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Applied Mathematics, 2011, 2, 752-756
doi:10.4236/am.2011.26100 Published Online June 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
On Exact Tr aveling Wave Solutions for (1 + 1) Dimensional
Kaup-Kupershmidt Equation
Dahe Feng1,2, Kezan Li1
1School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China
2School of Mathematical Sciences and Computing Technology, Central South University, Changsha, China
E-mail: [email protected], [email protected]
Received April 8, 2011; revised April 22, 2011; accepted April 25, 2011
Abstract
In this present paper, the Fan sub-equation method is used to construct exact traveling wave solutions of the
(1 + 1) dimensional Kaup-Kupershmidt equation. Many exact traveling wave solutions are successfully ob-
tained, which contain solitary wave solutions, trigonometric function solutions, hyperbolic function solutions
and Jacobian elliptic function periodic solutions with double periods.
Keywords: Fan Sub-Equation Method, Kaup-Kupershmidt Equation, Exact Traveling Wave Solutions
1. Introduction
Nonlinear partial differential equations are widely used
to describe complex phenomena in vary scientific fields
and especially in areas of physics such as plasma, fluid
mechanics, biology, solid state physics, nonlinear optics
and so on. Therefore the investigation of the exact solu-
tions to nonlinear equations plays an important role in
the study of nonlinear science. Up to now, many power-
ful methods to seek for exact solutions to the nonlinear
differential equations have been proposed. Among these
are inverse scattering method [1], Lie group method [2,3],
bifurcation method of dynamical systems [4-6], sine-
cosine method [7,8], tanh function method [9-11], ho-
mogenous balance method [12], Weierstrass elliptic
function method [13].
Recently, Fan [14] presented the Fan sub-equation
method which is a unified algebraic method to obtain
many types of traveling wave solutions based on an aux-
iliary nonlinear ordinary differential equation with con-
stant coefficients called Fan sub-equation. The important
feature of Fan’ method is to, without much extra effort
and without considering the integrability of nonlinear
equations, directly get a series of exact solutions in a
uniform way, which cover all results of tanh function
method, extended function method, F-expansion method,
etc. This method is a powerful technique to symbolically
compute traveling wave solutions of nonlinear evolution
equations and is widely used by many researcher such as
in [15-17] and by the references therein.
In this paper, we will use the Fan sub-equation method
to discuss the (1+1) dimensional Kaup-Kupershmidt equ-
ation [18] which can be shown in the form
2
35
25
55
2
txxxxxx
uuuuuuuu 0.
  (1.1)
2. The Fan Sub-Equation Method
For a given nonlinear partial differential equation
,, ,,,,0,
t xtt xt xx
Fuuu uuu (2.1)
where u = u(x,t) is an unknown function, F usually is a
polynomial in u(x,t).
To seek exact solutions of (2.1), we outline the Fan
sub-equation method. The main steps are given below
[14].
Step 1. By using the traveling wave transformation
,, uxtux ct

,
 (2.2)
where c is a wave speed, we can reduce (2.1) to an ordi-
nary differential equation in the form
,, ,,0,Fuuu u
   (2.3)
where the prime denotes the derivative with respect to
.
Step 2. Expand the solution of (2.3) in the form
 
0,
ni
i
i
ua

(2.4)
where ai (i = 1,2, ···, n) are constants to be determined
D. H. FENG ET AL.753
later and the new variable
satisfies the following
Fan sub-equation
 
4
0,
j
j
jc
 
(2.5)
where 1
 and are real constants.
i
Step 3. Determine n in (2.4) by substituting (2.4) and
(2.5) into (2.3) and balancing the highest order derivative
terms with the highest order nonlinear terms.
c
Step 4. Substituting (2.4) and (2.5) into (2.3) again and
collecting all coefficients like

4
0
l
k
j
jc
j
(l = 0, 1;
k = 0, 1, ···, n), then setting these coefficients to zero will
give a set of algebraic equations with respect to ai (i =
1,2, ···, n) and c.
Step 5. Solve these algebraic equations to obtain c and
i. Substituting these results into (2.4) yields to the gen-
eral form of traveling wave solutions.
a
Step 6. For each solution to (2.5) which depends on
the special conditions chosen for cj, it follows from (2.4)
that the corresponding exact traveling wave solution of
(2.1) can be constructed.
3. Exact Solutions for the (1 + 1) Dimensional
Kaup-Kupershmidt Equation
The fifth order Kaup-Kupershmidt Equation (1.1) is one
of the solitonic equations related to the integrable cases
of the Henon-Heiles system and belongs to the com-
pletely integrable hierarchy of higher order KdV equa-
tions. Moreover the equation has infinite sets of conser-
vation laws [19-22]. Let us find the exact traveling wave
solutions of the (1 + 1) dimensional Kaup-Kupershmidt
equation by using the Fan sub-equation method.
The traveling wave transformation (2.2) permits us to
reduce (1.1) to an ODE in the form

5
225
55
2
cuuuu uuuu
 
 0.
(3.1)
According to Steps 1 and 2 in Section 2, by balancing
and in (3.1), we obtain n + 3 = 3n − 1 and
therefore give n = 2. Thus we can suppose that (3.1) has
the following formal solutions

5
u2
uu
 
2
012 ,uaa a
 
  (3.2)
where

satisfies (2.5).
Substituting (3.2) and (2.5) into (3.1), collecting all
terms with the same power in 4
0
k
j
jcj
etting all their coefficients to zero yields a set of
tions.
(0 ≤ k ≤ 5),
then s
simultaneous algebraic equations omitted here for the
sake of brevity. Solving these algebraic equations with
the help of Maple, we get the following two sets of solu-
1) The first set of parameters is given by
2
3
1241 30
31
3
0, 3, ,
216
c
caca ca
 2
4
4
23422
23404 324
2
4
6,
240c c76845256,
256
cc
c
cc
cccc
cc

(3.3)
where 0234
, , , 0cccc
The second set of pa
are arbitrary constants.
2)rameters is given by

2
3243
124
2
4, 24,
cccc
cac

4
2
32
4
130
4
24 322
23 430424
2
4
8
31
6
12, ,
2
11 14421768256,
16
c
ccc
aca c
cccccccc
cc
 
 
(3.4)
where 0234
, , , 0cccc
ay obtain many
are arbitrary constants.
We m kinds of exact solutions de-
on the special vapending lues chosen for cj.
Case 1. If 013 2
0, 0ccc c
  and 40c, Equa-
tion (2.5) admits a triangle solution


2
2
sec .
cc
c4

 (3.5)
Substituting (3.5) along with (3.3)
respectively yields two triangle solutions of (1.1)
and (3.4) into (3.2)


22
12222
,3secuxtcccxct

 

(3.6)
and


22
22
,824 sec176
22 2
xtc ccxct

 

(3.7)
with c2 < 0 being an arbitrary constant.
u
Case 2. If 2
2
4
0
c
01
32
, 0,
4
cccc
c
  and ,
wo periodic solutions of (2.5)
40c
e can find tw

22
tan .
cc
4
22c


 

 (3.8)
Substituting (3.8), (3.3) and (3.4
tively yields two triangle solutions of (1.1)
) into (3.2) respec-

2
222
322
3
,tan
cc
uxt c cx


 
224
t



(3.9)
and


22
2
422
,812tan 44
2
c
uxtccx ct

 



2
(3.10)
with c2 > 0 being an arbitrary constant.
Copyright © 2011 SciRes. AM
D. H. FENG ET AL.
754
Case 3. For , we can
gain the followio
013 2 4
0, 0, 0ccc cc  
ng hyperbolic function solution t (2.5)


2
2
sech .
cc
c4

 (3.11)
e
solutions of (1.1)
Similarly, we obtain two peak-shaped solitary wav


2
ct
2
52222
,3sechuxtc ccx
 
an
(3.12)
d


22
62222
, 8uxtc 24 sech176.ccxct
 
(3.13)
with c2 > 0 being an arbitrary constant.
show the physical insight of these solitary wave
solutions, here we take u5 as an example. Figure
the wave plot of the solution u5 with c2 = 1 and th
ll-shaped solitary
w
To 1 shows
e initial
status of u5. Clearly the solution is a be
ave with peak form and describes the traveling of wave
in the negative x-direction.
Case 4. For
2
2
01324
4
,0,0,0
4
c
ccccc
c
, Equa-
tion (2.5) admits two following hyperbolic function solu-
tions

22
4
22ctanh .
cc

(3.14)

 


turn gives two peak-shaped solitary wave solu-
tions of (1.1)

This in

2
222
722
3
,tanh
cc
uxtccxt
224

 


and
(3.15)

