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Journal of Applied Mathematics and Physics, 2013, 1, 49-53
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15007
Open Access JAMP
Exact Solution and Conservation Laws for Fifth-Order
Korteweg-de Vr ies Equation
Elham M. Al-Ali
Mathematics Department, Faculty of Science, University of Tabuk, Tabuk, KSA
Email: [email protected]
Received September 11, 2013; revised October 11, 2013; accepted October 17, 2013
Copyright © 2013 Elham M. Al-Ali et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
With the aid of Mathematica, new exact travelling wave solutions for fifth-order KdV equation are obtained by using
the solitary wave ansatz method and the Wu elimination method. The derivation of conservation laws for a fifth-order
KdV equation is considered.
Keywords: Soliton Solutions; Conservation Laws; Nonlinear Evolution Equations
1. Introduction
It is well-known that nonlinear complex physical phe-
nomena are related to nonlinear partial differential equa-
tions (NLPDEs) which are involved in many fields from
physics to biology, chemistry, mechanics, etc. As mathe-
matical models of the phenomena, the investigation of
exact solutions to the NLPDEs reveals to be very important
for the understanding of these physical problems. Many
mathematicians and physicists have well understood this
importance when the importance of this so they decided to
pay special attention to the development of sophisticated
methods for constructing exact solutions to the NLPDEs.
Thus, a number of powerful methods have been presented.
We can cite the inverse scattering transform [1], the
Bäcklund and Darboux transform [2-5], Hirota’s bilinear
method [6], the homogeneous balance method [7], Jacobi
elliptic function method [8], the tanh-method and ex-
tended tanh-function method [9-15], F-expansion method
[16-18] and so on.
The notion of conservation laws is important in the
study of nonlinear evolution equations (NLEEs) appear-
ing in mathematical physics [19]. The mathematical ori-
gin of conservation laws results from the formulation of
familiar physical laws such as for mass, energy and mo-
mentum [20]. As is known, the investigation of conser-
vation laws of the Korteweg-de Vries (KdV) equation led
to the discovery of a number of techniques to solve
NLEEs [21], e.g., Miura transformation, Lax pair, in-
verse scattering technique and bi-Hamiltonian structures.
On the other hand, it is useful in the numerical integra-
tion of NLEEs [22] (e.g., to control numerical errors);
particularly with regard to integrability and linearization,
constants of motion, analysis of solutions, and numerical
solution methods [23]. Consider a dynamical system,
,, ,,,,
txt
ufxtuuuu
xt
(1)
where
,uuxt is a function of two independent
variables t and x. The functional

,
I
uxt
is said to
be a constant of motion or an integral of Equation (1), if
it satisfies

d,
dIuxt
t0.


Generally, we can derive
constants of motion from conservation law, which enjoys
the general form as [24]
 
dd
,,
dd
Vu xtG u xt
xt
0
 
 
(2)
while the components V and of the conserved
vector
G
,VG are functions of ,
x
tand derivatives of
. The equality (2) is assumed to be satisfied for any
solution of the corresponding system of equations, is
called conserved density and G is called conserved
flow. With the assumption that the function
uV
t,ux and
its derivatives with respect to
x
go to zero sufficiently
fast as x
 
,,d
I
uxtVxtx



(3)
is obtained to be a constant of motion. It has already been
proved that a large number of NLEEs possess an infinite
E. M. AL-ALI
50
number of conservation laws such as the fifth-order KdV
equation
2. Exact Solution for Fifth-Order KdV
Equation
With the rapid development of science and technology,
the study kernel of modern science is changed from lin-
ear to nonlinear step by step. Many nonlinear science
problems can simply and exactly be described by using
the mathematical model of nonlinear equation. Up to
now, many important physical nonlinear evolution equa-
tions are found, such as sin-Gordon equation, KdV equa-
tions, Schrodinger equation all possess solitary wave
solutions. There exist many methods to seek for the soli-
tary wave solutions, such as inverse scattering method,
Hopf-Cole transformation, Miura transformations, Dar-
boux transformation and Bäcklund transformation [2-5],
but solving nonlinear equations is still an important task.
In this paper, with the aid of Mathematica, a traveling
wave solution for a class of fifth-order KdV equation
2
53 2
10 2030
txx xx
uu uuuuuu x
(4)
In order to obtain the soliton solution of (4), the soli-
tary wave ansatz is assumed as

, sech,,
n
uxtABdx ct


(5)
where
A
is the soliton amplitude, is the width of
the soliton, is the soliton velocity and
d
c
B
is constant
to be determined later, the unknown index n will be de-
termined during the course of derivation of the solution
of Equation (4). From Equation (5), I obtain Equation (6).
With the aid of Mathematica or Maple, from (5) and
(6
uating the exponents
), we can get Equation (7).
Now, from Equation (7) eq
13n
and 32n
leads 1332nn, which gives
2.n
From (7) setting the coeffici
sinh
ents of
3
sech
, 5
sech sinh
and 7
sech sinh
to
zero, I get Equations (8)-(10).



 

2
21
5312 1
5
211
2
2333
23
,sech,sechsinh .
sech sinh41 sech sinh
4 1sechsinh6 12sechsinh
12sechsinh2 12sechsinh
3 112sech
nn
t
nn
x
nn
nn
n
uxtBAu Acdn
udAnAnn
An nAn nn
AnnnA nnn
Ann Annn

 
 
3




 
 
 
 







3
33
55
21 1
3
33
31
2
22
1
sinh
4 123sechsinh
1234 sechsinh
sech sinh1 sech sinh
12sech sinh
sechsinhsech sinh
1 sechsinh
sech si
n
n
nn
x
n
nn
xx
n
n
x
An nnn
An nnnn
uAn Ann
An nn
uu AdnAn
An n
uAdn



 








 

 nh .
(6)


2
532
233551
35325233
535455 3
5525354555
10 2030
3010sechsinh
208 302010
20102sech sinh
24503510sech sinh
60
txx xxx
n
n
n
uu uuuuuu
AB dnAcdnABdnAdn
ABd nAd nABd nAd nABd n
AdnAdn ABdn
Ad nAd nAd nAd nAd n


 

 




223312
23232233 32
313
30sechsinh
205030sech sinh
3020sechsinh 0
n
n
n
ABdn Adn
Adn AdnAdn
Adn



 

(7)
Open Access JAMP
E. M. AL-ALI 51
(8)
0,
(9)
0. (10)
Solving the above system by the aid of Wu elimination
method [25], I obtain the two solutions
4
(11)
and
224
304016 0,Bc Bdd
224
224Bd AB Add 
224
812AAd d
22
6,2,56,AdBdcd
22
4
2,, 16.
3
4
A
dB dcd
(12)
Then the soliton solutions of the fifth order KdV equa-
is given by tion

24
1,6 sech562,
n
uxtddxdtd
2
and
(13)


242
4
2,2 sech16.
3
uxtddx dtd
(14)
n
3. Systematic Construction Method
nfinitely Many Conservation L
ifth-Order KdV Equation
We recall the definition [16,23] of a differe
(DE
of
aws for I
F
ntial equation
) that describes a pss. Let 2
M
be a two dimensional
differentiable manifold with coordinates
,
x
t. A DE
for a real function a nec-
essary and sufficient conditifor the exi
unctions
(15)
depending on u and its derivatives such that the
one-forms
,

,uxt describes
on
pss if it is a
stence of dif-
ferentiable f
,1 3,12,
ij
fi j
1 11122212233132
dd,dd,dfxftfxftfx dft

  
(16)
isfy the structure equations of a pss, i.e.
sat ,
132 21312
d,d,d.
3

 (17)
As a consequence, each solution of the Dovides a
local metric on
E pr
2
M
, who
DE for is the integrability
condition for the problem [14,26]:
se Gaussian curvature is con-
stant, equal to −1. Moreover, the above definition is
equivalent to saying that u
1
dΩ,,
2




(18)

where denotes exterior differentiation,
d
is a col-
umr and the 2 × 2 matrix
n vecto
ΩΩ 1,2
ij ,,ij is
traceless
213
13 2

Take
1
Ω.
2



,
(19)
from Equations (18) and (19), we obtain


dddd
Ωdd
dd dd
xAt qxBtSx Tt
rx CtxAt






,,
xt
ST
 
(20)
where S and T are two 2 × 2 null-trace matrices
,
q
Sr



(21)
.
A
B
TCA



(22)
Here
is a paeter, independent of ram
x
and t,
while q and r are functions of
x
and t. Now

2
0d dΩΩddΩΩ ,Ω
 
 
which requires the vanishing of the two form
ΘdΩΩΩ 0,
 (23)
or in component form
0,
,
0,
22
x
AqCrB
qAqB B
–2
20
tx
tx
rC Ar C

 

(
or
24)
11,12,312221 32,
21,22,11 321231,
31,32,11 221221,
tx
tx
tx
ff ffff
ff ffff
ff ffff
 
 
 
(25)
where


22 12 3212 32
11 1
,,
11 3111 31
22
.
22 2
11
,qffrff
A
fBff cff
 
(26)
Chern and Tenenblat [27] obtain
rectly from the structure equations (17). By suitably
choosing and in (24), we shall obtain vari-
ous fifthV uation wh
Konno and Wadati introduced the function [28]
ed Equation (24) di-
,,rAB
order Kd
C
eqich q must satisfy.
1
Γ,
2
( 27)
this function first appeared used and explained in the
uations in
[11,13], and see also the classical pap
27]. Then Equati
:
geometric context of pseudo spherical eq
ers by Sasaki [29]
on (20) is reduced and Chern-Tenenblat [
to the Riccati equations
2,rq
x

  
(28)
2
2.
A
CB
t

 
(29)
Equations (28) and (29) imply that
Open Access JAMP
E. M. AL-ALI
52
 
ГГ 2Г.
xt
crcqrBcAr
 (30)
to both sides and using the expression
Гt
r Adding
xc
A
qrfrom (24), Equation (30) takerm s the fo
 
ГГ0,rAc
tx


 (31)
let us show how an infinite number of conservatio
result from these results. The Riccati equations for in
the
n laws
Г
-
x
variable can be rearranged to take the form
 
2Г
ГГ
r
rrqrr
x
r

 

(32)
A similar pair of equations can be obtained for the t
derivatives. Expand Гr into a power series in the
verse of
in-
so that

1
Г,, ,
n
n
rxt xt

n
(33)

the n
are unknown at this point, however a recursion
relation can be obtained for the n
by using (32), sub-
stituting (33) into the Г equation in (32), I find that



1
1,n
n
nxt

2
1
1
,
,
n
n
n
n
rq x t
xt
rr





a
j
n
n (34)
n
x
Appling the Cauchy product formul
2


1
12
,n
n
n
nnj
xt
n
j





in (34), then I obtain



 
1
1
2,n
n
nxt


 
12
1
2,
nn
jn
j
nj
rq x t


 

11
2
,,
.
nn
n
xx
xt xt
rr
rr











(35)
Now equate powers of
on both sides of this ex-
pression to produce the set of recursions,
 
12
1
1,,
. 2
x
n
n
nknk
x
rx
txtn
r


 


(36)
1
,
k
rq rq
 
Substituting (33) into (31), the followin
conservation laws appears
g system of
 
11
,,
.
nn
nn
This procedure generates an infinite number of con-
servation laws for the equation under examination. To
obtain conservation laws using (37) in a particular exam-
ple using this procedure, let us consider the fifth-order
KdV Equation (4), for Equation (4)
nn
xt xt
Ac
tx r




 




(37)
32
5
223
422
2
22
43 2 2
86
x
xxx
cu u uuu
33
224
1, ,6
3,
2
262,
662.
xx
x
x
x
xx
qruAu u uu
uuu
Buu u
u
uuuuuu
x



 
 



(38)
nto (24), I obtain the fifth-order
tting (38) into (37), it is found that

Substituting (38) i
KdV Equation (4). Pu

11
,,
nn
nn
nn
xt xt
Ac
tx r




 



Since
u
4.
ear wave that
possesses remarkable stability properties. Typically, prob-
lems that admit soliton solutions are in the form o
lution equations that describe how some variable or set of
variables evolve in time from a given state. The equa-
tio
tions, partial difference equa-
tions, and integro-differential equations, as well as cou-
pled ODEs of finite order.
In this paper, we considered the construction of exact
so
hysics and applied
mathematics. Solitons are found in various areas of
physics from hydrodynamics and plasma physics, non-
linear optics and solid state physics, to field theory and
gravitation. NLEEs which describe soliton ph
have a universal character.
A travelling wave of permanent form has already been
met; this is the solitary wave solution of the NLEE itself.
Such a wave is a special solution of the governing equa-
tio
The Soliton equations play a central role in the field of
integrable systems and also play a fundamental role in
several other areas of mathematics and physics.
REFERENCES
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1234
,0, ,,
x
uuu 
Conclusions
A soliton is a localized pulse-like nonlin
f evo-
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differential difference equa
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wave solutions for the above equation by using the soli-
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The soliton phenomena and conservation laws of
NLEEs represent an important and well established field
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enomena
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