Abundant Exact Traveling Wave Solutions of the (2+1)-Dimensional Couple Broer-Kaup Equations

Mohammad Mobarak Hossain, Harun-Or- Roshid, Md. Abu Naim Sheikh


To describe the propagation of small amplitude waves in nonlinear dispersive media, it is frequently necessary to take account of dissipative mechanisms to perfectly reflect real situations in many branches of physics like plasma physics, fluid dynamics and nonlinear optics. In this paper, the exp(-Fi(Eta))-expansion method is employed to solve the (2+1)-Dimensional couple Broer-Kaup equations as a model for wave propagation in nonlinear media with dispersive and dissipative effects. As a result, a number of exact traveling wave solutions including solitary wave and periodic wave have been found for the equation. Some representative 3D profiles and 2D profiles for different values of variables of the wave solutions are graphically displayed and discussed.


The exp(-phi(zi))-expansion method; traveling wave solutions; the (2+1)-dimensional couple Broer-Kaup equations; nonlinear evolution equations

Full Text:



E. Yomba, “The modified extended Fan sub-equation method and its application to the (2+1)-Dimensional Broer-Kaup-Kuperschmidt equations,” Chaos, Solitons and Fractals, 27, 1, 187-196 (2006)

E.M.E. Zayed, H.A. Zedan and K.A. Gepreel, On the solitary wave solutions for nonlinear Hirota-Sasuma coupled KDV equations, Chaos, Solitons and Fractals, 22, 285-303, (2010)

M.L. Wang, Exact solutions for a compound KdV-Burgers equation, Physics Letters A, 213, 279-287 (1996)

E.G. Fan, “Extended tanh-function method and its applications to non-linear equations”, Physics Letters A, 277, 4-5, 1619-1625,(2008)

S.A. El-Wakil, M.A. Abdou, “New exact travelling wave solutions using modified extended tanh-function method”, Chaos, Solitons and Fractals, 31(4), 840-852 (2007)

Zhenya Yan, “Abundant families of Jacobi elliptic function solutions of the (G’/G)-dimensional integrable Davey-Stewartson- type equation via a new method ”, Chaos, Solitons and Fractals, 18, 2, 299-309 (2003)

M. L. Wang and Y.B.Zhou, “The periodic wave solutions for the Klein-Gordon-Schrodinger equations”, Physics Letters A, 318, 84-92, (2003)

M.L. Wang and X.Z. Li, “Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations”, Physics Letters A, 343, 48-54 (2005)

M.R. Miura, Backlund transformation, Springer, Berlin, 1978.

V.B. Matveev and M.A. Salle, Darboux transformation and solitons, Springer, Berlin, 1991.

T. Y. Wu and J. E. Zhang, “On modeling nonlinear long waves”, in Mathematics is for Solving Problems, P. Cook, V. Roytburd, and M. Tulin, Eds., pp. 233-249, SIAM, Philadelphia, Pa, USA, 1996.

G. Adomain, “Solving frontier problems of physics: The decomposition method”, Kluwer Academic Publishers, Boston, (1994)

A.M. Wazwaz, Partial Differential equations: Method and Applications, Taylor and Francis, 2002.

Sirendaoreji,J.Sun, Auxiliary equation method for solving nonlinear partial differential equations, Physics Letters A, 309, 387-396 (2003).

Sirendaoreji, Auxiliary equation method and new solutions of Klein-Gordon equations, Chaos, Solitons and Fractals, 31, 943-950 (2007)

M.L. Wang, X.Z. Li, J. Zhang, The -expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372 417-423 (2008)

A. Bekir, Application of the -expansion method for nonlinear evolution equations, Physics Letters A, 372, 3400-3406 (2008)

H.O. Roshid, M.F. Hoque and M.A. Akbar, New extended -expansion method for traveling wave solutions of nonlinear partial differential equations (NPDEs) in mathematical physics, Italian J. of pure and applied math., 33, 175-190 ( 2014)

H.O. Roshid, M.N. Alam, M.F. Hoque and M.A. Akbar, A new extended -expansion method to find exact traveling wave solutions of nonlinear evolution equations, Mathematics and Statistics, 1(3), 162-16 ( 2013)

A. Neyrame, A. Roozi, S. S .Hosseini, S. M. Shafiof, Exact travelling wave solutions for some nonlinear partial differential equations, Journal of King Saud University (Science), 22, 275-278, (2012)

M.A. Akbar, N.H.M. Ali and E.M.E. Zayed, “A Generalized and improved -expansion method for nonlinear evolution equations”, Mathematical Problems in Engineering, 2012, 22 (2012)

H.O. Roshid, N. Rahman and M.A. Akbar, “Traveling wave solutions of nonlinear Klein-Gordon equation by extended -expansion method”, Annals of pure and applied math., 3(1), 10-16 (2013)

E. M. E. Zayed and Shorog Al-Joudi, “Applications of an Extended - Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics”, Mathematical Problems in Engineering, 2010, Article ID 768573, doi: 10.1155/2010/768573

M.M. Zhao and C. Li, “The -expansion method applied to nonlinear evolution equations”, http://www.Paper.Edu.Cn

H.O. Roshid and M. A. Rahman, “The -expansion method with application in the (1+1)-dimensional classical Boussinesq equations”, Results in Physics, 4, 150-155 (2014)


  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

ISSN: 2394-3688

© Science Front Publishers