Exact Solutions For Non Linear Partial Differential
Equations Using Hyperbolic-sine Function Method
M.F. El-Sabbagh, R. Zait And R.M. Abdelazeem
Mathematics Department, Faculty of science, Minia university, Egpt
Corresponding e-mail: rasha_math24@yahoo.com
Abstract
In this paper, we establish exact solutions for some nonlinear partial
differential equations. The Hyperbolic-sine method (ref. [1]) is used to
construct periodic and solitary wave solutions for some soliton equations
and systems such as the Klien-Gordon, the Kadomtsev-Petviashvili, and the
Zakharov-Kuznetsov (ZK) equations, the generalized coupled Drinfeld –
sokolov –wilso, and the general coupled Hirota-Satsuma Kdv systems.
Keywords: Nonlinear PDEs and systems, Exact Solutions, Nonlinear
Waves and Hyperbolic-sine function method.
1. Introduction
The aim of the present paper is to extend the cosine-function method
introduced to find new solitary solutions to the some soliton equations such
as the Klien-Gordon, the Kadomtsev-Petviashvili, and the ZakharovKuznetsov (ZK) equations, the generalized coupled Drinfeld –sokolov –
wilso, and the general coupled Hirota-Satsuma Kdv systems.
2. Hyperbolic-sine function method (ref. [1])
Consider the nonlinear partial differential equation in the form
𝐹(𝑢𝑡 , 𝑢𝑥 , 𝑢𝑛 𝑢𝑥 , 𝑢𝑥𝑥𝑥 , 𝑢𝑥𝑥𝑡 , … … … . )
(1)
where 𝑢(𝑥,𝑡) is the solution of (1); and 𝑢𝑡 , 𝑢𝑥 etc, are the partial derivatives
of 𝑢 with respect to t and x, respectively. We assume that equation (1)
admits travelling wave solution. We use the transformation
𝑢(𝑥, 𝑡) = 𝑓(𝜉),
𝜉 = 𝑥 − 𝑐𝑡 − 𝑑
(2)
Where c is the speed of the travelling wave and 𝑑 is a constant. This
enables us to use the following
1
𝜕
𝜕𝑡
𝑑
(·)=−c𝑑𝜉 (·),
𝜕
𝜕𝑥
𝑑
𝜕2
𝑑2
(·)=𝑑𝜉 (·), 𝜕𝑥 2 (·)=𝑑𝜉2 (·),……..
(3)
Using the above transformation the nonlinear partial differential equation
(1) is transformed to nonlinear ordinary differential equation:
𝑑𝑓
𝐺( ,
𝑑𝜉
n 𝑑𝑓 𝑑3 𝑓
𝑓 𝑑𝜉 , 3 , … … . . ) = 0
𝑑𝜉
(4)
By integrating equation (4) with respect to 𝜉, we obtain
𝐻 (𝑓, 𝑓
n+1
2
,
𝑑 𝑓
2 ,……..)
𝑑𝜉
=0
(5)
The solution of equation (2) can be expressed as
𝑢(𝑥, 𝑡) = 𝑓(𝜉)=𝜆𝑠𝑖𝑛ℎ𝛽 (𝜇𝜉)
(6)
Where 𝜆, 𝛽 and 𝜇 are unknown parameters which will be determined.
Then we have
𝑑𝑓
= −𝜆 𝛽 𝜇𝑠𝑖𝑛ℎ𝛽−1 (𝜇𝜉) csch(𝜇𝜉)
𝑑𝜉
𝑑2 𝑓
𝑑𝜉 2
= 𝜆 𝜇2 𝛽(𝛽 − 1)𝑠𝑖𝑛ℎ𝛽−2 (𝜇𝜉)+ 𝜆 𝜇2 𝛽(𝛽 − 1)𝑠𝑖𝑛ℎ𝛽 (𝜇𝜉)
+𝜆 𝜇2 𝛽 𝑠𝑖𝑛ℎ𝛽 (𝜇𝜉)
(7)
We substitute (6) and (7) in (4) to obtain an equation in different powers of
sine-hyperbolic functions. Now equating the coefficients of the same
powers of sine-hyperbolic functions we obtain a system of algebraic
equations in the parameters λ, β and μ. This system can be solved to obtain
the values of λ, β and μ. The exact analytical solution of NLPDE (2) is
then obtained by substituting the values of the parameters in (6).
