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Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.58 (Online Publication: Dec., 2013) 

 

 

BIBECHANA 
A Multidisciplinary Journal of Science, Technology and Mathematics 

ISSN 2091-0762 (online) 
Journal homepage: http://nepjol.info/index.php/BIBECHANA 

 

Application of the new approach of generalized )/( GG′ -expansion 

method to find exact solutions of nonlinear PDEs in mathematical 

physics 
 

Md. Nur Alam
1
 and M. Ali Akbar

2*  

1
Department of Mathematics, Pabna University of Science and Technology, Bangladesh 

2Department of Applied Mathematics, University of Rajshahi, Bangladesh 
*
Email: ali_math74@yahoo.com 

Article history: Received 27 October, 2013;  Accepted 17 November, 2013 

 

Abstract 

The exact solutions of nonlinear evolution equations (NLEEs) play a crucial role to make known the 
internal mechanism of complex physical phenomena. In this article, we construct the traveling wave 

solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation by means of the 

new approach of generalized )/( GG′ -expansion method. Abundant traveling wave solutions with 

arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in 
terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of 

generalized )/( GG′ -expansion method is a powerful and concise mathematical tool for solving 

nonlinear partial differential equations. 

 

Keywords: New approach of generalized )/( GG′ -expansion method, ZK-BBM equation; 

homogeneous balance, traveling wave solutions, solitary wave solutions, nonlinear evolution 

equation. 

 
 

1. Introduction 
 

Nonlinear phenomena arise in several aspects of physics as well as other natural and applied sciences. 

Essentially all the fundamental equations in physical sciences are nonlinear and, in general, such 

NLEEs are often very complicated to solve explicitly. The exact solutions of NLEEs play an 

important role in the study of nonlinear physical phenomena. Therefore, the powerful and efficient 
methods to find exact solutions of nonlinear equations still have drawn a lot of interest by diverse 

group of scientists. In the past three decades, there has been significant progress in the development of 

finding effective methods for obtaining exact solutions of NLEEs. These methods are the Exp-
function method [1-3], the generalized Riccati equation [4], the Miura transformation [5], the Jacobi 

elliptic function expansion method [6, 7], the Hirota’s bilinear method [8], the sine-cosine method [9], 

the tanh-function method [10], the extended tanh-function method [11-12], the homogeneous balance 

method [13], the modified Exp-function method [14], the )/( GG′ -expansion method [15-22], the  

 



 

 

Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.59 (Online Publication: Dec., 2013) 

 

improved )/( GG′ -expansion method [23], the modified simple equation method [24-28], the inverse 

scattering transform [29] and so on. 

Recently, Naher and Abdullah [30] established a highly effective extension of the )/( GG′  expansion 

method, called the new generalized )/( GG′  expansion method to obtain exact traveling wave 

solutions of NLEEs. The aim of this article is to look for new study relating to the new generalized 

)/( GG′  expansion method for solving the renowned ZK-BBM equation to make obvious the 

effectiveness and usefulness of the method. 

The outline of this paper is organized as follows: In Section 2, we give the description of the new 

generalized )/( GG′  expansion method. In Section 3, we apply this method to the ZK-BBM equation. 

In Section 4, Discussions are given. Conclusions are given in Section 5. 

2.  Description of the new generalized ( / )G G′ -expansion method 

Let us consider a general nonlinear PDE in the form 

),,,,,,( …
xtttxxxt

vvvvvvΦ ,         (1) 

where ),( txvv = is an unknown function, Φ  is a polynomial in ),( txv and its derivatives in which 

highest order derivatives and nonlinear terms are involved and the subscripts stand for the partial  

derivatives. 

Step 1: We combine the real variables x  and t  by a complex variable η  

)(),( ηvtxv = ,    tVx ±=η ,        (2) 

where V  is the speed of the traveling wave. The traveling wave transformation (2) converts Eq. (1) 

into an ordinary differential equation (ODE) for )(ηvv = : 

  ),,,,( ⋯vvvv ′′′′′′ψ ,         (3) 

where ψ  is a polynomial of v  and its derivatives and  the superscripts indicate the ordinary 

derivatives with respect to η . 

