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.
Citation: |
[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, 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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.
![]() ![]() |
[8] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, New York, 2003.
doi: 10.1090/cln/010.![]() ![]() ![]() |
[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] |
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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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] |
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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() |
[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.![]() ![]() ![]() |
[20] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001.
doi: 10.1090/gsm/014.![]() ![]() ![]() |
[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.![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() |
[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.
![]() ![]() |
[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.![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |
[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.
![]() ![]() |
[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.![]() ![]() ![]() |
[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.![]() ![]() ![]() |