Skip to main content
Log in

On the solitary wave solutions to the longitudinal wave equation in MEE circular rod

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

Abstract

This study investigates the longitudinal wave equation in a magneto-electro-elastic circular rod by using the extended sinh-Gordon equation expansion method. Topological, non-topological and singular soliton solutions are extracted. To illustrate the physical appearance of the obtained solutions, 2D, 3D and the contour graphs to some of the obtained solutions are plotted. The reported results may be useful in explaining the physical meaning of the studied models and other nonlinear physical models arising in nonlinear sciences.

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

Access this article

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Akbar, M.A., Ali, N.H.M.: The improved F-expansion method with Riccati equation and its applications in mathematical physics. Cogent Math. 4, 1282577 (2017)

    Google Scholar 

  • Akbar, N.S., Nadeem, S., Haq, R.U., Khan, Z.H.: Numerical solutions of Magnetohydrodynamic boundary layer flow of tangent hyperbolic fluid towards a stretching sheet. Indian J. Phys. 87(11), 1121–1124 (2013)

    Article  ADS  Google Scholar 

  • Alquran, M., Al-Khaled, K., Ananbeh, H.: New soliton solutions for systems of nonlinear evolution equations by the rational Sine–Cosine method. Stud. Math. Sci. 3(1), 1–9 (2011)

    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, pp. 24–26 (2016)

  • Baskonus, H.M., Bulut, H.: Exponential prototype structure for (2+1)-dimensional Boiti–Leon–Pempinelli systems in mathematical physics. Waves Random Complex Media 26(2), 189–196 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Baskonus, H.M., Bulut, H., Atangana, A.: On the complex and hyperbolic structures of longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Mater. Struct. 25(3), 035022 (2016)

    Article  ADS  Google Scholar 

  • Baskonus, H.M., Sulaiman, T.A., Bulut, H.: On the novel wave behaviors to the coupled nonlinear Maccari’s system with complex structure. Optik 131, 1036–1043 (2017)

    Article  ADS  Google Scholar 

  • Baskonus, H.M., Sulaiman, T.A., Bulut, H.: Investigations of dark, bright, combined dark-bright optical and other soliton solutions in the complex cubic nonlinear Schrödinger equation with \(\delta\)-potential. Superlattices Microstruct 115, 19–29(2018)

    Article  ADS  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), 1–14 (2016)

    Google Scholar 

  • Bulut, H., Sulaiman, T.A., Demirdag, B.: Dynamics of soliton solutions in the chiral nonlinear Schrödinger equations. Nonlinear Dyn. 1–7 (2017). https://doi.org/10.1007/s11071-017-3997-9

  • Bulut, H., Sulaiman, T.A., Baskonus, H.M., Sandulyak, A.A.: New solitary and optical wave structures to the (1+1)-dimensional combined KdV–mKdV equation. Optik 135, 327–336 (2017)

    Article  ADS  Google Scholar 

  • Bulut, H., Sulaiman, T.A., Baskonus, H.M.: Dark, bright and other soliton solutions to the Heisenberg ferromagnetic spin chain equation. Superlattices Microstruct. (2017). https://doi.org/10.1016/j.spmi.2017.12.009

  • Bulut, H., Sulaiman, T.A., Baskonus, H.M., Akturk, T.: Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media. Opt. Quant. Electron. 50, 19 (2018)

    Article  Google Scholar 

  • Cattani, C.: Harmonic wavelet solutions of the Schrodinger equation. Int. J. Fluid Mech. Res. 30(5), 1–11 (2003)

    Article  MathSciNet  Google Scholar 

  • Cattani, C., Rushchitskii, Y.Y.: Cubically nonlinear elastic waves: wave equations and methods of analysis. Int. Appl. Mech. 39(10), 1115–1145 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  • Dehghan, M., Shakeri, F.: Application of He’s variational iteration method for solving the Cauchy reaction–diffusion problem. J. Comput. Appl. Math. 214, 435–446 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Eslami, M., Neyrame, A., Ebrahimi, M.: Explicit solutions of nonlinear (2+1)-dimensional dispersive long wave equation. J. King Saud Univ. Sci. 24(1), 69–71 (2012)

    Article  Google Scholar 

  • Eslami, M., Rezazadeh, H., Rezazadeh, M., Mosavi, S.S.: Exact solutions to the space–time fractional Schrödinger–Hirota equation and the space-time modified KDV–Zakharov–Kuznetsov equation. Opt. Quant. Electron. 49(8), 279 (2017)

    Article  Google Scholar 

  • Fang, J.P., Ren, Q.B., Zheng, C.L.: New exact solutions and fractal localized structures for the (2+1)-dimensional Boiti–Leon–Pempinelli system. Z. Naturforsch. 60, 245–251 (2005)

    ADS  Google Scholar 

  • Haq, R.U., Soomro, F.A., Khan, Z.H., Al-Mdallal, Q.M.: Numerical study of streamwise and cross flow in the presence of heat and mass transfer. Eur. Phys. J. Plus 132, 214 (2017)

    Article  Google Scholar 

  • Inan, I.E., Kaya, D.: Exact solutions of some nonlinear partial differential equations. Phys. A 381, 104–115 (2007)

    Article  MathSciNet  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, 3, 724 (2014)

  • Khan, K., Koppelaar, H., Akbar, A.: Exact and numerical soliton solutions to nonlinear wave equations. Comput. Math. Eng. 2, 5–22 (2016)

    Google Scholar 

  • Ma, X., Pan, Y., Chang, L.: Explicit travelling wave solutions in a magneto-electro-elastic circular rod. Int. J. Comput. Sci. Issues 10(1), 62–68 (2013)

    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 

  • Momani, S., Abuasad, S.: Application of He’s variational iteration method to Helmholtz equation. Chaos Solitons Fractals 27(5), 1119–1123 (2006)

    Article  ADS  MathSciNet  MATH  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 

  • Nofal, T.A.: Simple equation method for nonlinear partial differential equations and its applications. J. Egypt. Math. Soc. 24, 204–209 (2016)

    Article  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 

  • Ozpinar, F., Baskonus, H.M., Bulut, H.: On the complex and hyperbolic structures for the (2+1)-dimensional Boussinesq water equation. Entropy 17(12), 8267–8277 (2015)

    Article  ADS  Google Scholar 

  • Rawashdeh, M.: Approximate solutions for coupled systems of nonlinear PDEs using the reduced differential transform method. Math. Comput. Appl. 19(2), 161–171 (2014)

    MathSciNet  Google Scholar 

  • Rizvi, S.T.R., Ali, K.: Jacobian elliptic periodic traveling wave solutions in the negative-index materials. Nonlinear Dyn. 87(3), 1967–1972 (2017)

    Article  Google Scholar 

  • Seadawy, A.R.: Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: part I. Comput. Math. Appl. 70, 345–352 (2015)

    Article  MathSciNet  Google Scholar 

  • Seadawy, A.R.: Ionic acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev–Petviashvili–Burgers equations in quantum plasma. Math. Methods Appl. Sci. 40, 1598–1607 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Lu, D., Khater, M.M.A.: 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 

  • Sulaiman, T.A., Akturk, T., Bulut, H., Baskonus, H.M.: Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation. J. Electromagn. Waves Appl. (2017). https://doi.org/10.1080/09205071.2017.1417919

    Google Scholar 

  • Wang, M., Li, X.: Simplified homogeneous balance method and its applications to the Whitham–Broer–Kaup model equations. J. Appl. Math. Phys. 2, 823–827 (2014)

    Article  Google Scholar 

  • Wazwaz, A.M.: New (3+1)-dimensional nonlinear evolution equations with Burgers and Sharma–Tosso–Olver equations constituting the main parts. Proc. Rom. Acad. Ser. A 16(1), 32–40 (2015)

    MathSciNet  Google Scholar 

  • Weisstein, E.W.: Concise Encyclopedia of Mathematics, 2nd edn. CRC Press, New York (2002)

    MATH  Google Scholar 

  • Xian-Lin, X., Jia-Shi, T.: Travelling wave solutions for Konopelchenko–Dubrovsky equation using an extended sinh-Gordon equation expansion method. Commun. Theor. Phys. 50, 1047 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  • Xue, C.X., Pan, E., Zhang, X.Y.: Solitary waves in a magneto-electro-elastic circular rod. Smart Mater. Struct. 20(10), 035022 (2011)

    Article  Google Scholar 

  • Yan, Z., Zhang, H.: New explicit and exact travelling wave solutions for a system of variant boussinesq equations in mathematical physics. Phys. Lett. A 252, 291–296 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Yokus, A., Baskonus, H.M., Sulaiman, T.A., Bulut, H.: Numerical simulations and solutions of the two component second order KdV evolutionary system. Numer. Methods Partial Differ. Equ. 34(1), 211–227 (2018)

    Article  MATH  Google Scholar 

  • Yokus, A., Sulaiman, T.A., Bulut, H.: On the analytical and numerical solutions of the Benjamin–Bona–Mahony equation. Opt. Quant. Electron. 50, 31 (2018)

    Article  Google Scholar 

  • Younis, M., Ali, S.: Bright, dark, and singular solitons in magneto-electro-elastic circular rod. Waves Random Complex Media 25(4), 549–555 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  • Zhang, W.: A generalized Tanh-function type method and the (G’/G)-expansion method for solving nonlinear partial differential equations. Appl. Math. 4, 11–16 (2013)

    Article  Google Scholar 

  • Zhao, Y.: F-expansion method and its application for finding new exact solutions to the Kudryashov–Sinelshchikov equation. J. Appl. Math. 2013, 895760 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Haci Mehmet Baskonus.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bulut, H., Sulaiman, T.A. & Baskonus, H.M. On the solitary wave solutions to the longitudinal wave equation in MEE circular rod. Opt Quant Electron 50, 87 (2018). https://doi.org/10.1007/s11082-018-1362-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-018-1362-y

Keywords

Navigation