Pascal Begout, Jesus Ildefonso Diaz
``Sharp localized'' solutions (i.e. with compact support for each given time t) of a singular nonlinear type Schr\"odinger equation in the whole space are constructed here under the assumption that they have a self-similar structure. It requires the assumption that the external forcing term satisfies that for some complex exponent and for some profile function which is assumed to be with compact support in . We show the existence of solutions of the form , with a profile , which also has compact support in . The proof of the localization of the support of the profile uses some suitable energy method applied to the stationary problem satisfied by after some unknown transformation.
Submitted December 9, 2013. Published April 2, 2014.
Math Subject Classifications: 35B99, 35A01, 35A02, 35B65, 35J60.
Key Words: Nonlinear self-similar Schrodinger equation; compact support; energy method.
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| Pascal Bégout |
Institut de Mathématiques de Toulouse & TSE
Université Toulouse I Capitole, Manufacture des Tabacs
21, Allée de Brienne, 31015 Toulouse Cedex 6, France
| Jesús Ildefonso Díaz |
Departamento de Matemática Aplicada
Instituto de Matemática Interdisciplinar
Universidad Complutense de Madrid
Plaza de las Ciencias, 3, 28040 Madrid, Spain
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