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Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media

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Abstract

This study acquires the wave solutions of the two well-known nonlinear models, namely; the modified Benjamin–Bona–Mahony and the coupled Klein–Gordon equations. The modified Benjamin–Bona–Mahony is a nonlinear model that describes the long surface gravity waves of small amplitude and the coupled Klein–Gordon equation describes the quantized version of the relativistic energy–momentum relation. We successfully acquire some new solutions to these models such as kink-type and soliton solutions in complex hyperbolic functions form. We plot the 3D and 2D surface of the all the obtained solutions in this study. The mathematical approach used in this study is the sine-Gordon expansion method.

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References

  • Abbasbandy, S., Shirzadi, A.: The first integral method for modified Benjamin–Bona–Mahony equation. Commun. Nonlinear Sci. Numer. Simul. 15, 1759–1764 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Abdou, M.A.: An extended riccati equation rational expansion method and its applications. Int. J. Nonlinear Sci. 7(1), 57–66 (2009)

    MathSciNet  MATH  Google Scholar 

  • Alam, Md.N., Belgacem, F.B.M.: Exact traveling wave solutions for the (1 + 1)-dim compound KdVB equation by the novel \((G^{^{\prime }}/G)\)-expansion method. Int. J. Mod. Nonlinear Theory Appl. 5(1), 28–39 (2016)

    Article  Google Scholar 

  • Ali, S., Rizvi, S.T.R., Younis, M.: Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients. Nonlinear Dyn. 82(4), 1755–1762 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Ali, A., Iqbal, M.A., Mohyud-din, S.T.: New analytical solutions for nonlinear physical models of the coupled Higgs equation and the Maccari system via rational exp (−φ(η))-expansion method. Pramana 87, 79 (2016)

    Article  ADS  Google Scholar 

  • Ali, A., Iqbal, M.A., Chen, Y.: Bright-dark mixed \(N\)-soliton solution of the two-dimensional Maccari system. Chin. Phys. Lett. 34(7), 070202 (2017)

    Article  Google Scholar 

  • Atangana, A., Alkahtani, B.S.T.: New model for process of phase separation in iron alloys. Iran. J. Sci. Technol. Trans. A: Sci. 1–6 (2016)

  • Atangana, A., Alqahtani, R.T.: Stability analysis of nonlinear thin viscous fluid sheet flow equation with local fractional variable order derivative. Open Phys. 14(1), 106–113 (2016)

    Article  Google Scholar 

  • Baskonus, H.M.: New acoustic wave behaviors to the Davey-Stewartson equation with power-law nonlinearity arising in fluid dynamics. Nonlinear Dyn. 86(1), 177–183 (2016)

    Article  MathSciNet  Google Scholar 

  • Baskonus, H.M.: New complex and hyperbolic function solutions to the generalized double combined Sinh–Cosh–Gordon equation. AIP Conf. Proc. 1798, 020018 (2017)

    Article  Google Scholar 

  • Baskonus, H.M., Askin, M.: Travelling wave simulations to the modified Zakharov–Kuzentsov model arising in plasma physics. 6th International Youth Science Forum ”LITTERIS ET ARTIBUS”’ Computer Science and Engineering, Lviv, Ukraine, 24–26 November (2016)

  • Baskonus, H.M., Bulut, H.: New hyperbolic function solutions for some nonlinear partial differential equation arising in mathematical physics. Entropy 17, 4255–4270 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Baskonus, H.M., Bulut, H.: Exponential prototype structures for (2 + 1)-dimensional Boiti–Leon–Pempinelli systems in mathematical physics. Waves Random Complex Media 26(2), 201–208 (2016a)

    Article  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 Med. 26(4), 613–625 (2016b)

    Article  MathSciNet  MATH  Google Scholar 

  • Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive system. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 272, 47–78 (1972)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Beyer, W.H.: CRC Standard Mathematical Tables, 28th edn. CRC Press, Boca Raton (1987)

    MATH  Google Scholar 

  • Bulut, H., Sulaiman, T.A., Baskonus, H.M.: New solitary and optical wave structures to the Korteweg–de Vries equation with dual-power law nonlinearity. Opt. Quant. Electron. 48, 564 (2016)

    Article  Google Scholar 

  • Chen, J., Yang, J., Yang, H.: Single soliton solutions of the coupled nonlinear Klein–Gordon equations with power law nonlinearity. Appl. Math. Comput. 246, 184–191 (2014)

    MathSciNet  MATH  Google Scholar 

  • Chen, Y., Liu, R.: Some new nonlinear wave solutions for two (3 + 1)-dimensional equations. Appl. Math. Comput. 260, 397–411 (2015)

    MathSciNet  Google Scholar 

  • Duran, S., Askin, M., Sulaiman, T.A.: New soliton properties to the ill-posed Boussinesq equation arising in nonlinear physical science. IJOCTA 7(3), 240–247 (2017)

    Google Scholar 

  • Khan, K., Akbar, M.A., Islam, S.M.R.: Exacts solutions for (1 + 1)-dimensional nonlinear dispersive modified Benjamin–Bona–Mahony equation and coupled Klein–Gordon equations. SpringerPlus (2014). https://doi.org/10.1186/2193-1801-3-724

  • Khusnutdinova, K.R., Pelinovsky, D.E.: On the exchange of energy in coupled Klein–Gordon equations. Wave Motion 38, 1–10 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Lee, J., Sakthivel, R.: Exact travelling wave solutions for some important nonlinear physical models. Pramana J. Phys. 80(5), 757–769 (2013)

    Article  ADS  Google Scholar 

  • Mirzazadeh, M.: Modified simple equation method and its applications to nonlinear partial differential equations. Inf. Sci. Lett. 3(1), 1–9 (2014)

    Article  Google Scholar 

  • Naher, H., Abdullah, F.A.: The modified Benjamin–Bona–Mahony equation via the extended generalized Riccati equation mapping method. Appl. Math. Sci. 6(111), 5495–5512 (2012)

    MathSciNet  MATH  Google Scholar 

  • Noor, M.A., Noor, K.I., Waheed, A., Al-Said, E.A.: Some new solitonary solutions of the modified Benjamin–Bona–Mahony equation. Comput. Math. Appl. 62, 2126–2131 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Priest, E.R.: Solar Magneto-Hydrodynamics. D. Reidel, Holland (1943)

    Google Scholar 

  • Seadawy, A.R., Sayed, A.: Traveling wave solutions of the Benjamin–Bona–Mahony water wave equations. Abst. Appl. Anal. 2014, 926838 (2014)

    MathSciNet  Google Scholar 

  • Seadawy, A.R., Lu, D., Khater, M.M.: Bifurcations of traveling wave solutions for Dodd–Bullough–Mikhailov equation and coupled Higgs equation and their applications. Chin. J. Phys. 55(4), 1310–1318 (2017)

    Article  Google Scholar 

  • Song, M., Ge, Y.: Application of the \((G^{^{\prime }}/G)\)-expansion method to (3 + 1)-dimensional nonlinear evolution equations. Comput. Math. Appl. 60, 1220–1227 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Tala-Tebue, E., Djoufack, Z.I., Tsobgni-Fozap, D.C., Kenfack-Jiotsa, A., Kapche-Tagne, F., Kofane, T.C.: Traveling wave solutions along microtubules and in the Zhiber–Shabat equation. Chin. J. Phys. 55(3), 939–946 (2017)

    Article  Google Scholar 

  • Tang, Y., Zai, W.: New exact periodic solitary-wave solutions for the (3 + 1)-dimensional generalized KP and BKP equations. Comput. Math. Appl. 70(10), 2432–2441 (2015)

    Article  MathSciNet  Google Scholar 

  • Wazwaz, A.M.: The tanh method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzeica–Dodd–Bullough equations. Chaos Solitons Fract 25, 55–63 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: The variational iteration method: a reliable analytic tool for solving linear and nonlinear wave equations. Comput. Math. Appl. 54, 926–932 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83(1), 591–596 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Yan, C.: A simple transformation for nonlinear waves. Phys. Lett. A 224, 77–84 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Zhao, Z., Zhang, Y., Rui, W.: Backlund transformation and solutions of a (3 + 1)-dimensional nonlinear evolution equation. Appl. Math. Comput. 248, 456–462 (2014)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Firat University, Elazig, Turkey

    Hasan Bulut & Tukur Abdulkadir Sulaiman

  2. Department of Mathematics, Federal University Dutse, Dutse, Jigawa, Nigeria

    Tukur Abdulkadir Sulaiman

  3. Department of Computer Engineering, Munzur University, Tunceli, Turkey

    Haci Mehmet Baskonus

  4. Department of Mathematics, Ordu University, Ordu, Turkey

    Tolga Akturk

Authors
  1. Hasan Bulut
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  2. Tukur Abdulkadir Sulaiman
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  3. Haci Mehmet Baskonus
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  4. Tolga Akturk
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Correspondence to Haci Mehmet Baskonus.

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Bulut, H., Sulaiman, T.A., Baskonus, H.M. et al. Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media. Opt Quant Electron 50, 19 (2018). https://doi.org/10.1007/s11082-017-1286-y

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