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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 |
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.
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 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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
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 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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
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