# American Institute of Mathematical Sciences

January  2022, 27(1): 45-72. doi: 10.3934/dcdsb.2021032

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

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

Received  June 2020 Revised  December 2020 Published  January 2022 Early access  February 2021

We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in
 $\mathbb{R}^2$
 $u_t+i\Delta u-i \left(\partial_y^4 u-\partial_y^6 u\right)+ig(|u|^2)u+\gamma u = f\,,\;\;(t,(x,y))\in \mathbb{R}\times \mathbb{R}^2\,.$
We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.
Citation: Brahim Alouini. Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $\mathbb{R}^2$. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 45-72. 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. [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.

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##### 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. [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.
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