Long-time behavior of solutions for a system of \(N\)-coupled nonlinear dissipative half-wave equations. (English) Zbl 1480.35026
Summary: In the current paper, we consider a system of \(N\)-coupled weakly dissipative fractional nonlinear Schrödinger equations. The well-posedness of the initial value problem is established by a refined analysis based on a limiting argument as well as the study of the asymptotic dynamics of the solutions. This asymptotic behavior is described by the existence of a compact global attractor in the appropriate energy space.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35R11 | Fractional partial differential equations |
| 35G55 | Initial value problems for systems of nonlinear higher-order PDEs |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
Keywords:
asymptotic behavior; global attractor; systems of nonlinear dispersive fractional Schrödinger-type equations; cubic nonlinearitiesReferences:
| [1] | R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Elsevier/Academic, Amsterdam, 2003. · Zbl 1098.46001 |
| [2] | B. Alouini, Asymptotic behavior of solutions for a class of two-coupled nonlinear fractional Schrödinger equations, Dyn. Partial Differ. Equ. 18 (2021), no. 1, 11-32. · Zbl 1462.35344 |
| [3] | J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), 31-52. · Zbl 1056.37084 |
| [4] | H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal. 4 (1980), no. 4, 677-681. · Zbl 0451.35023 |
| [5] | H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980), no. 7, 773-789. · Zbl 0437.35071 |
| [6] | J. Cai, C. Bai and H. Zhang, Decoupled local/global energy-preserving schemes for the N-coupled nonlinear Schrödinger equations, J. Comput. Phys. 374 (2018), 281-299. · Zbl 1416.65510 |
| [7] | T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, American Mathematical Society, Providence, 2003. · Zbl 1055.35003 |
| [8] | M. Cheng, The attractor of the dissipative coupled fractional Schrödinger equations, Math. Methods Appl. Sci. 37 (2014), no. 5, 645-656. · Zbl 1291.35336 |
| [9] | M. Cheng, The ground states for the N coupled nonlinear fractional Schrödinger equations, Complex Var. Elliptic Equ. 63 (2018), no. 3, 315-332. · Zbl 1387.35546 |
| [10] | K. W. Chow, Periodic waves for a system of coupled, higher order nonlinear Schrödinger equations with third order dispersion, Phys. Lett. A 308 (2003), no. 5-6, 426-431. · Zbl 1010.35095 |
| [11] | I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems. Vol. 19, AKTA, Kharkiv, 2002. · Zbl 1100.37047 |
| [12] | E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521-573. · Zbl 1252.46023 |
| [13] | A. Esfahani and A. Pastor, Sharp constant of an anisotropic Gagliardo-Nirenberg-type inequality and applications, Bull. Braz. Math. Soc. (N. S.) 48 (2017), no. 1, 171-185. · Zbl 1373.35242 |
| [14] | P. Gérard and S. Grellier, The cubic Szegő equation, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 5, 761-810. · Zbl 1228.35225 |
| [15] | P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE 5 (2012), no. 5, 1139-1155. · Zbl 1268.35013 |
| [16] | O. Goubet and E. Zahrouni, Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA Nonlinear Differential Equations Appl. 24 (2017), no. 5, Paper No. 59. · Zbl 1382.35266 |
| [17] | B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. Partial Differential Equations 36 (2011), no. 2, 247-255. · Zbl 1211.35268 |
| [18] | B. Guo and Q. Li, Existence of the global smooth solution to a fractional nonlinear Schrödinger system in atomic Bose-Einstein condensates, J. Appl. Anal. Comput. 5 (2015), no. 4, 793-808. · Zbl 1463.35459 |
| [19] | Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal. 14 (2015), no. 6, 2265-2282. · Zbl 1338.35466 |
