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Thirty traveling wave solutions to the systems of ion sound and Langmuir waves

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Abstract

Nine kinds of general solutions of certain popular first-order ordinary differential equation are obtained by direct calculations. Compared with complete discrimination system for polynomials, our classifications of solutions straightly depend on the coefficients of the ordinary differential equation and hence are easier to be employed. According to the nine kinds of solutions, we establish thirty traveling wave solutions to the coupled systems of ion sound and Langmuir waves, including rational function solution, exponential function solution, trigonometric function solutions, hyperbolic function solutions and Jacobi elliptic function solutions. To the best of our knowledge, many of them are new. Solutions discussed in two recent articles are showed to be equivalent to some special cases of our thirty traveling wave solutions. Our solutions include the kink and anti-kink soliton solutions, dark and bright soliton solutions as well as periodic soliton solutions and solitary wave solutions. We believe that these solutions will be of great use to researchers concerning with nonlinear physical phenomena. In addition, the famous (1+1)-dimensional dispersive long wave equations is taken to illustrate the applicability of these thirty solutions.

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References

  1. Baskonus, H.M., Bulut, H.: New wave behaviors of the system of equations for the ion sound and Langmuir waves. Wave. Random Complex 26, 613–625 (2016). https://doi.org/10.1080/17455030.2016.1181811

    Article  MathSciNet  MATH  Google Scholar 

  2. Manafian, J.: Application of the ITEM for the system of equations for the ion sound and Langmuir waves. Opt. Quant. Electron. 49, 17 (2017). https://doi.org/10.1007/s11082-016-0860-z

    Article  Google Scholar 

  3. Seadawy, A.R., Kumar, D., Hosseini, K., Samadani, F.: The system of equations for the ion sound and Langmuir waves and its new exact solutions. Results Phys. 9, 1631–1634 (2018). https://doi.org/10.1016/j.rinp.2018.04.064

    Article  Google Scholar 

  4. Ahmed, I., Seadawy, A.R., Lu, D.: Rogue waves generation and interaction of multipeak rational solitons in the system of equations for the ion sound and Langmuir waves. Int. J. Mod. Phys. B 33, 1950277 (2019). https://doi.org/10.1142/S0217979219502771

    Article  MathSciNet  MATH  Google Scholar 

  5. Seadawy, A.R., Ali, A., Lu, D.: Structure of system solutions of ion sound and Langmuir dynamical models and their applications. Pramana-J. Phys. 92, 88 (2019). https://doi.org/10.1142/S0217979219502771

    Article  Google Scholar 

  6. Mohammed, W.W., Abdelrahman, M.A.E., Inc, M., Hamza, A.E., Akinlar, M.A.: Soliton solutions for system of ion sound and Langmuir waves. Opt. Quant. Electron. 52, 460 (2020). https://doi.org/10.1007/s11082-020-02581-5

    Article  Google Scholar 

  7. Shakeel, M., Iqbal, M.A., Din, Q., Hassan, Q.M., Ayub, K.: New exact solutions for coupled nonlinear system of ion sound and Langmuir waves. Indian J. Phys. 94, 885–894 (2020). https://doi.org/10.1007/s12648-019-01522-7

    Article  Google Scholar 

  8. Ali, K.K., Yilmazer, R., Baskonus, H.M., Bulut, H.: Modulation instability analysis and analytical solutions to the system of equations for the ion sound and Langmuir waves. Phys. Scr. 95, 065602 (2020). https://doi.org/10.1088/1402-4896/ab81bf

    Article  Google Scholar 

  9. Tripathy, A., Sahoo, S.: Exact solutions for the ion sound Langmuir wave model by using two novel analytical method. Results Phys. 19, 103494 (2020). https://doi.org/10.1016/j.rinp.2020.103494

    Article  Google Scholar 

  10. Tahir, M., Awan, A.: Optical singular and dark solitons with Biswas-Arshed model by modified simple equation method. Optik 202, 163523 (2020). https://doi.org/10.1016/j.ijleo.2019.163523

    Article  Google Scholar 

  11. Mahak, N., Akram, G.: The modified auxiliary equation method to investigate solutions of the perturbed nonlinear Schrödinger equation with kerr law nonlinearity. Optik 207, 164467 (2020). https://doi.org/10.1016/j.ijleo.2020.164467

    Article  Google Scholar 

  12. Akram, G., Zainab, I.: Dark peakon, kink and periodic solutions of the nonlinear Biswas-Milovic equation with kerr law nonlinearity. Optik 208, 164420 (2020). https://doi.org/10.1016/j.ijleo.2020.164420

    Article  Google Scholar 

  13. Rehman, H., Ullah, N., Imran, M.: Optical solitons of Biswas-Arshed equation in birefringent fibers using extended direct algebraic method. Optik 226, 165378 (2021). https://doi.org/10.1016/j.ijleo.2020.165378

