April  2022, 11(2): 559-581. doi: 10.3934/eect.2021013

Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations

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 Published  April 2022 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.

Citation: Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations and Control Theory, 2022, 11 (2) : 559-581. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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]

E. Di NezzaG. 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.

[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.

[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.

[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]

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.

[16]

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

[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.

[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.

[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.

[20]

J. HuJ. 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.

[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.

[22]

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

[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.

[24]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001. doi: 10.1090/gsm/014.

[25]

M. LisakB. 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.

[26]

P. LiuZ. 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.

[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

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[35]

E. TimmermansP. TommasiniM. 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.

[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. 

[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.

[38]

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

[39]

W. YuW. LiuH. TrikiQ. ZhouA. 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.

[40]

Y. ZhangC. YangW. YuM. MirzazadehQ. 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.

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, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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]

E. Di NezzaG. 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.

[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.

[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.

[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]

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.

[16]

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

[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.

[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.

[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.

[20]

J. HuJ. 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.

[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.

[22]

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

[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.

[24]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001. doi: 10.1090/gsm/014.

[25]

M. LisakB. 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.

[26]

P. LiuZ. 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.

[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

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[35]

E. TimmermansP. TommasiniM. 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.

[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. 

[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.

[38]

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

[39]

W. YuW. LiuH. TrikiQ. ZhouA. 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.

[40]

Y. ZhangC. YangW. YuM. MirzazadehQ. 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.

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