22
2
822
,812 tanh44
2
c
uxtccx ct
2
 
c2 < 0 being an arbitrary constant.
Case 5. For
(3.16)
with
01 32424
(2.5)
0,2, 0, 0,cc ccccc  
one can find the following hyperbolic solutions of

2
2
11tanh c
c




4
.
22c




(3.17)
wave solutions.
Similar to Case 1, (1.1) has two peak-shaped solitary


2
22
22
9
3
,tanh16
2
c
cc
ux
t xct
24 32
 
and
(3.18)

2
2
102 22
,46tanh 11
2
c
uxtccxct
2
 
(3.19)
with c2 > 0 being an arbitrary constant.
Case 6. For
22 2
21 1
0132
0
2
41
1, 0,
ck k
cccc
cp
 
and
40c
, Equation (2.5) admits the fo
elliptic function solution
llowing Jacobian

2
12 2
1
41 1
cp p

where 2
11
21pk
cn,,
kc ck



(3

.20)
and

122,1k is an arbitrary
constant.
This in turn gives the following two
wave solution of (1.1)
doubly periodic

22
21 12
11 ,
ck c
uxt
22
2 1
3cn ,
c q
cx tk

 


(3.21)
2
11
1
pp
p



and

22
2
21 212
12 21
,,k
2
11
1
24 176
,8 cn
ck cqc
uxt cxt
pp
p


 





(3.22)
where 42
111
1qkk
.
emonstrate the phTo dysical insight of the new solu-
tions, we take u11 as an example. Obviously the solution
is a Jacobi elliptic function with two periods a
the traveling o the negative x-direction
nd de-
scribes f wave in
with the wave velocity 22
21 1
cqp . Figure 2 shows the
wave plot of the solution u11 to (1.1) with c2 = 0.5, k1 =
0.9 and the initial status of u11.
Case 7. For
,0,0,0,
1
4231
2
24
2
2
2
2
p
0
cccc
c
kc
c
we can obtain one Jacobian elliptic function solution of
(2.5)

22
2
42 2
cp p

where
dn, ,
cc
k



(3.23

)
2
22
2pk
and is an arbitrary con-
stant.
Thus we can give two corresponding
ing wave solutions of (1.1)

20, 1k
periodic travel-

2
2
2222
13
3n ,
ccqc
ux tk
2 2
2
,d
t cx
pp
22
2
p
 


and
(3.24)

2
2
22 22
14 22
,,tk
2
22
2
24 176
,8 dn
cc qc
uxt cx
pp
p


 





(3.25)
where 42
222
1qkk
.
Copyright © 2011 SciRes. AM
D. H. FENG ET AL.
Copyright © 2011 SciRes. AM
755
Figure 1. The plot of the peak-shaped solitary wave solution u5 to (1.1) with c2 = 1 and the initial status of u5.
Figure 2. The plot of the periodic traveling wave solution u11 to (1.1) with c2 = 0.5, k1 = 0.9 and the initial status of u11.
C2
ase 8. For 23
01
324
2,0,0,0,ccccc
cp
 (2.5)
43
ck
mits two kinds of Jacobian elliptic doubly periodic
w
ad
ave solutions

2
23 2
3
sn ,
ck ck
 

 


(3.26)
43 3
cp p

where and
2
33
1pk
30,1k is an arbitra
stant.
This in turn gives two corresponding periodic travel-
olutions)
ry con-
ing wave s of (1.1

22
2
23 32
2
15 23
2
3
3
,sn ,
ck qc
c
uxt cxtk
p


 


3
3
p
p



(3.27)
and

16
2
2
23 32
2
23
2
33
3
,
24 176
8sn ,
uxt
ck qc
c
cx
pp
p

2
,tk
 





where
4. Conclusions and Summary
In this paper, the Fan sub-equation method has been
successfully applied to obtain many traveling wave solu-
tions of the (1 + 1) dimensional Kaup-Kuper
equation. These rich results show that this method is ef-
fective and simple and a lot of solutions can be obtained
ame time. It is also a promising method to solve
) and Postdoctoral Science
oundation of Central South University.
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5. Acknowledgements
This work is supported by National Natural Science
Foundation of China (11061010, 61004101), Guangxi
Natural Science Foundation (2011GXNSFA018136 and
2011GXNSFB018059), China Postdoctoral Science
Foundation (20100480952
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(3.28)
 6
42
333
1qkk
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