Multiple Exact Travelling Solitary Wave Solutions of Nonlinear Evolution Equations

Main Article Content

M. M. El-Horbaty
F. M. Ahmed


An extended Tanh-function method with Riccati equation is presented for constructing multiple exact travelling wave solutions of some nonlinear evolution equations which are particular cases of a generalized equation. The results of solitary waves are general compact forms with non-zero constants of integration. Taking the full advantage of the Riccati equation improves the applicability and reliability of the Tanh method with its extended form.

Extended Tanh method, Riccati equation, solitary waves, evolution equations.

Article Details

How to Cite
El-Horbaty, M., & Ahmed, F. (2019). Multiple Exact Travelling Solitary Wave Solutions of Nonlinear Evolution Equations. Asian Research Journal of Mathematics, 14(2), 1-13.
Original Research Article


Malflient W. Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 1992;60(7):650–654.

M. W. The tanh method: A tool for solving certain classes of nonlinear evolution and wave equations. J. Comput. Appl. Math. 2004;164–165:529–541.

Seadawy AR, Alamri SZ. Mathematical methods via the nonlinear two-dimensional water waves of Olver dynamical equation and its exact solitary wave solutions. Results Phys. 2018;8:286–291.

Bibi S, Ahmed N, Khan U, Mohyud-Din ST. Some new exact solitary wave solutions of the van der Waals model arising in nature. Results Phys. 2018;9:648–655.

Lu D, Seadawy AR, Ali A. Dispersive traveling wave solutions of the Equal-Width and Modified Equal-Width equations via mathematical methods and its applications. Results Phys. 2018;9: 313–320.

Azmol Huda M, Akbar MA, Shanta SS. The new types of wave solutions of the Burger’s equation and the Benjamin–Bona–Mahony equation. J. Ocean Eng. Sci. 2017;3(1):1–10.

Elwakil SA, El-Labany SK, Zahran MA, Sabry R. Modified extended tanh-function method and its applications to nonlinear equations. Appl. Math. Comput. 2005;161(2):403–412.

Wazwaz AM. The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput. 2007;188(2):1467–1475.

Chen H, Zhang H. New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation. Chaos, Solitons and Fractals. 2004;19(1):71–76.

Albowits MJ, and Clarkson. Nonlinear evolution equations and Inverse scattering transformation. Cambridge: Cambridge University Press; 1991.

Miura MR. Backlund transformation. Berlin: Springerr; 1978.

Zhang ZY. Jacobi elliptic function expansion method for the modified Korteweg-de Vries-Zakharov-Kuznetsov and the Hirota equations. Rom. J. Phys. 2015;60(9–10):1384–1394.

El-Horbaty MM, Ahmed FM, Mansour M, Osamma AT. New optics solutions for the nonlinear (2+1)-dimensional generalization of complex nonlinear schr? dinger equation. Int. J. Sci. Res. 2017;6(2):608–613.

Chen L, Yang L, Zhang R, Cui J. Generalized (2 + 1)-dimensional mKdV-Burgers equation and its solution by modified hyperbolic function expansion method. Results Phys. 2019;13:102280.

Maliet W, Hereman W. 1996_Malfliet-Hereman-PhysicaScripta_The tanh method: Exact solutions of nonlinear evolution and wave equations.pdf. 1996;563–568.

Fan E. Extended tanh-method and its applications to nonlinear equations Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A. 2000;277:212–218.

Taghizadeh N, Mirzazadeh M. The modified extended tanh method with the riccati equation for solving nonlinear partial differential equations modified extended tanh method with the Riccati equation. Math. Aeterna. 2012;2(2):145–153.

Zheng XD, Xia TC, Zhang HQ. New exact traveling wave solutions for compound KdV-Burgers equations in mathematical physics. Appl. Math. E - Notes. 2002;2:45–50.

El-Horbaty MM, Ahmed FM. The solitary travelling wave solutions of some nonlinear partial differential equations using the modified extended tanh function method with riccati equation. Asian Res. J. Math. 2018;8(3):1–13.

Kudryashov NA. Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 2009;14(9–10):3507–3529.