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Short Time Behavior of Solutions to Linear and Nonlinear Schrödinger Equations

Published online by Cambridge University Press:  20 November 2018

Michael Taylor
Michael Taylor*
Affiliation:
Mathematics Deptartment, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, U.S.A. e-mail:[email protected]
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Abstract

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We examine the fine structure of the short time behavior of solutions to various linear and nonlinear Schrödinger equations ${{u}_{t}}=i\Delta u+q(u)$ on $I\times {{\mathbb{R}}^{n}}$, with initial data $u(0,x)=f(x)$. Particular attention is paid to cases where $f$ is piecewise smooth, with jump across an $(n-1)$-dimensional surface. We give detailed analyses of Gibbs-like phenomena and also focusing effects, including analogues of the Pinsky phenomenon. We give results for general $n$ in the linear case. We also have detailed analyses for a broad class of nonlinear equations when $n=1$ and 2, with emphasis on the analysis of the first order correction to the solution of the corresponding linear equation. This work complements estimates on the error in this approximation.

Keywords

35Q5535Q40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 5 , October 2008 , pp. 1168 - 1200
Copyright
Copyright © Canadian Mathematical Society 2008

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