Unique continuation

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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
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