Soliton collisions and integrable aspects of the fifth-order Korteweg-de Vries equation for shallow water with surface tension

  • Wen-Rong Sun
  • Wen-Rui Shan
  • Yan Jiang
  • Pan Wang
  • Bo Tian
Regular Article

Abstract

The fifth-order Korteweg-de Vries (KdV) equation works as a model for the shallow water waves with surface tension. Through symbolic computation, binary Bell-polynomial approach and auxiliary independent variable, the bilinear forms, N-soliton solutions, two different Bell-polynomial-type Bäcklund transformations, Lax pair and infinite conservation laws are obtained. Characteristic-line method is applied to discuss the effects of linear wave speed c as well as length scales τ and γ on the soliton amplitudes and velocities. Increase of τ, c and γ can lead to the increase of the soliton velocity. Soliton amplitude increases with the increase of τ. The larger-amplitude soliton is seen to move faster and then to overtake the smaller one. After the collision, the solitons keep their original shapes and velocities invariant except for the phase shift. Graphic analysis on the two and three-soliton solutions indicates that the overtaking collisions between/among the solitons are all elastic.

Graphical abstract

Keywords

Nonlinear Dynamics 

References

  1. 1.
    H.R. Dullin, G.A. Gottwald, D.D. Holm, Fluid Dyn. Res. 33, 73 (2003) CrossRefADSMATHMathSciNetGoogle Scholar
  2. 2.
    H.R. Dullin, G.A. Gottwald, D.D. Holm, Physica D 190, 1 (2004) CrossRefADSMATHMathSciNetGoogle Scholar
  3. 3.
    M. Kunze, G. Schneider, Lett. Math. Phys. 72, 17 (2005) CrossRefADSMATHMathSciNetGoogle Scholar
  4. 4.
    Y. Kodama, Phys. Lett. A 107, 245 (1985) CrossRefADSMATHMathSciNetGoogle Scholar
  5. 5.
    Y. Kodama, Phys. Lett. A 112, 193 (1985) CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Y. Kodama, Phys. Lett. A 123, 276 (1987) CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    M. Alber, R. Camassa, D.D. Holm, J.E. Marsden, Lett. Math. Phys. 32, 137 (1994) CrossRefADSMATHMathSciNetGoogle Scholar
  8. 8.
    M. Alber, R. Camassa, V.N. Fedorov, D.D. Holm, J.E. Marsden, Phys. Lett. A 264, 171 (1999) CrossRefADSMATHMathSciNetGoogle Scholar
  9. 9.
    M. Alber, R. Camassa, V.N. Fedorov, D.D. Holm, J.E. Marsden, Commun. Math. Phys. 221, 197 (2001) CrossRefADSMATHMathSciNetGoogle Scholar
  10. 10.
    R. Camassa, D.D. Holm, Phys. Rev. Lett. 71, 1661 (1993) CrossRefADSMATHMathSciNetGoogle Scholar
  11. 11.
    V. Busuioc, C.R. Acad. Sci. Paris Ser. I 328, 1241 (1999) CrossRefADSMATHMathSciNetGoogle Scholar
  12. 12.
    H. Dai, Acta Mech. 127, 193 (1998) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Y.J. Shen, Y.T. Gao, X. Yu, G.Q. Meng, Y. Qin, Appl. Math. Comput. 227, 502 (2014) CrossRefMathSciNetGoogle Scholar
  14. 14.
    D.W. Zuo, Y.T. Gao, G.Q. Meng, Y.J. Shen, X. Yu, Nonlinear Dyn. 75, 701 (2014) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Z.Y. Sun, Y.T. Gao, Y. Liu, X. Yu, Phys. Rev. E 84, 026606 (2011) CrossRefADSGoogle Scholar
  16. 16.
    M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge Univ. Press, Cambridge, 1992) Google Scholar
  17. 17.
    J.J. Nimmo, Phys. Lett. A 99, 279 (1983) CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    R. Hirota, The Direct Method in Soliton Theory (Springer, Berlin, 1980) Google Scholar
  19. 19.
    H.Q. Zhang, B. Tian, Eur. Phys. J. B 72, 233 (2009) CrossRefADSMATHMathSciNetGoogle Scholar
  20. 20.
    Y.S. Li, W.X. Ma, Phys. Lett. A 275, 60 (2000) CrossRefADSMATHMathSciNetGoogle Scholar
  21. 21.
    F. Lambert, S. Leble, J. Springael, Glasgow Math. J. 43A, 53 (2001) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    C. Gilson, F. Lambert, J. Nimmo, R. Willox, Proc. R. Soc. London A 452, 223 (1996) CrossRefADSMATHMathSciNetGoogle Scholar
  23. 23.
    F. Lambert, J. Springael, Acta Appl. Math. 102, 147 (2008) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    F. Lambert, I. Loris, J. Springael, Inv. Prob. 17, 1067 (2001) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    E.G. Fan, Phys. Lett. A 375, 493 (2011) CrossRefADSMATHMathSciNetGoogle Scholar
  26. 26.
    Y.J. Shen, Y.T. Gao, D.W. Zuo, Y.H. Sun, Y.J. Feng, L. Xue, Phys. Rev. E 89, 062915 (2014) CrossRefADSGoogle Scholar
  27. 27.
    D.W. Zuo, Y.T. Gao, Y.H. Sun, Y.J. Feng, L. Xue, Z. Naturforsch. A 69, 521 (2014) CrossRefGoogle Scholar
  28. 28.
    Z.Y. Sun, Y.T. Gao, X. Yu, Y. Liu, Phys. Lett. A 377, 3283 (2013) CrossRefADSMATHMathSciNetGoogle Scholar
  29. 29.
    Z.Y. Sun, Y.T. Gao, X. Yu, Y. Liu, Europhys. Lett. 93, 40004 (2011) CrossRefADSGoogle Scholar
  30. 30.
    B. Tian, Y.T. Gao, Phys. Plasmas 12, 070703 (2005) CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    B. Tian, Y.T. Gao, Eur. Phys. J. D 33, 59 (2005) CrossRefADSGoogle Scholar
  32. 32.
    Y.T. Gao, B. Tian, Phys. Lett. A 361, 523 (2007) CrossRefADSMATHGoogle Scholar
  33. 33.
    Y.T. Gao, B. Tian, Europhys. Lett. 77, 15001 (2007) CrossRefADSGoogle Scholar
  34. 34.
    B. Tian, Y.T. Gao, Phys. Plasmas 12, 054701 (2005) CrossRefADSMathSciNetGoogle Scholar
  35. 35.
    B. Tian, Y.T. Gao, Phys. Lett. A 340, 449 (2005) CrossRefADSMATHGoogle Scholar
  36. 36.
    B. Tian, Y.T. Gao, Phys. Lett. A 342, 228 (2005) CrossRefADSMATHGoogle Scholar
  37. 37.
    B. Tian, Y.T. Gao, Phys. Lett. A 359, 241 (2006) CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Wen-Rong Sun
    • 1
  • Wen-Rui Shan
    • 1
  • Yan Jiang
    • 1
  • Pan Wang
    • 1
  • Bo Tian
    • 1
  1. 1.State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and TelecommunicationsBeijingP.R. China

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