| [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, Comput. Math. Appl. 62 (2011), no. 3, 1510-1521. · Zbl 1228.35264 |
| [21] | S. Iula, A. Maalaoui and L. Martinazzi, A fractional Moser-Trudinger type inequality in one dimension and its critical points, Differential Integral Equations 29 (2016), no. 5-6, 455-492. · Zbl 1363.26011 |
| [22] | J. Krieger, E. Lenzmann and P. Raphaël, Nondispersive solutions to the L^2-critical half-wave equation, Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61-129. · Zbl 1288.35433 |
| [23] | N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), no. 4-6, 298-305. · Zbl 0948.81595 |
| [24] | N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3) 66 (2002), no. 5, Article ID 056108. |
| [25] | G. Li and C. Zhu, Global attractor for a class of coupled nonlinear Schrödinger equations, SeMA J. 60 (2012), 5-25. · Zbl 1260.35205 |
| [26] | E. H. Lieb and M. Loss, Analysis, Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001. · Zbl 0966.26002 |
| [27] | M. Lisak, B. Peterson and H. Wilhelmsson, Coupled nonlinear Schrödinger equations including growth and damping, Phys. Lett. A 66 (1978), no. 2, 83-85. |
| [28] | P. Liu, Z.-l. Li and S.-y. Lou, A class of coupled nonlinear Schrödinger equations: Painlevé property, exact solutions, and application to atmospheric gravity waves, Appl. Math. Mech. (English Ed.) 31 (2010), no. 11, 1383-1404. · Zbl 1203.86012 |
| [29] | S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Sov. Phys. JETP 38 (1974), 248-253. |
| [30] | A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Elsevier/North-Holland, Amsterdam (2008), 103-200. · Zbl 1221.37158 |
| [31] | T. Ozawa, On critical cases of Sobolev’s inequalities, J. Funct. Anal. 127 (1995), no. 2, 259-269. · Zbl 0846.46025 |
| [32] | O. Pocovnicu, First and second order approximations for a nonlinear wave equation, J. Dynam. Differential Equations 25 (2013), no. 2, 305-333. · Zbl 1270.65060 |
| [33] | J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theorie of Global Attractors, Cambridge Texts Appl. Math., Cambridge University, Cambridge, 2001. · Zbl 0980.35001 |
| [34] | E. Russ, Racines carrées d’opérateurs elliptiques et espaces de Hardy, Confluentes Math. 3 (2011), no. 1, 1-119. · Zbl 1221.42040 |
| [35] | X. Sha, H. Ge and J. Xin, On the existence and stability of standing waves for 2-coupled nonlinear fractional Schrödinger system, Discrete Dyn. Nat. Soc. 2015 (2015), Article ID 427487. · Zbl 1418.35367 |
| [36] | F. Takahashi, Critical and subcritical fractional Trudinger-Moser-type inequalities on \mathbb{R}, Adv. Nonlinear Anal. 8 (2019), no. 1, 868-884. · Zbl 1418.35012 |
| [37] | R. Temam, Navier-Stokes Equations, Stud. Math. Appl. 2, North-Holland, Amsterdam, 1977. · Zbl 0335.35077 |
| [38] | R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer, New York, 1997. · Zbl 0871.35001 |
| [39] | E. Timmermans, P. Tommasini, M. Hussein and A. Kerman, Feshbach resonances in atomic Bose-Einstein condensates, Phys. Rep. 315 (1999), 199-230. |
| [40] | M. V. Vladimirov, On the solvability of a mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR 275 (1984), no. 4, 780-783. · Zbl 0585.35019 |
| [41] | X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Phys. D 88 (1995), no. 3-4, 167-175. · Zbl 0900.35372 |
| [42] | G. Wei and J. Dong, Existence and uniqueness of the global smooth solution to the periodic boundary value problem of fractional nonlinear Schrödinger system, J. Partial Differ. Equ. 28 (2015), no. 2, 95-119. · Zbl 1340.35332 |
| [43] | 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 Dyn. 97 (2019), 471-483. · Zbl 1430.35218 |
| [44] | 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 Dyn. 94 (2018), 1351-1360. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
