Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 9, 405 - 426
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/nade.2016.6633
Conservation Laws and Multiple Simplest
Equation Methods to an Extended Quantum
Zakharov-Kuznetsov Equation
Shoukry El-Ganaini
Mathematics Department, Faculty of Science
Damanhour University, Bahira 22514, Egypt
M. Ali Akbar
Department of Applied Mathematics
University of Rajshahi, Bangladesh
Copyright © 2016 Shoukry El-Ganaini and M. Ali Akbar. This article is 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
The extended quantum Zakharov-Kuznetsov (QZK) equation, which arises in the
study of astrophysics is investigated. Exact traveling wave solutions are obtained
by using distinct types of the simplest equation method namely, the simplest
equation method, the extended simplest equation method and the modified
simplest equation method. In addition conservation laws via a new conservation
theorem are constructed for the underlying equation.
Mathematics Subject Classification: 35A20, 35D99, 35E05, 35Q40, 35Q51, 37J15
Keywords: simplest equation method, extended simplest equation method, modified
simplest equation method, conservation law, extended quantum Zakharov-Kuznetsov
(QZK) equation
1. Introduction
It is of great importance to extract exact solutions of nonlinear partial differential
equations (NPDEs) arising in mathematical physics, because many physical
phenomena are described by NPDEs. Great efforts have been developed to
different methods to solve NPDEs in the literature [1-9]. Also, Conservation laws
406 Shoukry El-Ganaini and M. Ali Akbar
have a significant role in the solution processes of NPDEs. By using the
relationship between Lie point symmetries and conservation laws, we can further
generate other conservation laws. Conservation laws describe physical conserved
quantities such as mass, energy, momentum and angular momentum as well as
charge and other constants of motion [10, 11]. A variety of powerful techniques
have been used to derive conservation laws [12-14]. The role conservation laws
have been recently used combined with double reduction theory to obtain exact
solutions of NPDEs [15-19], therefore it is very important to investigate
conservation laws for PDEs.
In this work, we consider the extended quantum Zakharov-Kuznetsov (QZK)
equation [20, 21] as
0)()( xxy
uxyy
ucyyy
uxxx
ubx
uuat
u (1)
where ba, and c are all constants, while ),,( tyxu represents the electrostatic wave
potential in plasmas. Eq.(1) supports soliton solutions that are the outcome of a
delicate balance between dispersion and nonlinearity.
The outline of this paper is as follows .In Sections 2, 3 and 4, a brief description
of the simplest equation methods for extracting traveling wave solutions of
nonlinear partial differential equations (NPDEs.) is presented. Thus, we construct
the conservation laws using a new conservation theorem in Sec.5 for the
underlying equation. Some conclusions are given in Sec.6.
2. Description of the simplest equation method [22-24]
Consider a general nonlinear PDE in the form
.0),...,,,,,,( xxx
uxt
utt
uxx
ux
ut
uuP (2)
where P is a polynomial in its arguments.
Using the traveling wave transformation 0,)(),( ctxutxu ,
where c is the wave speed ,and 0 is an arbitrary constant, to transform Eq.(2) to
a nonlinear ordinary differential equation (ODE) as
,0,...),,,( uuuuQ (3)
where the prime denotes the derivation with respect to .
Suppose the solution of Eq.(3) can be expressed as in the form
N
i
ii
u
0
))(()( , (4)
Conservation laws and multiple simplest equation methods 407
where )( satisfies either the Bernoulli or Riccati equation , N is a positive
integer that can be determined by balancing procedure and i
are parameters to
be determined , Ni ,...,2,1,0 .
Let us consider the Bernoulli equation in the following form
)(2)()( qp , (5)
where p and q are arbitrary constants. The solutions of Eq.(5) can be written as
,))
0((sinh))
0(cosh(
1
1)(
ppAq
Ap (6)
,))
0((sinh))
0(cosh(
2
)))0
((sinh))0
(cosh(()(
ppAq
ppp (7)
where 1
A ,2
A and 0 are arbitrary constants.
Also let us consider the Riccati equation as
,)())(2)( sqp (8)
where qp , and s are constants. It should be mentioned that Eq.(8) has the
following solutions [25]
,)]0
()2/tanh[()2/()2/()( ppq (9)
,))2/((sinh)/2())2/((cosh
))2/((sec
)])2/tanh[()2/()2/()(
pC
h
ppq
(10)
where spq 422 .
