Applied Mathematics, 2010, 1, 307-311
doi:10.4236/am.2010.14040 Published Online October 2010 (http://www.SciRP.org/journal/am)
New Periodic Solitary Wave Solutions for a
Variable-Coefficient Gardner Equation from Fluid
Dynamics and Plasma Physics
Mohamed Aly Abdou
Theoretical Re search Group, Physics Department, Faculty of Science, Mansoura University,
Mansoura, Egypt
Received May 6, 2010; revised August 17, 2010; accepted August 20, 2010
Abstract
The Gardner equation with a variable-coefficient from fluid dynamics and plasma physics is investigated.
Different kinds of solutions including breather-type soliton and two soliton solutions are obtained using bi-
linear method and extended homoclinic test approach. The proposed method can also be applied to solve
other types of higher dimensional integrable and non-integrable systems.
Keywords: Extended Homoclinic Test Approach, Bilinear Form, Gardner Equation with a
Variable-Coefficient, Periodic Solitary Wave Solutions
1. Introduction
In nonlinear science, many important phenomena in vari-
ous fields can be describe by the nonlinear evolution
equations. Seeking exact solutions of nonlinear partial
differential equations is of great significance as it ap-
pears that these (NLPDEs) are mathematical models of
complex physics phenomena arising in physics, mechan-
ics, biology, chemistry and engineers. In order to help
engineers and physicists to better understand the mecha-
nism that governs these physical models or to better pro-
vide knowledge to the physical problem and possible
applications, a vast variety of the powerful and direct
methods have been derived. Various powerful methods
for obtaining explicit travelling solitary wave solutions to
nonlinear equations have proposed such as [1-8].
One of the most exciting advances of nonlinear sci-
ence and theoretical physics has been a development of
methods to look for exact solutions for nonlinear partial
differential equations. A search of directly seeking for
exactly solutions of nonlinear equations has been more
interest in recent years because of the availability of
symbloic computation Mathematica or Maple. These
computer systems allow us to perform some complicated
and tedious algebraic and differential calculations on a
computer.
Much attention has been paid to the variable coeffi-
cients nonlinear equation which can describe many
nonlinear phenomena more realistically than their con-
stant coefficient ones [9]. The Gardner equation, or ex-
tended KdV equation can describe various interesting
physics phenomens, such as the internal waves in a stra-
tified Ocean [10], the long wave propagation in an inho-
mogeneous two-layer shallow liquid [11] and ion acous-
tic waves in plasma with negative ion [12], we consider a
generalized variable-coefficient Gardner equation [13]
2
()()()()() 0,
txxx xxx
ftgt htrtt  

 (1)
where (,)
x
t

is a function of
x
and t. The coeffi-
cients ()ft,()
g
t,()ht ,()rtand ()t

are differential func-
tions of t. Equaion (1) is not completely integrable in
the sense of the inverse scattering scheme it contains
some important special cases:
In case of () 0ht

,() 0rt

and ()0t

, Equation (1)
reduces to
()() 0,
txxx x
ft gt



 (2)
and
() ()()ftgta bftdt





. (3)
Equation (2) possesses the Painleve property [14,15].
The Bäcklund transformation, Lax pair, similarity reduc-
tion and special analytic solution of Equation (2) have
been obtained [16-20].
For ()6 ()
g
tat

,() 6ht r and ()() 0rt t


,

M. A. ABDOU 308
Equation (1) reduces to
2
6() 6()0,
txxxxx
atrf t   (4)
which describe strong and weak interactions of different
mode internal solitary waves, etc. When()
f
tr

,
() 6gt

,() 6ht

,()() 0rt t

, Equation (1) be-
comes the constant-coefficient Gardner equation
2
66 0,
tx xxxx
r     (5)
where r,

and

are constants. It is widely applied
to physics and quantum fields, such as solid state physics,
plasma physics, fluid dynamics and quantum field the-
ory.
When ()6gt ,() 1ft,()()0htrt, Equation (1)
reduces to constant coefficient KdV equation
6()0,
tx xxxt

    (6)
which possesses the Painleve property. If ()0t


or

0
1
() 2 ttt
 , it corresponds to the well known cy-
lindrical KdV equat i o n.
The structure of this paper is organized as follows; In
Section 2, with symbolic computation, the bilinear form
of Equation (1) are obtained. In order to illustrate the
proposed method, we consider for a variable-coefficient
Gardner equation from fluid dynamics and plasma phys-
ics and new periodic wave solutions are obtained which
included periodic two solitary solution, doubly periodic
solitary solution. Finally, conclusion and discussion are
given in Section 3.
2. Bilinear Form of the Gardner Equation
with Variable Coefficients
Making use the dependent variable transformation as
(,) ()(,),
x
tkt wxt
x


 (7)
into Equation (1) and integrating once with resp ect to
x
,
admits to [13]
22
33
1
()()() ()()()
2
1()()() ()() ()0
3
t xxxx
xx
ktwktwftktwgtk tw
htk twrtktwtktw 
 

, (8)
with the integration constant to zero. Then introducing
the transformation
(,)
(,) arctan,
(,)
vxt
wxt uxt



(9)
where (,)uxt and (,)vxt are differential functions of
x
and t into (8) yields [13]
22
32
22 222
2
32
22 22
3
3
22 2
. (,)
[()( )( )]arctan( )
(,)
..[(..)]
() ()3()
.. 1
8()( )()
2
.. 1()()() ()
3
t
xxx
xx
xx
Dvu vxt
kt tktkt
uxt vu
DvuDvuD uuvv
ftkt vu vu
Dvu Dvu
gtk t
vu vu
Dvu Dvu
htk trtkt
vu v



 







 






 20,
u




(10)
where the prime denotes the derivative with respect to t,
and t D,
x
D, 2
x
D and 3
x
D are the bilinear derivative
operators [7] d efined as
,
(,).(,)
[(,)(,)]
mn
xt
mn
x
xt t
DDfxtgxt
fxtgxt
xx tt




 
 
 


 

. (11)
Decopling Eq u at i on (1 0) , we obt ai n [1 3]
()()()0,kt tkt



 (12)
3
1
8()()() ()0,
3
ftkthtk t

 (13)
3
()(). 0,
txx
DftDrtDvu




(14)
21
3() (..]()().,
2
xx
f
tD uuvvgtktDvu (15)
Via Equations (12) and (13), we have the following
relations


 
2
01
,24
tdt
ktcefthtk t
 
 (16)
where 0
c is a nonzero arbitrary constant. That is to say,
through the dependent variable transformation
()
0(,)
(,)arctan (,)
tdt
x
vxt
xt ceuxt









. (17)
Equation (1) is transformed into its bilinear form, i.e.,
Equations (14) and (15) under constriant (16).To solve
the reduced Equations (14) and (15) using the extended
homoclinic test function [21-29], we suppose a solution
of Equations (14) and (15) as follows
11 11
1221
(,)cos[ ],
ax btax bt
vxtepax btqe

 (18)
and
11 11
2222
(,)cos[ ],
ax btax bt
uxtepax btqe

 (19)
where i p, i q, i a, i b (1,2)i

are parameters to be
determined later.
Substituting Equations (18) and (19) into Equations
(14) and (15), and equating all coefficients of 11
()
[
j
ax bt
e

M. A. ABDOU 309
(1,0,1),j 22 cos( ),ax bt 22 sin( )ax bt] to zero, we
get the set of algebraic equation for i p, i q, i a, i b
(1,2)i. Solve the set of algebraic equations with the
aid of Maple, we have many solutions, in which the fol-
lowing solutions are
Case (1):
3
221111
2211211
12 1
12
0, 0, 0, ()(),
()(), , , 0, ,
12( )()
() ()( )
bqqbftarta
rtrt pp ppaaa
fta pp
gt kt pp
 



(20)
Case (2):
3
22221111 1
212211
12 1
12
, , , 4()(),
()(), 0, 0, , ,
4()( )
() ()( )
bbqqqqbftarta
rtrtppaa aa
ftaq q
gt kt qq
 



(21)
Case (3):
212211
2211112
3
11 121
112
0, 0, , ,
, , 0,
4()24()()
(), ()()( )
ppqqqq
bbbbaaa
f
tabftaqq
rt gt
aktqq
 


 
(22)
Case (4):

22
21
112211222 2
1
32
22 12 2 21
21
11
22
1
32
22 12
21
2
()0, , , , , ,
4
2() 2()
, ,
4
()3 ()
(),
ap
gtppbbaa aaqa
ab ftaftaa
ap
qb a
a
bfta ftaa
rtp p
a




 
(23)
Case (5):
2
22222112 2
2
12 212
22 1
12
2222
11 12221 122
(4()()), , , ,
()(), (4()()), ,
24( )()
() ,
()( )
11 111 1
,
84 884 8
baftartaappp p
rtrtbiaftartaia
ift app
gt kt pp
qppppq pppp

 


  
(24)
Using Equation (20), Equations (18) and (19) can be
written as
3
111
[() ()]1
(,) ,
axf tartat
vxt ep
 
 (25)
and
3
111
[()()]2
(,) axf tartat uxt ep
 
. (26)
Inserting Equations (25) and (26) into Equation (17),
admits to the new solitary wave solution of Equation (1)
as
()
0(,)
(,)arctan (,)
tdt
x
vxt
xt ceuxt





(27)
With the aid of Equation (21), Equations (18) and (19)
yields
33
111 111
[ 4()()][ 4()()]
1
(,) ,
axftarta taxftarta t
vxt eqe
 
 (28)
and
33
111 111
[4()() ][4()() ]
2
(,) ,
axftarta taxftarta t
uxt eqe
 
 (29)
Knowing Equations (28) and (29) with Equation (17),
we have the solitary wave solution of Equation (1) as
()
0(,)
(,)arctan (,)
tdt
x
vxt
xt ceuxt





(30)
In view of case (3), Equations (18) and (19) reads
11 11
1
(,) ,
ax btax bt
vxt eqe

 (31)
and
11 11
2
(,) ax btaxbt uxt eqe

 . (32)
Inserting Equations (31) and (32) into Equation (17),
admits to the new solitary wave solution of Equation (1)
as
()
0(,)
(,)arctan (,)
tdt
x
vxt
xt ceuxt





(33)
Via Equation (23) with Equations (18) and (19), we
have
32
(2()2() )
12 2
21
12
32
(2()2() )
12 2
21
12
22
21 22
2
1
22
21
2
1
(,)cos[ ]
4
,
4
ab ftaftaa
a
ab ftaftaa
a
ax t
ax t
ap
vxteaxbt
a
ap e
a







 









(34)
and
32
(2()2() )
12 2
21
12
32
(2()2() )
12 2
21
12
22
21 22
2
1
22
21
2
1
(,)cos[]
4
,
4
ab fta ftaa
a
ab ftaftaa
a
ax t
ax t
ap
uxteaxbt
a
ap e
a







 









(35)
Using Equations (34) and (35), admits to the new soli-
tary wave solution of Equation (1) as

M. A. ABDOU 310
()
0(,)
(,)arctan (,)
tdt
x
vxt
xt ceuxt





(36)
According to case (5), Equations (18) and (19) be-
comes
2
22 2
2
22 2
(4 ()())
12
(4 ()())
2
22 1
(,) cos[
[(4()())]],
ia xiaft ar tt
ia xiaft ar tt
vxt epax
aftarttqe




 



(37)
and
2
22 2
2
22 2
(4 ()())
22
(4 ()())
2
22 2
(,) cos[
[(4()())]]
ia xiaft ar tt
ia xiaft ar tt
uxt epax
aftarttqe




 



.
(38)
By means of Equations (36) and (37) with Equation
(17) we have a new solitary wave solutions as
()
0(,)
(,)arctan (,)
tdt
x
vxt
xt ceuxt





(39)
3. Conclusions
In this paper, with the aid of two methods, namely, bi-
linear form and the extended homoclinic test approach,
we obtain breather-type soliton and two soliton solutions
for the Gardner equation with a variable-coefficient from
fluid dynamics and plasma physics. The results reported
here show that the extended homoclinic test approach is
very effective in finding exact solitary wave solutions for
nonlinear evolution equations with variable coefficients.
Finally, it is worthwhile to mention that, the proposed
method is reliable and effective can also be applied to
solve other types of higher dimensional integrable and
non-integrable systems.
4. Acknowledgements
The author would like to express sincerely thanks to the
referees for their useful comments and discussions.
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