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January/February 2020 Large time asymptotics for the fractional nonlinear Schrödinger equation
Nakao Hayashi, Pavel I. Naumkin
Adv. Differential Equations 25(1/2): 31-80 (January/February 2020). DOI: 10.57262/ade/1580958058

Abstract

We consider the Cauchy problem for the fractional nonlinear Schrödinger equation \[ \left\{ \begin{array} [c]{c}% i\partial_{t}u+\frac{1}{\alpha} \left| \partial_{x} \right| ^{\alpha }u=\lambda\left| u \right| ^{2}u,\text{ }t>0, \quad x\in\mathbb{R}% \mathbf{,}\\ u \left( 0,x \right) =u_{0} \left( x \right) , \quad x\in\mathbb{R}% \mathbf{,}% \end{array} \right. \] where $\lambda\in\mathbb{R},$ the order of the fractional derivative $\alpha\in\left( 1,\frac{3}{2} \right) .$ We obtain the large time asymptotic behavior of solutions which has a logarithmic phase modifications for a large time comparing with the linear problem.

Citation

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Nakao Hayashi. Pavel I. Naumkin. "Large time asymptotics for the fractional nonlinear Schrödinger equation." Adv. Differential Equations 25 (1/2) 31 - 80, January/February 2020. https://doi.org/10.57262/ade/1580958058

Information

Published: January/February 2020
First available in Project Euclid: 6 February 2020

zbMATH: 1303.35100
MathSciNet: MR4060443
Digital Object Identifier: 10.57262/ade/1580958058

Subjects:
Primary: 35B40 , 35Q92

Rights: Copyright © 2020 Khayyam Publishing, Inc.

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Vol.25 • No. 1/2 • January/February 2020
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