Nonlinear Dynamics

, Volume 85, Issue 3, pp 1665–1677 | Cite as

Some singular solutions and their limit forms for generalized Calogero–Bogoyavlenskii–Schiff equation

  • Shaoyong Li
  • Yin Li
  • Ben-gong Zhang
Original Paper


By using the bifurcation method of dynamical systems, we investigate the singular solutions and their limit forms for generalized Calogero–Bogoyavlenskii–Schiff equation. Via some special phase orbits, we obtain some new explicit singular wave solutions expressed in terms of elliptic integral and elliptic function. From their limit forms, we also obtain the hyperbolic function kink wave and trigonometric function singular solutions.


Generalized Calogero–Bogoyavlenskii–Schiff equation Bifurcation method Singular solutions Limit forms 



All authors wish to thank the editor and the anonymous referee for many valuable suggestions leading to the improvement of this paper.


  1. 1.
    Wazwaz, A.M.: New solutions of distinct physical structures to high-dimensional nonlinear evolution equations. Appl. Math. Comput. 196, 363–370 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Peng, Y.Z.: New types of localized coherent structures in the Bogoyavlenskii–Schiff equation. Int. J. Theor. Phys. 45, 1779–1783 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Kobayashi, T., Toda, K.: The Painlevé test and reducibility to the canonical forms for higher-dimensional soliton equations with variable-coefficients. Symmetry Integrability Geom. Methods Appl. 2, 1–10 (2006)MathSciNetMATHGoogle Scholar
  4. 4.
    Bruzón, M.S., Gandarias, M.L., Muriel, C., Ramírez, J., Saez, S., Romero, F.R.: The Calogero–Bogoyavlenskii–Schiff equation in 2+1 dimensions. Theor. Math. Phys. 137, 1367–1377 (2003)CrossRefMATHGoogle Scholar
  5. 5.
    Wazwaz, A.M.: Multiple-soliton solutions for the Calogero–Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equations. Appl. Math. Comput. 203, 592–597 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Li, B., Chen, Y.: Exact analytical solutions of the generalized Calogero–Bogoyavlenskii–Schiff equation using symbolic computation. Czechoslov. J. Phys. 54, 517–528 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zhang, H.P., Chen, Y., Li, B.: Infinitely many symmetries and symmetry reduction of the (2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation. Acta Phys. Sin. 58, 7393–7396 (2009)MathSciNetMATHGoogle Scholar
  8. 8.
    Wang, J.M., Yang, X.: Quasi-periodic wave solutions for the (2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff (CBS) equation. Nonlinear Anal. 75, 2256–2261 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ayati, Z., Biazar, J.: Application of Exp-function method to the (2+1)-dimensional Calogero–Bogoyavlanskii–Schiff equation. Iranian J. Optim. 1, 173–187 (2009)Google Scholar
  10. 10.
    Wazwaz, A.M.: The (2+1) and (3+1)-dimensional CBS equations: multiple soliton solutions and multiple singular soliton solutions. Zeitschrift für Naturforschung A 65, 173–181 (2010)Google Scholar
  11. 11.
    Moatimid, G.M., El-Shiekh, Rehab M., Al-Nowehy, Abdul-Ghani A.A.H.: Exact solutions for Calogero–Bogoyavlenskii–Schiff equation using symmetry method. Appl. Math. Comput. 220, 455–462 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gardner, C.S., et al.: Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)CrossRefMATHGoogle Scholar
  13. 13.
    Miura, M.R.: Bäcklund Transformation. Springer, Berlin (1978)Google Scholar
  14. 14.
    Liu, S.K., et al.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A. 289, 69–74 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wang, M.L., Li, X.Z.: Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos Solitons. Fractals 24, 1257–1268 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Wang, M.L., Li, X.Z.: Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Phys. Lett. A 343, 48–54 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Song, M., Ge, Y.L.: Application of the \(\frac{G^{\prime }}{G}\)-expansion method to (3+1)-dimensional nonlinear evolution equations. Comput. Math. Appl. 60, 1220–1227 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Rui, W.G.: The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation. Nonlinear Dyn. 76, 1529–1542 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, J.B., Liu, Z.R.: Smooth and non-smooth traveling waves in a nonlinearly dispersive equation. Appl. Math. Model. 25, 41–56 (2000)CrossRefMATHGoogle Scholar
  20. 20.
    Li, S.Y., Liu, Z.R.: Kink-like wave and compacton-like wave solutions for generalized KdV equation. Nonlinear Dyn. 79, 903–918 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Liu, Z.R., Liang, Y.: The explicit nonlinear wave solutions and their bifurcations of the generalized Camassa–Holm equation. Int. J. Bifur. Chaos 21, 3119–3136 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Wen, Z.S.: Bifurcation of solitons, peakons, and periodic cusp waves for \(\theta \)-equation. Nonlinear Dyn. 77, 247–253 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Song, M.: Nonlinear wave solutions and their relations for the modified Benjamin–Bona–Mahony equation. Nonlinear Dyn. 80, 431–446 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Song, M., Liu, Z.R., Biswas, A.: Soliton solution and bifurcation analysis of the KP–Benjamin–Bona–Mahoney equation with power law nonlinearity. Nonlinear Anal. Model. Control 20, 417–427 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Song, M., Liu, Z.R.: Periodic wave solutions and their limits for the ZK–BBM equation. Appl. Math. Comput. 232, 9–26 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShaoguan UniversityShaoguanPeople’s Republic of China
  2. 2.School of Mathematical and Computer ScienceWuhan Textile UniversityWuhanPeople’s Republic of China

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