Substituting (4) into (3) with (5) (or (8)), then the left-hand side of (3) is reduced
to a polynomial in )( . Setting each coefficient of this polynomial to zero,
yields a system of algebraic equations which can be solved for qpi
,, . Hence,
substituting these values into (4), the exact traveling wave solutions are reported
then for Eq.(2).
Note. If we put Ap and 1q in Bernoulli equation (5), we obtain
)(2)()( A , (11)
which has the exact solution
408 Shoukry El-Ganaini and M. Ali Akbar
,))]0
()2/tanh((1[)2/()( AA (12)
provided 0A , and the other exact solution
,))]0
()2/tanh((1[)2/()( AA (13)
if 0A .
Application of the simplest equation method using Bernoulli equation
Introducing a wave variable
0 vtrykx , to convert (1) an ordinary differential equation(ODE)
0))()33(( urkrkcrkbuakuuv (14)
Integrating (2) with respect to and setting the integration constant to zero, yields
0))()33((2)2
1( urkrkcrkbuakvu . (15)
Thus, we are concerned to solve the equation (15).
Considering the homogeneous balance between u and 2u , we get 2N ,
therefore , the solution of (24) can be written as
)(22
)(10
)( u (16)
Substituting (16) into (15) and making use of the Bernoulli equation (5) and then
equating the coefficients of the functions )( i to zero, we obtain the following
system of algebraic equations
)21(0202
1
0:)(0
)20(01
2101
:)(1
)19(0)1
32
24(2
)20
221
(2
1:)(2
)18(0)2
101
22(21
:)(3
)17(02
26222
1:)(4
kav
Mpkav
Mqppvka
Mqpqka
Mqka
.))()33(( rkrkcrkbM (22)
Conservation laws and multiple simplest equation methods 409
On solving the above system of over-determined algebraic equations, we obtain
two sets of solution,
,,)/12(,)/12(,)/2( 2221
20 MpvMkaqMkaqpMkap
(23)
.,)/12(,)/12(,0 22210 MpvMkaqMkaqp (24)
Therefore, using solutions (6) and (7) of (5) and the values of 210 ,, from
(23) into solution ansatz (16), we obtain the exact solutions of (1) in the form:
,
2
))0
((sinh))0
(cosh(1
1)/212(
))0
((sinh))0
(cosh(1
1)/12(
)/22(),,(1
ppAq
ApMkaq
ppAq
ApMkaqp
Mkaptyxu
(25)
2
))0
((sinh))0
(cosh(2
)))0
((sinh))0
(cosh(()/212(
))0
((sinh))0
(cosh(2
)))0
((sinh))0
(cosh(()/12(
)/22(),,(2
ppAq
pppMkaq
ppAq
pppMkaqp
Mkaptyxu
(26)
where .))()33((,0
)2( rkrkcrkbMtMpyrxk
In the same way, using solutions (6) and (7) of (5) and the values of 210 ,, ααα
from (24) into solution ansatz (16), we obtain the following exact solutions of (1):
,
2
))0
((sinh))0
(cosh(1
1)/212(
))0
((sinh))0
(cosh(1
1)/12(),,(3
ppAq
ApMkaq
ppAq
ApMkaqptyxu
(27)
410 Shoukry El-Ganaini and M. Ali Akbar
2
))0
((sinh))0
(cosh(2
)))0
((sinh))0
(cosh(()/212(
,))
0((sinh))
0(cosh(
2
)))0
((sinh))0
(cosh(()/12(),,(
4
ppAq
pppMkaq
ppAq
pppMkaqptyxu
(28)
where .))()33((,)( 2 rkrkcrkbMtMpyrxk
and 1,,,,,,, Akrqpcba and 2A are arbitrary constants.
Application of the simplest equation method using Riccati equation
In the case of Riccati equation, the solution ansatz can be written as,
)(22
)(10
)( u (29)
Substituting (29) into (15) and making use of the Riccati equation (8) and then
equating the coefficients of the functions )( i to zero, we thus obtain a system
of algebraic equations and by solving it, we obtain
)30(,2/2
,)(212
2)33(212
,23,1
)/(2
,11
,00
sqp
rkprk
karkpbcMqvqp
where .))()33(( rkrkcrkbM
Thus, using solutions (9) and (10) of (8), (30) and (16)), yields following exact
solutions of Eq. (1):
)32(
2)))2/((sinh)/2())2/((cosh
))2/((sec
)])2/tanh[()2/()2/(()/(
))2/((sinh)/2())2/((cosh
))2/((sec
)])2/tanh[()2/()2/
1),,(
6
)31(2)])
0()2/tanh[()2/()2/(()/(
)]0
()2/tanh[()2/()2/(
1),,(
5
pC
h
ppqqp
pC
h
ppq
tyxu
ppqqp
ppq
tyxu
Conservation laws and multiple simplest equation methods 411
where
spqrkrkcrkbMtMqyrxk 42.,))()33((,0
)23( , c
and p are determined in (39) , Cksrqpba ,1
,,,,,,, are arbitrary constants.
3 The modified simplest equation method [26,27]
Step1.We suppose that (15) admits the formal solution
,
0
)(
iN
ii
a
(33)
where i
a are constants to be determined such that 0N
a Ni ,...,1,0, .
The function )( is an unknown function to be determined later provided that
0)( .
Step 2. We substitute (33) into (15), we calculate all the necessary derivatives
,...,, and then we determine the function )( .
As a result, we get a polynomial of ,...)1,0()(
jj , with the derivatives of
)( .
Setting all the coefficients of ,...)1,0()(
jj , to zero yields a system of
algebraic equations which can be solved to obtain i
a and )( .
Step3. Substituting the values of i
a and )( and its derivative )( into (33)
leads to exact traveling wave solutions of (1).
Application of the modified simplest equation method
Suppose that (15) admits the formal solution
,
2
)(
)(
2)(
)(
10)(
aaau (34)
where .10
aa and 2
a are constants to be determined such that 02a .
Substituting (34) into (15) and equating all the coefficients of
,4
,3
,2
,1
,0 to zero, we respectively obtain
,0202
1
0:0 akaav (35)
412 Shoukry El-Ganaini and M. Ali Akbar
,01101
:1
aMaakaav (36)
,022
22
2
1322
12
1220
22
:2
aMMa
aMakaaakaav (37)
022
1031
2321
:3
aMMaaaka (38)
042
64222
1:
4
Makaa (39)
where .))()33(( rkrkcrkbM
From (35) and (39), we deduce that
kaMakavaa /122
,/20
,00
, (40)
provided 0ka . Now we consider the following cases:
Case 1. When 00a , 0
1a , and 0 , then we deduce from Eqs.(36-38) that
,0)( 0 Makav (41)
0)2()/224(1
32)212
1/12( kaMMaakakaMv (42)
0121
Maka (43)
From (41) and (43) we have
10
12
aka
M
akav
M
(44)
and consequently we get
112 akav (45)
Integrating (45), we get
1
12
1
aka
v
eC , (46)
and substituting from (46) into (44), we get
Conservation laws and multiple simplest equation methods 413
)
1
12(
11
12)(
aka
v
eCaka
M (47)
Integrating (47), we obtain
,
)
1
12(
12)(
aka
v
eCv
MC (48)
where 1C and 2C are arbitrary constants of integration.
Substituting (44) into (42), we have
,)/12(1 Mvkaa (49)
provided 0ka .
Thus the exact solution of (1) in this case has the form
2
)()
1
12(
)/1
(2
)()
1
12(
)21
33/321
1728(
)()
1
12(
)/1
(2
)()
1
12(
)/1
12()(
0
0
0
0
tvyrxkaka
v
evMCC
tvyrxkaka
v
eakaMC
tvyrxkaka
v
evMCC
tvyrxkaka
v
ekaMCu
(50)
where the value of 1a is provided in (49).
Since 1C and 2C are constants of integration, we might arbitrarily choose their
values. Therefore, if we choose ,/1
,12
MvCC we have respectively the
solitary wave solution
2
))(
1
6(tanh1)2
133/2432(
))(
1
6(tanh1)/6()(
1
0
0
tvyrxkaka
vakaMv
tvyrxkaka
vkavu
(51)
414 Shoukry El-Ganaini and M. Ali Akbar
2
))(
1
6(coth1)2
133/2432(
))(
1
6(coth1)/6()(
2
0
0
tvyrxkaka
vakaMv
tvyrxkaka
vkavu
(52)
Case 2. When 00a , 0
1a , and 0 , then we deduce from Eqs.(36) to (38)
that
,0 Mv (53)
0)2(2241
32)212
112( 22 MkaMaakaMv (54)
0121
Maka (55)
Consequently, we deduce from Eqs. (53) and (55) that
(56)
and thus, we get
112 akav (57)
Integrating (57), we then have
)
1
12(
1
aka
v
eC
. (58)
From (58) into (56), we get
)
1
12(
11
12)(
aka
v
eCaka
M
(59)
Integrating (59), we obtain
,
)
1
12(
12)(
aka
v
eCv
MC
(60)
where 1C and 2C are an arbitrary constants of integration.
Consequently, we conclude from (54) that
1
12
aka
M
v
M
Conservation laws and multiple simplest equation methods 415
1,3)/12(1
iMvkaia (61)
provided 0ka .
Thus, the exact solution of (1) in this case has the form
2
)()
1
12(
)/1
(2
)()
1
12(
)21
33/321
1728(
)()
1
12(
)/1
(2
)()
1
12(
)/1
12()/2()(
0
0
0
0
tvyrxkaka
v
evMCC
tvyrxkaka
v
eakaMC
tvyrxkaka
v
evMCC
tvyrxkaka
v
ekaMCkavu
(62)
where 1
a is as given in (61).
The value of 1C and 2C can be selected randomly as they are integration
constant. Therefore, if we select ,/1
,12
MvCC we have the solitary wave
solution
)63(,2
))(
1
6(coth1)2
122/216(
))(
1
6(coth34
)/2()(1
0
0
tvyrxkaka
vakavM
tvyrxkaka
v
kavu
4. The extended simplest equation method [28, 29]
Suppose the solution of Eq.(5) can be expressed as another
ansatz in the following form
)64(2
))(
1
6(tanh1)2
122/216(
))(
1
6(tanh34
)/2()(2
0
0
tvyrxkaka
vakavM
tvyrxkaka
v
kavu
416 Shoukry El-Ganaini and M. Ali Akbar
,11
00
)(
jN
jj
iN
ii
u (65)
where i
, j
are constants to be determined later, and
01
NN , 1,...,2,1,0,,...,2,1,0 NjNi .
The function )( satisfies the second order linear ODE
, (66)
where and are constants.
Eq.(66) has the following three types of general solutions with double arbitrary
parameters, namely,
0,21
)2/(
0,/)(sin2
)(cos1
0,/)(sinh2
)(cosh1
)(
2
AA
AA
AA
, (67)
0,/2
21
)22
221
(
0,/2
21
)/222
21
(
0,/2
21
)/222
21
(
2
AA
AA
AA
(68)
where 1
A and 2
A are arbitrary constants.
By substituting (65) into (3) and using the second order linear ODE (66) and
Eq.(68 ), the left-hand side of (3) is transformed to a polynomial in i/1 and
)/()/1( i . Setting each coefficient of this polynomial to zero , yields a set of
algebraic equations which can be solved for the constants
,0
,)1,...,2,1,0,,...,2,1,0(, vNjNiji
and .Therefore, substituting the
resultant values of these constants and the general solutions (67) of (66) into (5),
we can obtain the exact traveling wave solutions of Eq.(4) in a concise manner.
Conservation laws and multiple simplest equation methods 417
Application of the extended simplest equation method Suppose the solution of (3) is of the form
,1
4
1
3
2
210)(
u (69)
where 0
, 1
, 2
, 3
, and 4
are constants to be determined later and the
function )( satisfies the linear ODE (66).
By substituting (69) into (3) and using (66)) and (68), the left-hand side of (3)
become a polynomial in i/1 and )/()/1( i . Putting every coefficient of this
polynomial as zero, yields a system of algebraic equations where by solving it via
Mathematica 9, we therefore have the following cases:
If 0 , we obtain
12
,12
,04
,03
,00
AAi
, , 0,
2 akav , (70)
δ
AδAi
2
129
2,
223
3,0
4,0
1,0
0
2
,
rk
rkrkbc
)22( ,
22
1kav , .
23
i (71)
Substituting (70) and (71) into (69) and making use of solutions (67) of (66), then
the exact traveling wave solutions which expressed by hyperbolic functions are
obtained and can be written as
(72),0
)1
(
,)(sinh)(cosh
)(cosh)(sinh1
,)(sinh)(cosh
)(cosh)(sinh
1)(
1
tkaiyrxk
u
)73(0
)2
1(
,
2/3)(sinh2
129
)(cosh1
23
2
2/3)(sinh2
129
)(cosh1
)(cosh2
129
)(sinh1
2)(
2
2
2
2
2
tkayrxk
iδ
AδA
i
iδ
AδA
δ
AδA
u
418 Shoukry El-Ganaini and M. Ali Akbar
where 2
,12
,1
AA , and 0 are an arbitrary constants.
If 0 , we obtain
12
,12
,04
,03
,00
AAi
, , 0,
2 kav , (74)
δ
AδAi
2
129
2,
223
3,0
4,0
1,0
0
2
,
rk
rkrkbc
)22( ,
22
1kav , .
23
i (75)
Substituting (74) and (75) into (69) and making use of solutions (67) of (66), the
exact traveling wave solutions expressed by trigonometric functions are obtained
as
(76),0
)1
(
,)(sin)(cos
)(cos)(sin1
,)(sin)(cos
)(cos)(sin
1)(
1
tkaiyrxk
i
iiu
where 2
,12
,1
AA , and 0 are an arbitrary constants.
If 0 , we obtain
1
3
22
,04
,01
,00
AiA
, 2
22/2
1AA (78)
Substituting (78) into (69) and making use of solutions (67) of (66), we obtain the
rational function solutions of Eq.(1) and can be written as
)77(0
)2
1(
,
2/3)(sin2
129
)(cos1
23
2
2/3)(sin2
129
)(cos1
)(cos2
129
)(sin1
2)(
2
2
2
2
2
tkayrxk
iδ
AδA
i
iδ
AδA
δ
AδA
u
Conservation laws and multiple simplest equation methods 419
,2
23/
242
242
1
)1
/22
(3
42
223
/2
422
421
22
21
24)(
AiAA
AA
AiAA
AAu
.0 tvyrxk (79)
where 2
,13
,2
AA , and 0 are an arbitrary constants.
5. Conservation laws for the extended quantum Zakharov-
Kuznetsov (qZK) equation
Preliminaries[30]
Consider an thr -order partial differential equation
0))(
,...,)1(
,,( r
uuuxF (80)
with independent variables ),...,1( nxxx and one dependent variable u , where
},...{)2(
},{)1( ij
uui
uu denote the set of partial derivatives of the first order,
second order ,etc. , and ./2,/j
xixuij
uixui
u
The adjoint equation to (80) is
0))(
,)(
,...,)1(
,)1(
,,,( r
vr
uvuvuxF (81)
where
uvFr
ur
uvuvuxF )())(
,)(
,...,)1(
,)1(
,,,(* (82)
and
1 ...1
...
1
)1(
rrii
uri
Di
Dr
uu
(83)
denote the Euler-Lagrange operator, and v is a new dependent variable. Here
...
juij
uui
uix
iD (84)
are the total differentiations with respect to niix ,...,1, .
Eq.(80) is called self-adjoint if the equation
0))(
,)(
,...,)1(
,)1(
,,,(* r
ur
uuuuuxF (85)
420 Shoukry El-Ganaini and M. Ali Akbar
obtained from the adjoint equation (82) by the substitution uv ,
is identical to the original equation (81). This concept can be written in an
equivalent form as
))(
,...,)1(
,,(,...))1(
,,())(
,)(
,...,)1(
,)1(
,,,(*r
uuuxFuuxr
ur
uuuuuxF (86)
Theorem of Ibragimov on conservation laws [31]:
Every Lie point, Lie –Backlund or non-local symmetry
uuux
ixuuxiX
,...)
)1(,,(,...)
)1(,,( (87)
of Eq.(80) provides a conservation law 0)( iCi
D , for the system of equations
(81) and (82).
