Applied Mathematics, 2011, 2, 343-347
doi:10.4236/am.2011.23040 Published Online March 2011 (http://www.scirp.org/journal/am)
The Traveling Wave Solutions for Some Nonlinear
PDEs in Mathematical Physics
Khaled A. Gepreel1,2, Saleh Omran1,3, Sayed K. Elagan1,4
1Mathematics Department, Faculty of Science, Taif University, El-Taif, Kingdom of Saudi Arabia
2Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
3Mathematics Department, Faculty of Science, South Valley University, Egypt
4Mathematics Department, Faculty of Science,Menofia University, Egypt
Received December 4, 2010; revised January 17, 2011; accepted January 21, 201 1
Abstract
In the present article, we construct the exact traveling wave solutions of some nonlinear PDEs in the mathe-
matical physics via (1 + 1) dimensional Kaup Kupershmit equation, the (1 + 1) dimensional seventh order
KdV equation and (1 + 1) dimensional Kersten-Krasil Shchik equations by using the modified truncated ex-
pansion method. New exact solutions of these equations are found.
Keywords: Conjugate Modified Truncated Expansion Method, Traveling Wave Solutions, Nonlinear
Evolution Equations
1. Introduction
Nonlinear partial differential equations are known to
describe a wide variety of phen omena not only in physics
-where applications extend over magneto fluid dynamics,
water surface gravity waves, electromagnetic radiation
reactions, and ion acoustic waves in plasma, just to name
a few- but also in biology, chemistry and several other
fields. It is one of the important tasks in the study of
nonlinear partial differential equations to seek exact and
explicit solutions. In the past several decades, both ma-
thematicians and physicists have made many attempts in
this direction. Various methods for obtaining exact solu-
tions to nonlinear partial differential equations have been
proposed. These are some of them: the truncated expan-
sion method [1-4], the simplest equation method [5], an
automated tanh-function method [6], the polygons me-
thod [7], and the Clarkson-Kruskal direct method [8]. In
this paper, we use a modification of the truncated expan-
sion method introduced in [9,10] to find the exact solu-
tions of the following nonlinear partial differential equa-
tions in mathematical physics.
1) The (1 + 1) di mensional Kaup Kupershm it equation [11]:
2235
25
550,
2
 
txxxxx
uuuuuuuu (1.1)
2) The (1 + 1) dimensional seventh order KdV Equa-
tion [12]:
33 2
23
23457 0
txxxx x
xxxxx x
uauubucuuu duu
euuf uuguuu
 


(1.2)
where ,,,,,abcde f and
g
are arbitrary constants.
3) The (1 + 1) dimensional Kersten-Krasil Shchik eq-
uations [13,14]:
2
332
2
3
6333 60,
3330.

 
 
txxxxx xx
txx x x
uuuuWWW Wu WuWW
WWWWuW Wu
(1.3)
The modification of this method allows us to trans-
form this system of differential equations to a system of
algebraic equations. As a result we have essential simpli-
fication of solutions construction procedure.
2. The Modification of the Truncated
Expansion Method
Let us present the modification of the truncated expan-
sion method [9,10].
Suppose we have the following nonlinear partial dif-
ferential equation


,,,,, ,0,
txxxxttt
Fuuu uuu (2.1)
where


,uuxt is an unknown function,
F
is a
polynomial in


,uxtand its partial derivatives in which
the highest order derivatives and the nonlinear terms are
involved.

K. A. GEPREEL ET AL. 344
The traveling wave variable
 
,, ,uxtYzz kxt

 (2.2)
where k and

are constants, permits us to reduce
Equation (2.1) to an ODE for

uYzin the form


,,,,0,PYY YY
   (2.3)
where 'd
dz
. The modification of the truncated expan-
sion method contains the following steps [9,10].
Step 1. Determination of the dominant term with
highest order of singularity. To find the dominant terms
we substitute

,
p
Yz z

 (2.4)
into all terms of Equation (2.3). Then we should compare
the degrees of all the terms of Equation (2.3) and choose
two or more with the lowest degree. The maximum value
of p is the pole of Equation (2.3) and we denote it by
N . It should be noted that this method can be applied
when N is an integer. In order to apply this method
when N is equal to a fraction or negative integer, we
make the following transformation:
1) When q N
g
 , where q
g
is a fraction in lowest
term we take the transformation
 
.
qg
Yz z  
2) When N is negative integer, we take the trans-
formation
 
.
N
Yz z   and return to determine the
value of N again from the new equation
Step 2. We look for the exact solution of Equation
(2.3) in the form
 
2
01 2,
N
N
YzaaQza QzaQz   
 

(2.5)
where

0,1, ,
i
ai N are arbitrary constants to be
determined later and

Qz equals the following func-
tion:

1
1
z
Qz e  (2.6)
Step 3. We calculate the necessary number of deriva-
tives of the function

Yz (using Maple or Mathema-
tica for example). Using the case 2N, we get some
derivatives of the fu nct i o n


Yz as follows:
 
 
 
2
01 2
23
1122
234
121122
,
Q 2Q 2Q,
432106,
YzaaQza Qz
Yzaa aa
Yz aQzaaQaaQaQ
 

 
  
(2.7)
and so on.
Step 4. We substitute expressions (2.5)-(2.7) to Equa-
tion (2.1). Then we collect all terms with the same power
in the function

Qz and equate the expressions to zero.
As a result we obtain an algebraic system of equations.
Solving this system we get the values of the unknown
parameters.
This algorithm can be easily generalized to po lynomial
differential equations of any order.
3. Exact Solution of the (1 + 1) Dimensional
Kaup Kupershmit Equation
The fifth order Kaup-Kupershmit equation is one of the
solitonic equations related to the integrable cases of the
Henon-Heiles system. Let us find the exact solutions of
the (1 + 1) dimensional Kaup Kupershmit equation by
using the modified truncated expansion method. The
traveling wave variable (2.2) permits us to reduce Equa-
tion (1.1) to an ODE for

uYz in the form
  
 


2
33
5
35
1
515
34
50,
YzkYzkYz
kY zYzkYzC








(3.1)
where 1 C is the integration constant. The pole order
of Equation (3.1) is 2N

. So we look for the solution
of Equation (3.1) in the form:
 
2
01 2,YzaaQza Qz 

(3.2)
where 01
,aaand 2 a are arbitrary constants. We substi-
tute Equation(3.2), and the derivative equations (such as
Equation(2.6) and Equations (2.7)) into Equation (3.1)
and collect all terms with the same power in

,
i
Qz






0,1,2,i. Equating each coefficient of the poly-
nomial to zero yields a set of simultaneous algebraic eq-
uations omitted here for the sake of brevity. Solving
these algebraic equations by either Maple or Mathemati-
ca, we get the formulae for the solutions of system (3.1)
as follows:
Case 1.
257
22
210 1
3, 3,,,,
41696
kkk
akaka C  
(3.3)
where k is an arbitrary constant. In this case the soli-
trary wave solution takes the following form:


22 2
12
33
,
411
zz
kk k
Yz ee  
 (3.4)
where 5.
16
tk
zkx
Case 2.
7
22 2 5
210 1
26
24 ,24 ,2 ,11,,
3
k
akakak kC   
(3.5)

K. A. GEPREEL ET AL. 345
where k is an arbitrary constant. In this case the soli-
trary wave solution takes the following form:


22
2
22
24 24
2,
11
zz
kk
Yz kee  
 (3.6)
where 5
11 .zkx kt
4. Exact Solution of the (1 + 1) Dimensional
Seventh Order KdV Equation
The seventh-order KdV Equation (1.2) deals with the
structural stability of the KdV equation under a singular
perturbation. The traveling wave variable (2.2) permits
us to reduce Equation (1.2) to an ODE for


uYz in
the form

 
3
3333 32
3457 5557
0,
YkaYYbkY ckYYYdkYY
ek YYfk YYgk YYk Y

  
 

(4.1)
The pole order of Equation (4.1) is 2N. We subs-
titute Equation (3.2), and the derivative equations (such
as Equation (2.6) and Equations (2.7)) into Equation (4. 1)
and collect all terms with the same power

,
i
Qz





0,1,2,i. Equating each coefficient of the poly-
nomial to zero yields a set of simultaneous algebraic eq-
uations omitted here for the sake of brevity. Solving
these algebraic equations by either Maple or Mathemati-
ca, we get the formulae for the solutions of system (4.1)
as follows:
Case 1.

