# Tri-Integrable Couplings of the Giachetti-Johnson Soliton Hierarchy as well as Their Hamiltonian Structure

Authors: Lei Wang; Ya-Ning Tang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Based on zero curvature equations from semidirect sums of Lie algebras, we construct tri-integrable couplings of the Giachetti-Johnson (GJ) hierarchy of soliton equations and establish Hamiltonian structures of the resulting tri-integrable couplings by the variational identity.


---

## Body

## 1. Introduction

Soliton theory is a power tool in expanding and describing the nonlinear phenomena in the fields of nonlinear optics, plasma physics, magnetic fluid, and so on. Searching for new integrable systems is an interesting and significant event and the subject of the integrable coupling is a new and important direction in soliton theory. Recently, various examples of bi-integrable couplings and tri-integrable couplings were introduced, which bring us inspiring thoughts and ideas to classify integrable systems with multicomponents and can generate even more diverse recursion operators in block matrix form.For a given integrable system of evolution type [1]:
(1)
u
t
=
K
(
u
)
=
K
(
x
,
t
,
u
,
u
x
,
u
x
x
,
…
)
,
where u is a column vector of dependent variables. It has an integrable coupling as follows:
(2)
u
-
t
=
[
u
t
v
t
]
=
K
-
(
u
-
)
=
[
K
(
u
)
S
(
u
,
v
)
]
,
where u and v denote two column vectors of additional dependent variables. The earliest paper on integrable couplings obtained by Lie algebras and the Tu scheme is the [2], which gave a direct method for establishing integrable couplings and the integrable couplings of TD hierarchy. Many papers have been dedicated to this topic [3–10]. And there are other ways to construct integrable couplings such as by using perturbations [11], enlarging spectral problems [12], and creating new loop algebras [13]. Professor Yu, especially, shows that the Kronecker product is an important and effective method to construct the discrete integrable couplings in [14] and presents a scheme for constructing real nonlinear integrable couplings of continuous soliton hierarchy in [15]. In 2012, we know that bi-integrable couplings were introduced and developed in [16]. Recently, bi-integrable couplings were further extended to tri-integrable couplings. The following enlarged triangular integrable system:
(3)
u
-
t
=
[
u
t
u
1
,
t
u
2
,
t
u
3
,
t
]
=
K
-
(
u
-
)
=
[
K
(
u
)
S
1
(
u
,
u
1
)
S
2
(
u
,
u
1
,
u
2
)
S
3
(
u
,
u
1
,
u
2
,
u
3
)
]
,
is called a tri-integrable coupling of the system (1) in [17, 18]. If at least one of S
1
(
u
,
u
1
), S
2
(
u
,
u
1
,
u
2
), and S
3
(
u
,
u
1
,
u
2
,
u
3
) is nonlinear with respect to any subvectors u
1, u
2, and u
3 of new dependent variables, we call this system (3) a nonlinear integrable coupling.To construct tri-integrable couplings, we need a class of triangular4
×
4 block matrices M
(
A
1
,
A
2
,
A
3
,
A
4
) with A
i
(
i
=
1
,
…
,
4
) being square matrices of the same order. Therefore the Lie algebra g
- has a semidirect sum decomposition:
(4)
g
-
=
g
⊕
g
c
,
in which g
=
{
M
(
A
1
,
0,0
,
0
)
∣
A
1
-arbitrary
}, g
c
=
{
M
(
0
,
A
2
,
A
3
,
A
4
)
∣
A
2
,
A
3
,
A
4
-arbitrary
}. g
- is non-semisimple because of g
c being a nontrivial ideal of g
-. The block A
1 corresponds to the original integrable system, and the other three blocks A
2, A
3, and A
4 are used to generate the supplementary vector fields S
1, S
2, and S
3 in (3) that we are looking for. Such presented Lie algebras establish a basis for generating nonlinear Hamiltonian tri-integrable couplings, while many other existing Lie algebras lead to linear Hamiltonian integrable couplings [5, 19–22].Four classes of block matrices were introduced in [17] and the Hamiltonian tri-integrable couplings of the AKNS hierarchy were constructed based on one of the four triangular block matrices. While in this paper, we would like to construct tri-integrable couplings of the Giachetti-Johnson (GJ) hierarchy based on other triangular block matrices as follows:
(5)
M
(
A
1
,
A
2
,
A
3
,
A
4
)
=
[
A
1
A
2
A
3
A
4
0
A
1
α
A
2
β
A
2
+
μ
A
3
0
0
A
1
μ
A
2
0
0
0
A
1
]
,
where A
i
(
i
=
1
,
…
,
4
) are square matrices of the same order and α, β, μ are arbitrary constants. Moreover, we also hope to generate the Hamiltonian structure of the resulting tri-integrable couplings.The rest of the paper is organized as follows. In the next section, we first recall the GJ soliton hierarchy; then we construct a kind of tri-integrable couplings of the Giachetti-Johnson (GJ) soliton hierarchy and furnish Hamiltonian structures for the resulting tri-integrable couplings by the corresponding variational identity. Moreover, we will show that the resulting tri-integrable couplings have a recursion relation. In the final section, conclusions will be given.


## 2. Tri-Integrable Couplings of the Giachetti-Johnson (GJ) Hierarchy

### 2.1. The Giachetti-Johnson (GJ) Hierarchy

We first recall the GJ soliton hierarchy as follows [23]:
(6)
ϕ
x
=
U
ϕ
,
U
=
U
(
u
,
λ
)
=
[
-
λ
+
s
q
r
λ
-
s
]
,
u
=
(
q
r
s
)
,
where λ is the spectral parameter, q, r, and s are three dependent variables. Upon setting
(7)
W
=
[
a
b
c
-
a
]
=
∑
i
≥
0
W
i
λ
-
i
=
∑
i
≥
0
(
a
i
b
i
c
i
-
a
i
)
λ
-
i
,
and choosing the initial data a
0
=
-
1, b
0
=
c
0
=
0, the stationary zero curvature equation W
x
=
[
U
,
W
] generates
(8)
b
i
+
1
=
-
1
2
b
i
,
x
+
s
b
i
-
q
a
i
,
c
i
+
1
=
1
2
c
i
,
x
+
s
c
i
-
r
a
i
,
a
i
+
1
,
x
=
-
r
b
i
+
1
+
q
c
i
+
1
,
hhhhhhhhhhhh
i
≥
0
.Using the compatibility conditions(9)
U
t
m
-
V
x
[
m
]
+
[
U
,
V
[
m
]
]
=
0
,
m
≥
0
,
with
(10)
V
[
m
]
=
(
λ
m
W
)
+
+
Δ
m
=
∑
i
≥
0
[
a
i
b
i
c
i
-
a
i
]
λ
m
-
i
+
[
a
m
+
1
0
0
-
a
m
+
1
]
,
we have the GJ hierarchy of soliton equations:
(11)
u
t
m
=
(
q
r
s
)
t
m
=
K
m
(
u
)
=
(
2
q
a
m
+
1
-
2
b
m
+
1
-
2
r
a
m
+
1
+
2
c
m
+
1
a
m
+
1
,
x
)
=
J
δ
H
m
δ
u
=
J
L
m
(
r
q
0
)
,
m
≥
0
.
The Hamiltonian operator J, the recursion operator L, and the Hamiltonian functionals in (11) are given by
(12)
J
=
(
0
-
2
q
2
0
-
r
-
q
r
∂
)
,
∂
=
∂
∂
x
,
(13)
L
=
(
1
2
∂
+
s
0
-
1
2
r
0
-
1
2
∂
+
s
-
1
2
q
∂
-
1
q
∂
+
2
∂
-
1
q
s
∂
-
1
r
∂
-
2
∂
-
1
r
s
0
)
,
(14)
H
m
=
∫
2
a
m
+
2
m
+
1
d
x
,
m
≥
0
.
Note J
L is not antisymmetric; therefore, the system (11) does not possess bi-Hamiltonian structures (the method of the verification is the same as the Appendix A of [24]) and is not Liouville integrable.


