Unique continuation: Difference between revisions
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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 [[Schrodinger equations|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? | A question arising by analogy from the theory of '''unique continuation''' in elliptic equations, and also in control theory, is the following: if <math>u\,</math> is a solution to a [[Schrodinger equations|nonlinear Schrodinger equation]], and <math>u(t_0)\,</math> and <math>u(t_1)\,</math> is specified on a domain <math>D\,</math> at two different times <math>t_0, t_1\,</math>, 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 the 1D cubic NLS, with <math>D\,</math> equal to a half-line, and u assumed to vanish on <math>D\,</math>, this is in [[Zg1997]]. | ||
* For general NLS with analytic non-linearity, and with u assumed compactly supported, this is in [[Bo1997b]]. | * 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 <math>D\,</math> 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]] | * For <math>D\,</math> 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]] | * A local unique continuation theorem (asserting that a non-zero solution cannot vanish on an open set) is in [[Isk1993]] | ||
[[Category:Schrodinger]] | [[Category:Schrodinger]] | ||
[[Category:concept]] | [[Category:concept]] | ||
Latest revision as of 18:58, 7 August 2006
A question arising by analogy from the theory of unique continuation in elliptic equations, and also in control theory, is the following: if is a solution to a nonlinear Schrodinger equation, and and is specified on a domain at two different times , does this uniquely specify the solution everywhere at all other intermediate times?
- For the 1D cubic NLS, with equal to a half-line, and u assumed to vanish on , this is in Zg1997.
- For general NLS with analytic non-linearity, and with u assumed compactly supported, this is in Bo1997b.
- For the complement of a convex cone, and arbitrary NLS of polynomial growth with a bounded potential term, this is in KnPoVe2003
- For 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
