In this section, we will investigate the selfadjointness and conservation laws of (1) by Ibragimov’s theorem.
3.1. SelfAdjointness for (<xref reftype="dispformula" rid="EEq1">1</xref>) and (<xref reftype="dispformula" rid="EEq2">2</xref>)
Let
(20)
H
=
u
t
+
u
x
+
a
u
m
u
x
+
b
u
x
x
t
;
then we have the following formal Lagrangian for (1):
(21)
ℒ
1
=
w
(
u
t
+
u
x
+
a
u
m
u
x
+
b
u
x
x
t
)
,
where
w
=
w
(
x
,
t
)
is a new function. Computing the variational derivative of this formal Lagrangian, we obtain the following adjoint equation of (1):
(22)
H
*
=

w
t

w
x

a
w
x
u
m

b
w
x
x
t
=
0
.
Assume that
H
*

w
=
ϕ
(
t
,
x
,
u
)
=
λ
H
, for a certain function
λ
. Then it is easy to obtain that
ϕ
=
c
1
u
+
c
2
and
λ
=

ϕ
u
=

c
1
, where
c
1
,
c
2
are two arbitrary constants. This implies that (1) is quasiselfadjoint; especially, if
c
1
=
1
and
c
2
=
0
; that is,
ϕ
=
u
, then (1) is strictly selfadjoint (see [16] for the definitions). Thus, we have demonstrated the following statement.
Theorem 3.
Equation (1) is quasiselfadjoint with the substitution
w
=
c
1
u
+
c
2
, where
c
1
≠
0
,
c
2
are two arbitrary constants. Especially, (1) is strictly selfadjoint with the substitution
w
=
u
.
Next, we discuss the selfadjointness of (2), and then by the nonlinear selfadjointness and Ibragimov’s theorem on conservation laws we construct some conserved quantities of (1).
Similar to the above, we set
(23)
E
=
v
x
t
+
v
x
x
+
a
v
x
m
v
x
x
+
b
v
x
x
x
t
;
then the formal Lagrangian for (2) is
(24)
ℒ
2
=
w
(
v
x
t
+
v
x
x
+
a
v
x
m
v
x
x
+
b
v
x
x
x
t
)
and the adjoint equation of (1) is
(25)
E
*
=
w
x
t
+
w
x
x
+
a
w
x
x
v
x
m
+
a
m
w
x
v
x
m

1
v
x
x
+
b
w
x
x
x
t
=
0
.
According to [16], (2) will be nonlinearly selfadjoint if there exists a differential substitution
(26)
w
=
h
(
x
,
t
,
v
,
v
x
,
v
t
,
v
x
x
,
…
,
)
,
h
≠
0
,
involving a finite number of partial derivatives of
v
with respect to
x
and
t
, such that the equation
(27)
E
*

w
=
h
=
μ
0
E
+
μ
1
D
x
E
+
μ
2
D
t
E
+
μ
3
D
x
2
E
+
⋯
holds identically in the variables
t
,
x
,
u
,
u
x
,
u
t
,
…
, where
μ
0
,
μ
1
,
…
are undetermined variable coefficients different from
∞
on the solutions of (2). The highest order of derivatives involved in
u
is called the order of the differential substitution (26). The calculation provides the following result.
Theorem 4.
Equation (2) is nonlinearly selfadjoint. The minimal order of the differential substitution (26) satisfying (27) is equal to one and is given by the function
h
=
c
1
v
x
+
c
2
v
t
+
g
(
t
)
, where
c
1
and
c
2
are two constants and
g
(
t
)
is a differentiable function of
t
.
3.2. Conservation Laws of (<xref reftype="dispformula" rid="EEq1">1</xref>)
Now, we study the conservation laws of (1). We will first construct the conservation laws of (2) and then reduce them to the ones of (1). From the classical Lie group theory [17], we assume that a Lie point symmetry of (2) is a vector field
(28)
X
=
ξ
x
(
t
,
x
,
v
)
∂
∂
x
+
ξ
t
(
t
,
x
,
v
)
∂
∂
t
+
ρ
(
t
,
x
,
v
)
∂
∂
v
on
ℝ
+
×
ℝ
×
ℝ
such that
X
(
4
)
E
=
0
when
E
=
0
, where
E
is given by (23). Taking into account (2), the operator
X
(
4
)
is given as follows:
(29)
X
(
4
)
=
X
+
ρ
x
(
1
)
∂
∂
v
x
+
ρ
x
t
(
2
)
∂
∂
v
x
t
+
ρ
x
x
(
2
)
∂
∂
v
x
x
+
ρ
x
x
x
t
(
4
)
∂
∂
v
x
x
x
t
,
where
(30)
ρ
x
(
1
)
=
D
x
ρ

