Korean J. Math. Vol. 23 No. 1 (2015) pp.11-27
DOI: https://doi.org/10.11568/kjm.2015.23.1.11

Travelling wave solutions for some nonlinear evolution equations

Main Article Content

Hyunsoo Kim
Jin Hyuk Choi


Nonlinear partial differential equations are more suitable to model many physical phenomena in science and engineering. In this paper, we consider three nonlinear partial differential equations such as Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation which serves as a model for the unidirectional propagation of the shallow water waves over a flat bottom. The main objective in this paper is to apply the generalized Riccati equation mapping method for obtaining more exact traveling wave solutions of Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation. More precisely, the obtained solutions are expressed in terms of the hyperbolic, the trigonometric and the rational functional form. Solutions obtained are potentially significant for the explanation of better insight of physical aspects of the considered nonlinear physical models.

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[1] AH.Bhrawy, MA.Abdelkawy and A.Biswas, Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 915–925. Google Scholar

[2] AH.Bhrawy, A.Biswas, M.Javidi, WX.Ma, Z.Pinar and A.Yildirim, New solu- tions for (1+1)-dimensional and (2+1)-dimensional Kaup-Kupershmidt equa- tions, Results Math. 63 (2013) 675–686. Google Scholar

[3] A.-M.Wazwaz, Multiple soliton solutions and rational solutions for the (2+1)- dimensional dispersive long water-wave system, Ocean Engineering 60 (2013), 95–98. Google Scholar

[4] R.Rach, J.-S.Duan and A.-M.Wazwaz, Solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method, J. Math. Chem. 52 (2014), 255–267. Google Scholar

[5] H.Triki and A.M.Wazwaz, A variety of exact periodic wave and solitary wave solutions for the coupled higgs equation, Z.Naturforsch.A, (Journal of Physical Sciences) 67 (2012), 545–549. Google Scholar

[6] H.Triki and A.M.Wazwaz, Traveling wave solutions for fifth-order KdV type equations with time-dependent coefficients, Commun. Nonlinear Sci. Numer. Simulat. 19 (2014), 404–408. Google Scholar

[7] H.Naher and F.A.Abdullah, New traveling wave solutions by the extended gen- eralized riccati equation mapping method of the (2+1) dimensional evolution equation, J. Appl. Math. 2012 (2012), 1–18. Google Scholar

[8] M.Dehghan and A.Nikpour, The solitary wave solution of coupled Klein-Gordon- Zakharov equations via two different numerical methods, Comput. Phys. Comm. 184 (2013), 2145–2158. Google Scholar

[9] M.K.Elboree, Hyperbolic and trigonometric solutions for some nonlinear evolution equation, Commun. Nonlinear Sci. Numer. Simulat. 17 (2010) 4085–4096. Google Scholar

[10] B.Salim Bahrami, H.Abdollahzadeh, I.M.Berijani, D.D.Ganji and M.Abdollahazdeh, Exact travelling solutions for some nonlinear physicalmodels by (G’/G)-expansion method, Pramana Journal of Physics 77 (2011), 263–275. Google Scholar

[11] A.Biswas, G.Ebadi, H.Triki, A.Yildirim and N.Yousefzadeh, Topological soliton and other exact solutions to KdV-Cauch-Dodd-Gibbon equation, Results Math. 63 (2013), 687–703. Google Scholar

[12] A.J.M.Jawad, M.D.Petkovic and A.Biswas, Soliton solutions to a few coupled nonlinear wave equations by tanh method, Iran. J. Sci. Technol. 37 (2013), 109–115. Google Scholar

[13] S.Zhou and C.Mu, Global conservative solutions for a model equation for shallow water waves of moderate amplitude, J.Differental Equations 256 (2014), 1793–1816. Google Scholar

[14] A.Biswas, A.J.M.Jawad, W.N.Manrakhan, A.K.Sarma and K.R.Khan, Optical solitons and complexitons of the Schrodinger-Hirota equation, Optics and Laser Technology 44 (2012), 2265–2269. Google Scholar

[15] H.Kim and R.Sakthivel, New exact travelling wave solutions of some nonlinear higher dimensional physical models, Rep. Math. Phys. 70 (2012), 39–50. Google Scholar

[16] H.Kim, J.-H.Bae and R.Sakthivel, Exact travelling wave solutions of two important nonlinear partial differential equations, Z.Naturforsch.A 69a (2014), 155–162. Google Scholar

[17] S.D.Zhu, The generalized Riccati equation mapping method in nonlinear evolution equation: application to (2+1)-dimensional Boitilion-Pempinelle equation, Chaos Solitons Fractals 37 (2008), 1335–1342. Google Scholar

[18] E.M.E.Zayed and A.H.Arnous, Many exact solutions for nonlinear dynamics of DNA model using the generalized Riccati equation mapping method, Sci. Res. Ess 8 (2013), 340–346. Google Scholar

[19] V.Novikov, Generalizations of the Cmassa-Holm equation, J.Phys. A 42 (2009) 342002. Google Scholar

[20] X.Geng and B.Xue, An extension of integrable peakon equations with cubic non-linearity, Nonolinearity 22 (2009), 1847–1856. Google Scholar