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Pramana

, 92:31 | Cite as

All single travelling wave patterns to fractional Jimbo–Miwa equation and Zakharov–Kuznetsov equation

  • Xin WangEmail author
  • Yang Liu
Article
  • 19 Downloads

Abstract

By the complete discrimination system of polynomial method, we obtain the classification and representation of all single travelling wave solutions to \((3+1)\)-dimensional conformal fractional Jimbo–Miwa equation and fractional Zakharov–Kuznetsov equation. These solutions show rich evolution patterns of models described by these two equations.

Keywords

Conformal fractional derivative complete discrimination system for polynomial method travelling wave solution fractional Jimbo–Miwa equation fractional Zakharov–Kuznetsov equation 

PACS Nos

02.30.Jr 05.45.Yv 

References

  1. 1.
    M J Ablowitz and P A Clarkson, Solitons, nonlinear evolutions and inverse scattering (Cambridge University Press, Cambridge, 1991)CrossRefGoogle Scholar
  2. 2.
    R M Miura (Ed.), Bachlund transformation, in: Lecture notes in mathematics (Springer-Verlag, New York, 1976), Vol. 515Google Scholar
  3. 3.
    R Hirota, Direct method in soliton theory, Solitons, in: Topics in current Physics edited by R K Bullough and P J Caudrey (Springer-Verlag, New York, 1980) Vol. 17, pp. 157–176Google Scholar
  4. 4.
    E G Fan and H Q Zhang, Phys. Lett. A 246, 403 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    M L Wang, Phys. Lett. A 213, 279 (1996)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    C S Liu, Chaos Solitons Fractals 42, 441 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    C Q Dai and J F Zhang, Chaos Solitons Fractals 27, 1042 (2006)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    S Liu, Z Fu, S Liu and Q Zhao, Phys. Lett. A 289, 69 (2001)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    C S Liu, Chin. Phys. Lett. 21, 2369 (2004)ADSCrossRefGoogle Scholar
  10. 10.
    C S Liu, Commun. Theor. Phys. 44, 799 (2005)ADSCrossRefGoogle Scholar
  11. 11.
    C S Liu, Commun. Theor. Phys. 43, 787 (2005)ADSCrossRefGoogle Scholar
  12. 12.
    C S Liu, Commun. Theor. Phys. 45, 991 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    C S Liu, Commun. Theor. Phys. 49, 291 (2008)ADSCrossRefGoogle Scholar
  14. 14.
    C S Liu, Commun. Theor. Phys. 49, 153 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    C S Liu, Comput. Phys. Commun. 181, 317 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    A M Wazwaz, Appl. Math. Comput. 190, 633 (2007)MathSciNetGoogle Scholar
  17. 17.
    A Biswas, H Triki and M Labidi, Phys. Wave Phenom. 19, 24 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    A Bekir, E Aksoy and O Guner, J. Nonlinear Opt. Phys. Mater. 22, 1350015 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    H Triki and A M Wazwaz, Nonlinear Anal.: Real World Appl. 12, 2822 (2011)Google Scholar
  20. 20.
    W Malfliet and W Hereman, Phys. Scr. 54, 569 (1996)ADSCrossRefGoogle Scholar
  21. 21.
    M Inc and B Kilic, Waves Rand. Complex Media 25, 334 (2015)ADSCrossRefGoogle Scholar
  22. 22.
    C S Liu, Acta Phys. Sin. 54, 2505 (2005)Google Scholar
  23. 23.
    C S Liu, Acta Phys. Sin. 54, 4506 (2005)Google Scholar
  24. 24.
    C S Liu, Chaos Solitons Fractals 40, 708 (2009)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    C S Liu, Found. Phys. 41, 793 (2011)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    X H Du, Pramana – J. Phys. 75, 415 (2010)ADSCrossRefGoogle Scholar
  27. 27.
