Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 598570, 10 pages
http://dx.doi.org/10.1155/2013/598570
Research Article
Self-Consistent Sources and Conservation Laws for Nonlinear
Integrable Couplings of the Li Soliton Hierarchy
Han-yu Wei1,2 and Tie-cheng Xia1
1
2
Department of Mathematics, Shanghai University, Shanghai 200444, China
Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou 466001, China
Correspondence should be addressed to Han-yu Wei; weihanyu8207@163.com
Received 24 November 2012; Accepted 24 January 2013
Academic Editor: Changbum Chun
Copyright © 2013 H.-y. Wei and T.-c. Xia. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
New explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Li soliton hierarchy are obtained.
Then, the nonlinear integrable couplings of Li soliton hierarchy with self-consistent sources are established. Finally, we present the
infinitely many conservation laws for the nonlinear integrable coupling of Li soliton hierarchy.
1. Introduction
Soliton theory has achieved great success during the last
decades, it is being applied to many fields. The diversity
and complexity of soliton theory enables investigators to do
research from different views, such as binary nonlinearization
of soliton hierarchy [1] and Bäcklund transformations of
soliton systems from symmetry constraints [2].
Recently, with the development of integrable systems,
integrable couplings have attracted much attention. Integrable couplings [3, 4] are coupled systems of integrable
equations, which have been introduced when we study of
Virasoro symmetric algebras. It is an important topic to
look for integrable couplings because integrable couplings
have much richer mathematical structures and better physical
meanings. In recent years, many methods of searching for
integrable couplings have been developed [5–13], but all
the integrable couplings obtained are linear for the V =
(V1 , . . . , V𝑚 )𝑇 . As for how to generate nonlinear integrable
couplings, Ma proposed a general scheme [14]. Suppose that
an integrable system
𝑢𝑡 = 𝐾 (𝑢)
(1)
has a Lax pair 𝑈 and 𝑉, which belong to semisimple matrix
Lie algebras. Introduce an enlarged spectral matrix
𝑈 = 𝑈 (𝑢) = [
𝑈 (𝑢)
0
]
𝑈𝑎 (V) 𝑈 (𝑢) + 𝑈𝑎 (V)
(2)
from a zero curvature representation
𝑈𝑡 − 𝑉𝑥 + [𝑈, 𝑉] = 0,
(3)
where
𝑉 = 𝑉 (𝑢) = [
𝑉 (𝑢)
0
],
𝑉𝑎 (𝑢) 𝑉 (𝑢) + 𝑉𝑎 (𝑢)
(4)
then we can give rise to
𝑈𝑡 − 𝑉𝑥 + [𝑈, 𝑉] = 0,
𝑈𝑎,𝑡 − 𝑉𝑎,𝑥 + [𝑈, 𝑉𝑎 ] + [𝑈𝑎 , 𝑉] + [𝑈𝑎 , 𝑉𝑎 ] = 0.
(5)
This is an integrable coupling of (1), and it is a nonlinear
integrable coupling because the commutator [𝑈𝑎 , 𝑉𝑎 ] can
generate nonlinear terms.
Soliton equation with self-consistent sources (SESCS)
[15–22] is an important part in soliton theory. Physically,
2
Abstract and Applied Analysis
the sources may result in solitary waves with a nonconstant
velocity and therefore lead to a variety of dynamics of physical
models. For applications, these kinds of systems are usually
used to describe interactions between different solitary waves
and are relevant to some problems of hydrodynamics, solid
state physics, plasma physics, and so forth. How to obtain an
integrable coupling of the SESCS is an interesting topic; in
this paper, we will use new formula [23] presented by us to
generalize soliton hierarchy with self-consistent sources.
The conservation laws play an important role in discussing the integrability for soliton hierarchy. An infinite
number of conservation laws for KdV equation was first
discovered by Miura et al. [24]. The direct construction
method of multipliers for the conservation laws was presented [25], the Lagrangian approach for evolution equations was considered in [26], Wang and Xia established the
infinitely many conservation laws for the integrable super
𝐺-𝐽 hierarchy [27], and the infinite conservation laws of the
generalized quasilinear hyperbolic equations were derived in
[28]. Comparatively, the less nonlinear integrable couplings
of the soliton equations have been considered for their
conservation laws.
This paper is organized as follows. In Section 2, a kind of
explicit Lie algebras with the forms of blocks is introduced
to generate nonlinear integrable couplings of Li soliton
hierarchy. In Section 3, a new nonlinear integrable coupling
of Li soliton hierarchy with self-consistent sources is derived.
In Section 4, we obtain the conservation laws for the nonlinear integrable couplings of Li hierarchy. Finally, some
conclusions are given.
2. Lie Algebras for Constructing Nonlinear
Integrable Couplings of Li Soliton Hierarchy
Tu [29] presented a base of the Li algebra sl(2) as follows:
𝐺1 = span {𝑒1 , 𝑒2 , 𝑒3 } ,
(6)
where
1 0
𝑒1 = (
),
0 −1
0 1
𝑒2 = (
),
1 0
0 1
𝑒3 = (
) , (7)
−1 0
which have the commutative relations
[𝑒1 , 𝑒2 ] = 2𝑒2 ,
[𝑒1 , 𝑒3 ] = −2𝑒3 ,
[𝑒2 , 𝑒3 ] = 𝑒1 .
[𝑎, 𝑏] = 𝑎𝑏 − 𝑏𝑎,
(9)
where
𝑎, 𝑏 ∈ 𝐺.
(11)
A direct verification exhibits that
[𝑔1 , 𝑔2 ] = 2𝑔3 ,
[𝑔1 , 𝑔3 ] = 2𝑔2 ,
[𝑔2 , 𝑔3 ] = −2𝑔1 ,
[𝑔1 , 𝑔5 ] = 2𝑔6 ,
[𝑔1 , 𝑔6 ] = 2𝑔5 ,
[𝑔2 , 𝑔4 ] = −2𝑔6 ,
[𝑔2 , 𝑔6 ] = −2𝑔4 ,
(12)
[𝑔3 , 𝑔4 ] = −2𝑔5 ,
[𝑔3 , 𝑔5 ] = 2𝑔4 ,
[𝑔4 , 𝑔5 ] = 2𝑔6 ,
[𝑔4 , 𝑔6 ] = 2𝑔5 ,
[𝑔5 , 𝑔6 ] = −2𝑔4 ,
[𝑔1 , 𝑔4 ] = [𝑔3 , 𝑔6 ] = 0.
Set
̃1 = span {𝑔1 , 𝑔2 , 𝑔3 } ,
𝐺
̃2 = span {𝑔4 , 𝑔5 , 𝑔6 } ,
𝐺
(13)
then we find that
̃1 ⊕ 𝐺
̃2 ,
𝐺=𝐺
̃1 ≅ 𝐺1 ,
𝐺
̃1 , 𝐺
̃2 ] ⊆ 𝐺
̃2 ,
[𝐺
(14)
̃2 are all simple Lie subalgebras.
̃1 and 𝐺
and 𝐺
While we use Lie algebras to generate integrable hierarchies of evolution equations, we actually employ their loop
̃ = 𝐺 ⊗ 𝐶(𝜆, 𝜆−1 ) to establish Lax pairs, where
algebras 𝐺
−1
𝐶(𝜆, 𝜆 ) represents a set of Laurent ploynomials in 𝜆 and 𝐺
is a Lie algebra. Based on this, we give the loop algebras of (9)
as follows:
̃ = span {𝑔1 (𝑛) , . . . , 𝑔6 (𝑛)} ,
𝐺
(15)
where 𝑔𝑖 (𝑛) = 𝑔𝑖 𝜆𝑛 , [𝑔𝑖 (𝑚), 𝑔𝑗 (𝑛)] = [𝑔𝑖 , 𝑔𝑗 ]𝜆𝑚+𝑛 , 1 ≤ 𝑖, 𝑗 ≤ 6,
𝑚, 𝑛 ∈ 𝑍.
We consider an auxiliary linear problem as follows:
𝜑1
𝜑1
𝜑2
𝜑
( ) = 𝑈 (𝑢, 𝜆) ( 2 ) ,
𝜑3
𝜑3
𝜑4 𝑥
𝜑4
(8)
Let us introduce a Lie algebra with matrix blocks by using
𝐺1 in order to get nonlinear couplings of soliton hierarchy as
follows:
𝐺 = span {𝑔1 , . . . , 𝑔6 } ,
Define a commutator as follows:
6
𝑈 (𝑢, 𝜆) = 𝑅1 + ∑𝑢𝑖 𝑔𝑖 (𝜆) ,
(16)
𝑖=1
𝜑1
𝜑1
𝜑2
𝜑
( ) = 𝑉𝑛 (𝑢, 𝜆) ( 2 ) ,
𝜑3
𝜑3
𝜑4 𝑡
𝜑4
𝑛
𝑒 0
𝑔1 = ( 1 ) ,
0 𝑒1
𝑒 0
𝑔2 = ( 2 ) ,
0 𝑒2
𝑔3 = (
𝑒3 0
),
0 𝑒3
0 0
𝑔4 = (
),
𝑒1 𝑒1
𝑔5 = (
0 0
),
𝑒2 𝑒2
0 0
𝑔6 = (
).
𝑒3 𝑒3
(10)
where 𝑢 = (𝑢1 , . . . , 𝑢𝑠 )𝑇 , 𝑈𝑛 = 𝑅1 + 𝑢1 𝑔1 + ⋅ ⋅ ⋅ + 𝑢6 𝑔6 , 𝑅1 is a
pseudoregular element, 𝑢𝑖 (𝑛, 𝑡) = 𝑢𝑖 (𝑖 = 1, 2, . . . , 6), and 𝜑𝑖 =
̃
𝜑(𝑥, 𝑡) are field variables defined on 𝑥 ∈ 𝑅, 𝑡 ∈ 𝑅, 𝑔𝑖 (𝜆) ∈ 𝐺.
The compatibility of (16) gives rise to the well-known zero
curvature equation
𝑈𝑡 − 𝑉𝑥 + [𝑈, 𝑉] = 0,
𝜆 𝑡 = 0.
(17)
Abstract and Applied Analysis
3
The general scheme of searching for the consistent 𝑉𝑛 ,
and generating a hierarchy of zero curvature equations was
proposed in [30]. Solving the following equation:
𝑉𝑥 = [𝑈, 𝑉] ,
∞
𝑉 = ∑ 𝑉𝑛 𝜆−𝑛
𝑛=0
𝑎
𝑏+𝑐
0
0
𝑏 − 𝑐 −𝑎
0
0
= (
),
𝑒
𝑓+𝑔
𝑎+𝑒
𝑏+𝑐+𝑓+𝑔
𝑓 − 𝑔 −𝑒 𝑏 − 𝑐 + 𝑓 − 𝑔 − (𝑎 + 𝑒)
(18)
̃ the new 𝑉𝑛 can be constructed by
then we sesrch for 󳵻𝑛 ∈ 𝐺,
𝑛
𝑉𝑛 = ∑ 𝑉𝑚 (𝑢) 𝜆𝑛−𝑚 + 󳵻𝑛 (𝑢, 𝜆) .
(19)
𝑚=0
the spectral problem (21) is firstly solved, we assume that a
solution 𝑉 is given by the following:
𝑎
𝑏+𝑐
0
0
𝑏 − 𝑐 −𝑎
0
0
𝑉=(
)
𝑒
𝑓+𝑔
𝑎+𝑒
𝑏+𝑐+𝑓+𝑔
𝑓 − 𝑔 −𝑒 𝑏 − 𝑐 + 𝑓 − 𝑔 − (𝑎 + 𝑒)
∞
= ∑ 𝑉𝑛 𝜆−𝑛
𝑛=0
∞
=∑
𝑛=0
𝑎𝑛
𝑏𝑛 + 𝑐𝑛
−𝑎𝑛
𝑏𝑛 − 𝑐𝑛
𝑓𝑛 + 𝑔𝑛
× ( 𝑒𝑛
𝑓𝑛 − 𝑔𝑛
Solving zero curvature (17), we could get evolution equation
as follows:
𝑢𝑡 = 𝐾 (𝑢, 𝑢𝑥 , . . . ,
𝑝
𝜕 𝑢
).
𝜕𝑥𝑝
(20)
0
0
𝑏𝑛 + 𝑐𝑛 + 𝑓𝑛 + 𝑔𝑛 )𝜆−𝑛 .
𝑏𝑛 − 𝑐𝑛 + 𝑓𝑛 − 𝑔𝑛
− (𝑎𝑛 + 𝑒𝑛 )
(23)
Therefore, the condition (18) becomes the following recursion
relation:
𝑎𝑛,𝑥 = 2V𝑏𝑛 − 2𝑢𝑐𝑛 ,
𝑏𝑛,𝑥 = −2𝑐𝑛+1 + 2V𝑐𝑛 − 2V𝑎𝑛 ,
Now, we consider Li soliton hierarchy [31]. In order to set
up nonlinear integrable couplings of the Li soliton hierarchy
with self-consistent sources, we first consider the following
matrix spectral problem:
𝑐𝑛,𝑥 = −2𝑏𝑛+1 + 2V𝑏𝑛 − 2𝑢𝑎𝑛 ,
𝑒𝑛,𝑥 = − 2𝑢𝑔𝑛 + 2V𝑓𝑛 − 2𝑝1 𝑐𝑛
+ 2𝑝2 𝑏𝑛 − 2𝑝1 𝑔𝑛 + 2𝑝2 𝑓𝑛 ,
𝜑𝑥 = 𝑈 (𝑢, 𝜆) 𝜑,
𝑈 (𝑢, 𝜆) = − 𝑔1 (1) + V𝑔1 (0) + 𝑢𝑔2 (0) + V𝑔3 (0)
−𝑒𝑛
0
0
𝑎𝑛 + 𝑒𝑛
(21)
(24)
𝑓𝑛,𝑥 = − 2𝑔𝑛+1 + 2V𝑔𝑛 − 2V𝑒𝑛
− 𝑔4 (1) + 𝑝2 𝑔4 (0) + 𝑝1 𝑔5 (0) + 𝑝2 𝑔6 (0) ,
+ 2𝑝2 𝑐𝑛 − 2𝑝2 𝑎𝑛 + 2𝑝2 𝑔𝑛 − 2𝑝2 𝑒𝑛 ,
𝑔𝑛,𝑥 = − 2𝑓𝑛+1 + 2V𝑓𝑛 − 2𝑢𝑒𝑛 + 2𝑝2 𝑏𝑛
that is,
− 2𝑝1 𝑎𝑛 + 2𝑝2 𝑓𝑛 − 2𝑝1 𝑒𝑛 .
𝑈 (𝑢, 𝜆)
Choose the initial data
−𝜆 + V 𝑢 + V
0
0
𝑢−V
𝜆−V
0
0
=(
)
−𝜆 + 𝑝2 𝑝1 + 𝑝2 −2𝜆 + V + 𝑝2 𝑢 + V + 𝑝1 + 𝑝2
𝑝1 − 𝑝2 𝜆 − 𝑝2 𝑢 − V + 𝑝1 − 𝑝2 2𝜆 − V − 𝑝2
0
𝑈
=( 1
),
𝑈0 𝑈1 + 𝑈0
𝑎0 = 𝑒0 = 𝛽,
where 𝜆 is a spectral parameter and 𝑈1 satisfies 𝜑𝑥 = 𝑈1 𝜑
which is matrix spectral problem of the Li soliton hierarchy
[31].
To establish the nonlinear integrable coupling system of
the Li soliton hierarchy, the adjoint equation 𝑉𝑥 = [𝑈, 𝑉] of
(25)
we see that all sets of functions 𝑎𝑛 , 𝑏𝑛 , 𝑐𝑛 , 𝑒𝑛 , 𝑓𝑛 , and 𝑔𝑛 are
uniquely determined. In particular, the first few sets are as
follows:
𝑎1 = 0,
(22)
𝑏0 = 𝑐0 = 𝑓0 = 𝑔0 = 0,
𝑓1 = −𝑝1 𝛽,
𝑏1 = −𝑢𝛽,
𝑔1 = −𝑝2 𝛽,
1
𝑏2 = ( V𝑥 − 𝑢V) 𝛽,
2
𝑐1 = −V𝛽,
𝑎2 =
𝑒1 = 0,
1 2
(V − 𝑢2 ) ,
2
1
𝑐2 = ( 𝑢𝑥 − V2 ) 𝛽,
2
1
1
1
1
1
1
𝑒2 = ( V𝑝2 − 𝑢𝑝1 + 𝑢2 − V2 − 𝑝12 + 𝑝22 ) 𝛽,
2
2
4
4
4
4
4
Abstract and Applied Analysis
1
1
1
1
1
1
𝑓2 = ( 𝑝2,𝑥 − V𝑥 + 𝑢V − V𝑝1 − 𝑢𝑝2 − 𝑝1 𝑝2 ) 𝛽,
4
4
2
2
2
2
1
1
1
1
𝑔2 = ( 𝑝1,𝑥 − 𝑢𝑥 + V2 − 𝑝22 − V𝑝2 ) 𝛽, . . . .
4
4
2
2
(26)
where
1
1
1
𝑀1 = − 𝜕 − 𝑝1 + 𝑢,
4
2
2
1
1
1
𝑀2 = 𝜕 − 𝑝1 − 𝑢,
4
2
2
1
1
1
𝑀3 = − 𝜕−1 𝑢 − 𝜕−1 𝑝1 − 𝜕−1 𝑝1 𝜕 − 𝜕,
2
2
4
1
1
1
1
𝑀4 = − 𝜕−1 V𝜕 + 𝜕−1 𝑝2 𝜕 + 𝜕−1 𝑝2 𝑢 − V + 𝑝2 ,
2
2
2
2
Considering
1
1
1
𝑀5 = 𝜕−1 𝑢𝜕 + 𝜕−1 𝑝1 𝜕 + 𝜕,
2
2
4
𝑉𝑛 = 𝑉 + 󳵻𝑛 ,
3
1
𝑀6 = − 2𝜕−1 𝑢V − 𝜕−1 𝑝1 V + 𝜕−1 V𝜕
2
2
󳵻𝑛
− (𝑎𝑛 − 𝑐𝑛 )
0
0
= (− (𝑒 − 𝑔 )
𝑛
𝑛
0
0
0
𝑎𝑛 − 𝑐𝑛
0
0
0
− (𝑎𝑛 − 𝑐𝑛 ) − (𝑒𝑛 − 𝑔𝑛 )
𝑒𝑛 − 𝑔𝑛
0
).
𝑎𝑛 − 𝑐𝑛 + 𝑒𝑛 − 𝑔𝑛
(27)
From the zero curvature equation 𝑈𝑡 − 𝑉𝑥 + [𝑈, 𝑉] = 0, we
obtain the nonlinear integrable coupling system
𝑛
0
−𝜕
0
0
(28)
0
0
𝜕
0
1
V𝑡2 = ( 𝑢𝑥𝑥 − 3VV𝑥 + 𝑢𝑢𝑥 ) 𝛽,
2
(31)
1
1
1
3
3
𝑝2,𝑡2 = ( 𝑝1,𝑥 − 𝑢𝑥 + V2 − 𝑝22 − V𝑝2
4
4
2
4
2
1
1
1
1
+ 𝑢𝑝1 − 𝑢2 + V2 + 𝑝12 ) 𝛽.
