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1. CJM Online first
On the Dispersive Estimate for the Dirichlet SchrÃ¶dinger Propagator and Applications to Energy Critical NLS |
On the Dispersive Estimate for the Dirichlet SchrÃ¶dinger Propagator and Applications to Energy Critical NLS We consider the obstacle problem for the SchrÃ¶dinger evolution
in the exterior of the unit ball with Dirichlet boundary condition. Under
the radial symmetry we compute explicitly the fundamental solution
for the linear Dirichlet SchrÃ¶dinger
propagator $e^{it\Delta_D}$
and give a robust algorithm to prove sharp $L^1 \rightarrow
L^{\infty}$ dispersive estimates. We showcase the analysis in
dimensions $n=5,7$. As an application, we obtain global
well-posedness and scattering for defocusing energy-critical NLS on
$\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary
condition and radial data in these dimensions.
Keywords:Dirichlet SchrÃ¶dinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy critical Categories:35P25, 35Q55, 47J35 |
2. CJM 2011 (vol 63 pp. 1201)
Resonant Tunneling of Fast Solitons through Large Potential Barriers We rigorously study the resonant tunneling of fast solitons through large
potential barriers for the nonlinear SchrÃ¶dinger equation in
one dimension. Our approach covers the case of general nonlinearities,
both local and Hartree (nonlocal).
Keywords:nonlinear Schroedinger equations, external potential, solitary waves, long time behavior, resonant tunneling Categories:37K40, 35Q55, 35Q51 |
3. CJM 2008 (vol 60 pp. 1168)
Short Time Behavior of Solutions to Linear and Nonlinear Schr{Ã¶dinger Equations 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.
Categories:35Q55, 35Q40 |
4. CJM 2002 (vol 54 pp. 1065)
Large Time Behavior for the Cubic Nonlinear SchrÃ¶dinger Equation We consider the Cauchy problem for the cubic nonlinear Schr\"odinger
equation in one space dimension
\begin{equation}
\begin{cases}
iu_t + \frac12 u_{xx} + \bar{u}^3 = 0,
& \text{$t \in \mathbf{R}$, $x \in \mathbf{R}$,} \\
u(0,x) = u_0(x), & \text{$x \in \mathbf{R}$.}
\end{cases}
\label{A}
\end{equation}
Cubic type nonlinearities in one space dimension heuristically appear
to be critical for large time. We study the global existence and
large time asymptotic behavior of solutions to the Cauchy problem
(\ref{A}). We prove that if the initial data $u_0 \in
\mathbf{H}^{1,0} \cap \mathbf{H}^{0,1}$ are small and such that
$\sup_{|\xi|\leq 1} |\arg \mathcal{F} u_0 (\xi) - \frac{\pi n}{2}|
< \frac{\pi}{8}$ for some $n \in \mathbf{Z}$, and $\inf_{|\xi|\leq
1} |\mathcal{F} u_0 (\xi)| >0$, then the solution has an additional
logarithmic time-decay in the short range region $|x| \leq
\sqrt{t}$. In the far region $|x| > \sqrt{t}$ the asymptotics have
a quasi-linear character.
Category:35Q55 |