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Pramana

, 92:25 | Cite as

Testing efficiency of the generalised \(\left( {{{G}'}/G} \right) \)-expansion method for solving nonlinear evolution equations

  • G C PaulEmail author
  • A H M Rashedunnabi
  • M D Haque
Article
  • 17 Downloads

Abstract

In this investigation, we employ the generalised \((G'{/}G)\)-expansion method to test its efficiency in extracting travelling wave solutions of nonlinear evolution equations (NLEEs). As test cases, the modified Kuramoto–Sivashinsky (mK-S) and the modified Burgers–Korteweg–de Vries (mB-KdV) equations are considered because of their importance in soliton theory. The general solutions are obtained in hyperbolic, trigonometric and rational function forms for both the equations. Taking specific parametric values in the corresponding general solutions, some new exact travelling waves in trigonometric and hyperbolic forms and only in hyperbolic form are obtained for the mK-S and mB-KdV equations, respectively. The obtained results are checked to see whether the criticism made by Parkes (Comput. Fluids 42, 108 (2011)), that the so-called ‘new’ solutions derived by the \((G'{/}G)\)-expansion method are often erroneous and are merely disguised versions of previously known solutions, is justified also for the generalised \((G'{/}G)\)-expansion method. The solutions were checked with Maple by putting them back into their corresponding equations. With specific values of parameters, some of our obtained solutions satisfied directly and some solutions never satisfied the considered NLEEs. Among the satisfactory solutions, some are found to be in disguised versions of some results obtained in this study.

Keywords

Modified Kuramoto–Sivashinsky equation modified Burgers–Korteweg–de Vries equation generalised (\({G'}{/}{G}\) )-expansion method nonlinear evolution equations travelling wave homogeneous balance 

PACS Nos

05.45.Yv 02.30.Jr 02.30.Ik 

Notes

Acknowledgements

The authors are grateful to the two anonymous reviewers for their thoughtful comments and suggestions which helped to improve this paper. The comments and suggestions of the editor for improving the quality of this paper are gratefully acknowledged.

References

  1. 1.
    E J Parkes, Comput. Fluids 42, 108 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J H He and X H Wu, Chaos Solitons Fractals 29, 108 (2006)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    E J Parkes and B R Duffy, Comput. Phys. Comm. 98, 288 (1996)ADSCrossRefGoogle Scholar
  4. 4.
    H Zhang, Chaos Solitons Fractals 32, 653 (2007)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    J H He and X H Wu, Chaos Solitons Fractals 30, 700 (2006)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Y Yildirim, E Yasar and A R Adem, Nonlinear Dyn. 89, 2291 (2017)CrossRefGoogle Scholar
  7. 7.
    M L Wang, Y B Zhou and Z B Li, Phys. Lett. A 216, 67 (1996)ADSCrossRefGoogle Scholar
  8. 8.
    W X Ma, Phys. Lett. A 301, 35 (2002)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    R Hirota, The direct method in soliton theory (Cambridge University Press, Cambridge, 2004)CrossRefGoogle Scholar
  10. 10.
    Y Yildirim and E Yasar, Nonlinear Dyn. 90, 1571 (2017)CrossRefGoogle Scholar
  11. 11.
    Sirendaoreji, Chaos Solitons Fractals 31, 943 (2007)Google Scholar
  12. 12.
    M A Abdou, Chaos Solitons Fractals 31, 95 (2007)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    K Ayub, M Y Khan, Q Mahmood-ul-Hassan and J Ahmad, Pramana – J. Phys. 89: 45 (2017)ADSCrossRefGoogle Scholar
  14. 14.
    A Ali, M A Iqbal and S T Mohyud-Din, Pramana – J. Phys. 87: 79 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    P J Olver, Applications of Lie groups to differential equations (Springer, New York, USA, 1986)CrossRefGoogle Scholar
  16. 16.
    E Yasar and I B Giresunlu, Acta Phys. Pol. A 128, 243 (2015)CrossRefGoogle Scholar
  17. 17.
    S Y Lou, Stud. Appl. Math. 134, 372 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    V B Matveev and M A Salle, Darboux transformations and solitons (Springer, Berlin, 1991)CrossRefGoogle Scholar
  19. 19.
    W X Ma, Chaos Solitons Fractals 26, 1453 (2005)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    X P Cheng, C L Chen and S Y Lou, Wave Motion 51, 1298 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    R Conte and M Musette, J. Phys. A 22, 169 (1989)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    N A Kudryashov and N B Loguinova, Commun. Nonlinear Sci. Numer. Simul. 14, 1881 (2009)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    J H He, Int. J. Nonlinear Sci. Numer. Simul. 14, 363 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    O Guner and A Bekir, Nonlinear Sci. Lett. A 8, 41(2017)Google Scholar
  25. 25.
    M L Wang, X Z Li and J L Zhang, Phys. Lett. A 372, 417 (2008)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    J Manafian and M Lakestani, Pramana – J. Phys. 85, 31 (2015)ADSCrossRefGoogle Scholar
  27. 27.
    S Zhang, L Dong, J M Ba and Y N Siu, Phys. Lett. A 373, 905 (2009)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    A Malik, F Chand, H Kumar and S C Mishra, Pramana – J. Phys. 78, 513 (2012)ADSCrossRefGoogle Scholar
  29. 29.
    A Sohail, A M Siddiqui and M Iftikhar, Nonlinear Sci. Lett. A 8, 228 (2017)Google Scholar
  30. 30.
    N A Kudryashov, Appl. Math. Comput. 217, 1755 (2010)MathSciNetGoogle Scholar
  31. 31.
    J Zhang, X Wei and Y Lu, Phys. Lett. A 372, 3653 (2008)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    H Naher and F A Abdullah, J. Egypt. Math. Soc. 22, 390 (2014)CrossRefGoogle Scholar
  33. 33.
    E M E Zayed, J. Phys. A 42, 195202 (2009)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    S Saprykin, E A Demekhin and S Kalliadasis, Phys. Fluids 17, 117105 (2005)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    J Topper and T Kawahara, J. Phys. Soc. Jpn 44, 663 (1978)ADSCrossRefGoogle Scholar
  36. 36.
    T Tatsumi (Ed.), Proceedings IUTAM Symposium (North Holland, Kyoto, 1983) pp. 1–10Google Scholar
  37. 37.
    Z J Yang, J. Phys. A. 27, 2837 (1994)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    H I Abdel-Gawad and H A Abdusalam, Chaos Solitons Fractals 12, 2039 (2001)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    L van Wijngaarden, Ann. Rev. Fluid Mech. 4, 369 (1972)ADSCrossRefGoogle Scholar
  40. 40.
    G Gao, Sci. Sin. A 28, 616 (1985)Google Scholar
  41. 41.
    H Segur, Tsunami and nonlinear waves edited by A Kundu, (Springer, Berlin, 2007)Google Scholar
  42. 42.
    R S Johnson, J. Fluid Mech. 42, 49 (1970)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    M S Ruderman, J. Appl. Math. Mech. 39, 656 (1975)MathSciNetCrossRefGoogle Scholar
  44. 44.
    S D Liu and S K Liu, Sci. Sin. A 35, 576 (1992)Google Scholar
  45. 45.
    G W Griffiths and W E Schiesser, Traveling wave analysis of partial differential equations: Numerical and analytical methods with MATLAB and MAPLE (Academic Press, New York, 2012)zbMATHGoogle Scholar
  46. 46.
    H Kheiri, M R Moghaddam and V Vafaei, Pramana – J. Phys. 76, 831 (2011)ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RajshahiRajshahiBangladesh

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