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Nonlinear Dynamics

, Volume 67, Issue 1, pp 619–627 | Cite as

Numerical studies of the cubic non-linear Schrodinger equation

  • Talaat S. El-Danaf
  • Mohamed A. Ramadan
  • Faisal E. I. Abd Alaal
Original Paper

Abstract

In this paper, we are concerned with the problem of applying cubic non-polynomial spline functions to develop a numerical method for obtaining approximation for the solution for cubic non-linear Schrodinger equation. The truncation error of the method is theoretically analyzed. Using the Von Neumann method, the proposed method is shown to be unconditionally stable. The linearization technique is carried out to solve the system and to prove that the method is unconditionally stable. Two numerical examples are included to illustrate the practical implementation of the proposed method.

Keywords

Non-polynomial spline Non-linear Schrodinger equation Von Neumann stability 

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References

  1. 1.
    Bratsos, A.G.: A linearized finite-difference scheme for the numerical solution of the nonlinear cubic Schrodinger equation. Korean J. Comput. Appl. Math. 8(3), 459–467 (2001) MathSciNetMATHGoogle Scholar
  2. 2.
    Bratsos, A., Ehrhardt, M., Famelis, I.T.: A discrete Adomian decomposition method for discrete nonlinear Schrödinger equations. Appl. Math. Comput. 197, 190–205 (2008) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Daele, M.V.G., Berghe, V., Meyer, H.D.: A smooth approximation for the solution of a fourth-order boundary value problem based on non-polynomial splines. J. Comput. Appl. Math. 51, 383–394 (1994) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Griffiths, D.F., Mitchell, A.R., Morris, J.L.I.: A numerical study of the non-linear Schrödinger equation. Comput. Math. Appl. Mech. Eng. 45, 177–215 (1984) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hasegawa, A.: Solitons in Optical Communications. Clarendon, Oxford (1995) MATHGoogle Scholar
  6. 6.
    Islam, S.U., Khan, M.A., Tirmizi, I.A., Twizell, E.H.: Non-polynomial spline approach to the solution of a system of third-order boundary-value problems. Appl. Math. 168, 152–163 (2005) MATHGoogle Scholar
  7. 7.
    Khan, M.A., Siraj-ul-Islam, Tirmizi, I.A., Twizell, E.H., Ashraf, S.: A class of methods based on non-polynomial sextic spline functions for the solution of a special fifth order boundary-value problems. J. Math. Anal. Appl. 321, 651–660 (2006) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kivshar, Yu.S., Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego (2003) Google Scholar
  9. 9.
    Korkmaz, A., Dağ, İ.: A differential quadrature algorithm for nonlinear Schrodinger equation. Nonlinear Dyn. 56, 69–83 (2009) MATHCrossRefGoogle Scholar
  10. 10.
    Ramadan, M.A., Lashien, I.F., Zahra, W.K.: Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems. Appl. Math. Comput. 184, 476–484 (2007) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Rashidinia, J., Mohammadi, R.: Non-polynomial cubic spline methods for the solution of parabolic equations. Int. J. Comput. Math. 85(5), 843–850 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Sweilam, N.H.: Variational iteration method for solving cubic nonlinear Schrodinger equation. Comput. Appl. Math. 207, 155–163 (2007) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain non-linear evolution equations. II. Numerical non-linear Schrodinger equation. Int. J. Comput. Phys. 55, 203–230 (1984) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Twizell, E.H., Bratsos, A.G., Newby, J.C.: A finite-difference method for cubic Schrodinger equation. Math. Comput. Simul. 43, 67–75 (1997) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35, 908–914 (1972) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Talaat S. El-Danaf
    • 1
  • Mohamed A. Ramadan
    • 1
  • Faisal E. I. Abd Alaal
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceMenoufia UniversityShebeen El-KoomEgypt

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