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Large Time Behavior for the Cubic Nonlinear Schrödinger Equation


Published:20021001
Printed: Oct 2002
Nakao Hayashi
Pavel I. Naumkin
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Abstract
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 timedecay in the short range region $x \leq
\sqrt{t}$. In the far region $x > \sqrt{t}$ the asymptotics have
a quasilinear character.