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Nonlinear Dynamics

, Volume 51, Issue 4, pp 585–594 | Cite as

New explicit and exact traveling wave solutions for two nonlinear evolution equations

  • S. A. El-Wakil
  • M. A. Abdou
Original Paper

Abstract

In this paper, with the aid of computer symbolic computation system such as Maple, an algebraic method is firstly applied to two nonlinear evolution equations, namely, nonlinear Schrodinger equation and Pochhammer–Chree (PC) equation. As a consequence, some new types of exact traveling wave solutions are obtained, which include bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.

Keywords

Generalized algebraic method Nonlinear evolution equations Exact solutions 

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References

  1. 1.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)MATHGoogle Scholar
  2. 2.
    Matveev, V.B., Salle, M.A.: Darboux Transformation and Solitons. Springer, Berlin (1991)Google Scholar
  3. 3.
    Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformation and Solitons Theory and Its Geometric Applications. Shanghai Scientific and Technical, Shanghai, P.R. China (1999)Google Scholar
  4. 4.
    Rogers, C., Schief, W.K.: Backlund and Darboux Transformation, Geometry and Modern Applications in Soliton Theory. Cambridge University Press, Cambridge, UK (2002)Google Scholar
  5. 5.
    Abdou, M.A.: On the variational iteration method. Phys. Lett. A (2007) in pressGoogle Scholar
  6. 6.
    Abdou, M.A. The extended F-expansion method and its applications for nonlinear evolution equations. Chaos, Solitons Fractals 31, 95–104 (2007)CrossRefMathSciNetGoogle Scholar
  7. 7.
    El-Wakil, S.A., Abdou, M.A.: The extended mapping method and its applications for nonlinear evolution equations. Phys. Lett. A 358, 275–282 (2006)CrossRefMathSciNetGoogle Scholar
  8. 8.
    El-Wakil, S.A., Abdou, M.A.: New exact travelling wave solutions using modified extended tanh function method. Choas, Solitons Fractals 31, 840–852 (2007)CrossRefMathSciNetGoogle Scholar
  9. 9.
    El-Wakil, S.A., Abdou, M.A.: Modified extended tanh function method for solving nonlinear partial differential equations. Choas, Solitons Fractals 31, 1256–1264 (2007)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Yan, C.T.: A simple transformation for nonlinear waves. Phys. Lett. A 224, 77 (1996)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    El-Wakil, S.A., Abdou, M.A., Elhanbaly, A.: New soliton and periodic wave solutions for nonlinear evolution equations. Phys. Lett. A 353, 40 (2006)CrossRefGoogle Scholar
  12. 12.
    Chen, Y.-Z., Ding, X.-W.: Exact travelling wave solutions of nonlinear evolution equations in (1+1) and (2+1) dimensions. Nonlinear Anal. 61, 1005 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fan, E.G.: Uniformly constructing a series of explicit exact solutions to nonlinear equation in mathematical physics. Chaos, Solitons Fractals 16, 819 (2003)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fan, E.G.: Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method. J. Phys. A 35, 6853 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Huang, D.J., Zhang, H.Q.: Link between travelling waves and first kind order nonlinear ordinary differential equation with a sixth degree nonlinear term. Chaos, Solitons Fractals 29, 828 (2006)MathSciNetGoogle Scholar
  16. 16.
    Huang, D.J., Zhang, H.Q.: The extended first kind elliptic sub equation method and its application to the generalized reaction Duffing model. Phys. Lett. A 344, 229 (2005)CrossRefGoogle Scholar
  17. 17.
    Feng, Z.: On explicit exact solutions for the Lienard equation and its applications. Phys. Lett. A 293, 50 (2002)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Bololubsky, I.L.: Some examples of inelastic soliton interaction. Comput. Phys. Commun. 13, 149 (1977)CrossRefGoogle Scholar
  19. 19.
    Zhang, W.G., Chang, Q.S.: Methods of judging shape of solitary wave ans solutions formulae for evolution equation with nonlinear terms of higher order. J. Math. Ana. Appl. 287, 1 (2003)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Saarloos, W.V., Hohenberg, P.C.: Fronts pulses sources and sinks in generalized Ginzburg Landou equations. Phys. D 56, 303 (1992)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Wu, W.T.: Algarithms and Computation. Springer, Berlin (1994) Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Theoretical Research Group, Physics Department, Faculty of ScienceMansoura UniversityMansouraEgypt

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