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CAF 2936

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Computers & Fluids xxx (2015) xxx–xxx
1

Contents lists available at ScienceDirect

Computers & Fluids
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
6
7

5

Lie symmetry analysis, nonlinear self-adjointness and conservation laws
to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers
equation q

8

Gangwei Wang a,⇑, K. Fakhar b

3
4

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12
1
2 4
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15
16
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19

a
b

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China
Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, Canada

a r t i c l e

i n f o

Article history:
Received 6 July 2014
Received in revised form 28 May 2015
Accepted 26 June 2015
Available online xxxx

a b s t r a c t
This paper addresses an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers (ZKB) equation. The
Lie symmetry analysis leads to many plethora of solutions to the equation. The nonlinear self-adjointness
condition for the ZKB equation is established and subsequently used to construct simplified but infinitely
many nontrivial and independent conserved vectors. In particular, we also get conservation laws of the
equation with the corresponding Lie symmetry.
Ó 2015 Elsevier Ltd. All rights reserved.

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Keywords:
(2+1)-dimensional Zakharov–Kuznetsov–
Burgers equation
Lie symmetry analysis
Nonlinear self-adjointness
Conservation laws

37

1. Introduction

38

The nonlinear evolution equations (NLEEs) are encountered in a
variety of scientific fields, such as physics, chemistry, engineering
and others. A vast amount of research work has been investigated
in the study of exact solutions of the NLEEs, in particular, the solitary wave solutions largely due to their frequent occurrence in nature. Besides their physical relevances, they serve as a bench mark
for accuracy of numerical schemes as well as prove to be quiet
handy in testing of computer algorithms. Due to the increased
interest in the NLEEs, a broad range of analytical and numerical
methods have been developed to construct exact solutions to
NLEEs. Some of these efficient methods are the Lie symmetry
method [1–3], Darboux transformation method [4], Jacobi elliptic
method [5], Painleve analysis [6], the inverse scattering method
[7], the Baklund transformation method [8], the conservation law
method [9], the Hirota bilinear method [10], the ansatz method
[11] and many other methods.
Zakharov and Kuznetsov [12] established an equation which is
related to nonlinear ion-acoustic waves (IAWs) in magnetized

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45
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49
50
51
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q

The project is supported by the Research Project of China Scholarship Council
(No. 201406030057), National Natural Science Foundation of China (NNSFC) (Grant
No. 11171022), Graduate Student Science and Technology Innovation Activities of
Beijing Institute of Technology (No. 2014cx10037).
⇑ Corresponding author.
E-mail address: pukai1121@163.com (G. Wang).

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31
32
33
34

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plasma including cold ions and hot isothermal electrons. Few
related studies concerning Zakharov and Kuznetsov and its generalized form are discussed in [13–19]. The quantum hydrodynamic
(QHD) model is a generalization of the classical fluid model of plasmas where QHD transport equations are expressed with reference
to conservation laws for particle momentum and energy. Several
authors for instance, Stenflo et al. [20], Khan et al. [21] etc., have
used QHD model to study the linear and nonlinear waves in quantum plasma. Further, the existence of a small number of ions along
with the electrons and positron in many astrophysical environments, attracted much attention and consequently lot of work
available in the literature (e.g.,[22,23]). It is well know that the
small amplitude wave propagation in a medium which posses both
the characteristics dispersive and dissipative terms can be best
modeled by Korteweg–deVries–Burgers (KdVB) equation. Mamun
and Shukla [24] and Shukla and Mamun [25] in their study
observed that the dissipative Burger term in KdVB was due to
the presence of kinematic viscosity in the plasma. Also, they were
able to generate dispersive shock wave in the plasma. El-Bedwehy
and Moslem derived the Zakharov Kuznetsov Burgers (ZKB) equation [26] in an electron–positron–ion (e–p–i) plasma and applied
their numerical results to the electrostatic fluctuations in the interstellar medium. Masood et al. [27] studies the obliquely propagating nonlinear quantum ion acoustic shock wave in a viscous
quantum (e–p–i) magnetoplasma. They have used QHD model
with the small amplitude expansion method to independently

