American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2021030
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line

Abderrazak Chrifi , , Mostafa Abounouh and  Hassan Al Moatassime

Department of Mathematics, Faculty of Science and Technology, Cadi Ayyad University, B.P. 549, Av. Abdelkarim Elkhattabi, Guéliz, Marrakesh, 40000, Morocco

* Corresponding author: Abderrazak Chrifi (abderrazak.chrifi@gmail.com)

Received  August 2020 Revised  January 2021 Early access March 2021

We consider a weakly damped cubic nonlinear Schrödinger equation with Dirac interaction defect in a half line of $ \mathbb{R} $. Endowed with artificial boundary condition at the point $ x = 0 $, we discuss the global existence and uniqueness of solution of this equation by using Faedo–Galerkin method.

Keywords: Nonlinear Schrödinger equation, Artificial boundary condition, Fractional derivative, Galerkin method.
Mathematics Subject Classification: Primary: 35Q55, 49K40; Secondary: 37L65.
Citation: Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021030
References:
[1]

M. Abounouh, H. Al Moatassime and A. Chrifi, Artificial boundary condition for one-dimensional nonlinear Schrödinger problem with Dirac interaction: Existence and uniqueness results, Boundary Value Problems, 2018 (2018), 16. doi: 10.1186/s13661-018-0935-9.  Google Scholar

[2]

M. Abounouh, H. Al Moatassime and A. Chrifi, Existence of global attractor for one-dimensional weakly damped nonlinear Schrödinger equation with Dirac interaction and artificial boundary condition in half-line, Advances in Difference Equations, 2017 (2017), 137. doi: 10.1186/s13662-017-1194-2.  Google Scholar

[3]

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., (2008), 729–796.  Google Scholar

[4]

X. Antoine, C. Besse and P. Klein, Absorbing boundary conditions for the one-dimensional schrödinger equation with an exterior repulsive potential, Journal of Computational Physics, 228 (2009), 312-335.  doi: 10.1016/j.jcp.2008.09.013.  Google Scholar

[5]

X. Antoine, C. Besse and P. Klein, Absorbing boundary conditions for schrödinger equations with general potentials and nonlinearities, SIAM Journal on Scientific Computing, 33 (2011), 1008-1033.  doi: 10.1137/090780535.  Google Scholar

[6]

A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, (1998). Google Scholar

[7]

W. Bao, The nonlinear Schrödinger equation and applications in Bose-Einstein condensation and plasma physics, in Dynamics in Models of Coarsening, Coagulation, Condensation and Quantization, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 9, World Sci. Publ., Hackensack, NJ, 2007,141–239. doi: 10.1142/9789812770226_0003.  Google Scholar

[8]

L. Burgnies, O. Vanbésien and D. Lippens, Transient analysis of ballistic transport in stublike quantum waveguides, Applied Physics Letters, (1997). Google Scholar

[9]

A. Chrifi, Analyse des schémas numériques et comportement asymptotique de certaines EDP dispersives, Ph.D thesis, Cadi Ayyad University, 2017. Google Scholar

[10]

J. F. Claerbout, Coarse grid calculation of waves in inhomogeneous media with application to delineation of complicated seismic structure, Geophysics, (1970). Google Scholar

[11]

J.-M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Annales de l'I.H.P. Analyse non Linéaire, 5 (1988), 365-405.  doi: 10.1016/S0294-1449(16)30343-2.  Google Scholar

[12]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.  doi: 10.1063/1.523491.  Google Scholar

[13]

R. H. Goodman, P. J. Holmes and M. I. Weinstein, Strong NLS soliton–defect interactions, Phys. D, 192 (2004), 215-248.  doi: 10.1016/j.physd.2004.01.021.  Google Scholar

[14]

E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-477.  doi: 10.1007/BF02731494.  Google Scholar

[15]

W. Kechiche, Systèmes d'équations de Schrödinger non linéaires, Ph.D thesis, University of Monastir, Tunisia, 2012. Google Scholar

[16]

M. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation, IEE electromagnetic waves series, 45, Institution of Electrical Engineers, London, 2000. doi: 10.1049/PBEW045E.  Google Scholar

[17]

L. P. Pitaevskiĭ, Vortex lines in an imperfect bose gas, Soviet Physics JETP, 13 (1961), 451-454.   Google Scholar

[18]

F. Schmidt and P. Deuflhard, Discrete transparent boundary conditions for the numerical solution of Fresnel's equation, Computers & Mathematics with Applications, 29 (1995), 53-76.  doi: 10.1016/0898-1221(95)00037-Y.  Google Scholar

[19]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, 139, Springer Verlag, New York, New York, 1999.  Google Scholar

[20]

F. D. Tappert, The Parabolic Approximation Method, Springer Berlin Heidelberg, Berlin, Heidelberg, 1977,224–287.  Google Scholar

show all references

References:
[1]

