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Large Time Behavior for the Cubic Nonlinear Schrödinger Equation

Published online by Cambridge University Press:  20 November 2018

Nakao Hayashi  and
Pavel I. Naumkin
Nakao Hayashi
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Osaka Toyonaka 560-0043, Japan, email: nhayashi@math.wani.osaka-u.ac.jp
Pavel I. Naumkin
Affiliation:
Instituto de Física y Matemáticas, Universidad Michoacana, AP 2-82, Morelia, CP 58040, Michoacán, Mexico email: pavelni@ccu.umich.mx
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Abstract

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We consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimension

1

$$\left\{ \begin{align} & i{{u}_{t}}\,+\,\frac{1}{2}{{u}_{xx}}\,+\,{{{\bar{u}}}^{3}}\,=\,0,\,\,\,\,\,t\,\in \,\mathbf{R},\,x\,\in \,\mathbf{R}, \\ & u(0,\,x)\,=\,{{u}_{0}}(x),\,\,\,\,\,\,\,\,\,\,\,x\,\in \,\mathbf{R}. \\ \end{align} \right.$$

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 (1). We prove that if the initial data ${{u}_{0}}\,\in \,{{\mathbf{H}}^{1,\,0}}\,\cap \,{{\mathbf{H}}^{0,\,1}}$ are small and such that ${{\sup }_{|\xi |\le 1}}\left| \arg \mathcal{F}{{u}_{0}}(\text{ }\xi \text{ )}\text{0}\frac{\pi n}{2} \right|<\frac{\pi }{8}$ for some n ∈ Z, and ${{\inf }_{\left| \xi \right|\le 1}}\,\left| \mathcal{F}{{u}_{0}}(\xi ) \right|\,>\,0$, then the solution has an additional logarithmic timedecay in the short range region $\left| x \right|\,\le \,\sqrt{t}$. In the far region $\left| x \right|\,>\,\sqrt{t}$ the asymptotics have a quasilinear character.

Keywords

35Q55
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 5 , 01 October 2002 , pp. 1065 - 1085
Copyright
Copyright © Canadian Mathematical Society 2002

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