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Unique continuation properties of the nonlinear Schrödinger equation

Published online by Cambridge University Press:  14 November 2011

Bing-Yu Zhang
Bing-Yu Zhang
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A
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Abstract

Consider the unique continuation problem for the nonlinear Schrödinger (NLS) equation

By using the inverse scattering transform and some results from the Hardy function theory, we prove that if u ∈ C(R; H1(R)) is a solution of the NLS equation, then it cannot have compact support at two different moments unless it vanishes identically. In addition, it is shown under certain conditions that if u is a solution of the NLS equation, then u vanishes identically if it vanishes on two horizontal half lines in the x–t space. This implies that the solution u must vanish everywhere if it vanishes in an open subset in the x–t space.

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Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 1 , 1997 , pp. 191 - 205
Copyright
Copyright © Royal Society of Edinburgh 1997

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