Step 2: According to possibility, Eq. (3) can be integrated term by term one or more times, yields 

constant(s) of integration. The integral constant may be zero, for simplicity. 

Step 3. Suppose the traveling wave solution of Eq. (3) can be expressed as follows: 

i

N

i

i

i

N

i

i
MdMdv

−

==

+++= ∑∑ )()()(
10

βαη ,                (4) 

where either 
N

α  or 
N

β  may be zero, but could be zero simultaneously, 
i

α  ),,2,1,0( Ni ⋯=  and 

i
β  ),,2,1( Ni ⋯=  and d  are arbitrary constants to be determined and  )(ηM  is 

  )/()( GGM ′=η          (5) 

where )(ηGG =  satisfies the following auxiliary nonlinear ordinary differential equation: 

  0)( 22 =′−−′−′′ GCGEGBGGAG ,       (6) 

where the prime stands for derivative with respect to η ; A , B ,C  and E  are real parameters. 

Step 4: To determine the positive integer N , taking the homogeneous balance between the highest 

order nonlinear terms and the derivatives of the highest order appearing in Eq. (3). 

 



 

 

Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.60 (Online Publication: Dec., 2013) 

 

Step 5: Substitute Eq. (4) and Eq. (6) including Eq. (5) into Eq. (3) with the value of N  obtained in 

Step 4, we obtain polynomials in 
N

Md )( + ),2,1,0( ⋯=N  and 
N

Md
−+ )(  ),2,1,0( ⋯=N . 

Subsequently, we collect each coefficient of the resulted polynomials to zero, yields a set of algebraic 

equations for 
i

α  ),,2,1,0( Ni ⋯=  and 
i

β  ),,2,1( Ni ⋯= , d  and V . 

Step 6: Suppose that the value of the constants 
i

α  ),,2,1,0( Ni ⋯= ,
i

β  ),,2,1( Ni ⋯= , d  and V  

can be found by solving the algebraic equations obtained in Step 5. Since the general solutions of Eq. 

(6) are known to us, inserting the values of 
i

α  ),,2,1,0( Ni ⋯= , 
i

β  ),,2,1( Ni ⋯=  , d  and V  

into Eq. (4), we obtain more general type and new exact traveling wave solutions of the nonlinear 

partial differential equation (1). 

Step 7: Using the general solution of Eq. (6), we have the following solutions of Eq. (5): 

Family 1: When ,0≠B  CA −=ω  and ,0)(42 >−+=Ω CAEB  

 










 Ω
+








 Ω










 Ω
+








 Ω

Ω
+=







 ′
=

ηη

ηη

ωω
η

A
C

A
C

A
C

A
C

B

G

G
M

2
sinh

2
cosh

2
cosh

2
sinh

22
)(

21

21

    (7) 

Family 2: When ,0≠B  CA −=ω  and ,0)(42 <−+=Ω CAEB  

 










 Ω−
+








 Ω−










 Ω−
+








 Ω−
−

Ω−
+=







 ′
=

ηη

ηη

ωω
η

A
C

A
C

A
C

A
C

B

G

G
M

2
sin

2
cos

2
cos

2
sin

22
)(

21

21

    (8) 

Family 3: When ,0≠B  CA −=ω  and ,0)(42 =−+=Ω CAEB  

 
ηω

η
21

2

2
)(

CC

CB

G

G
M

+
+=







 ′
=                    (9) 

Family 4: When ,0=B  CA −=ω  and ,0>=∆ Eω  

 

)sinh()cosh(

)cosh()sinh(

)(

21

21

ηη

ηη

ω
η

A
C

A
C

A
C

A
C

G

G
M

∆
+

∆

∆
+

∆
∆

=






 ′
=               (10) 

Family 5: When ,0=B  CA −=ω  and ,0<=∆ Eω  

 

)sin()cos(

)cos()sin(

)(

21

21

ηη

ηη

ω
η

A
C

A
C

A
C

A
C

G

G
M

∆−
+

∆−

∆−
+

∆−
−∆−

=






 ′
=               (11) 

