Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition

Konan Firmin N'gohisse; Diabate Nabongo; Lassane Traoré

doi:doi:10.11648/j.ijamtp.20190503.13

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Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition

Konan Firmin N'gohisse, Diabate Nabongo, Lassane Traoré

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 5, Issue 3)

Received: 17 July 2019 Accepted: 12 August 2019 Published: 26 August 2019

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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.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 5, Issue 3)
DOI	10.11648/j.ijamtp.20190503.13
Page(s)	66-71
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Semidiscretization, Blow-up, Schrödinger Equations

References

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Konan Firmin N'gohisse, Diabate Nabongo, Lassane Traoré. (2019). Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition. International Journal of Applied Mathematics and Theoretical Physics, 5(3), 66-71. https://doi.org/10.11648/j.ijamtp.20190503.13

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ACS Style

Konan Firmin N'gohisse; Diabate Nabongo; Lassane Traoré. Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition. Int. J. Appl. Math. Theor. Phys. 2019, 5(3), 66-71. doi: 10.11648/j.ijamtp.20190503.13

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AMA Style

Konan Firmin N'gohisse, Diabate Nabongo, Lassane Traoré. Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition. Int J Appl Math Theor Phys. 2019;5(3):66-71. doi: 10.11648/j.ijamtp.20190503.13

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@article{10.11648/j.ijamtp.20190503.13,
  author = {Konan Firmin N'gohisse and Diabate Nabongo and Lassane Traoré},
  title = {Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {5},
  number = {3},
  pages = {66-71},
  doi = {10.11648/j.ijamtp.20190503.13},
  url = {https://doi.org/10.11648/j.ijamtp.20190503.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20190503.13},
  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.},
 year = {2019}
}

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TY  - JOUR
T1  - Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition
AU  - Konan Firmin N'gohisse
AU  - Diabate Nabongo
AU  - Lassane Traoré
Y1  - 2019/08/26
PY  - 2019
N1  - https://doi.org/10.11648/j.ijamtp.20190503.13
DO  - 10.11648/j.ijamtp.20190503.13
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 66
EP  - 71
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20190503.13
AB  - 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.
VL  - 5
IS  - 3
ER  -

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Author Information

Konan Firmin N'gohisse

Department of Biology, University of Peleforo Gon Coulibaly, Korhogo, C?te d'Ivoire
Diabate Nabongo

Departement of Economy, University of Alassane Ouattara, Bouaké, C?te d'Ivoire
Lassane Traoré

Department of Biology, University of Peleforo Gon Coulibaly, Korhogo, C?te d'Ivoire

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Konan Firmin N'gohisse, Diabate Nabongo, Lassane Traoré. (2019). Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition. International Journal of Applied Mathematics and Theoretical Physics, 5(3), 66-71. https://doi.org/10.11648/j.ijamtp.20190503.13

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ACS Style

Konan Firmin N'gohisse; Diabate Nabongo; Lassane Traoré. Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition. Int. J. Appl. Math. Theor. Phys. 2019, 5(3), 66-71. doi: 10.11648/j.ijamtp.20190503.13

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AMA Style

Konan Firmin N'gohisse, Diabate Nabongo, Lassane Traoré. Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition. Int J Appl Math Theor Phys. 2019;5(3):66-71. doi: 10.11648/j.ijamtp.20190503.13

Copy | Download

@article{10.11648/j.ijamtp.20190503.13,
  author = {Konan Firmin N'gohisse and Diabate Nabongo and Lassane Traoré},
  title = {Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {5},
  number = {3},
  pages = {66-71},
  doi = {10.11648/j.ijamtp.20190503.13},
  url = {https://doi.org/10.11648/j.ijamtp.20190503.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20190503.13},
  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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Blow-up for Semidiscretisations of a Semilinear Schrodinger Equation with Dirichlet Condition
AU  - Konan Firmin N'gohisse
AU  - Diabate Nabongo
AU  - Lassane Traoré
Y1  - 2019/08/26
PY  - 2019
N1  - https://doi.org/10.11648/j.ijamtp.20190503.13
DO  - 10.11648/j.ijamtp.20190503.13
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 66
EP  - 71
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20190503.13
AB  - 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.
VL  - 5
IS  - 3
ER  -

Copy | Download