# A New Soliton Hierarchy Associated withso

Authors: Hanyu Wei; Tiecheng Xia; Guoliang He
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

Based on the three-dimensional real special orthogonal Lie algebras
o
(
3
,
R
), we construct a new hierarchy of soliton equations by zero curvature equations and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure and thus integrable in the Liouville sense. Furthermore, we present the infinitely many conservation laws for the new soliton hierarchy.


---

## Body

## 1. Introduction

Conservation laws are widespread in applied mathematics, which reflect a phenomenon that some physical quantities do not change with time. In soliton theory, they play an important role in the study of integrability of soliton equations. An infinite number of conservation laws for KdV equation was first discovered by Miura in 1968 [1]; then many methods have been developed to find them. This may be mainly due to the contributions of Wadati et al. [2–4]. Conservation laws also play an important role in mathematics and engineering. Many papers dealing with symmetries and conservation laws have been presented. The direct construction method of multipliers for the conservation laws was presented in [5]; the Lagrangian approach for evolution equations was considered in [6, 7].It is known that soliton hierarchies such as the Ablowitz-Kaup-Newell-Segur hierarchy and the Kaup-Newell hierarchy are generated from spectral problems associated with matrix Lie algebras [8]. The trace identity is used to find Hamiltonian structures of soliton hierarchies [9]. When associated matrix Lie algebras are non-semisimple, we can obtain integrable couplings [10, 11] and generate their Hamiltonian structures by the variational identity [12, 13].We first recall a standard procedure for building soliton hierarchies. Starting from a spectral problem(1)
φ
x
=
U
φ
,
U
=
U
u
,
λ
∈
g
~
, where λ is the spectral parameter and u is a dependent variable, based on a matrix loop algebra g
~ associated with a given matrix Lie algebra g, often being simple. We take solution (2)
W
=
W
u
,
λ
=
∑
n
≥
0
W
0
n
λ
-
n
,
W
0
n
∈
g
,
n
≥
0 to the stationary zero curvature equation(3)
W
x
=
U
,
W
. We introduce the Lax matrices(4)
V
n
=
V
n
u
,
λ
=
λ
n
W
+
+
Δ
n
∈
g
~
,
n
≥
0
, where P
+ denotes the polynomial part of P in λ and Δ
n is the modification terms, to formulate the temporal spectral problems(5)
φ
t
n
=
V
n
φ
,
n
≥
0
. The zero curvature equations(6)
U
t
n
-
V
x
n
+
U
,
V
n
=
0
,
n
≥
0 can engender a hierarchy of soliton equations with Hamiltonian structures(7)
U
t
n
=
K
n
u
=
J
δ
H
n
δ
u
,
n
≥
0
. Those Hamiltonian functionals H
n can often be generated by applying the trace identity [9, 14](8)
δ
δ
u
∫
t
r
V
∂
U
∂
λ
d
x
=
λ
-
γ
∂
∂
λ
λ
γ
t
r
V
∂
U
∂
u
, where the constant γ is determined by(9)
γ
=
λ
2
d
d
λ
ln
⁡
t
r
V
V
.We will make use of the three-dimensional real special orthogonal Lie algebras
o
(
3
,
R
); this Lie algebra is simple and has a basis [15](10)
e
1
=
0
0
-
1
0
0
0
1
0
0
,
e
2
=
0
0
0
0
0
-
1
0
1
0
,
e
3
=
0
-
1
0
1
0
0
0
0
0
, along with the communicative operation(11)
e
1
,
e
2
=
e
3
,
e
2
,
e
3
=
e
1
,
e
1
,
e
3
=
-
e
2
. The derived algebra [
s
o
(
3
,
R
)
,
s
o
(
3
,
R
)
] is s
o
(
3
,
R
) itself, and s
o
(
3
,
R
) is one of the only two three-dimensional real Lie algebras with a three-dimensional derived algebra.The matrix loop algebra we will adopt in what follows is(12)
so
~
3
,
R
=
M
∈
s
o
3
,
R
∣
entries of
M
-
Laurent series in
λ
. The algebra s
o
(
3
,
R
) contains matrices of the form λ
m
e
1
+
λ
n
e
2
+
λ
l
e
3 with arbitrary integers m
,
n
,
l. This matrix loop algebra lays a foundation for our study of soliton equations.In this paper, we would like to from a spectral problem, based on the matrix loop algebras
o
(
3
,
R
), and construct a new soliton hierarchy from associated zero curvature equations. The trace identity is used to furnish the corresponding Hamiltonian structures and so all equations in the resulting soliton hierarchy are Liouville integrable. Additionally, we introduce two variables E and F to construct conservation laws of the equation hierarchy and the first two conserved densities and fluxes are listed.


