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On the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities and its application to plasma physics: semi-analytical solution and moving boundary method

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Abstract

In this paper, a strongly nonlinear oscillator equation with higher-order nonlinear (cubic) restoring force, namely damped Duffing equation with higher-order nonlinearities, has been solved and analyzed analytically and numerically using some effectiveness and high-accuracy methods. Firstly, the physical oscillator system consisting of a spring–mass moves under cubic resorting force with constant spring is considered for getting damped Duffing equation. After that a semi-analytical solution is obtained in terms of the Jacobian elliptic function cn. Also, the numerical approximate solution using Runge–Kutta-4th (RK4) procedure is carried out and compared to the semi-analytical solution. It is found that the agreement between both solutions is very good especially if the coefficient of the cubic nonlinear term becomes larger than the coefficient of the linear one. However, we made some improvements to the semi-analytical solution using the numerical moving boundary method to get higher accuracy solution. Furthermore, the error analysis is estimated for both the semi-analytical solution, improved semi-analytical solution, and RK4 solution. The method that is used for getting the semi-analytical solution is elementary, i.e., allows to obtain only integrable case without using any Lie group theory. In another meaning, our approach gives exact/analytical solution for the integrable case such as undamped Duffing equation. Moreover, the t-intercepts and amplitude points are estimated precisely. Furthermore, the plasma oscillations are discussed in the framework of damped Duffing equation. Firstly, we reduced the fluid basic equations of pair-ion plasmas with Maxwellian electrons using reductive perturbation technique to a modified korteweg de-Vries Burgers (mKdVB) equation, and by using traveling wave transformation the mKdVB equation is converted to the damped Duffing equation. After that we compared between all mentioned solutions using plasma parameters.

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References

  1. A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory (Higher Education Press, Beijing, 2009)

    Book  Google Scholar 

  2. A.M. Wazwaz, Partial Differential Equations : Methods and Applications (Balkema, cop, Lisse, 2002)

    MATH  Google Scholar 

  3. X. Lü, Nonlinear Dyn. 81, 239 (2015)

    Article  ADS  Google Scholar 

  4. X. Lü, F. Lin, Commun. Nonlinear Sci. Numer. Simul. 241, 32 (2016)

    Google Scholar 

  5. M.S. Ruderman, Eur. Phys. J. Spec. Top. 185, 57 (2010)

    Article  Google Scholar 

  6. M.S. Ruderman, T. Talipova, E. Pelinovsky, J. Plasma Phys. 74, 639 (2008)

    Article  ADS  Google Scholar 

  7. Q. Xu, W. Fan, Y. Luo, S. Wang, H. Jiang, Am. J. Phys. 87, 116 (2019)

    Article  ADS  Google Scholar 

  8. T.S. Gill, A.S. Bains, C. Bedi, Phys. Plasmas 17, 013701 (2010)

    Article  ADS  Google Scholar 

  9. N. Akhtar, S. Mahmood, N. Jehan, A.M. Mirza, Phys. Plasmas 24, 113707 (2017)

    Article  ADS  Google Scholar 

  10. S.A. El-Tantawy, Phys. Lett. A 381, 787 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  11. S.A. El-Tantawy, Astrophys. Space Sci. 361, 164 (2016)

    Article  ADS  Google Scholar 

  12. H.G.E. Kadji, B.R.N. Nbendjo, J.B.C. Orou, P.K. Talla, Phys. Plasmas 15, 032308 (2008)

    Article  ADS  Google Scholar 

  13. C.H. Miwadinou, L.A. Hinviy, A.V. Monwanouzand, J.B. Chabi Orou, Physics. Fluid Dyn. arXiV: 1308.6132vl

  14. A.E.-Zúñiga, Appl. Math. Model. 37, 2574 (2013)

  15. Z. Petrova, T. Puleva, AIP Conf. Proc. 2048, 040004 (2018). https://doi.org/10.1063/1.5082076

    Article  Google Scholar 

  16. D.S. Ricketts, D. Ham, Electrical Solitons: Theory, Design, and Applications (CRC Press Taylor & Francis Group, Boca Raton, 2010)

