# The Semidirect Sum of Lie Algebras and Its Applications to C-KdV Hierarchy

Authors: Xia Dong; Tiecheng Xia; Desheng Li
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

By use of the loop algebraG
-
~, integrable coupling of C-KdV hierarchy and its bi-Hamiltonian structures are obtained by Tu scheme and the quadratic-form identity. The method can be used to produce the integrable coupling and its Hamiltonian structures to the other integrable systems.


---

## Body

## 1. Introduction

Integrable coupling is a new topic of the Soliton theory; especially, looking for the new Hamiltonian structure of integrable coupling is more important. The integrable coupling of some known integrable hierarchies is obtained. But their Hamiltonian structure has not been presented because there exists a limitation in trace identity till the quadratic-form identity [1] and the variational identity [2] are proposed. In this paper, a higher-dimensional Lie algebra G
- and the loop algebra G
-
~ are constructed [3, 4]. With the help of Tu scheme [5] and the quadratic-form identity, the integrable coupling of C-KdV hierarchy as well as its bi-Hamiltonian structures is produced.


## 2. Basic Principle of the Semidirect Sum of Lie Algebras

LetG be a linear space over real or complex number field F together with multiplication, for any x
,
y
,
z
∈
G
,
c
∈
F, if G satisfy(1)
distributive law(1)
(
x
+
y
)
z
=
x
z
+
y
z
,
z
(
x
+
y
)
=
z
x
+
z
y
;
(2)
multiplication commutativity(2)
c
(
x
y
)
=
(
c
x
)
y
=
x
(
c
y
)
.Then, G is called algebra.Lie algebraG is an algebra over number field F, if its multiplication satisfies the following:(1)
bilinearity(3)
[
c
x
+
c
′
y
,
z
]
=
c
[
x
,
z
]
+
c
′
[
y
,
z
]
;
[
z
,
c
x
+
c
′
y
]
=
c
[
z
,
x
]
+
c
′
[
z
,
y
]
;
(2)
anticommutative(4)
[
x
,
y
]
=
-
[
y
,
x
]
;
(3)
the Jacobi identity(5)
[
[
x
,
y
]
,
z
]
+
[
[
y
,
z
]
,
x
]
+
[
[
z
,
x
]
,
y
]
=
0
,where [
·
,
·
] denote the multiplication of G,  x
,
y
,
z
∈
G
,
c
,
c
′
∈
K. The multiplication of Lie algebra is called Lie product. One kind of the most important Lie algebras on integrable systems is A
n
-
1
=
{
M
n
×
n
∣
tr
⁡
M
n
×
n
=
0
}, where M
n
×
n denote matrix order n over number field K.I satisfies [
I
,
A
]
⊆
I for arbitrary Lie algebra A; then I is called Lie ideal.Lie algebraA is called simple Lie algebra if A has A and 0 as Lie ideal and without other Lie ideal. Semisimple Lie algebra R can be written as
(6)
R
=
⨁
i
A
i
,
where A
i is simple Lie algebra. We have already known that A
n
,
B
n
,
C
n
,
D
n, E
6,7
,
8
,
F
4, and G
2 are all semisimple Lie algebras which has been studied by Cartan long ago [5]. We also know that Lie algebra R can be written as
(7)
R
=
R
1
⨄
R
2
,
where R
1 is semisimple Lie algebras and R
2 is solvable Lie algebras [3, 6, 7] and ⨄ denote the semidirect sum. So we can apply the above basic principle to integrable coupling systems.


## 3. C-KdV Hierarchy

Firstly, let us recall the construction of the C-KdV hierarchy [8, 9]. Consider the basis of G
(8)
e
1
=
(
1
0
0
-
1
)
,
e
2
=
(
0
1
0
0
)
,
e
3
=
(
0
0
1
0
)
.
The loop algebra G
~ is presented as e
i
(
n
)
=
e
i
λ
n.The C-KdV spectral problem reads as(9)
ψ
x
=
U
ψ
,
λ
t
=
0
U
=
(
q
-
λ
2
-
r
1
-
q
+
λ
2
)
,
u
=
(
q
r
)
.
Upon setting V
=
Σ
m
≥
0
(
a
m
e
1
(
-
m
)
+
b
m
e
2
(
-
m
)
+
c
m
e
3
(
-
m
)
), solving the stationary zero curvature equation,
(10)
V
x
=
[
U
,
V
]
,
engenders
(11)
a
n
x
=
-
r
c
m
-
b
m
,
b
m
x
=
-
b
m
+
1
+
q
b
m
+
2
r
a
m
,
c
m
x
=
c
m
+
1
-
q
c
m
+
2
a
m
,
a
0
=
1
2
,
b
0
=
c
0
=
0
,
a
1
=
0
,
b
1
=
r
,
c
1
=
-
1
.
The compatibility conditions of the spectral problems
(12)
ψ
x
=
U
ψ
,
ψ
t
=
V
(
n
)
ψ
;
V
(
n
)
=
(
λ
n
V
)
+
+
1
2
c
n
+
1
e
1
(
0
)
,
n
≥
0
,
determine the C-KdV hierarchy of Soliton equations
(13)
u
t
=
(
q
r
)
t
=
(
0
-
∂
-
∂
0
)
(
a
n
+
1
-
c
n
+
1
)
=
J
δ
H
n
δ
u
,
where
(14)
H
n
=
a
n
+
1
n
,
(
n
≥
1
)
,
H
1
=
r
,
H
2
=
1
2
(
-
r
x
+
2
q
r
)
.


## 4. A New Integrable Coupling of the C-KdV Hierarchy

In what follows, we expand Lie algebraG into a bigger one as the following Lie algebra G
-:
(15)
h
1
=
(
e
1
0
0
e
1
)
,
h
2
=
(
e
2
0
0
e
2
)
,
h
3
=
(
e
3
0
0
e
3
)
,
h
4
=
(
0
e
1
0
0
)
,
h
5
=
(
0
e
2
0
0
)
,
h
6
=
(
0
e
3
0
0
)
.
We do this along with the following commutative relations:
(16)
[
h
1
,
h
2
]
=
2
h
2
,
[
h
1
,
h
3
]
=
-
2
h
3
,
[
h
1
,
h
5
]
=
2
h
5
,
[
h
1
,
h
6
]
=
-
2
h
6
,
[
h
2
,
h
3
]
=
h
1
,
[
h
2
,
h
4
]
=
-
2
h
5
,
[
h
2
,
h
6
]
=
h
4
,
[
h
3
,
h
4
]
=
2
h
6
,
[
h
3
,
h
5
]
=
-
h
4
,
[
h
1
,
h
4
]
=
[
h
2
,
h
5
]
=
[
h
3
,
h
6
]
=
[
h
4
,
h
5
]
=
[
h
4
,
h
6
]
=
[
h
5
,
h
6
]
=
0
.
Taking G
-
1
=
span
⁡
{
h
1
,
h
2
,
h
3
} and G
-
2
=
span
⁡
{
h
4
,
h
5
,
h
6
}, it is easy to verify that
(17)
G
-
=
G
-
1
⨄
G
-
2
,
G
≅
G
-
1
,
[
G
-
1
,
G
-
2
]
⊆
G
-
2
,
where G
-
1 is semisimple Lie algebras and G
-
2 is solvable Lie algebras [3, 6, 7].In terms of the Lie algebraG
-, we constructed the loop algebra G
-
~ as follows h
k
(
i
,
n
)
=
h
k
λ
2
n
+
i [4, 10], with the following commutative relations:
(18)
[
h
1
(
i
,
m
)
,
h
2
(
j
,
n
)
]
=
2
h
2
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
1
(
i
,
m
)
,
h
3
(
j
,
n
)
]
=
-
2
h
3
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
1
(
i
,
m
)
,
h
5
(
j
,
n
)
]
=
2
h
5
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
1
(
i
,
m
)
,
h
6
(
j
,
n
)
]
=
-
2
h
6
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
2
(
i
,
m
)
,
h
3
(
j
,
n
)
]
=
h
1
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
2
(
i
,
m
)
,
h
4
(
j
,
n
)
]
=
-
2
h
5
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
2
(
i
,
m
)
,
h
4
(
j
,
n
)
]
=
-
2
h
5
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
2
(
i
,
m
)
,
h
4
(
j
,
n
)
]
=
-
2
h
5
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
2
(
i
,
m
)
,
h
6
(
j
,
n
)
]
=
h
4
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
3
(
i
,
m
)
,
h
4
(
j
,
n
)
]
=
2
h
6
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
3
(
i
,
m
)
,
h
5
(
j
,
n
)
]
=
-
h
4
(
δ
i
j
,
m
+
n
+
ρ
i
j
)
[
h
1
(
i
,
m
)
,
h
4
(
j
,
n
)
]
=
[
h
2
(
i
,
m
)
,
h
5
(
j
,
n
)
]
=
0
[
h
3
(
i
,
m
)
,
h
6
(
j
,
n
)
]
=
[
h
4
(
i
,
m
)
,
h
5
(
j
,
n
)
]
=
0
[
h
