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Large Time Behavior for the Cubic Nonlinear Schrödinger Equation

 Printed: Oct 2002
  • Nakao Hayashi
  • Pavel I. Naumkin
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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.
MSC Classifications: 35Q55 show english descriptions NLS-like equations (nonlinear Schrodinger) [See also 37K10] 35Q55 - NLS-like equations (nonlinear Schrodinger) [See also 37K10]

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