Semiclassical nonlinear Schrödinger equations with potential and focusing initial data
type: Item
instance of: scholarly article
title: Semiclassical nonlinear Schrödinger equations with potential and focusing initial data
zbMATH Open document ID: 1155.35090
author: Rémi Carles
author: Luc Miller
published in: Osaka Journal of Mathematics
publication date: 2005-02-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0212263
review text: The authors study the semiclassical limit for solutions \(u^{\varepsilon}\) to a Schrödinger equation with repulsive subcritical nonlinearity and potential, with concentrating initial data. The potential \(V\) is assumed to be polynomial of degree 2, and the eigenvalues of the quadratic part of \(V\) are allowed to be negative. We stress that in such a case the energy of \(u^{\varepsilon}\) contains negative terms which are not a priori controlled by the positive ones, so global existence in \(H^1\) is not trivial. For any \(T>0\), the authors show that \(u^{\varepsilon}\) cannot blow up for \(0<| t| < T\). Moreover they prove the asymptotics for the semiclassical limit: the behavior near \(t=0\) is governed by the nonlinear term where the potential is negligible, while for \(| t| \) large we have the reversed situation. The transition between these two regimes is measured by the scattering operator associated to the potential \(V\).
reviewed by: Matteo Franca
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2134875
zbMATH Keywords: semiclassical limit
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: subquadratic potential
zbMATH Keywords: asymptotics
MaRDI profile type: Publication
arXiv ID: math/0212263
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description: scientific article