3. Applications
In order to illustrate the effectiveness of the proposed method
examples of mathematical interest are chosen as follows:
2
3.1.The Klien-Gordon equation
In this section we introduce solitary exact solution for a generalized
Klien-Gordon equation which is as follows:
𝑢𝑡𝑡 -𝑢𝑥𝑥 −𝑢 + 𝑢𝑝 = 0
(8)
from equations (3), we have:
c2
𝑑 2 𝑓(𝜉)
𝑑𝜉 2
𝑑 2 𝑓(𝜉)
−
𝑑𝜉 2
− 𝑓(𝜉 ) − (𝑓(𝜉))𝑝 = 0
(9)
From equations (6) and (7) we have:
(c 2 − 1)[ 𝜆 𝜇2 𝛽 (𝛽 − 1)sinh𝛽−2 (𝜇𝜉 ) + 𝜆 𝜇2 𝛽 (𝛽 − 1)sinh𝛽 (𝜇𝜉 ) +
𝜆 𝜇2 𝛽sinh𝛽 (𝜇𝜉 )] − 𝜆 sinh𝛽 (𝜇𝜉 ) −𝜆 𝑝 sinh𝑝𝛽 (𝜇𝜉 ) = 0
(10)
By balancing the exponents of each pair of sinsh we have:
(c 2 − 1)𝜆 𝜇2 𝛽(𝛽 − 1) + (c 2 − 1)𝜆 𝜇2 𝛽 − 𝜆 = 0,
𝛽−
2
1−𝑝
=0
(c 2 − 1)𝜆 𝜇2 𝛽(𝛽 − 1) − 𝜆 𝑝 =0
(11)
Using MATHEMATICA package software for solving the system
equation (11) we obtain:
𝛽=
2
1−𝑝
,
𝜇=−
1
√1−2𝑝+𝑝2
,
2√𝑐 2 −1
𝜆 = 21−𝑝 (
1
1+𝑝
1
)1−𝑝
(12)
Thus we obtain a new exact solution of the general Klien-Gordon
equation in the form
1
1−𝑝
𝑢(𝑥, 𝑡) = 2
1
2
2
√1−2𝑝+𝑝
1 1−𝑝
(1+𝑝) 𝑠𝑖𝑛𝑠ℎ1−𝑝 ((−
2√𝑐2 −1
) (𝑥 − 𝑐𝑡))
(13)
As special case if p=3 we get the Klien-Gordon equation (ref. [2]) in
the following form:
𝑢𝑡𝑡 -𝑢𝑥𝑥 −𝑢 + 𝑢3 = 0
(14)
and thus its exact soliton solution is
2
𝑢(𝑥, 𝑡) = √ √−1 + 𝑐 2 𝑠𝑖𝑛ℎ−1 (
3
ⅈ
√3
(𝑥 − 𝑐𝑡))
3
(15)
3.2. The Kadomtsev-Petviashvili (KP) equation
Consider the following nonlinear partial differential equation (refs.
