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doi: 10.3934/eect.2021013
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Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations

Brahim Alouini

Research lab: Analysis, Probability and Fractals, University of Monastir, Faculty of Sciences, Monastir, Tunisia, I.P.E.I. Monastir, Ibn el Jazzar street, 5019 Monastir, Tunisia

Received  September 2020 Revised  January 2021 Early access March 2021

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.

Keywords: Schrödinger equation, asymptotic behavior, global attractor, fractal dimension.
Mathematics Subject Classification: Primary: 35B40, 35Q55; Secondary: 76B03, 37L30.
Citation: Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, doi: 10.3934/eect.2021013
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, 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

E. Di Nezza, G. 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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[16]

L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, 249, Springer, New-York, 2014. doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Hu, J. 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.  Google Scholar

[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.  Google Scholar

[22]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 56108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

M. Lisak, B. 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.  Google Scholar

[26]

P. Liu, Z. 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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

E. Russ, Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confluente Mathematici, 3 (2011), 1-119.  doi: 10.1142/S1793744211000278.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

E. Timmermans, P. Tommasini, M. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[38]

T. H. Wolff, Lectures On Harmonic Analysis, University Lecture Series, 29, American Mathematical Society, 2003. doi: 10.1090/ulect/029.  Google Scholar

[39]

W. Yu, W. Liu, H. Triki, Q. Zhou, A. 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.  Google Scholar

[40]

Y. Zhang, C. Yang, W. Yu, M. Mirzazadeh, Q. 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.  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, 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

E. Di Nezza, G. 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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[16]

L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, 249, Springer, New-York, 2014. doi: 10.1007/978-1-4939-1194-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. Hu, J. 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.  Google Scholar

[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.  Google Scholar

[22]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 56108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

M. Lisak, B. 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.  Google Scholar

[26]

P. Liu, Z. 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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

E. Russ, Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confluente Mathematici, 3 (2011), 1-119.  doi: 10.1142/S1793744211000278.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

E. Timmermans, P. Tommasini, M. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[38]

T. H. Wolff, Lectures On Harmonic Analysis, University Lecture Series, 29, American Mathematical Society, 2003. doi: 10.1090/ulect/029.  Google Scholar

[39]

W. Yu, W. Liu, H. Triki, Q. Zhou, A. 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.  Google Scholar

[40]

Y. Zhang, C. Yang, W. Yu, M. Mirzazadeh, Q. 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.  Google Scholar

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