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Nonlinear Dynamics

, Volume 87, Issue 3, pp 1917–1922 | Cite as

Bifurcations and exact traveling wave solutions of a new two-component system

  • Zhenshu Wen
Original Paper

Abstract

In this paper, we study the bifurcations and exact traveling wave solutions of a new two-component system from the perspective of the theory of dynamical systems. We obtain all possible bifurcations of phase portraits of the system under various conditions about the parameters associated with the planar dynamical system. Then, we show the existence of traveling wave solutions including solitary waves, periodic waves and periodic blow-up waves, and give their exact expressions. These results can help understand the dynamical behavior of the traveling wave solutions of the system.

Keywords

A new two-component system Traveling waves Solitary waves Periodic waves Periodic blow-up waves 

Notes

Acknowledgments

This research is supported by the Natural Science Foundation of Fujian Province (No. 2015J05008), and Science and Technology Program (Class A) of the Education Department of Fujian Province (No. JA14023).

References

  1. 1.
    Byrd, P., Friedman, M.: Handbook of Elliptic Integrals for Engineers and Scientists, vol. 33. Springer, Berlin (1971)CrossRefMATHGoogle Scholar
  2. 2.
    Chen, A., Wen, S., Tang, S., Huang, W., Qiao, Z.: Effects of quadratic singular curves in integrable equations. Stud. Appl. Math. 134, 24–61 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, Y., Song, M., Liu, Z.: Soliton and riemann theta function quasi-periodic wave solutions for a (2+ 1)-dimensional generalized shallow water wave equation. Nonlinear Dyn. 82, 333–347 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dutykh, D., Ionescu-Kruse, D.: Travelling wave solutions for some two-component shallow water models. J. Differ. Equ. 262, 1099–1114 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    El-Wakil, S., Abdou, M.: New explicit and exact traveling wave solutions for two nonlinear evolution equations. Nonlinear Dyn. 51(4), 585–594 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ionescu-Kruse, D.: A new two-component system modelling shallow-water waves. Q. Appl. Math. 73, 331–346 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Li, C., Wen, S., Chen, A.: Single peak solitary wave and compacton solutions of the generalized two-component Hunter–Saxton system. Nonlinear Dyn. 79, 1575–1585 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Li, J.: Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions. Science Press, Beijing (2013)Google Scholar
  9. 9.
    Li, J., Dai, H.: On the Study of Singular Nonlinear Traveling Wave Equations: Dynamical System Approach. Science Press, Beijing (2007)Google Scholar
  10. 10.
    Li, J., Qiao, Z.: Bifurcations and exact traveling wave solutions of the generalized two-component Camassa–Holm equation. Int. J. Bifurcat. Chaos. 22, 1250305 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Liu, Z., 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
  12. 12.
    Morris, R.M., Kara, A.H., Biswas, A.: An analysis of the Zhiber–Shabat equation including lie point symmetries and conservation laws. Collect. Math. 67, 55–62 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Song, M.: Nonlinear wave solutions and their relations for the modified Benjamin–Bona–Mahony equation. Nonlinear Dyn. 80, 431–446 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Wang, Y., Bi, Q.: Different wave solutions associated with singular lines on phase plane. Nonlinear Dyn. 69(4), 1705–1731 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wen, Z.: Bifurcation of traveling wave solutions for a two-component generalized \(\theta \)-equation. Math. Probl. Eng. 2012, 1–17 (2012)MathSciNetGoogle Scholar
  16. 16.
    Wen, Z.: Extension on bifurcations of traveling wave solutions for a two-component Fornberg–Whitham equation. Abstr. Appl. Anal. 2012, 1–15 (2012)MathSciNetMATHGoogle Scholar
  17. 17.
    Wen, Z.: Bifurcation of solitons, peakons, and periodic cusp waves for \(\theta \)-equation. Nonlinear Dyn. 77, 247–253 (2014)Google Scholar
  18. 18.
    Wen, Z.: New exact explicit nonlinear wave solutions for the broer-kaup equation. J. Appl. Math. 2014, 1–7 (2014)MathSciNetGoogle Scholar
  19. 19.
    Wen, Z.: Several new types of bounded wave solutions for the generalized two-component Camassa–Holm equation. Nonlinear Dyn. 77, 849–857 (2014)Google Scholar
  20. 20.
    Wen, Z.: Bifurcations and nonlinear wave solutions for the generalized two-component integrable Dullin-Gottwald-Holm system. Nonlinear Dyn. 82, 767–781 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Wen, Z.: Extension on peakons and periodic cusp waves for the generalization of the Camassa-Holm equation. Math. Meth. Appl. Sci. 38, 2363–2375 (2015)Google Scholar
  22. 22.
    Wen, Z., Liu, Z.: Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa–Holm equation. Nonlinear Anal. 12, 1698–1707 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Wen, Z., Liu, Z., Song, M.: New exact solutions for the classical Drinfel’d–Sokolov–Wilson equation. Appl. Math. Comput. 215, 2349–2358 (2009)MathSciNetMATHGoogle Scholar
  24. 24.
    Zhang, L., Chen, L.Q., Huo, X.: The effects of horizontal singular straight line in a generalized nonlinear Klein-Gordon model equation. Nonlinear Dyn. 72, 789–801 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Zhou, Q., Liu, L., Zhang, H., Mirzazadeh, M., Bhrawy, A., Zerrad, E., Moshokoa, S., Biswas, A.: Dark and singular optical solitons with competing nonlocal nonlinearities. Opt. Appl. 46, 79–86 (2016)Google Scholar
  26. 26.
    Zhou, Q., Mirzazadeh, M., Zerrad, E., Biswas, A., Belic, M.: Bright, dark, and singular solitons in optical fibers with spatio-temporal dispersion and spatially dependent coefficients. J. Modern Opt. 63, 950–954 (2016)CrossRefGoogle Scholar
  27. 27.
    Zhou, Q., Zhong, Y., Mirzazadeh, M., Bhrawy, A., Zerrad, E., Biswas, A.: Thirring combo-solitons with cubic nonlinearity and spatio-temporal dispersion. Waves in Random and Complex Media 26, 204–210 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesHuaqiao UniversityQuanzhouPR China

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