Small solutions to nonlinear Schrödinger equations
type: Item
instance of: scholarly article
title: Small solutions to nonlinear Schrödinger equations
zbMATH Open document ID: 0786.35121
DOI: 10.1016/S0294-1449(16)30213-X
author: C. E. Kenig
author: G. Ponce
author: L. Vega
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1993-12-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78303
review text: A local existence theorem is proved for solutions of the Cauchy problem for the nonlinear Schrödinger equation of the following form: \[ iu_ t=-\Delta u+iP(u,\nabla u,\overline u,\nabla \overline u), \] where \(u\) is a complex function of time and \(n\) spatial variables, and \(P\) is a polynomial function (without constant or linear terms) of a degree \(s\). The existence of solutions at finite values of time is proved, using assumptions which establish boundedness of the initial data. The main technical ingredient in proving the theorem is the so-called smoothing effect of Kato type. The theorem is proved separately for four different cases corresponding, respectively, to \(n=1\) and \(n \geq 2\), and \(s=2\) and \(s \geq 3\). In each case, different specific boundedness conditions should be imposed on the initial data in order to prove the local existence of the solution.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 438585
zbMATH Keywords: local existence
zbMATH Keywords: Cauchy problem
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: boundedness of the initial data
zbMATH Keywords: smoothing effect of Kato type
reviewed by: B. A. Malomed
MaRDI profile type: Publication
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OpenAlex ID: W2278997551
description: scientific article; zbMATH DE number 438585