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Proceedings Mathematical Sciences

, Volume 117, Issue 4, pp 555–574 | Cite as

Dynamics and bifurcations of travelling wave solutions of R(m, n) equations

  • Dahe Feng
  • Jibin Li
Article

Abstract

By using the bifurcation theory and methods of planar dynamical systems to R(m, n) equations, the dynamical behavior of different physical structures like smooth and non-smooth solitary wave, kink wave, smooth and non-smooth periodic wave, and breaking wave is obtained. The qualitative change in the physical structures of these waves is shown to depend on the systemic parameters. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above waves are given. Moreover, some explicit exact parametric representations of travelling wave solutions are listed.

Keywords

R(m, n) equations solitary wave periodic wave breaking wave solitary cusp wave periodic cusp wave 

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References

  1. [1]
    Benjamin T B, Bona J L and Mahoney J J, Model equations for long waves in nonlinear dispersive media, Philos. Trans. R. Soc. London. A272 (1972) 47–78Google Scholar
  2. [2]
    Boussinesq J, Théorie des ondes et des remous qui se propagent le long dun canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pure. Appl. 17 (1872) 55–108Google Scholar
  3. [3]
    Guckenheimer J and Holmes P J, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (1983) (New York: Springer-Verlag) pp. 117–165zbMATHGoogle Scholar
  4. [4]
    Inc M, New compacton solutions of nonlinearly dispersive R(m, n) equations, Commun. Theor. Phys. (Beijing, China) 45 (2006) 389–394CrossRefGoogle Scholar
  5. [5]
    Inc M, New exact solitary pattern solutions of the nonlinearly dispersive R(m, n) equations, Chaos, Solitons and Fractals. 29 (2006) 499–505zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Korteweg D J and de Vries G, On the change of form of long waves advancing in a rectangular canal and a new type of long stationary wave, Philos. Mag. 39 (1895) 422–443Google Scholar
  7. [7]
    Li J B and Liu Z R, Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Appl. Math. Model. 25 (2000) 41–56zbMATHCrossRefGoogle Scholar
  8. [8]
    Li J B and Liu Z R, Travelling wave solutions for a class of nonlinear dispersive equations, Chin. Ann. Math. 23B(3) (2002) 397–418CrossRefGoogle Scholar
  9. [9]
    Li Y A, Olver P J and Rosenau P, Non-analytic solutions of nonlinear wave models, Technical Report 1591, at Institute of Mathematics and Applications, Minnesota (1998)Google Scholar
  10. [10]
    Luo D J et al, Bifurcation theory and methods of dynamical systems (1997) (Singapore: World Scientific Publishing Co. Pte. Ltd.) pp. 1–89zbMATHGoogle Scholar
  11. [11]
    Parker A, On exact solutions of the regularized long-wave equation: A direct approach to partially integrable equations. I. Solitary wave and solitions, J. Math. Phys. 36 (1995) 3498–3505zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Parker A, On exact solutions of the regularized long-wave equation: A direct approach to partially integrable equations. II. Periodic solutions, J. Math. Phys. 36 (1995) 3506–3519zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Perko L, Differential equations and dynamical systems (1991) (New York: Springer-Verlag) pp. 20–28zbMATHGoogle Scholar
  14. [14]
    Russell J S, On waves, in: Report of 14th Meeting of the British Association for the Advancement of Science, York (1844) pp. 311–390.Google Scholar
  15. [15]
    Yan Z Y, New soliton solutions with compact support for a family of two-parameter regularized long-wave Boussinesq equations, Commun. Theor. Phys. (Beijing, China) 37 (2002) 641–644Google Scholar
  16. [16]
    Yan Z Y, New families of exact solitary patterns solutions for the nonlinearly dispersive R (m, n) equations, Chaos, Solitons and Fractals 15 (2003) 891–896zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2007

Authors and Affiliations

  1. 1.School of Mathematics and Computing ScienceGuilin University of Electronic TechnologyGuilinPeople’s Republic of China
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaPeople’s Republic of China

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