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Lax pair, conservation laws and N-soliton solutions for the extended Korteweg-de Vries equations in fluids

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Abstract.

Under investigation in this paper are two extended Korteweg-de Vries (eKdV) equations in fluids with the second-order nonlinear and dispersive terms. Based on the Ablowitz-Kaup-Newell-Segur system, the Lax pair and infinitely many conservation laws are derived. By virtue of the Hirota method and symbolic computation, the bilinear forms and N-soliton solutions for the two eKdV equations are obtained, respectively. Relevant propagation properties and interaction behaviors of the solitons are illustrated graphically. The collisions for the η profile are proved to be elastic through the asymptotic analysis. Types of collisions (head-on or overtaking collisions) can be controlled when we adjust the sign of the velocity v. Velocities of solitons are related to c 4 and α during the collisions. Moreover, there is not a direct proportion relationship between the velocity v and amplitude a during the collisions. On the one hand, the soliton with the larger amplitude travels faster and catches up with the smaller one. On the other hand, the soliton with the smaller amplitude travels faster and catches up with the larger one.

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References

  1. S. Balberg, S.L. Shapiro, Phys. Rev. Lett. 88, 101301 (2002)

    Article  ADS  Google Scholar 

  2. M. Barriola, A. Vilenkin, Phys. Rev. Lett. 63, 341 (1989)

    Article  ADS  Google Scholar 

  3. N. Turok, D. Spergel, Phys. Rev. Lett. 64, 2736 (1990)

    Article  ADS  Google Scholar 

  4. M. Gleiser, Phys. Rev. Lett. 63, 1199 (1989)

    Article  ADS  Google Scholar 

  5. C.M. Bender, S.A. Orszag, Advanced mathematical methods for scientists and engineers (Springer, New York, 1999)

  6. F.V. Kusmartsev, E.W. Mielke, F.E. Schunck, Phys. Rev. D 43, 3895 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  7. V.A. Brazhnyi, V.V. Konotop, Mod. Phys. Lett. B 18, 627 (2004)

    Article  ADS  Google Scholar 

  8. J. Sarma, Chaos Solit. Fract. 42, 1599 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. M.P. Barnett, J.F. Capitani, J. Von Zur Gathen, J. Gerhard, Int. J. Quant. Chem. 100, 80 (2004)

    Article  Google Scholar 

  10. B. Tian, Y.T. Gao, Phys. Lett. A 340, 243 (2005)

    Article  ADS  MATH  Google Scholar 

  11. B. Tian, Y.T. Gao, Phys. Lett. A 340, 449 (2005)

    Article  ADS  MATH  Google Scholar 

  12. B. Tian, Y.T. Gao, Phys. Lett. A 362, 283 (2007)

    Article  ADS  MATH  Google Scholar 

  13. B. Tian, Y.T. Gao, Phys. Plasmas 12, 054701 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  14. B. Tian, G.M. Wei, C.Y. Zhang, W.R. Shan, Y.T. Gao, Phys. Lett. A 356, 8 (2006)

    Article  ADS  MATH  Google Scholar 

  15. G. Das, J. Sarma, Phys. Plasmas 6, 4394 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  16. W.P. Hong, Phys. Lett. A 361, 520 (2007)

    Article  ADS  MATH  Google Scholar 

  17. Y.T. Gao, B. Tian, Phys. Lett. A 349, 314 (2006)

    Article  ADS  Google Scholar 

  18. Y.T. Gao, B. Tian, Phys. Plasmas 13, 112901 (2006)

    Article  ADS  Google Scholar 

  19. Y.T. Gao, B. Tian, Phys. Plasmas 13, 120703 (2006)

    Article  ADS  Google Scholar 

  20. B. Tian, W.R. Shan, C.Y. Zhang, G.M. Wei, Y.T. Gao, Eur. Phys. J. B 47, 329 (2005)

    Article  ADS  Google Scholar 

  21. B. Tian, Y.T. Gao, H.W. Zhu, Phys. Lett. A 366, 223 (2007)

    Article  ADS  MATH  Google Scholar 

  22. B. Tian, Y.T. Gao, Phys. Plasmas 12, 070703 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  23. B. Tian, Y.T. Gao, Eur. Phys. J. D 33, 59 (2005)