[3], [4], [5], [6] and [7])
(𝑢𝑡 + 𝜀𝑢𝑝 𝑢𝑥 + 𝑢𝑥𝑥𝑥 )𝑥 + 𝛼𝑢𝑦𝑦 = 0
(16)
From equations (3), we have:
𝑑
𝑑𝑓(𝜉)
𝑑𝑓(𝜉) 𝑑 3 𝑓(𝜉)
𝑑 2 𝑓(𝜉)
𝑝
+ 𝜀𝑓(𝜉)
+
=0
(−𝑐
)+𝛼
𝑑𝜉
𝑑𝜉
𝑑𝜉
𝑑𝜉 3
𝑑𝜉 2
(17)
By integration twice we have:
(𝛼 − 𝑐 )𝑓 (𝜉 ) +
𝑝+1
𝜀
𝑝+1
(𝑓(𝜉 ))
+
𝑑 2 𝑓(𝜉)
=0
𝑑𝜉 2
(18)
[[[
From equations (6) and (7) we have:
𝜀
𝜆 𝑝+1 sinh(𝑝+1)𝛽 (𝜇𝜉)
𝑝+1
2
𝛽−2 (𝜇𝜉)
+ 𝜆 𝜇 𝛽(𝛽 − 1)sinh𝑠
+ 𝜆 𝜇2 𝛽(𝛽 − 1)sinh𝛽 (𝜇𝜉) +
𝜆 𝜇2 𝛽sinh𝛽 (𝜇𝜉) = 0
(𝛼 − 𝑐)𝜆 sinh𝛽 (𝜇𝜉) +
(19)
By balancing the exponents of each pair of sinsh we have:
2
𝜀
𝑝
𝑝+1
𝛽 + = 0,
𝜆 𝑝+1 + 𝜆 𝜇2 𝛽(𝛽 − 1) = 0,
(𝛼 − 𝑐)𝜆 + 𝜆 𝜇2 𝛽(𝛽 − 1) + 𝜆 𝜇2 𝛽 = 0
(20)
Using MATHEMATICA package software for solving the system
equation (20) we obtain:
2
𝛽=− ,
𝑝
−1
𝑝
𝑝
(𝛼−𝑐)(𝑝2 +3𝑝+2)
𝜇 = ± √𝑐 − 𝛼, 𝜆 = 2 (
2
𝜀
1
𝑝
)
(21)
Thus we now have new exact solution of the Kadomtsev-petviashvili
equation is given by
−1
𝑝
𝑢(𝑥, 𝑡 ) = 2 (
1
(𝛼−𝑐)(𝑝2 +3𝑝+2) 𝑝
𝜀
) 𝑠𝑖𝑛ℎ
−
2
𝑝
𝑝
(± √𝑐 − 𝛼(𝑥 + 𝑦 − 𝑐𝑡 ))
2
4
(22)
3.3.The Zakharov-Kuznetsov (ZK) equation
This ZK appears in many areas of physics, applied Mathematics, and
Engineering. In particular, it shows up in the areas of Plasma Physics.
The ZK govern the behaviour of weakly nonlinear ion-acoustics waves
in a plasma comprising of cold ion and hot isothermal electron in the
presence of a uniform magnetic field.
The ZK equation (refs. [8], [9], [10], [11], [12] and [13]) is given by
𝑢𝑡 + 𝑎𝑢𝑛 𝑢𝑥 + 𝑏(𝑢𝑥𝑥 + 𝑢𝑦𝑦 )𝑥 = 0
(23),
where
𝑢(𝑥, 𝑦, 𝑡) = 𝑓(𝜉),
𝜉 = 𝑥 + 𝑦 − 𝑐𝑡
(24)
From equations (3), we have:
𝑑𝑓(𝜉)
𝑑𝑓(𝜉)
𝑑
𝑑 2 𝑓(𝜉)
𝑛
−𝑐
+ 𝑎(𝑓(𝜉))
+ 𝑏 (2
)
𝑑𝜉
𝑑𝜉
𝑑𝜉
𝑑𝜉 2
(25)
By integration we have:
−𝑐 𝑓(𝜉) +
𝑎
𝑑 2 𝑓(𝜉)
𝑛+1
(𝜉)
+
2𝑏
(𝑓 )
𝑛+1
𝑑𝜉 2
(26)
From equations (6) and (7) we have:
−𝑐𝜆sinh𝛽 (𝜇𝜉 ) +
𝑎
𝑛+1
𝜆 𝑛+1 sinh(𝑛+1)𝛽 (𝜇𝜉 ) + 2𝑏𝜆 𝜇2 𝛽 (𝛽 −
1)sinh𝛽−2 (𝜇𝜉 ) + 2𝑏𝜆 𝜇2 𝛽 (𝛽 − 1)sinh𝛽 (𝜇𝜉 ) + 2𝑏𝜆 𝜇2 𝛽sinh𝛽 (𝜇𝜉 ) =
0
(27)
By balancing the exponents of each pair of sinsh we have:
−𝑐𝜆 + 2𝑏𝜆 𝜇2 𝛽(𝛽 − 1) + 2𝑏𝜆 𝜇2 𝛽 = 0,
𝛽+
2
= 0,
𝑛
𝑎
𝜆 𝑛+1 + 2𝑏𝜆 𝜇2 𝛽(𝛽 − 1) = 0
𝑛+1
Using MATHEMATICA package software for solving the system
equation (28) we obtain:
5
(28)
−1
𝜆 = 2𝑛(
−2𝑐−2𝑐𝑛−𝑐𝑛2 1
)𝑛 , 𝛽 =
𝑎
−2
𝑛
, 𝜇=±
√𝑐𝑛
2√2𝑏
(29)
Thus we obtain new exact solution of the ZK equation in the form:
𝑢(𝑥, 𝑡 ) = 2
−1
𝑛
1
−2
−2𝑐−2𝑐𝑛−𝑐𝑛2 𝑛
√𝑐𝑛
𝑛 (±
(
)
𝑠𝑖𝑛𝑠ℎ
𝑎
2√2𝑏
(𝑥 − 𝑐𝑡))
(30)
3.4. The generalized coupled Drinfeld –sokolov –wilso system
This system ( refs. [14], [15], [16], [17], [18]) is given by
𝑢𝑡 − 3𝑣𝑣𝑥 = 0,
𝑣𝑡 − 3𝑣𝑥𝑥𝑥 − 𝑎(𝑢𝑣)𝑥 = 0
(31)
We assume the solution of the system (30) in the form:
𝑢(𝑥, 𝑡) = 𝑓(𝜉),
𝑣(𝑥, 𝑡) = 𝑔(𝜉),
𝜉 = 𝑥 − 𝑐𝑡
(32)
From equations (2) and (3), we have:
−𝑐
𝑑𝑓(𝜉)
𝑑𝑔(𝜉)
− 3𝑔 𝑑𝜉
𝑑𝜉
= 0, −𝑐
𝑑𝑔(𝜉)
𝑑3 𝑔(𝜉)
𝑑
−
3
3 − a 𝑑𝜉 (𝑓 (𝜉 )𝑔 (𝜉 ))
𝑑𝜉
𝑑𝜉
=0
(33)
By integration we have:
[[
𝑓(𝜉) = −
3
2𝑐
(𝑔(𝜉))2 ,
−𝑐𝑔(𝜉) − 3
𝑑 2 𝑔(𝜉)
𝑑𝜉 2
− a(𝑓(𝜉)𝑔(𝜉)) = 0 (34)
Thus from (33) and (34) we have:
−𝑐𝑔(𝜉) − 3
𝑑 2 𝑔(𝜉)
𝑑𝜉 2
−
3a
2c
(𝑔(𝜉))3 = 0
(35),
We assume 𝑣𝑖𝑎 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝑠𝑖𝑛𝑒 𝑚𝑒𝑡ℎ𝑜𝑑𝑠 𝑡ℎ𝑎𝑡:
𝑔(𝜉) = 𝜆 sinh𝛽 (𝜇𝜉)
(36)
Thus we have
−𝑐𝜆 sinh𝛽 (𝜇𝜉 ) − 3[𝜆𝜇2 𝛽 (𝛽 − 1)sinh𝛽−2 (𝜇𝜉 ) + 𝜆 𝜇2 𝛽 (𝛽 −
1)sinh𝛽 (𝜇𝜉 ) + 𝜆 𝜇2 𝛽sinh𝛽 (𝜇𝜉 )] +
3𝑎
2𝑐
𝜆3 𝛽sinh3𝛽 (𝜇𝜉 ) = 0.
