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January  2022, 15(1): 79-93. doi: 10.3934/dcdss.2021030

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 Published  January 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (1) : 79-93. 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[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.

[19]

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

[20]

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

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[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.

[19]

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

[20]

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

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