Nonlinear Dyn
DOI 10.1007/s11071-015-2480-8

pro
of

ORIGINAL PAPER

Nonlocal symmetry analysis and conservation laws
to an third-order Burgers equation

Received: 28 March 2015 / Accepted: 30 October 2015
© Springer Science+Business Media Dordrecht 2015

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Abstract The nonlocal residual symmetry related to
truncated Painlevé expansion of third-order Burgers
equation is performed. Then, the potential symmetries
of the equation are derived; on the basis of nonlocal symmetries, we linearize the nonlinear third-order
Burgers equation to a linear third-order PDE. Furthermore, the exact solutions of the potential equation are
presented in terms of the symmetries. In particular, conservation laws are constructed of the equations.
Keywords Third-order Burger equation · Nonlocal
symmetry analysis · Linearization · The multiplier
approach · Conservation laws
Mathematics Subject Classification
70S10

1 Introduction

15

It is known that the Burgers hierarchy is of the form
[1–7]

u t = K m (u) = (D + u + u x D

−1 m−1

)

A. H. Kara
School of Mathematics, University of the Witwatersrand,
Private Bag 3, Wits, Johannesburg 2050, South Africa
A. H. Kara · K. Fakhar
Department of Mathematics, University of British
Columbia, Vancouver V6T 1Z2, Canada

K. Fakhar
Department of Mathematics, Faculty of Science and Horticulture, Kwantlen Polytechnic University, Surrey, BC V3W 2MB,
Canada

21

ut = u x ,

u t = 2uu x + u x x ,

u t = 3uu x x + u x x x + 3u 2x + 3u 2 u x ,

TYPESET

DISK

LE

22

(3)

23

(4)

24
25

(5)

They are of first order, second order, third order
and fourth order, respectively. The third-order Burgers equation, similar to the second-order ones, appear
in many physical and engineering fields [8–10], such
as the plasma physics and fluid mechanics. In addition, these equations play a key role in nonlinear theory and mathematical physics, in particular in the integrable system, soliton theory, nonlinear wave theory,
and so on. It is to be noted that Eq. (3) is the dissipative
Burgers equation; by using the Hopf–Cole transformation, this equation can be reduced the heat equation
u t − u x x = 0. Moreover, the third equation in Eq. (4)
is the well-known Sharma–Tasso–Olver (STO) equation. The Burgers equation and the STO equation were
investigated in many papers such as [1–19]. In paper

CP Disp.:2015/11/6 Pages: 12 Layout: Medium

20

(2)

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Journal: 11071 MS: 2480

18

Substituting n = 1, 2, 3, 4 into (1), one can get few
elements of the hierarchy (1), which are

+ 12uu 2x + 4u 3 u x + 6u 2 u x x .

G. Wang (B)
School of Mathematics and Statistics, Beijing Institute of
Technology, Beijing 100081, People’s Republic of China
e-mail: pukai1121@163.com; wanggangwei@bit.edu.cn

17

19

u t = u x x x x + 10u x u x x + 4uu x x x

22E70 ·

u x , m = 1, 2, . . .

16

(1)

orre

1

unc

1

cted

Author Proof

Gangwei Wang · A. H. Kara · K. Fakhar

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G. Wang et al.

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Author Proof

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u 1t − 3u 1 u 1x x − u 1x x x − 3u 21x − 3u 21 u 1x

83


+ φ (−1) − 6u 0 u 1 u 1x − 3u 21 u 0x − 3u 0 u 1x x

− 3u 1 u 0x x − 6u 0x u 1x + u 0t − u 0x x x

+ φ (−2) 3φx u 0 u 21 + 6φx u 0 u 1x + 6φx u 1 u 0x
TYPESET

DISK

u 0 = φx ,

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(7)

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(8)

or

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u 0 = 2φx .

(9)

94

The compatibility conditions at j = 1, 3 and this equation possesses the Painlevé property [7,11]. Moreover,
from the coefficients of φ 0 and φ 1 , one can obtain

96

Lemma 1 u 0 = φx is a symmetry of (4) with a solution
of u 1 .
In order to proof this Lemma, we first give the definition of symmetry [12,13] and the Lemma 2.
Definition 1 [12,13] Given an evolution equation
u t = K [u],

(10)

here K [u] = K (x, t, u x , u x x . . .), if σ (x, t, u x , u x x . . .)
is a symmetry of Eq. (14), then σ (x, t, u x , u x x . . .) satisfy
(11)

where K ′ means the Fréchet derivative.

σt − 6σ uu x − 3u 2 σx − 3σ u x x − 3uσx x
− 6σx u x − σx x x = 0,

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105
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107

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Lemma 2 σ = σ (x, t, u) is a symmetry of Eq. (4) if
and only if

109
110

111

(12)

112

where u is a solution of (4). This lemma is easily proved
by Definition 1. From the coefficients of φ 0 and φ 1 , we
can get

114

u 1t − 3u 1 u 1x x − u 1x x x − 3u 21x − 3u 21 u 1x = 0,

−6u 0 u 1 u 1x − 3u 21 u 0x
− 3u 0 u 1x x − 3u 1 u 0x x − 6u 0x u 1x

LE

86

Vanishing the coefficients of φ −4 , one can get

123
Journal: 11071 MS: 2480

85

+ 6φx u 20 u 1 − 6φx2 u 0x − 6φx φx x u 0

+ 12φx u 0 u 0x + 3φx x u 20 − 3u 20 u 0x


+ φ (−4) 6φx3 u 0 − 9φx2 u 20 + 3φx u 30 .

dσ
= K ′ σ,
dt

The STO equation (4) is Painlevé integrable [7,11].
In order to get residual symmetry via Painlevé analysis, we use the following truncated Painlevé expansion
form
u0
+ u1.
(6)
u=
φ
Plugging (7) into (4), one can have

82

− u o φt + 3φx u 0x x + 3φx x u 0x

+ φx x x u 0 − 3u 0 u 0x x − 3u 20x

+ φ (−3) − 6φx2 u 0 u 1

2 Residual symmetry via Painlevé analysis

79

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pro
of

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cted

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+ 3φx x u 1 u 0 − 3u 20 u 1x − 6u 0 u 1 u 0x

orre

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[1], the authors considered the famous Burgres’ equation and gave infinite series of flows. The authors in
[4,5] presented exact solutions, which included selfsimilar solutions for Burgers hierarchy. Also, multiple kink solutions and multiple singular kink solutions
of Burgers hierarchy were constructed in [6]. In Ref.
[7], the authors considered the Painlev test, generalized
symmetries, Bäklund transformations and exact solutions of the STO equation. In paper [18], the authors
discussed group classification and exact solutions of
generalized Burgers equations with linear damping and
variable coefficients in detail. The authors gave the
results about self-adjointness and conservation laws of
a generalized Burgers equation in [19]. In paper [20],
the authors considered the nonclassical potential symmetries of the Burgers equation. The authors analyzed
the connection between symmetries and linearization
in [21,22].
In this paper, we consider the nonlocal symmetries,
exact solutions and derive the conservation laws of
third-order Burgers equation (4) and its potential form.
The remainder of this paper is organized as follows:
In Sect. 2, nonlocal residual symmetry related to truncated Painlevé expansion of this equation is obtained.
In Sect. 3, we employ potential symmetry on this equation and present all of the geometric vector fields are
constructed. In Sect. 4, linearization by nonlocal symmetries is derived. In Sect. 5, similarity reductions and
explicit solutions are presented. In Sect. 6, conservation laws are given. Finally, the main findings of the
paper are recapitulated in Sect. 7.

unc

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Nonlocal symmetry analysis and conservation laws

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Substituting u 0 = φx and u 1 into (16) and (4), one can
get the Lemma 1.
It is noted that u 0 = φx is a nonlocal symmetry with
regard to Bäcklund transformation (6).
3 Point symmetry of the potential system and
potential equation
In order to get more nonlocal symmetries, we use potential symmetry analysis to deal with (4). PDE (4) is
equivalent to the potential system
vx = u,

130

vt = 3uu x + u x x + u 3 ,

131

and the potential equation

132

vt = 3vx vx x + vx x x + (vx )3 .

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The point symmetry

134

t ∗ = t + ετ (x, t, u, v) + O(ε2 ),

135
136

(14)
(15)

u ∗ = u + εη(x, t, u, v) + O(ε2 ),

v ∗ = v + εψ(x, t, u, v) + O(ε2 ),

138

where ε is the group parameter, and

139

X (3) (vx − u) = 0,

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X (3) (vt − 3uu x − u x x − u 3 ) = 0,

Theorem 1 The potential system (18) admits the point
symmetry (20)

(16)

(17)

for any (u, v) that solves system (18);
∂
∂
X = τ (x, t, u, v) + ξ(x, t, u, v)
∂t
∂x
∂
∂
+ ψ(x, t, u, v) ,
(18)
+ η(x, t, u, v)
∂u
∂v
is the infinitesimal generator of the point symmetry
(20);
∂
∂
X (3) = X + η x
+ ηt
∂u x
∂u t
∂
∂
+ ψx
+ ψt
+ ···
(19)
∂vx
∂vt
with
t

∂
∂
, X2 = ,
∂x
∂t
x ∂
u ∂
∂
∂
, X4 =
+
−
,
X3 =
∂v
∂t
3 ∂x
3 ∂u

 ∂
∂
X F = e−v u F − e−v Fx
− e−v F ,
∂u
∂v
X1 =

η = Dt (η) − u x Dt (ξ ) − u t Dt (τ ),
x

(23)

ψ = Dt (ψ) − vx Dt (ξ ) − vt Dt (τ ),
···

(20)

(24)

here F = F(x, t), and F satisfies Ft − Fx x x = 0, and
hence, the scalar equation (4) admits the corresponding potential symmetry.
Proof: One can get the determining equations
τu = 0,

DISK

LE

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τx = 0,

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τv = 0,

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ψu = 0,

177

ξu = 0,

178

ξv − ηuu = 0,

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2

η − uψv + u ξv − ψx + uξx = 0,

180

ψv + uξv − ηu − τt + 2ξx = 0,

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3η + ηv − 3uψv + uψvv + 3uηu + uηuv

182
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(25)

Solving these equations, one can obtain (27).

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TYPESET

160

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154

Journal: 11071 MS: 2480

159

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+ 3uηx + u 2 ψx − u 3 ξx + 2uηxv + ηx x = 0.