### 2.2. Tri-Integrable Couplings

Based on the special non-semisimple Lie algebrag
-, we choose the enlarged spectral matrix
(15)
U
-
=
U
-
(
u
-
,
λ
)
=
M
(
U
,
U
1
,
U
2
,
U
3
)
∈
g
-
,
u
-
=
(
u
T
,
u
1
T
,
u
2
T
,
u
3
T
)
T
,
with U being defined as in (6) and
(16)
U
1
=
[
v
1
v
2
v
3
-
v
1
]
,
U
2
=
[
v
4
v
5
v
6
-
v
4
]
,
U
3
=
[
v
7
v
8
v
9
-
v
7
]
,
u
1
=
(
v
2
,
v
3
,
v
1
)
T
,
u
2
=
(
v
5
,
v
6
,
v
4
)
T
,
u
3
=
(
v
8
,
v
9
,
v
7
)
T
,
where v
i, 1
≤
i
≤
9 are new dependent variables.To solve the enlarged stationary zero curvature equation(17)
W
-
x
=
[
U
-
,
W
-
]
,
we take a solution of the following type:
(18)
W
-
=
W
-
(
u
-
,
λ
)
=
M
(
W
,
W
1
,
W
2
,
W
3
)
∈
g
-
,
where W is defined by (7), and
(19)
W
1
=
W
1
(
u
,
u
1
,
λ
)
=
[
e
f
g
-
e
]
=
∑
i
≥
0
W
1
,
i
λ
-
i
,
W
2
=
W
2
(
u
,
u
1
,
u
2
,
λ
)
=
[
e
′
f
′
g
′
-
e
′
]
=
∑
i
≥
0
W
2
,
i
λ
-
i
,
W
3
=
W
3
(
u
,
u
1
,
u
2
,
u
3
,
λ
)
=
[
e
′′
f
′′
g
′′
-
e
′′
]
=
∑
i
≥
0
W
3
,
i
λ
-
i
.
Then, from (17), we immediately get
(20)
W
x
=
[
U
,
W
]
,
W
1
x
=
[
U
,
W
1
]
+
[
U
1
,
W
]
,
W
2
x
=
[
U
,
W
2
]
+
[
U
2
,
W
]
+
α
[
U
1
,
W
1
]
,
W
3
x
=
[
U
,
W
3
]
+
[
U
3
,
W
]
+
β
[
U
1
,
W
1
]
+
μ
[
U
1
,
W
2
]
+
μ
[
U
2
,
W
1
]
,
with the help of Maple, which leads to
(21)
b
x
=
-
2
(
λ
-
s
)
b
-
2
q
a
,
c
x
=
2
(
λ
-
s
)
c
+
2
r
a
,
a
x
=
-
r
b
+
q
c
,
f
x
=
-
2
(
λ
-
s
)
f
-
2
q
e
-
2
v
2
a
+
2
v
1
b
,
g
x
=
2
(
λ
-
s
)
g
+
2
r
e
+
2
v
3
a
-
2
v
1
c
,
e
x
=
-
r
f
+
q
g
-
v
3
b
+
v
2
c
,
f
x
′
=
2
α
v
1
f
-
2
α
v
2
e
-
2
(
λ
-
s
)
f
′
-
2
q
e
′
-
2
v
5
a
+
2
v
4
b
,
g
x
′
=
-
2
α
v
1
g
+
2
α
v
3
e
+
2
(
λ
-
s
)
g
′
+
2
r
e
′
+
2
v
6
a
-
2
v
4
c
,
e
x
′
=
-
α
v
3
f
+
α
v
2
g
-
r
f
′
+
q
g
′
-
v
6
b
+
v
5
c
,
f
x
′′
=
2
(
β
v
1
+
μ
v
4
)
f
-
2
(
β
v
2
+
μ
v
5
)
e
+
2
μ
v
1
f
′
-
2
μ
v
2
e
′
-
2
(
λ
-
s
)
f
′′
-
2
q
e
′′
-
2
v
8
a
+
2
v
7
b
,
g
x
′′
=
-
2
(
β
v
1
+
μ
v
4
)
g
+
2
(
β
v
3
+
μ
v
6
)
e
-
2
μ
v
1
g
′
+
2
μ
v
3
e
′
+
2
(
λ
-
s
)
g
′′
+
2
r
e
′′
+
2
v
9
a
-
2
v
7
c
,
e
x
′′
=
-
(
β
v
3
+
μ
v
6
)
f
+
(
β
v
2
+
μ
v
5
)
g
-
μ
v
3
f
′
+
μ
v
2
g
′
-
r
f
′′
+
q
g
′′
-
v
9
b
+
v
8
c
.
The corresponding recursion relations are
(22)
f
i
,
x
=
-
2
f
i
+
1
+
2
s
f
i
-
2
q
e
i
-
2
v
2
a
i
+
2
v
1
b
i
,
g
i
,
x
=
2
g
i
+
1
-
2
s
g
i
+
2
r
e
i
+
2
v
3
a
i
-
2
v
1
c
i
,
e
i
,
x
=
-
r
f
i
+
q
g
i
-
v
3
b
i
+
v
2
c
i
;
f
i
,
x
′
=
2
α
v
1
f
i
-
2
α
v
2
e
i
-
2
f
i
+
1
′
+
2
s
f
i
′
-
2
q
e
i
′
-
2
v
5
a
i
+
2
v
4
b
i
,
g
i
,
x
′
=
-
2
α
v
1
g
i
+
2
α
v
3
e
i
+
2
g
i
+
1
′
-
2
s
g
i
′
+
2
r
e
i
′
+
2
v
6
a
i
-
2
v
4
c
i
,
e
i
,
x
′
=
-
α
v
3
f
i
+
α
v
2
g
i
-
r
f
i
′
+
q
g
i
′
-
v
6
b
i
+
v
5
c
i
;
f
i
,
x
′′
=
2
(
β
v
1
+
μ
v
4
)
f
i
-
2
(
β
v
2
+
μ
v
5
)
e
i
+
2
μ
v
1
f
i
′
-
2
μ
v
2
e
i
′
-
2
f
i
+
1
′′
+
2
s
f
i
′′
-
2
q
e
i
′′
-
2
v
8
a
i
+
2
v
7
b
i
,
g
i
,
x
′′
=
-
2
(
β
v
1
+
μ
v
4
)
g
i
+
2
(
β
v
3
+
μ
v
6
)
e
i
-
2
μ
v
1
g
i
′
+
2
μ
v
3
e
i
′
+
2
g
i
+
1
′′
-
2
s
g
i
′′
+
2
r
e
i
′′
+
2
v
9
a
i
-
2
v
7
c
i
,
e
i
,
x
′
=
-
(
β
v
3
+
μ
v
6
)
f
i
+
(
β
v
2
+
μ
v
5
)
g
i
-
μ
v
3
f
i
′
+
μ
v
2
g
i
′
-
r
f
i
′′
+
q
g
i
′′
-
v
9
b
i
+
v
8
c
i
,
together with (8), where i
≥
0. We select the initial data to be
(23)
e
0
=
e
0
′
=
e
0
′′
=
-
1
,
f
0
=
g
0
=
f
0
′
=
g
0
′
=
f
0
′′
=
g
0
′′
=
0
.
Then the recursion relations (22) uniquely determine the sequence of f
i, g
i, e
i, f
i
′, g
i
′, e
i
′, f
i
′′, g
i
′′, e
i
′′, i
≥
1, recursively. It is direct to compute the first two sets of functions by Maple:
(24)
b
1
=
q
,
c
1
=
r
,
a
1
=
0
;
b
2
=
-
1
2
q
x
+
s
q
,
c
2
=
1
2
r
x
+
s
r
,
a
2
=
1
2
q
r
;
f
1
=
q
+
v
2
,
g
1
=
r
+
v
3
,
e
1
=
0
;
f
2
=
-
1
2
(
q
+
v
2
)
x
+
s
(
q
+
v
3
)
+
v
1
q
,
g
2
=
1
2
(
r
+
v
3
)
x
+
s
(
r
+
v
3
)
+
v
1
r
,
e
2
=
1
2
[
(
r
+
v
3
)
q
+
r
v
2
]
;
f
1
′
=
q
+
α
v
2
+
v
