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On the fuzzy stability results for fractional stochastic Volterra integral equation
Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line
Department of Mathematics, Faculty of Science and Technology, Cadi Ayyad University, B.P. 549, Av. Abdelkarim Elkhattabi, Guéliz, Marrakesh, 40000, Morocco |
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.
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). 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. |
[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. |
[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. 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. |
[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. |
[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). 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. |
[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. |
[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. 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. |
[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. |
[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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