    Article  Google Scholar 

  14. Zayed, E.M.E., Alngar, M.E.M.: Optical solitons in birefringent fibers with Biswas-Arshed model by generalized Jacobi elliptic function expansion method. Optik 203, 163922 (2020). https://doi.org/10.1016/j.ijleo.2019.163922

    Article  Google Scholar 

  15. Yıldırım, Y.: Optical solitons with Biswas-Arshed equation by F-expansion method. Optik 227, 165788 (2021). https://doi.org/10.1016/j.ijleo.2020.165788

    Article  Google Scholar 

  16. Pandir, Y., Turhan, N.: Multiple Soliton Solutions for Nonlinear Differential Equations with a New Version of Extended F-Expansion Method. Proc. Nati. Acad. Sci. India. Sect. A Phys, Sci (2020). https://doi.org/10.1007/s40010-020-00687-9

  17. Kong, Y., Xin, L., Qiu, Q., Han, L.: Exact periodic wave solutions for the modified Zakharov equations with a quantum correction. Appl. Math. Lett. 94, 140–148 (2019). https://doi.org/10.1016/j.aml.2019.01.009

    Article  MathSciNet  MATH  Google Scholar 

  18. Lu, D., Seadawy, A.R., Ali, A.: Dispersive traveling wave solutions of the Equal-Width and Modified Equal-Width equations via mathemathical methods and its applications. Results Phys. 9, 313–320 (2018). https://doi.org/10.1016/j.rinp.2018.02.036

    Article  Google Scholar 

  19. Özkan, Y.S., Yasar, E., Seadawy, A.R.: A third-order nonlinear Schrödinger equation:the exact solutions, group-invariant solutions and conservation laws. J. Taibah Univ. Sci. 14, 585–597 (2020). https://doi.org/10.1080/16583655.2020.1760513

    Article  Google Scholar 

  20. Cheemaa, N., Chen, S., Seadawy, A.R.: Propagation of isolated waves of coupled nonlinear (2+1)-dimensional Maccari System in plasma physics. Results Phys. 17, 102987 (2020). https://doi.org/10.1016/j.rinp.2020.102987

    Article  Google Scholar 

  21. Seadawy, A.R., Cheemaa, N.: Propagation of nonlinear complex waves for the coupled nonlinear Schrödinger Equations in two core optical fibers. Physica A 529, 121330 (2019). https://doi.org/10.1016/j.physa.2019.121330

    Article  MathSciNet  Google Scholar 

  22. Cheemaa, N., Seadawy, A.R., Chen, S.: More general families of exact solitary wave solutions of the nonlinear Schrödinger equation with their applications in nonlinear optics. Eur. Phys. J. Plus 133, 547 (2018). https://doi.org/10.1140/epjp/i2018-12354-9

    Article  Google Scholar 

  23. Liu, H.Z., Sun, X.Q., Chen, L.J.: Comment on: An observation on the periodic solutions to nonlinear physical models by means of the auxiliary equation with a sixth-degree nonlinear term. Commun Nonlinear Sci Numer Simulat 18, 2177–2187 (2013)

    Article  Google Scholar 

  24. Liu, H.Z., Sun, X.Q., Chen, L.J.: Comment on: “An observation on the periodic solutions to nonlinear physical models by means of the auxiliary equation with a sixth-degree nonlinear term’’. Commun. Nonlinear Sci. Numer. Simul. 19, 2553–2557 (2014). https://doi.org/10.1016/j.cnsns.2013.11.018

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, H.Z., Zhu, G.Q.: Comment on the solitons and periodic travelling wave solutions for Dodd–Bullough–Mikhailov and Tzitzeica–Dodd–Bullough equations in quantum field theory. Optik 168, 807–816 (2018)

    Article  Google Scholar 

  26. Liu, H.Z., Zhu, G.Q.: Comment on “the solitons and periodic travelling wave solutions for Dodd–Bullough–Mikhailov and Tzitzeica–Dodd–Bullough equations in quantum field theory, Optik 203, 163870 (‘). https://doi.org/10.1016/j.ijleo.2019.163870

  27. Liu, C.S.: Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations. Comput. Phys. Commun. 181, 317–324 (2010). https://doi.org/10.1016/j.cpc.2009.10.006

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, Z.X., Guo, D.R.: Special Functions Generality. Peking University Press, Beijing (2000)

    Google Scholar 

  29. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972)

    MATH  Google Scholar 

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Acknowledgements

The author would like to thank the anonymous referee for their helpful comments and suggestions.

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  1. Zhijiang College, Zhejiang University of Technology, Shaoxing, 312030, People’s Republic of China

    Hong-Zhun Liu

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Liu, HZ. Thirty traveling wave solutions to the systems of ion sound and Langmuir waves. Japan J. Indust. Appl. Math. 38, 877–902 (2021). https://doi.org/10.1007/s13160-021-00465-z

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  • DOI: https://doi.org/10.1007/s13160-021-00465-z

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