5
,
g
1
′
=
r
+
α
v
3
+
v
6
,
e
1
′
=
0
;
f
2
′
=
-
1
2
(
q
+
α
v
2
+
v
5
)
x
+
α
v
1
(
q
+
v
2
)
+
s
(
q
+
α
v
2
+
v
5
)
+
v
4
q
,
g
2
′
=
1
2
(
r
+
α
v
3
+
v
6
)
x
+
α
v
1
(
r
+
v
3
)
+
s
(
r
+
α
v
3
+
v
6
)
+
v
4
r
,
e
2
′
=
1
2
[
r
(
q
+
α
v
2
+
v
5
)
+
(
α
v
3
+
v
6
)
q
+
α
v
2
v
3
]
;
f
1
′′
=
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
+
q
,
g
1
′′
=
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
,
e
1
′′
=
0
;
f
2
′′
=
-
1
2
[
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
+
q
]
x
+
(
β
v
1
+
μ
v
4
)
(
q
+
v
2
)
+
μ
v
1
(
q
+
α
v
2
+
v
5
)
+
s
[
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
+
q
]
+
v
7
q
,
g
2
′′
=
1
2
[
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
]
x
+
(
β
v
1
+
μ
v
4
)
(
r
+
v
3
)
+
μ
v
1
(
r
+
α
v
3
+
v
6
)
+
s
[
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
]
+
v
7
r
,
e
2
′′
=
1
2
{
q
[
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
]
+
r
[
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
]
+
(
β
+
α
μ
)
v
2
v
3
+
μ
(
v
2
v
6
+
v
3
v
5
)
}
.For each integerm
≥
0, let us further introduce the enlarged Lax matrices:
(25)
V
-
[
m
]
=
M
(
V
[
m
]
,
V
1
[
m
]
,
V
2
[
m
]
,
V
3
[
m
]
)
∈
g
-
,
with V
[
m
] being defined as in (10), and
(26)
V
i
[
m
]
=
(
λ
m
W
i
)
+
+
Δ
m
i
,
i
=
1,2
,
3
,
in which
(27)
Δ
m
1
=
[
e
m
+
1
0
0
-
e
m
+
1
]
,
Δ
m
2
=
[
e
m
+
1
′
0
0
-
e
m
+
1
′
]
,
Δ
m
3
=
[
e
m
+
1
′′
0
0
-
e
m
+
1
′′
]
.
Then the enlarged zero curvature equation
(28)
U
-
t
m
-
V
-
x
[
m
]
+
[
U
-
,
V
-
[
m
]
]
=
0
,
generates
(29)
U
1
,
t
m
-
V
1
,
x
[
m
]
+
[
U
,
V
1
[
m
]
]
+
[
U
1
,
V
[
m
]
]
=
0
,
U
2
,
t
m
-
V
2
,
x
[
m
]
+
[
U
,
V
2
[
m
]
]
+
[
U
2
,
V
[
m
]
]
+
α
[
U
1
,
V
1
[
m
]
]
=
0
,
U
3
,
t
m
-
V
3
,
x
[
m
]
+
[
U
,
V
3
[
m
]
]
+
[
U
3
,
V
[
m
]
]
+
β
[
U
1
,
V
1
[
m
]
]
+
μ
[
U
1
,
V
2
[
m
]
]
+
μ
[
U
2
,
V
1
[
m
]
]
=
0
,
together with (9). This presents the supplementary systems:
(30)
v
-
t
m
=
S
m
(
u
-
)
=
[
S
1
,
m
(
u
,
u
1
)
S
2
,
m
(
u
,
u
1
,
u
2
)
S
3
,
m
(
u
,
u
1
,
u
2
,
u
3
)
]
,
v
-
=
(
v
2
,
v
3
,
v
1
,
v
5
,
v
6
,
v
4
,
v
8
,
v
9
,
v
7
)
T
,
m
≥
0
,
where
(31)
S
1
,
m
(
u
,
u
1
)
=
[
-
2
f
m
+
1
+
2
q
e
m
+
1
+
2
v
2
a
m
+
1
2
g
m
+
1
-
2
r
e
m
+
1
-
2
v
3
a
m
+
1
e
m
+
1
,
x
]
,
(32)
S
2
,
m
(
u
,
u
1
,
u
2
)
=
[
-
2
f
m
+
1
′
+
2
q
e
m
+
1
′
+
2
α
v
2
e
m
+
1
+
2
v
5
a
m
+
1
2
g
m
+
1
′
-
2
r
e
m
+
1
′
-
2
α
v
3
e
m
+
1
-
2
v
6
a
m
+
1
e
m
+
1
,
x
′
]
,
(33)
S
3
,
m
(
u
,
u
1
,
u
2
,
u
3
)
=
[
-
2
f
m
+
1
′′
+
2
q
e
m
+
1
′′
+
2
μ
v
2
e
m
+
1
′
+
2
(
β
v
2
+
μ
v
5
)
e
m
+
1
+
2
v
8
a
m
+
1
2
g
m
+
1
′′
-
2
r
e
m
+
1
′′
-
2
μ
v
3
e
m
+
1
′
-
2
(
β
v
3
+
μ
v
6
)
e
m
+
1
-
2
v
9
a
m
+
1
e
m
+
1
,
x
′′
]
.
In this way, the hierarchy from enlarged zero curvature equations can be written as
(34)
u
-
t
m
=
K
-
m
(
u
-
)
=
(
q
t
m
,
r
t
m
,
s
t
m
,
v
2
,
t
m
,
v
3
,
t
m
,
v
1
,
t
m
,
v
5
,
t
m
,
v
6
,
t
m
,
v
4
,
t
m
,
v
8
,
t
m
,
v
9
,
t
m
,
v
7
,
t
m
)
T
=
[
K
m
(
u
)
S
1
,
m
(
u
,
u
1
)
S
2
,
m
(
u
,
u
1
,
u
2
)
S
3
,
m
(
u
,
u
1
,
u
2
,
u
3
)
]
=
[
2
q
a
m
+
1
-
2
b
m
+
1
-
2
r
a
m
+
1
+
2
c
m
+
1
a
m
+
1
,
x
-
2
f
m
+
1
+
2
q
e
m
+
1
+
2
v
2
a
m
+
1
2
g
m
+
1
-
2
r
e
m
+
1
-
2
v
3
a
m
+
1
e
m
+
1
,
x
-
2
f
m
+
1
′
+
2
q
e
m
+
1
′
+
2
α
v
2
e
m
+
1
+
2
v
5
a
m
+
1
2
g
m
+
1
′
-
2
r
e
m
+
1
′
-
2
α
v
3
e
m
+
1
-
2
v
6
a
m
+
1
e
m
+
1
,
x
′
-
2
f
m
+
1
′′
+
2
q
e
m
+
1
′′
+
2
μ
v
2
e
m
+
1
′
+
2
(
β
v
2
+
μ
v
5
)
e
m
+
1
+
2
v
8
a
m
+
1
2
g
m
+
1
′′
-
2
r
e
m
+
1
′′
-
2
μ
v
3
e
m
+
1
′
-
2
(
β
v
3
+
μ
v
6
)
e
m
+
1
-
2
v
9
a
m
+
1
e
m
+
1
,
x
′′
]
,
for the given hierarchy (11).Obviously, takingv
i
=
0
(
i
=
1
,
…
,
9
), the system (34) reduces to the system (11). Therefore, the system (34) is a tri-integrable coupling of the system (11).


