Volume 5, Issue 3, September 2019, Page: 66-71
Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition
Konan Firmin N'gohisse, Department of Biology, University of Peleforo Gon Coulibaly, Korhogo, Côte d'Ivoire
Diabate Nabongo, Departement of Economy, University of Alassane Ouattara, Bouaké, Côte d'Ivoire
Lassane Traoré, Department of Biology, University of Peleforo Gon Coulibaly, Korhogo, Côte d'Ivoire
Received: Jul. 17, 2019;       Accepted: Aug. 12, 2019;       Published: Aug. 26, 2019
DOI: 10.11648/j.ijamtp.20190503.13      View  128      Downloads  41
Abstract
Theoretical study of the phenomenon of blow-up solutions for semilinear Schrödinger equations has been the subject of investigations of many authors. It is said that the maximal time interval of existence of the solution blows up in a finite time when this time is finite, and the solution develops a singularity in a finite time. In fact, semilinear Schrhödinger equation models a lot of physical phenomenon such as nonlinear optics, energy transfer in molecular systems, quantum mechanics, seismology, plasma physics. In the past, certain authors have used numerical methods to study the phenomenon of blow-up for semilinear Schrödinger equations. They have considered the same problem and one proves that the energy of the system is conserved, and the method used to show blow-up solutions are based on the energy's method. This paper proposes a method based on a modification of the method of Kaplan using eigenvalues and eigenfunctions to show that the semidiscrete solution blows up in a finite time under some assumptions. The semidiscrete blow-up time is also estimate. Similar results are obtain replacing the reaction term by another form to generalise the result. Finally, this paper propose two schemes for some numerical experiments and a graphics is given to illustrate the analysis.
Keywords
Semidiscretization, Blow-up, Schrödinger Equations
To cite this article
Konan Firmin N'gohisse, Diabate Nabongo, Lassane Traoré, Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition, International Journal of Applied Mathematics and Theoretical Physics. Vol. 5, No. 3, 2019, pp. 66-71. doi: 10.11648/j.ijamtp.20190503.13
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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