4
(
i
,
m
)
,
h
6
(
j
,
n
)
]
=
[
h
5
(
i
,
m
)
,
h
6
(
j
,
n
)
]
=
0
.
δ
i
j
=
i
+
j, ρ
i
j
=
0, when i
+
j
<
2, and δ
i
j
=
0
,
ρ
i
j
=
1, when i
+
j
=
2. With the help of above equations, we consider an isospectral problem:
(19)
ψ
x
=
U
ψ
,
λ
t
=
0
.
U
=
-
1
2
h
1
(
1,0
)
+
q
2
h
1
(
0,0
)
-
r
h
2
(
0,0
)
+
h
3
(
0,0
)
+
u
1
h
4
(
0,0
)
+
u
2
h
5
(
0,0
)
Set
(20)
V
=
∑
m
≥
0
‍
∑
i
=
0
1
‍
(
a
(
i
,
m
)
h
1
(
i
,
-
m
)
+
b
(
i
,
m
)
h
2
(
i
,
-
m
)
hhhhihhhh
+
c
(
i
,
m
)
h
3
(
i
,
-
m
)
+
d
(
i
,
m
)
h
4
(
i
,
-
m
)
hhhhiihhhh
+
e
(
i
,
m
)
h
5
(
i
,
-
m
)
+
f
(
i
,
m
)
h
6
(
i
,
-
m
)
)
.
Solving the stationary zero curvature equation (10) permits that
(21)
a
x
(
0
,
m
)
=
-
r
c
(
0
,
m
)
-
b
(
0
,
m
)
,
a
x
(
1
,
m
)
=
-
r
c
(
1
,
m
)
-
b
(
1
,
m
)
,
b
x
(
0
,
m
)
=
-
b
(
1
,
m
+
1
)
+
q
b
(
0
,
m
)
+
2
r
a
(
0
,
m
)
,
b
x
(
1
,
m
)
=
-
b
(
0
,
m
)
+
q
b
(
1
,
m
)
+
2
r
a
(
1
,
m
)
,
c
x
(
0
,
m
)
=
c
(
1
,
m
+
1
)
-
q
c
(
0
,
m
)
+
2
a
(
0
,
m
)
,
c
x
(
1
,
m
)
=
c
(
0
,
m
)
-
q
c
(
1
,
m
)
+
2
a
(
1
,
m
)
,
d
x
(
0
,
m
)
=
-
r
f
(
0
,
m
)
-
e
(
0
,
m
)
+
u
2
c
(
0
,
m
)
,
d
x
(
1
,
m
)
=
-
r
f
(
1
,
m
)
-
e
(
1
,
m
)
+
u
2
c
(
1
,
m
)
,
e
x
(
0
,
m
)
=
-
e
(
1
,
m
+
1
)
+
q
e
(
0
,
m
)
+
2
r
d
(
0
,
m
)
+
2
u
1
b
(
0
,
m
)
-
2
u
2
a
(
0
,
m
)
,
e
x
(
1
,
m
)
=
-
e
(
0
,
m
)
+
q
e
(
1
,
m
)
+
2
r
d
(
1
,
m
)
+
2
u
1
b
(
1
,
m
)
-
2
u
2
a
(
1
,
m
)
,
f
x
(
0
,
m
)
=
f
(
1
,
m
+
1
)
-
q
f
(
0
,
m
)
+
2
d
(
0
,
m
)
-
2
u
1
c
(
0
,
m
)
,
f
x
(
1
,
m
)
=
f
(
0
,
m
)
-
q
f
(
1
,
m
)
+
2
d
(
1
,
m
)
-
2
u
1
c
(
1
,
m
)
,
a
(
0,0
)
=
v
1
,
d
(
0,0
)
=
v
2
,
b
(
0,0
)
=
c
(
0,0
)
=
e
(
0,0
)
=
f
(
0,0
)
=
0
,
a
(
1,0
)
=
b
(
1,0
)
=
c
(
1,0
)
=
d
(
1,0
)
=
e
(
1,0
)
=
f
(
1,0
)
=
0
,
a
(
0,1
)
=
2
v
1
r
,
b
(
0,1
)
=
2
v
1
q
r
+
2
v
3
r
-
2
v
1
r
x
,
c
(
0,1
)
=
-
2
v
1
q
-
2
v
3
,
d
(
0,1
)
=
2
v
2
r
-
2
v
1
u
2
,
e
(
0,1
)
=
2
v
2
q
r
-
2
v
1
u
2
q
+
2
v
4
r
+
4
v
1
r
u
1
-
2
v
3
u
2
-
2
v
2
r
x
+
2
v
1
u
2
x
,
f
(
0,1
)
=
-
2
v
2
q
-
2
v
4
-
4
v
1
u
1
,
a
(
1,1
)
=
v
3
,
b
(
1,1
)
=
2
v
1
r
,
c
(
1,1
)
=
-
2
v
1
,
(22)
d
(
1,1
)
=
v
4
,
e
(
1,1
)
=
2
v
2
r
-
2
v
1
u
2
,
f
(
1,1
)
=
-
2
v
2
,
where v
1, v
2, v
3, and v
4 are nonzero constants.Assume thatV
+
(
n
)
=
∑
m
=
0
n
‍
∑
i
=
0
1
‍
(
a
(
i
,
m
)
h
1
(
i
,
n
-
m
)
+
b
(
i
,
m
)
h
2
(
i
,
n
-
m
)
+
c
(
i
,
m
)
h
3
(
i
,
n
-
m
)
+
d
(
i
,
m
)
h
4
(
i
,
n
-
m
)
+
e
(
i
,
m
)
h
5
(
i
,
n
-
m
)
+
f
(
i
,
m
)
h
6
(
i
,
n
-
m
)
)
=
λ
2
n
V
-
V
-
(
n
); then (10) may be written as
(23)
-
V
+
x
(
n
)
+
[
U
,
V
+
(
n
)
]
=
V
-
x
(
n
)
-
[
U
,
V
-
(
n
)
]
.
A direct calculation reads
(24)
-
V
+
x
(
n
)
+
[
U
,
V
+
(
n
)
]
=
b
(
1
,
n
+
1
)
h
2
(
0,0
)
-
c
(
1
,
n
+
1
)
h
3
(
0,0
)
+
e
(
1
,
n
+
1
)
h
5
(
0,0
)
-
f
(
1
,
n
+
1
)
h
6
(
0,0
)
.
Take V
(
n
)
=
V
+
(
n
)
+
1
/
2
c
(
1
,
n
+
1
)
h
1
(
0,0
)
+
1
/
2
f
(
1
,
n
+
1
)
h
4
(
0,0
); then the zero curvature equation
(25)
U
t
-
V
x
(
n
)
+
[
U
,
V
(
n
)
]
=
0
is equivalent to(26)
u
t
=
(
q
r
u
1
u
2
)
t
=
(
0
0
0
∂
0
0
-
1
2
∂
0
0
-
1
2
∂
0
-
1
2
∂
∂
0
-
1
2
∂
0
)
·
(
a
(
1
,
n
+
1
)
+
d
(
1
,
n
+
1
)
-
c
(
1
,
n
+
1
)
-
f
(
1
,
n
+
1
)
2
a
(
1
,
n
+
1
)
c
(
1
,
n
+
1
)
)
=
J
1
(
a
(
1
,
n
+
1
)
+
d
(
1
,
n
+
1
)
-
c
(
1
,
n
+
1
)
-
f
(
1
,
n
+
1
)
2
a
(
1
,
n
+
1
)
c
(
1
,
n
+
1
)
)
=
(
0
0
-
∂
∂
2
+
∂
q
0
0
1
2
(
∂
2
-
q
∂
)
r
∂
+
∂
r
-
∂
-
1
2
(
∂
2
+
∂
q
)
1
2
∂
-
1
2
(
∂
2
+
∂
q
)
+
∂
u
1
-
∂
2
+
q
∂
r
∂
+
∂
r
1
2
(
∂
2
-
q
∂
)
+
u
1
∂
r
∂
+
∂
r
+
u
2
∂
+
∂
u
2
)
·
(
a
(
0
,
n
)
+
d
(
0
,
n
)
-
c
(
0
,
n
)
-
f
(
0
,
n
)
2
a
(
0
,
n
)
c
(
0
,
n
)
)
=
J
2
(
a
(
0
,
n
)
+
d
(
0
,
n
)
-
c
(
0
,
n
)
-
f
(
0
,
n
)
2
a
(
0
,
n
)
c
(
0
,
n
)
)
,where J
1 and J
2 are Hamiltonian operators.From (22), a recurrence operator L is obtained, which satisfies
(27)
(
a
(
1
,
n
+
1
)
+
d
(
1
,
n
+
1
)
-
c
(
1
,
n
+
1
)
-
f
(
1
,
n
+
1
)
2
a
(
1
,
n
+
1
)
c
(
1
,
n
+
1
)
)
=
L
(
a
(
0
,
n
)
+
d
(
0
,
n
)
-
c
(
0
,
n
)
-
f
(
0
,
n
)
2
a
(
0
,
n
)
c
(
0
,
n
)
)
,
where
(28)
L
=
(
-
∂
+
∂
-
1
q
∂
∂
-
1
r
∂
+
r
∂
-
1
u
1
∂
∂
-
1
u
2
∂
+
u
2
2
q
+
∂
0
-
2
u
1
0
0
-
∂
-
∂
-
1
q
∂
-
2
∂
-
1
r
∂
-
2
r
0
0
-
1
∂
+
q
)
.
It is easy to verify that
(29)
J
1
L
=
L
*
J
1
=
J
2
.
Therefore, the hierarchy (26) is Liouville integrable. Taking q
=
r
=
0, u
1
=
q
/
2, and u
2
=
-
r, (26) reduces to (13). According to the integrable theory, the hierarchy (26) is the integrable coupling of the C-KdV hierarchy. Furthermore, in the following part we will point out that there exist bi-Hamiltonian structures from constructing of Lie loop algebras.


