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

| Peer-Reviewed

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

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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), 2019. Published by Science Publishing Group

Keywords

Semidiscretization, Blow-up, Schrödinger Equations

References

[1]	R. Adami, G. D. Antonio, R. Figari and A. Teta, Blow-up solution for the Schrodinger equation in dimension three with a concentation nonlinearity, Ann. I. H. Poincaré, 21 (2004), 121-137.
[2]	A. Ambrosetti, E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London. Math. Soc., 75 (2007), 67-82.
[3]	G. D Akrivis, V. A. Dougalis, O. A. Krakashian and W. R. McKinney, Numerical approximation of blow-up of radially symmetric solutions of the nonliear Schrödinger, SIAM. J. Sci. Comput., 25 (2003), 186-212.
[4]	O. Bang, P. L. Christiansen, K. O. Rasmussen and Yu. B. Gaididei, The role of nonlinearity in modeling enrgy transfer in scheibe aggregates in nonlinear excitation in Bimoleculear, M. Peyrard, ed., Les Editions de Physique, Springer, Berlin, (1995), 317-336.
[5]	C. Besse, N. J. Mauser, H. P. Stimming, Numerical study of the Davey-Stewartson system, Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 6, p. 1035-1054.
[6]	T. K. Boni and F. K. N'gohisse, Numerical blow-up for a nonlinear heat equation, In Acta Math. Sinica English serie, Vol. 27 (5) (2011), 845-862.
[7]	J. Colliander, M. Grillakis, and N. Tzirakis, Remarks on global a priori estimates for the nonlinear Schrodinger equation, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4359-4371.
[8]	W. Dai, An unconditionally stable three-level explicit difference scheme for the Schrödinger equation with a variable coefficient, SIAM. J. Numer. Anal., 29 (1992), 174-181.
[9]	V. D. Dinh, Blowup of H¹ solutions for a class of the focusing inhomogeneous nonlinear Schrodinger equation, Nonl. Analysis, 174 (2018), 169-188.
[10]	W. Dai, Absolute stable explicit and semi-explicit schemes for Schrödinger equations, Math. Num. Sinica, 11 (1989), 128-131.
[11]	V. V. Drits, Conservative difference schemes for problems of nonlinear optics, 1, Differential Equation, 27 (1991), 1153-1161.
[12]	R. ~ Ferreira, P. ~ Groisman and J. ~ D. ~ Rossi, Numerical blow-up for a nonlinear problem with a nonlinear boundary condition, Math. Models Methods. Appl. Sci., 12 (2002), 461--483.
[13]	F. Ivanauskas and M. Radziunas, On convergence and stability of the explicit difference method for solution of nonlinear Schrödinger equations, SIAM J. Numer. Anal., 36 (1999), 1466-1481.
[14]	S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Communs pure Appl. Math., 16 (1963), 305-330.
[15]	O. Kavian, A remark on the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. Amer. Math. Soc., 193 (1987), 193-203.
[16]	S. Klainerman and G. Ponce, Global, small amplitude solutions to nonlinear evolution equations, Communs pure Appl. Math., 36 (1983), 133-141.
[17]	Z. Lu and Z. Liu, L² -concentration of blow-up solutions for two-coupled nonlinear Schrodinger equations, J. Math. Anal. Appl., 380 (2) (2011), 531-539.
[18]	F. Merle, P. Raphael,} Profiles and quantization of the blow up mass for critical nonlinear Schrdinger equation, Comm. Math. Phys., 235 (2005), 675-704.
[19]	T. Nakagawa, Blowing up on the finite difference solution to parabolic equation, Appl. Math. Optim., 2 (1976), 337-350.
[20]	K. N'guessan, N. Diabate and A. K. Toure, Blow-up for semidiscratization of some nonlinear parabolic equation with a convection term, In J. of progrssive reachach in Math. Vol. 5 (2) (2015), 499-518.
[21]	B. E. A. Saleh and M. C. Teich, Fundamentals of fotonics, Interscience Publication, New-York., (1991).
[22]	T. R. Taha, A numerical scheme for the nonlinear Schrödinger equation, Comput. Math. Optim., 2 (1976), 337-350.
[23]	Y. Tourigny and J. M. Sanz Serna, The numerical study of blow-up with application to a nonlinear Schrödinger equation, J. Comp. Phys., 102 (1992), 407-416.
[24]	M. Tsutsumi, On global solutions to the initial-boundary value problem for the nonlinear Schrödinger equations in exterior domains, Comm. Part. Diff. Equat., 16 (1991), 885-907.
[25]	L. Wu, Dufort-Frankel-type methods for linear and nonlinear Schrödinger equations, SIAM. J. Numer. Anal., 33 (1996), 1526-1533.
[26]	J. Zhang, On an initial-boundary value problem for a class of nonlinear Schrödinger equations, Comm. Part. Diff. Equat., 21 (1996), 687-692.
[27]	S. Zhu, Blow-up solutions for inhomogeneous Schrodinger equation with L² supercritical nonlinearity, J. Math. Anal. Appl., 409 (2) (2014), 760-776.
[28]	V. E. Zakharov and A. B. Shabat, An L^∞ Exact theory of two dimensional self focusing and one dimensional self modulation of waves in nonlinear media, Soviet Physics JETP.}, 34 (1972), 62-69.

Cite This Article

Plain Text BibTeX RIS

APA Style

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

Copy | Download

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

Copy | Download

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

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

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

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

Copy | Download

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

Copy | Download

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