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On the number of limit cycles of a quartic polynomial system
Global attractor for a one dimensional weakly damped half-wave equation
Laboratoire de Recherche: Analyse, Probabilité et Fractals, Faculté des Sciences de Monastir, Avenue de l'environnement, 5019 Monastir, Tunisie |
$ u_t-iD u+ig(|u|^2)u+\gamma u = f\, . $ |
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 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. |
[3] |
A. Babin and M. 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. |
[4] |
Y. Bahri, S. Ibrahim and H. Kikuchi, Remarks on solitary waves and Cauchy problem for a half-wave Schrödinger equations, (2018), 985–989. arXiv: math/1810.01385 Google Scholar |
[5] |
F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero,
Recent developments in dynamical systems: three prespectives, International Journal of Bifurcation and Chaos, 20 (2010), 2591-2636.
doi: 10.1142/S0218127410027246. |
[6] |
H. Brezis and T. Gallouet,
Nonlinear Schrödinger evolution equations, Nonlinear Analysis, 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[7] |
H. Brezis and S. Wainger,
A note on limiting cases of sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
[8] |
D. Cai, A. Majda, D. McLaughlin and E. Tabak,
Spectrat bifurcation in dispersive wave turbulence, PNAS, 96 (1999), 14216-14221.
doi: 10.1073/pnas.96.25.14216. |
[9] |
C. Calgaro, O. Goubet and E. Zahrouni,
Finite dimensional global attractor for a semi-discrete fractional nonlinear Schrödinger equation, Math. Methods Appl. Sci., 40 (2017), 5563-5574.
doi: 10.1002/mma.4409. |
[10] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, New York, 2003.
doi: 10.1090/cln/010. |
[11] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete and Continuous Dynamical Systems - A, 35 (2015), 2863-2880.
doi: 10.3934/dcds.2015.35.2863. |
[12] |
Y. Cho, T. Ozawa and S. Xia,
Remarks on some dispersive estimates, Communications on Pure and Applied Analysis, 10 (2011), 1121-1128.
doi: 10.3934/cpaa.2011.10.1121. |
[13] |
A. Choffrut and O. Pocovnicu,
Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Not., 2018 (2018), 699-738.
doi: 10.1093/imrn/rnw246. |
[14] |
I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, 19, ACTA, 1999. Available from: http://www.emis.de/monographs/Chueshov/book.pdf |
[15] |
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. |
[16] |
V. Dinh, Well-posedness of nolinear fractional Schrödinger and wave equations in Sobolev spaces, arXiv: math/1609.06181v3. Google Scholar |
[17] |
V. Dinh,
On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Communications on Pure and Applied Analysis, 18 (2019), 689-708.
doi: 10.3934/cpaa.2019034. |
[18] |
E. Elgart and B. Schlein,
Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2017), 500-545.
doi: 10.1002/cpa.20134. |
[19] |
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. |
[20] |
P. Gérard and S. Grellier,
The cubic Szegö equation, Ann. Sc. de L'école Normale Supérieure, 43 (2010), 761-810.
doi: 10.24033/asens.2133. |
[21] |
P. Gérard and S. Grellier,
Effective integrable dynamics for a certain nonlinear wave equation, Analysis and PDE, 5 (2012), 1139-1155.
doi: 10.2140/apde.2012.5.1139. |
[22] |
P. Gérard and S. Grellier, The cubic Szegö equation and Hankel operators, Société Mathématiques de France Astérisques, 389 (2017), vi+112 pp, Available from: https://hal.archives-ouvertes.fr/hal-01187657 |
[23] |
O. Goubet and E. Zahrouni,
Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA, 24 (2017), 59-74.
doi: 10.1007/s00030-017-0482-6. |
[24] |
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. |
[25] |
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. |
[26] |
N. Karachalios and N. M. Stavrakakis,
Global attractor for the weakly damped driven Schrödinger equation in $H^2(\mathbb{R})$, NoDEA, 9 (2002), 347-360.