(37)
6
By balancing the exponents of each pair of sinsh we have:
−𝑐𝜆 − 3𝜆𝜇2 𝛽(𝛽 − 1) − 3𝜆𝜇2 𝛽 = 0,
−3𝜆𝜇2 𝛽(𝛽 − 1) +
3𝑎
2𝑐
𝛽 + 1 = 0,
𝜆3 =0
(38)
Using MATHEMATICA package software for solving the system
equation (38) we obtain:
𝑐
𝜇 = ±√ ,
𝜆=±
3
2𝑐
√3𝑎
,
𝛽 = −1
(39)
Thus the exact solution of the generalized coupled Drinfeld –sokolov –
wilso system is given as follows:
𝑣(𝑥, 𝑡) = ±
𝑢(𝑥, 𝑡) = ±
2𝑐
√3𝑎
𝑐
sinh−1 (±√ (𝑥 − 𝑐𝑡)),
3
2𝑐
𝑐
sinh−2 (±√ (𝑥 − 𝑐𝑡))
𝑎
3
(40)
3.5. The general couplied Hirota-Satsuma Kdv system
The general couplied Hirota-Satsuma Kdv system (refs. [18], [19],
[20]) is given as follows:
𝑢𝑡 − 𝑎𝑢𝑥𝑥𝑥 − 3𝑢𝑢𝑥 + 6𝑣𝑣𝑥 = 0,
𝑣𝑡 + 𝑏𝑣𝑥𝑥𝑥 + 3(𝑢𝑣)𝑥 = 0
(41),
To obtain the travelling wave solutions we use the following
transformations:
𝑢(𝑥, 𝑡) = 𝑓(𝜉),
𝑣(𝑥, 𝑡) = 𝑔(𝜉),
𝜉 = 𝑥 − 𝑐𝑡
(42)
From equations (2) and (3), we have:
𝑑𝑓(𝜉)
𝑑 3 𝑓(𝜉)
𝑑𝑓(𝜉)
𝑑𝑔(𝜉)
−𝑐
−𝑎
−
3𝑓(𝜉)
+
6𝑔(𝜉)
= 0,
𝑑𝜉
𝑑𝜉 3
𝑑𝜉
𝑑𝜉
𝑑𝑔(𝜉)
𝑑 3 𝑔(𝜉)
𝑑
−𝑐
+𝑏
+
3
(𝑓(𝜉)𝑔(𝜉)) = 0
𝑑𝜉
𝑑𝜉 3
𝑑𝜉
7
(43)
By integration we have
𝑑 2 𝑓(𝜉) 3
2
2
(𝜉)
−𝑐 𝑓(𝜉) − 𝑎
−
+
3(𝑔(𝜉))
= 0,
(𝑓
)
𝑑𝜉 2
2
𝑑 2 𝑔(𝜉)
−𝑐 𝑔(𝜉) + 𝑎
+ 3𝑓𝑔 = 0
𝑑𝜉 2
(44)
If we take via hyperbolic sine method the following forms:
𝑓(𝜉) = 𝜆1 sinh𝛽1 (𝜇𝜉),
𝑔(𝜉) = 𝜆2 sinh𝛽2 (𝜇𝜉)
(45)
from equation (45), equation (44) becomes in the following form:
[
𝑐𝜆1 sinh𝛽1 (𝜇𝜉) − 𝑎[𝜆1 𝜇2 𝛽1 (𝛽1 − 1)sinh𝛽1−2 (𝜇𝜉)
+ 𝜆1 𝜇2 𝛽1 (𝛽1 − 1)sinh𝛽1 (𝜇𝜉) + 𝜆1 𝜇2 𝛽1 sinh𝛽1 (𝜇𝜉)
3
− 𝜆1 2 sinh2𝛽1 (𝜇𝜉)+3𝜆2 2 sinh2𝛽2 (𝜇𝜉) = 0,
2
−𝑐𝜆2 sinh𝛽2 (𝜇𝜉) + 𝑏[𝜆2 𝜇2 𝛽2 (𝛽2 − 1)sinh𝛽2−2 (𝜇𝜉)
+ 𝜆2 𝜇2 𝛽2 (𝛽2 − 1)sinh𝛽2 (𝜇𝜉) + 𝜆2 𝜇2 𝛽2 sinh𝛽2 (𝜇𝜉)
+ 3𝜆1 𝜆2 sinh𝛽1+𝛽2 (𝜇𝜉) = 0
(46)
By balancing the exponents of each pair of sinsh we have:
−𝑐𝜆1 − 𝑎𝜆1 𝜇2 𝛽1 (𝛽1 − 1) − 𝑎𝜆1 𝜇2 𝛽1 = 0, 2𝛽1 = 2𝛽2 = 𝛽1 − 2,
3
−𝑎𝜆1 𝜇2 𝛽1 (𝛽1 − 1) − 𝜆1 2 +3𝜆2 2 = 0,
2
−𝑐𝜆2 + 𝑏𝜆2 𝜇2 𝛽2 (𝛽2 − 1) + 𝑏𝜆2 𝜇2 𝛽2 = 0,
