Skip to main content
SpringerLink
Log in

A new sensitive visualization, solitary wave profiles and conservation laws of ion sound waves arising in plasma

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

The current study covers the traveling wave structures of a fourth-order nonlinear symmetric regularized long-wave (SRLW) equation which emerges in a few actual applications incorporating particle sound waves in plasma using the new extended direct algebraic method. This model depicts nonlinear particle acoustic and space-charge waves. This model admits the two-dimensional Lie algebra. We have computed the optimal system and reduced our given model corresponding to each element of the optimal system. After that, by using a proposed method, we obtained the solitary wave profiles of the model under investigation. By using the concept of nonlinear self-adjointness, we have shown that our given model is not self-adjoint. Moreover, we have computed the conserved quantities corresponding to each element of the Lie algebra. The obtained solutions are in different classes of solitary wave solutions with dark, combined dark-bright, singular, periodic-singular, combined dark-singular, combined singular, dark-singular combo solitons, and rational solutions. Furthermore, existing constraints for the resultant solutions are reported. Also, a graphical interpretation of some particular solutions is shown by considering the specific values of the parameters. In the end, the sensitive visualization of the model is done by considering different initial states. We are confident that these outstanding results will offer insightful information and further the investigation of other evolutionary mechanisms connected to the equation under study.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References

  • Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  • Akinyemi, L., Rezazadeh, H., Yao, S.W., Akbar, M.A., Khater, M.M., Jhangeer, A., Ahmad, H.: Nonlinear dispersion in parabolic law medium and its optical solitons. Results Phys. 26, 104411 (2021)

    Article  Google Scholar 

  • Ali, K.K., Nuruddeen, R.I., Raslan, K.R.: New structures for the space-time fractional simplified MCH and SRLW equations. Chaos, Solitons Fractals 106, 304–309 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  • Almusawa, H., Jhangeer, A., Munawar, M.: Analytical analyses for a fractional low-pass electrical transmission line model with dynamic transition. Symmetry 14(7), 1377 (2022)

    Article  ADS  Google Scholar 

  • Bekir, A., Cevikel, A.C.: New exact traveling wave solutions of nonlinear physical models. Chaos, Solitons Fractals 41(4), 1733–1739 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  • Bluman, G., Anco, S.: Symmetry and Integration Methods for Differential Equations, vol. 154. Springer, Berlin (2008)

    Google Scholar 

  • Bulut, H., Baskonus, H.M., Cuvelek, E.: On the prototype solutions of symmetric regularized long wave equation by generalized Kudryashov method. Math. Lett. 1(2), 10–16 (2015)

    Google Scholar 

  • Bulut, H., Sulaiman, T.A., Erdogan, F., Mehmet Baskonus, H.: On the new hyperbolic and trigonometric structures to the simplified MCH and SRLW equations. Eur. Phys. J. Plus 132, 1–12 (2017)

    Article  Google Scholar 

  • Cantwell, B.J.: Introduction to Symmetry Analysis. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  • Demiray, H.: An application of modified reductive perturbation method to symmetric regularized-long-wave. TWMS J. Appl. Eng. Math. 1(1), 49–57 (2011)

    MathSciNet  Google Scholar 

  • Faridi, W.A., Bakar, M.A., Akgul, A., Abd El-Rahman, M., El Din, S.M.: Exact fractional soliton solutions of thin-film ferroelectric material equation by analytical approaches. Alex. Eng. J. 78, 483–497 (2023)

    Article  Google Scholar 

  • Faridi, W.A., Bakar, M.A., Myrzakulova, Z., Myrzakulov, R., Akgül, A., El Din, S.M.: The formation of solitary wave solutions and their propagation for Kuralay equation. Results Phys. 52, 106774 (2023)

    Article  Google Scholar 

  • Fatima, S., Abbas, N., Munawar, M., Eldin, S.M.: Ion-acoustic wave dynamics and sensitivity study in a magnetized Auroral phase plasma. Math. Open 2, 2350003 (2023)

    Article  Google Scholar 

  • Fatima, S., Abbas, N., Muhammad, S.: Dynamical features and sensitivity visualization of thin-film Polarisation equation. Physica Scr. 98, 115248 (2023)

    Article  ADS  Google Scholar 

  • Gandarias, M.L.: Weak self-adjoint differential equations. J. Phys. A Math. Theor. 44(26), 262001 (2011)

    Article  ADS  Google Scholar 

  • Heris, J.M., Zamanpour, I.: Exact traveling wave solutions of the symmetric regularized long wave (SRLW) using analytical methods. Stat. Optim. Inf. Comput. 2(1), 47–55 (2014)

    MathSciNet  Google Scholar 

  • Hosseini, K., Mirzazadeh, M., Salahshour, S., Baleanu, D., Zafar, A.: Specific wave structures of a fifth-order nonlinear water wave equation. J. Ocean Eng. Sci. 7(5), 462–466 (2022)

    Article  Google Scholar 

  • Hussain, A., Usman, M., Zaman, F.D., Eldin, S.M.: Symmetry analysis and invariant solutions of Riabouchinsky Proudman Johnson equation using optimal system of Lie subalgebras. Results Phys. 49, 106507 (2023)

    Article  Google Scholar 

  • Hussain, A., Usman, M., Zaman, F.D., Eldin, S.M.: Double reductions and traveling wave structures of the generalized Pochhammer-Chree equation. Partial Differ. Equ. Appl. Math. 7, 100521 (2023)

    Article  Google Scholar 

  • Hussain, A., Ali, H., Zaman, F., Abbas, N.: New closed form solutions of some nonlinear pseudo-parabolic models via a new extended direct algebraic method. Int. J. Math. Comput. Eng. 2(1), 35–58 (2023)

    Article  Google Scholar 

  • Hussain, A., Chahlaoui, Y., Zaman, F.D., Parveen, T., Hassan, A.M.: The Jacobi elliptic function method and its application for the stochastic NNV system. Alex. Eng. J. 81, 347–359 (2023)

    Article  Google Scholar 

  • Hussain, A., Chahlaoui, Y., Usman, M., Zaman, F.D., Park, C.: Optimal system and dynamics of optical soliton solutions for the Schamel KdV equation. Sci. Rep. 13(1), 15383 (2023)

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  • Hussain, A., Kara, A.H., Zaman, F.D.: An invariance analysis of the Vakhnenko-Parkes Equation. Chaos Solitons Fractals 171, 113423 (2023)

    Article  MathSciNet  Google Scholar 

  • Hussain, A., Kara, A.H., Zaman, F.D.: Symmetries, associated first integrals and successive reduction of Schrödinger type and other second order difference equations. Optik 287, 171080 (2023)

    Article  ADS  CAS  Google Scholar 

  • Hussain, A., Usman, M., Al-Sinan, B.R., Osman, W.M., Ibrahim, T.F.: Symmetry analysis and closed-form invariant solutions of the nonlinear wave equations in elasticity using optimal system of Lie subalgebra. Chin. J. Phys. 83(1), 1–13 (2023)

    Article  MathSciNet  Google Scholar 

  • Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333(1), 311–328 (2007)

    Article  MathSciNet  Google Scholar 

  • Ibragimov, N.H.: Quasi-self-adjoint differential equations. Arch. ALGA 4, 55–60 (2007)

    Google Scholar 

  • Ibragimov, N.H.: Nonlinear self-adjointness and conservation laws. J. Phys. A Math. Theor. 44(43), 432002 (2011)

    Article  ADS  Google Scholar 

  • Jhangeer, A., Munawar, M., Riaz, M.B., Baleanu, D.: Construction of traveling waves patterns of (1+ n)-dimensional modified Zakharov-Kuznetsov equation in plasma physics. Results Phys. 19, 103330 (2020)

    Article  Google Scholar 

  • Jhangeer, A., Raza, N., Rezazadeh, H., Seadawy, A.: Nonlinear self-adjointness, conserved quantities, bifurcation analysis and traveling wave solutions of a family of long-wave unstable lubrication model. Pramana 94, 1–9 (2020)

    Article  Google Scholar 

  • Jhangeer, A., Munawar, M., Atangana, A., Riaz, M.B.: Analysis of electron acoustic waves interaction in the presence of homogeneous unmagnetized collision-free plasma. Physica Scr. 96(7), 075603 (2021)

    Article  ADS  Google Scholar 

  • Jhangeer, A., Munawar, M., Atangana, A., Riaz, M.B.: Analysis of electron acoustic waves interaction in the presence of homogeneous unmagnetized collision-free plasma. Physica Scr. 96(7), 075603 (2021)

    Article  ADS  Google Scholar 

  • Jhangeer, A., Rezazadeh, H., Seadawy, A.: A study of traveling, periodic, quasiperiodic and chaotic structures of perturbed Fokas-Lenells model. Pramana 95, 1–11 (2021)

    Article  ADS  Google Scholar 

  • Khater, M.M.: Abundant and accurate computational wave structures of the nonlinear fractional biological population model. Int. J. Modern Phys. B 37(18), 2350176 (2023)

    Article  ADS  Google Scholar 

  • Khater, M.M.: Hybrid accurate simulations for constructing some novel analytical and numerical solutions of three-order GNLS equation. Int. J. Geom. Methods Modern Phys. 20, 2350159 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  • Khater, M.M.: Physics of crystal lattices and plasma; analytical and numerical simulations of the Gilson-Pickering equation. Results Phys. 44, 106193 (2023)

    Article  Google Scholar 

  • Khater, M.M.: Novel computational simulation of the propagation of pulses in optical fibers regarding the dispersion effect. Int. J. Modern Phys. B 37(09), 2350083 (2023)

    Article  ADS  Google Scholar 

  • Liu, J.G., Yang, X.J.: Symmetry group analysis of several coupled fractional partial differential equations. Chaos, Solitons Fractals 173, 113603 (2023)

    Article  MathSciNet  Google Scholar 

  • Liu, J.G., Yang, X.J., Geng, L.L., Yu, X.J.: On fractional symmetry group scheme to the higher-dimensional space and time fractional dissipative Burgers equation. Int. J. Geom. Methods Mod. Phys. 19(11), 2250173 (2022)

    Article  MathSciNet  Google Scholar 

  • Mhlanga, I.E., Khalique, C.M.: Exact solutions of the symmetric regularized long wave equation and the Klein-Gordon-Zakharov equations. Abstract and Applied Analysis, vol. 2014. Hindawi, London (2014)

    Google Scholar 

  • Munawar, M., Jhangeer, A., Pervaiz, A., Ibraheem, F.: New general extended direct algebraic approach for optical solitons of Biswas-Arshed equation through birefringent fibers. Optik 228, 165790 (2021)

    Article  ADS  Google Scholar 

  • Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, Berlin (1993)

    Book  Google Scholar 

  • Ovsyannikov, L.V.: Lectures on the Theory of Group Properties of Differential Equations. World Scientific Publishing Company, Singapore (2013)

    Book  Google Scholar 

  • Raza, N., Jhangeer, A., Arshed, S., Butt, A.R., Chu, Y.M.: Dynamical analysis and phase portraits of two-mode waves in different media. Results Phys. 19, 103650 (2020)

    Article  Google Scholar 

  • Raza, N., Jhangeer, A., Rahman, R.U., Butt, A.R., Chu, Y.M.: Sensitive visualization of the fractional Wazwaz-Benjamin-Bona-Mahony equation with fractional derivatives: a comparative analysis. Results Phys. 25, 104171 (2021)

    Article  Google Scholar 

  • Riaz, M.B., Jhangeer, A., Atangana, A., Awrejcewicz, J., Munawar, M.: Supernonlinear wave, associated analytical solitons, and sensitivity analysis in a two-component Maxwellian plasma. J. King Saud Univ. Sci. 34(5), 102108 (2022)

    Article  Google Scholar 

  • Seadawy, A.R., Iqbal, M., Lu, D.: Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev-Petviashvili modified equal width dynamical equation. Comput. Math. Appl. 78(11), 3620–3632 (2019)

    Article  MathSciNet  Google Scholar 

  • Seadawy, A.R., Iqbal, M., Lu, D.: Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg-de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma. Physica A Stat. Mech. Appl. 544, 123560 (2020)

    Article  Google Scholar 

  • Senol, M.: New analytical solutions of fractional symmetric regularized-long-wave equation. Rev. Mex. Fis. 66(3), 297–307 (2020)

    Article  MathSciNet  Google Scholar 

  • Seyler, C.E., Fenstermacher, D.L.: A symmetric regularized-long-wave equation. Phys. Fluids 27(1), 4–7 (1984)

    Article  ADS  Google Scholar 

  • Tian, B., Gao, Y.T.: Symbolic computation and observable effect for the (2+ 1)-dimensional symmetric regularized-long-wave equation from strongly magnetized cold-electron plasmas. Comput. Math. Appl. 45(4–5), 731–735 (2003)

    Article  MathSciNet  Google Scholar 

  • Ugurlu, Y., Dogan, K.A.Y.A.: Generalized Jacobi elliptic function method for periodic wave solutions of SRLW equation and \((1+1)\)-dimensional dispersive long wave equation. Cankaya Univ. J. Sci. Eng., 8(2) (2011)

  • Usman, M., Hussain, A., Zaman, F.D., Eldin, S.M.: Symmetry analysis and exact Jacobi elliptic solutions for the nonlinear couple Drinfeld Sokolov Wilson dynamical system arising in shallow water waves. Results Phys. 51, 106613 (2023)

    Article  Google Scholar 

  • Usman, M., Hussain, A., Zaman, F.D., Eldin, S.M.: Group invariant solutions of wave propagation in phononic materials based on the reduced micromorphic model via optimal system of Lie subalgebra. Results Phys. 48, 106413 (2023)

    Article  Google Scholar 

  • Usman, M., Hussain, A., Zaman, F.D.: Invariance analysis of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported Graphene sheets. Physica Scr. 98(9), 095205 (2023)

    Article  ADS  Google Scholar 

  • Wang, D., Zhang, H.Q.: Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equation. Chaos, Solitons Fractals 25(3), 601–610 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  • Wang, M., Li, X., Zhang, J.: The (\(G^\prime /G\))-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372(4), 417–423 (2008)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  • Wazwaz, A.M.: A sine-cosine method for handlingnonlinear wave equations. Math. Comput. Model. 40(5–6), 499–508 (2004)

    Article  Google Scholar 

  • Wazwaz, A.M.: The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput. 188(2), 1467–1475 (2007)

    MathSciNet  Google Scholar 

  • Wazwaz, A.M.: Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh-coth method. Appl. Math. Comput. 190(1), 633–640 (2007)

    MathSciNet  Google Scholar 

  • Xu, F.: Application of Exp-function method to symmetric regularized long wave (SRLW) equation. Phys. Lett. A 372(3), 252–257 (2008)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  • Yong, C., Biao, L.: traveling wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms. Chin. Phys. 13(3), 302 (2004)

    Article  Google Scholar 

  • Zill, D.G.: A First Course in Differential Equations with Modeling Applications. Cengage Learning, Boston (2012)

    Google Scholar 

Download references

Acknowledgements

Guangdong basic and applied basic research foundation (2021A1515110566), Guangdong philosophy and social science planning project (GD22YGL03).

Funding

No funding available.

Author information

Authors and Affiliations

  1. International Business School, Guangdong University of Finance & Economics, Foshan, 528100, China

    Renfei Luo

  2. Department of Mathematics, Quaid-i-Azam University 45320, Islamabad, 44000, Pakistan

    Naseem Abbas

  3. Abdus Salam School of Mathematical Sciences, Government College University, 68-B New Muslim Town, Lahore, 54600, Pakistan

    Akhtar Hussain

  4. Department of Mathematics, Quaid-i-Azam University 45320, Islamabad, 44000, Pakistan

    Shahbaz Ali

Authors
  1. Renfei Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Naseem Abbas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Akhtar Hussain
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shahbaz Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Naseem Abbas.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

Not applicable.

Consent for publication

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, R., Abbas, N., Hussain, A. et al. A new sensitive visualization, solitary wave profiles and conservation laws of ion sound waves arising in plasma. Opt Quant Electron 56, 415 (2024). https://doi.org/10.1007/s11082-023-06033-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-023-06033-8

Keywords

Search

Navigation