doi: 10.1007/s00030-002-8132-y. |
[27] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[28] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[29] |
N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002), 56108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[30] |
S. Lula, A. Maalaoui and L. Martinazzi,
A fractional Moser-Trudinger type inequality in one dimension and its critical points, Differential Integral Equations, 29 (2016), 455-492.
|
[31] |
A. Majda, D. McLaughlin and E. Tabak,
A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.
doi: 10.1007/BF02679124. |
[32] |
V. S. Melnik and J. Valero,
On attractors of multivalued semi-flows and differential inclusions, Set-Valued Analysis, 6 (1998), 83-111.
doi: 10.1023/A:1008608431399. |
[33] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001.
doi: 10.1090/gsm/014. |
[34] |
A. Ouled Elmounir and F. Simondon,
Attracteurs compacts pour des problèmes d'évolutions sans unicité, Annales de la Faculté des Sciences de Toulouse, 9 (2000), 631-654.
doi: 10.5802/afst.975. |
[35] |
T. Ozawa,
On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[36] |
O. Pocovnicu,
First and second order approximations for a nonlinear wave equation, J. Dyn. Diff. Equa., 25 (2013), 305-333.
doi: 10.1007/s10884-013-9286-5. |
[37] |
G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885–982.
doi: 10.1016/S1874-575X(02)80038-8. |
[38] |
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. |
[39] |
E. Russ,
Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confluente Mathematici, 3 (2011), 1-119.
doi: 10.1142/S1793744211000278. |
[40] |
F. Takahashi,
Critical and subcritical fractional Trudinger-Moser type inequalities on $\mathbb{R}$, Advances in Nonlinear Analysis, 8 (2019), 868-884.
doi: 10.1515/anona-2017-0116. |
[41] |
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. |
[42] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and Its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
doi: 10.1115/1.3424338. |
[43] |
M. V. Vladimirov,
On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR, 275 (1984), 780-783.
|
[44] |
H. Xu,
Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z., 286 (2017), 443-489.
doi: 10.1007/s00209-016-1768-9. |
[45] |
Y. Zhang, H. Zhong, M. Belieć, N. Ahmed, Y. Zhang and M. Xiao,
Diffraction free beams in fractional Schrödinger equation, Sci. Rep., 6 (2016), 1-8.
doi: 10.1038/srep23645. |
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 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. |
[3] |
A. Babin and M. 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. |
[4] |
Y. Bahri, S. Ibrahim and H. Kikuchi, Remarks on solitary waves and Cauchy problem for a half-wave Schrödinger equations, (2018), 985–989. arXiv: math/1810.01385 Google Scholar |
[5] |
F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero,
Recent developments in dynamical systems: three prespectives, International Journal of Bifurcation and Chaos, 20 (2010), 2591-2636.
doi: 10.1142/S0218127410027246. |
[6] |
H. Brezis and T. Gallouet,
Nonlinear Schrödinger evolution equations, Nonlinear Analysis, 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[7] |
H. Brezis and S. Wainger,
A note on limiting cases of sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
[8] |
D. Cai, A. Majda, D. McLaughlin and E. Tabak,
Spectrat bifurcation in dispersive wave turbulence, PNAS, 96 (1999), 14216-14221.
doi: 10.1073/pnas.96.25.14216. |
[9] |
C. Calgaro, O. Goubet and E. Zahrouni,
Finite dimensional global attractor for a semi-discrete fractional nonlinear Schrödinger equation, Math. Methods Appl. Sci., 40 (2017), 5563-5574.
doi: 10.1002/mma.4409. |
[10] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, New York, 2003.
doi: 10.1090/cln/010. |
[11] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete and Continuous Dynamical Systems - A, 35 (2015), 2863-2880.
doi: 10.3934/dcds.2015.35.2863. |
[12] |
Y. Cho, T. Ozawa and S. Xia,
Remarks on some dispersive estimates, Communications on Pure and Applied Analysis, 10 (2011), 1121-1128.
doi: 10.3934/cpaa.2011.10.1121. |
[13] |
A. Choffrut and O. Pocovnicu,
Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Not., 2018 (2018), 699-738.
