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Soliton solutions for system of ion sound and Langmuir waves

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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.

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References

  • Abdelrahman, M.A.E.: Global solutions for the ultra-relativistic Euler equations. Nonlinear Anal. 155, 140–162 (2017a)

    MathSciNet  MATH  Google Scholar 

  • Abdelrahman, M.A.E.: On the shallow water equations. Z. Naturforsch. 72(9a), 873–879 (2017b)

    ADS  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Abdelrahman, M.A.E., Kunik, M.: The ultra-relativistic Euler equations. Math. Methods Appl. Sci. 38, 1247–1264 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    ADS  Google Scholar 

  • 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)

    ADS  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    ADS  MathSciNet  MATH  Google Scholar 

  • Bhrawy, A.H.: An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl. Math. Comput. 247, 30–46 (2014)

    MathSciNet  MATH  Google Scholar 

  • Biswas, A., Mirzazadeh, M.: Dark optical solitons with power law nonlinearity using \(G^{\prime }/G\)-expansion. Optik 125, 4603–4608 (2014)

    ADS  Google Scholar 

  • Dai, C.Q., Zhang, J.F.: Jacobian elliptic function method for nonlinear differential difference equations. Chaos Solut. Fractals 27, 1042–1049 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  Google Scholar 

  • 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)

    ADS  MathSciNet  MATH  Google Scholar 

  • Fan, E., Zhang, H.: A note on the homogeneous balance method. Phys. Lett. A 246, 403–406 (1998)

    ADS  MATH  Google Scholar 

  • Fan, E., Zhang, J.: Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. A 305, 383–392 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    ADS  MathSciNet  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    ADS  MathSciNet  Google Scholar 

  • He, J.H.: Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons Fractals 19(4), 847–851 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  • He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20, 1141–1199 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  • He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700–708 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  • 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)

    ADS  MathSciNet  MATH  Google Scholar 

  • Malfliet, W., Hereman, W.: The tanh method: exact solutions of nonlinear evolution and wave equations. Phys. Scr. 54, 563–568 (1996)

    ADS  MathSciNet  MATH  Google Scholar 

  • Manafian, J.: Application of the ITEM for the system of equations for the ion sound and Langmuir waves. Opt. Quant. Electron. 49, 17 (2017)

    Google Scholar 

  • Mohammed, W.W.: Approximate solution of the Kuramoto–Shivashinsky equation on an unbounded domain. Chin. Ann. Math. Ser. B 39(1), 145–162 (2018)

    MathSciNet  MATH  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    MathSciNet  Google Scholar 

  • 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)

    ADS  MathSciNet  MATH  Google Scholar 

  • Rizvi, S.T.R., Ali, K., Ahmad, M.: Optical solitons for Biswas–Milovic equation by new extended auxiliary equation method. Optik 204, 164181 (2020a)

    ADS  Google Scholar 

  • 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)

    ADS  MathSciNet  Google Scholar 

  • 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)

    Google Scholar 

  • Wang, M.L.: Exact solutions for a compound KdV–Burgers equation. Phys. Lett. A 213, 279–287 (1996)

    ADS  MathSciNet  MATH  Google Scholar 

  • 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)

    ADS  MATH  Google Scholar 

  • Wazwaz, A.M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. Math. Comput. 154, 714–723 (2004a)

    Google Scholar 

  • Wazwaz, A.M.: A sine–cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499–508 (2004b)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Appl. Math. Comput. 187, 1131–1142 (2007)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    ADS  MathSciNet  MATH  Google Scholar 

  • 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)

    ADS  MATH  Google Scholar 

  • 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)

    ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research has been funded by Scientific Research Deanship at University of Ha’il - Saudi Arabia through Project Number RG-191207.

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Correspondence to Mustafa Inc.

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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

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