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2015

Symmetry Analysis and Conservation Laws for the
Class of Time-Fractional Nonlinear Dispersive
Equation
Kamran Fakhar
Kwantlen Polytechnic University

Gangwei Wang
A. H. Kara

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Original Publication Citation
Wang, G., Kara, A.H. & Fakhar, K. (2015). Symmetry analysis and conservation laws for the class of time-fractional nonlinear
dispersive equation. Nonlinear Dynamics, May 2015. DOI 10.1007/s11071-015-2156-4

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Nonlinear Dyn
DOI 10.1007/s11071-015-2156-4

of

ORIGINAL PAPER

pro

Symmetry analysis and conservation laws for the class
of time-fractional nonlinear dispersive equation

Received: 31 December 2014 / Accepted: 10 May 2015
© Springer Science+Business Media Dordrecht 2015

2
3

1

4

2
3

6

5

7

Abstract In this paper, we derive the complete
algebra of Lie point symmetries for the class of
(((
time
fractionaltime-fractionalnonlinear
dispersive equa(((
tion. By means of the classical Lie symmetry method,
the associated vector fields are obtained which in turn
are utilized for the reduction of the equation. In particular, the conservation laws of the equation are obtained.

9

Keywords Time-fractional nonlinear dispersive
equation · Lie symmetry method · Conservation laws

10

1 Introduction

11
12

Differential equations play an important and central
role in many fields. It is well known that Lie theory of

unc

8

symmetry group provides a systemic, general and effi- 13
cient method to deal with differential equations. This 14
theory is mainly used for the construction of similarity 15
reductions, group invariant solutions and the conserva- 16
tions laws. In general, Lie symmetries can be used to 17
reduce the order as well number of independent vari- 18
ables of original equation ( system of equations ) . For 19
further details, readers are referred to [1–11] .
20
((
Unlike the case of (
21
integral
orderintegral-orderpartial
(((
(
((
differential equations ( PDEs ) , symmetries of (
22
fractional
((( orderfractio
orderpartial differential equations ( FPDEs ) have not 23
been investigated extensively. The study of FPDEs 24
through symmetries is quite interesting and signifi- 25


cant [12–28] . Therefore, in our present 
studystudy,we
26
investigate the symmetries of FPDEs and thereby do the 27
analysis. We successfully obtain the reduction in inde- 28
pendent and dependent variables. Moreover, we look 29
into the key issue of whether we can identify the FPDEs 30
from which the Lie point symmetries are inherited.
31
In this paper, we will investigate the class of 32
(((
time
fractionaltime-fractionalnonlinear
dispersive equa- 33
((
(
tion
34
  
1  a
α
m
b a
b
u (u u u
u t + ε(u )x +
= 0,
(1) 35
xx x
b 

where u(x, t) represents the wave 
profileprofile,while
36
a, b, ε and m are constants. Some special cases of ( 37
1 ) have been used to describe physical situations in 38
various fields. If α = 1, a = 0, one can get the gen- 39
eralization of the KdV equation. In the special case if 40
m = 2, a = 0, b = 1, Eq. ( 1 ) reduces to the classical 41
KdV equation, while m = 3, a = 0, b = 1, Eq. ( 1 42

orre

1

cted

Gangwei Wang · A. H. Kara · K. Fakhar

G. Wang (B)
School of Mathematics and Statistics, Beijing Institute of
Technology, Beijing 100081, Republic of China
e-mail: pukai1121@163.com; wanggangwei@bit.edu.cn
G. Wang · A. H. Kara · K. Fakhar
Department of Mathematics, University of British
Columbia, Vancouver V6T 1Z2, Canada

A. H. Kara
School of Mathematics, University of the Witwatersrand,
Private Bag 3, Johannesburg Wits 2050, South Africa
K. Fakhar
Department of Mathematics, Faculty of Science
and Horticulture, Kwantlen Polytechnic University,
Surrey, BC V3W 2MB, Canada

123
Journal: 11071 MS: 2156

TYPESET

DISK

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CP Disp.:2015/5/15 Pages: 7 Layout: Medium

G. Wang et al.

48
49
50
51
52
53

54
55
56

dt ∗

dx ∗

|ε=0 , ξ(x, t, u) =
|ε=0 ,
dε
dε
∗
du
|ε=0 .
η(x, t, u) =
(5)
dε
On the basis of the infinitesimal invariance criterion,
one can get
τ (x, t, u) =

pr (α,3) V (

1 )|

80

81

82
83

= 0,

(6)