3. Application of the method 

In this section, we will put forth the new generalized )/( GG′  expansion method to construct many 

new and more general traveling wave solutions of the ZK-BBM equation. Let us consider the ZK-
BBM equation, 



 

 

Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.61 (Online Publication: Dec., 2013) 

 

  02 =−−+
txxxxt

bvvvavv .                   (12) 

We will use the traveling wave transformation Eq. (2) into the Eq. (12), which yields: 

  02)1( =′′′−′−′+ vVbvvavV .       (13) 

Integrating Eq. (13) with respect to η  once yields 

           ,0)1( 2 =+′′−−+ KvVbvavV        (14) 

where K  is an integration constant which is to be determined. 

Taking the homogeneous balance between highest order nonlinear term 
2
v  and linear term of the 

highest order v ′′  in Eq. (14), we obtain 2=N . Therefore, the solution of Eq. (14) is of the form: 

,)()()()()( 2

2

1

1

2

210

−− ++++++++= MdMdMdMdv ββαααη     (15) 

where 21210 ,,,, ββααα  and d  are constants to be determined. 

Substituting Eq. (15) together with Eqs. (5) and (6) into Eq. (14), the left-hand side is converted into 

polynomials in ( )NMd + ,.......)2,1,0( =N  and ( ) N

Md
−+ ),2,1( ⋯=N . We collect each 

coefficient of these resulted polynomials to zero yields a set of simultaneous algebraic equations (for 

simplicity, the equations are not presented here) for 0α , 1α ,  2α , 1β , 2β  d , K  and V . Solving 

these algebraic equations with the help of computer algebra, we obtain following: 

Set 1: )2816(
4

1 44242242222222

4
VAAVBVbAEBVbEVb

aA
K ++−+−−−= ωω ,  

)81212(
2

1 22222

20 bVBVAAbVEbVBdbVd
aA

+−−−+−= ωωωα , ,VV =         (16) 

),232(
6 2223

21 EBdBEdBdd
aA

bV
−+−+= ωωωβ  ,dd =  ,01 =α  ,02 =α  

)222(
6 2222324

22 EdBBdEEdbdd
aA

bV
++−−+−= ωωωβ . 

where ,CA −=ω ,V ,d ,A ,B ,C E  are free parameters. 

Set 2: )2816(
4

1 44242242222222

4
VAAVBVbAEBVbEVb

aA
K ++−+−−−= ωω , 

)81212(
2

1 22222

20 bVBVAAbVEbVBdbVd
aA

+−−−+−= ωωωα , ,VV =   (17) 

)2(
6 2

21 ωωα Bd
aA

bV
+= , 

2

2

2

6

aA

bVω
α −= , ,01 =β 02 =β . 

where ,CA −=ω ,V ,d ,A ,B ,C E  are free parameters. 

Set 3: )16128256)1((
4

1 4222222222242

4
BVbEBVbEAVbAV

aA
K −−−+−= ωω , 

)28)1((
2

1 22

20 bVBbVEAV
aA

+++= ωα , 
2

2

2

6

aA

bVω
α −= , ,01 =α ,VV =  



 

 

Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.62 (Online Publication: Dec., 2013) 

 

),816(
8

3 4222

222 BEBE
aA

bV
++−= ωω

ω
β ,01 =β

ω2
B

d −=                  (18) 

where ,CA −=ω ,V ,A ,B ,C E  are free parameters. 

For set 1, substituting Eq. (16) into Eq. (15), along with Eq. (7) and simplifying, yields following 

traveling wave solutions, if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

2
2

1
101 ))

2
coth(

22
())

2
coth(

22
()(

1

−− ΩΩ
+++

ΩΩ
+++= η

ωω
βη

ωω
βαη

A

B
d

A

B
dv . 

2
2

1
101 ))

2
tanh(

22
())

2
tanh(

22
()(

2

−− ΩΩ
+++

ΩΩ
+++= η

ωω
βη

ωω
βαη

A

B
d

A

B
dv . 

