Advanced Search
Article Contents
Article Contents

Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations

Abstract Full Text(HTML) Related Papers Cited by
  • In the current issue, we consider a general class of two coupled weakly dissipative fractional Schrödinger-type equations. We will prove that the asymptotic dynamics of the solutions for such NLS system will be described by the existence of a regular compact global attractor in the phase space that has finite fractal dimension.

    Mathematics Subject Classification: Primary: 35B40, 35Q55; Secondary: 76B03, 37L30.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
    [2] 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.
    [3] B. Alouini, Global attractor for a one dimensional weakly damped Half-Wave equation, Discrete Continuous Dynamical Systems - S, 2020. doi: 10.3934/dcdss.2020410.
    [4] A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 116 (1990), 221-243.  doi: 10.1017/S0308210500031498.
    [5] 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.
    [6] B. J. Benny and A. C. Newell, The propagation of nonlinear wave envelopes, Journal of Mathematical Physics, 46 (1967), 133-139.  doi: 10.1002/sapm1967461133.
    [7] M. Cheng, The attractor of the dissipative coupled fractional Schrödinger equations, Math. Meth. Appl. Sci., 37 (2014), 645-656.  doi: 10.1002/mma.2820.
    [8] K. W. Chow, Periodic waves for a system of coupled higher order nonlinear Schrödinger equations with third order dispersion, Physics Letters A, 203 (2003), 426-431.  doi: 10.1016/S0375-9601(03)00108-7.
    [9] I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, 19, ACTA, 2002.
    [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.
    [11] E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.
    [12] A. Esfahani and A. Pastor, Sharp constant of an anisotropic Gagliardo-Nirenberg type inequality and applications, Bull. Braz. Math. Soc. (New Series), 48 (2017), 171-185.  doi: 10.1007/s00574-016-0017-5.
    [13] J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. Henri Poincaré: Anal. Non Linéaire, 5 (1988), 365-405.  doi: 10.1016/S0294-1449(16)30343-2.
    [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. 
    [15] O. Goubet and E. Zahrouni, Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA, 24 (2017), 59. doi: 10.1007/s00030-017-0482-6.
    [16] L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, 249, Springer, New-York, 2014. doi: 10.1007/978-1-4939-1194-3.
    [17] B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Communications in Partial Differential Equations, 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.
    [18] B. Guo and Q. Li, Existence of the global smooth solution to a fractional nonlinear Schrödinger system in atomic Bose-Einstein condensates, Journal of Applied Analysis and Computation, 5 (2015), 793-808.  doi: 10.11948/2015060.
    [19] Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Communications on Pure and Applied Analysis, 14 (2015), 2265-2282.  doi: 10.3934/cpaa.2015.14.2265.
    [20] J. HuJ. Xin and H. Lu, The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition, Computers and Mathematics with Applications, 62 (2011), 1510-1521.  doi: 10.1016/j.camwa.2011.05.039.
    [21] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.
    [22] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 56108. doi: 10.1103/PhysRevE.66.056108.
    [23] G. Li and C. Zhu, Global attractor for a class of coupled nonlinear Schrödinger equations, SeMA Journal, 60 (2012), 5-25.  doi: 10.1007/BF03391708.
    [24] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001. doi: 10.1090/gsm/014.
    [25] M. LisakB. Peterson and H. Wilhelmsson, Coupled nonlinear Schrödinger equations including growth and damping, Physics Letters A, 66 (1978), 83-85.  doi: 10.1016/0375-9601(78)90002-6.
    [26] P. LiuZ. Li and S. Lou, A class of coupled nonlinear Schrödinger equations: Painlevé property, exact solutions, and application to atmospheric gravity waves, Appl. Math. Mech.(Eng. Ed.), 31 (2010), 1383-1404.  doi: 10.1007/s10483-010-1370-6.
    [27] S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys. JETP, 38 (1974), 248–253. Available from: http://www.jetp.ac.ru/cgi-bin/dn/e_038_02_0248.pdf
    [28] C. R. Menyuk, Application of multiple-length-scale methods to the study of optical fiber transmission, Journal of Engineering Mathematics, 36 (1999), 113-136.  doi: 10.1023/A:1017255407404.
    [29] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, 4 (2008), 103-200.  doi: 10.1016/S1874-5717(08)00003-0.
    [30] J. C. Robinson, Infinite-Dimensionel Dynamical Systems, An Introduction To Dissipative Parabolic PDEs And The Theorie Of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.
    [31] E. Russ, Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confluente Mathematici, 3 (2011), 1-119.  doi: 10.1142/S1793744211000278.
    [32] X. Sha, H. Ge and J. Xin, On the existence and stability of standing waves for 2-coupled nonlinear fractional Schrödinger system, Discrete Dynamics in Nature and Society, 2015 (2015). doi: 10.1155/2015/427487.
    [33] B. K. Tan, Collision interactions of envelope Rossby solitons in barotropic atmosphere, Journal of the Atmospheric Sciences, 53 (1996), 1604-1616.  doi: 10.1175/1520-0469(1996)053<1604:CIOERS>2.0.CO;2.
    [34] R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics, Springer Applied Mathmatical Sciences, 68, Springer-Verlag, 1997. doi: 10.1007/978-1-4612-0645-3.
    [35] E. TimmermansP. TommasiniM. Hussein and A. Kerman, Feshbach resonances in atomic Bose-Einstein condensates, Physics Reports, 315 (1999), 199-230.  doi: 10.1016/S0370-1573(99)00025-3.
    [36] M. V. Vladimirov, On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR, 275 (1984), 780-783. 
    [37] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D: Nonlinear Phenomena, 88 (1995), 167-175.  doi: 10.1016/0167-2789(95)00196-B.
    [38] T. H. Wolff, Lectures On Harmonic Analysis, University Lecture Series, 29, American Mathematical Society, 2003. doi: 10.1090/ulect/029.
    [39] W. YuW. LiuH. TrikiQ. ZhouA. Biswas and J. R. Belić, Control of dark and anti-dark solitons in the (2+1)-dimensional coupled nonlinear Schrödinger equations with perturbed dispersion and nonlinearity in a nonlinear optical system, Nonlinear Dynamics, 97 (2019), 471-483.  doi: 10.1007/s11071-019-04992-w.
    [40] Y. ZhangC. YangW. YuM. MirzazadehQ. Zhou and W. Liu, Interactions of vector anti-dark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers, Nonlinear Dynamics, 94 (2018), 1351-1360.  doi: 10.1007/s11071-018-4428-2.
  • 加载中

Article Metrics

HTML views(453) PDF downloads(381) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint