On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension
type: Item
instance of: scholarly article
title: On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension
zbMATH Open document ID: 0617.35025
DOI: 10.1007/BF01162710
author: Nakao Hayashi
author: Kuniaki Nakamitsu
author: Masayoshi Tsutsumi
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173763
review text: The authors show that the nonlinear Schrödinger equation \(iu_ t+u_{xx}=\lambda | u|^{p-1} u\), (t,x)\(\in {\mathbb{R}}\times {\mathbb{R}}\), \(\lambda\in {\mathbb{R}}\), \(p>1\), with the initial condition \(u(0,x)=\phi (x)\), \(x\in {\mathbb{R}}\) has a regularizing effect on its solution. More precisely, if the initial function \(\phi\) is in \(H^ 1({\mathbb{R}})\) (or in \(L^ 2({\mathbb{R}}))\) and decays sufficiently rapidly as \(| x| \to +\infty\), then u(t,x) is smooth for \(t\neq 0\).
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4000493
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: regularizing effect
zbMATH Keywords: decays
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4000493