Substituting Eq. (16) into Eq. (15), along with Eq. (8) and simplifying, our exact solutions become, if 

01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

        
2

2
1

101 ))
2

cot(
22

())
2

cot(
22

()(
3

−− Ω−Ω−
+++

Ω−Ω−
+++= η

ωω
βη

ωω
βαη

A

B
d

A

B
dv . 

       
2

2
1

101 ))
2

tan(
22

())
2

tan(
22

()(
4

−− Ω−Ω−
−++

Ω−Ω−
−++= η

ωω
βη

ωω
βαη

A

B
d

A

B
dv . 

Substituting Eq. (16) into Eq. (15), together with Eq. (9) and simplifying, our obtained solution 

becomes: 

.)
2

()
2

()( 2

21

2
2

1

21

2
1015

−−

+
+++

+
+++=

ηω
β

ηω
βαη

CC

CB
d

CC

CB
dv  

Substituting Eq. (16) into Eq. (15), along with Eq. (10) and simplifying, we obtain following traveling 

wave solutions, if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

.))coth(())coth(()( 2
2

1
1016

−− ∆∆
++

∆∆
++= η

ω
βη

ω
βαη

A
d

A
dv  

2
2

1
101 ))tanh(())tanh(()(

7

−− ∆∆
++

∆∆
++= η

ω
βη

ω
βαη

A
d

A
dv . 

Substituting Eq. (16) into Eq. (15), together with Eq. (11) and simplifying, our obtained exact 

solutions become, if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

2
2

1
101 ))cot(())cot(()(

8

−− ∆−∆−
++

∆−∆−
++= η

ω
βη

ω
βαη

A
d

A
dv  

2
2

1
101 ))tan(())tan(()(

9

−− ∆−∆−
−+

∆−∆−
−+= η

ω
βη

ω
βαη

A
d

A
dv , 

where .Vtx −=η  

Again for set 2, substituting Eq. (17) into Eq. (15), along with Eq. (7) and simplifying, our traveling 

wave solutions become, if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 



 

 

Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.63 (Online Publication: Dec., 2013) 

 

)),
2

(coth3)1()4(2(
2

1
)( 222

221
ηωη

A
bVVAEBbV

aA
v

Ω
Ω−+++=  

)),
2

(tanh3)1()4(2(
2

1
)( 222

222
ηωη

A
bVVAEBbV

aA
v

Ω
Ω−+++=  

Substituting Eq. (17) into Eq. (15), along with Eq. (8) and simplifying yields exact solutions, if 

01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

)),
2

(cot3)1()4(2(
2

1
)( 222

223
ηωη

A
bVVAEBbV

aA
v

Ω−
Ω++++=  

))
2

(tan3)1()4(2(
2

1
)( 222

224
ηωη

A
bVVAEBbV

aA
v

Ω−
Ω++++= . 

Substituting Eq. (17) into Eq. (15), along with Eq. (9) and simplifying, our obtained solution 

becomes: 

),)(12)1()4(2(
2

1
)( 2

21

2222

225 η
ωωη

CC

C
bVVAEBbV

aA
v

+
−+++=  

Substituting Eq. (17) into Eq. (15), together with Eq. (10) and simplifying, yields following traveling 

wave solutions, if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

  )))(coth)coth((12)1()8((
2

1
)( 222

226
ηηωη

AA
BbVVAEBbV

aA
v

∆
∆−

∆
∆++++−= . 

  )))(tanh)tanh((12)1()8((
2

1
)( 222

227
ηηωη

AA
BbVVAEBbV

aA
v

∆
∆−

∆
∆++++−= . 

