Abstract
We study on constructing new traveling wave solution of a system known as ion sound and Langmuir waves by using method of extended Jacobian elliptic function expansion and He’s semi-inverse technique. Some new solutions will be beneficial for researchers concerning with nonlinear physical phenomena. The proposed methods may serve as the framework for solutions of various equations in applied science. Graphical simulations are provided to illustrate the behavior of these solutions.
Similar content being viewed by others
References
Abdelrahman, M.A.E.: Global solutions for the ultra-relativistic Euler equations. Nonlinear Anal. 155, 140–162 (2017a)
Abdelrahman, M.A.E.: On the shallow water equations. Z. Naturforsch. 72(9a), 873–879 (2017b)
Abdelrahman, M.A.E.: Numerical investigation of the wave-front tracking algorithm for the full ultra-relativistic Euler equations. Int. J. Nonlinear Sci. Numer. Simul. 19, 223–229 (2018). https://doi.org/10.1515/ijnsns-2017-0121
Abdelrahman, M.A.E., Kunik, M.: The ultra-relativistic Euler equations. Math. Methods Appl. Sci. 38, 1247–1264 (2015)
Abdelrahman, M.A.E., Sohaly, M.A.: Solitary waves for the nonlinear Schrödinger problem with the probability distribution function in stochastic input case. Eur. Phys. J. Plus 132, 339 (2017)
Abdelrahman, M.A.E., Sohaly, M.A.: The development of the deterministic nonlinear PDEs in particle physics to stochastic case. Results Phys. 9, 344–350 (2018)
Ali, I., Ali, K., Rizvi, S.T.R.: Conserved quantities for compressional dispersive Alfvén and soliton dynamics with non-local nonlinearity. Phys. Scr. 95(4), 045209 (2020)
Aminikhad, H., Moosaei, H., Hajipour, M.: Exact solutions for nonlinear partial differential equations via Exp-function method. Numer. Methods Partial Differ. Equ. 26, 1427–1433 (2009)
Baskonus, H.M., Bulut, H.: New wave behaviors of the system of equations for the ion sound and Langmuir waves. Waves Random Complex Media 26, 613–625 (2016)
Bhrawy, A.H.: An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl. Math. Comput. 247, 30–46 (2014)
Biswas, A., Mirzazadeh, M.: Dark optical solitons with power law nonlinearity using \(G^{\prime }/G\)-expansion. Optik 125, 4603–4608 (2014)
Dai, C.Q., Zhang, J.F.: Jacobian elliptic function method for nonlinear differential difference equations. Chaos Solut. Fractals 27, 1042–1049 (2006)
Demiray, S.T., Bulut, H.: New soliton solutions of the system of equations for the ion sound and Langmuir waves. Int. J. Opt. Control Theor. Appl. 7(1), 42–49 (2017)
Dubey, V.P., Kumar, R., Kumar, D., Khan, I., Singh, J.: An efficient computational scheme for nonlinear time fractional systems of partial differential equations arising in physical sciences. Adv. Differ. Equ. 2020, 46 (2020)
EL-Wakil, S.A., Abdou, M.A.: New exact travelling wave solutions using modified extented tanh-function method. Chaos Solitons Fractals 31, 840–852 (2007)
Fan, E., Zhang, H.: A note on the homogeneous balance method. Phys. Lett. A 246, 403–406 (1998)
Fan, E., Zhang, J.: Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. A 305, 383–392 (2002)
Goswami, A., Singh, J., Kumar, D.: Numerical simulation of fifth order KdV equations occurring in magneto-acoustic waves. Ain Shams Eng. J. 9, 2265–2273 (2018)
Goswami, A., Singh, J., Kumar, D., Sushila: An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Physica A 524, 563–575 (2019a)
Goswami, A., Singh, J., Kumar, D., Gupta, S., Sushila: An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma. J. Ocean Eng. Sci. 4(2), 85–99 (2019b)
He, J.H.: Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics. Int. J. Turbo Jet-Engines 14(1), 23–28 (1997)
He, J.H.: Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons Fractals 19(4), 847–851 (2004)
He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20, 1141–1199 (2006)
He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700–708 (2006)
Liu, S., Fu, Z., Liu, S., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289, 69–74 (2001)
Malfliet, W., Hereman, W.: The tanh method: exact solutions of nonlinear evolution and wave equations. Phys. Scr. 54, 563–568 (1996)
Manafian, J.: Application of the ITEM for the system of equations for the ion sound and Langmuir waves. Opt. Quant. Electron. 49, 17 (2017)
Mohammed, W.W.: Approximate solution of the Kuramoto–Shivashinsky equation on an unbounded domain. Chin. Ann. Math. Ser. B 39(1), 145–162 (2018)
Mohammed, W.W.: Modulation equation for the stochastic Swift–Hohenberg equation with cubic and quintic nonlinearities on the real line. Mathematics 6(12), 1–12 (2020)
Razborova, P., Ahmed, B., Biswas, A.: Solitons, shock waves and conservation laws of Rosenau–KdV–RLW equation with power law nonlinearity. Appl. Math. Inf. Sci. 8(2), 485–491 (2014)
Ren, Y.J., Zhang, H.Q.: A generalized F-expansion method to find abundant families of Jacobi elliptic function solutions of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation. Chaos Solitons Fractals 27, 959–979 (2006)
Rizvi, S.T.R., Ali, K., Ahmad, M.: Optical solitons for Biswas–Milovic equation by new extended auxiliary equation method. Optik 204, 164181 (2020a)
Rizvi, S.T.R., Afzal, I., Ali, K.: Dark and singular optical solitons for Kundu–Mukherjee–Naskar model. Mod. Phys. Lett. B 34(6), 2050074 (2020b)
Veeresha, P., Prakasha, D.G., Kumar, D., Baleanu, D., Singh, J.: An efficient computational technique for fractional model of generalized Hirota–Satsuma coupled KdV and coupled mKdV equations. J. Comput. Nonlinear Dyn. 15, 071003 (2020)
Wang, M.L.: Exact solutions for a compound KdV–Burgers equation. Phys. Lett. A 213, 279–287 (1996)
Wang, Q., Chen, Y., Zhang, H.: An extended Jacobi elliptic function rational expansion method and its application to (2 + 1)-dimensional dispersive long wave equation. Phys. Lett. A 289, 411–426 (2005)
Wazwaz, A.M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. Math. Comput. 154, 714–723 (2004a)
Wazwaz, A.M.: A sine–cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499–508 (2004b)
Wazwaz, A.M.: Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE method. Comput. Math. Appl. 50, 1685–1696 (2005)
Wazwaz, A.M.: The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Appl. Math. Comput. 187, 1131–1142 (2007)
Yang, X.F., Deng, Z.C., Wei, Y.: A Riccati–Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Adv. Differ. Equ. 1, 117–133 (2015)
Younis, M., Ali, S., Mahmood, S.A.: Solitons for compound KdV Burgers equation with variable coefficients and power law nonlinearity. Nonlinear Dyn. 81, 1191–1196 (2015)
Younis, M., Bilal, M., Rehman, S., Younas, U., Rizvi, S.T.R.: Investigation of optical solitons in birefringent polarization preserving fibers with four-wave mixing effect. Int. J. Mod. Phys. B 34(11), 2050113 (2020)
Zhang, J.L., Wang, M.L., Wang, Y.M., Fang, Z.D.: The improved F-expansion method and its applications. Phys. Lett. A 350, 103–109 (2006)
Zhao, X.Q., Zhi, H.Y., Zhang, H.Q.: Improved Jacobi-function method with symbolic computation to construct new double-periodic solutions for the generalized Ito system. Chaos Solitons Fractals 28, 112–126 (2006)
Acknowledgements
This research has been funded by Scientific Research Deanship at University of Ha’il - Saudi Arabia through Project Number RG-191207.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mohammed, W.W., Abdelrahman, M.A.E., Inc, M. et al. 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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-020-02581-5