The conserved vector is given by
(88)......)(
...)()(...)()(
ijku
LW
kD
jD
ijku
L
kD
iju
LW
jD
ijku
L
kD
jD
iju
L
jD
iu
LWLiiT
where W (the characteristic function) and L (the Lagrangian) are defined as
.,...,1),)(
,...,)1(
,,(, njr
uuuxFvLj
uj
W (89)
Let us recall the symmetry algebra of Eq.(1) as given in [29] which is spanned by
the five vector fields
(90))./)(/1())/(5
),/(2)/()/()/(34
,/3
,/1
,/1
uaxtX
uuyyxxttXyXxXtX
Now, the extended quantum Zakharov-Kuznetsov equation (1) together with its
adjoint equation are given by
0)()( xxy
uxyy
ucyyy
uxxx
ubx
uuat
uF (91)
0)()(* xxy
vxyy
vcyyy
vxxx
vbx
vuat
vF (92)
The third-order Lagrangian L for the system of equations (81) and (82) is given
by
))()((xxy
uxyy
ucyyy
uxxx
ubx
uuat
uvFvL (93)
Conservation laws and multiple simplest equation methods 421
Therefore, we must consider the following five cases:
(i) Considering the vector field tX /1
, for Eq.(1) .By using (88) and taking
uv ,due to self-adjointness of Eq.(1), the resultant conserved vector is given by
xxycuu
xyycuu
yyybuu
xxxuub
xuuatC 2
1,
yytcuu
xytcuu
xxtbuu
ytu
yuc
ytu
xuc
xtu
ycu
xtu
xbu
yyu
tcu
xyu
tcu
xxu
tbu
tuuaxC
2
221
,
yytubu
xytucu
xxtucu
ytu
yub
ytu
xuc
xtu
yuc
xtu
xuc
yyu
tbu
xyu
tcu
xxu
tcuyC
2
21
(94)
(ii) Using the Lie point symmetry generator xX /2
, for Eq.(1) and using
(88) , the components of the conserved vector are given by
xuutC
2
xyu
ycu
yu
xxuc
yyu
xcu
xyu
xcu
xxycuu
yyybuu
tuuxC
2
xyybuu
xxycuu
xxxcuu
xyu
ybu
yyu
xbu
xyu
xcu
xxu
ycuyC 2
2
(95)
(iii) Using yX /3
, for Eq.(1) and (88) , the component of the conserved
vector are obtained and can be written as
yuutC
3,
yyycuu
xyycuu
xxybuu
yyu
xcu
xyu
xbu
xyu
ycu
xxu
ybu
yuuaxC 22
3
yyu
xcu
xyu
xcu
xyu
ycu
xxu
ycu
xyycuu
xxxbuu
xuau
tuuyC 2
3
(96)
(iv) The Lie point symmetry generator
)/(2)/()/()/(34
uuyyxxttX
For Eq.(1), leads to a conserved vector with component given by
422 Shoukry El-Ganaini and M. Ali Akbar
yuyu
xuxuu
xxyctuu
xyyctuu
yyybtuu
xxxbtuu
xuuattC 22333323
4
.326
33333
262336
3612
63223234
yyycyuu
yytctuu
xyycyuu
xxyxcuu
xytctuu
xxyybuu
xxtbtuu
yyyxbuu
ytu
yctu
ytu
xctu
xtu
yctu
xtu
xbtu
yuayu
yu
xcu
tuatu
yyu
tctu
yyu
xcyu
yyu
xcxu
yycuu
xxu
ycxu
xxu
yybu
xxu
tbtu
xxbuu
xycuu
xyu
xbyu
xyu
ycxu
xyu
tctu
xyu
ycyu
xyu
xcxuau
ycu
xbu
tuxuxC
(97).3
263
33363
36612
36232324
xyyxbuu
yytbtuu
xxycxuu
xytctuu
xxxcxuu
xxtctuu
xyycyuu
xxxybuu
ytu
ybtu
ytu
xctu
xtu
yctu
yu
xcu
xtu
xctu
yyu
xcyu
yyu
xxbu
yyu
tbtu
yybuu
xyu
yxbu
xyu
xcyu
xyu
tctu
xyu
ycyu
xyu
xcxu
xycuu
xxu
ycxu
xxu
ycyu
xxu
tctu
xxcuu
ybu
xcu
xuuay
tuyuyC
(v) The last symmetry generator )/)(/1())/(5
uaxtX , give the
following conseved quantities
xutuatC )/1(
5,
,
)/()/2()/(25
xxyctuu
yyybtuu
xyu
yctu
xxu
yctu
yyu
xctu
xyu
xctu
yyuac
xyuac
xxuab
ttuuuxC
)98(.2
)/()/2()/(5
xyybtuu
xxyctuu
xxxctuu
xyu
ybtu
xxu
yctu
yyu
xbtu
xyu
xctu
yyuab
xyuac
xxuacyC
6. Conclusion
In this article, we have obtained wider classes of exact traveling wave solutions of
the extended qZK equation by using distinct types of the simplest equation
methods. In addition, we have constructed conservation laws for the extended
qZK equation via the new conservation theorem. The conservation laws can be
associated with Lie symmetry generators of NPDEs to yield another conservation
laws and more exact solutions of these nonlinear equations with the aid of double
reduction theory and Kara and Mahomed results. These useful ideas are in urgent
need to be conducted in a separate study in a future work.
Conservation laws and multiple simplest equation methods 423
Competing interests Authors have declared that no competing interests exist.
Authors’ contributions This research work was carried out in collaboration between the authors. Both
authors have a good contribution to design the study and to perform the analysis
of this research work. Both the authors read and approved the final manuscript.
References
[1] M.A. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and
Inverse Scattering, Cambridge University Press, Cambridge, 1991.
http://dx.doi.org/10.1017/cbo9780511623998
[2] V.B. Matveev, M.A. Salle, Darboux Transformation and Solitons, Springer,
Berlin, 1991.
[3] R.M. Miura, Backlund Transformation, Springer, Berlin, 1978.
[4] M. A. Abdou, Generalized solitonary and periodic solutions for nonlinear
partial differential equations by the Exp-function method, Nonlinear Dyn., 52
(2008), 1-9. http://dx.doi.org/10.1007/s11071-007-9250-1
[5] M. Mirzazadeh, M. Eslami, A. Biswas, Soliton solutions of the generalized
Klein-Gordon equation by using )/( GG -expansion method, Comput. Appl.
Math., 33 (2014), no. 3, 831-839. http://dx.doi.org/10.1007/s40314-013-0098-3
[6] Z.Y. Yan, Generalized method and its application in the higher-order
nonlinear Schrodinger equation in nonlinear optical fibres, Chaos Solitons Fract.,
16 (2003), 759-766. http://dx.doi.org/10.1016/s0960-0779(02)00435-6
[7] Shoukry Ibrahim Atia El-Ganaini, Travelling Wave Solutions to the
Generalized Pochhammer-Chree (PC) Equations Using First Integral Method,
Math. Prob. Eng., 2011 (2011), 1-13. http://dx.doi.org/10.1155/2011/629760
[8] Shoukry Ibrahim Atia El-Ganaini, New Exact Solutions of Some Nonlinear
Systems of Partial Differential Equations Using the First Integral Method, Abst.
Appl. Anal., 2013 (2013), 1-13. http://dx.doi.org/10.1155/2013/693076
[9] Shoukry Ibrahim Atia El-Ganaini, The First Integral Method to the Nonlinear
Schrodinger Equations in Higher Dimensions, Abstract and Applied Analysis,
2013 (2013), 1-10. http://dx.doi.org/10.1155/2013/349173
424 Shoukry El-Ganaini and M. Ali Akbar
[10] G.W. Bluman, A.F. Cheviakov, S.C. Anco, Applications of Symmetry
Methods to Partial Differential Equations, Applied Mathematical Sciences, Vol.
168, Springer, New York, NY, USA, 2010.
http://dx.doi.org/10.1007/978-0-387-68028-6
[11] B. Muatjetjeja and C.M. Khalique, Conservation Laws for a Variable
Coefficient Variant Boussinesq System, Abst. Appl. Anal., 2014 (2014), 1-5.
http://dx.doi.org/10.1155/2014/169694
[12] E. Noether, Invariant Variation problemes, Nacr. Konig. Cesell. Wissen.,
Gottingen, Math-Phys. Kl. Heft, 1918-2-235-57, English translation in Transport
Theory and Statistical Physics, 1 (1971), no. 3, 186-207.
http://dx.doi.org/10.1080/00411457108231446
[13] H. Steudel, Uber die Zuordnung zwischen invarianzeigenschaften und
Erhaltungssatzen, Zeitschrift fur Naturforschung, 17A (1962), 129-132.
http://dx.doi.org/10.1515/zna-1962-0204
[14] A.H. Kara, F.M. Mahomed, Noether-type-symmetries and conservation laws
via partial Lagrangians, Nonlinear Dyn., 45 (2006), no. 3-4, 367-383.
http://dx.doi.org/10.1007/s11071-005-9013-9
[15] A. Sjoberg, Double reduction of PDEs from the association of symmetries
with conservation laws with applications, Appl. Math. Comput., 184 (2007), 608-
616. http://dx.doi.org/10.1016/j.amc.2006.06.059
[16] B. Muatjetjeja and C.M. Khalique, Lie group classification of a generalized
coupled Lane-Emden system in dimension one, East Asian J. Appl. Math., 4
(2014), 301-311. http://dx.doi.org/10.4208/eajam.080214.230814a
[17] S. San and E. Yasar, On the Conservation Laws and Exact Solutions of a
Modified Hunter-Saxton Equation, Adv. Math. Phys., 2014 (2014), 1-6.
http://dx.doi.org/10.1155/2014/349059
[18] R. Naz, Z. Ali and I. Naeem, Reductions and New Exact Solutions of ZK,
Gardner KP, and Modified KP Equations via generalized Double Reduction
Theorem, Abst. Appl. Anal., 2013 (2013), 1-11.
http://dx.doi.org/10.1155/2013/340564
[19] R. Tracina, M.S. Bruzon, M.L. Grandarias and M. Torrisi, Nonlinear self-
adjointness, conservation laws, exact solutions of a system of dispersive evolution
equations, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 3036-3043.
http://dx.doi.org/10.1016/j.cnsns.2013.12.005
Conservation laws and multiple simplest equation methods 425
[20] A.M. Wazwaz, Solitary waves solutions for extended forms of quantum
Zakharov-Kuznetsov equations, Phys. Scr., 85 (2012), 025006.
http://dx.doi.org/10.1088/0031-8949/85/02/025006
[21] G.-W. Wang, T.-Z.Xu, S. Johnson and A. Biswas, Solitons and Lie group
analysis to an extended quantum Zakharov-Kuznetsov equation, Astrophys. and
Space Science, 349 (2014), 317-327.
http://dx.doi.org/10.1007/s10509-013-1659-z
[22] N.A. Kudryashov, Simplest equation method to look for exact solutions of
nonlinear differential equations, Chaos, Solitons & Fractals, 24 (2005), no. 5,
1217-1231. http://dx.doi.org/10.1016/j.chaos.2004.09.109
[23] N.A. Kudryashov, Exact solitary waves of the Fisher equation, Phys. Lett. A,
342 (2005), 99-106. http://dx.doi.org/10.1016/j.physleta.2005.05.025
[24] N.A. Kudryashov, One method for finding exact solutions of nonlinear
differential equations, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2248-
2253. http://dx.doi.org/10.1016/j.cnsns.2011.10.016
[25] H. Naher and F.A. Abdullah, New traveling wave solutions by the extended
generalized Riccati equation mapping method of the (2+1)-dimensional evolution
equation, J. Appl. Math., 2012 (2012), Article ID486458, 1-18.
http://dx.doi.org/10.1155/2012/486458
[26] E.M.E. Zayed and S.A.H. Ibrahim, Exact solutions of nonlinear evolution
equations in mathematical physics using the modified simple equation method,
Chinese Phys. Lett., 29 (2012), no. 6, Article ID060201.
http://dx.doi.org/10.1088/0256-307x/29/6/060201
[27] S. El-Ganaini, Solutions of some class of nonlinear PDEs in mathematical
physics, J. Egypt. Math. Soc., 24 (2016), 214-219.
http://dx.doi.org/10.1016/j.joems.2015.02.005
[28] S. Bilige and T. Chaolu, An extended simplest equation method and its
application to several forms of the fifth-order KdV equation, Appl. Math.
Comput., 216 (2010), 3146-3153. http://dx.doi.org/10.1016/j.amc.2010.04.029
[29] S. Bilige and T. Chaolu and X. Wang, Application of the extended simplest
equation method to the coupled Schrodinger-Boussinesq equation, Appl. Math.
Comput., 224 (2013), 517-523. http://dx.doi.org/10.1016/j.amc.2013.08.083
[30] N.H. Ibragimov, Quasi-self-adjoint differential equations, Archives of Alga, 4
(2007), 55-60.
426 Shoukry El-Ganaini and M. Ali Akbar
[31] N.H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333
(2007), 311-328. http://dx.doi.org/10.1016/j.jmaa.2006.10.078
Received: June 20, 2016; Published: July 25, 2016