 

b ua ub
u a
.
where 1 = u αt + ε(u m )x + b1 
(u
xx
x
The prolongation operator pr (α,3) V is
pr

2 Lie symmetry analysis of the class of the
((((
time
fractionaltime-fractionalnonlinear
((
(
dispersive equation

79

of

47

Here,

pro

45
46

) becomes the famous mKdV equation. Further more
descriptions of ( 1 ) and its applications can be found
in [29–32] and references therein.
Sect. 2 ,

The paper is divided as follows. In 
Sec.
some definitions and properties of Lie group method


to 
analysisanalyzethe
FPDEs are given. Moreover, the
infinitesimal operators of the 
Lie-pointLie
 pointsymmetries

admitted by Eq. ( 1 ) are also constructed. In 
Sec.Sect.
3
, the conservation laws of the equation are obtained. The
main results of the paper are summarized and discussed
in the last section.

(α,3)

1 =0



V = V + ηα0 ∂∂tα u + η x ∂u x + η x x ∂u x x
+ η x x x ∂u x x x ,

84

85
86

87

(7)

88

where

cted

43
44

89

η = ηx + (ηu − ξx )u x − τx u t − ξu u 2x − τu u x u t ,
η x x = ηx x + (2ηxu − ξx x )u x − τx x u t
x

58
59
60
61
62
63

64

In this section, we employ Lie symmetry method to
deal with the fractional nonlinear dispersive equation.
We first briefly recall the concept of fractional derivative ( [33–36] and(
references therein ) . In particular, the
(((
Riemann-LiouvilleRiemann–Liouvillefractional
deriv(((
(
ative is defined by
Dtα f (t)
 ∂nu
=

(
(((

, ( α = n,
α = n ∈ N,
∂t n (
 t u(θ,x)
1
∂n
Γ (n−α) ∂t n 0 (t−θ)α+1−n dθ, n −1 < α < n, n ∈ N ,

68

69

70

71

72

73

74

75

76
77

78

− 3ξu u x x u x − τu u x x u t − 2τu u xt u x ,

xxx

ut 
= ηx x x + (3ηx xu − ξx x x )u x − τx x x
+
3u t
+ 3(ηxuu−ξx xu )u 2x − 3τx xu u x u t + u x u t

where Γ (z) is the Euler gamma function.
Assume that ( 1 ) is invariant under the one parameter
Lie group of point transformations
 

2

t ∗ = t + ετ (x, t, u)
O ε2 ,
+O(ε
 

2

x ∗ = x + εξ(x, t, u)
+ O ε2 ,
+O(ε
 

2

u ∗ = u + εη(x, t, u)
+ O ε2 ,
+O(ε
 
∂ α ū ∂ α u
(3)
= α + εηα0 (x, t, u) + O ε2 ,
α
∂ t¯
∂t
 
∂ ū ∂u

2

=
+ εη x (x, t, u)
+ O ε2 ,
+O(ε
∂ x̄ ∂ x
 
∂ 2 ū ∂ 2 u

2

= 2 + εη x x (x, t, u)
+ O ε2 ,
+O(ε
2
∂ x̄
∂x
 
∂ 3 ū ∂ 3 u
xxx
2
=
+
εη
(x,
t,
u)
+O(ε
),
+
O
ε2 ,

∂ x̄ 3 ∂ x 3
where ε is the group parameter, and its associated Lie
algebra is spanned by the following vector fields
∂
∂
∂
+ η(x, t, u) . (4)
V = τ (x, t, u) + ξ(x, t, u)
∂t
∂x
∂u

TYPESET

DISK

LE

90

− 3τx x u xt − 3τxuu u 2x u t +3(ηuu −3ξxu )u x u x x
− 3τxu u x x u t − 6τxu u xt u x −3τx u x xt +
+ (ηu − 3ξx )u x x x − ξx x x u 4x


− 6ξuu u x x u 2x − 3τuu u 2x u t x −τuuu
ut 
− 3u 3x u t
u 3x

− 3ξu u 2x x − 3τu u x xt u x − 3τu u xt u x x
−3
− 3τuu u t u x u x x −4ξu u x u x x x − τu u t u x x x .