Substituting Eq. (17) into Eq. (15), along with Eq. (11) and simplifying, our exact solutions become, 

if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

))).(cot)cot((12)1()8((
2

1
)( 222

228
ηηωη

AA
iBbVVAEBbV

aA
v

∆−
∆+

∆−
∆++++−=

 

))(tan)tan((12)1()8((
2

1
)( 222

229
ηηωη

AA
iBbVVAEBbV

aA
v

∆−
∆−

∆−
∆−+++−= , 

where .Vtx −=η  

Similarly, for set 3, substituting Eq. (18) into Eq. (15), together with Eq. (7) and simplifying, yields 

following traveling wave solutions, if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

)
2

(tanh
4

)
2

(coth
2

3
)( 2

2
22

2031
η

ω
ηαη

A

b

AaA

bV
v

Ω
Ω

+
ΩΩ

−= . 

)
2

(coth
4

)
2

(tanh
2

3
)( 2

2
22

2032
η

ω
ηαη

A

b

AaA

bV
v

Ω
Ω

+
ΩΩ

−= . 



 

 

Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.64 (Online Publication: Dec., 2013) 

 

Substituting Eq. (18) into Eq. (15), along with Eq. (8) and simplifying, we obtain following solutions, 

if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

)
2

(tan
4

)
2

(cot
2

3
)( 2

2
22

2033
η

ω
ηαη

A

b

AaA

bV
v

Ω−
Ω

−
Ω−Ω

+= . 

)
2

(cot
4

)
2

(tan
2

3
)( 2

2
22

2034
η

ω
ηαξ

A

b

AaA

bV
v

Ω−
Ω

−
Ω−Ω

+= . 

Substituting Eq. (18) into Eq. (15), along with Eq. (9) and simplifying, our obtained solution 

becomes: 

.)()(
6

)( 2

21

2
2

2

21

2

2

2

035

−

+
+

+
−=

η
β

η
ω

αη
CC

C

CC

C

aA

bV
v  

Substituting Eq. (18) into Eq. (15), along with Eq. (10) and simplifying, yields  following exact 

traveling wave  solutions, if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

2
2

2

2

2

03 ))coth(
2

())coth(
2

(
6

)(
6

−∆∆
+

−
+

∆∆
+

−
−= η

ωω
βη

ωω
ω

αη
A

B

A

B

aA

bV
v . 

2
2

2

2

2

03 ))tanh(
2

())tanh(
2

(
6

)(
7

−∆∆
+

−
+

∆∆
+

−
−= η

ωω
βη

ωω
ω

η
A

B

A

B

aA

bV
av . 

Substituting Eq. (18) into Eq. (15), along with Eq. (11) and simplifying, our obtained exact solutions 

become, if 01 =C  but ;02 ≠C 02 =C  but 01 ≠C  respectively: 

2
2

2

2

2

03 ))cot(
2

())cot(
2

(
6

)(
8

−∆−∆−
+

−
+

∆−∆−
+

−
−= η

ωω
βη

ωω
ω

αη
A

B

A

B

aA

bV
v . 

2
2

2

2

2

03 ))tan(
2

())tan(
2

(
6

)(
9

−∆−∆−
−

−
+

∆−∆−
−

−
−= η

ωω
βη

ωω
ω

αη
A

B

A

B

aA

bV
v . 

where .Vtx −=η  

 

4. Discussions 

The advantages and validity of the method over the basic )/( GG′ -expansion method and the 

generalized and improved )/( GG′ -expansion method have been discussed in the following. 

Advantages: The crucial advantage of the new approach against the basic )/( GG′ -expansion 

method and the generalized and improved )/( GG′ -expansion method is that the method provides 

more general and abundant new exact traveling wave solutions with many real parameters. The exact 

solutions have its great importance to expose the inner mechanism of the complex physical 

phenomena. Apart from the physical application, the close-form solutions of nonlinear evolution 

equations assist the numerical solvers to compare the correctness of their results and help them in the 

stability analysis. 

 

 



 

 

Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.65 (Online Publication: Dec., 2013) 

 

Validity: In Ref. [31] Zhang et al. used the linear ordinary differential equation (LODE) as auxiliary 

equation and traveling wave solutions presented in the form ∑
−=

′=
m

mi

i

i
GGau ,)/()(ξ  where 0≠

m
a . 