𝛽2 − 2 = 𝛽1 + 𝛽2 ,
(47)
𝑏𝜆2 𝜇2 𝛽2 (𝛽2 − 1) + 3𝜆1 𝜆2 = 0
Using MATHEMATICA package software for solving the system
equation we have
𝛽1 = 𝛽2 = −2, 𝑏 = −𝑎,
𝑐
𝑐
3
1
𝑐
𝜆1 = , 𝜆2 = ± √ , 𝜇 = ± √
2
2 2
2 𝑎
8
(48)
Thus the exact solution of the general couplied Hirota-Satsuma Kdv
system (41) is given as
𝑐
1
𝑐
𝑢(𝑥, 𝑡) = sinh−2 (± √ (𝑥 − 𝑐𝑡)),
2
2 𝑎
𝑐
3
1
𝑐
𝑣(𝑥, 𝑡) = ± √ sinh−2 (± √ (𝑥 − 𝑐𝑡))
2 2
2 𝑎
(49),
where a is constant.
References
[1] R. Arora, A. kumar, Soliton Solution of GKDV, RLW, GEW and
GRLW Equations by Sine-hyperbolic Function Method, American
Journal of Computational and Applied Mathematics. 2011; 1(1): 1-4,
doi: 10. 5923/j.ajcam.20110101.01.
[2] A. Neirameh, New solutions for some time fractional differential
equations, International journal of computing science and mathematics ,
03 Oct 2012.
[3] A. Borhanifa, R. Abazari, General Solution of Generalized (2+1)–
Dimensional Kadomtsev-Petviashvili (KP) Equation by Using the –
Expansion Method, American Journal of Computational Mathematics,
Vol. 1, No. 4, December 2011.
[4] C. Liu, New exact periodic solitary wave solutions for Kadomtsev–
Petviashvili equation, Applied Mathematics and Computation, Vol. 217,
Issue 4, 15 October 2010, pp. 1350–1354.
[5] M. Jie-Jian , Y. Jian-Rong, A new Method Of New Exact Solutions
and Solitary Wave-Like Solutions For Generalized Variable Cofficients
Kadomtsev-Petviashvili Equation, Chinese Phys. 15 2804, Vol. 15 No. 12,
doi:10.1088/1009-1963/15/12/007, 2006.
[6] S. A. Mohammad-Abadi, Analytic Solutions of the KadomtsevPetviashvili Equation with Power Law Nonlinearity Using the Sine-Cosine
Method, American Journal of Computational and Applied Mathematics
2011; 1(2): 63-66 DOI: 10.5923/j.ajcam.20110102.12.