### 2.3. Hamiltonian Structures

As we all know, when an integrable system is generated, one of our primary tasks is to construct Hamiltonian structures of the resulting integrable system. In this subsection, we will generate Hamiltonian structures for the tri-integrable couplings (34) by applying the associated variational identity [25]:
(35)
δ
δ
u
-
∫
〈
∂
U
-
∂
λ
,
W
-
〉
d
x
=
λ
-
γ
∂
∂
λ
λ
γ
〈
∂
U
-
∂
u
-
,
W
-
〉
.
For the sake of convenience, we transform the Lie algebra g
- into a vector form by the mapping:
(36)
δ
:
g
-
⟶
R
12
,
A
⟼
(
a
1
,
…
,
a
12
)
T
,
A
=
M
(
A
1
,
A
2
,
A
3
,
A
4
)
∈
g
-
,
A
i
=
[
a
3
i
-
2
a
3
i
-
1
a
3
i
-
a
3
i
-
2
]
,
1
≤
i
≤
4
.
The mapping δ induces a Lie algebraic structure and the commutator [
·
,
·
] on R
12 reads
(37)
[
a
,
b
]
T
=
a
T
R
(
b
)
,
a
=
(
a
1
,
…
,
a
12
)
T
,
b
=
(
b
1
,
…
,
b
12
)
T
∈
R
12
,
where
(38)
R
(
b
)
=
M
(
R
1
,
R
2
,
R
3
,
R
4
)
,
R
i
=
[
0
2
b
3
i
-
1
-
2
b
3
i
b
3
i
-
2
b
3
i
-
2
0
-
b
3
i
-
1
0
2
b
3
i
-
2
]
,
1
≤
i
≤
4
.
A bilinear form on R
12 can be defined by 〈
a
,
b
〉
=
a
T
F
b, where F is a constant matrix. The symmetric property 〈
a
,
b
〉
=
〈
b
,
a
〉 and the Lie product 〈
a
,
[
b
,
c
]
〉
=
〈
[
a
,
b
]
,
c
〉 mean that F
T
=
F and F
(
R
(
b
)
)
T
=
-
R
(
b
)
F for all b
∈
R
12. This matrix equation leads to a linear system of equations on the elements of F. Solving the resulting system by Maple yields
(39)
F
=
[
η
1
η
2
η
3
η
4
η
2
α
η
3
+
β
η
4
μ
η
4
0
η
3
μ
η
4
0
0
η
4
0
0
0
]
⊗
[
2
0
0
0
0
1
0
1
0
]
,
where η
i, 1
≤
i
≤
4, are arbitrary constants. Therefore, a bilinear form on the semidirect sum g
- of Lie algebras can be determined by
(40)
〈
A
,
B
〉
g
-
=
〈
δ
(
A
)
,
δ
(
B
)
〉
R
12
=
(
a
1
,
a
2
,
…
,
a
12
)
F
(
b
1
,
b
2
,
…
,
b
12
)
T
,
with A
,
B
∈
g
-. It is easy to compute det
⁡
(
F
)
=
16
η
4
12
μ
6; obviously, when η
4 and μ are nonzero constants, the bilinear form (40) is nondegenerate. But η
1, η
2, η
3 can be arbitrary constants. Simply, we take η
1
=
η
2
=
η
3
=
0; therefore, to apply the variational identity (35), we compute that
(41)
〈
W
-
,
∂
U
-
∂
λ
〉
=
-
2
e
′′
η
4
,
〈
W
-
,
∂
U
-
∂
u
-
〉
=
(
g
′′
η
4
,
f
′′
η
4
,
2
e
′′
η
4
,
(
β
g
+
μ
g
′
)
η
4
,
(
β
f
+
μ
f
′
)
η
4
,
hhhhh
2
(
β
e
+
μ
e
′
)
η
4
,
μ
g
η
4
,
μ
f
η
4
,
2
μ
e
η
4
,
c
η
4
,
b
η
4
,
2
a
η
4
)
T
,
γ
=
-
λ
2
d
d
λ
ln
⁡
|
〈
W
-
,
W
-
〉
|
=
0
.Thus by (35), we obtain
(42)
δ
δ
u
-
∫
2
e
m
+
1
′′
η
4
m
d
x
=
(
g
m
′′
η
4
,
f
m
′′
η
4
,
2
e
m
′′
η
4
,
(
β
g
m
+
μ
g
m
′
)
η
4
,
(
β
f
m
+
μ
f
m
′
)
η
4
,
h
h
h
h
h
l
2
(
β
e
m
+
μ
e
m
′
)
η
4
,
μ
g
m
η
4
,
μ
f
m
η
4
,
h
h
h
h
h
l
2
μ
e
m
η
4
,
c
m
η
4
,
b
m
η
4
,
2
a
m
η
4
)
T
.Therefore, the tri-integrable couplings of the GJ hierarchy in (34) possess the following Hamiltonian structures:
(43)
u
-
t
m
=
K
-
m
(
u
-
)
=
J
-
δ
H
-
m
δ
u
-
,
m
≥
0
,
where the Hamiltonian operator J
- is given by
(44)
J
-
=
[
0
0
0
1
η
4
J
0
0
1
μ
η
4
J
J
1
0
1
μ
η
4
J
J
2
J
3
1
η
4
J
J
1
J
3
J
4
]
,
with J being the same as (12), and
(45)
J
1
=
1
η
4
[
0
0
v
2
0
0
-
v
3
-
v
2
v
3
0
]
,
J
2
=
1
μ
2
η
4
[
0
2
β
α
μ
v
2
-
β
q
-
2
β
0
-
(
α
μ
v
3
-
β
r
)
-
(
α
μ
v
2
-
β
q
)
α
μ
v
3
-
β
r
-
β
∂
]
,
J
3
=
1
η
4
[
0
0
v
5
0
0
-
v
6
-
v
5
v
6
0
]
,
J
4
=
1
η
4
[
0
0
v
8
0
0
-
v
9
-
v
8
v
9
0
]
,
and the Hamiltonian functionals are determined by
(46)
H
-
m
=
∫
2
e
m
+
2
′′
η
4
m
+
1
d
x
,
m
≥
0
.The hierarchy (43) can be rewritten as
(47)
u
-
t
m
=
K
-
m
=
J
-
δ
H
-
m
δ
u
-
=
J
-
L
-
δ
H
-
m
-
1
δ
u
-
=
J
-
L
-
m
[
[
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
]
η
4
[
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
+
q
]
η
4
0
[
β
(
r
+
v
3
)
+
μ
(
r
+
α
v
3
+
v
6
)
]
η
4
[
β
(
q
+
v
2
)
+
μ
(
q
+
α
v
2
+
v
5
)
]
η
4
0
μ
(
r
+
v
3
)
η
4
μ
(
q
+
v
2
)
η
4
0
r
η
4
q
η
4
0
]
,
m
≥
1
,
where the recursion operator L
- is given by
(48)
L
-
=
[
L
L
1
L
2
L
3
0
L
α
L
1
β
L
1
+
μ
L
2
0
0
L
μ
L
1
0
0
0
L
]
,
with L being the same as (13), and(49)
L
1
=
[
v
1
0
-
1
2
v
3
0
v
1
-
1
2
v
2
2
∂
-
1
(
q
v
1
+
v
2
s
)
+
∂
-
1
v
2
∂
-
2
∂
-
1
(
s
v
3
+
r
v
1
)
+
∂
-
1
v
3
∂
0
]
,
L
2
=
[
v
4
0
-
1
2
v
6
0
v
4
-
1
2
v
5
2
∂
-
1
(
α
v
1
v
2
+
q
v
4
+
s
v
5
)
+
∂
-
1
v
5
∂
-
2
∂
-
1
(
α
v
1
v
3
+
r
v
4
+
s
v
6
)
+
∂
-
1
v
6
∂
0
]
,
L
3
=
[
v
7
0
-
1
2
v
9
0
v
7
-
1
2
v
8
L
3
,
1
L
3
,
2
0
]
,with
(50)
L
3
,
1
=
2
∂
-
1
[
v
1
(
β
v
2
+
μ
v
5
)
+
μ
v
2
v
4
+
q
v
7
+
s
v
8
]
+
∂
-
1
v
8
∂
,
L
3
,
2
=
-
2
∂
-
1
[
v
1
(
β
v
3
+
μ
v
6
)
+
μ
v
3
v
4
+
r
v
7
+
s
v
9
]
+
∂
-
1
v
9
∂
.
Therefore, the hierarchy (34) possesses a recursion relation:
(51)
K
m
+
1
=
Φ
K
m
,
m
≥
0
,
where Φ
=
J
-
L
-
J
-
-
1. But J
-
L
- is not antisymmetric; therefore, the system (11) does not have bi-Hamiltonian structures (the way of the verification is the same as the Appendix A in [24]) and is not Liouville integrable.


## 2.1. The Giachetti-Johnson (GJ) Hierarchy

We first recall the GJ soliton hierarchy as follows [23]:
(6)
ϕ
x
=
U
ϕ
,
U
=
U
(
u
,
λ
)
=
[
-
λ
+
s
q
r
λ
-
s
]
,
u
=
(
q
r
s
)
,
where λ is the spectral parameter, q, r, and s are three dependent variables. Upon setting
(7)
W
=
[
a
b
c
-
a
]
=
∑
i
≥
0
W
i
λ
-
i
=
∑
i
≥
0
(
a
i
b
i
c
i
-
a
i
)
λ
-
i
,
and choosing the initial data a
0
=
-
1, b
0
=
c
0
=
0, the stationary zero curvature equation W
x
=
[
U
,
W
] generates
(8)
b
i
+
1
=
-
1
2
b
i
,
x
+
s
b
i
-
q
a
i
,
c
i
+
1
=
1
2
c
i
,
x
+
s
c
i
-
r
a
i
,
a
i
+
1
,
x
=
-
r
b
i
+
1
+
q
c
i
+
1
,
hhhhhhhhhhhh
i
≥
0
.Using the compatibility conditions(9)
U
t
m
-
V
x
[
m
]
+
[
U
,
V
[
m
]
]
=
0
,
m
≥
0
,
with
(10)
V
[
m
]
=
(
λ
m
W
)
+
+
Δ
m
=
∑
i
≥
0
[
a
i
b
i
c
i
-
a
i
]
λ
m
-
i
+
[
a
m
+
1
0
0
-
a
m
+
1
]
,
we have the GJ hierarchy of soliton equations:
(11)
u
t
m
=
(
q
r
s
)
t
m
=
K
m
(
u
)
=
(
2
q
a
m
+
1
-
2
b
m
+
1
-
2
r
a
m
+
1
+
2
c
m
+
1
a
m
+
1
,
x
)
=
J
δ
H
m
δ
u
=
J
L
m
(
r
q
0
)
,
m
≥
0
.
The Hamiltonian operator J, the recursion operator L, and the Hamiltonian functionals in (11) are given by
(12)
J
=
(
0
-
2
q
2
0
-
r
-
q
r
∂
)
,
∂
=
∂
∂
x
,
(13)
L
=
(
1
2
∂
+
s
0
-
1
2
r
0
-
1
2
∂
+
s
-
1
2
q
∂
-
1
q
∂
+
2
∂
-
1
q
s
∂
-
1
r
∂
-
2
∂
-
1
r
s
0
)
,
(14)
H
m
=
∫
2
a
m
+
2
m
+
1
d
x
,
m
≥
0
.
Note J
L is not antisymmetric; therefore, the system (11) does not possess bi-Hamiltonian structures (the method of the verification is the same as the Appendix A of [24]) and is not Liouville integrable.


## 2.2. Tri-Integrable Couplings

Based on the special non-semisimple Lie algebrag
-, we choose the enlarged spectral matrix
(15)
U
-
=
U
-
(
u
-
,
λ
)
=
M
(
U
,
U
1
,
U
2
,
U
3
)
∈
g
-
,
u
-
=
(
u
T
,
u
1
T
,
u
2
T
,
u
3
T
)
T
,
with U being defined as in (6) and
(16)
U
1
=
[
v
1
v
2
v
3
-
v
1
]
,
U
2
=
[
v
4
v
5
v
6
-
v
4
]
,
U
3
=
[
v
7
v
8
v
9
-
v
7
]
,
u
1
=
(
v
2
,
v
3
,
v
1
)
T
,
u
2
=
(
v
5
,
v
6
,
v
4
)
T
,
u
3
=
(
v
8
,
v
9
,
v
7
)
T
,
where v
i, 1
≤
i
≤
9 are new dependent variables.To solve the enlarged stationary zero curvature equation(17)
W
-
x
=
[
U
-
,
W
-
]
,
we take a solution of the following type:
(18)
W
-
=
W
-
(
u
-
,
λ
)
=
M
(
W
,
W
1
,
W
2
,
W
3
)
∈
g
-
,
where W is defined by (7), and
(19)
W
1
=
W
1
(
u
,
u
1
,
λ
)
=
[
e
f
g
-
e
]
=
∑
i
≥
0
W
1
,
i
λ
-
i
,
W
2
=
W
2
(
u
,
u
1
,
u
2
,
λ
)
=
[
e
′
f
′
g
′
-
e
′
]
=
∑
i
≥
0
W
2
,
i
λ
-
i
,
W
3
=
W
3
(
u
,
u
1
,
u
2
,
u
3
,
λ
)
=
[
e
′′
f
′′
g
′′
-
e
′′
]
=
∑
i
≥
0
W
3
,
i
λ
-
i
.
Then, from (17), we immediately get
(20)
W
x
=
[
U
,
W
]
,
W
1
x
=
[
U
,
W
1
]
+
[
U
1
,
W
]
,
W
2
x
=
[
U
,
W
2
]
+
[
U
2
,
W
]
+
α
[
U
1
,
W
1
]
,
W
3
x
=
[
U
,
W
3
]
+
[
U
3
,
W
]
+
β
[
U
1
,
W
1
]
+
μ
[
U
1
,
W
2
]
+
μ
[
U
2
,
W
1
]
,
with the help of Maple, which leads to
(21)
b
x
=
-
2
(
λ
-
s
)
b
-
2
q
a
,
c
x
=
2
(
λ
-
s
)
c
+
2
r
a
,
a
x
=
-
r
b
+
q
c
,
f
x
=
-
2
(
λ
-
s
)
f
-
2
q
e
-
2
v
2
a
+
2
v
1
b
,
g
x
=
2
(
λ
-
s
)
g
+
2
r
e
+
2
v
3
a
-
2
v
1
c
,
e
x
=
-
r
f
+
q
g
-
v
3
b
+
v
2
c
,
f
x
′
=
2
α
v
1
f
-
2
α
v
2
e
-
2
(
λ
-
s
)
f
′
-
2
q
e
′
-
2
v
5
a
+
2
v
4
b
,
g
x
′
=
-
2
α
v
1
g
+
2
α
v
3
e
+
2
(
λ
-
s
)
g
′
+
2
r
e
′
+
2
v
6
a
-
2
v
4
c
,
e
x
′
=
-
α
v
3
f
+
α
v
2
g
-
r
f
′
+
q
g
′
-
v
6
b
+
v
5
c
,
f
x
′′
=
2
(
β
v
1
+
μ
v
4
)
f
-
2
(
β
v
2
+
μ
v
5
)
e
+
2
μ
v
1
f
′
-
2
μ
v
2
e
′
-
2
(
λ
-
s
)
f
′′
-
2
q
e
′′
-
2
v
8
a
+
2
v
7
b
,
g
x
′′
=
-
2
(
β
v
1
+
μ
v
4
)
g
+
2
(
β
v
3
+
μ
v
6
)
e
-
2
μ
v
1
g
′
+
2
μ
v
3
e
′
+
2
(
λ
-
s
)
g
′′
+
2
r
e
′′
+
2
v
9
a
-
2
v
7
c
,
e
x
′′
=
-
(
β
v
3
+
μ
v
6
)
f
+
(
β
v
2
+
μ
v
5
)
g
-
μ
v
3
f
′
+
μ
v
2
g
′
-
r
f
′′
+
q
g
′′
-
v
9
b
+
v
8
c
.
The corresponding recursion relations are
(22)
f
i
,
x
=
-
2
f
i
+
1
+
2
s
f
i
-
2
q
e
i
-
2
v
2
a
i
+
2
v
1
b
i
,
g
i
,
x
=
2
g
i
+
1
-
2
s
g
i
+
2
r
e
i
+
2
v
3
a
i
-
2
v
1
c
i
,
e
i
,
x
=
-
r
f
i
+
q
g
i
-
v
3
b
i
+
v
2
c
i
;
f
i
,
x
′
=
2
α
v
1
f
i
-
2
α
v
2
e
i
-
2
f
i
+
1
′
+
2
s
f
i
′
-
2
q
e
i
′
-
2
v
5
a
i
+
2
v
4
b
i
,
g
i
,
x
′
=
-
2
α
v
1
g
i
+
2
α
v
3
e
i
+
2
g
i
+
1
′
-
2
s
g
i
′
+
2
r
e
i
′
+
2
v
6
a
i
-
2
v
4
c
i
,
e
i
,
x
′
=
-
α
v
3
f
i
+
α
v
2
g
i
-
r
f
i
′
+
q
g
i
′
-
v
6
b
i
+
v
5
c
i
;
f
i
,
x
′′
=
2
(
β
v
1
+
μ
v
4
)
f
i
-
2
(
β
v
2
+
μ
v
5
)
e
i
+
2
μ
v
1
f
i
′
-
2
μ
v
2
e
i
′
-
2
f
i
+
1
′′
+
2
s
f
i
′′
-
2
q
e
i
′′
-
2
v
8
a
i
+
2
v
7
b
i
,
g
i
,
x
′′
=
-
2
(
β
v
1
+
μ
v
4
)
g
i
+
2
(
β
v
3
+
μ
v
6
)
e
i
-
2
μ
v
1
g
i
′
+
2
μ
v
3
e
i
′
+
2
g
i
+
1
′′
-
2
s
g
i
′′
+
2
r
e
i
′′
+
2
v
9
a
i
-
2
v
7
c
i
,
e
i
,
x
′
=
-
(
β
v
3
+
μ
v
6
)
f
i
+
(
β
v
2
+
μ
v
5
)
g
i
-
μ
v
3
f
i
′
+
μ
v
2
g
i
′
-
r
f
i
′′
+
q
g
i
′′
-
v
9
b
i
+
v
8
c
i
,
together with (8), where i
≥
0. We select the initial data to be
(23)
e
0
=
e
0
′
=
e
0
′′
=
-
1
,
f
0
=
g
0
=
f
0
′
=
g
0
′
=
f
0
′′
=
g
0
′′
=
0
.
Then the recursion relations (22) uniquely determine the sequence of f
i, g
i, e
i, f
i
′, g
i
′, e
i
′, f
i
′′, g
i
′′, e
i
′′, i
≥
1, recursively. It is direct to compute the first two sets of functions by Maple:
(24)
b
1
=
q
,
c
1
=
r
,
a
1
=
0
;
b
2
=
-
1
2
q
x
+
s
q
,
c
2
=
1
2
r
x
+
s
r
,
a
2
=
1
2
q
r
;
f
1
=
q
+
v
2
,
g
1
=
r
+
v
3
,
e
1
=
0
;
f
2
=
-
1
2
(
q
+
v
2
)
x
+
s
(
q
+
v
3
)
+
v
1
q
,
g
2
=
1
2
(
r
+
v
3
)
x
+
s
(
r
+
v
3
)
+
v
1
r
,
e
2
=
1
2
[
(
r
+
v
3
)
q
+
r
v
2
]
;
f
1
′
=
q
+
α
v
2
+
v