Akbar et al. [17] used same auxiliary equation (LODE) and the solution is presented in the form 

∑
−=

−

′+
=

m

mn

n

n

GGd

e
u ,

))/((
)(ξ  where either 

m
e−  or 

m
e  may be zero, but both  

m
e−  or 

m
e  cannot be 

zero together. It is noteworthy to point out that some of our solutions are coincided with previously 

published results, if the parameters are taken particular values which validate our solutions. Moreover, 

In Ref. [31] Zhang et al. investigated the ZK-BBM equation to obtain exact solutions via the 

improved )/( GG′ -expansion method and achieved only six solutions (A.1)-(A.6) (see appendix A). 

On the other hand, by using the generalized and improved )/( GG′ -expansion method, Akbar et al. 

[17] obtained nine solutions (B.1)-(B.9) (see appendix B) of the ZK-BBM equation. But in this 

article twenty seven exact solutions including (A.1)-(A.6) and (B.1)-(B.2) of the ZK-BBM equation 

are constructed by applying the new approach of generalized )/( GG′ -expansion method. 

5. Conclusion 
 

In this study, we considered the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation. 

We employed the new approach of generalized )/( GG′ -expansion method for the exact solution to 

this equation and constructed some new solutions which are not found in the previous literature. The 

method offers solutions with free parameters that might be imperative to explain some intricate 

physical phenomena. This study shows that the new generalized )/( GG′ -expansion method is quite 

efficient and practically well suited to be used in finding exact solutions of NLEEs. Also, we observe 

that the new generalized )/( GG′ -expansion method is straightforward and can be applied to many 

other nonlinear evolution equations. 
 

References  
 

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10.1155/2012/459879. 

18. A. Bekir, Phys. Lett. A, 372 (2008) 3400. 



 

 

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19. E.M.E. Zayed, J. Appl. Math. Comput., 30 (2009) 89. 

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pages doi: 10.1155/2013/146704. 

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Appendix A: Zhang et al. solutions [31] 

Zhang et al. [31] established exact solutions of the well-known the ZK-BBM equation by using the improved 

)/( GG′ -expansion method which are as follows: 

When ,042 >− µλ  

2

1
2 2

2 1
2 2

2 2
2 1

2 2
2 1

2

2 2
2 1

2

1

6bV
u

1 1
C sinh 4 C cosh 4

1 2 2a( ( ) ( 4 )( ))
1 12 2

C cosh 4 C sinh 4
2 2

6bV

1 1
C sinh 4 C cosh 4

1 2 2a( ( ) ( 4 )( ))
1 12 2

C cosh 4 C sinh 4
2 2

bV 8bV V 1
,

2a

− µ
= −

λ − µξ+ λ − µξλ
− + λ − µ

λ − µξ + λ − µξ

− µλ
−

λ − µξ + λ − µξλ
− + λ − µ

λ − µξ+ λ − µξ

λ + µ − −

(A.1) 

where 21 ,CC  are arbitrary constants. 

When ,042 <− µλ  

   

2

1
2 2

2 1
2 2

2 2
2 1

2 2
2 1

2

2 2
2 1

2

2

6bV
u

1 1
C sin 4 C cos 4

1 2 2a( ( ) ( 4 )( ))
1 12 2

C cos 4 C sin 4
2 2

6bV

1 1
C sin 4 C cos 4

1 2 2a( ( ) ( 4 )( ))
1 12 2

C cos 4 C sin 4
2 2

bV 8bV V 1
,

2a

− µ
= −

− µ −λ ξ + µ − λ ξλ
− + µ −λ

µ − λ ξ + µ − λ ξ

− µλ
−

− µ − λ ξ + µ − λ ξλ
− + µ − λ

µ −λ ξ + µ −λ ξ

λ + µ − −

(A.2) 

where  21 ,CC  are arbitrary constants. 



 

 

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When ,042 =− µλ  

2

1
22 2

1 2 1 2

2

2

6bV 6bV
u

C C
a( ( ) ( )) a( ( ) a( ( ) ( ))

2 C C 2 2 C C

bV 8bV V 1
,

2a

− µ − µλ
= − −

λ λ λ
− + − + − +

+ ξ + ξ

λ + µ − −

 (A.3) 

where  21 ,CC  are arbitrary constants. 

When ,042 >− µλ  

2 2
2 2 1

2
2

2 2
2 1

2

1

1 1
C sinh 4 C cosh 4

3bV( 4 ) 2 2u ( ))
1 12a

C cosh 4 C sinh 4
2 2

bV 8bV V 1
,

2a

λ − µξ + λ − µξ− λ − µ
=

λ − µξ + λ − µξ

λ + µ − −
−

            (A.4) 

where  21 ,CC  are arbitrary constants. 

For ,01 >C
2

2

2

1 CC <  above, then the solutions (A.4) turns into 

22 2
2

2 01

43bV( 4 ) bV 4bV V 1
u sech ( ) ,

2a 2 2a

λ − µξ− λ − µ λ − µ − −
= + ξ −   ),(tanh

2

11

0
C

C−=ξ  

When ,042 <− µλ  

2 2
2 2 1

2
2

2 2
2 1

2

2

1 1
C sin 4 C cos 4

3bV( 4 ) 2 2u ( ))
1 12a

C cos 4 C sin 4
2 2

2bV 8bV V 1
,

2a

− µ −λ ξ + µ − λ ξλ − µ
=

µ − λ ξ + µ − λ ξ

λ − µ + +
+

            (A.5) 

where 21 ,CC  are arbitrary constants. 

When ,042 =− µλ  

2 2
2

2 2
1 2

2

6bVC 2bV 8bV V 1
u ,

2aa(C C )

λ − µ + +
= +

+ ξ
    (A.6) 

where 21 ,CC  are arbitrary constants. 

Appendix B: Akbar et al. [17] 

Akbar et al. [17] established a generalized and improved )/( GG′ -expansion method and studied ZK-BBM 

equation, and obtained following solutions: 

When ,042 >− µλ  

 



 

 

Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.68 (Online Publication: Dec., 2013) 

 

2 2 3 2

1
2 2

2 2

2 2

2 2

2 2

2

2

1

6bV{ d( d d 2 d 2d 2 )}
u

1 1
Bsinh 4 Acosh 4

1 2 2a(d ( 4 )( ))
1 12 2

Bcosh 4 Asinh 4
2 2

6bV{ d( 2d 2 d 2 )}

1 1
Bsinh 4 Acosh 4

1 2 2a(d ( 4 )(
1 12 2

Bcosh 4 Asinh
2 2

− µ + λ + − λ + µ − λµ
= −

λ − µξ + λ − µξλ
− + λ − µ

λ − µξ + λ − µξ

− µλ − λ + − λ + µ

λ − µξ + λ − µξλ
− + λ − µ

λ − µξ+ 2

2

))

4

{(bV 8bV V 1) 12bVd(d )
,

2a

−

λ − µξ

λ + µ − − + −λ

  (B.1) 

where BA,  are arbitrary constants. 

When ,042 <− µλ  

2 2 3 2

1
2 2

2 2

2 2

2 2

2 2

2

2

2

6bV{ d( d d 2 d 2d 2 )}
u

1 1
Bsinh 4 Acosh 4

1 2 2a(d ( 4 )( ))
1 12 2

Bcosh 4 Asinh 4
2 2

6bV{ d( 2d 2 d 2 )}

1 1
Bsinh 4 Acosh 4

1 2 2a(d ( 4 )(
1 12 2

Bcosh 4 Asinh
2

− µ + λ + − λ + µ − λµ
= −

− µ −λ ξ + µ − λ ξλ
− + µ − λ

µ − λ ξ + µ −λ ξ

µλ − λ + − λ + µ

− µ − λ ξ + µ −λ ξλ
− + µ −λ

µ −λ ξ + 2

2

))

4
2

{(bV 8bV V 1) 12bVd(d )
,

2a

−

µ − λ ξ

λ + µ − − + −λ

(B.2) 

where BA,  are arbitrary constants. 

