Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations

Bégout, Pascal and Diaz, Jesus Ildefonso (2014) Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations. Electronic Journal of Differential Equations, vol. 90. pp. 1-15.

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Official URL: http://tse-fr.eu/pub/28119

Abstract

This paper deals with the study of "\textit{sharp localized}" solutions of a nonlinear type Schr\"odinger equation in the whole space $\R^N,$ $N\ge1,$ with a zero order term, in modulus, like a power $m$ less than one of the modulus of the solution, and with a non zero external forcing term $\f.$ Our fundamental assumption is that such an exponent $m$ verifies $m\in (0,1).$ The self-similar structure of the solution is justified from the assumption that the external forcing term satisfies that $\f(t,x)=t^{-(\vp-2)/2}\F(t^{-1/2}x)$ for some complex exponent $\vp$ and for some profile function $\F$ which is assumed to be with compact support in $\R^N.$ We show the existence of solutions $\vu(t,x)=t^{\vp/2}\U(t^{-1/2}x),$ with a profile $\U,$ which also have compact support in $\R^N,$ reason why we call as "\textit{sharp localized}" solutions to this type of solutions. The proof of the localization of the support of the profile $\U$ uses some suitable energy method applied to the stationary problem satisfied by $\U$ after some unknown transformation.

Item Type: Article
Language: English
Date: 2 April 2014
Refereed: Yes
Subjects: B- ECONOMIE ET FINANCE
Divisions: Institut de mathématiques de Toulouse, TSE-R (Toulouse)
Site: UT1
Date Deposited: 09 Jul 2014 17:43
Last Modified: 12 Mar 2018 11:12
OAI ID: oai:tse-fr.eu:28119
URI: http://publications.ut-capitole.fr/id/eprint/15883

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