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

http://dx.doi.org/10.4153/CJM-2008-051-3
Canad. J. Math. 60(2008), 1168-1200

  Published:2008-10-01
 Printed: Oct 2008

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.