    Article  ADS  Google Scholar 

  24. Y.T. Gao, B. Tian, Phys. Lett. A 361, 523 (2007)

    Article  ADS  MATH  Google Scholar 

  25. Y.T. Gao, B. Tian, Europhys. Lett. 77, 15001 (2007)

    Article  ADS  Google Scholar 

  26. W.J. Liu, B. Tian, L.L. Li, X. Yu, Z.Y. Sun, J. Mod. Opt. 56, 1151 (2009)

    Article  ADS  MATH  Google Scholar 

  27. G.B. Whitham, Linear and nonlinear waves (Wiley, New York, 1974)

  28. D.J. Benny, J. Math. Phys. 45, 52 (1966)

    Google Scholar 

  29. N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15, 240 (1965)

    Article  ADS  MATH  Google Scholar 

  30. L.M. Tolbert, Acc. Chem. Res. 25, 561 (1992)

    Article  Google Scholar 

  31. S. Yitzchaik, T.J. Marks, Acc. Chem. Res. 29, 197 (1996)

    Article  Google Scholar 

  32. C.Y. Liu, A.J. Bard, Acc. Chem. Res. 32, 235 (1999)

    Article  Google Scholar 

  33. R.M. Metzger, Acc. Chem. Res. 32, 950 (1999)

    Article  Google Scholar 

  34. J.M. Tour, Acc. Chem. Res. 33, 791 (2000)

    Article  Google Scholar 

  35. R. Radhakrishnan, M. Lakshmanan, J. Hietarinta, Phys. Rev. E 56, 427 (1997)

    Article  Google Scholar 

  36. R. Sahadevan, K.M. Tamizhmani, M. Lakshmanan, J. Phys. A 19, 1783 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. S.M. Ulam, Proc. Symp. Appl. Math. 14, 215 (1962)

    Google Scholar 

  38. A. Bekir, Chaos Solit. Fract. 32, 449 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. M.J. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, New York, 1991)

  40. M. Wadati, K. Konno, Y.H. Ichikawa, J. Phys. Soc. Jpn 53, 2642 (1983)

    Article  Google Scholar 

  41. F. Caruello, M. Tabor, Physica D 39, 77 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  42. V.B. Matveev, M.A. Salle, Darboux transformations and solitons (Springer, Berlin, 1991)

  43. M. Wadati, J. Phys. Soc. Jpn 38, 673 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  44. R. Hirota, The direct method in soliton theory (Springer, Berlin, 1980)

  45. E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its, V.B. Matveev, Algebro-geometrical approach to nonlinear integrable equations (Springer-Verlag, Berlin, 1994)

  46. C.W. Cao, X.G. Geng, H.Y. Wang, J. Math. Phys. 43, 621 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. R. Hirota, Prog. Theor. Phys. 52, 1498 (1974)

    Article  ADS  MATH  Google Scholar 

  48. J. Satsuma, Prog. Theor. Phys. 52, 1396 (1974)

    Article  ADS  Google Scholar 

  49. J.D. Gibbon, P. Radmore, M. Tabor, D. Wood, Stud. Appl. Math. 72, 39 (1985)

    MathSciNet  MATH  Google Scholar 

  50. J. Hietarinta, M.D. Kruskal, Painlevé transcendents (Plenum, New York, 1992)

  51. D.K. Ludlow, P.A. Clarkson, Applications of analytic and geometric methods to nonlinear differential equations (Kluwer, Dordrecht, 1993)

  52. T.R. Marchant, Proc. R. Soc. Lond. A 456, 433 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  53. T.R. Marchant, ANZIAM J. 44, 95 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  54. Y. Kodama, Phys. Lett. A 107, 245 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. A.S. Fokas, Q.M. Liu, Phys. Rev. Lett. 77, 2347 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. T.R. Marchant, N.F. Smyth, IMA J. Appl. Math. 56, 157 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  57. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Phys. Rev. Lett. 31, 2 (1973)

    Article  MathSciNet  Google Scholar 

  58. K. Konno, H. Sanuki, Y.H. Ichikawa, Prog. Theor. Phys. 52, 886 (1974)

    Article  ADS  Google Scholar 

  59. H. Sanuki, K. Konno, Phys. Lett. A 48, 221 (1974)

    Article  ADS  Google Scholar 

  60. M. Wadati, H. Sanuki, K. Konno, Prog. Theor. Phys. 53, 419 (1975)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34, 62 (1972)

    MathSciNet  ADS  Google Scholar 

  62. M. Wadati, K. Konno, Y. Ichikawa, J. Phys. Soc. Jpn 46, 1965 (1979)

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. School of Science, P.O. Box 122, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Pan Wang, Bo Tian, Wen-Jun Liu, Qi-Xing Qu, Min Li & Kun Sun

  2. State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Bo Tian

  3. Key Laboratory of Information Photonics and Optical Communications (BUPT), Ministry of Education, P.O. Box 128, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Bo Tian

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  1. Pan Wang
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Correspondence to Bo Tian.

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Wang, P., Tian, B., Liu, WJ. et al. Lax pair, conservation laws and N-soliton solutions for the extended Korteweg-de Vries equations in fluids. Eur. Phys. J. D 61, 701–708 (2011). https://doi.org/10.1140/epjd/e2010-10357-x

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  • DOI: https://doi.org/10.1140/epjd/e2010-10357-x

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