M. Abounouh, H. Al Moatassime and A. Chrifi, Artificial boundary condition for one-dimensional nonlinear Schrödinger problem with Dirac interaction: Existence and uniqueness results, Boundary Value Problems, 2018 (2018), 16. doi: 10.1186/s13661-018-0935-9.  Google Scholar

[2]

M. Abounouh, H. Al Moatassime and A. Chrifi, Existence of global attractor for one-dimensional weakly damped nonlinear Schrödinger equation with Dirac interaction and artificial boundary condition in half-line, Advances in Difference Equations, 2017 (2017), 137. doi: 10.1186/s13662-017-1194-2.  Google Scholar

[3]

X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., (2008), 729–796.  Google Scholar

[4]

X. Antoine, C. Besse and P. Klein, Absorbing boundary conditions for the one-dimensional schrödinger equation with an exterior repulsive potential, Journal of Computational Physics, 228 (2009), 312-335.  doi: 10.1016/j.jcp.2008.09.013.  Google Scholar

[5]

X. Antoine, C. Besse and P. Klein, Absorbing boundary conditions for schrödinger equations with general potentials and nonlinearities, SIAM Journal on Scientific Computing, 33 (2011), 1008-1033.  doi: 10.1137/090780535.  Google Scholar

[6]

A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, (1998). Google Scholar

[7]

W. Bao, The nonlinear Schrödinger equation and applications in Bose-Einstein condensation and plasma physics, in Dynamics in Models of Coarsening, Coagulation, Condensation and Quantization, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 9, World Sci. Publ., Hackensack, NJ, 2007,141–239. doi: 10.1142/9789812770226_0003.  Google Scholar

[8]

L. Burgnies, O. Vanbésien and D. Lippens, Transient analysis of ballistic transport in stublike quantum waveguides, Applied Physics Letters, (1997). Google Scholar

[9]

A. Chrifi, Analyse des schémas numériques et comportement asymptotique de certaines EDP dispersives, Ph.D thesis, Cadi Ayyad University, 2017. Google Scholar

[10]

J. F. Claerbout, Coarse grid calculation of waves in inhomogeneous media with application to delineation of complicated seismic structure, Geophysics, (1970). Google Scholar

[11]

J.-M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Annales de l'I.H.P. Analyse non Linéaire, 5 (1988), 365-405.  doi: 10.1016/S0294-1449(16)30343-2.  Google Scholar

[12]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.  doi: 10.1063/1.523491.  Google Scholar

[13]

R. H. Goodman, P. J. Holmes and M. I. Weinstein, Strong NLS soliton–defect interactions, Phys. D, 192 (2004), 215-248.  doi: 10.1016/j.physd.2004.01.021.  Google Scholar

[14]

E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-477.  doi: 10.1007/BF02731494.  Google Scholar

[15]

W. Kechiche, Systèmes d'équations de Schrödinger non linéaires, Ph.D thesis, University of Monastir, Tunisia, 2012. Google Scholar

[16]

M. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation, IEE electromagnetic waves series, 45, Institution of Electrical Engineers, London, 2000. doi: 10.1049/PBEW045E.  Google Scholar

[17]

L. P. Pitaevskiĭ, Vortex lines in an imperfect bose gas, Soviet Physics JETP, 13 (1961), 451-454.   Google Scholar

[18]

F. Schmidt and P. Deuflhard, Discrete transparent boundary conditions for the numerical solution of Fresnel's equation, Computers & Mathematics with Applications, 29 (1995), 53-76.  doi: 10.1016/0898-1221(95)00037-Y.  Google Scholar

[19]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, 139, Springer Verlag, New York, New York, 1999.  Google Scholar

[20]

F. D. Tappert, The Parabolic Approximation Method, Springer Berlin Heidelberg, Berlin, Heidelberg, 1977,224–287.  Google Scholar

[1]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878
[2]

Vyacheslav A. Trofimov, Evgeny M. Trykin. A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation. Conference Publications, 2015, 2015 (special) : 1070-1078. doi: 10.3934/proc.2015.1070
[3]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021028
[4]

Razvan Mosincat, Haewon Yoon. Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 47-80. doi: 10.3934/dcds.2020003
[5]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[6]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93
[7]

Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102
[8]

Kazumasa Fujiwara, Tohru Ozawa. On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 275-280. doi: 10.3934/eect.2018013
[9]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
[10]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1447-1478. doi: 10.3934/cpaa.2021028
[11]

Masoumeh Hosseininia, Mohammad Hossein Heydari, Carlo Cattani. A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2273-2295. doi: 10.3934/dcdss.2020295
[12]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615
[13]

Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237
[14]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[15]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[16]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[17]

Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188
[18]

Songbai Peng, Aliang Xia. Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021128
[19]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3589-3610. doi: 10.3934/dcdss.2021021
[20]

Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (100)
  • HTML views (263)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences