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Numerical approximation of solution for the coupled nonlinear Schrödinger equations

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Abstract

In this article, a compact finite difference scheme for the coupled nonlinear Schrödinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ2 + h4) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.

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References

  1. Dehghan, M., Mohebbi, A., Asgari, Z. Fourth-order compact solution of the nonlinear Klein-Gordon equation. Numer. Algorithms, 52: 523–540 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Gao, Z., Xie, S. Fourth-order alternating direction implicit compact finite difference schemes for twodimensional Schrödinger equation. Appl. Numer. Math., 61: 593–614 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gopaul, A., Bhuruth, M. Analysis of a fourth-order scheme for a three-dimensional convection-diffusion model problem. SIAM J. Sci. Comput., 28: 2075–2094 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hu, X., Zhang, L. Conservative compact difference schemes for the coupled nonlinear Schrödinger equation system. Numer. Methods Partial Differ. Eq., 30: 749–772 (2014)

    Article  MATH  Google Scholar 

  5. Ismail, M., Taha T. Numerical simulation of coupled nonlinear Schrödinger equation. Math. Comput. Sim., 56: 547–562 (2001)

    Article  MATH  Google Scholar 

  6. Ismail, M., Taha T. A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation. Math. Comput. Sim., 74: 302–311 (2007)

  7. Ismail, M., Alamri, S. Highly accurate finite difference method for coupled nonlinear Schrödinger equation. Comput. Math. Appl., 81: 333–351 (2004)

    MATH  Google Scholar 

  8. Kurtinaitis, A., Ivanauska F. Finite difference solution methods for a system of the nonlinear Schrödinger equations. Nonlinear Anal. Model. Control, 9 (3): 247–258 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Li, J., Sun, Z., Zhao, X. A three level linearized compact difference scheme for the Cahn-Hilliard equation. Sci. China Ser. A, 55: 805–826 (2012)

  10. Menyuk, C. Stability of solitons in birefringent optical fibers. J. Opt. Soc. Amer. B, 5: 392–402 (1998)

    Article  Google Scholar 

  11. Pan, X., Zhang, L. High-order linear compact conservative method for the nonlinear Schrödinger equation coupled with the nonlinear Klein-Gordon equation. Nonlinear Anal., 92: 108–118 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Reichel, B., Leble, S. On convergence and stability of a numerical scheme of coupled nonlinear Schrödinger equations. Comput. Math. Appl., 55: 745–759 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sun, J., Gu, X., Ma, Z. Numerical study of the soliton waves of the coupled nonlinear Schrödinger system. Phys. D, 196: 311–328 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sepulveda, M., Vera, O. Numerical methods for a coupled nonlinear Schrödinger system. Bol. Soc. Esp. Mat. Apl., 43: 95–102 (2008)

    MATH  Google Scholar 

  15. Sun, Z., Zhao, D. On the L8 convergence of a difference scheme for coupled nonlinear Schrödinger equations. Comput. Math. Appl., 59: 3286–3300 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sun, Z. Numerical methods of the partial differential equations. Science Press, Beijing, 2005

  17. Wadati, M., Izuka, T., Hisakado, M. A coupled nonlinear Schrödinger equation and optical solitons. J. Phys. Soc. Japan, 61: 2241–2245 (1992)

    Article  MathSciNet  Google Scholar 

  18. Wang, T., Nie, T., Zhang, L. Analysis of a symplectic difference scheme for a coupled nonlinear Schrödinger system. J. Comput. Appl. Math., 231: 745–759 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, T., Zhang, L., Chen, F. Numerical approximation for a coupled Schrödinger system. Chinese J. Comput. Phys., 25: 179–185 (2008)

    Google Scholar 

  20. Wang, T., Zhang, L., Chen, F. On Sonnier-Christov’s difference scheme for the nonlinear coupled Schrödinger system. Acta Math. Sci., 30: 114–125 (2010) (in Chinese)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, T., Zhang, L., Jiang, Y. Convergence of an efficient and compact finite difference scheme for the Klein-Gordon-Zakharov equation. Appl. Math. Comput., 221: 433–443 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Wang, T. Optimal point-wise error estimate of a compact difference scheme for the coupled Gross-Pitaevskii equations in one dimension. J. Sci. Comput., 59 (1): 158–186 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, T., Guo, B. Unconditional convergence of two conservative compact difference schemes for non-linear Schrödinger equation in one dimension (in Chinese). Sci. Sin. Math., 41: 1–27 (2011)

    Article  Google Scholar 

  24. Xie, S., Li, G., Yi, S. Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödinger equation. Comput. Methods Appl. Mech. Engrg., 198: 1052–1060 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhou, Y. Application of discrete functional analysis to the finite difference method. International Academic Publishers, 1990

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Changshu Institute of Technology, Changshu, 215500, China

    Juan Chen

  2. Department of Mathematics, Nanjing university of Aeronautics and Astronautics, Nanjing, 210016, China

    Juan Chen & Lu-ming Zhang

Authors
  1. Juan Chen
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  2. Lu-ming Zhang
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Corresponding author

Correspondence to Juan Chen.

Additional information

Supported by the National Natural Science Foundation of China (No. 11201041).

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Chen, J., Zhang, Lm. Numerical approximation of solution for the coupled nonlinear Schrödinger equations. Acta Math. Appl. Sin. Engl. Ser. 33, 435–450 (2017). https://doi.org/10.1007/s10255-017-0672-3

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  • DOI: https://doi.org/10.1007/s10255-017-0672-3

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