Browse > Article
http://dx.doi.org/10.4134/BKMS.b190154

ERROR ESTIMATES FOR A GALERKIN METHOD FOR A COUPLED NONLINEAR SCHRÖDINGER EQUATIONS  

Omrani, Khaled (Institut Superieur des Sciences Appliquees et de Technologie de Sousse Universite de Sousse)
Rahmeni, Mohamed (Ecole Superieure des Sciences et de Technologie de Hammam Sousse Universite de Sousse)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.1, 2020 , pp. 219-244 More about this Journal
Abstract
In this paper, we approximate the solution of the coupled nonlinear Schrödinger equations by using a fully discrete finite element scheme based on the standard Galerkin method in space and implicit midpoint discretization in time. The proposed scheme guarantees the conservation of the total mass and the energy. First, a priori error estimates for the fully discrete Galerkin method is derived. Second, the existence of the approximated solution is proved by virtue of the Brouwer fixed point theorem. Moreover, the uniqueness of the solution is shown. Finally, convergence orders of the fully discrete Crank-Nicolson scheme are discussed. The end of the paper is devoted to some numerical experiments.
Keywords
Coupled $Schr{\ddot{o}}dinger$ equations; Galerkin finite element scheme; conservation laws; unique solvability; convergence;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Z. Sun and D. Zhao, On the $L_{\infty}$ convergence of a difference scheme for coupled nonlinear Schrodinger equations, Comput. Math. Appl. 59 (2010), no. 10, 3286-3300. https://doi.org/10.1016/j.camwa.2010.03.012   DOI
2 V. Thomee, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, 1054, Springer-Verlag, Berlin, 1984.
3 H. Triki, A. Choudhuri, K.Porsezian, and P.Tchofo Dinda, Dark solitons in an extended nonlinear Schrodinger equation with higher-order odd and even terms, Optik - International Journal for Light and Electron Optics. 164 (2018), 661-670.   DOI
4 H. Triki, T. Hayat, O. M. Aldossary, and A. Biswas, Bright anddark solitons for the resonant nonlinear Schrodinger's equationwith time-dependent coefficients. Optics and Laser Technology. 44 (2012), no. 7, 2223-2231.   DOI
5 H. Triki, K. Porsezian, A. Choudhuri, and P. T. Dinda, Chirped solitary pulses for a nonic nonlinear Schrodinger equation on a continuous-wave background, Phys. Rev. A. 93 (2016), no. 6, 063810.   DOI
6 T. Wang, Maximum norm error bound of a linearized difference scheme for a coupled nonlinear Schrodinger equations, J. Comput. Appl. Math. 235 (2011), no. 14, 4237- 4250. https://doi.org/10.1016/j.cam.2011.03.019   DOI
7 T. Wang, Optimal point-wise error estimate of a compact finite difference scheme for the coupled nonlinear Schrodinger equations, J. Comput. Math. 32 (2014), no. 1, 58-74. https://doi.org/10.4208/jcm.1310-m4340   DOI
8 T. Wang, T. Nie, and L. Zhang, Analysis of a symplectic difference scheme for a coupled nonlinear Schrodinger system, J. Comput. Appl. Math. 231 (2009), no. 2, 745-759. https://doi.org/10.1016/j.cam.2009.04.022   DOI
9 C. Sulem and P.-L. Sulem, The Nonlinear Schrodinger equation, Applied Mathematical Sciences, 139, Springer-Verlag, New York, 1999.
10 A.-M. Wazwaz, Exact solutions for the fourth order nonlinear Schrodinger equations with cubic and power law nonlinearities, Math. Comput. Modelling 43 (2006), no. 7-8, 802-808. https://doi.org/10.1016/j.mcm.2005.08.010   DOI
11 T. Wang, L. M. Zhang, and F. Q. Chen, Numerical approximation for a coupled Schrodinger system, Chinese J. Comput. Phys. 25 (2008), 179-185.   DOI
12 A.-M. Wazwaz, A study on linear and nonlinear Schrodinger equations by the variational iteration method, Chaos Solitons Fractals 37 (2008), no. 4, 1136-1142. https://doi.org/10.1016/j.chaos.2006.10.009   DOI
13 A.-M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science, Higher Education Press, Beijing, 2009. https://doi.org/10.1007/978-3-642-00251-9
14 C. Yang, Q. Zhou, H. Triki, M. Mirzazadeh, M. Ekici, W. J. Liu, A. Biswas, and M. Belic, Bright soliton interactions in a (2 + 1)-dimensional fourth-order variable-coefficient nonlinear Schrodinger equation for the Heisenberg ferromagnetic spin chain, Nonlinear Dyn. 95 (2019).