## 5. The Bi-Hamiltonian Structures of the Hierarchy (26)

Let(30)
a
=
∑
i
=
1
6
‍
a
i
h
i
,
b
=
∑
i
=
1
6
‍
b
i
h
i
.
We have [
a
,
b
]=(
a
2
b
3
-
a
3
b
2
,
2
a
1
b
2
-
2
a
2
b
1
,
2
a
3
b
1
-
2
a
1
b
3
,
a
2
b
6
-
a
6
b
2
+
a
5
b
3
-
a
3
b
5
,
2
a
1
b
5
-
2
a
5
b
1
+
2
a
4
b
2
-
2
a
2
b
4
,
2
a
3
b
4
-
2
a
1
b
6
+
2
a
6
b
1
-
2
a
4
b
3
)
T.In what follows, from[
a
,
b
]
T
=
a
T
R
(
b
), we get
(31)
R
(
b
)
=
(
0
2
b
2
-
2
b
3
0
2
b
5
-
2
b
6
b
3
-
2
b
1
0
b
6
-
2
b
4
0
-
b
2
0
2
b
1
-
b
5
0
2
b
4
0
0
0
0
2
b
2
-
2
b
3
0
0
0
b
3
-
2
b
1
0
0
0
0
-
b
2
0
2
b
1
)
.
Solving the matrix equation R
(
b
)
F
=
-
(
R
(
b
)
F
)
T for F gives rise to
(32)
F
=
(
2
0
0
2
0
0
0
0
1
0
0
1
0
1
0
0
1
0
2
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
)
.
So we have {
a
,
b
}
=
a
T
F
b
=
(
2
a
1
+
2
a
4
)
b
1
+
(
a
3
+
a
6
)
b
2
+
(
a
2
+
a
5
)
b
3
+
2
a
1
b
4
+
a
3
b
5
+
a
2
b
6.A direct calculation reads(33)
{
V
,
∂
U
∂
λ
}
=
-
a
(
0
)
-
a
(
1
)
-
d
(
0
)
-
d
(
1
)
{
V
,
∂
U
∂
q
}
=
a
(
0
)
+
a
(
1
)
+
d
(
0
)
+
d
(
1
)
{
V
,
∂
U
∂
r
}
=
-
c
(
0
)
-
c
(
1
)
-
f
(
0
)
-
f
(
1
)
{
V
,
∂
U
∂
u
1
}
=
2
a
(
0
)
+
2
a
(
1
)
{
V
,
∂
U
∂
u
2
}
=
c
(
0
)
+
c
(
1
)
,
where a
(
0
)
=
∑
m
≥
0
‍
a
(
0
,
m
)
λ
-
2
m and a
(
1
)
=
∑
m
≥
0
‍
a
(
1
,
m
)
λ
-
2
m
+
1
⋯.Substituting the above formulas into the quadratic-form identity yields(34)
δ
δ
u
(
-
a
(
0
)
-
a
(
1
)
-
d
(
0
)
-
d
(
1
)
)
=
λ
-
γ
∂
∂
λ
λ
γ
(
a
(
0
)
+
a
(
1
)
+
d
(
0
)
+
d
(
1
)
-
c
(
0
)
-
c
(
1
)
-
f
(
0
)
-
f
(
1
)
2
a
(
0
)
+
2
a
(
1
)
c
(
0
)
+
c
(
1
)
)
.
Comparison of coefficients of λ
-
2
n
-
2 of both sides of the above equations leads to
(35)
δ
δ
u
(
-
a
(
0
,
n
+
1
)
-
d
(
0
,
n
+
1
)
)
=
(
-
2
n
-
1
+
γ
)
(
a
(
1
,
n
+
1
)
+
d
(
1
,
n
+
1
)
-
c
(
1
,
n
+
1
)
-
f
(
1
,
n
+
1
)
2
a
(
1
,
n
+
1
)
c
(
1
,
n
+
1
)
)
.
To fix the γ we take n
=
0 into the above equation and find γ
=
1.So(36)
(
a
(
1
,
n
+
1
)
+
d
(
1
,
n
+
1
)
-
c
(
1
,
n
+
1
)
-
f
(
1
,
n
+
1
)
2
a
(
1
,
n
+
1
)
c
(
1
,
n
+
1
)
)
=
δ
H
(
1
,
n
)
δ
u
,
H
(
1
,
n
)
=
a
(
0
,
n
+
1
)
+
d
(
0
,
n
+
1
)
2
n
.
Comparison of coefficients of λ
-
2
n
-
1 of both sides of the above equations gives
(37)
δ
δ
u
(
-
a
(
1
,
n
+
1
)
-
d
(
1
,
n
+
1
)
)
=
(
-
2
n
+
γ
)
(
a
(
0
,
n
)
+
d
(
0
,
n
)
-
c
(
0
,
n
)
-
f
(
0
,
n
)
2
a
(
0
,
n
)
c
(
0
,
n
)
)
.
In this situation, we have γ
=
0.So(38)
(
a
(
0
,
n
)
+
d
(
0
,
n
)
-
c
(
0
,
n
)
-
f
(
0
,
n
)
2
a
(
0
,
n
)
c
(
0
,
n
)
)
=
δ
H
(
2
,
n
)
δ
u
,
(39)
H
(
2
,
n
)
=
a
(
1
,
n
+
1
)
+
d
(
1
,
n
+
1
)
2
n
.
Thus the bi-Hamiltonian structures of the system (26) are given by
(40)
u
t
=
J
1
δ
H
(
1
,
n
)
δ
u
=
J
2
δ
H
(
2
,
n
)
δ
u
.
From the system (26), we easily give the following equations:
(41)
q
t
=
-
2
v
1
q
x
x
-
4
v
1
q
q
x
-
2
v
3
q
x
-
4
v
1
r
x
,
r
t
=
-
4
v
1
(
q
r
)
x
x
-
2
v
3
r
x
-
2
v
1
r
x
x
.


## 6. Conclusion

On the one hand, we obtain a new integrable coupling of C-KdV hierarchy by expanding a bigger Lie algebra. On the other hand, the bi-Hamiltonian structures of the integrable coupling of C-KdV hierarchy are observed by use of the quadratic-form identity.



---

*Source: 295068-2014-05-27.xml*