(
D
x
ξ
x
)
v
x

(
D
x
ξ
t
)
v
t
,
ρ
x
t
(
2
)
=
D
t
ρ
x
(
1
)

(
D
t
ξ
x
)
v
x
x

(
D
t
ξ
t
)
v
x
t
,
ρ
x
x
(
2
)
=
D
x
(
ρ
x
(
1
)
)

(
D
x
ξ
x
)
v
x
x

(
D
x
ξ
t
)
v
x
t
,
ρ
x
x
x
t
(
4
)
=
D
x
(
ρ
x
x
x
(
3
)
)

(
D
x
ξ
x
)
v
x
x
x
x

(
D
x
ξ
t
)
v
x
x
x
t
.
The condition
X
(
4
)
E

E
=
0
=
0
yields determining equations. Solving these determining equations, we can obtain the symmetries of (5) as follows:
(31)
X
1
=
∂
∂
x
,
X
2
=
∂
∂
t
,
X
3
=
f
(
t
)
∂
∂
v
with
f
(
t
)
as an arbitrary function. Ibragimov’s theorem on conservation laws yields the following conserved quantities of (2) for a general Lie symmetry
X
=
ξ
x
(
∂
/
∂
x
)
+
ξ
t
(
∂
/
∂
t
)
+
ρ
(
∂
/
∂
v
)
. Consider the following:
(32)
C
t
=
ξ
t
ℒ
2

W
(
w
x
2
+
b
w
x
x
x
4
)
+
W
x
(
w
2
+
b
w
x
x
4
)

W
x
x
(
b
w
x
4
)
+
W
x
x
x
(
b
w
4
)
,
C
x
=
ξ
x
ℒ
2
+
W
(
w
t
2

3
b
w
x
x
t
4
a
m
v
x
m

1
v
x
x
w

D
x
[
w
(
1
+
a
v
x
m
)
]
h
h
h
h
h
h
h
h
h
h
h
h

w
t
2

3
b
w
x
x
t
4
)
+
W
x
(
(
1
+
a
v
x
m
)
w
+
b
w
x
t
2
)
+
W
t
(
w
2
+
b
w
x
x
4
)

W
x
x
(
b
w
t
4
)

W
x
t
b
w
x
2
+
W
x
x
t
(
3
b
w
4
)
,
where
W
=
ρ

ξ
x
v
x

ξ
t
v
t
is the Lie characteristic function and
w
is given by the differential substitution
w
=
c
1
v
x
+
c
2
v
t
+
g
(
t
)
. Let us construct the conserved vector corresponding to the space translation group with the generator
(33)
X
1
=
∂
∂
x
.
In this case
W
=

v
x
. Taking account of the differential substitution
w
=
c
1
v
x
+
c
2
v
t
+
g
(
t
)
and
v
x
=
u
, from (32), we obtain the conserved quantities of (2), which can be transformed to the ones of (1) as follows:
(34)
C
1
t
=
1
4
c
2
(
2
u
u
t
+
b
u
u
x
x
t

2
u
x
∫
u
t
d
x
h
h
h
h
h
h

b
u
x
u
x
t
+
b
u
t
u
x
x

b
u
x
x
x
∫
u
t
d
x
)

1
4
b
g
(
t
)
u
x
x
x

1
2
g
(
t
)
u
x
=
1
2
c
2
(
2
u
u
t
+
b
u
u
x
x
t
+
b
u
t
u
x
x
)

