Solitary Wave solutions of the BK equation and ALWW system by using the first integral method

Document Type : Research Paper


Department of mathematics,Gonbad University


Solitary wave solutions to the Broer-Kaup equations and approximate long water wave equations are considered challenging by using the rst integral method.The exact solutions obtained during the present investigation are new. This method can be applied to nonintegrable equations as well as to integrable ones.


[1] W.X. Ma, Travelling wave solutions to a seventh order generalized KdV equation, Phys.
Lett. A. 180 (1993) 221 􀀀 224:
[2] W. Mal iet, Solitary wave solutions of nonlinear wave equations, Amer. J. Phys.
60(7) (1992) 650 􀀀 654:
[3] A.H. Khater, W. Mal iet, D.K. Callebaut and E.S. Kamel, The tanh method, a simple
transformation and exact analytical solutions for nonlinear reactiondi usion equations,
Chaos Solitons Fractals 14(3) (2002) 513 􀀀 522:
[4] A.M. Wazwaz, Two reliable methods for solving variants of the KdV equation with
compact and noncompact structures. Chaos Solitons Fractals, 28(2) (2006) 454 􀀀 462:
[5] W. X. Ma and B. Fuchssteiner, Explicit and exact solutions to a Kolmogorov- Petrovskii-
Piskunov equation, Int. J. Non-Linear Mech. 31 (1996) 329-338.
[6] S.A. El-Wakil and M.A. Abdou , New exact travelling wave solutions using modi ed
extended tanh-function method, Chaos Solitons Fractals, 31(4) (2007) 840-852.
[7] E. Fan, Extended tanh-function method and its applications to nonlinear equations,
Phys. Lett. A. 277(4 -5) (2000) 212 -218.
[8] A.M.Wazwaz, The tanh-function method: Solitons and periodic solutions for the Dodd-
Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos Solitons and
Fractals 25(1) (2005) 55-63.
[9] T.C. Xia ,B. Li and H.Q. Zhang, New explicit and exact solutions for the Nizhnik-
Novikov-Vesselov equation, Appl. Math. E-Notes 1, (2001) 139-142.
[10] A.M. Wazwaz, The sine-cosine method for obtaining solutions with compact and non-
compact structures, Appl. Math. Comput. 159(2) (2004) 559-576:
[11] A.M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math.
Comput. Modelling, 40(5-6) (2004) 499-508:
[12] E. Yusufoglu and A. Bekir, Solitons and periodic solutions of coupled nonlinear evolution
equations by using Sine-Cosine method, Internat. J. Comput. Math. 83(12) (2006) 915-
[13] M. Inc and M. Ergut, Periodic wave solutions for the generalized shallow water
wave equation by the improved Jacobi elliptic function method, Appl. Math. E-Notes
5 (2005) 89 􀀀 96:
[14] Zhang Sheng, The periodic wave solutions for the (2 + 1) dimensional Konopelchenko-
Dubrovsky equations, Chaos Solitons Fractals, 30 (2006) 1213-1220.
[15] W. X. Ma and J.-H. Lee, A transformed rational function method and exact solutions to
the (3+1)-dimensional Jimbo-Miwa equation, Chaos Solitons Fractals, 42 (2009) 1356-
[16] Z.S Feng, X.H Wang, The rst integral method to the two-dimensional Burgers-KdV
equation, Phys. Lett. A. 308 (2002) 173-178.
[17] T.R. Ding and C.Z. Li, Ordinary di erential equations. Peking University Press, Peking,
[18] Z.S. Feng, X.H Wang, The rst integral method to the two-dimensional Burgers-KdV
equation, Phys. Lett. A. 308 (2002) 173 - 178.
[19] K.R. Raslan, The rst integral method for solving some important nonlinear partial
di erential equations, Nonlinear Dynam 53 (2008) 281:
[20] D.J. Kaup, A higher order water wave equation and method for solving it, Progress of
Theoretical physics 54 (1975) 396 - 408.
[21] Mingliang Wang, Jinliang Zhang, Xiangzheng Li, Application of the (G0
G )-expansion
to travelling wave solutions of the Broer-Kaup and the approximate long water wave
equations, Appl. Math. Comput. 206 (2008) 321 - 326:
[22] G.B. Whitham, Variational methods and application to water waves, Proceedings of the
Royal Society of London Series A 299 (1967) 6-25:
[23] L.J.F. Broer, Approximate equations for long water waves, Applied Scienti c Research
31 (1975) 377-395.