In particular,
ηα0 =

∂αη
∂t α

91

+ (ηu − α Dt (τ ))
∞

+μ +
n=1

× Dtα−n (u)−

∂αu
∂t α

−u

∂αη

∞
n=1

a
Dtn (ξ )Dtα−n (u x ),
n

where
μ
=μ =


u
92

∂t α



a ∂ α ηu
a
−
D n+1 (τ )
n ∂t α
n+1 t

93

(9)

94

95

∞

n

m k−1

a
n

n
m

k 1
t n−α
96
r k! Γ (n + 1 − α)

n=2 m=2 k=2 r =0
∂m
∂ n−m+k η
× [−u]r m [u k−r ] n−m k ,
∂t
∂t
∂u

123
Journal: 11071 MS: 2156

(8)

+ (ηuuu − 3ξxuu ) u 3x + 3(ηxu − ξx x )u x x

(2)

unc

67

− τuu u 2x u t + (ηu − 2ξx )u x x − 2τx u xt

η

65
66

+ (ηuu − 2ξxu )u 2x − 2τxu u x u t −ξuu
−u 3x
u 3x

orre

57

CP Disp.:2015/5/15 Pages: 7 Layout: Medium

(10)
97

Time-fractional nonlinear dispersive equation

101
102
103
104
105
106
107

108
109

110

111

112
113
114

115

of

100

here c1 and c2 are arbitrary constants. Thus, the corresponding vector fields are
τ (x, t, u)|t=0 = 0.
(11)
c1 (a + b − 3m + 2) t ∂
(( V =
Compared with the Lie symmetry method to (
integral
orderintegral(((
α
∂t
 ∂

∂
orderdifferential equations, it is can be easily seen that
+ 2c1 u . (16)
+ c1 (a + b − m) x + c2
constraint condition ( 11 ) and formula ( 9 ) are critical
∂x
∂u
or
to FPDEs.
(((
∂
(a + b − 3m + 2) t ∂
Now, we will study the class of (
time
fractionaltime((
V1 =
, V2 =
fractionalnonlinear dispersive equation using above
∂x
α
∂t

∂
∂
Lie symmetry group theory. First, one can get the fol+ 2u ,
(17)
+ (a + b − m)x
lowing assertion
∂x
∂u
which complete the proof.


Theorem 1 The symmetry group of the equation is
In particular, for the symmetry V2 , we have the charspanned by the following vector fields
acteristic equation
∂
(a + b − 3m + 2) t ∂
dx
αdt
du
, V2 =
V1 =
,
(18)
=
=
∂x
α
∂t
2u
(a + b − m) x
(a + b − 3m + 2) t
∂
∂
which leads to the following similarity variable and the
+ 2u .
(12)
+ ((a + b − m)x)
∂x
∂u
similarity transformation
α(a+b−m)
2α
Proof By assuming that Eq. ( 1 ) is invariant under the
ξ = xt − a+b−3m+2 , u = t a+b−3m+2 g(ξ ),
(19)
transformation group ( 3 ) , one can get the symmetry



as 
requiredrequired.
equation as follows

pro

99

and additional constraint condition is

cted

98

+ η x x x u a+b−1 + εm(m − 1)ηu m−2 u x

117

+ εmη x u m−1 + (a +3b−3)(a +b−2)ηu a+b−3 u x u x x

118

+ (a + 3b − 3)η u

119

+ (a + 3b − 3)η x x u a+b−2 u x

120
121

orre

116

uxx

+ (b − 1)(a + b − 2)(a + b − 3)ηu a+b−4 u 3x
+ 3(b − 1)(a + b − 2)η x u a+b−3 u 2x = 0.

(13)

124

ξu = τu = ξt = ξx x = τx = ηuu = 0,

125

(τt α − 3ξx )u + (a + b − 1)η = 0,

126

(τt α + ηu − 3ξx )u + (a + b − 2)η = 0,

127

(τt α + 2ηu − 3ξx )u + (a + b − 3)η = 0,

129

130

unc

123

128

1−α+

2α

a+b−3m+2
P a+b−3m+2

,α

α(a+b−m)

Substituting ( 8 ) -–(
 11 ) into ( 13 ) , and letting all of
the powers of derivatives of u to zero, one can have

122

(τt α − ξx )u + (m − 1)η = 0,
a n
a
∂ (ηu ) −
D n+1 (τ ) = 0,
n t
n+1 t
for n = 1, 2, 
·
· ·. . .

