Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Unique continuation properties of the nonlinear Schrödinger equation

Bing-Yu Zhanga1

a1 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A

Abstract

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

S0308210500023581_eqnU1

By using the inverse scattering transform and some results from the Hardy function theory, we prove that if uC(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.

(Received June 12 1995)