Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in <inline-formula><tex-math id="M1">\begin{document} \mathbb{R}^2 \end{document}</tex-math></inline-formula>

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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 $

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 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, 2^nd 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.

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.
[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, 2^nd 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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