 


22
02
7
2265 129105
,,
2138 196813819
19192624247159,
616835 97395
2 1381972635054,
5130225
120542092014 ,
817 13819
kk
aa
g
fgf
egf
gfgf
d
kgf
gf


 
 
 


 
 
 
 
2
1
138197985302728321666 ,
1397131275
136 13819846129284582,
9493507013625
13819268357809147586870 ,
15927296535
0,
gfg f
c
gfg f
a
g
fgf
b
a
 

 

 


(4.2)
where ,kg and
f
are arbitrary constants. In this case
the solitary wave solution of Equation (1.2) takes the
following form
 


22
2
2265129105 ,
2 1381968138191z
kk
Yz gf gf e 
 
(4.3)
where



7
120542092014 .
817 13819
kt gf
zkx gf

 
Case 2.

22
21
7
6895 6895
,,
223 623 6
364823 ,,
985 985
kk
aa
g
fgf
egfk 
 

 




 
 
 
0
2
223 6 13679228,
6791575
23628817910938 ,0,
6791575
23693187978,
2716630
8 23640219271,
46827909625
gfgf
d
gfgf
ca
gfg f
b
gf fg
a









(4.4)
where ,kg and
f
are arbitrary constants. In this case
the solitary wave solution of Equation (1.2) takes the
following form
 



22
2
68956895 ,
2361 22361
zz
kk
Yz gf egf e 
 
(4.5)
where 7.zkxkt
Case 3.

1
210
2224 222
1111
4
1
323 7
11
,,
12
1248412096 ,
48
,
144 1728
a
aaa
aadk afk ack ak
eak
dk aakak

 



63 224
111
22
1
40321296 ,0,
48
kaa dkafka
bg
ka
 

 (4.6)
where 1,,,,aakdc and
f
are arbitrary constants. In this
case the solitary wave solution of Equation (1.2) takes
the following form




2
1
2
110 ,
12 1
zz
z
aee
Yz e
 
 (4.7)
where 323 7
11.
144 1728
dk aaka
zkxt k


 



 Similarly, we can

K. A. GEPREEL ET AL. 346
write down the other families of exact solutions of Eq-
uation (1.2) which are omitted for convenience.
5. Exact Solution of the (1 + 1) Dimensional
Kersten-Krasil Shchik Equations
In this section, we find the exact solutions to the (1 + 1)
dimensional Kersten- Krasil Shchik Equations (1.3) by
the modified truncated expansion method. The traveling
wave variables
 
,uYzWHz and zkx



permit us to reduce Equations (1.3) to ODEs for

uYz and

WHz in the form
322 2
1
33 2
33 30,
30,
YkYkYkHHkYH C
HkH kHkYHC


 
 

  (5.1)
where 1 C and 2 C are the integration constants. To find
dominant terms we substitute

1
p
Yz z

and

2
p
H
zz

 into all terms of Equation
(5.1). Then we should compare degrees of all terms of
Equations (5.1) and choose two or more with the lowest
degree we have
 
2
01 201
,,YzaaQza QzHzbbQz  

(5.2)
where 0011 ,,,ababand 2 a are arbitrary constants. We
substitute Equations (5 .2) into Equation (5.1) and collect
all terms with the same power in
 
,0,1,2,
i
Qz i 
 . Equating each coefficient of
these polynomials to zero yields a set of simultaneous
algebraic equations omitted here for the sake of brevity.
Solving these algebraic equations by either Maple or
Mathematica, we get the formulae for the solutions of
system (5.1) as follows:
Case 1.
22
201010
3
00120
2, 2,2,
,, 0,
ababbb
kib ibCCa 
 
   (5.3)
where 0 b is an arbitrary constant and 1i. In this
case the solitary wave solution takes the following form:




2
00
0
2
2,,
1
1
z
z
z
be b
YzHz be
e


 (5.4)
where 3
00
.zibxibt 
Case 2.
22
201010 0
23
000100 2
4,4,2,2 ,
2( 3),4,0,
ababbbkib
ibbaCib aC

 
 
 (5.5)
where 0 a and 0 b are arbitrary constants. In this case
the solitary wave solution takes the following form:


22
00
02
44
,
11
zz
bb
Yz aee  


0
02,
1z
b
Hz be 
 (5.6)
where


2
0000
22 3.zibxibbat 
Similarly, we can write down the other families of ex-
act solutions of Equation (1.3) which are omitted for
convenience.
6. Conclusions
In summary, we may conclude that this method is relia-
ble and straightforward solution method to find the trav-
eling waves of nonlinear partial differential equations.
This method allowed us to construct the exact solution
for some complicated nonlinear evolution equation than
exp-function method. The performance of this method is
simple, direct and gives more new exact solutions com-
pared to other method.
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