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On the convergence to equilibria of a sequence defined by an implicit scheme
Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point
Laboratoire de recherche: Analyse, Probabilité et Fractales, Faculté des Sciences de Monastir, Av. de l'environement, 5000 Monastir, Tunisie |
We consider the nonlinear Schrödinger equation in dimension one with a nonlinearity concentrated in one point. We prove that this equation provides an infinite dimensional dynamical system. We also study the asymptotic behavior of the dynamics. We prove the existence of a global attractor for the system.
References:
[1] |
R. Adami and A. Teta,
A simple model of concentrated nonlinearity: Operator theory, Mathematical Results in Quantum Mechanics, 108 (1999), 183-189.
doi: 10.1007/978-3-0348-8745-8_13. |
[2] |
N. Akroune,
Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on $\mathbb R$, Appl. Math. Lett., 12 (1999), 45-48.
doi: 10.1016/S0893-9659(98)00170-0. |
[3] |
J. M. Ball,
Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[4] |
O. M. Bulashenko, V. A. Kochelap and L. L. Bonilla, Coherent patterns and self-indiced diffraction of electrons on a thin nonlinear layer, Phys.Rev B, 54 (1996), 1537–1540. arXiv: cond-mat/9604164.
doi: 10.1103/PhysRevB.54.1537. |
[5] |
T. Cazenave, Semilinear Schrödinger Equations, , Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[6] |
R. H. Goodman, P. J. Holmes and M. I. Wenstein, Strong NLS soliton-defect interactions, Physica D, 192 (2004), 215–248. arXiv: nlin/0203057
doi: 10.1016/j.physd.2004.01.021. |
[7] |
O. Goubet,
Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Appl. Anal., 60 (1996), 99-119.
doi: 10.1080/00036819608840420. |
[8] |
J. Holmer and C. Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory, J. Math. Anal. Appl., 483 (2020), 123522, 20 pp. arXiv: 1510.03491
doi: 10.1016/j.jmaa.2019.123522. |
[9] |
G. Jona-Lasinio, C. Presilla and J. Sjöstrand, On Schrödinger equations with concentrated nonlinearities, Ann. Phys., 240 (1995), 1–21. arXiv: cond-mat/9501037
doi: 10.1006/aphy.1995.1040. |
[10] |
W. Kechiche,
Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect, Commun. Pure Appl. Anal., 16 (2017), 1233-1252.
doi: 10.3934/cpaa.2017060. |
[11] |
W. Kechiche, Systemes d'Equations de Schrödinger non Lin aires, Ph. D thesis, Université de Monastir et Universit'e de Picardie Jules Vernes, 2012 (To appear). |
[12] |
P. Laurençot,
Long-time behavior for weakly damped driven nonlinear Schrödinger equation in $\mathbb R^{N}, \; N\leq 3$, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 357-369.
doi: 10.1007/BF01261181. |
[13] |
K. Lu and B. Wang,
Global attractor for the Klein-Gordon-Schrödinger equations in unbounded domains, J. Differential Equations, 170 (2001), 281-316.
doi: 10.1006/jdeq.2000.3827. |
[14] |
B. A. Malomed and M. Ya. Azbel, Modulational instability of a wave scattered by a nonlinear center, Phys. Rev. B, 47 (1993), 10402.
doi: 10.1103/PhysRevB.47.10402. |
[15] |
A. Miranville and S. Zelik,
Attractors for dissipative partial differential equation in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, 4 (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[16] |
I. Moise, R. Rosa and X. Wang, Attractors for noncompact semigroups via energy equations, Nonlinearity, 11 (1998), 1369–1393. https://pdfs.semanticscholar.org/cfbf/a1fb70b618f40193593a93d1b39f551a772c.pdf
doi: 10.1088/0951-7715/11/5/012. |
[17] |
F. Nier,
The dynamics of some quantum open systems with short-rang nonlinearities, Nonlinearity, 11 (1998), 1127-1172.
doi: 10.1088/0951-7715/11/4/022. |
[18] |
G. Raugel,
Global attractor in partial differential equations, Handbook of Dynamical Systems, 2 (2002), 885-982.
doi: 10.1016/S1874-575X(02)80038-8. |
[19] |
R. Rosa, The global attractor of weakly damped forced Korteweg-De Vries equation in $H^1(\mathbb R)$, Mat. Contemp. 19 (2000), 129–152. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.407.421&rep=rep1&type=pdf |
[20] |
C. Sulem and P.-L. Sulem,
Focusing nonlinear Schrödinger equation and wave-packet collapse, Nonlinear Analysis, 30 (1997), 833-844.
doi: 10.1016/S0362-546X(96)00168-X. |
[21] |
R. Temam, Infinite-Dimentional Dynamical Systems in Mecanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[22] |
X. Wang,
An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D, 88 (1995), 167-175.
doi: 10.1016/0167-2789(95)00196-B. |
show all references
References:
[1] |
R. Adami and A. Teta,
A simple model of concentrated nonlinearity: Operator theory, Mathematical Results in Quantum Mechanics, 108 (1999), 183-189.
doi: 10.1007/978-3-0348-8745-8_13. |
[2] |
N. Akroune,
Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on $\mathbb R$, Appl. Math. Lett., 12 (1999), 45-48.
doi: 10.1016/S0893-9659(98)00170-0. |
[3] |
J. M. Ball,
Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[4] |
O. M. Bulashenko, V. A. Kochelap and L. L. Bonilla, Coherent patterns and self-indiced diffraction of electrons on a thin nonlinear layer, Phys.Rev B, 54 (1996), 1537–1540. arXiv: cond-mat/9604164.
doi: 10.1103/PhysRevB.54.1537. |
[5] |
T. Cazenave, Semilinear Schrödinger Equations, , Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[6] |
R. H. Goodman, P. J. Holmes and M. I. Wenstein, Strong NLS soliton-defect interactions, Physica D, 192 (2004), 215–248. arXiv: nlin/0203057
doi: 10.1016/j.physd.2004.01.021. |
[7] |
O. Goubet,
Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Appl. Anal., 60 (1996), 99-119.
doi: 10.1080/00036819608840420. |
[8] |
J. Holmer and C. Liu, Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory, J. Math. Anal. Appl., 483 (2020), 123522, 20 pp. arXiv: 1510.03491
doi: 10.1016/j.jmaa.2019.123522. |
[9] |
G. Jona-Lasinio, C. Presilla and J. Sjöstrand, On Schrödinger equations with concentrated nonlinearities, Ann. Phys., 240 (1995), 1–21. arXiv: cond-mat/9501037
doi: 10.1006/aphy.1995.1040. |
[10] |
W. Kechiche,
Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect, Commun. Pure Appl. Anal., 16 (2017), 1233-1252.
doi: 10.3934/cpaa.2017060. |
[11] |
W. Kechiche, Systemes d'Equations de Schrödinger non Lin aires, Ph. D thesis, Université de Monastir et Universit'e de Picardie Jules Vernes, 2012 (To appear). |
[12] |
P. Laurençot,
Long-time behavior for weakly damped driven nonlinear Schrödinger equation in $\mathbb R^{N}, \; N\leq 3$, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 357-369.
doi: 10.1007/BF01261181. |
[13] |
K. Lu and B. Wang,
Global attractor for the Klein-Gordon-Schrödinger equations in unbounded domains, J. Differential Equations, 170 (2001), 281-316.
doi: 10.1006/jdeq.2000.3827. |
[14] |
B. A. Malomed and M. Ya. Azbel, Modulational instability of a wave scattered by a nonlinear center, Phys. Rev. B, 47 (1993), 10402.
doi: 10.1103/PhysRevB.47.10402. |
[15] |
A. Miranville and S. Zelik,
Attractors for dissipative partial differential equation in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, 4 (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[16] |
I. Moise, R. Rosa and X. Wang, Attractors for noncompact semigroups via energy equations, Nonlinearity, 11 (1998), 1369–1393. https://pdfs.semanticscholar.org/cfbf/a1fb70b618f40193593a93d1b39f551a772c.pdf
doi: 10.1088/0951-7715/11/5/012. |
[17] |
F. Nier,
The dynamics of some quantum open systems with short-rang nonlinearities, Nonlinearity, 11 (1998), 1127-1172.
doi: 10.1088/0951-7715/11/4/022. |
[18] |
G. Raugel,
Global attractor in partial differential equations, Handbook of Dynamical Systems, 2 (2002), 885-982.
doi: 10.1016/S1874-575X(02)80038-8. |
[19] |
R. Rosa, The global attractor of weakly damped forced Korteweg-De Vries equation in $H^1(\mathbb R)$, Mat. Contemp. 19 (2000), 129–152. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.407.421&rep=rep1&type=pdf |
[20] |
C. Sulem and P.-L. Sulem,
Focusing nonlinear Schrödinger equation and wave-packet collapse, Nonlinear Analysis, 30 (1997), 833-844.
doi: 10.1016/S0362-546X(96)00168-X. |
[21] |
R. Temam, Infinite-Dimentional Dynamical Systems in Mecanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[22] |
X. Wang,
An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D, 88 (1995), 167-175.
doi: 10.1016/0167-2789(95)00196-B. |
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