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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 & 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.  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

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