## 2. A New Spectral Problem and Soliton Hierarchy

To construct a soliton hierarchy from the matrix loop algebras
o
~
(
3
,
R
), we consider an isospectral problem(13)
φ
x
=
U
φ
=
U
u
,
λ
φ
,
u
=
q
r
,
φ
=
φ
1
φ
2
φ
3
, where(14)
U
u
,
λ
=
e
1
+
q
e
1
+
r
e
2
+
λ
e
3
=
0
-
λ
-
1
-
q
λ
0
-
r
1
+
q
r
0
. This isospectral problem is of the same type as the Broer-Kaup-Kupershmidt one [16], but its underlying loop algebra is different.Starting from the stationary zero curvature equation(15)
V
x
=
U
,
V
, we have(16)
a
x
=
r
c
-
λ
b
,
b
x
=
λ
a
-
c
-
q
c
,
c
x
=
b
+
q
b
-
r
a
. Setting(17)
V
=
a
e
1
+
b
e
2
+
c
e
3
=
0
-
c
-
a
c
0
-
b
a
b
0
∈
s
o
~
3
,
R
, further letting(18)
a
=
∑
m
≥
0
a
m
λ
-
m
,
b
=
∑
m
≥
0
b
m
λ
-
m
,
c
=
∑
m
≥
0
c
m
λ
-
m
, and choosing the initial data(19)
a
0
=
b
0
=
0
,
c
0
=
1
, thus, substituting (18) into system (16), we obtain(20)
a
m
x
=
r
c
m
-
b
m
+
1
,
b
m
x
=
a
m
+
1
-
c
m
-
q
c
m
,
m
≥
0
,
c
m
x
=
b
m
+
q
b
m
-
r
a
m
. We impose the following conditions on constants of integration(21)
a
m
u
=
0
=
b
m
u
=
0
=
c
m
u
=
0
=
0
,
m
≥
0
. From the recursion relations in (20), the first few results can be obtained as follows:(22)
a
1
=
1
+
q
,
b
1
=
r
,
c
1
=
0
,
a
2
=
r
x
,
b
2
=
-
q
x
,
c
2
=
-
q
-
1
2
q
2
-
1
2
r
2
,
a
3
=
-
q
x
x
-
q
-
1
2
r
2
-
3
2
q
2
-
1
2
q
3
-
1
2
q
r
2
,
b
3
=
-
r
x
x
-
q
r
-
1
2
q
2
r
-
1
2
r
3
,
c
3
=
-
r
x
+
r
q
x
-
r
x
q
,
a
4
=
-
r
x
x
x
-
3
r
x
q
-
3
2
q
2
r
x
-
3
2
r
2
r
x
-
r
x
,
b
4
=
q
x
x
x
+
q
x
+
3
q
q
x
+
3
2
q
2
q
x
+
3
2
r
2
q
x
,
c
4
=
q
x
x
+
q
q
x
x
+
r
r
x
x
-
1
2
q
x
2
-
1
2
r
x
2
+
q
+
2
q
2
+
3
2
q
3
+
1
2
r
2
+
3
2
q
r
2
+
3
4
q
2
r
2
+
3
8
q
4
+
3
8
r
4
,
…
.Then, we consider the auxiliary spectral problem(23)
φ
t
n
=
V
n
φ
=
λ
n
V
+
φ
,
n
≥
0
, where (24)
V
n
=
∑
m
=
0
n
0
-
c
m
-
a
m
c
m
0
-
b
m
a
m
b
m
0
λ
n
-
m
. Substituting (23) into the zero curvature equation(25)
U
t
n
-
V
x
n
+
U
,
V
n
=
0
, we get a new soliton hierarchy(26)
u
t
n
=
K
n
=
q
r
t
n
=
0
-
1
1
0
a
n
+
1
b
n
+
1
=
J
a
n
+
1
b
n
+
1
.Whenn
=
2, hierarchy (26) can be reduced to the second-order nonlinear integrable equations(27)
q
t
2
=
r
x
x
+
q
r
+
1
2
q
2
r
+
1
2
r
3
,
r
t
2
=
-
q
x
x
-
q
-
1
2
r
2
-
3
2
q
2
-
1
2
q
3
-
1
2
q
r
2
.Asn
=
3, hierarchy (26) is reduced to the third-order nonlinear integrable equations(28)
q
t
3
=
-
q
x
x
x
-
q
x
-
3
q
q
x
-
3
2
q
2
q
x
-
3
2
r
2
q
x
,
r
t
3
=
-
r
x
x
x
-
3
r
x
q
-
3
2
q
2
r
x
-
3
2
r
2
r
x
-
r
x
.