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http://dx.doi.org/10.1016/j.compfluid.2015.06.033
0045-7930/Ó 2015 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Wang G, Fakhar K. Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers equation. Comput Fluids (2015), http://dx.doi.org/10.1016/j.compfluid.2015.06.033

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G. Wang, K. Fakhar / Computers & Fluids xxx (2015) xxx–xxx

derive the ZKB equation. They have used tanh method to obtain the
results.
In this paper, we consider the (2+1)-dimensional ZKB equation
and investigate it from the point view of Lie symmetries and conservation laws. By omitting the details of derivation, we directly
write ZKB equation in the form

ut þ auux þ buxxx þ cuxyy  duxx  euyy ¼ 0;

103

104

2. Similarity reductions and exact solutions

92
93
94
95
96
97
98
99
100
101
102

h ¼ t;

105

108
109
110

@
;
@x

V2 ¼

@
;
@y

V3 ¼

@
;
@t

V4 ¼ t

@ 1 @
þ
:
@x a @u

154

ð11Þ

ð12Þ

It is should been noted that all above reduced equation are
(1+1)-dimensional PDEs. It is also difficult to get solutions of them.
In order to get their solutions, once again, by using Lie symmetry
method. For the sake of simplicity, in what follows, we only consider (12) in details.
The symmetry algebra of (12) is generated by the vector fields

@
!1 ¼ ;
@h

@
!2 ¼ ;
@s

@ 1 @
!3 ¼ h þ
:
@s a @f

ð13Þ

The combination !1 þ k!2 of the two symmetries !1 and !2 yields
the following invariants

f ¼ WðhÞ;

h ¼ s  kh;

ð14Þ

and using (14) and (12) is transformed to the nonlinear ODE
0

000

00

kW þ aWW þ ðb þ cÞW  ðd þ eÞW ¼ 0:

The corresponding vector fields of (1) are

V1 ¼

u ¼ f ðh; sÞ:

f h þ affs þ ðb þ cÞf sss  ðd þ eÞf ss ¼ 0:

0

106

s ¼ x  y;

153

Treating f as the new dependent variable and h and s as new
independent variables, the ZKB Eq. (1) transforms to

ð1Þ

where a, b, c, d and e, with d and e being positive, are constant quantities which involves the physical quantities like: mass; density;
magnetic field; kinematic viscosity; plasma frequency; superthermality; ion gyrofrequency, etc., while x; y and t are the independent
variables that represent the spatial and temporal variables respectively, where as uðx; y; tÞ is the dependent variable that represents
the wave profile. For detail discussions reader is refer to [26,27].
The paper is organized in the following manner. In Section 2,
similarity reductions and explicit solutions are derived. In
Section 3, we will show that this equation is nonlinearly
self-adjoint. On the basis of the point symmetries, conservation
laws are constructed. Finally, the main findings of the paper are
recapitulated in Section 4.

91

(V) V 1 þ V 2
The symmetry V 1 þ V 2 yields the following invariants

ð2Þ

We make some discussions on the ZKB equation based on the
vector fields.

ð15Þ

ð16Þ

n¼0

Substituting (16) into (15), one can get
111
112

113
115
116
117

118
120
121
122
123
124

125
127
128
129

130
132
133
134

135
137
138
139

140
142
143
144

145
147
148
149

150
152

(I) V 1
For the generator V 1 , we have

u ¼ f ðg; hÞ;

where g ¼ y; h ¼ t are the group-invariants. Substituting (3)
into (1), one can get

ð4Þ

It is important to note that (4) is the celebrated Heat
equation.
(II) V 2
For the generator V 2 , we get

u ¼ f ðr; hÞ;

ð5Þ

where r ¼ x; h ¼ t are the group-invariants. Substituting (5)
into (1), we reduce it to the following PDE

f h þ affr þ bfrrr  dfrr ¼ 0:

ð6Þ

(III) V 3
For the generator V 3 , we get

u ¼ f ðr; gÞ;

ð8Þ

(IV) V 4
In this case, one can obtain

x
;
at

163
164
165
166
167
168
169

170
172
173
174

175
177
179
181
182
183

184
186

n¼1 j¼0

1
X
þ 2ðd þ eÞc2 þ ðd þ eÞ ðn þ 1Þðn þ 2Þcnþ2 hn þ 6ðb þ cÞc3
n¼1
1
X
þ ðb þ cÞ ðn þ 3Þðn þ 2Þðn þ 1Þcnþ3 hn ¼ 0:

ð17Þ

n¼1

Next, from (17), for the case of n ¼ 0, one gets

kc1  ac0 c1  2ðd þ eÞc2
c3 ¼
:
6ðb þ cÞ

ð9Þ

ð10Þ

190
191
192

ð18Þ

Generally, for n P 1, one obtains

194
195

196

1
kðn þ 1Þcnþ1  ðd þ eÞðn þ 1Þðn þ 2Þ
cnþ3 ¼
ðb þ cÞðn þ 1Þðn þ 2Þðn þ 3Þ
!
n
X
ð19Þ
cnþ2  a ðn þ 1  jÞcj cnþ1j :
198
199

200

1
X
WðhÞ ¼ c0 þ c1 h þ c2 h2 þ c3 h3 þ cnþ3 hnþ3
n¼1

¼ c 0 þ c 1 h þ c 2 h2 þ

þ

where h ¼ t; g ¼ y are the group-invariants. Substituting (9)
into (1), one can obtain

hfh þ f  ehfgg ¼ 0:

n¼1

Thus, the power series solution of (16) is as follows

where r ¼ x; g ¼ y are the group-invariants. Putting (7) into
(1), one can obtain

u ¼ f ðh; gÞ þ

160
162

187
188

j¼0

ð7Þ

affr þ bfrrr þ cfrgg  dfrr  efgg ¼ 0:

159

1
1 X
n
X
X
 kc1  k ðn þ 1Þcnþ1 hn þ ac0 c1 þ a
ðn þ 1  jÞcj cnþ1j hn

ð3Þ

f h  efgg ¼ 0:

158

178

Now, we search a solution of (15) in a power series of the form
[28,29]
1
X
W¼
c n hn :

155
157

kc1  ac0 c1  2ðd þ eÞc2 3
h
6ðb þ cÞ

1
X

1
kðn þ 1Þcnþ1
ðb
þ
cÞðn
þ
1Þðn
þ 2Þðn þ 3Þ
n¼1

!
n
X
ðd þ eÞðn þ 1Þðn þ 2Þcnþ2  a ðn þ 1  jÞcj cnþ1j hnþ3 : ð20Þ
j¼0

Consequently, the exact power series solution of (1) can be
written as follows

Please cite this article in press as: Wang G, Fakhar K. Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers equation. Comput Fluids (2015), http://dx.doi.org/10.1016/j.compfluid.2015.06.033

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CAF 2936

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205

uðx;y;tÞ ¼ c0 þ c1 ðx  y  ktÞ þ c2 ðx  y  ktÞ2 þ c3 ðx  y  ktÞ3
1
X
þ
cnþ3 ðx  y  ktÞnþ3 ¼ c0 þ c1 ðx  y  ktÞ

In particular, if we let C 1 ¼ 1; C 2 ¼ 1; p ¼ 0, and 4rq  p2 > 0,
one can get

n¼1

kc1  ac0 c1  2ðd þ eÞc2
ðx  y  ktÞ3
þ c2 ðx  y  ktÞ þ
6ðb þ cÞ
1
X
1
þ
ðkðn þ 1Þcnþ1
ðb þ cÞðn þ 1Þðn þ 2Þðn þ 3Þ
n¼1
!
n
X
ðd þ eÞðn þ 1Þðn þ 2Þcnþ2  a ðn þ 1  jÞcj cnþ1j ðx  y  ktÞnþ3 ;
2

j¼0

207

ð21Þ

208

where ci ði ¼ 0; 1; 2; 3Þ are arbitrary constants, the other coefficients
cn ðn P 3Þ also can derived.

209
210
211
212
213
214

215
217
218
219
220

221

223
224
225
226

227
229
230
231
232
233

234

Remark 1. The explicit solutions of other equations can also be
derived in the same way. Here we do not list them for simplicity.
In order to search for others explicit solutions of Eq. (1), we
make use of the auxiliary equation, i.e., the Riccati equation of
the following ’’general form’’

u0 ¼ r þ pu þ qu2 ;

ð22Þ

and use its solution to construct the solutions for ZKB equation.
Here, p; q; r are real constant. By omitting the details, we directly
write the general solution of (22) as

pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
h
2
2
p
4rq  p2 C 1 e2 4rqp  C 2 e2 4rqp
p
ffiffiffiffiffiffiffiffiffiffiffi
p
ffiffiffiffiffiffiffiffiffiffiffi
 ;
u¼
h
h
2
2
2q
2q
C e2 4rqp þ C e2 4rqp
1

237

238

ð23Þ

2

248
249
250

252
253

263

3. Conservation laws of (1)

264

In this section, we obtain conservation laws for the ZKB
equation.

265

Theorem 1 ([30,31]). The system and its adjoint equation

267

266

268

F  ðx; u; v ; uð1Þ ; v ð1Þ ; uð2Þ ; v ð2Þ ; . . . ; uðsÞ ; v ðsÞ Þ ¼ 0;

ð29Þ

270
271

L ¼ v Fðx; u; uð1Þ ; uð2Þ ; . . . ; uðsÞ Þ:

ð30Þ

272
274

Definition 1. [30,31] The first equation of (29) is said to be
nonlinearly self-adjoint if for some arbitrary function /ðx; uÞ – 0,
we have

277

q2 ðb þ cÞ
12 ð5 bp þ 5 cp  d  eÞq
; a1 ¼ 
;
a
5
a
5aa0  6pðd þ eÞ þ 60qrðb þ cÞ
k¼
;
5

where k is an indeterminate variable coefficient.

2

2

276

2

2

pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi
!2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
h
2
2
4rq  p2 C 1 e2 4rqp  C 2 e2 4rqp
q 2 ðb þ c Þ
p
p
ffiffiffiffiffiffiffiffiffiffiffi
p
ffiffiffiffiffiffiffiffiffiffiffi
uðx;y;tÞ ¼ 12

h
h
2
2
2q
a
C e2 4rqp þ C e2 4rqp 2q

1

F jv ¼/ ¼ kðx; u; uð1Þ ; . . .ÞF;

ð31Þ

279

280
282
283
284

ð26Þ

2

12 ð5bp þ 5cp  d  eÞq

5
a pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi
!
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
h
2
2
4rq  p2 C 1 e2 4rqp  C 2 e2 4rqp
p
pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi 

þ a0 ;
h
h
2
2
2q
C e2 4rqp þ C e2 4rqp 2q



278

ð25Þ

where a0 is a arbitrary constant. Also,

Theorem 2 [32]. Every Lie point, Lie-Bäcklund and non-local symmetry of Eq. (1) provides a conservation law for Eq. (1) and the adjoint

285

equation. Then the elements of conservation vector ðC 1 ; C 2 ; C 3 Þ are
given by

287

"

@L
@L
C ¼nLþW
 Dj
@uai
@uaij
"
!
@L
a
 Dk
þ Dj ðW Þ
@uaij
i

!

a

i

a

where W ¼ g

þ Dj Dk
@L
@uaijk

!!

@L
@uaijk

!#

286

288

289

#

þ ... ;

ð32Þ
291

a  n j ua .
j

292

ð27Þ

2

here h ¼ x  y  kt, and k is given by (25).
The solution (27) is the general solution of ZKB equation and
therefore several independently real solutions can be obtained.
For instance, the exact solution (21) which was obtained by ’’tanh
method’’ in Ref. [27], can be retrieved from (27) by choosing
2

251

261

!2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4rq  p2
q ðb þ cÞ
h
p
4rq  p2 
uðx;y;tÞ ¼ 12
tan h
a
2
2q
2q
 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


12 ð5bp þ 5cp  d  eÞq
h
p

tan h
4rq  p2 
þ a0 :
5
a
2
2q
2

ð24Þ

1

246

260

where a0 ; a1 ; a2 are constants to be determined.
Substituting the ansatz (24) along with (22) into (15), collecting
coefficients of monomials of u with the aid of Maple, and then setting each coefficients equal to zero, one gets

f ðhÞ ¼ a0 þ a1 u þ a2 u2 ;

From the asatz (24) and making use of Eqs. (25) and (23), one
can get the explicit solution of (1)

247

259

275

241

245

In particular, if we set C 1 ¼ 1; C 2 ¼ 1, and 4rq  p2 > 0, one can
derive

258

In the following we recall the ’’new conservation theorem’’
given by Ibragimov in [31].

ðd þ eÞ ¼ 25p ðb þ cÞ þ 100qrðb þ cÞ :

243

256

q2 ð b þ c Þ
uðx;y;tÞ ¼ 12
a

has a formal Lagrangian, namely

where C 1 ; C 2 are arbitrary constants. By balancing the highest
derivative and nonlinear terms in (15), we assume the solution of
(1) of the form

240

242

255

Fðx; u; uð1Þ ; uð2Þ ; . . . ; uðsÞ Þ ¼ 0;

a2 ¼ 12
236

!2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4rq  p2
h
p
4rq  p2 
cot h
2
2q
2q
 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


12 ð5bp þ 5cp  d  eÞq
h
p

cot h
4rq  p2 
þ a0 : ð28Þ
5
a
2
2q

254

2

9ðdþeÞ
ðdþeÞ
C 1 ¼ C 2 ¼ 1; p ¼ 0; a0 ¼ 25aðbþcÞ
, and qr ¼ 100ðbþcÞ
2 . Similarly, one can

compare the accuracy of the numerical results obtained in [26]
via (27).

3.1. Nonlinear self-adjointness

293

For (1), the adjoint equation has the form


F ¼ ðv t þ auv x þ bv xxx þ cv xyy þ dv xx þ ev yy Þ ¼ 0;

294

ð33Þ

and the Lagrangian in the symmetrized form




L ¼ v ut þ auux þ buxxx þ cuxyy  duxx  euyy :

295
297
298

ð34Þ

If we substitute u instead of v in Eq. (33), one can find that Eq. (1) is
not recovered. Therefore, we can say Eq. (1) is not self adjoint. Next,

Please cite this article in press as: Wang G, Fakhar K. Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers equation. Comput Fluids (2015), http://dx.doi.org/10.1016/j.compfluid.2015.06.033

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we look for an explicit form of /ðx; uÞ – 0 for Eq. (1) that holds Eq.
(33).




1 
L ¼ v ut þ auux þ buxxx þ c uxyy þ uyxy þ uyyx  duxx  euyy :
3
ð43Þ

349

Consider Theorem 2, the corresponding vector fields is given by

350

v t þ auv x þ bv xxx þ cv xyy þ dv xx þ ev yy jv ¼/ðx;uÞ
308
309

310





¼ k ut þ auux þ buxxx þ cuxyy  duxx  euyy :

ð35Þ

@
@
@
V ¼ n1 ðx; y; t; uÞ þ n2 ðx; y; t; uÞ þ n3 ðx; y; t; uÞ
@t
@x
@y
@
þ /ðx; y; t; uÞ :
@u

If we set v ¼ /ðt; x; y; uÞ, one can get

v t ¼ /t þ /u ut ;
v x ¼ /x þ /u ux ;
v xx ¼ /xx þ 2/xu ux þ /uu u2x þ /u uxx ;
v yy ¼ /yy þ 2/yu uy þ /uu u2y þ /u uyy ;

The conservation law is decided by
1

314

315

318

319
321

322
323

324

ð36Þ

Solve them, one can get the solution

c3

331

335
336

/¼

 pffiffiffiffi 

2
e k1 y c 1 þ c 2
pffiffiffiffi
:
e k1 y ek1 et

ð39Þ

ð40Þ

In particular, (39) has a special solution

/ ¼ c 1 y þ c2 ;

ð41Þ

where c1 ; c2 and c3 are constants. We, thus, have the following
statement.