1
4
c
2
D
x
(
b
u
u
x
t
+
2
u
∫
u
t
d
x
+
b
u
x
x
∫
u
t
d
x
)

1
4
D
x
[
b
g
(
t
)
u
x
x
+
2
g
(
t
)
u
]
,
C
1
x
=
1
4
c
2
(
3
b
u
u
x
t
t

2
b
u
x
u
t
t
+
b
u
t
u
x
t
+
b
u
x
x
∫
u
t
t
d
x
h
h
h
h
h
h
h
+
2
u
∫
u
t
t
d
x
+
4
a
u
m
+
1
u
t
h
h
h
h
h
h
h
+
4
u
u
t
+
2
u
t
∫
u
t
d
x
+
b
u
x
x
t
∫
u
t
d
x
)
+
b
4
[
g
(
t
)
u
x
x
t
+
g
′
(
t
)
u
x
x
]
+
c
1
u
(
u
x
+
u
t
+
a
u
m
u
x
+
b
u
x
x
t
)
+
1
2
[
u
g
′
(
t
)
+
g
(
t
)
u
t
]
=
c
1
u
(
u
x
+
u
t
+
a
u
m
u
x
+
b
u
x
x
t
)
+
1
2
c
2
(
b
u
u
x
t
t

b
u
x
u
t
t
+
2
a
u
m
+
1
u
t
+
2
u
u
t
)
+
1
4
c
2
D
t
(
b
u
u
x
t
+
2
u
∫
u
t
d
x
+
b
u
x
x
∫
u
t
d
x
)
+
1
4
D
t
[
b
g
(
t
)
u
x
x
+
2
g
(
t
)
u
]
,
which can be reduced to the form
(35)
C
1
t
=
1
2
c
2
(
2
u
u
t
+
b
u
u
x
x
t
+
b
u
t
u
x
x
)
,
C
1
x
=
c
2
2
(
b
u
u
x
t
t

b
u
x
u
t
t
+
2
a
u
m
+
1
u
t
+
2
u
u
t
)
.
Here we have used (1).
By the procedure analogous to that used above, we obtain the conserved quantities of (2) corresponding to the generators
X
2
and
X
3
, which can be reduced by the differential substitutions
w
=
c
1
v
x
+
c
2
v
t
+
g
(
t
)
and
v
x
=
u
to the conserved quantities of (1) as follows:
(36)
C
2
t
=
c
1
b
2
(
u
u
x
x
t

u
t
u
x
x
)
+
a
g
(
t
)
u
m
u
x
,
C
2
x
=
c
1
b
2
(
u
x
u
t
t

u
u
x
t
t
)

a
g
(
t
)
u
m
u
t
+
g
′
(
t
)
(
∫
u
t
d
x
+
u
+
b
u
x
t
)
,
C
3
t
=
f
(
t
)
(
c
1
u
x
+
c
2
u
t
)
,
C
3
x
=
f
(
t
)
(
c
1
u
x
+
c
2
u
t
)
(
1
+
a
u
m
)
+
b
f
(
t
)
(
c
1
u
x
x
t
+
c
2
u
x
t
t
)

f
′
(
t
)
(
c
1
u
+
c
2
∫
u
t
d
x
+
g
(
t
)
)
.
Here we have used (1).
Remark 5.
In fact, (2) admits a Lagrangian
(37)
L
2
=

1
2
v
x
v
t

1
2
v
x
2

a
(
m
+
1
)
(
m
+
2
)
v
x
m
+
2
+
b
2
v
x
t
v
x
x
and has the following Noether symmetry generators:
(38)
X
5
=
∂
∂
x
,
X
6
=
∂
∂
t
,
X
7
=
f
(
t
)
∂
∂
v
with
gangue
function
A
t
=
0
,
h
h
h
h
h
h
h
h
h
h
h
h
h
A
x
=

1
2
f
′
(
t
)
v
,
where
f
(
t
)
is a differentiable function in
t
. Noether’s theorem on conservation laws [18] can yield the conserved quantities of (2), which are the same as the ones in [11]. Those were constructed by the socalled “partial Noether approach.”