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Original Articles

Numerical analysis of a conservative linear compact difference scheme for the coupled Schrödinger–Boussinesq equations

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Pages 961-978 | Received 19 Nov 2016, Accepted 23 Feb 2017, Published online: 17 Mar 2017

References

  • D.M. Bai and J.L. Wang, The time-splitting Fourier spectral method for the coupled Schrödinger–Boussinesq equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), pp. 1201–1210. doi: 10.1016/j.cnsns.2011.08.012
  • D.M. Bai and L.M. Zhang, The quadratic B-spline finite element method for the coupled Schrödinger–Boussinesq equations, Int. J. Comput. Math. 88 (2011), pp. 1714–1729. doi: 10.1080/00207160.2010.522234
  • W.Z. Bao and Y.Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal. 50 (2012), pp. 492–521. doi: 10.1137/110830800
  • W.Z. Bao and Y.Y. Cai, Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comput. 281 (2013), pp. 99–128.
  • S. Bilige, T. Chaolu, and X.M. Wang, Application of the extended simplest equation method to the coupled Schrödinger–Boussinesq equation, Appl. Math. Comput. 224 (2013), pp. 517–523. doi: 10.1016/j.amc.2013.08.083
  • L.G. Farah and A. Pastor, On the periodic Schrödinger–Boussinesq system, J. Math. Anal. 368 (2010), pp. 330–349. doi: 10.1016/j.jmaa.2010.03.007
  • B.L. Guo, The global solution of the system of equations for complex Schrödinger field coupled with Boussinesq type self-consistent field, Acta Math. Sin. 26 (1983), pp. 295–306 (in Chinese). doi: 10.1007/s10114-010-8034-6
  • B.L. Guo and X.Y. Du, Existence of the periodic solution for the weakly damped Schrödinger–Boussinesq equation, J. Math. Anal. Appl. 262 (2001), pp. 453–472. doi: 10.1006/jmaa.2000.7455
  • B.L. Guo and X.Y. Du, The behavior of attractors for the weakly damped Schrödinger–Boussinesq equation, Commun. Nonlinear Sci. Numer. Simul. 6 (2001), pp. 54–60. doi: 10.1016/S1007-5704(01)90030-9
  • L.Y. Huang, Y.D. Jiao, and D.M. Liang, Multi-sympletic scheme for the coupled Schrödinger–Boussinesq equations, Chin. Phys. B. 22 (2013), pp. 1–5.
  • A. Kilicman and R. Abazari, Travelling wave solutions of the Schrödinger–Boussinesq system, Abst. Appl. Anal. 2012 (2012), pp. 1–11. doi: 10.1155/2012/198398
  • S. Li and L.V. Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal. 32 (1995), pp. 1839–1875. doi: 10.1137/0732083
  • Y.S. Li and Q.Y. Chen, Finite dimensional global attractor for dissipative Schrödinger–Boussinesq equations, J. Math. Anal. Appl. 205 (1997), pp. 107–132. doi: 10.1006/jmaa.1996.5148
  • F. Liao and L.M. Zhang, Conservative compact finite difference scheme for the coupled Schrö dinger–Boussinesq equation, Numer. Methods Partial Differential Eqs. 32 (2016), pp. 1667–1688. doi: 10.1002/num.22067
  • V.G. Makhankov, On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying Boussinesq's equation, Phys. Lett. A. 50 (1974), pp. 42–44. doi: 10.1016/0375-9601(74)90344-2
  • N.N. Rao, Exact solutions of coupled scalar field equations, J. Phys. A: Math. General. 22 (1989), pp. 4813–4825. doi: 10.1088/0305-4470/22/22/012
  • N.N. Rao, Coupled scalar field equations for nonlinear wave modulations in dispersive media, Pramana. J. Phys. 46 (1991), pp. 161–202. doi: 10.1007/BF02846945
  • Z.Z. Sun, Numerical Solution for Partial Differential Equation, Science Press, Beijing, 2005. (in Chinese).
  • T.C. Wang, B.L. Guo, and L.M. Zhang, New conservative difference schemes for a coupled nonlinear Schrödinger system, Appl. Math. Comput. 217 (2010), pp. 1604–1619. doi: 10.1016/j.amc.2009.07.040
  • T.C. Wang, Optimal point-wise error estimate of a compact difference scheme for the coupled Gross-Pitaevskii equations in one dimension, J. Sci. Comput. 41 (2014), pp. 158–186. doi: 10.1007/s10915-013-9757-1
  • T.C. Wang and X.F. Zhao, Optimal l∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions, Sci. China Math. 57 (2014), pp. 2189–2214. doi: 10.1007/s11425-014-4773-7
  • Y.R. Xia and L.Z. Bin, Exact explicit solutions of the nonlinear Schrödinger equation coupled to the Boussinesq equation, Acta Math. Scientia. 23B (2003), pp. 453–460.
  • N. Yajima and J. Satsuma, Soliton solutions in a diatomic lattice system, Prog. Theor. Phys. 62 (1979), pp. 370–378. doi: 10.1143/PTP.62.370
  • L.M. Zhang, D.M. Bai, and S.S. Wang, Numerical analysis for a conservative difference scheme to solve the Schrödinger–Boussinesq equation, J. Comput. Appl. Math. 235 (2011), pp. 4899–4915. doi: 10.1016/j.cam.2011.04.001
  • F. Zhang, V.M.P. Garcia, and L. Vazquez, Numerical simulation of nonlinear Schrödinger systems:a new conservative scheme, Appl. Math. Comput. 71 (1995), pp. 165–177. doi: 10.1016/0096-3003(94)00152-T
  • Y.L. Zhou, Application of Discrete Functional Analysis to the Finite Difference Method, International Academic Publishers, Beijing, 1990.

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