Nonlinear self-adjointness, conserved quantities, bifurcation analysis and travelling wave solutions of a family of long-wave unstable lubrication model

Abstract

The paper investigates a class of long-wave unstable lubrication model using Lie theory. A nonlinear self-adjoint classification of the considered equation is carried out. Without having to go into microscopic details of the physical aspects, non-trivial conservation laws are computed. Then, minimal set of Lie point symmetries of the discussed model is classified up to one-dimensional conjugacy classes which are further utilised one by one to construct the similarity variables to reduce the dimension of the considered model. After that, all possible phase trajectories are classified with respect to the parameters of the equation. Some travelling wave and kink-wave solutions are also showed and graphical representations are displayed to depict their propagation.

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References

  1. 1.

    A L Bertozzi and M C Pugh, Commun. Pure Appl. Math. 51, 625 (1998)

    Article  Google Scholar 

  2. 2.

    A L Bertozzi and M C Pugh, Indiana Univ. Math. J. 49(4), 1323  (2000)

    MathSciNet  Article  Google Scholar 

  3. 3.

    P Constantin, T F Dupont, R E Goldstein, L P Kadanoff, M J Shelley and S Zhou, Phys. Rev. E 47(6), 4169 (1993)

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    S Ulusoy, Nonlinearity 20, 685 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    D Lu, A R Seadawy, J Wang, M Arshad and U Farooq, Pramana – J. Phys. 93(3): 44 (2019)

    ADS  Article  Google Scholar 

  6. 6.

    A R Seadawy, Pramana – J. Phys. 89: 49 (2017)

    ADS  Article  Google Scholar 

  7. 7.

    A R Seadawy, D Lu and M Iqbal, Pramana – J. Phys. 93: 10 (2019)

    ADS  Article  Google Scholar 

  8. 8.

    K U Tariq, A R Seadawy and S Z Alamri, Pramana – J. Phys. 91: 68 (2018)

    ADS  Article  Google Scholar 

  9. 9.

    P J Olver, Applications of Lie groups to differential equations (Springer-Verlag, New York, 1986)

    Google Scholar 

  10. 10.

    W Malfliet, J. Comput. Appl. Math. 164, 529 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

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

    Google Scholar 

  12. 12.

    D Wang and H Q Zhang, Chaos Solitons Fractals 25, 601 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    S Liu, Z Fu, S D Liu and Q Zhao, Phys. Lett. A 289, 69 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    M Wang, X Li and J Zhang, Phys. Lett. A 372, 417 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    G Akram and N Mahak, Eur. Phys. J. Plus 133, 212 (2018)

    Article  Google Scholar 

  16. 16.

    A Biswas, Optik 171, 217 (2018)

    ADS  Article  Google Scholar 

  17. 17.

    A Biswas and S Arshed, Optik 172, 847 (2018)

    ADS  Article  Google Scholar 

  18. 18.

    O González-Gaxiola and A Biswas, Optik 179, 434 (2019)

    ADS  Article  Google Scholar 

  19. 19.

    A Biswas, M Ekici, A Sonmezoglu and M R Belic, Optik 185, 456 (2019)

    ADS  Article  Google Scholar 

  20. 20.

    A Biswas, M Ekici, A Sonmezoglu and M R Belic, Optik 186, 431 (2019)

    ADS  Article  Google Scholar 

  21. 21.

    E Bessel-Hagen, Math. Ann. 84, 258 (1921)

    MathSciNet  Article  Google Scholar 

  22. 22.

    N H Ibragimov, J. Math. Anal. Appl. 333, 311 (2007)

    MathSciNet  Article  Google Scholar 

  23. 23.

    N H Ibragimov, Preprint Archives of ALGA 4, 55 (2007)

    Google Scholar 

  24. 24.

    E Nother, Gott. Nachr. 2, 235 (1918), (English translation in Transp. Theory Statist. Phys. 1(3), 186 (1971)

  25. 25.

    H Steudel, Z. Naturforsch. A 17, 129 (1962)

    ADS  MathSciNet  Article  Google Scholar 

  26. 26.

    J C S Sampaio and I L Freire, Abs. Appl. Anal. 2014, 804703 (2014)

    Google Scholar 

  27. 27.

    M S Bruźon, M L Gandarias and N H Ibragimov, J. Math. Anal. Appl. 357, 307 (2009)

    MathSciNet  Article  Google Scholar 

  28. 28.

    R Tracinà, M S Bruzón and M L Gandarias, Appl. Math. Comput. 275, 299 (2016)

    MathSciNet  Google Scholar 

  29. 29.

    I L Freire, Appl. Math. Comput. 217, 9467 (2011)

    MathSciNet  Google Scholar 

  30. 30.

    M N Ali, S M Husnine, A Saha, S K Bhowmik, S Dhawan and T Ak, Nonlinear Dyn. 94, 1791 (2018)

    Article  Google Scholar 

  31. 31.

    T Ak, A Saha and S Dhawan, Int. J. Mod. Phys. C 30(4),1950028 (2019)

    ADS  Article  Google Scholar 

  32. 32.

    A E Dubinov, D Yu Kolotkov and M A Sazonkin, Plasma Phys. Rep. 38(10), 833 (2012)

    ADS  Article  Google Scholar 

  33. 33.

    J Tamang and A Saha, Z. Naturforsch. A  74(6), 499 (2019)

    ADS  Article  Google Scholar 

  34. 34.

    R Cherniha, P Broadbridge and L Myroniuk, arXiv:1003.2532v1

  35. 35.

    Z S Feng, J. Phys. A35(2), 343 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  36. 36.

    N H Ibragimov, J. Phys. A 44, 432002 (2011)

    ADS  Article  Google Scholar 

  37. 37.

    M L Gandarias, J. Phys. A 44, 262001 (2011)

    ADS  Article  Google Scholar 

  38. 38.

    S C Anco and G Bluman, Euro. J. Appl. Math. 41, 567 (2002)

    Article  Google Scholar 

  39. 39.

    Y Li, W Shan, T Shuai and K Rao, Math. Prob. Eng. 2015, 408586 (2015)

    Google Scholar 

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Correspondence to Adil Jhangeer.

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Jhangeer, A., Raza, N., Rezazadeh, H. et al. Nonlinear self-adjointness, conserved quantities, bifurcation analysis and travelling wave solutions of a family of long-wave unstable lubrication model. Pramana - J Phys 94, 87 (2020). https://doi.org/10.1007/s12043-020-01961-6

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Keywords

  • Nonlinear self-adjointness
  • bifurcation analysis
  • analytic solutions
  • long-wave unstable lubrication model

PACS Nos

  • 02.20.Sv
  • 02.30.Jr
  • 11.30.−i
  • 47.20.Ky