doi: 10.1093/imrn/rnw246. |
[14] |
I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, 19, ACTA, 1999. Available from: http://www.emis.de/monographs/Chueshov/book.pdf |
[15] |
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. |
[16] |
V. Dinh, Well-posedness of nolinear fractional Schrödinger and wave equations in Sobolev spaces, arXiv: math/1609.06181v3. Google Scholar |
[17] |
V. Dinh,
On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Communications on Pure and Applied Analysis, 18 (2019), 689-708.
doi: 10.3934/cpaa.2019034. |
[18] |
E. Elgart and B. Schlein,
Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2017), 500-545.
doi: 10.1002/cpa.20134. |
[19] |
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. |
[20] |
P. Gérard and S. Grellier,
The cubic Szegö equation, Ann. Sc. de L'école Normale Supérieure, 43 (2010), 761-810.
doi: 10.24033/asens.2133. |
[21] |
P. Gérard and S. Grellier,
Effective integrable dynamics for a certain nonlinear wave equation, Analysis and PDE, 5 (2012), 1139-1155.
doi: 10.2140/apde.2012.5.1139. |
[22] |
P. Gérard and S. Grellier, The cubic Szegö equation and Hankel operators, Société Mathématiques de France Astérisques, 389 (2017), vi+112 pp, Available from: https://hal.archives-ouvertes.fr/hal-01187657 |
[23] |
O. Goubet and E. Zahrouni,
Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA, 24 (2017), 59-74.
doi: 10.1007/s00030-017-0482-6. |
[24] |
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. |
[25] |
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. |
[26] |
N. Karachalios and N. M. Stavrakakis,
Global attractor for the weakly damped driven Schrödinger equation in $H^2(\mathbb{R})$, NoDEA, 9 (2002), 347-360.
doi: 10.1007/s00030-002-8132-y. |
[27] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[28] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[29] |
N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002), 56108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[30] |
S. Lula, A. Maalaoui and L. Martinazzi,
A fractional Moser-Trudinger type inequality in one dimension and its critical points, Differential Integral Equations, 29 (2016), 455-492.
|
[31] |
A. Majda, D. McLaughlin and E. Tabak,
A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.
doi: 10.1007/BF02679124. |
[32] |
V. S. Melnik and J. Valero,
On attractors of multivalued semi-flows and differential inclusions, Set-Valued Analysis, 6 (1998), 83-111.
doi: 10.1023/A:1008608431399. |
[33] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001.
doi: 10.1090/gsm/014. |
[34] |
A. Ouled Elmounir and F. Simondon,
Attracteurs compacts pour des problèmes d'évolutions sans unicité, Annales de la Faculté des Sciences de Toulouse, 9 (2000), 631-654.
doi: 10.5802/afst.975. |
[35] |
T. Ozawa,
On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[36] |
O. Pocovnicu,
First and second order approximations for a nonlinear wave equation, J. Dyn. Diff. Equa., 25 (2013), 305-333.
doi: 10.1007/s10884-013-9286-5. |
[37] |
G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885–982.
doi: 10.1016/S1874-575X(02)80038-8. |
[38] |
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. |
[39] |
E. Russ,
Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confluente Mathematici, 3 (2011), 1-119.
doi: 10.1142/S1793744211000278. |
[40] |
F. Takahashi,
Critical and subcritical fractional Trudinger-Moser type inequalities on $\mathbb{R}$, Advances in Nonlinear Analysis, 8 (2019), 868-884.
doi: 10.1515/anona-2017-0116. |
[41] |
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. |
[42] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and Its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
doi: 10.1115/1.3424338. |
[43] |
M. V. Vladimirov,
On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR, 275 (1984), 780-783.
|
[44] |
H. Xu,
Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z., 286 (2017), 443-489.
doi: 10.1007/s00209-016-1768-9. |
[45] |
Y. Zhang, H. Zhong, M. Belieć, N. Ahmed, Y. Zhang and M. Xiao,
Diffraction free beams in fractional Schrödinger equation, Sci. Rep., 6 (2016), 1-8.
doi: 10.1038/srep23645. |
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