339

3.2. Conservation laws of (1)

340

We now construct the conservation laws by using the adjoint
equation and symmetries of (1). For (1), the adjoint equation is
given by

342

343
345
346

347

þ W yy

@L
;
@uxyy

F ¼ v t þ auv x þ bv xxx þ cv xyy þ dv xx þ ev yy ;
and the Lagrangian in the symmetrized form

ð47Þ

365

@L
@L
@L
C ¼ n L þ W Dy
þ Dxy
þ Dyx
@uyy
@uyxy
@uyyx
þ W x Dy
þ W yx

@L
@L
@L
@L
þ Wy
þ W xy
 Dx
@uyxy
@uyy
@uyyx
@uyxy

@L
;
@uyyx

ð42Þ

ð48Þ

that is

368
369

370
1



1
C ¼ n L þ W auv  Dx ðdv Þ þ Dxx ðbv Þ þ Dyy
cv
3



1
þ W x ðdv  Dx ðbv ÞÞ þ W y Dy
þ W xx bv
cv
3
1
þ W yy cv
3


1
2
¼ n L þ W auv þ dv x þ bv xx þ cv yy  W x ðdv þ bv x Þ
3


1
1
cv y þ bv W xx þ cv W yy ;
 Wy
3
3
2

Theorem 3. The ZKB Eq. (1) is nonlinearly self-adjoint with the
substitution v ¼ / and / given by (40) or (41).

341

@L
@L
@L
@L
þ W y Dy
þ W xx
 Dx
@uxx
@uxxx
@uxyy
@uxxx

C ¼ n L þ Wv;

338

337

þ Wx

3

1

3b/xu þ 2d/u ¼ 0; c/xu þ 2e/u ¼ 0;

327

362

2

ð38Þ

Note that the other coefficients of u and all of the derivative
yields

326

360

@L
@L
@L
@L
C ¼n LþW
 Dx
þ Dxx
þ Dyy
@ux
@uxx
@uxxx
@uxyy
2

From the coefficient of the term ut , one can get

k ¼ /u :

ð46Þ

359

366

þ2/yuu ux uy þ2/yu uxy þ/uux u2y þ/uuu ux u2y þ2/uu uy uyx
 

þ/ux uyy þ/uu ux uyy þ/u uxyy ¼ k ut þauux þbuxxx þcuxyy duxx euyy : ð37Þ

3b/xuu þ d/uu ¼ 0; c/xuu þ e/uu ¼ 0;
/t þ au/x þ b/xxx þ d/xx þ e/yy þ c/xyy ¼ 0:

332
334

1

3

3b/xxu þ 2d/xu þ c/yyu ¼ 0; 2ec/yu þ 2c/xyu ¼ 0;

330

358

Plugging them into (35), one can arrive at the following
self-adjointness condition

b/uuu ¼ 0; c/uuu ¼ 0; 3b/uu ¼ 0; 2c/yuu ¼ 0; /yu ¼ 0;

328

Here the conserved vector C ¼ ðC ; C ; C Þ are given by (32) and
the components given by

3

363

/t þ/u ut þauð/x þ/u ux Þþdð/xx þ2/xu ux þ/uu u2x þ/u uxx Þ

 
þe /yy þ2/yu uy þ/uu u2y þ/u uyy þb /xxx þ3/xxu ux þ3/xuu u2x
 
þ3/xu uxx þ/uuu u3x þ3/uu ux uxx þ/u uxxx þc /xyy þ/yyu ux þ2/xyu uy
317

357

2

@L
C ¼n LþW
;
@ut

þ /uux u2y þ /uuu ux u2y þ 2/uu uy uyx þ /ux uyy

313

355

ð45Þ

1

v xyy ¼ /xyy þ /yyu ux þ 2/xyu uy þ 2/yuu ux uy þ 2/yu uxy

353
354

3

1

þ 3/uu ux uxx þ /u uxxx ;

þ /uu ux uyy þ /u uxyy :

2

Dt ðC Þ þ Dx ðC Þ þ Dy ðC Þ ¼ 0:

v xxx ¼ /xxx þ 3/xxu ux þ 3/xuu u2x þ 3/xu uxx þ /uuu u3x

312

ð44Þ

351

ð49Þ

372
373

2





2
1
þ W x Dy
C 3 ¼ n3 L þ W Dy ðev Þ þ Dxy
cv
cv
3
3


1
2
þ W y ev  Dx
þ W xy cv
cv
3
3




2
1
1
3
¼ n L þ W ev y þ cv xy  cv y W x  W y ev þ cv x
3
3
3
2
þ cv W xy ;
3

ð50Þ

376

ð51Þ

with

W ¼ /  n1 ut  n2 ux  n3 uy :