    Y Liu, Appl. Math. Comput. 217, 5866 (2011)MathSciNetGoogle Scholar
  28. 28.
    Y Gurefe, A Sonmezoglu and E Misirli, Pramana – J. Phys. 77, 1023 (2011)ADSCrossRefGoogle Scholar
  29. 29.
    Y Gurefe, E Misirli, A Sonmezoglu and M Ekici, Appl. Math. Comput. 219, 5253 (2013)MathSciNetGoogle Scholar
  30. 30.
    H Bulut, Y Pandir and S Tuluce Demiray, Waves Rand. Complex Media 24, 439 (2014)Google Scholar
  31. 31.
    Y Kai, Pramana – J. Phys. 87: 59 (2016)ADSCrossRefGoogle Scholar
  32. 32.
    R Khalil, M Al Horani, A Yousef and M Sababheh, J. Comput. Appl. Math. 264, 65 (2014)Google Scholar
  33. 33.
    T Abdeljawad, J. Comput. Appl. Math. 279, 57 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    D R Anderson and D J Ulness, J. Math. Phys. 56, 063502 (2015)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    M Eslami, Appl. Math. Comput. 285, 141 (2016)MathSciNetGoogle Scholar
  36. 36.
    A Korkmaz and K Hosseini, Opt. Quantum Electron. 49, 278 (2017)CrossRefGoogle Scholar
  37. 37.
    Y Cenesiz, D Baleanu, A Kurt and O Tasbozan, Waves Rand. Complex Media 27, 103 (2017)ADSCrossRefGoogle Scholar
  38. 38.
    A Korkmaz, Commun. Theor. Phys. 67, 479 (2017)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    M Eslami and H Rezazadeh, Calcolo 53, 475 (2016)MathSciNetCrossRefGoogle Scholar
  40. 40.
    M Jimbo and T Miwa, Publ. Res. Inst. Math. Sci. 19, 943 (1983)MathSciNetCrossRefGoogle Scholar
  41. 41.
    B Cao, Acta Appl. Math. 112, 181 (2010)MathSciNetCrossRefGoogle Scholar
  42. 42.
    W X Ma, T Huang and Y Zhang, Phys. Scr. 82, 065003 (2010)ADSCrossRefGoogle Scholar
  43. 43.
    Y Tang, W X Ma, W Xu and L Gao, Appl. Math. Comput. 217, 8722 (2011)MathSciNetGoogle Scholar
  44. 44.
    Z Xu and H Chen, Int. J. Numer. Methods Heat Fluid Flow 25, 19 (2015)Google Scholar
  45. 45.
    J F Zhang and F M Wu, Chin. Phys. 11, 425 (2002)ADSCrossRefGoogle Scholar
  46. 46.
    S H Ma, J P Fang and C L Zheng, Chaos Solitons Fractals 40, 1352 (2009).ADSCrossRefGoogle Scholar
  47. 47.
    W Hong and K S Oh, Comput. Math. Appl. 39, 29 (2000)MathSciNetCrossRefGoogle Scholar
  48. 48.
    T Ozis and I Aslan, Phys. Lett. A 372, 7011 (2005)ADSCrossRefGoogle Scholar
  49. 49.
    B B Kadomtsev and V I Petviashvili, Sov. Phys. Dokl. 15, 539 (1970)ADSGoogle Scholar
  50. 50.
    V E Zakharov and E A Kuznetsov, Z. Eksp. Teoret. Fiz. 66, 594 (1974)ADSGoogle Scholar
  51. 51.
    A Mushtaq and H A Shah, Phys. Plasmas 12, 072306 (2005)ADSCrossRefGoogle Scholar
  52. 52.
    B Li, Y Chen and H Zhang, Appl. Math. Comput. 146, 653 (2003)MathSciNetGoogle Scholar
  53. 53.
    A M Wazwaz, Commun. Nonlinear Sci. Numer. Simul. 10, 597 (2005)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    A M Wazwaz, Commun. Nonlinear Sci. Numer. Simul. 13, 1039 (2008)ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    Z Yan and X Liu, Appl. Math. Comput. 180, 288 (2006)MathSciNetGoogle Scholar
  56. 56.
    Z Li and X Zhang, Commun. Nonlinear Sci. Numer. Simul. 15, 3418 (2010)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    B K Shivamoggi and D K Rollins, Phys. Lett. A 161, 263 (1991)ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    A M Hamza, Phys. Lett. A 190, 309 (1994)ADSCrossRefGoogle Scholar
  59. 59.
    A R Seadawy, Pramana – J. Phys. 89: 49 (2017)ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.College of Petroleum EngineeringNortheast Petroleum UniversityDaqingChina
  2. 2.No. 3 Oil Production Plant, Daqing Oilfield Company Ltd.DaqingChina

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