American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2021032

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 Published  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]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[2]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[3]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[4]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[5]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298
[6]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[7]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[8]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[9]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[10]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[11]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[12]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[13]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[14]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[15]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[16]

Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511
[17]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
[18]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[19]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[20]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (12)
  • HTML views (41)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences