Skip to main content

Exact and numerical solutions for the nanosoliton of ionic waves propagating through microtubules in living cells


In this article, the Paul–Painleve approach (PPA) which was formulated recently and built on the balance role has been used perfectly to achieve new impressive solitary wave solutions to the nanosoliton of ionic waves (NSOIW) propagating along the microtubules in the living cells. In addition, variational iteration method (VIM) has been applied in the same vein and parallel to establish numerical solutions of this model.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. N A Kudryashov, Optik. 183, 642 (2019)

    Article  ADS  Google Scholar 

  2. H Fröhlich, Int. J. Quantum Chem. 2641, 153 (1968)

    Google Scholar 

  3. M V Sataric, J A Tuszyosky and R B Zakula, Phys. Rev. E 48, 589 (1993)

    Article  ADS  Google Scholar 

  4. M V Sataric and J A Tuszynski, J. Biol. Phys. 31, 487 (2005)

    Article  Google Scholar 

  5. M V Sataric, M Dragic and D Sejulic, Rom. Rep. Phys. 63, 624 (2011)

  6. A M Wazwaz, Appl. Math. Comput. 212, 120 (2009)

    MathSciNet  Google Scholar 

  7. S S Ray and A K Gupta, J. Math. Chem. 52, 1066 (2014)

    Article  MathSciNet  Google Scholar 

  8. S S Ray, Numer. Meth. Partial Diff. Eqs 37, 341 (2021)

    Article  Google Scholar 

  9. N Raza, M S Osman, A-H Abdel-Aty and H R Besbes, Adv. Diff. Eqs 2020, 517 (2020)

    Article  Google Scholar 

  10. N Raza, S Arshed and A Javid, Int. J. Nonlinear Sci. Numer. Simul. 21, 855 (2020)

    Article  MathSciNet  Google Scholar 

  11. Md A Kayum, S Ara, H K Barman and M Ali Akbar, Results Phys. 18, 103269 (2020)

  12. H K Barman, A R Seadawy, M Ali Akbar and D Baleanu, Results Phys. 17, 103131 (2020)

    Article  Google Scholar 

  13. H M E Zahran, World J. Nano Sci. Eng. 5, 78 (2015)

    Article  ADS  Google Scholar 

  14. H M E Zahran, Adv. Nanopart. 4, 25 (2015)

    Article  Google Scholar 

  15. A H Abdel Aty, M A Khater, R A M Attia and H Eleuch, Mathematics 8, 697 (2020)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ahmet Bekir.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bekir, A., Zahran, E.H.M. Exact and numerical solutions for the nanosoliton of ionic waves propagating through microtubules in living cells. Pramana - J Phys 95, 158 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: