# Solitary Wave Solutions for the Shallow Water Wave Equations and the Generalized Klein-Gordon Equation Using Exp(-ϕ(η) )-Expansion Method

## Main Article Content

## Abstract

In this article, we investigate the exact and solitary wave solutions for the shallow water wave equations and the generalized Klein-Gordon equation using the exp -expansion method. A wave transformation is applied to convert the problem into the form of an ordinary differential equation. By using this method, we found the explicit solitary wave solutions in terms of the hyperbolic functions, trigonometric functions, exponential functions and rational functions. The extracted solution plays a significant role in many physical phenomena such as electromagnetic waves, nonlinear lattice waves, ion sound waves in plasma, nuclear physics, shallow water waves and so on. It is noted that the method is reliable, straightforward and an effective mathematical tool for analytic treatment of nonlinear systems of partial differential equation in mathematical physics and engineering.

## Article Details

*Journal of Advances in Mathematics and Computer Science*,

*35*(4), 72-86. https://doi.org/10.9734/jamcs/2020/v35i430272

## References

Jawad AJM, Petkovic MD, Laketa P, Biswas A. Dynamics of shallow water waves with Boussinesq equation. Scientia Iranica. 2013;20(1):179-184.

Azari R, Jamshidzadeh S, Biswas AA. Solitary wave solutions of coupled Boussinesq equation. Wiley Periodicals, Inc; 2016.

DOI: 10.1002/cplx.21791

Darvishi MT, Najafi M, Wazwaz AM. Traveling solutions for Boussinesq-like equations with spatial and spatial-temporal dispersion. Rom. Rep. Phy, Romanian Academy Publishing House; 2017. ISSN: 1221-1451.

Elgrayhi A. New periodic wave solutions for the shallow water equations and the generalized Klein-Gordon equation. Communications in Nonlinear Science and Numerical Simulation. 2016;13:877-888.

Wazwaz AM. Partial differential equations: Method and applications. Taylor and Francis; 2002.

Biswas A, Kara AH, Moraru L, Rriki H. Shallow water waves modeled by the Boussinesq equation having logarithmic nonlinearity. Proceeding of the Romanian Academy, Series A. 2017;18(2):144-149.

Biswas A, Zony C, Zerrad E. Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation. Applied Mathematics and Computation. 2008;203(1):153-156.

Akbar MA, Ali NHM. New solitary and periodic solutions of nonlinear evolution equation by Exp-function method. World Appl. Sci. Journal. 2012;17(12):1603-1610.

Bekir A, Boz A. Exact solutions for nonlinear evolution equations using exp-function method. Phys. Lett. A. 2008;372:1619–1625.

Jawad AJM, Petkovic A, Biswas A. Modified simple equation method for nonlinear evolution equations. Appl. Math. Computation. 2010;217:869-877.

Liu D. Jacobi elliptic function solutions for two variant Boussinesq equations. Chaos Solitons and Fract. 2005;24:1373-85.

Chen Y, Wang Q. Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation. Chaos Solitons and Fract. 2005;24:745-57.

Adomain G. Solving frontier problems of physics: The decomposition method. Kluwer Academic Publishers, Boston; 1994.

Wang ML, Li X. Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Phys. Lett. A. 2005;343:48-54.

Fan E, Zhang H. A note on the homogenous balance method. Physics Letters A. 1998;246:403-406.

Available:http://dx.doi.org/10.1016/S0375-9601(98)00547-7

Alam MN, Stepanyants YA. New generalized -expansion method in investigating the traveling wave solutions to the typical breaking soliton and the Benjamin-Bona-Mahony equations. Int. J. Math. Comput. 2016;27(3):69-82.

Alam MN, Akbar MA, Mohyud-Din ST. A novel -expansion method and its application to the Boussinesq equation. Chin. Phys. B. 2014;23(2):020203-020210.

Akbar MA, Alam MN, Hafez MG. Application of the Novel -expansion method to traveling wave solutions for the positive Gardner-KP equation. Indian Journal of Pure and Applied Mathematics. 2016;47:85-96.

Naher H, Abdullah FA. New approach of -expansion method and new approach of generalized -expansion method for nonlinear evolution equation. AIP Advances. 2016;3: 032116.

Zhang J, Jiang F, Zhao X. An improved -expansion method for solving nonlinear evolution equations. Int. J. Com. Math. 2010;87(8):1716-1725.

Zhang J, Wei X, Lu Y. A generalized -expansion method and its applications. Phys. Lett. A. 2008;372:3653-3658.

Alam MN, Stepanyants YA. New generalized -expansion method in investigating the traveling wave solutions to the typical breaking soliton and the Benjamin-Bona-Mahony equations. International Journal of Mathematics and Computation. 2016;27(3):69-82.

Islam SMR, Khan K, Akbar MA. Study of exp (Φ(η))-expansion method for solving nonlinear partial differential equations. British Journal of Mathematics and Computer Science. 2015;5(3):397-407.

Chen G, Xin X, Liu H. The improved (-(η) )-expansion method and new exact solutions of nonlinear evolution equation in mathematical physics. Advances in Mathematical Physics. 2019;Id 4354310:1-8.

Rahid H, Azizur R. The exp(-ϕ(η))-expansion method with application in the (1+1)-dimensional classical Boussinesq equations. Results in Physics. 2014;4:150-155.

Akbar MA, Ali NHM. Solitary wave solutions of the fourth order Boussinesq equation through the exp(-ϕ(η))-expansion. Springer Plus. 2014;3(344):1-6.

Nematollah K, Feckan M, Khalili Y. Application of the exp(-ϕ)-expansion method to the Pochhammer-Chree equation. Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia. 2018;3347-3354.

Rashid MM, Khatun W, Rabbani G. Exact traveling wave solutions for the (2+1)-dimensional Burgers equation using exp(-ϕ(η))-expansion method. Journal of Mathematics. 2020;16(2):29-34.

Dolapci IT, Yildirim A. Some exact solutions to the generalized Korteweg-de Vries equation and the system of shallow water wave equation. Nonlinear Analysis, Modeling and Control. 2013;18:27-36.

Zahran EHM, Khater MMA. Exact traveling wave solutions for the system of shallow water wave equations and modified Liouville equation using extended Jacobian elliptic function expansion method. American Journal of Computational Mathematics. 2014;4:455-463.

Fu Z, Liu S, Liu S. New transformation and new approach to find exact solutions to nonlinear equation. Physics Letters A. 2002;299:507-512.

Fan E, Hon YC. A series of traveling wave solutions for two variant Boussinesq equation in shallow water waves. Chaos, Solitons and Fractals. 2003;15:559-566.

Jesmin A, Ali MM. Solitary wave solutions of two nonlinear evolution equations via the modified simple equation method. New Trends in Mathematical Sciences. NTMSCI. 2016;4(4):12-16.

Ablowitz MJ. Soliton, nonlinear evolution equations and inverse scatting. Cambridge University Press, New York; 1999.

Xie F, Yan Z, Zhang H. Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shall water equations. Physical Letters A. 2001;285:76-80.

Yan Z, Zhang H. New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics. Physics Letters A. 1999;252:291-296.

Lu D, Wang ML, Li W. Solving generalized Klein-Gordon equation by using modified (G'/G) expansion method. 2011 Fourth International Conference on Information and Computing, Phuket Island. 2011;249-252.

Yasuk F, Durmus A, Boztosun I. Exact analytical solution to the relativistic Klein-Gordon equation with noncentral equal scalar and vector potentials. Journal of Mathematical Physics. 2006;47(8): 082302.

Zheng Y, Lai S. A study on three types of nonlinear Klein-Gordon equations. Dynamics of Continuous, Discrete and Impulsive Systems, Series B. 2009;16(2):271-279.

Hafez MG, Alam MN, Akbar MA. Exact traveling wave solutions to the Klein-Gordon equation using the novel (G' /G)-expansion method. Results in Physics. 2014;4:177-184.

Zayed EME, Gepreel KA. The (G^'/G) -expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. Journal of Mathematical Physics. 2012;50:013502.