American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces
  • DCDS-B Home
  • This Issue
  • Next Article
    Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China
doi: 10.3934/dcdsb.2021032
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $

Brahim Alouini

University of Monastir, Faculty of Sciences, Research Laboratory: Analysis, Probability and Fractals, The Environment Avenue, 5019 Monastir, Tunisia

Received  June 2020 Revised  December 2020 Early access February 2021

We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in
$ \mathbb{R}^2 $
 that reads
$ u_t+i\Delta u-i \left(\partial_y^4 u-\partial_y^6 u\right)+ig(|u|^2)u+\gamma u = f\,,\;\;(t,(x,y))\in \mathbb{R}\times \mathbb{R}^2\,. $
We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.
Keywords: Schrödinger equation, global attractor, fractal dimension, asymptotic behaviour.
Mathematics Subject Classification: Primary: 35B40, 35Q55; Secondary: 76B03, 37L30.
Citation: Brahim Alouini. Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021032
References:
[1]

B. Alouini, Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Commun. Pure Appl. Anal., 14 (2015), 1781-1801.  doi: 10.3934/cpaa.2015.14.1781.  Google Scholar

[2]

B. Alouini, Finite dimensional global attractor for a dissipative anisotropic fourth order Schrödinger equation, Journal of Differential Equations, 266 (2019), 6037-6067.  doi: 10.1016/j.jde.2018.10.044.  Google Scholar

[3]

B. Alouini, A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger-type equations, Math. Meth. Appl. Sci., 44 (2021), 91-103.  doi: 10.1002/mma.6709.  Google Scholar

[4]

B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Discrete and Continuous Dynamical Systems - B, 19 (2014), 651-677.  doi: 10.3934/dcdsb.2014.19.651.  Google Scholar

[5]

A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow and N. Akhmediev, Infinite hierarchy of nonlinear Schrödinger equations and their solutions, Phys. Rev. E, 93 (2016), 012206. doi: 10.1103/PhysRevE.93.012206.  Google Scholar

[6]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynam. Systems - A, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[7]

O. V. Besov, V. P. Il'in and S. M. Nikol'ski ĭ, Integral Representations of Functions and Imbedding Theorems, Scripta Series in Mathematics, I, 1978.  Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[9]

I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, 19, ACTA, 2002.  Google Scholar

[10]

I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping, Memoirs of the American Mathematical Society, 195, American Mathematical Society, 2008. doi: 10.1090/memo/0912.  Google Scholar

[11]

S. Cui and S. Tao, Strichartz estimates for dispersive equations and solvability of the Kawahara equation, J. Math. Anal. Appl., 304 (2005), 683-702.  doi: 10.1016/j.jmaa.2004.09.049.  Google Scholar

[12]

G. Fibich and G. Papanicolao, A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation, Phys. Lett. A, 239 (1998), 167-173.  doi: 10.1016/S0375-9601(97)00941-9.  Google Scholar

[13]

G. Fibich, B. Ilan and S. Schochet, Critical exponents and collapse of nonlinear Schrödinger equations with anisotropic fourth-order dispersion, Nonlinearity, 16 (2003), 1809-1821.  doi: 10.1088/0951-7715/16/5/314.  Google Scholar

[14]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $ \mathbb{R}^2$, Advances in Differential Equations, 3 (1998), 337-360.   Google Scholar

[15]

C. Guo and S. Cui, Solvability of the Cauchy problem of non-isotropic Schrödinger equations in Sobolev spaces, Nonlinear Analysis, 68 (2008), 768-780.  doi: 10.1016/j.na.2006.11.033.  Google Scholar

[16]

C. Guo, X. Zhao and X. Wei, Cauchy problem for higher-order Schrödinger equations in aniosotropic Sobolev space, App. Anal., 88 (2009), 1329-1338.  doi: 10.1080/00036810903277127.  Google Scholar

[17]

V. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), R1336–R1339. doi: 10.1016/0375-9601(95)00752-0.  Google Scholar

[18]

Z. Lan and B. Guo, Conservation laws, modulation instability and solitons interactions for a nonlinear Schrödinger equation with the sextic operators in an optical fiber, Optical and Quantum Electronics, 50 (2018). doi: 10.1007/s11082-018-1597-7.  Google Scholar

[19]

P. Laurençot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $\mathbb{R}^N, \; N\leq 3$, NoDEA, 2 (1995), 357-369.  doi: 10.1007/BF01261181.  Google Scholar

[20]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001. doi: 10.1090/gsm/014.  Google Scholar

[21]

P. V. Mamyshev and S. V. Chernikov, Ultrashort pulse propagation in optics fibers, Optics Letters, 15 (1990), 1076-1078.  doi: 10.1364/OL.15.001076.  Google Scholar

[22]

B. Pausader, Global wellposedness and scattering for the defocusing energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

[23]

B. Pausader, The cubic fourth-order Schrödinger equation, J. of Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

[24]