ψ x = Dx (ψ) − vx Dx (ξ ) − vt Dx (τ ),

158

166

c2
τ = c1 + c2 t, ξ = x + c3 ,
3

1 −v  v
e uc2 − 3u F + 3Fx ,
η=− e
3
ψ = c4 − e−v F,

2u 2 η + 3u 2 ηv + u 2 ηvv − ψt + uξt + u 3 τt

t

157

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+ 3uτt − uτtv − 3uξx + 2ηxu − ξx x = 0,

η = Dx (η) − u x Dx (ξ ) − u t Dx (τ ),

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156

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with

x ∗ = x + εξ(x, t, u, v) + O(ε2 ),

137

140

∂
∂
∂
∂
∂
+ u xt
+ ux
+ vx
+ uxx
∂x
∂u
∂v
∂u x
∂u t
∂
∂
+ vx x
+ vxt
+ ···
(21)
∂vx
∂vt
∂
∂
∂
∂
∂
Dt =
+ u tt
+ ut
+ vt
+ u xt
∂t
∂u
∂v
∂u x
∂u t
∂
∂
+ vtt
+ ···
(22)
+ vxt
∂vx
∂vt

Dx =

unc

Author Proof

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Here, Dx and Dt denote the total derivative operator
and are defined by

pro
of

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(13)

cted

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+ u 0t − u 0x x x ) = 0.

orre

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G. Wang et al.

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vx = u,

c2
x + c3 , ψ = c4 − e−v F,
3

3

vt = 3uu x + u x x + u ,

(27)

admits the infinite-parameter Lie group of point transformations with infinitesimal generator


 ∂
1 ∂
−v
1
2
,
(30)
−F
uF − F
XF = e
∂u
∂v

here F = F(x, t), and F also satisfies Ft − Fx x x = 0.
Proof: One can get the determining equations

∂ F1
= F 2,
∂x
∂ F2
∂ F1
.
=
∂x2
∂t
Then, one has

τv = 0,

τx = 0,

ξxvv = 0,

ψv + ψvv = 0,
τt − 3ξx = 0,

ψx + ψxv − ξx x = 0.

(28)

Notes

1. In the above, the vector fields X F are not recoverable point symmetries of the original equation and
are potential symmetries (nonlocal).
2. The conservation laws used to construct the potential system are not unique if the there exists more
than such conserved forms. Moreover, not all the
potential forms may produce potential symmetries.
If the respective conserved form is physical, it is
purely coincidental.

4 Linearization (4) using nonlocal symmetries

One of the approaches that involve invariance deal with
derivative or integral dependent coefficients in the vector fields/symmetries. The latter is somewhat complicated but may be constructed via the conserved form
of the differential equation often referred to as potential form. The symmetries of these systems are nonlocal/potential symmetries of the original equation (see
[21–24]).

DISK

(32)

LE

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240

(33)

In this way, one can get

241

242

∂Ψ 1
∂Ψ 1
∂Ψ 2
− e−v
= 1, ue−v
∂u
∂v
∂u
2
1
−v ∂Ψ
−v ∂Ψ
= 0, −e
= 1.
−e
∂u
∂u

ue−v

− e−v

∂Ψ 2
= 0,
∂v

243

(34)

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It is easy to get a particular solution, which is
Ψ 1 = −ev , Ψ 2 = −uev .

245

(35)

Then, the invertible mapping µ is given by
v

2

(36)

which transforms (29) to the following linear system
∂w 1
= w2 ,
∂z 1
∂ 2 w2
∂w 1
.
=
2
∂z 2
∂z 1

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v

z 1 = x, z 2 = t, w = −e , w = −ue ,

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250

(37)

Consequently, the noninvertible mapping

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∂w1

u=

w2
∂z
= 11 ,
w1
w

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TYPESET

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We now identify

1

228

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α11 = α12 = α21 = β22 = 0, β11 = ue−v ,
β12 = −e−v , β21 = −e−v .

Solving these equations, one can obtain (30).

Journal: 11071 MS: 2480

(31)

ψ = e−v F 1 .

ξt + 2ξx x x = 0,

225
226

234



τ = 0, ξ = 0, η = e−v u F 1 − F 2 ,

ψt − ψx x x = 0,

(29)

where F = (F 1 , F 2 ) is an arbitrary solution of the
linear system of equations

ξv = 0,

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(26)

with
τ = c1 + c2 t, ξ =

pro
of

Author Proof

191

X1 =

cted

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∂
∂
∂
, X2 = , X3 =
,
∂x
∂t
∂v
x ∂
∂
∂
, X F = −e−v F ,
X4 = t +
∂t
3 ∂x
∂v

In order to linearize (4), we use the Theorems as
shown in [21,22]. In the previous section, we showed
that the system (17)

orre

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Theorem 2 The potential equation (19) admits the
point symmetry (20)

unc

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(38)

253

Nonlocal symmetry analysis and conservation laws

Author Proof

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=

∂z 2

c2 =

(39)

.

ut = u x x x .

(40)

In this equation, we deal with the invariant solutions
of the potential equation (19). Now we consider the
following cases
Case 1 X 1 .
For this case, the invariants is ξ = t, and we get the
trivial solution v = c1 .
Case 2 X 2 .
For this case, the invariants is ξ = x, and one can
obtain

C1 C2
(C1 C2 )2 + 8c1
C1
, c3 = −
, c4 =
.
2
2
2C1
(48)

λ f ′ + 3 f ′ f ′′ + f ′′′ + ( f ′ )3 = 0.

It should be noted that the mapping (38) is the famous
Hopf–Cole transformation. Also, we note that this
result is the same as in [4,7]. It should be stressed that,
however, we analyzed the results from another perspective.

5 Reduction equations and similarity solutions
related to potential equation

3 f ′ f ′′ + f ′′′ + ( f ′ )3 = 0.

(41)

For this case, we get the integrating factor

1 
Λ = C1 e2 f + e f C2 ξ 2 + 2C3 ξ + 2C4 ,
2

(42)

where C1 , C2 , C3 and C4 are arbitrary constants.
In this way, (41) can be reduced into following cases:

1 2 f  ′2
e
f + 2 f ′′ = c1 .
(43)
2 

e f f ′2 + f ′′ = c2 .
(44)


e f ξ f ′2 + ξ f ′′ − f ′ = c3 .
(45)


1 f 2 ′2
(46)
e ξ f + ξ 2 f ′′ − 2ξ f ′ + 2 = c4 .
2
Solving the first one, one can obtain
 2 2
C1 C2 + 2 C1 2 C2 ξ + C1 2 ξ 2 + 8 c1
f = ln
4C1

.
(47)

286

287

Case 3 λX 1 + X 2 .
For the linear combination, the invariants is ξ =
x − λt and f (ξ ), and one can have

For brevity, we rewrite the (39) as follows

unc

257

∂z 13

∂w 1

pro
of

∂ 3 w1
256

Putting (47) into (44), (45), (46), one can get

embeds third-order Burgers equation (4) in the following linear equation

For this case, we get the integrating factor

√ 
Λ = C1 e2 f + C2 + C3 sin
λξ
√ 
+ C4 cos
λξ e f ,

(49)

(50)

Solving (51), one can get

TYPESET

DISK

LE

290
291
292

293

296

297
298

299

300

301

302

303

304

305

√
√  √
⎞
λx
λC1 − C2 cos
λx
λ − c2
⎠.
λ

306

Substituting (55) into (52), (53) and (54), one can get

308

⎛

f = ln ⎝−

sin

(55)

√
C1 2 λ + C2 2 λ − c2 2
, c3 = C1 λ,
2λ
√
c4 = −C2 λ.
c1 = −

307

309

(56)

310

Case 4 X 4 .
1
One can get the invariants and function are ξ = xt − 3
and f (ξ ), and one can have

312

1 ′
ξ f + 3 f ′ f ′′ + f ′′′ + ( f ′ )3 = 0.
3
We get the integrating factor




ξ3
4 5
,−
Λ = e f C1 ξ 2 hypergeom [1], ,
3 3
27

(57)

123
Journal: 11071 MS: 2480

289

295

where C1 , C2 , C3 and C4 are arbitrary constants.
In this way, one can have

1 2 f  ′2
e
f + 2 f ′′ + λ = c1 .
(51)
2 

(52)
e f f ′2 + f ′′ + λ = c2 .

√ √
√
−e f cos( λξ ) λ f ′ − sin( λξ ) f ′2

√
− sin( λξ ) f ′′ = c3 .
(53)

√
√
√
e f cos( λξ ) f ′2 + λ sin( λξ ) f ′

√
+ cos( λξ ) f ′′ = c4 .
(54)
√

288

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cted

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pro
of

Author Proof

320

(59)



√ 
√

√
−1 2 3 −ξ 3
3 f
3 61
e (−ξ )
,
3ξ I
f ′2
3ξ
3
9


√ 
3

−1
2
3
−ξ
,
− −ξ 3 I
f′
3
9


√ 
√
−1 2 3 −ξ 3
(60)
+ 3ξ I
f ′′ = c2 .
,
3
9


√
√ 


√
−1 2 3 −ξ 3
3 f
3 13
,
3I
e (−ξ )
f ′2 −ξ 3
2
3ξ
3
9


√ 
−1 2 3 −ξ 3
,
+I
f ′ξ 2
3
9


√ 

√
−1 2 3 −ξ 3
+ 3I
f ′′ −ξ 3 = c3 . (61)
,
3
9

Therefore, one can obtain




ξ3
10 11
ef
6
3ξ hy [3],
,−
,
1680
3 3
27



3
ξ
7 8
+ 84ξ 4 hy [2], ,
,−
f′
3 3
27



ξ3
4 5
2
,−
f ′2
+ 1680ξ hy [1], ,
3 3
27



ξ3
4 5
,−
f ′′
+ 1680ξ 2 hy [1], ,
3 3
27



ξ3
4 5
,−
+ 560ξ 3 hy [1], ,
3 3
27



ξ3
7 8
,−
− 504ξ 3 hy [2], ,
3 3
27

cted

319

(58)

326

Solving (60), one can get

1 ⎝
6
×
6
−ξ 3
×







−3 ξ 3

−3 ξ 3



⎞−1

5 2
,
+I
−3 ξ 3 ⎠ dξ
3 9


√
√ √
3
−ξ 3 I 23 , 29 −3 ξ 3

−c2

e

1
3

6

I 23 , 9
√

1
−ξ 3

−3 ξ 3

+I



5 2
3, 9

√

−3 ξ 3



−1

I

⎝6



2 2
3, 9





√
2



−3 ξ 3

−3 ξ 3

−3 ξ 3

−3 ξ 3

+I





+I



5 2
3, 9

√

−3 ξ 3



−1

⎞−1

5 2
,
−3 ξ 3 ⎠
3 9

TYPESET

DISK

LE

332

333

338

dξ C1

339

ξ −1 dξ

dξ

340

ξ −1 dξ
341

⎞

⎟
dξ + C2 ⎠ ,

123
Journal: 11071 MS: 2480

331

dξ c2

−3 ξ



√
√ √
I 2,
3
−ξ 3 I 23 , 29 −3 ξ 3 6 3 √9

×
6

−3 ξ 3



 


√



 −1
√
√
√ √
I 2 , 2 −3 ξ 3
5 2
3
ξ −1 dξ
,
3
−ξ 3 I 23 , 29 −3 ξ 3 6 3 √9
+I
−3
ξ
3 9
3


⎛

√
2



330

337

−3 ξ




329

336



1
3

e

 

e

2 2
3, 9

 


√



 −1
√
√
√ √
I 2 , 2 −3 ξ 3
5 2
3
,
3
−ξ 3 I 23 , 29 −3 ξ 3 6 3 √9
+I
−3
ξ
ξ −1 dξ
3 9
3

e

− 13



1
3



−

I

328

335

 


√
⎛
2 2



 −1
3
 − 1 √3  √−ξ 3 I 2 , 2 √−3 ξ 3 6 I 3 ,√9 −3 ξ +I 5 , 2 √−3 ξ 3
ξ −1 dξ
3
3 9
3 9
⎜
−3 ξ 3
f = ln ⎝ e

⎛

327

334

orre

318




ξ3
4 5
,−
f′
− 3360ξ hy [1], ,
3 3
27




ξ3
4 5
,−
= c1 .
+ 3360hy [1], ,
3 3
27

unc

317


√ 
1
−1 2 3 −ξ 3
,
+ C2 I
(−ξ 3 ) 6
3
9



√ 
−1 2 3 −ξ 3
3 − 16
,
+ C3 ξ I
.
(−ξ )
3
9


CP Disp.:2015/11/6 Pages: 12 Layout: Medium

(62)

342

343

Nonlocal symmetry analysis and conservation laws

where I is the Bessel functions of the first kind and hy
is the hypergeom functions.

346

6 Conservation law

Author Proof

349

350

351
352

353
354

355

356

357
358
359
360
361

362

363
364
365

366

367
368
369

370

372
373

374

Fι (x, u, u (1) , u (2) , . . . , u (k) ) = 0,
ι = 1, 2, . . . , r,
∂ 2 uι

∂u ι

{u iι }, u (2)

ι
∂ xi , u i j = ∂ xi ∂ x j , . . ..

(63)

{u iι j }, . . ., and u iι

=

=

1. In this paper, the total derivative operators Di are
defined by (21) and (22).
2. Multipliers for PDE system (63) are undetermined
functions {Λα [U ]}, which satisfies the following
equation
Λι [U ]Fι [U ] = Di T i [U ],

(64)

for a set of functions {T i [U ]} [22–24].
If U σ = u σ (x) is a solution of PDE system, then
one can get the conservation law
i

Di T [U ] = 0,

(65)

of PDE system (63), where T i [U ] are the conserved
densities.
3. The Euler operators defined by [22–25]
∂
∂
− Di j + · · ·
Eu j =
j
∂u
∂u
+ (−1)s Di1 · · · Dis

∂
j

∂u i1 ···is

+ ··· ,

(66)

for each j = 1, 2, . . . , m. By employing the following equation,
EU j (Λι [U ]Fι [U ]) ≡ 0, j = 1, . . . , N .

376

6.2 Conservation laws of (4)

379

x

Given a system of r PDEs of k-order with x and u
[22–25]

one can get a set of multipliers.

377

T = u,

3

T = −u − 3uu x − u x x .

375

378

t

(67)

From the above, we suppose the conservation law is
given by Dx T x + Dt T t = 0 on the solutions of (4).
For the fourth-order multiplier, one can get

(69)

TYPESET

DISK

LE

383

384

385

(70)

386

6.3 Conservation laws of (40)

387

Applying the previous steps, for the fourth-order multiplier, one can get

389

x 2u
9t 2 u x x x x
+ 3xtu x x + 3tu x +
,
2
2
Λ2 = xu + 3tu x x ,
Λ1 =

388

390

391

Λ3 = u,

392

Λ4 = xu x x + 3tu x x x x + u x ,

393

Λ5 = u x x ,

394

Λ6 = u x x x x ,

(71)

395

for which the respective conserved quantities are

396

u(u 2x + 6u x xt x + 9t 2 u x x x x + 6u xt )

397

T1t =

4

,

xuu x
9tuu x x x
x 2 uu x x
3tu x u x x
−
−
+
2
2
2
2
9t 2 uu t x x x
9t 2 u x u t x x
3xtu 2x x
−
+
−
2
4
4
9t 2 u x x u t x
9t 2 u t u x x x
u2
x 2 u 2x
−
+
−
+
4
4
2
4
2
2
3xtu x u t
3xtuu t x
9t u x x x
+
−
.
(72)
−
4
2
2
u(xu + 3tu x x )
T2t =
,
2
xu 2
3tu 2x x
uu x
+ x −
T2x = −xuu x x −
2
2
2
3tu t u x
3tuu t x
+
.
(73)
−
2
2
u2
,
T3t =
2
ux
T3x = −uu x x + .
(74)
2
u(xu x x + 3tu x x x x + u x )
,
T4t =
2
3tu x x x u t
xuu t x
3tuu t x x x
xu x u t
+
−
−
T4x =
2
2
2
2
T1x = −

123
Journal: 11071 MS: 2480

381

382

Therefore, one can yield the fluxes

6.1 Basic concepts

i

371

Λ = C1 .

In this section, using the multipliers method [23,24],
we will handle the conservation law of the (4), (40) and
(15).

where u (1) =

(68)

Solving them, one can get

cted

348

380

Λu x x = 0, Λu x x x = 0, Λu x x x x = 0.

orre

347

Λt = 0, Λx = 0, Λu = 0, Λu x = 0,

unc

344

pro
of

345

CP Disp.:2015/11/6 Pages: 12 Layout: Medium

398

399

400

401

402

403

404

405

406

407

408

G. Wang et al.

Author Proof

413

414

415

416

417
418

419

421

423

424

425

(76)

(77)

Once again applying the previous steps, for the fourthorder multiplier, one can get

420

422

(75)

6.4 Conservation laws of (15)

Λ1 =

e2u  2 4
9t u x + 54t 2 u 2x u x x + 6t xu 2x
18
+ 36t 2 u x u x x x + 27t 2 u 2x x


+ 6t xu x x + 9t 2 u x x x x + 6tu x + x 2 ,

e2u  4
3tu x + 18tu 2x u x x + xu 2x + 12tu x u x x x
3

+ 9tu 2x x + xu x x + 3tu x x x x + u x ,


Λ3 = e2u u 4x + 6u 2x u x x + 4u x u x x x + 3u 2x x + u x x x x ,

Λ2 =


e2u  2
Λ4 =
3tu x + 3tu x x + x ,
3
T1t =

Λ6 = e ,

pro
of

412

2u

7 Conclusions

430

In the present paper, we studied the nonlocal symmetries, similarity reduction and explicit solutions of
the third-order Burgers equation. In particular, we linearized the equation to the third-order linear PDE via
symmetries. Also, some explicit solutions of potential
equation are constructed. Furthermore, we also give the
conservation laws of the third-order Burgers equation,
the reduced equation and the potential equation. In the
future, we will deal with the high-order equation, such
as the fourth-order Burgers equation, even more highorder equation and then with variable coefficients.
Acknowledgments This work was supported by International
Graduate Exchange Program of Beijing Institute of Technology
(1320012341502), 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). The authors thank the reviewers for the useful comments and suggestions which have been
incorporated into the text.

Appendix

123
TYPESET

DISK

LE

(78) 427

429

27t 2 u 4x e2u + 2x 2 u 4 e2u + 6tu 3 u x − 27t 2 u 2 u 2x x + 9t 2 u 3 u x x x x − 2x 2 u 4
72u 4
2u
27t 2 e u 2 u 2x x − 27t 2 u 4x − 9t 2 e2u u 3 u x x x x + 12te2u u x u 4 + 54t 2 e2u u 4 u 2x x
+
72u 4
2
2u
4
2
2u
3
2
18t e u u x x x x − 54t e u u x x + 18t 2 e2u u 4 u 4x − 36t 2 e2u u 3 u 4x + 54t 2 e2u u 2 u 4x
+
72u 4
3
2
2u
4
2
2
6tu u x x x − 54t e uu x − 36t u u x u x x x + 81t 2 uu 2x u x x − 6t xu 2 u 2x − 6te2u u 3 u x
+
72u 4
2
2u
4
2
2u
4
2
72t e u u x u x x x + 108t e u u x u x x − 162t 2 e2u u 3 u 2x u x x + 12t xe2u u 4 u 2x
+
72u 4
2u
4
2
2u
3
12t xe u u x x − 72t e u u x u x x x + 162t 2 e2u u 2 u 2x u x x − 12xte2u u 3 u 2x
+
72u 4
2
2u
2
2u
3
36t e u u x u x x x − 6t xe u u x x − 81t 2 e2u uu x x u 2x + 6t xe2u u 2 u 2x
,
+
72u 4

Journal: 11071 MS: 2480

426

for which the respective conserved quantities are given
in Appendix.

cted

411

Λ5 = e2u (u 2x + u x x ),

orre

410

3tu t x u x x
3uu x x x
3tu x u t x x
−
−
2
2
2
ux uxx
xu 2x x
3tu 2x x x
+
−
.
−
2
2
2
uu x x
T5t =
,
2
uu t x
u x ut
u2
+
.
T5x = − x x −
2
2
2
uu x x x x
T6t =
,
2
u x ut x x
u x x ut x
uu t x x x
T6x = −
+
−
2
2
2
u 2x x x
u x x x ut
−
.
+
2
2
+

unc

409

CP Disp.:2015/11/6 Pages: 12 Layout: Medium

428

431
432
433
434
435
436
437
438
439
440
441

442
443
444
445
446
447
448
449
450

451

452

453

454

455

456

457

458

459

Nonlocal symmetry analysis and conservation laws

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

pro
of

Author Proof

463

cted

462

orre

461

4e2u u 4 − 4u 4 + 27t 2 e2u u 2 u x u t x x + 27t 2 e2u u 2 u x x u t x + 54te2u u 2 u x u x x
−72u 4
2u
4
2
2u
4
12t xe u u x x + 96te u u x u x x − 108te2u u 3 u x u x x + 36t 2 e2u u 4 u x u t x x
+
−72u 4
2
2u
4
4
2
2u
3
108t e u u x u x x − 18t e u u t u x x x − 54t 2 e2u u 3 u x u t x x − 54t 2 e2u u 3 u t x u x x
+
−72u 4
12t xe2u u 4 u 4x + 162t 2 e2u u 4 u 2x u 2x x + 36t 2 e2u u 4 u 3x u x x x − 54t 2 e2u u 3 u 2x u t x
+
−72u 4
2
2u
2
2
2
2u
4
54t e u u t x u x + 72t e u u t x u x x + 12t xe2u u 4 u t x − 36t 2 e2u u 3 u t u 3x
+
−72u 4
2
2u
2
3
2
2u
u
3
54t e u u t u x − 54t e u t u x + 54t 2 uu t u x u x x − 6xtu 2 u t u x
+
−72u 4
2
2u
2
2
2u
9t e u u t u x x x − 27t e uu t x u 2x − 6t xe2u u 3 u t x + 4x 2 e2u u 4 u x x
+
−72u 4
3
2
2
6t xu u t x − 9t u u t u x x x − 27t 2 u 2 u t x u x x − 27t 2 u 2 u x u t x x − 54tu 2 u x u x x
+
−72u 4
2
2
2u
4
27t uu t x u x + 36te u u x x x + 24te2u u 4 u 3x + 18t 2 e2u u 4 u 6x + 27t 2 e2u u t u 3x
+
−72u 4
2
2u
4
2
2u
3
3
2
2x e u u x − 36te u u x + 18t e2u u 4 u 2x x x + 36te2u u 2 u 3x − 6xe2u u 3 u x
+
−72u 4
2u
4
2u
3
2u
4xe u u x − 18te uu x − 18te u 3 u x x x − 9t 2 e2u u 3 u t x x x + 18t 2 e2u u 4 u t x x x
+
−72u 4
18tu 3 u x x x + 9t 2 u 3 u t x x x + 6xu 3 u x − 27t 2 u t u 3x + 108t 2 e2u u 4 u x u x x u x x x
+
−72u 4
3
2
2u
18tuu x − 54t e uu t u x u x x + 6t xe2u u 2 u t u x − 108t 2 e2u u 3 u t u x u x x
+
−72u 4
2
2u
2
2u
108t e u u t u x u x x − 12t xe u 3 u t u x + 24t xe2u u 4 u 2x u x x
+
.
−72u 4
−3te2u u 3 u x x x x + xe2u u 2 u 2x u x x + 9te2u u 2 u 2x x − xe2u u 3 u x x + 2xe2u u 4 u 2x
T2t =
12u 4
2u
4
2u
4
2
24te u u x u x x x + 36te u u x u x x − 54te2u u 3 u 2x u x x − 24te2u u 3 u x u x x x
+
12u 4
−27te2u uu x x u 2x + 12te2u u 2 u x u x x x + u 3 u x + 18te2u u 4 u 2x x − 18te2u u 3 u 2x x
+
12u 4
2u
4
4
2u
3
4
2u
6te u u x − 12te u u x + 18te u 2 u 4x − 18te2u uu 4x + 6te2u u 4 u x x x x
+
12u 4
2
4
2u
4
−12tu u x u x x x − 9tu x + 2e u u x + 9te2u u 4x − e2u u 3 u x + 3tu 3 u x x x x − xu 2 u 2x
+
12u 4
2u
2
2
2u
4
54te u u x u x x + 2xe u u x x + 27tuu 2x u x x − 9tu 2 u 2x x + xu 3 u x x
+
,
12u 4

T1x =

unc

460

480

123
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TYPESET

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G. Wang et al.

485

486

487

488

489

490

491

492

493

494

495

496

unc

Author Proof

484

pro
of

483

cted

482

36te2u u 4 u x u x x u x x x − 18te2u uu t u x u x x − 36te2u u 3 u t u x u x x + 36te2u u 2 u t u x u x x
−12u 4
3u 3 u x x x + 3uu 3x − xu t u x u 2 − 3tu 2 u t u x x x − 9tu 2 u t x u x x − 9tu 2 u t x x u x
+
−12u 4
2xe2u u 4 u 4x + 6tu 4 e2u u 6x + 9te2u u t u 3x + 2xe2u u 4 u 2x x + 16e2u u 4 u x u x x
+
−12u 4
2u
3
2u
2
−xe u u t x + 9e u u x u x x − 3te2u u 3 u t x x x − 18e2u u 3 u x u x x + 6te2u u 4 u t x x x
+
−12u 4
2u
4
2u
4
2xe u u t x + 24te u u t x u x x + 12te2u u 4 u x u t x x − 6te2u u t u 3 u x x x − 2xe2u u 3 u x u t
+
−12u 4
2
2u
3
2u
4
2
54te u u x u x x − 18te u u t x x u x − 18te2u u 3 u t x u x x + 12te2u u 4 u 3x u x x x
+
−12u 4
2u
2
2
2u
3
2
18te u u x u t x − 18te u u x u t x + 4xe2u u 4 u x x u 2x − 12te2u u 3 u t u 3x
+
−12u 4
2u
2
3
2u
3
18te u u t u x − 18te uu t u x + 18tuu t u x u x x − 9te2u uu t x u 2x + xe2u u 2 u t u x
+
−12u 4
2u
2
2u
2
9te u u x u t x x + 9te u u x x u t x + 3te2u u 2 u t u x x x − 3e2u u 3 u x x x − 3e2u uu 3x
+
−12u 4
6e2u u 4 u x x x − 6e2u u 3 u 3x + 6e2u u 2 u 3x + 4e2u u 4 u 3x + 12te2u u 4 u 2x u t x + 6te2u u 4 u 2x x x
+
−12u 4
2u
4
4
3
2
36te u u x u x x 3tu u t x x x − 9u u x u x x + xu 3 u t x − 9tu t u 3x + 9tuu t x u 2x
+
.
−12u 4
2e2u u 4 u 4x + 12e2u u 4 u 2x u x x − 4e2u u 3 u 4x + 8e2u u 4 u x u x x x + 6e2u u 4 u 2x x
T3t =
4u 4
2u
3
2
2u
2
4
−18e u u x u x x + 6e u u x + 2e2u u 4 u x x x x − 8e2u u 3 u x u x x x − 6e2u u 3 u 2x x
+
,
4u 4
18e2u u 2 u 2x u x x − 6e2u uu 4x − e2u u 3 u x x x x + 4e2u u 2 u x u x x x + 3e2u u 2 u 2x x
,
+
4u 4
9e2u uu 2x u x x + 3e2u uu 4x + u 3 u x x x x − 4u 2 u x u x x x − 3u 2 u 2x x + 9uu 2x u x x − 3u 4x
+
,
4u 4

T2x =

orre

481

123
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TYPESET

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Nonlocal symmetry analysis and conservation laws

501

502

503

504

505

506

507

508

509

510

511

512

513

514

pro
of

Author Proof

500

cted

499

orre

498

−3u t u 3x + u 3 u t x x x − 4e2u u 3 u t u 3x + 4e2u u 4 u t x u 2x + 12e2u u 4 u 4x u x x + 18e2u u 4 u 2x u 2x x
4u 4
2u
3
2u
−6e u x u t uu x x − 12u e u x u t u x x + 12u 2 e2u u x u t u x x + 12e2u u 4 u x u x x u x x x
+
4u 4
2
2u
3
2u
3
3uu t x u x − e u u t x x x + 3e u t u x + 2e2u u 4 u 2x x x + 2e2u u 4 u t x x x + 2e2u u 4 u 6x
+
4u 4
2
2
2
−3u u t x x u x − 3u u t x u x x − u u t u x x x + 4e2u u 4 u 3x u x x x − 6e2u u 3 u t x u 2x
+
4u 4
2u
2
3
2u
4
6e u u t u x + 4e u u x u t x x + 8e2u u 4 u t x u x x − 3e2u uu t x u 2x + 3e2u u 2 u x u t x x
+
4u 4
3e2u u 2 u t x u x x + e2u u 2 u t u x x x − 6e2u uu t u 3x − 6e2u u 3 u t x x u x − 6e2u u 3 u t x u x x
+
4u 4
2u
3
2u
2
2
−2e u u t u x x x + 6e u u t x u x + 6uu x u t u x x
.
+
4u 4
6te2u u 2 u 2x + 6te2u u 2 u x x − 6te2u uu 2x + 2xe2u u 2 − 3tue2u u x x
T4t =
12u 2
2u
2
2
3te u x − 2xu + 3tuu x x − 3tu 2x
+
,
12u 2
2xe2u u 2 u 2x + 12te2u u 2 u 2x u x x − 3te2u uu t x + 6te2u u 2 u t x − 3tu t u x
T4x =
12u 2
2u
3uu x + 3tuu t x + 3tu t u x e − 6tue2u u x u t + 6te2u u 2 u 2x x
+
12u 2
2u
2
2u
4
2
4e u u x x x + 6te u x u + 2e2u u 2 u x − 3e2u uu x
+
.
12u 2
2e2u u 2 u 2x + 2e2u u 2 u x x − 2e2u uu 2x − e2u uu x x + e2u u 2x + uu x x − u 2x
,
T5t =
4u 2
2e2u u 2 u 4x + 4e2u u 2 u 2x u x x + 2e2u u 2 u 2x x + 2e2u u 2 u t x
T5x = −
4u 2
2u
2u
−2e uu t u x − e uu t x + e2u u x u t + uu t x − u x u t
+
.
4u 2
e2u − 1
,
T6t =
2

e2u  2
u x + 2u x x .
T6x = −
2
T3x =

unc

497

123
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G. Wang et al.

519
520
521
522
523

Author Proof

524
525
526
527
528
529
530
531
532
533
534
535
536

2

537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552

pro
of

518

cted

517

1. Olver, P.J.: Evolution equations possessing infinitely many
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2. Gerdjikov, V.S., Vilasi, G., Yanovski, A.B.: Integrable
Hamiltonian Hierarchies. Springer, Berlin (2008)
3. Sharma, A.S., Tasso, H.: Connection between wave envelope and explicit solution of a nonlinear dispersive equation.
Report IPP 6/158 (1977)
4. Kudryashov, N.A., Sinelshchikov, D.I.: Exact solutions of
equations for the Burgers hierarchy. Appl. Math. Comput.
215, 1293–1300 (2009)
5. Kudryashov, N.A.: Self-similar solutions of the Burgers
hierarchy. Appl. Math. Comput. 215, 1990–1993 (2009)
6. Wazwaz, A.M.: Burgers hierarchy: multiple kink solutions
and multiple singular kink solutions. J. Franklin Inst. 347,
618–626 (2010)
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