131

By solving these equations, we have

132

ξ = c1 (a + b−m)x +c2 , τ =

133

η = 2c1 u,

(14)

c1 (a +b−3m +2) t
,
α
(15)

g(ξ )

TYPESET

DISK

137

138

139

140

141

142
143

144

145
146

147
148

152

+ εmg m−1 gξ + (b − 1)(a + b − 2)g a+b−3 gξ3

153

+ (a + b − 3)g

154

a+b−2

gξ ξ gξ + g

a+b−1

gξ ξ ξ = 0,

(20)
(
(
(
with the (
Erdelyi-KoberErdelyi–Koberfractional
differ((
ential operator Pβτ,α of order [33]
(Pβτ,α g):=

n−1

j=0

τ+j−

1 d
ξ
β dξ




K βτ +α,n−α gg

 (ξ ),
(21)



[α] + 1, α ∈
/ N,
(22)
α
α ∈ N,
where


K βτ,α gg

 (ξ )
⎧
⎨ 1  ∞ (u − 1)α−1 u −(τ +α) g(ξ u β1 )du, α > 0,
:= Γ (α) 1
⎩ g(ξ ), α = 0,
n=

LE

155

156
157

158

159

160

161

162

163

(23) 164
is the Erdé lyi–Kober fractional integral operator.

123
Journal: 11071 MS: 2156

136

Theorem 2 The transformation ( 19 ) reduces ( 1 ) to 149
(((
the following nonlinear ODE of (
fractional
((( orderfractional-150
order
151

ηα0 + (a + b − 1)ηu a+b−2 u x x x

x a+b−2

134
135

CP Disp.:2015/5/15 Pages: 7 Layout: Medium

165

G. Wang et al.

169

170

∂αu
∂t α

==


∂n

1
∂t n Γ (n − α)

 t
0

174

1+

=

∂t n

2α

t n−α+ a+b−3m+2

K

2α
1+ a+b−3m+2
,n−α
a+b−3m+2
α(a+b−m)


g (ξ ) .

(25)

176

α(a+b−m)

179

180

By using the relation ξ = xt − a+b−3m+2 , Eq. ( 25 )
further simplifies to

t

α(a+b−m)
∂
α(a +b−m)
t − a+b−3m+2 −1 φ  (ξ )
φ(ξ ) = t x −
∂t
a +b−3m +2
α(a + b − m) d
ξ φ(ξ ).
(26)
=−
a + b − 3m + 2 dξ

181

Thus, one can get

182

∂ n n−α+ 2α
a+b−3m+2
t
∂t n

183

2α
1+ a+b−3m+2
,n−α

K a+b−3m+2
α(a+b−m)

2α

a+b−3m+2
× K a+b−3m+2

,n−α

α(a+b−m)

185

186

187



g (ξ )



gg

 (ξ )

∂ n−1 n−α+ 2α −1
a+b−3m+2
t
(n − α
∂t n−1
α(a + b − m) d
2α
−
ξ
+
a + b − 3m + 2 a + b − 3m + 2 dξ

2α
1+ a+b−3m+2
,n−α
× K a+b−3m+2
g (ξ ) .

=

α(a+b−m)

(27)

2α

= · · · = t n−α+ a+b−3m+2
+

n−1
j=0

TYPESET

DISK

LE



191

192

193

194

(1 − α

195

α(a + b − m) d
2α
+j−
ξ
a + b − 3m + 2
a + b − 3m + 2 dξ
1+

2α

a+b−3m+2
× K a+b−3m+2

,n−α

α(a+b−m)

g (ξ ).

196

(28)

So we have
2α
2α
∂αu
1−α+ a+b−3m+2
,α
−α+ a+b−3m+2
P
=
t
g (ξ ). (29)
a+b−3m+2
α
∂t
α(a+b−m)
Hence, it is easily found that ( 1 ) reduces into the
(
((
following (
fractional
((( orderfractional-orderODE
P

2α
1−α+ a+b−3m+2
,α
a+b−3m+2
α(a+b−m)

g(ξ ) + εmg m−1 gξ

+ (a + b − 3)g

a+b−2

gξ ξ gξ + g

198

199

200
201

203

a+b−1

gξ ξ ξ = 0.

204

(30)

205




206
207

In this section, we study the conservation laws of the
class of time-fractional nonlinear
dispersive equa((((
tion. The (
Riemann-LiouvilleRiemann–Liouvilleleft(((
sided time-fractional derivative will be used as
α
n
n−α
u),
(31)
0 D t u = D t (0 I t



in Eq. ( 1 ) . HereHere,Dt is the operator of differentiation with respect to t, n = [α] + 1, and 0 Itn−α u is
the left-sided time-fractional integral of order n − α
defined by [25]
 t
u(θ, x)
1
(0 Itn−α u)(x, t) =
dθ,
Γ (n − α) 0 (t − θ )1−n+α
(32)
where Γ (z) is the Gamma function.