G. Raugel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, 2, North-Holland, (2002), 885–982. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[25] J. C. Robinson, Infinite Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and The Theorie of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001.  doi: 10.1115/1.1579456.  Google Scholar
[26]

Y. V. Sedletsky and I. S. Gandzha, A sixth-order nonlinear Schrödinger equation as a reduction of the nonlinear Klein–Gordon equation for slowly modulated wave trains, Nonlinear Dyamics, 94 (2018), 1921-1932.  doi: 10.1007/s11071-018-4465-x.  Google Scholar

[27] E. M. Stein and T. S. Murphy, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Monographs in Harmonic Analysis, 43, Princeton University Press, New Jersey, 1993.   Google Scholar
[28]

J. Su and Y. Gao, Bilinear forms and solitons for a generalized sixth-order nonlinear Schrödinger equation in an optical fiber, The European Physical Journal Plus, 132 (2017). doi: 10.1140/epjp/i2017-11308-1.  Google Scholar

[29]

H. Su and C. Guo, The solution of anisotropic sixth-order Schrödinger equation, Math. Meth. Appl. Sci., 43 (2020), 1868-1891.  doi: 10.1002/mma.6009.  Google Scholar

[30]

W. Sun, Breather-to-soliton transitions and nonlinear wave interactions for the nonlinear Schrödinger equation with the sextic operators in optical fibers, Annalen der Physik, 529 (2017), 1600227. doi: 10.1002/andp.201600227.  Google Scholar

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer Applied Mathmatical Sciences, 68, Springer-Verlag, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[32]

P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 81 (1975), 477-478.  doi: 10.1090/S0002-9904-1975-13790-6.  Google Scholar

[33]

M. V. Vladimirov, On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk. SSSR, 275 (1984), 780-783.   Google Scholar

[34]

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

[35]

Y. Yue, L. Huang and Y. Chen, Modulation instability, rogue waves and spectral analysis for the sixth-order nonlinear Schrödinger equation, (2019). Available from: https://arXiv.org/pdf/1908.04941.pdf doi: 10.1016/j.cnsns.2020.105284.  Google Scholar

show all references

References:
[1]

B. Alouini, Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Commun. Pure Appl. Anal., 14 (2015), 1781-1801.  doi: 10.3934/cpaa.2015.14.1781.  Google Scholar

[2]

B. Alouini, Finite dimensional global attractor for a dissipative anisotropic fourth order Schrödinger equation, Journal of Differential Equations, 266 (2019), 6037-6067.  doi: 10.1016/j.jde.2018.10.044.  Google Scholar

[3]

B. Alouini, A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger-type equations, Math. Meth. Appl. Sci., 44 (2021), 91-103.  doi: 10.1002/mma.6709.  Google Scholar

[4]

B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Discrete and Continuous Dynamical Systems - B, 19 (2014), 651-677.  doi: 10.3934/dcdsb.2014.19.651.  Google Scholar

[5]

A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow and N. Akhmediev, Infinite hierarchy of nonlinear Schrödinger equations and their solutions, Phys. Rev. E, 93 (2016), 012206. doi: 10.1103/PhysRevE.93.012206.  Google Scholar

[6]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynam. Systems - A, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[7]

O. V. Besov, V. P. Il'in and S. M. Nikol'ski ĭ, Integral Representations of Functions and Imbedding Theorems, Scripta Series in Mathematics, I, 1978.  Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[9]

I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, 19, ACTA, 2002.  Google Scholar

[10]

I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping, Memoirs of the American Mathematical Society, 195, American Mathematical Society, 2008. doi: 10.1090/memo/0912.  Google Scholar

[11]

S. Cui and S. Tao, Strichartz estimates for dispersive equations and solvability of the Kawahara equation, J. Math. Anal. Appl., 304 (2005), 683-702.  doi: 10.1016/j.jmaa.2004.09.049.  Google Scholar

[12]

G. Fibich and G. Papanicolao, A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation, Phys. Lett. A, 239 (1998), 167-173.  doi: 10.1016/S0375-9601(97)00941-9.  Google Scholar

[13]

G. Fibich, B. Ilan and S. Schochet, Critical exponents and collapse of nonlinear Schrödinger equations with anisotropic fourth-order dispersion, Nonlinearity, 16 (2003), 1809-1821.  doi: 10.1088/0951-7715/16/5/314.  Google Scholar

[14]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $ \mathbb{R}^2$, Advances in Differential Equations, 3 (1998), 337-360.   Google Scholar

[15]

C. Guo and S. Cui, Solvability of the Cauchy problem of non-isotropic Schrödinger equations in Sobolev spaces, Nonlinear Analysis, 68 (2008), 768-780.  doi: 10.1016/j.na.2006.11.033.  Google Scholar

[16]

C. Guo, X. Zhao and X. Wei, Cauchy problem for higher-order Schrödinger equations in aniosotropic Sobolev space, App. Anal., 88 (2009), 1329-1338.  doi: 10.1080/00036810903277127.  Google Scholar

[17]

V. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), R1336–R1339. doi: 10.1016/0375-9601(95)00752-0.  Google Scholar

[18]

Z. Lan and B. Guo, Conservation laws, modulation instability and solitons interactions for a nonlinear Schrödinger equation with the sextic operators in an optical fiber, Optical and Quantum Electronics, 50 (2018). doi: 10.1007/s11082-018-1597-7.  Google Scholar

[19]

P. Laurençot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $\mathbb{R}^N, \; N\leq 3$, NoDEA, 2 (1995), 357-369.  doi: 10.1007/BF01261181.  Google Scholar

[20]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001. doi: 10.1090/gsm/014.  Google Scholar

[21]

P. V. Mamyshev and S. V. Chernikov, Ultrashort pulse propagation in optics fibers, Optics Letters, 15 (1990), 1076-1078.  doi: 10.1364/OL.15.001076.  Google Scholar

[22]

B. Pausader, Global wellposedness and scattering for the defocusing energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

[23]

B. Pausader, The cubic fourth-order Schrödinger equation, J. of Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

[24]

G. Raugel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, 2, North-Holland, (2002), 885–982. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[25] J. C. Robinson, Infinite Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and The Theorie of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001.  doi: 10.1115/1.1579456.  Google Scholar
[26]

Y. V. Sedletsky and I. S. Gandzha, A sixth-order nonlinear Schrödinger equation as a reduction of the nonlinear Klein–Gordon equation for slowly modulated wave trains, Nonlinear Dyamics, 94 (2018), 1921-1932.  doi: 10.1007/s11071-018-4465-x.  Google Scholar

[27] E. M. Stein and T. S. Murphy, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Monographs in Harmonic Analysis, 43, Princeton University Press, New Jersey, 1993.   Google Scholar
[28]

J. Su and Y. Gao, Bilinear forms and solitons for a generalized sixth-order nonlinear Schrödinger equation in an optical fiber, The European Physical Journal Plus, 132 (2017). doi: 10.1140/epjp/i2017-11308-1.  Google Scholar

[29]

H. Su and C. Guo, The solution of anisotropic sixth-order Schrödinger equation, Math. Meth. Appl. Sci., 43 (2020), 1868-1891.  doi: 10.1002/mma.6009.  Google Scholar

[30]

W. Sun, Breather-to-soliton transitions and nonlinear wave interactions for the nonlinear Schrödinger equation with the sextic operators in optical fibers, Annalen der Physik, 529 (2017), 1600227. doi: 10.1002/andp.201600227.  Google Scholar

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer Applied Mathmatical Sciences, 68, Springer-Verlag, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[32]

P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 81 (1975), 477-478.  doi: 10.1090/S0002-9904-1975-13790-6.  Google Scholar

[33]

M. V. Vladimirov, On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk. SSSR, 275 (1984), 780-783.   Google Scholar

[34]

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

[35]

Y. Yue, L. Huang and Y. Chen, Modulation instability, rogue waves and spectral analysis for the sixth-order nonlinear Schrödinger equation, (2019). Available from: https://arXiv.org/pdf/1908.04941.pdf doi: 10.1016/j.cnsns.2020.105284.  Google Scholar

[1]

Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060
[2]

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
[3]

Rolci Cipolatti, Otared Kavian. On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompact global attractor. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 121-132. doi: 10.3934/dcds.2007.17.121
[4]

Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206
[5]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887
[6]

Maria Rosaria Lancia, Paola Vernole. The Stokes problem in fractal domains: Asymptotic behaviour of the solutions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1553-1565. doi: 10.3934/dcdss.2020088
[7]

Olivier Goubet, Ezzeddine Zahrouni. Global attractor for damped forced nonlinear logarithmic Schrödinger equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2933-2946. doi: 10.3934/dcdss.2020393
[8]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[9]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[10]

Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939
[11]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013
[12]

Kazuhiro Kurata, Tatsuya Watanabe. A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 597-610. doi: 10.3934/cpaa.2006.5.597
[13]

Thierry Cazenave, Zheng Han. Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4801-4819. doi: 10.3934/dcds.2020202
[14]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
[15]

Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
[16]

Haoyue Cui, Dongyi Liu, Genqi Xu. Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback. Mathematical Control & Related Fields, 2018, 8 (2) : 383-395. doi: 10.3934/mcrf.2018015
[17]

Xinmin Xiang. The long-time behaviour for nonlinear Schrödinger equation and its rational pseudospectral approximation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 469-488. doi: 10.3934/dcdsb.2005.5.469
[18]

Jason Murphy, Fabio Pusateri. Almost global existence for cubic nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2077-2102. doi: 10.3934/dcds.2017089
[19]

Delin Wu and Chengkui Zhong. Estimates on the dimension of an attractor for a nonclassical hyperbolic equation. Electronic Research Announcements, 2006, 12: 63-70.

[20]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (63)
  • HTML views (258)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences