounded potential term, this is in KnPoVe2003
• For $D\,$ 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