CP Disp.:2015/5/15 Pages: 7 Layout: Medium

197

202

+ (b − 1)(a + b − 2)g a+b−3 gξ3

123
Journal: 11071 MS: 2156

190

gg)(ξ
)



This
completes the
proof.
3 Conservation
laws

∂ n−1 ∂ n−α+ 2α
a+b−3m+2
t
= n−1
∂t
∂t
1+

184

,n−α

α(a+b−m)

orre

178

=

unc

177

189

∂ n−1 n−α+ 2α −1
a+b−3m+2
t
(n − α
∂t n−1
α(a + b − m) d
2α
−
ξ
+
a + b − 3m + 2 a + b − 3m + 2 dξ

2α
1+ a+b−3m+2
,n−α
× K a+b−3m+2
g (ξ )

(24)

 ∞
∂αu
1
∂ n n−α+ 2α
a+b−3m+2
=
(v−1)n−α−1
t
∂t α
∂t n
Γ (n−α) 1



2α
α(a+b−m)
− n−α+ a+b−3m+2
+1
×v
g(ξ v a+b−3m+2 )dv

175

2α

α(a+b−m)

Under the assumption v = st , Eq. ( 24 ) reduces

∂n

∂ n−1 ∂ n−α+ 2α
a+b−3m+2
t
∂t n−1 ∂t

a+b−3m+2
× K a+b−3m+2



α(a+b−m)

2α

173

(t − s)n−α−1

× s a+b−3m+2 g(xs − a+b−3m+2 )ds .

171

172

=

of

168

188

pro

167


RepeatRepeatingthe
previous step, one has


n
2α
2α
∂
1+ a+b−3m+2
,n−α
n−α+ a+b−3m+2
t
K
g
(ξ
)
a+b−3m+2
∂t n
α(a+b−m)

Proof We first let n − 1 < α < n, n =(1, 2, 3, . . ..
(((
Then, in light of the (
Riemann-LiouvilleRiemann–
(((
Liouvillefractional derivative and the similarity transformation, we get

cted

166

208
209
210
211
212
213
214
215
216

217

218
219

Time-fractional nonlinear dispersive equation

226

227

228
229
230

231

232
233

234

235

236
237
238
239

240

241
242
243
244

245

246
247
248

249

250
251
252
253
254

255

256

W = η − τ ut − ξ u x .

of

(41)

258

(((
(33)
(((
(
For the (
259
case
Riemann-Liouvillecase,
Riemann–Liouvilletime(
(
where C t = C t (t, x, u, 
.... . .), C x = C x (t, x, 
u,
...u, . . .). fractional derivative is used in Eq. ( 1 ) , the operator 260
Eq. ( 33 ) is called a conservation law for Eq. ( 1 ) .
N t is given by [25,26]
261
A formal Lagrangian for ( 1 ) can be introduced as
   
 m
1  a
n−1
α
b a
u (u u u b
(34)
.
L = v(x, t) u t + ε u x +
∂
t
xx x
b 
N = τl +
(−1)k 0 Dtα−1−k (W )Dtk
262
α
∂
D
0 t u



k=0
HereHere,v(x,
t) is a new dependent variable. Con∂
sidering the formal Lagrangian, an action integral is
,
(42) 263
− (−1)n J W, Dtn
given by
∂ 0 Dtα u
 T

L x, t, u, v, Dtα u, u x · · · dxdt.
(35)
where J is the integral [25,26]
264
Ω
0
(
(
(
(
(
The (
Euler-LagrangeEuler–Lagrangeoperator
is defined
(
 t T
by [25,26]
f (τ, x)g(μ, x)
1
J ( f, g) =
dμdt.
265
∂
δ
∂
∂
∂
Γ
(n
−
α)
(μ
− τ )α+1−n
0 t
=
+ (Dtα )∗
− Dx
+ Dx2
α
δu
∂u
∂ Dt u
∂u x
∂u x x
(43) 266
3 ∂
− Dx
,
(36)
∂u x x x
The operator N x is defined by
267
where (Dtα )∗ is the adjoint operator of (Dtα ).
((((
Note that Eq. ( 1 ) with the (
Riemann-LiouvilleRiemann–
(((
∂
∂
∂
N x = ξl + W
− Dx
+ Dx2
268
Liouvillefractional derivative can be rewritten in the
∂u x
∂u x x
∂u x x x
form of conservation law form ( 33 ) with
∂
∂
2


+D
+ Dx (W )
− Dx
269
1 a b
t
n−1
n−α
x
m
∂u x x
∂u x x x  x
C = Dt (0 It u), C = εu + u (u )x x . (37)
b
∂
The adjoint equation is similarly to the case of integer+ Dx2 (W )
.
(44) 270
∂u x x x
order nonlinear differential equations [25,26] , so we
have the adjoint equation to the nonlinear TFDE ( 1 )
For any generator X admitted by Eq. ( 1 ) and any 271
as Euler–Lagrange equation
solution
of this equation, we have:
272
δL
= 0.
(38)
δu

Considered the case of two independent variables
X̄ L + Dt (τ )L + Dx (ξ )L |(1) = 0.
(45) 273
t, x, and one dependent variable u(t, x), this fundaThis equality yields the conservation law
274
mental identity can be written as
δ
Dt (N t L) + Dx (N x L) = 0.
(46) 275
+ Dt N t + D x N x ,
X̄ + Dt (τ )l + Dx (ξ )l = W
δu
(39)
(
(
δ
(
(
(
where l is the identity operator, δu is the (
Euler-LagrangeEuler–
(
Lagrangeoperator, N t and N x are the Noether operators, X̄ is an appropriate prolongation for the Lie point
generator
3.2 Conservation laws
276
∂
∂
∂
∂
∂
+η
+ ηα0
+ ηx
X̄ = τ + ξ
In the previous subsection, we gave some basic defini- 277
∂t
∂x
∂u
∂ Dtα u
∂u x
tions. In this subsection, we will present the conserva- 278
∂
∂
+ ηx x
+ ηx x x
,
(40)
tion laws.
279
∂u x x
∂u x x x
Dt (C t ) + Dx (C x ) = 0,

pro

224
225

A conserved vector satisfies the following conservation
equation

257

cted

223

and

orre

221
222

3.1 Necessary preliminaries

unc

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297

− (−1)1 J

m−1

+ 3(b − 1)(a + b − 2)u a+b−3 u 2x
+ε+εmu

299

300

301
302

303

304

305







− Dx v(a + b−3)u a+b−2 u x + Dx2 vu a+b−1

+ Dx (Wi ) v(a + 3b − 3)u a+b−2
u x
−
D
x ux

− Dx vu a+b−1



+Dx2 (Wi ) vu a+b−1 ,

where i = 1, 2 and functions Wi are
(a + b − 3m + 2) t
ut
W1 = −u x , W2 = 2u −
α


− (a + b − m) x u x .

(48)

− (−1)1 J

(a + b − 3m + 2) t
ut
W1 = −u x , W2 = 2u −
α


− (a + b − m) x u x .

(49)

∂L
 α 
∂ 0 Dt u


∂L
Wi , Dt1  α 
∂ 0 Dt u

∂L
+ (−1)1 0 D α−2
(Wi )Dt1  α 
t
∂ 0 Dt u


2
2  ∂L 
− (−1) J Wi , Dt
∂ 0 Dtα u
= v 0 Dtα−1 (Wi ) + J (Wi , vt )
− vt 0 Dtα−2 (Wi ) − J (Wi , vtt ),

(50)

∂L
∂L
∂L
− Dx
+ Dx2
∂u x
∂u x x
∂u x x x
∂L
∂L
∂L
+ Dx2 (Wi )
+ Dx (Wi )
− Dx
∂u x x
∂u x x x
∂u x x x
 
= Wi v (a + b − 3)u a+b−2 u x x

Cix = ξ L + Wi

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312

(52)

(((
In this paper, we investigated (
time
fractionaltime((
fractionalnonlinear dispersive equation via Lie symmetries and conservation laws. We firstly obtained the
Lie point symmetries and perform symmetry reductions. Furthermore, the conservation laws are constructed for the first time in this paper. The obtained
results will serve as benchmark in the accuracy testing, comparison of numerical results. There are several
issues which need to be pursued furthers. For example, here we have used classical Lie symmetry method
for only two independent x, t and one dependent u
variables. It is not clear that, how to derive similar
results in the case of time FPDEs with more independent and dependent variables.
In addition, for thecon

b ua ub
servative form (u)αt + εu m + b1
(u
=
u a
xx x
0, one can write the potential system u = vx and
bua ub
εu m + b1
(u
= −vtα , it is also not clear that
u a
xx
whether there exists nonlocal symmetry. It is of interest
in general to study whether the method of investigating
PDEs can be extended to FPDEs. It is worthy of investigating further and these topics will be reported in the
future series of research works.
Acknowledgments The authors are thankful for the referees
useful suggestions which help us to improve the manuscript. This
work 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 ) .

