In this work, by implementing the generalized -expansion method, we present exact solutions of a completely integrable model namely; Whitham-Broer-Kaup (WBK) equations. The considered equation describes the small amplitude regime for dispersive long waves in shallow water. The solutions are obtained in terms of exponential functions, trigonometric functions and rational functions. Moreover, by selecting arbitrary constants appropriately in the solutions, we discovered various interesting periodic soliton structures. Also, innovative as well as distinct exact solutions raised in this paper may hold powerful applications in various fields of mathematical physics.
REFERENCES
1.
C.
Zhang
, B.
Tian
, X. H.
Meng
, X.
Lu
, K. J.
Cai
, and T.
Geng
, Z. Naturforsch. A
63
, 253
–260
(2008
).2.
L.
Ji
, X. Y.
Sheng
, and W. F.
Min
, Chinese Phys.
12
, 1049
–1053
(2003
).3.
S.
Murata
, Int. J. Non Linear Mech.
41
, 242
–246
(2006
).4.
F.
Engui
and Z.
Hongqing
, Appl. Math. Mech.
19
, 713
–716
(1998
).5.
Y.
Chen
, Q.
Wang
, and B.
Li
, Chaos Solitons Fract.
26
, 231
–246
(2005
).6.
Z.
Zhang
, X.
Yong
, and Y.
Chen
, J. Nonlinear Math. Phys.
15
, 383
–397
(2008
).7.
M. A.
Abdou
, Nonlinear Dyn.
52
, 277
–288
(2008
).8.
L.
Wang
, Y. T.
Gao
, X. L.
Gai
, X.
Yu
, and Z. Y.
Sun
, J. Nonlinear Math. Phys.
17
, 197
–211
(2010
).9.
S.
Guo
and Y.
Zhou
, Appl. Math. Comput.
215
, 3214
–3221
(2010
).10.
G. D.
Lin
, Y. T.
Gao
, X. L.
Gai
, and D. X.
Meng
, Nonlinear Dyn.
64
, 197
–206
(2011
).11.
M.
Wang
and X.
Li
, JAMP
2
, 823
–827
(2014
).12.
S.
Zhang
, M.
Liu
, and B.
Xu
, Therm. Sci.
21
, 137
–144
(2017
).13.
M.
Arshad
, A. R.
Seadawy
, D.
Lu
, and J.
Wang
, Chinese J. Phys.
55
, 780
–797
(2017
).14.
C.
Gu
, Soliton theory and its applications
(Zhejiang Publishing House of Science and Technology
, Hangzhou
, 1990
).15.
R. M.
Miura
, J. Math. Phys.
9
, 1202
–1204
(1968
).16.
R.
Conte
and M.
Musette
, J. Phys. A: Mathematical and General
25
, 5609
–5623
(1992
).17.
E. J.
Parkes
, B. R.
Duffy
, and P. C.
Abbott
, Phys. Lett. A
295
, 280
–286
(2002
).18.
Y. Z.
Peng
, Phys. Lett. A
314
, 401
–408
(2003
).19.
J. H.
He
, Chaos Solitons Fract.
26
, 695
–700
(2005
).20.
S.
Zhang
and T. C.
Xia
, J. Phys. A: Mathematical and Theoretical
40
, 227
–248
(2007
).21.
T.
Ozis
and A.
Yildirim
, Comput. Math. Appl.
54
, 1039
–1042
(2007
).22.
E. M. E.
Zayed
and K. A.
Gepreel
, J. Math. Phys.
50
, p. 013502
(2009
).23.
H.
Zhang
, Chaos Solitons Fract.
39
, 1020
–1026
(2009
).24.
L. W.
An
, C.
Hao
, and Z. G.
Cai
, Chinese Phys. B
18
, 400
–404
(2009
).25.
A. J. M.
Jawad
, M. D.
Petkovic
, and A.
Biswas
, Appl. Math. Comput.
217
, 869
–877
(2010
).26.
M. M.
Miah
, H. M. S.
Ali
, M. A.
Akbar
, and A. M.
Wazwaz
, Eur. Phys. J. Plus
132
, p. 252
(2017
).27.
Y. J.
Yang
and Y. M.
Zhao
, Adv. Mat. Res.
1044-1045
, 1110
–1112
(2014
).28.
K. A.
Gepreel
and T. A.
Nofal
, IJPAM
106
, 1003
–1016
(2016
).29.
Y.
Zhang
, L.
Zhang
, and J.
Pang
, Advances in Applied Mathematics
6
, 212
–217
(2017
).30.
G.
Akram
and N.
Mahak
, Eur. Phys. J. Plus
133
, p. 212
(2018
).31.
S.
Sirisubtawee
and S.
Koonprasert
, Adv. Math. Phys.
2018
, p. 7628651
(2018
).
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