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August  2021, 14(8): 3027-3042. doi: 10.3934/dcdss.2021031

Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point

Wided Kechiche

Laboratoire de recherche: Analyse, Probabilité et Fractales, Faculté des Sciences de Monastir, Av. de l'environement, 5000 Monastir, Tunisie

Received  December 2019 Revised  January 2021 Published  August 2021 Early access  March 2021

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.

Keywords: Global attractor, nonlinear Schrödinger equation, impurity concentrated in one point, global solution, infinite dimensional dynamical system.
Mathematics Subject Classification: Primary: 37L30, 37L50, 35Q55; Secondary: 35A01.
Citation: Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Appl. Anal., 60 (1996), 99-119.  doi: 10.1080/00036819608840420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Appl. Anal., 60 (1996), 99-119.  doi: 10.1080/00036819608840420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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