    Google Scholar 

  17. S.A. El-Tantawy, T. Aboelenen, S.M.E. Ismaeel, Phys. Plasmas 26, 022115 (2019)

    Article  ADS  Google Scholar 

  18. N. Nasreen, D. Lu, M. Arshad, Optik 161, 221 (2018)

    Article  ADS  Google Scholar 

  19. S. Arshed, L. Arshad, Optik 195, 163077 (2019)

    Article  ADS  Google Scholar 

  20. B.M. Deb, P.K. Chattaraj, Generalized nonlinear Schrödinger equations in quantum fluid dynamics, in Solitons. Springer Series in Nonlinear Dynamics, ed. by M. Lakshmanan (Springer, Berlin, 1988)

    Google Scholar 

  21. L. Cveticanin, G.M. Ismail, Eur. Phys. J. Plus 134, 266 (2019)

    Article  Google Scholar 

  22. H. Trikia, A. Biswasb, Q. Zhou, S.P. Moshokoac, M. Belic, Optik 177, 1 (2019)

    Article  ADS  Google Scholar 

  23. K. Johannesen, Eur. J. Phys. 36, 065020 (2015)

    Article  Google Scholar 

  24. K. Johannesen, Int. J. Appl. Comput. Math. 3, 3805 (2017)

    Article  MathSciNet  Google Scholar 

  25. D.E. Panayotoukanos, N.D. Panayotounakou, A.F. Vakakis, Nonlinear Dyn. 28, 1 (2002)

    Article  Google Scholar 

  26. I. Kovacic, M.J. Brennan, The Duffing Equation: Nonlinear Oscillators and Their Behaviour (Wiley, Hoboken, 2011)

    Book  Google Scholar 

  27. A.H. Salas, J.E. Castillo, Appl. Math. Sci. 8, 8781 (2014)

    Google Scholar 

  28. S.S. Ganji, D.D. Ganji, H. Babazadeh, S. Karimpour, Prog. Electromagn. Res. 4, 23 (2008)

    Article  Google Scholar 

  29. A.M. El-Naggara, G.M. Ismail, Alex. Eng. J. 55, 1581 (2016)

    Article  Google Scholar 

  30. T. Pirbodaghi, S.H. Hoseini, M.T. Ahmadian, G.H. Farrahi, Comput. Math. Appl. 57, 500 (2009)

    Article  MathSciNet  Google Scholar 

  31. D.D. Ganji, M. Gorji, S. Soleimani, M. Esmaeilpour, J. Zhejiang Univ. Sci. A 10, 1263 (2009)

    Article  Google Scholar 

  32. S.K. Lai, C.W. Lim, B.S. Wu, C. Wang, Q.C. Zeng, X.F. He, Appl. Math. Model. 13, 852 (2009)

    Article  Google Scholar 

  33. H. Babazadeh, D.D. Ganji, M. Akbarzade, Prog. Electromag. Res. 4, 143 (2008)

    Article  Google Scholar 

  34. L.B. Ibsen, A. Barari, A. Kimiaeifar, Sadhana 35, 433 (2010)

    Article  Google Scholar 

  35. J.H. He, Appl. Math. Comput. 135, 73 (2003)

    MathSciNet  Google Scholar 

  36. J.H. He, Int. J. Non-Linear Mech. 34, 699 (1999)

    Article  ADS  Google Scholar 

  37. F. Ozen Zengin, M.O. Kaya, S.A. Demirbag, Nonlinear Anal. Real World Appl. 10, 2177 (2009)

Download references

Acknowledgements

This project was supported financially by the Academy of Scientific Research and Technology (ASRT), Egypt, Grant No. (6616).

Author information

Authors and Affiliations

  1. Department of Mathematics, Universidad Nacional de Colombia, Sede Bogotá, 111321, Colombia

    Alvaro H. Salas

  2. Department of Mathematics and Statistics FIZMAKO Research Group, Universidad Nacional de Colombia-Nubia Campus, Manizales, Colombia

    Alvaro H. Salas

  3. Department of Physics, Faculty of Science, Port Said University, Port Said, 42521, Egypt

    S. A. El-Tantawy

  4. Academy of Scientific Research and Technology (ASRT), Cairo, Egypt

    S. A. El-Tantawy

  5. Research Center for Physics (RCP), Department of Physics, Faculty of Science and Arts, Al-Mikhwah, Al-Baha University, Al-Baha, Saudi Arabia

    S. A. El-Tantawy

Authors
  1. Alvaro H. Salas
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  2. S. A. El-Tantawy
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Corresponding author

Correspondence to S. A. El-Tantawy.

Appendix: The appendix for getting system (7)–(9)

Appendix: The appendix for getting system (7)–(9)

Let us rewrite Eqs. (2) and (6) as

$$\begin{aligned} R(t)=x(t)+2\nu ~x(t)+\alpha x(t)+\varepsilon x^{3}(t)\text { : Residual Function,} \end{aligned}$$
(A1)

and

$$\begin{aligned} x(t)=A\exp (-\rho t)\text {cn}(f(t),m(t)). \end{aligned}$$
(A2)

Also, the following relations are defined

$$\begin{aligned} \text {cn}=\text {cn}(f(t),m(t))\text {, sn}=\text {sn}(f(t),m(t))\text {, dn}=\text {dn}(f(t),m(t))\text {, etc.} \end{aligned}$$
(A3)

and

$$\begin{aligned} \text {ea}=E(\text {am}(f(t)|m(t))|m(t))\text { and fa}=F(\text {am} (f(t)|m(t))|m(t)). \end{aligned}$$
(A4)

Taking into account that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\text {cn}(f(t),m(t))=\frac{1}{2}\text {dn}\text {sn}\left( \frac{m^{\prime }(t)\left( -\frac{\text {cn}\text {sn}}{\text {dn}} m(t)+\text {ea}+f(t)(m(t)-1)\right) }{(1-m(t))m(t)}-2f^{\prime }(t)\right) .\nonumber \\ \end{aligned}$$
(A5)

and

$$\begin{aligned} \frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\text {cn}(f(t),m(t))=\frac{\varphi (t)}{4\text {dn} (m(t)-1)^{2}m(t)^{2}}, \end{aligned}$$
(A6)

where

$$\begin{aligned} \varphi (t)&=-4\text {cn}\text {dn}f^{\prime }(t)^{2}\text {sn} ^{2}m(t)^{5}+2\text {cn}^{2}f^{\prime }(t)m^{\prime }(t)\text {sn} ^{3}m(t)^{4}-2f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}{m(t) }^{4}\\&+\, 4\text {cn}\text {dn}^{3}f^{\prime }(t)^{2}m(t) ^{4}+8\text {cn}\text {dn}f^{\prime }(t)^{2}\text {sn}^{2}m(t) ^{4}-4\text {cn}\text {dn}f(t)f^{\prime }(t)m^{\prime }(t)\hbox {sn}^{2}{m(t) }^{4}\\&+\, 4\text {dn}^{2}f^{\prime \prime }(t)\text {sn}m(t)^{4}+\text {cn} ^{2}f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{3} -f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{3}\\&-\, 2\text {cn} ^{2}f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t)^{3}\\&+\, 2\text {dn}^{2}f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t) ^{3}+4f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t)^{3} -8\text {cn}\text {dn}^{3}f^{\prime }(t)^{2}m(t)^{3}\\&-\, 4\text {cn}\text {dn}f^{\prime }(t)^{2}\text {sn}^{2}m(t)^{3}\\&-\, \text {cn}\text {dn}f(t)^{2}\text {m}^{\prime }(t)^{2}\text {sn} ^{2}m(t)^{3}-4\text {cn}\text {dn}\text {ea}f^{\prime }(t)m^{\prime } (t)\hbox {sn}^{2}m(t)^{3}\\&+\, 8\text {cn}\text {dn}f(t)f^{\prime }(t)m^{\prime }(t)\hbox {sn}^{2}m(t)^{3}\\&-\, 2\text {cn}\text {dn}m^{\prime \prime }(t)\hbox {sn}^{2}m(t)^{3} +4\text {cn}\text {dn}^{3}f(t)f^{\prime }(t)m^{\prime }(t)m(t) ^{3}-8\text {dn}^{2}f^{\prime \prime }(t)\text {sn}m(t)^{3}\\&-\, 6\text {cn}^{2}\text {dn}^{2}f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t) ^{3}+2\text {dn}^{2}f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)^{3} +2\text {dn}^{2}f(t)m^{\prime \prime }(t)\hbox {sn}m(t)^{3}\\&-\, \text {cn}\text {dn}m^{\prime }(t)^{2}\text {sn}^{4}m(t) ^{2}+\text {cn}^{2}\text {ea}m^{\prime }(t)^{2}\text {sn}^{3}m(t) ^{2}-\text {ea}m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{2}\\&-\, \text {cn} ^{2}f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{2}\\&+\, \text {dn}^{2}f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t) ^{2}+2f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{2} -2\text {dn}^{2}f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t)^{2}\\&-\, 2f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t)^{2}+4\text {cn} \text {dn}^{3}f^{\prime }(t)^{2}m(t)^{2}+\text {cn} \text {dn}^{3}f(t)^{2}\text {m}^{\prime }(t)^{2}m(t) ^{2} \\&+\, 2\text {cn}\text {dn}f(t)^{2}\text {m}^{\prime }(t)^{2}\text {sn} ^{2}m(t)^{2}\\&+\, 2\text {cn}^{3}\text {dn}m^{\prime }(t)^{2}\text {sn}^{2}m(t) ^{2}+2\text {cn}\text {dn}m^{\prime }(t)^{2}\text {sn}^{2}m(t) ^{2}-2\text {cn}\text {dn}\text {ea}f(t)m^{\prime }(t)^{2}\text {sn} ^{2}m(t)^{2}\\&+\, 4\text {cn}\text {dn}\text {ea}f^{\prime }(t)m^{\prime }(t)\hbox {sn}^{2}m(t) ^{2}-4\text {cn}\text {dn}f(t)f^{\prime }(t)m^{\prime }(t) \hbox {sn}^{2}m(t)^{2}+2\text {cn}\text {dn}m^{\prime \prime }(t)\hbox {sn}^{2}m(t)^{2}\\&+\, 4\text {cn}\text {dn}^{3}\text {ea}f^{\prime }(t)m^{\prime }(t)m(t) ^{2}-8\text {cn}\text {dn}^{3}f(t)f^{\prime }(t)m^{\prime }(t)m(t) ^{2}-3\text {cn}^{2}\text {dn}^{2}f(t)m^{\prime }(t)^{2}\text {sn}m(t) ^{2}\\&-\, 2\text {dn}^{2}f(t)m^{\prime }(t)^{2}\text {sn}m(t)^{2} +4\text {dn}^{2}f^{\prime \prime }(t)\text {sn}m(t)^{2}+2\text {dn} ^{4}{f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)}^{2}\\&+\, 6\text {cn}^{2}\text {dn}^{2}{f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)} ^{2}-4\text {dn}^{2}{f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)}^{2} +2\text {dn}^{2}\text {ea}m^{\prime \prime }(t)\hbox {sn}m(t)^{2}\\&-\, 4\text {dn}^{2}{f(t)m^{\prime \prime }(t)\hbox {sn}m(t)}^{2}+\text {dn} ^{2}\text {ea}m^{\prime }(t)^{2}\text {sn}^{3}m(t)+\text {ea}m^{\prime }(t)^{2}\text {sn}^{3}m(t)\\&-\, \text {dn}^{2}{f(t)m^{\prime }(t) }^{2}\text {sn}^{3}m(t)\\&-\, f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)-2\text {cn} \text {dn}^{3}f(t)^{2}\text {m}^{\prime }(t)^{2}{m(t) }+2\text {cn}\text {dn}^{3}\text {ea}f(t)m^{\prime }(t)^{2}{m(t)}\\&-\, \text {cn}\text {dn}^{3}\text {m}^{\prime }(t)^{2}\text {sn}^{2}{m(t) }\\&-\, \text {cn}\text {dn}\text {ea}^{2}\text {m}^{\prime }(t)^{2}\text {sn} ^{2}m(t)-\text {cn}\text {dn}f(t)^{2}\text {m}^{\prime }(t) ^{2}\text {sn}^{2}m(t)+2\text {cn}\text {dn}\text {ea}f(t)m^{\prime }(t) ^{2}\text {sn}^{2}m(t)\\&-\, 4\text {cn}\text {dn}^{3}\text {ea}f^{\prime }(t)m^{\prime }(t)m(t) +4\text {cn}\text {dn}^{3}f(t)f^{\prime }(t)m^{\prime }(t)m(t) -3\text {cn}^{2}\text {dn}^{2}\text {ea}m^{\prime }(t)^{2}\text {sn}m(t)\\&-\, 3\text {dn}^{2}\text {ea}m^{\prime }(t)^{2}\text {sn}m(t)-\text {dn} ^{2}\text {ef}m^{\prime }(t)^{2}\text {sn}m(t)+\text {dn}^{4} f(t)m^{\prime }(t)^{2}\text {sn}m(t)\\&+\, 3\text {cn}^{2}\text {dn} ^{2}f(t)m^{\prime }(t)^{2}\text {sn}m(t)\\&+\, 4\text {dn}^{2}f(t)m^{\prime }(t)^{2}\text {sn}m(t)-2\text {dn} ^{4}{f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)}+2\text {dn}^{2}{f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)}\\&-\, 2\text {dn}^{2}\text {ea}m^{\prime \prime } (t)\hbox {sn}m(t)\\&+\, 2\text {dn}^{2}{f(t)m^{\prime \prime }(t)\hbox {sn}m(t)}+\text {cn} \text {dn}^{3}\text {ea}^{2}\text {m}^{\prime }(t)^{2}+\text {cn}\text {dn} ^{3}f(t)^{2}\text {m}^{\prime }(t)^{2}-2\text {cn}\text {dn} ^{3}\text {ea}f(t)m^{\prime }(t)^{2}\\&+\, \text {dn}^{4}\text {ea}m^{\prime }(t)^{2}\text {sn}+\text {d}n^{2} \text {ea}m^{\prime }(t)^{2}sn+dn^{2}efm^{\prime }(t)^{2}sn-dn^{4}f(t)m^{\prime }(t)^{2}sn\\&-\,2dn^{2}f(t)m^{\prime }(t)^{2}sn. \end{aligned}$$

We may write

$$\begin{aligned} R(t)=\frac{1}{4\text {dn}(m(t)-1)^{2}m(t)^{2}}e^{-3\rho t}\tilde{R}(t)\text {,} \end{aligned}$$
(A7)

where

$$\begin{aligned} \tilde{R}(t)&=4\text {cn}\text {dn}f^{\prime }(t)^{2}\text {sn} ^{2}m(t)^{5}-2\text {cn}^{2}f^{\prime }(t)m^{\prime }(t)\text {sn} ^{3}m(t)^{4}+2f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}{m(t) }^{4}\\&-\, 4\text {cn}\text {dn}^{3}f^{\prime }(t)^{2}m(t) ^{4}-8\text {cn}\text {dn}f^{\prime }(t)^{2}\text {sn}^{2}m(t) ^{4}+4\text {cn}\text {dn}f(t)f^{\prime }(t)m^{\prime }(t)\hbox {sn}^{2}{m(t) }^{4}\\&-\, 4\text {dn}^{2}f^{\prime \prime }(t)\text {sn}m(t)^{4}-4\text {cn} \text {dn}f(t)f^{\prime }(t)m^{\prime }(t)\hbox {sn}^{2}m(t)^{4} -4\text {dn}^{2}f^{\prime \prime }(t)\text {sn}m(t)^{4}\\&-\, \text {cn}^{2}f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t) ^{3}+f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{3} +2\text {cn}^{2}f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t)^{3}\\&-\, 2\text {dn}^{2}f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t) ^{3}-4f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t)^{3} +8\text {cn}\text {dn}^{3}f^{\prime }(t)^{2}m(t)^{3}\\&+\, 4\text {cn}\text {dn}f^{\prime }(t)^{2}\text {sn}^{2}m(t) ^{3}+\text {cn}\text {dn}f(t)^{2}\text {m}^{\prime }(t)^{2}\text {sn} ^{2}m(t)^{3}+4\text {cn}\text {dn}\text {ea}f^{\prime }(t)m^{\prime } (t)\hbox {sn}^{2}m(t)^{3}\\&-\, 8\text {cn}\text {dn}f(t)f^{\prime }(t)m^{\prime }(t)\hbox {sn}^{2}m(t) ^{3}+2\text {cn}\text {dn}m^{\prime \prime }(t)\hbox {sn}^{2}m(t) ^{3}-4\text {cn}\text {dn}^{3}f(t)f^{\prime }(t)m^{\prime }(t)m(t)^{3}\\&+\, 8\text {dn}^{2}f^{\prime \prime }(t)\text {sn}m(t)^{3}+6\text {cn} ^{2}\text {dn}^{2}f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)^{3} -2\text {dn}^{2}{f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)}^{3}\\&-\, 2\text {dn}^{2}f(t)m^{\prime \prime }(t)\hbox {sn}m(t)^{3}+\text {cn} \text {dn}m^{\prime }(t)^{2}\text {sn}^{4}m(t)^{2}-\text {cn} ^{2}\text {ea}m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{2}\\&+\, \text {ea}m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{2}+\text {cn} ^{2}f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{2}-\text {dn} ^{2}f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{2}\\&-\, 2f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)^{2} +2\text {dn}^{2}f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t) ^{2}+2f^{\prime }(t)m^{\prime }(t)\text {sn}^{3}m(t)^{2}\\&-\, 4\text {cn}\text {dn}^{3}f^{\prime }(t)^{2}m(t) ^{2}-\text {cn}\text {dn}^{3}f(t)^{2}\text {m}^{\prime }(t) ^{2}m(t)^{2}-2\text {cn}\text {dn}f(t)^{2}\text {m}^{\prime }(t) ^{2}\text {sn}^{2}m(t)^{2}\\&-\, 2\text {cn}^{3}\text {dn}m^{\prime }(t)^{2}\text {sn}^{2}m(t) ^{2}-2\text {cn}\text {dn}m^{\prime }(t)^{2}\text {sn}^{2}m(t) ^{2}+2\text {cn}\text {dn}\text {ea}f(t)m^{\prime }(t)^{2}\text {sn} ^{2}m(t)^{2}\\&-\, 4\text {cn}\text {dn}\text {ea}f^{\prime }(t)m^{\prime }(t)\hbox {sn}^{2}{m(t) }^{2}+4\text {cn}\text {dn}f(t)f^{\prime }(t)m^{\prime }(t)\hbox {sn}^{2}{m(t) }^{2}-2\text {cn}\text {dn}m^{\prime \prime }(t)\hbox {sn}^{2}m(t)^{2}\\&-\, 4\text {cn}\text {dn}^{3}\text {ea}f^{\prime }(t)m^{\prime }(t)m(t) ^{2}+8\text {cn}\text {dn}^{3}f(t)f^{\prime }(t)m^{\prime }(t)m(t) ^{2}+3\text {cn}^{2}\text {dn}^{2}f(t)m^{\prime }(t)^{2}\text {sn}m(t) ^{2}\\&+\, 2\text {dn}^{2}f(t)m^{\prime }(t)^{2}\text {sn}m(t)^{2} -4\text {dn}^{2}f^{\prime \prime }(t)\text {sn}m(t)^{2}-2\text {dn} ^{4}f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)^{2}\\&-\, 6\text {cn}^{2}\text {dn}^{2}{f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)} ^{2}+4\text {dn}^{2}{f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)}^{2} -2\text {dn}^{2}\text {ea}m^{\prime \prime }(t)\hbox {sn}m(t)^{2}\\&+\, 4\text {dn}^{2}f(t)m^{\prime \prime }(t)\hbox {sn}m(t)^{2}-\text {dn} ^{2}\text {ea}m^{\prime }(t)^{2}\text {sn}^{3}m(t)-\text {ea}m^{\prime }(t)^{2}\text {sn}^{3}m(t)\\&+\, \text {dn}^{2}f(t)m^{\prime }(t) ^{2}\text {sn}^{3}m(t)\\&+\, f(t)m^{\prime }(t)^{2}\text {sn}^{3}m(t)+2\text {cn} \text {dn}^{3}f(t)^{2}\text {m}^{\prime }(t)^{2}{m(t) }-2\text {cn}\text {dn}^{3}\text {ea}f(t)m^{\prime }(t)^{2}{m(t)}\\&+\,\text {cn}\text {dn}^{3}\text {m}^{\prime }(t)^{2}\text {sn}^{2}{m(t) }\\&+\, \text {cn}\text {dn}\text {ea}^{2}\text {m}^{\prime }(t)^{2}\text {sn} ^{2}m(t)+\text {cn}\text {dn}f(t)^{2}\text {m}^{\prime }(t) ^{2}\text {sn}^{2}m(t)-2\text {cn}\text {dn}\text {ea}f(t)m^{\prime }(t) ^{2}\text {sn}^{2}m(t)\\&+\, 4\text {cn}\text {dn}^{3}\text {ea}f^{\prime }(t)m^{\prime }(t)m(t) -4\text {cn}\text {dn}^{3}f(t)f^{\prime }(t)m^{\prime }(t)m(t) +3\text {cn}^{2}\text {dn}^{2}\text {ea}m^{\prime }(t)^{2}\text {sn}m(t)\\&+\, 3\text {dn}^{2}\text {ea}m^{\prime }(t)^{2}\text {sn}m(t)+\text {dn} ^{2}\text {ef}m^{\prime }(t)^{2}\text {sn}m(t)-\text {dn}^{4} f(t)m^{\prime }(t)^{2}\text {sn}m(t)\\&-\, 3\text {cn}^{2}\text {dn} ^{2}f(t)m^{\prime }(t)^{2}\text {sn}m(t)\\&-\, 4\text {dn}^{2}f(t)m^{\prime }(t)^{2}\text {sn}m(t)+2\text {dn} ^{4}{f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)}-2\text {dn}^{2}f^{\prime }(t)m^{\prime }(t)\hbox {sn}m(t)\\&+\, 2\text {dn}^{2}\text {ea}m^{\prime \prime } (t)\hbox {sn}m(t)\\&-\, 2\text {dn}^{2}{f(t)m^{\prime \prime }(t)\hbox {sn}m(t)}-\text {cn} \text {dn}^{3}\text {ea}^{2}\text {m}^{\prime }(t)^{2}-\text {cn}\text {dn} ^{3}f(t)^{2}\text {m}^{\prime }(t)^{2}+2\text {cn}\text {dn} ^{3}\text {ea}f(t)m^{\prime }(t)^{2}\\&-\, \text {dn}^{4}\text {ea}m^{\prime }(t)^{2}\text {sn}-\text {dn}^{2} \text {ea}m^{\prime }(t)^{2}\text {sn}-\text {dn}^{2}\text {ef}m^{\prime }(t) ^{2}\text {sndn}^{4}f(t)m^{\prime }(t)^{2}\text {sn}+2\text {dn} ^{2}f(t)m^{\prime }(t)^{2}\text {sn.} \end{aligned}$$

Equating to zero the coefficients of the Jacobian elliptic functions as well as the coefficients of ea and fa give the following system of algebraic-differential equations

$$\begin{aligned}&4m(t)^{2}\left( A^{2}\varepsilon +e^{2\rho t}\left( \alpha -{f^{\prime }(t)}^{2}+\rho (\rho -2\nu )\right) \right) =f(t){m^{\prime }(t)}e^{2\rho t}(f\text {m}^{\prime }(t)+4f^{\prime }(t)m(t)), \end{aligned}$$
(A8)
$$\begin{aligned}&m(t)(m(t)(-2f(t)(2\text {m}^{\prime }(t)(\nu -\rho )+{m^{\prime \prime }(t)})+ \end{aligned}$$
(A9)
$$\begin{aligned}&4f^{\prime }(t)(2\nu +\text {m}^{\prime }(t)-2\rho )+4f^{\prime \prime }(t))+f(t)(\text {m}^{\prime }(t)(4\nu +5\text {m}^{\prime }(t)-4\rho )+2m^{\prime \prime }(t)))\nonumber \\&=3f\text {m}^{\prime }(t)^{2}+4m(t)^{3}(2f^{\prime }(t)(\nu -\rho )+f^{\prime \prime }(t)).\nonumber \\&\text {m}^{\prime }(t)((3m-1)\text {m}^{\prime }(t)-2(m(t)-1)m(t)(\nu -\rho ))=(m(t)-1)m(t){m^{\prime \prime }(t),} \end{aligned}$$
(A10)
$$\begin{aligned}&4A^{2}\varepsilon (m(t)-1)^{2}m(t)\nonumber \\&=e^{2\rho t}\left( \begin{array} [c]{c} 2m(t)^{2}\left( \text {m}^{\prime }(t)\left( f(t)^{2}{m^{\prime }(t)}+2\nu -2\rho \right) -8f(t){f^{\prime }(t)m^{\prime }(t)} +4f^{\prime }(t)^{2}+m^{\prime \prime }(t)\right) \\ -\,4m\text {m}^{\prime }(t)\left( \left( f(t)^{2}+1\right) {m^{\prime }(t)}-2f(t)f^{\prime }(t)+\nu -\rho \right) \\ +\,\left( 2f(t)^{2}+1\right) \text {m}^{\prime }(t)^{2}+8{f^{\prime }(t)}m(t)^{3}(f\text {m}^{\prime }(t)-2f^{\prime }(t))+8{f^{\prime }(t)} m(t)^{4}-2m(t)m^{\prime \prime }(t), \end{array} \right) \end{aligned}$$
(A11)
$$\begin{aligned}&m(t)^{2}(-2f(2\text {m}^{\prime }(t)(\nu -\rho )+{m^{\prime \prime }(t) })+4f^{\prime }(t)(2\nu +\text {m}^{\prime }(t)-2\rho )+4f^{\prime \prime }(t))\nonumber \\&+\, m(t)(\text {m}^{\prime }(t)(f(t)(4\nu +5\text {m}^{\prime }(t) -4\rho )+8f^{\prime }(t))+2fm^{\prime \prime }(t) )+f\text {m}^{\prime }(t)^{2}\nonumber \\&=4m(t)^{3}(2f^{\prime }(t)(\nu -\rho )+f^{\prime \prime }(t)), \end{aligned}$$
(A12)
$$\begin{aligned}&\text {m}^{\prime }(t)((3m(t)+1)\text {m}^{\prime }(t)-2(m(t)-1)m(t)(\nu -\rho ))=(m(t)-1)m(t)m^{\prime \prime }(t). \end{aligned}$$
(A13)

Now, using Mathematica Eliminate command, we get

$$\begin{aligned} A\varepsilon m(t)\left( A^{2}\varepsilon +e^{2\rho t}\left( \alpha -f^{\prime }(t)^{2}+\rho (\rho -2\nu )\right) \right)&=0,\nonumber \\ (m(t)-1)m(t)\left( 2f^{\prime }(t)^{2}m(t)e^{2\rho t}-A^{2}\varepsilon \right)&=0,\nonumber \\ \text {m}^{\prime }(t)&=0,\nonumber \\ (m(t)-1)m(t)(2f^{\prime }(t)(\nu -\rho )+f^{\prime \prime }(t))&=0,\nonumber \\ (m(t)-1)m(t)m^{\prime \prime }(t)&=0,\nonumber \\ m(t)m^{\prime \prime }(t)\left( A^{2}\varepsilon +e^{2\rho t}\left( \alpha -f^{\prime }(t)^{2}+\rho (\rho -2\nu )\right) \right)&=0. \end{aligned}$$
(A14)

According to the first two equations and with the help of \(f(0)=0\), we obtain

$$\begin{aligned} f(t)=\frac{1}{\rho }\left\{ \begin{array} [c]{c} \sqrt{\alpha +A^{2}\varepsilon -2\nu \rho +\rho ^{2}}-\sqrt{\alpha +A^{2}\varepsilon e^{-2\rho t}+\rho (\rho -2\nu )}\\ +\sqrt{\alpha +\rho (\rho -2\nu )}\left[ \begin{array} [c]{c} \tanh ^{-1}\left( \frac{\sqrt{\alpha +A^{2}\varepsilon e^{-2\rho t}-2\nu \rho +\rho ^{2}}}{\sqrt{\alpha -2\nu \rho +\rho ^{2}}}\right) \\ -\tanh ^{-1}\left( \frac{\sqrt{\alpha +A^{2}\varepsilon -2\nu \rho +\rho ^{2}} }{\sqrt{\alpha -2\nu \rho +\rho ^{2}}}\right) \end{array} \right] , \end{array} \right\} \end{aligned}$$
(A15)

and

$$\begin{aligned} m(t)=\frac{A^{2}\varepsilon }{2f^{\prime }(t)^{2}e^{2\rho t}}=\frac{A^{2}\varepsilon e^{\rho t}}{2A^{2}\varepsilon +2e^{2\rho t}(\alpha +\rho (\rho -2\nu ))}. \end{aligned}$$
(A16)

These are precisely the expressions for (8) and (9) in the manuscript.

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Salas, A.H., El-Tantawy, S.A. On the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities and its application to plasma physics: semi-analytical solution and moving boundary method. Eur. Phys. J. Plus 135, 833 (2020). https://doi.org/10.1140/epjp/s13360-020-00829-3

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