5
,
g
1
′
=
r
+
α
v
3
+
v
6
,
e
1
′
=
0
;
f
2
′
=
-
1
2
(
q
+
α
v
2
+
v
5
)
x
+
α
v
1
(
q
+
v
2
)
+
s
(
q
+
α
v
2
+
v
5
)
+
v
4
q
,
g
2
′
=
1
2
(
r
+
α
v
3
+
v
6
)
x
+
α
v
1
(
r
+
v
3
)
+
s
(
r
+
α
v
3
+
v
6
)
+
v
4
r
,
e
2
′
=
1
2
[
r
(
q
+
α
v
2
+
v
5
)
+
(
α
v
3
+
v
6
)
q
+
α
v
2
v
3
]
;
f
1
′′
=
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
+
q
,
g
1
′′
=
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
,
e
1
′′
=
0
;
f
2
′′
=
-
1
2
[
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
+
q
]
x
+
(
β
v
1
+
μ
v
4
)
(
q
+
v
2
)
+
μ
v
1
(
q
+
α
v
2
+
v
5
)
+
s
[
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
+
q
]
+
v
7
q
,
g
2
′′
=
1
2
[
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
]
x
+
(
β
v
1
+
μ
v
4
)
(
r
+
v
3
)
+
μ
v
1
(
r
+
α
v
3
+
v
6
)
+
s
[
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
]
+
v
7
r
,
e
2
′′
=
1
2
{
q
[
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
]
+
r
[
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
]
+
(
β
+
α
μ
)
v
2
v
3
+
μ
(
v
2
v
6
+
v
3
v
5
)
}
.For each integerm
≥
0, let us further introduce the enlarged Lax matrices:
(25)
V
-
[
m
]
=
M
(
V
[
m
]
,
V
1
[
m
]
,
V
2
[
m
]
,
V
3
[
m
]
)
∈
g
-
,
with V
[
m
] being defined as in (10), and
(26)
V
i
[
m
]
=
(
λ
m
W
i
)
+
+
Δ
m
i
,
i
=
1,2
,
3
,
in which
(27)
Δ
m
1
=
[
e
m
+
1
0
0
-
e
m
+
1
]
,
Δ
m
2
=
[
e
m
+
1
′
0
0
-
e
m
+
1
′
]
,
Δ
m
3
=
[
e
m
+
1
′′
0
0
-
e
m
+
1
′′
]
.
Then the enlarged zero curvature equation
(28)
U
-
t
m
-
V
-
x
[
m
]
+
[
U
-
,
V
-
[
m
]
]
=
0
,
generates
(29)
U
1
,
t
m
-
V
1
,
x
[
m
]
+
[
U
,
V
1
[
m
]
]
+
[
U
1
,
V
[
m
]
]
=
0
,
U
2
,
t
m
-
V
2
,
x
[
m
]
+
[
U
,
V
2
[
m
]
]
+
[
U
2
,
V
[
m
]
]
+
α
[
U
1
,
V
1
[
m
]
]
=
0
,
U
3
,
t
m
-
V
3
,
x
[
m
]
+
[
U
,
V
3
[
m
]
]
+
[
U
3
,
V
[
m
]
]
+
β
[
U
1
,
V
1
[
m
]
]
+
μ
[
U
1
,
V
2
[
m
]
]
+
μ
[
U
2
,
V
1
[
m
]
]
=
0
,
together with (9). This presents the supplementary systems:
(30)
v
-
t
m
=
S
m
(
u
-
)
=
[
S
1
,
m
(
u
,
u
1
)
S
2
,
m
(
u
,
u
1
,
u
2
)
S
3
,
m
(
u
,
u
1
,
u
2
,
u
3
)
]
,
v
-
=
(
v
2
,
v
3
,
v
1
,
v
5
,
v
6
,
v
4
,
v
8
,
v
9
,
v
7
)
T
,
m
≥
0
,
where
(31)
S
1
,
m
(
u
,
u
1
)
=
[
-
2
f
m
+
1
+
2
q
e
m
+
1
+
2
v
2
a
m
+
1
2
g
m
+
1
-
2
r
e
m
+
1
-
2
v
3
a
m
+
1
e
m
+
1
,
x
]
,
(32)
S
2
,
m
(
u
,
u
1
,
u
2
)
=
[
-
2
f
m
+
1
′
+
2
q
e
m
+
1
′
+
2
α
v
2
e
m
+
1
+
2
v
5
a
m
+
1
2
g
m
+
1
′
-
2
r
e
m
+
1
′
-
2
α
v
3
e
m
+
1
-
2
v
6
a
m
+
1
e
m
+
1
,
x
′
]
,
(33)
S
3
,
m
(
u
,
u
1
,
u
2
,
u
3
)
=
[
-
2
f
m
+
1
′′
+
2
q
e
m
+
1
′′
+
2
μ
v
2
e
m
+
1
′
+
2
(
β
v
2
+
μ
v
5
)
e
m
+
1
+
2
v
8
a
m
+
1
2
g
m
+
1
′′
-
2
r
e
m
+
1
′′
-
2
μ
v
3
e
m
+
1
′
-
2
(
β
v
3
+
μ
v
6
)
e
m
+
1
-
2
v
9
a
m
+
1
e
m
+
1
,
x
′′
]
.
In this way, the hierarchy from enlarged zero curvature equations can be written as
(34)
u
-
t
m
=
K
-
m
(
u
-
)
=
(
q
t
m
,
r
t
m
,
s
t
m
,
v
2
,
t
m
,
v
3
,
t
m
,
v
1
,
t
m
,
v
5
,
t
m
,
v
6
,
t
m
,
v
4
,
t
m
,
v
8
,
t
m
,
v
9
,
t
m
,
v
7
,
t
m
)
T
=
[
K
m
(
u
)
S
1
,
m
(
u
,
u
1
)
S
2
,
m
(
u
,
u
1
,
u
2
)
S
3
,
m
(
u
,
u
1
,
u
2
,
u
3
)
]
=
[
2
q
a
m
+
1
-
2
b
m
+
1
-
2
r
a
m
+
1
+
2
c
m
+
1
a
m
+
1
,
x
-
2
f
m
+
1
+
2
q
e
m
+
1
+
2
v
2
a
m
+
1
2
g
m
+
1
-
2
r
e
m
+
1
-
2
v
3
a
m
+
1
e
m
+
1
,
x
-
2
f
m
+
1
′
+
2
q
e
m
+
1
′
+
2
α
v
2
e
m
+
1
+
2
v
5
a
m
+
1
2
g
m
+
1
′
-
2
r
e
m
+
1
′
-
2
α
v
3
e
m
+
1
-
2
v
6
a
m
+
1
e
m
+
1
,
x
′
-
2
f
m
+
1
′′
+
2
q
e
m
+
1
′′
+
2
μ
v
2
e
m
+
1
′
+
2
(
β
v
2
+
μ
v
5
)
e
m
+
1
+
2
v
8
a
m
+
1
2
g
m
+
1
′′
-
2
r
e
m
+
1
′′
-
2
μ
v
3
e
m
+
1
′
-
2
(
β
v
3
+
μ
v
6
)
e
m
+
1
-
2
v
9
a
m
+
1
e
m
+
1
,
x
′′
]
,
for the given hierarchy (11).Obviously, takingv
i
=
0
(
i
=
1
,
…
,
9
), the system (34) reduces to the system (11). Therefore, the system (34) is a tri-integrable coupling of the system (11).