When ,042 =− µλ  

2 2 3 2

1
2

2

3

6bV{ d( d d 2 d 2d 2 )}
u

B
a(d ( ))

2 A B

{(bV 8bV V 1) 12bVd(d )
,

2a

− µ + λ + − λ + µ − λµ
= −

λ
− +

+ ξ

λ + µ − − + −λ

              (B.3) 

where BA,  are arbitrary constants. 

When ,042 >− µλ  

2 2
2

2
2

2 2

2

1

1 1
Bsinh ( 4 ) Acosh ( 4 )

3bV( 4 ) 2 2u ( ))
1 12a

Bcosh ( 4 ) Asinh ( 4 )
2 2

2bV 8bV V 1
,

2a

λ − µξ + λ − µξλ − µ
= −

λ − µξ + λ − µξ

λ − µ + +
−

              (B.4) 

where BA,  are arbitrary constants. 



 

 

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If ,0>B 22
BA <  above, then the solutions (B.4) turns into 

     
22 2

2
2 01

43bV( 4 ) bV 4bV V 1
u sech ( ) ,

2a 2 2a

λ − µξλ − µ λ − µ − −
= + ξ −   ),(tanh 1

0
B

A−=ξ  

When ,042 <− µλ  

,
2

182

))

4
2

1
sin4

2

1
cos

4
2

1
cos4

2

1
sin

(
2

)4(3

2

2

22

22
2

22

a

VbVbV

AB

AB

a

bV
u

++−
+

−+−

−+−−−
−=

µλ

ξλµξλµ

ξλµξλµλµ

  (B.5) 

where BA,  are arbitrary constants. 

When ,042 =− µλ  

2 2

2 22

6bVB 2bV 8bV V 1
u ,

2aa(A B )

λ − µ + +
= − +

+ ξ
   (B.6) 

where BA,  are arbitrary constants. 

When ,042 >− µλ  

2 2

3
2 2

2 2

2 2

2 2

2 2

2 2

2

1

3bV( 4 )
u

1 1
Bsinh 4 Acosh 4

1 2 28a(d ( 4 )( ))
1 12 2

Bcosh 4 Asinh 4
2 2

1 1
Bsinh 4 Acosh 4

6bV 1 2 2(d ( 4 )( ))
1 1a 2 2

Bcosh 4 Asinh 4
2 2

(2bV 8bV V V 1)
,

2a

− λ − µ
=

λ − µξ + λ − µξλ
− + λ − µ

λ − µξ + λ − µξ

λ − µξ + λ − µξλ
− − + λ − µ

λ − µξ + λ − µξ

λ − µ + +
−

     (B.7) 

where BA,  are arbitrary constants. 

When ,042 <− µλ  

2 2

3
2 2

2 2

2 2

2 2

2 2

2 2

2

2

3bV( 4 )
u

1 1
Bsin 4 Acos 4

1 2 28a(d ( 4 )( ))
1 12 2

Bcos ( 4 ) Asin ( 4 )
2 2

1 1
Bsin ( 4 ) Acos ( 4 )

6bV 1 2 2(d ( 4 )( ))
1 1a 2 2

Bcos ( 4 ) Asin ( 4 )
2 2

(2bV 8bV V V 1)
,

2a

λ − µ
= −

− µ − λ ξ + µ − λ ξλ
− + µ −λ

µ − λ ξ + µ − λ ξ

− µ −λ ξ + µ −λ ξλ
− − + µ −λ

µ − λ ξ + µ − λ ξ

λ − µ + +
−

(B.8) 



 

 

Md. Nur Alam and M. Ali Akbar/ BIBECHANA 10 (2014) 58-70 : BMHSS, p.70 (Online Publication: Dec., 2013) 

where BA,  are arbitrary constants. 

When ,042 =− µλ  

2 2
2

3
2

2

3

3bV( 4 ) 6bV B
u (d ( ))

B a 2 A B
8a(d ( ))

2 A B

(V 1 2bV 8bV V)
,

2a

− λ − µ λ
= − − +

λ + ξ− +
+ ξ

+ + λ − µ
+

                       (B.9) 

where BA,  are arbitrary constants.