15 S. Boulaaras and M. Haiour, The finite element approximation of evolutionary Hamilton-Jacobi-Bellman equations with nonlinear source terms, Indag. Math. (N.S.) 24 (2013), no. 1, 161-173. https://doi.org/10.1016/j.indag.2012.07.005   DOI
16 R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
17 H. P. Bhatt and A. Q. M. Khaliq, Higher order exponential time differencing scheme for system of coupled nonlinear Schrodinger equations, Appl. Math. Comput. 228 (2014), 271-291. https://doi.org/10.1016/j.amc.2013.11.089   DOI
18 A. Biswas, Soliton solutions of the perturbed resonant nonlinear Schrodinger's equation with full nonlinearity by semi-inverse variational principle, Quant. Phys. Lett. 1 (2012), no. 2, 79-86.
19 S. Boulaaras, M. A. B. Le Hocine, and M. Haiour, $L^{\infty}$-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms, J. Appl. Math. Comput. Mech. 16 (2017), no. 1, 19-31. https://doi.org/10.17512/jamcm.2017.1.02   DOI
20 F. E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, in Proc. Sympos. Appl. Math., Vol. XVII, 24-49, Amer. Math. Soc., Providence, RI, 1965.
21 J. Cai, Y. Wang, and H. Liang, Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrodinger system, J. Comput. Phys. 239 (2013), 30-50. https://doi.org/10.1016/j.jcp.2012.12.036   DOI
22 Y. Chen, H. Zhu, and S. Song, Multi-symplectic splitting method for the coupled nonlinear Schrodinger equation, Comput. Phys. Comm. 181 (2010), no. 7, 1231-1241. https://doi.org/10.1016/j.cpc.2010.03.009   DOI
23 M. S. Ismail and T. R. Taha, A linearly implicit conservative scheme for the coupled nonlinear Schrodinger equation, Math. Comput. Simulation 74 (2007), no. 4-5, 302-311. https://doi.org/10.1016/j.matcom.2006.10.020   DOI
24 R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, Inc., London, 1982.
25 T. Geyikli and S. B. G. Karakoc, Subdomain finite element method with quartic Bsplines for the modified equal width wave equation, Comput. Math. Math. Phys. 55 (2015), no. 3, 410-421. https://doi.org/10.1134/S0965542515030070   DOI
26 X. Hu and L. Zhang, Conservative compact difference schemes for the coupled nonlinear Schrodinger system, Numer. Methods Partial Differential Equations 30 (2014), no. 3, 749-772. https://doi.org/10.1002/num.21826   DOI
27 L. Kong, J. Hong, L. Ji and P. Zhu, Compact and efficient conservative schemes for coupled nonlinear Schrodinger equations, Numer. Methods Partial Differential Equations 31 (2015), no. 6, 1814-1843. https://doi.org/10.1002/num.21969   DOI
28 A. S. Davydov, Solitons in molecular systems, translated from the Russian by Eugene S. Kryachko, Mathematics and its Applications (Soviet Series), 4, D. Reidel Publishing Co., Dordrecht, 1985. https://doi.org/10.1007/978-94-017-3025-9
29 N. Atouani and K. Omrani, Galerkin finite element method for the Rosenau-RLW equation, Comput. Math. Appl. 66 (2013), no. 3, 289-303. https://doi.org/10.1016/j.camwa.2013.04.029   DOI
30 S. B. G. Karakoc, A quartic subdomain finite element method for the modified KdV equation, Stat. Optim. Inf. Comput. 6 (2018), no. 4, 609-618. https://doi.org/10.19139/soic.v6i4.485
31 Y. Ma, L. Kpng, and J. Hong, High-order compact splitting multisymplectic method for the coupled nonlinear Schrodinger equations, Comput. Math. Appl. 61 (2011), no. 2, 319-333. https://doi.org/10.1016/j.camwa.2010.11.007   DOI
32 K. Omrani, On the numerical approach of the enthalpy method for the Stefan problem, Numer. Methods Partial Differential Equations 20 (2004), no. 4, 527-551. https://doi.org/10.1002/num.10105   DOI
33 W. J. Sonnier and C. I. Christov, Strong coupling of Schrodinger equations: conservative scheme approach, Math. Comput. Simulation 69 (2005), no. 5-6, 514-525. https://doi.org/10.1016/j.matcom.2005.03.016   DOI