Nonlinear Dynamics

, Volume 88, Issue 4, pp 2797–2805 | Cite as

Solitary wave solutions of three special types of Boussinesq equations

  • Shabnam Jamshidzadeh
  • Reza Abazari
Original Paper


In this paper, the \((G'/G)\)-expansion method is employed to construct more general solitary wave solutions of three special types of Boussinesq equation, namely Boussinesq equation, improved Boussinesq equation and variant Boussinesq equation, where the French scientist Joseph Valentin Boussinesq (1842–1929) described in the 1870s model equations for the propagation of long waves on the surface of water with a small amplitude. Our work is motivated by the fact that the \((G'/G)\)-expansion method provides not only more general forms of solutions but also periodic, solitary waves and rational solutions. The method appears to be easier and faster by means of a symbolic computation.


Boussinesq equations Soliton Solitary wave solution Periodic wave solution 


  1. 1.
    Russell, J.S.: Report on Waves. Report of the Fourteenth Meeting of the British Association for the Advancement of Science, Plates XLVII–LVII, pp. 311–390. John Murray, London (1844)Google Scholar
  2. 2.
    Ablowtiz, M.J., Ladik, J.F.: On the solution of a class of nonlinear partial difference equations. Stud. Appl. Math. 57, 1–12 (1977)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Boussinesq, J.: Thorie de l’intumescence liquide, applele onde solitaire ou de translation, se propageant dans un canal rectangulaire. Comptes Rendus de l’Acad. Sci. 72, 755–0759 (1871)MATHGoogle Scholar
  4. 4.
    Boussinesq, J.: Thorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. Deux. Srie. 17, 55–108 (1872)MathSciNetMATHGoogle Scholar
  5. 5.
    Kaptsov, G.V.: Construction of exact solutions of the Boussinesq equation. J. Appl. Mech. Tech. Phys. 39, 389–392 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bogolubsky, I.L.: Some examples of inelastic soliton interaction. Comput. Phys. Commun. 13, 149–155 (1977)CrossRefGoogle Scholar
  7. 7.
    Makhankov, V.G.: Dynamics of classical solitons (in nonintegrable systems). Phys. Rep. A 35(1), 1–128 (1978)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bona, J., Sachs, R.: Global existence of smooth solution and stability waves for a generalized Boussinesq equation. Commun. Math. Phys. 118, 12–29 (1988)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Clarkson, P.: New exact solution of the Boussinesq equation. Eur. J. Appl. Math. 1, 279–300 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hirota, R.: Solutions of the classical Boussinesq equation and the spherical Boussinesq equation: the Wronskian technique. J. Phys. Soc. Jpn. 55, 2137–2150 (1986)CrossRefGoogle Scholar
  11. 11.
    Liu, Y.: Instability and blow-up of solutions to a generalized Boussinesq equation. SIAM J. Math. Anal. 26, 1527–1546 (1995)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Yazhima, N.: On a growing mode of the Boussinesq equation. Progr. Theor. Phys. 69, 678–680 (1983)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wang, S.B., Chen, G.W.: Small amplitude solutions of the generalized IMBq equation. J. Math. Anal. Appl. 264, 846–866 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Nakamura, A.: Exact solitary wave solution of the spherical Boussinesq equation. J. Phys. Soc. Jpn. 54, 4111–4114 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sachs, R.L.: On the integrable variant of the Boussinesq system: Painlev property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy. Phys. D 30, 1–27 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Yao, R., Li, Z.: New exact solutions for three nonlinear evolution equations. Phys. Lett. A. 297(3–4), 196–204 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fu, Z., Liu, S., Liu, S.: New transformations and new approach to find exact solutions to nonlinear equations. Phys. Lett. A 299, 507–512 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Zhang, J.F.: Multi-solitary wave solutions for variant Boussinesq equations and Kupershmidt equations. Appl. Math. Mech 21(2), 171–175 (2000)MathSciNetGoogle Scholar
  19. 19.
    Singh, K., Gupta, R.K.: Exact solutions of a variant Boussinesq system. Int. J. Eng. Sci. 44, 1256–1268 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wu, X.-H.B., He, J.-H.: EXP-function method and its application to nonlinear equations. Chaos Solitons Fractals 38, 903–910 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Wang, M., Li, X., Zhang, J.: The \((G^{\prime }/G)\)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372, 417–423 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Talati, D., Wazwaz, A.-M.: Some new integrable systems of two-component fifth-order equations. Nonlinear Dyn. (2016). doi: 10.1007/s11071-016-3101-x Google Scholar
  23. 23.
    Khan, K., Akbar, M.A., Bekir, A.: Solitary wave solutions of the (2\(+\)1)-dimensional Zakharov-Kuznetsevmodified equal-width equation. J. Inf. Optim. Sci. 37(4), 569–589 (2016)MathSciNetGoogle Scholar
  24. 24.
    Wang, D.-S., Li, H.: Single and multi-solitary wave solutions to a class of nonlinear evolution equations. J. Math. Anal. Appl. 343, 273–298 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wang, G.W., Fakhar, K., Kara, A.H.: Soliton solutions and group analysis of a new coupled (2\(+\)1)-dimensional Burgers equations. Acta Phys. Polonica B 46(5), 923–930 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Abazari, R., Abazari, R.: Hyperbolic, trigonometric and rational function solutions of Hirota-Ramani equation via \((\frac{G^{\prime }}{G})\)-expansion method. Math. Prob. Eng. 2011, 11 (2011). doi: 10.1155/2011/424801. [Article ID 424801]MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Abazari, R.: Solitary wave solutions of Klein–Gordon equation with quintic nonlinearity. J. Appl. Mech. Tech. Phys 54(3), 397–403 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Abazari, R., Jamshidzadeh, S.: Exact solitary wave solutions of the complex Klein–Gordon equation. Optik Int. J. Light Electron Optik 126, 1970–1975 (2015)Google Scholar
  29. 29.
    Kılıcman, A., Abazari, R.: Traveling wave solutions of the Schrodinger–Boussinesq system. Abstr. Appl. Anal. 2012, 11 (2012). doi: 10.1155/2012/198398. [Article ID 198398]MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Young Researchers and Elite Club, Ardabil BranchIslamic Azad UniversityArdabilIran

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