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Short Time Behavior of Solutions to Linear and Nonlinear Schr{ödinger Equations

Open Access article
 Printed: Oct 2008
  • Michael Taylor
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We examine the fine structure of the short time behavior of solutions to various linear and nonlinear Schr{\"o}dinger equations $u_t=i\Delta u+q(u)$ on $I\times\RR^n$, 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.
MSC Classifications: 35Q55, 35Q40 show english descriptions NLS-like equations (nonlinear Schrodinger) [See also 37K10]
PDEs in connection with quantum mechanics
35Q55 - NLS-like equations (nonlinear Schrodinger) [See also 37K10]
35Q40 - PDEs in connection with quantum mechanics

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