9
[7] W. Yun-Hu, C. Yong, Bäcklund Transformations and Solutions of a
Generalized Kadomtsev—Petviashvili Equation, Commun. Theor. Phys.
57 217, Vol. 57, No. 2, 2012.
[8] W. Zhang, J. Zhou, Traveling Wave Solutions of a Generalized
Zakharov-Kuznetsov Equation, Journal of Applied Mathematics, Vol.
2012 (2012), Article ID 560531, 8 pages.
[9] Y. Zheng, Ma Song-Hua, Fang Jian-Ping, Exact solutions and
soliton structures of (2+1)-dimensional Zakharov-Kuznetsov equation,
Acta Phys. Sin., Vol. 60, Issue (4): 040508,2011.
[10] Yan-Ze Peng, Exact travelling wave solutions for the Zakharov–
Kuznetsov equation, Applied Mathematics and Computation, Vol. 199,
Issue 2, 1 June 2008, pp. 397–405.
[11] F. Awawdeh, New exact solitary wave solutions of the Zakharov–
Kuznet sove quation in the electron–positron–ion plasmas, Applied
Mathematics and Computation, Vol. 218, Issue 13, 1 March 2012, pp.
7139–7143.
[12] K. Batiha, Approximate analytical solution for the ZakharovKuznetsov equations with fully nonlinear dispersion, Journal of
Computational and Applied Mathematics , Vol. 216 Issue 1, 2008.
[13] S. I. El – Ganaini, Travelling Wave Solutions of the ZakharovKuznetsov Equation in Plasmas with Power Law Nonlinearity, Int. J.
Contemp. Math. Sciences, Vol. 6, 2011, No. 48, pp.2353 – 2366.
[14] Z. Xue-Qin1, Z. Hong-Yan, An Improved F-Expansion Method and
Its Application to Coupled Drinfel’d–Sokolov–Wilson Equation, Commun.
Theor. Phys. (Beijing, China) 50 (2008) pp. 309–314, Vol. 50, No. 2,
August 15, 2008.
[15] A. H. A. Ali, K. R. Raslan, The First Integral Method for Solving a
System of Nonlinear Partial Differential Equations, International Journal
of Nonlinear Science, ISSN 1749-3889 (print), 1749-3897 (online), Vol.
5(2008) No.2, pp.111-119.
10
[16] M. A. Akbar, Norhashidah H. M. Ali, The modified alternative
(G//G)-expansion method for finding the exact solutions of nonlinear
PDEs in mathematical physics, International Journal of the Physical
Sciences Vol. 6(35), pp. 7910 - 7920, 23 December, 2011.
[17] W. Zhang, Solitary Solutions and Singular Periodic Solutions of the
Drinfeld-Sokolov-WilsonEquationby Variational Approach, Applied
Mathematical Sciences, Vol. 5, 2011, No. 38, 1887 – 1894.
[18] M. M. El-Borai, A. A. Zaghrout and A. M. Elshaer, Exact
Solutions For Nonlinear Patial Differential Equations By Using CosineFunction Method, IJRRAS 9 (3), Vol. 9, Issue3, Desember 2012.
[19] C. Yong, Y. Zhen-Ya, L. Biao, and Z. Hong-Qing, New Explicit
Solitary Wave Solutions and Periodic Wave Solutions for the Generalized
Coupled Hirota Satsuma KdV System, Commun. Theor. Phys. (Beijing,
China) 38 (2002) pp. 261–266, Vol. 38, No. 3, September 15, 2002.
[20] Z. Yu-Feng, Z. Hong-Qing, Solitary Wave Solutions for the Coupled
Ito System and a Generalized Hirota-Satsuma Coupled Kdv System,
Commun. Theor. Phys. (Beijing, China) 36 (2001), pp. 657-660, Vol. 36,
No. 6, December 15, 2001.
11