CP Disp.:2015/5/15 Pages: 7 Layout: Medium

310

311

4 Concluding remarks and discussion

123
Journal: 11071 MS: 2156

309

where i = 1, 2 and functions Wi have the form

Also, when α ∈ (1, 2), we get the components of
conserved vectors
Cit = τ L + (−1)0 0 Dtα−1 (Wi )Dt0

308





+ Dx2 (Wi ) vu a+b−1 , (51)
− Dx vu a+b−1

= v0 Dtα−1 (Wi ) + J (Wi , vt ),
(47)
∂
L
∂
L
∂
L
Cix = ξ L + Wi
− Dx
+ Dx2
∂u x
∂u x x
∂u x x x
∂L
∂L
∂L
+ Dx2 (Wi )
+ Dx (Wi )
− Dx
∂u x x
∂u x x x
∂u x x x
 
= Wi v (a + b − 3)u a+b−2 u x x



307


+ Dx (Wi ) v(a + 3b − 3)u a+b−2
u x
−
D
x ux

∂L
Wi , Dt1  α 
∂ 0 Dt u


298

of

283

∂L
 α 
∂ 0 Dt u


pro



306

cted

282

Cit = τ L + (−1)0 0 Dtα−1 (Wi )Dt0

+ 3(b − 1)(a + b − 2)u a+b−3 u 2x



+ εmu m−1 − Dx v(a + b − 3)u a+b−2 u x


2
a+b−1
+ Dx vu

orre

281

For the case, when α ∈ (0, 1), using ( 42 ) and ( 44
) , one can get the components of conserved vectors

unc

280

313

314

315
316
317
318
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320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336

337
338
339
340
341
342
343

Time-fractional nonlinear dispersive equation

351
352
353
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356
357
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369
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of

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347

1. Bluman, G.W., Kumei, S.: Symmetries and Differential
Equations. Springer, New York ( 1989 )
2. Ibragimov, N.H. ( ed. ) : CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1–3. CRC Press, Boca
(
((FLRaton
Raton,
( 1994 )
(
3. Bluman, G.W., Cheviakov, A., Anco, S.: Applications
of Symmetry Methods to Partial Differential Equations.
Springer, New York ( 2010 )
4. Bluman, G.W., Anco, S.: Symmetry and Integration Methods for Differential Equations. Springer, New York ( 2002
)
5. Olver, P.J.: Application of Lie Group to Differential Equation. Springer, New York ( 1986 )
6. Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York ( 1982 )
7. Gang, G.W., et al.: Singular solitons, shock waves, and other
solutions to potential KdV equation. Nonlinear Dyn. 76 ,
1059–1068 ( 2014 )
8. Biswas, A., et al.: Solitons and conservation laws of
((((
Klein-GordonKlein–Gordonequation
with power law and
(
log law nonlinearities. Nonlinear Dyn. 73 , 2191–2196 (
2013 )
9. Biswas, A.: Solitary waves for power-law regularized longwave equation and R ( m, n ) equation. Nonlinear Dyn. 59 ,
423–426 ( 2010 )
10. Razborova, P., Kara, A.H., Biswas, A.:(Additional con(((
servation laws for (
RosenauCKdVCRLWRosenau–KdV–
(((
RLWequation with power law nonlinearity by Lie symmetry.
Nonlinear Dyn. 79 , 743–748 ( 2015 )
11. Gazizov, R.K., Kasatkin, A.A.: Construction of exact solutions for fractional order differential equations by the invariant subspace method. Comput. Math. Appl. 66 , 576–584 (
2013 )
12. Liu, H.Z.: Complete group classifications and symmetry
reductions of the fractional fifth-order KdV types of equations. Stud. Appl. Math. 131 , 317–330 ( 2013 )