375

378
379

380

ð52Þ

Next, we consider following cases.
Case 1.
For the operator V ¼ @t@ , we have

Please cite this article in press as: Wang G, Fakhar K. Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers equation. Comput Fluids (2015), http://dx.doi.org/10.1016/j.compfluid.2015.06.033

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385

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388

W ¼ ut ;

389

we can get the conservation vector of (1)

392

C 1 ¼ ut v ;

393



390

395
396

398

ð53Þ

ð54Þ


1
C 2 ¼ ut auv þ dv x þ bv xx þ cv yy þ utx ðdv þ bv x Þ
3
1
1
þ cuty v y  bv utxx  cv utyy ;
3
3




2
1
1
C 3 ¼ ut ev y þ cv xy þ cv y utx þ uty ev þ cv x
3
3
3
2
 cv utxy :
3

400

401
403

W ¼ ux ;

404

one can obtain the conservation vector of (1)

ð56Þ

ð57Þ

405
1

407

C ¼ ux v ;

408



1
C 2 ¼ ux auv þ dv x þ bv xx þ cv yy þ uxx ðdv þ bv x Þ
3
1
1
þ cuxy v y  bv uxxx  cv uxyy ;
3
3




2
1
1
C 3 ¼ ux ev y þ cv xy þ cv y uxx þ uxy ev þ cv x
3
3
3
2
 cv uxxy :
3

410
411

413
414
415

W ¼ uy ;

419

we can reach the conservation vector of (1)

422

C 1 ¼ uy v ;

423



1
C 2 ¼ uy auv þ dv x þ bv xx þ cv yy þ uyx ðdv þ bv x Þ
3
1
1
þ cuyy v y  bv uyxx  cv uyyy ;
3
3




2
1
1
C 3 ¼ uy ev y þ cv xy þ cv y uyx þ uyy ev þ cv x
3
3
3
2
 cv uyxy :
3

420

425
426

428
429
430

431

ð58Þ

ð59Þ

ð60Þ

Case 3.
@
For the Lie algebra V ¼ @y
, one can arrive at

416
418

ð61Þ

ð62Þ

ð63Þ

ð64Þ

Case 4.
@
@
For the operator V ¼ t @x
þ 1a @u
, we have

1
 tux ;
a

433

W¼

434

we derive the conservation vector of (1)

ð65Þ

435
437
438

440

1
v  tux v ;
ð66Þ
a



1
1
auv þ dv x þ bv xx þ cv yy þ tuxx ðdv þ bv x Þ
C2 ¼
 tux
a
3
1
1
ð67Þ
þ ctuxy v y  btv uxxx  ctv uxyy ;
3
3
C1 ¼

441

ð68Þ

443

It is clear that they are involves an arbitrary solution v of the
adjoint Eq. (33), and they presents an infinite number of the conservation laws.

444

4. Conclusions

447

In the present paper, by using the Lie symmetry groups, we
studied the symmetry properties, similarity reduction forms and
explicit solutions of the (2+1)-dimensional ZKB equation.
Moreover, we also constructed the power series solution of the
equation. At last, we derived the nonlinear self-adjointness of Eq.
(1), by virtue of this fact, infinitely many conservation laws of
the equation are exhibited. Furthermore, the results obtained here
can be useful in enhancing the understanding of nonlinear propagation of small amplitude electrostatic structures in dense, dissipative (e–p–i) magnetoplasmas.

448

Acknowledgement

458

We are extremely thankful for the reviewers for their many
valuable suggestions that greatly improved this paper.

459

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461

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445
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ð55Þ

Case 2.
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For the generator V ¼ @x
, we get

399



1
2
1
ev y þ cv xy þ ctv y uxx
C3 ¼
 tux
a
3
3


1
2
þ tuxy ev þ cv x  ctv uxyx :
3
3

5



Please cite this article in press as: Wang G, Fakhar K. Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers equation. Comput Fluids (2015), http://dx.doi.org/10.1016/j.compfluid.2015.06.033

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Please cite this article in press as: Wang G, Fakhar K. Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov–Kuznetsov–Burgers equation. Comput Fluids (2015), http://dx.doi.org/10.1016/j.compfluid.2015.06.033