## 3. Hamiltonian Structures of the New Soliton Hierarchy

In this section, we will establish Hamiltonian structures of the new soliton hierarchy by trace identity [9, 14](29)
δ
δ
u
∫
t
r
V
∂
U
∂
λ
d
x
=
λ
-
γ
∂
∂
λ
λ
γ
t
r
V
∂
U
∂
u
, where the constant γ is determined as in (9).A direct calculation reads(30)
t
r
V
∂
U
∂
λ
=
-
2
c
,
t
r
V
∂
U
∂
q
=
-
2
a
,
t
r
V
∂
U
∂
r
=
-
2
b
. Substitute the above formula into the trace identity (29) yields(31)
δ
δ
u
∫
c
d
x
=
λ
-
γ
∂
∂
λ
λ
γ
a
b
. Comparing the coefficients of λ on both sides of (31) gives rise to(32)
δ
δ
u
∫
c
n
+
1
d
x
=
γ
-
n
a
n
b
n
. By employing the computing formula (9) on the constant γ, we obtain γ
=
0. Therefore, we conclude that(33)
δ
H
n
δ
u
=
a
n
+
1
b
n
+
1
,
H
n
=
∫
-
c
n
+
2
n
+
1
d
x
,
n
⩾
0
.So, the new soliton hierarchy (26) possesses the following Hamiltonian structures:(34)
u
t
n
=
K
n
u
=
J
δ
H
n
δ
u
,
n
≥
0
.From the recursion relations in (20), we can obtain the hereditary recursion operator Φ which satisfies that(35)
δ
H
n
δ
u
=
Φ
δ
H
n
-
1
δ
u
,where(36)
Φ
=
-
1
+
q
∂
-
1
r
∂
+
1
+
q
∂
-
1
1
+
q
-
∂
-
r
∂
-
1
r
r
∂
-
1
1
+
q
.


## 4. Liouville Integrability

It is a direct but lengthy computation to verify thatJ defined by (26) and(37)
M
=
Φ
J
=
∂
+
1
+
q
∂
-
1
1
+
q
1
+
q
∂
-
1
r
r
∂
-
1
1
+
q
∂
+
r
∂
-
1
r constitute a Hamiltonian pair [17, 18]; that is, any linear combination N of J and M satisfies(38)
∫
K
T
N
′
u
N
S
T
d
x
+
cycle
⁡
K
,
S
,
T
=
0
, for all vector fields K
,
S, and T. This implies that operator Φ defined by (36) is hereditary [19], which satisfies(39)
Φ
′
u
Φ
K
S
-
Φ
Φ
′
u
K
S
=
Φ
′
u
Φ
S
K
-
Φ
Φ
′
u
S
K
, for all vector fields K and S.The hereditary property (39) is equivalent to(40)
L
Φ
K
Φ
=
Φ
L
K
Φ
, where K is an arbitrary vector field. The Lie derivative L
K
Φ above is defined by(41)
L
K
Φ
S
=
Φ
K
,
S
-
K
,
Φ
S
, where [
·
,
·
] is the Lie bracket of vector fields. It is known that an autonomous operator Φ
=
Φ
(
u
,
u
x
,
…
) is a recursion operator of an evolution equation u
t
=
K
(
u
) if operator Φ needs to satisfy(42)
L
K
Φ
=
0
. Because operator Φ (36) satisfies(43)
L
K
0
Φ
=
0
,
K
0
=
-
r
1
+
q
, thus(44)
L
K
m
Φ
=
L
Φ
K
m
-
1
Φ
=
Φ
L
K
m
-
1
Φ
=
0
,
m
≥
1
,where K
m
′
s are defined by (26). This implies that operator Φ defined by (36) is a common hereditary recursion operator for the soliton hierarchy (26).The new soliton hierarchy (26) has the following bi-Hamiltonian structures [17, 20]:(45)
u
t
n
=
K
n
=
J
δ
H
n
δ
u
=
M
δ
H
n
-
1
δ
u
,
n
≥
1
, where J, M, and H
m are defined by (26), (37), and (33), respectively, and thus, the hierarchy is Liouville integrable; that is, it possesses infinitely many commuting symmetries and conservation laws. In particular, we have the Abelian symmetry algebra(46)
K
k
,
K
l
=
K
K
′
u
K
l
-
K
l
′
u
K
k
=
0
,
k
,
l
≥
0 and the Abelian algebras of conserved functionals(47)
H
k
,
H
l
J
=
∫
δ
H
k
δ
u
T
J
δ
H
l
δ
u
d
x
=
0
,
k
,
l
≥
0
,
H
k
,
H
l
M
=
∫
δ
H
k
δ
u
T
M
δ
H
l
δ
u
d
x
=
0
,
k
,
l
≥
0
.The first two nonlinear integrable systems in hierarchy (26) are as follows:(48)
u
t
1
=
K
1
=
q
r
t
1
=
q
x
r
x
=
J
δ
H
1
δ
u
=
M
δ
H
0
δ
u
,
u
t
2
=
K
2
=
q
r
t
2
=
r
x
x
+
q
r
+
1
2
q
2
r
+
1
2
r
3
-
q
x
x
-
q
-
1
2
r
2
-
3
2
q
2
-
1
2
q
3
-
1
2
q
r
2
=
J
δ
H
2
δ
u
=
M
δ
H
1
δ
u
, where the Hamiltonian functionals H
0, H
1, and H
2 are given by(49)
H
0
=
1
2
∫
2
q
+
q
2
+
r
2
d
x
,
H
1
=
-
1
2
∫
-
r
x
+
r
q
x
-
q
r
x
d
x
,
H
2
=
-
1
3
∫
q
x
x
+
q
q
x
x
+
r
r
x
x
-
1
2
q
x
2
-
1
2
r
x
2
+
q
+
2
q
2
+
3
2
q
3
+
1
2
r
2
+
3
2
q
r
2
+
3
4
q
2
r
2
+
3
8
q
4
+
3
8
r
4
d
x
.


## 5. Conservation Laws of the New Soliton Hierarchy

In the following, we will construct conservation laws of the new soliton hierarchy. We introduce the variables(50)
E
=
φ
1
φ
3
,
F
=
φ
2
φ
3
. From (13), we have(51)
E
x
=
-
1
+
q
-
λ
F
-
1
+
q
E
2
-
r
E
F
,
F
x
=
-
r
-
λ
E
-
1
+
q
E
F
-
r
F
2
. Expand E
,
F in the power of λ:(52)
E
=
∑
j
=
1
∞
e
j
λ
-
j
,
F
=
∑
j
=
1
∞
f
j
λ
-
j
. Substituting (52) into (51) and comparing the coefficients of the same power of λ, we obtain(53)
e
1
=
-
r
,
f
1
=
-
1
-
q
,
e
2
=
q
x
,
f
2
=
r
x
,
e
3
=
-
r
x
x
-
2
r
1
+
q
2
,
f
3
=
-
q
x
x
-
2
1
+
q
r
2
,
… and a recursion formula for e
j and f
j:(54)
e
j
+
1
=
-
f
j
x
-
1
+
q
∑
l
=
1
j
-
1
e
l
f
j
-
l
-
r
∑
l
=
1
j
-
1
f
l
f
j
-
l
,
f
j
+
1
=
-
e
j
x
-
1
+
q
∑
l
=
1
j
-
1
e
l
e
j
-
l
-
r
∑
l
=
1
j
-
1
e
l
f
j
-
l
,
j
≥
2
.Because(55)
∂
∂
t
E
+
q
E
+
r
F
=
∂
∂
x
a
E
+
b
F
, where(56)
a
=
1
+
q
λ
+
r
x
,
b
=
r
λ
-
q
x
,in order to obtain the conservation laws for integrable hierarchy, we define(57)
θ
=
E
+
q
E
+
r
F
,
δ
=
a
E
+
b
F
. Then (55) can be written as θ
t
=
δ
x, which is just the formal definition of conservation laws. We expand θ and δ as series in powers of λ with the coefficients, which are called conserved densities and fluxes, respectively:(58)
θ
=
∑
j
=
1
∞
θ
j
λ
-
j
,
δ
=
∑
j
=
1
∞
δ
j
λ
-
j
; the first two conserved densities and fluxes are read as follows:(59)
θ
1
=
-
2
r
1
+
q
,
δ
1
=
2
1
+
q
q
x
-
2
r
r
x
,
θ
2
=
1
+
q
q
x
+
r
r
x
,
δ
2
=
r
q
x
x
-
1
+
q
r
x
x
-
2
r
1
+
q
3
+
2
1
+
q
r
3
.The recursion relation forθ
j and δ
j is(60)
θ
j
=
1
+
q
e
j
+
r
f
j
,
δ
j
=
1
+
q
e
j
+
1
+
r
x
e
j
-
r
f
j
+
1
-
q
x
f
j
, where e
j and f
j can be calculated from (54). We can display the first two conservation laws as(61)
θ
1
t
=
δ
1
x
,
θ
2
t
=
δ
2
x
, where θ
1, δ
1, θ
2, and δ
2 are defined in (59). Then the infinitely many conservation laws of (26) can be easily obtained from (50)–(61), respectively.


## 6. Conclusions

Based on the real loop algebras
o
(
3
,
R
), we introduced a spectral problem and generated a new hierarchy of soliton equations from the associated zero curvature equations. The Liouville integrability of the resulting soliton equations has been shown upon furnishing a bi-Hamiltonian formulation by the trace identity. Finally, the conservation laws of the new equation hierarchy are also obtained. As its reduction, we gain the nonlinear integrable equations. The solution of reduced equations is a very important and difficult work, we will plan to study and discuss this problem in the near future.



---

*Source: 3682579-2016-07-13.xml*