13. Gazizov, 
R. 
K.,R.K.,Kasatkin,
A. 
A.,A.A.,Yu,
S.:

Lukashchuk, Continuous transformation groups of fractional differential equations, Vestnik, USATU 9 125–135 (
in Russian ) ( 2007 )
14. Buckwar, E., Luchko, Y.: Invariance of a partial differential
equation of fractional order under the Lie group of scaling
transformations. J. Math. Anal. Appl. 227 , 81–97 ( 1998 )
15. Djordjevic,
V.D.,
Atanackovic,
T.M.:
Similarity solutions to nonlinear heat conduction and
(
((
(((
Burgers/Korteweg-deVriesBurgers/Korteweg–
(((
(
deVriesfractional equations. J. Comput. Appl. Math.
212 , 701–714 ( 2008 )
16. Gazizov, R.K., Kasatkin, A.A., Lukashchuk, SYu.: Symmetry properties of fractional diffusion equations. Phys. Scr.
T136 , 014016 ( 2009 )
17. Sahadevan, R., Bakkyaraj, T.: Invariant analysis of time
fractional generalized Burgers and (
Korteweg-deKorteweg–
((((
deVries equations. J. Math. Anal. Appl. 393 , 341–347 (
2012 )
18. Wang, G.W., Liu, X.Q., Zhang, Y.Y.: Lie symmetry analysis
to the time fractional generalized fifth-order KdV equation.


Commun. Nonlinear Sci. Numer. 
Simulat.Simul.18
, 2321–
2326 ( 2013 )
19. Wang, G.W., Xu, T.Z., Feng, T.: Lie symmetry analysis
and explicit solutions of the time fractional fifth-Order KdV
equation. Plos One 9 ( 2 ) , e88336 ( 2014 )
20. Wang, G.W., Xu, T.Z.: Invariant analysis and exact solutions
(
(((
of nonlinear time fractional (
Sharma-Tasso-OlverSharma–
(((
Tasso–Olverequation by Lie group analysis. Nonlinear Dyn.
76 , 571–580 ( 2014 )



21. Wang, 
G. 
W.,G.W.,Xu,
T. Z.:T.Z.:Symmetry
properties and
explicit solutions of the nonlinear time fractional KdV equation, Bound. Value Probl. ( 2013 )
22. Huang, Q., Zhdanov, R.: Symmetries and exact solutions
(
(((
of the time fractional (
Harry-DymHarry–Dymequation
with

Riemann-Liouville derivative. 
PhysicaPhys.A
409 , 110–
118 ( 2014 )
23. Hu, J., Ye, Y., Shen, S., Zhang, J.: Lie symmetry analysis of
the time fractional KdV-type equation. Appl. Math. Comput.
233 , 439–444 ( 2014 )
24. Hashemi, M.S.: Group analysis and exact solutions of
(
(
(((
the time fractional (
Fokker-PlanckFokker–Planckequation.


PhysicaPhys.A
417 , 141–149 ( 2015 )

25. Lukashchuk, S.Y.: Conservation laws for time-fractional
subdiffusion and diffusion-wave equations. Nonlinear Dyn.
( 2015 ) . doi:10.1007/s11071-015-1906-7
26. Gazizov, R.K., Ibragimov, N.H., Lukashchuk, S.Y.: Nonlinear self-adjointness, conservation laws and exact solutions
of time-fractional Kompaneets equations. Commun. Non
linear Sci. Numer. 
Simulat.Simul.
( 2014 ) . doi:10.1016/j.
cnsns.2014.11.010
:M.S.:Lie symmetry


27. Wang, 
G. 
W.,G.W.,Hashemi,
M.S.

analysis and solitons solutions of time fractional K ( m, n )


equation,equation
(submitteds*)s*

:A.H.:Group analysis and

28. Wang, 
G. 
W.,G.W.,Kara,
A. 
H.

soliton solution of the time fractional generalized Burgers


equation,equation
(submitteds*)s*

29. Rosenau, P.: On nonalnalytic solitary waves formed by a
nonlinear dispersion. Phys. Lett. A 230 , 305–318 ( 1997 )
30. Rosenau, P.: Compact and noncompact dispersive patterns.
Phys. Lett. A 275 , 193–203 ( 2000 )
31. Rosenau, P.: On a model equation of traveling and stationary
compactons. Phys. Lett. A 356 , 44–50 ( 2006 )
32. Charalambous, K., Sophocleous, C.: Symmetry analysis for
a class of nonlinear dispersive equations. Commun. Nonlin
ear Sci. Numer. 
Simulat.Simul.22
, 1275–1287 ( 2015 )
33. Kiryakova, V. : Generalised Fractional Calculus and Applications. In: Pitman Res. Notes in Math., vol. 301, ( 1994
)
34. Liu, C.S.: Counterexamples on Jumarie’s two basic fractional calculus formulae. Commun. Nonlinear Sci. Numer.

Simulat.Simul.22
, 92–94 ( 2015 )

35. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. World Scientific, Singapore ( 2010 )
36. Podlubny, 
I
. :I.:Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their
applications,applications.Academic
press, New York ( 1998
((((
(
)

orre

345

References

unc

344

123
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425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
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444
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446
447
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450
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453
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