## 2.3. Hamiltonian Structures

As we all know, when an integrable system is generated, one of our primary tasks is to construct Hamiltonian structures of the resulting integrable system. In this subsection, we will generate Hamiltonian structures for the tri-integrable couplings (34) by applying the associated variational identity [25]:
(35)
δ
δ
u
-
∫
〈
∂
U
-
∂
λ
,
W
-
〉
d
x
=
λ
-
γ
∂
∂
λ
λ
γ
〈
∂
U
-
∂
u
-
,
W
-
〉
.
For the sake of convenience, we transform the Lie algebra g
- into a vector form by the mapping:
(36)
δ
:
g
-
⟶
R
12
,
A
⟼
(
a
1
,
…
,
a
12
)
T
,
A
=
M
(
A
1
,
A
2
,
A
3
,
A
4
)
∈
g
-
,
A
i
=
[
a
3
i
-
2
a
3
i
-
1
a
3
i
-
a
3
i
-
2
]
,
1
≤
i
≤
4
.
The mapping δ induces a Lie algebraic structure and the commutator [
·
,
·
] on R
12 reads
(37)
[
a
,
b
]
T
=
a
T
R
(
b
)
,
a
=
(
a
1
,
…
,
a
12
)
T
,
b
=
(
b
1
,
…
,
b
12
)
T
∈
R
12
,
where
(38)
R
(
b
)
=
M
(
R
1
,
R
2
,
R
3
,
R
4
)
,
R
i
=
[
0
2
b
3
i
-
1
-
2
b
3
i
b
3
i
-
2
b
3
i
-
2
0
-
b
3
i
-
1
0
2
b
3
i
-
2
]
,
1
≤
i
≤
4
.
A bilinear form on R
12 can be defined by 〈
a
,
b
〉
=
a
T
F
b, where F is a constant matrix. The symmetric property 〈
a
,
b
〉
=
〈
b
,
a
〉 and the Lie product 〈
a
,
[
b
,
c
]
〉
=
〈
[
a
,
b
]
,
c
〉 mean that F
T
=
F and F
(
R
(
b
)
)
T
=
-
R
(
b
)
F for all b
∈
R
12. This matrix equation leads to a linear system of equations on the elements of F. Solving the resulting system by Maple yields
(39)
F
=
[
η
1
η
2
η
3
η
4
η
2
α
η
3
+
β
η
4
μ
η
4
0
η
3
μ
η
4
0
0
η
4
0
0
0
]
⊗
[
2
0
0
0
0
1
0
1
0
]
,
where η
i, 1
≤
i
≤
4, are arbitrary constants. Therefore, a bilinear form on the semidirect sum g
- of Lie algebras can be determined by
(40)
〈
A
,
B
〉
g
-
=
〈
δ
(
A
)
,
δ
(
B
)
〉
R
12
=
(
a
1
,
a
2
,
…
,
a
12
)
F
(
b
1
,
b
2
,
…
,
b
12
)
T
,
with A
,
B
∈
g
-. It is easy to compute det
⁡
(
F
)
=
16
η
4
12
μ
6; obviously, when η
4 and μ are nonzero constants, the bilinear form (40) is nondegenerate. But η
1, η
2, η
3 can be arbitrary constants. Simply, we take η
1
=
η
2
=
η
3
=
0; therefore, to apply the variational identity (35), we compute that
(41)
〈
W
-
,
∂
U
-
∂
λ
〉
=
-
2
e
′′
η
4
,
〈
W
-
,
∂
U
-
∂
u
-
〉
=
(
g
′′
η
4
,
f
′′
η
4
,
2
e
′′
η
4
,
(
β
g
+
μ
g
′
)
η
4
,
(
β
f
+
μ
f
′
)
η
4
,
hhhhh
2
(
β
e
+
μ
e
′
)
η
4
,
μ
g
η
4
,
μ
f
η
4
,
2
μ
e
η
4
,
c
η
4
,
b
η
4
,
2
a
η
4
)
T
,
γ
=
-
λ
2
d
d
λ
ln
⁡
|
〈
W
-
,
W
-
〉
|
=
0
.Thus by (35), we obtain
(42)
δ
δ
u
-
∫
2
e
m
+
1
′′
η
4
m
d
x
=
(
g
m
′′
η
4
,
f
m
′′
η
4
,
2
e
m
′′
η
4
,
(
β
g
m
+
μ
g
m
′
)
η
4
,
(
β
f
m
+
μ
f
m
′
)
η
4
,
h
h
h
h
h
l
2
(
β
e
m
+
μ
e
m
′
)
η
4
,
μ
g
m
η
4
,
μ
f
m
η
4
,
h
h
h
h
h
l
2
μ
e
m
η
4
,
c
m
η
4
,
b
m
η
4
,
2
a
m
η
4
)
T
.Therefore, the tri-integrable couplings of the GJ hierarchy in (34) possess the following Hamiltonian structures:
(43)
u
-
t
m
=
K
-
m
(
u
-
)
=
J
-
δ
H
-
m
δ
u
-
,
m
≥
0
,
where the Hamiltonian operator J
- is given by
(44)
J
-
=
[
0
0
0
1
η
4
J
0
0
1
μ
η
4
J
J
1
0
1
μ
η
4
J
J
2
J
3
1
η
4
J
J
1
J
3
J
4
]
,
with J being the same as (12), and
(45)
J
1
=
1
η
4
[
0
0
v
2
0
0
-
v
3
-
v
2
v
3
0
]
,
J
2
=
1
μ
2
η
4
[
0
2
β
α
μ
v
2
-
β
q
-
2
β
0
-
(
α
μ
v
3
-
β
r
)
-
(
α
μ
v
2
-
β
q
)
α
μ
v
3
-
β
r
-
β
∂
]
,
J
3
=
1
η
4
[
0
0
v
5
0
0
-
v
6
-
v
5
v
6
0
]
,
J
4
=
1
η
4
[
0
0
v
8
0
0
-
v
9
-
v
8
v
9
0
]
,
and the Hamiltonian functionals are determined by
(46)
H
-
m
=
∫
2
e
m
+
2
′′
η
4
m
+
1
d
x
,
m
≥
0
.The hierarchy (43) can be rewritten as
(47)
u
-
t
m
=
K
-
m
=
J
-
δ
H
-
m
δ
u
-
=
J
-
L
-
δ
H
-
m
-
1
δ
u
-
=
J
-
L
-
m
[
[
(
β
+
μ
)
v
3
+
μ
v
6
+
v
9
+
r
]
η
4
[
(
β
+
μ
)
v
2
+
μ
v
5
+
v
8
+
q
]
η
4
0
[
β
(
r
+
v
3
)
+
μ
(
r
+
α
v
3
+
v
6
)
]
η
4
[
β
(
q
+
v
2
)
+
μ
(
q
+
α
v
2
+
v
5
)
]
η
4
0
μ
(
r
+
v
3
)
η
4
μ
(
q
+
v
2
)
η
4
0
r
η
4
q
η
4
0
]
,
m
≥
1
,
where the recursion operator L
- is given by
(48)
L
-
=
[
L
L
1
L
2
L
3
0
L
α
L
1
β
L
1
+
μ
L
2
0
0
L
μ
L
1
0
0
0
L
]
,
with L being the same as (13), and(49)
L
1
=
[
v
1
0
-
1
2
v
3
0
v
1
-
1
2
v
2
2
∂
-
1
(
q
v
1
+
v
2
s
)
+
∂
-
1
v
2
∂
-
2
∂
-
1
(
s
v
3
+
r
v
1
)
+
∂
-
1
v
3
∂
0
]
,
L
2
=
[
v
4
0
-
1
2
v
6
0
v
4
-
1
2
v
5
2
∂
-
1
(
α
v
1
v
2
+
q
v
4
+
s
v
5
)
+
∂
-
1
v
5
∂
-
2
∂
-
1
(
α
v
1
v
3
+
r
v
4
+
s
v
6
)
+
∂
-
1
v
6
∂
0
]
,
L
3
=
[
v
7
0
-
1
2
v
9
0
v
7
-
1
2
v
8
L
3
,
1
L
3
,
2
0
]
,with
(50)
L
3
,
1
=
2
∂
-
1
[
v
1
(
β
v
2
+
μ
v
5
)
+
μ
v
2
v
4
+
q
v
7
+
s
v
8
]
+
∂
-
1
v
8
∂
,
L
3
,
2
=
-
2
∂
-
1
[
v
1
(
β
v
3
+
μ
v
6
)
+
μ
v
3
v
4
+
r
v
7
+
s
v
9
]
+
∂
-
1
v
9
∂
.
Therefore, the hierarchy (34) possesses a recursion relation:
(51)
K
m
+
1
=
Φ
K
m
,
m
≥
0
,
where Φ
=
J
-
L
-
J
-
-
1. But J
-
L
- is not antisymmetric; therefore, the system (11) does not have bi-Hamiltonian structures (the way of the verification is the same as the Appendix A in [24]) and is not Liouville integrable.


## 3. Conclusions

In this paper, tri-integrable couplings for the Giachetti-Johnson hierarchy of continuous soliton equations were generated by using semidirect sums of Lie algebras. Moreover, we established their Hamiltonian structures through the variational identities. Clearly, mathematical structures behind integrable couplings are indeed rich and interesting, though complicated. It is worthy to mention that the method proposed in this paper can also be applied to other soliton hierarchy.Note that we can generate more diverse tri-integrable couplings because the enlarged spectral matrixU
- has more other forms. For instance, we can specify it in either one of the following forms:
(52)
U
-
=
[
U
U
1
U
2
U
3
0
U
α
U
1
+
β
U
3
0
0
0
U
0
0
0
0
U
]
,
U
-
=
[
U
U
1
U
2
U
3
0
U
0
α
U
1
+
β
U
2
0
0
U
ζ
U
1
+
μ
U
2
0
0
0
U
]
,
which were introduced in [17].



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*Source: 627924-2014-06-26.xml*