2
4
4
4
𝑥
So, we can say that the system in (28) with 𝑛 ≥ 2
provides a hierarchy of nonlinear integrable couplings for the
Li hierarchy of the soliton equation.
𝑛 ≥ 0,
3. Self-Consistent Sources for the Nonlinear
Integrable Couplings of Li Soliton Hierarchy
According to (16), now we consider a new auxiliary linear
problem. For 𝑁 distinct 𝜆 𝑗 , 𝑗 = 1, 2, . . . , 𝑁 and the systems
of (16) become in the following form:
0
0
),
0
−𝜕
1
𝜕−𝑢
0 0
2
1
−1
−1
𝐿 = (𝜕 𝑢𝜕 + 𝜕 𝜕 V𝜕 + V 0 0 ) ,
2
0
𝑀1
0 𝑀2
𝑀3
𝑀4
𝑀5 𝑀6
0
1
𝑢𝑡2 = (− V𝑥𝑥 − 𝑢𝑥 V − 𝑢V𝑥 ) 𝛽,
2
1
1
− 𝑢𝑝2 − 𝑝1 𝑝2 ) 𝛽,
2
2
𝑥
with the Hamiltonian operator 𝐽 and the hereditary recursion
operator 𝐿, respectively, as follows:
𝜕
0
𝐽=(
0
0
1
1
1
+ 𝜕−1 𝑝2 𝜕 − V + 𝑝2 .
2
2
2
Obviously, when 𝑝1 = 𝑝2 = 0 in (28), the above results
become Li soliton hierarchy. So, we can say that (28) is
integrable coupling of the Li soliton hierarchy.
Taking 𝑛 = 2, we get that the nonlinear integrable
couplings of Li soliton hierarchy are as follows:
1
1
1
1
𝑝1,𝑡2 = ( 𝑝2,𝑥 − V𝑥 + 𝑢V − V𝑝1
4
4
2
2
𝑢
𝑏𝑛,𝑥
−(𝑎𝑛 − 𝑐𝑛 )𝑥
V
𝑢𝑡𝑛 = 𝐾𝑛 = ( ) = (
)
𝑓𝑛,𝑥
𝑝1
𝑝2 𝑡
−(𝑒𝑛 − 𝑔𝑛 )𝑥
𝑏𝑛
0
𝑎𝑛 − 𝑐𝑛
𝛽
𝑛
= 𝐽(
) = 𝐽𝐿 ( ) ,
𝑓𝑛
0
𝑒𝑛 − 𝑔𝑛
𝛽
(30)
(29)
𝜑1𝑗
𝜑1𝑗
𝜑2𝑗
𝜑
( ) = 𝑈 (𝑢, 𝜆 𝑗 ) ( 2𝑗 )
𝜑3𝑗
𝜑3𝑗
𝜑4𝑗 𝑥
𝜑4𝑗
𝜑1𝑗
𝜑
= ∑𝑢𝑖 𝑔𝑖 (𝜆) ( 2𝑗 ) ,
𝜑3𝑗
𝑖=1
𝜑4𝑗
6
𝑗 = 1, . . . , 𝑁,
Abstract and Applied Analysis
5
𝜑1𝑗
𝜑1𝑗
𝜑2𝑗
𝜑
( ) = 𝑉𝑛 (𝑢, 𝜆 𝑗 ) ( 2𝑗 )
𝜑3𝑗
𝜑3𝑗
𝜑4𝑗 𝑡
𝜑4𝑗
Therefore, according to formulas (34) and (36), we have the
following results by direct computations:
𝑛
𝛿𝜆 𝑗
𝑛
= [ ∑ 𝑉𝑚 (𝑢) 𝜆𝑛−𝑚
+ Δ 𝑛 (𝑢, 𝜆 𝑗 )]
𝑗
𝑚=0
𝜑1𝑗
𝜑
× ( 2𝑗 ) ,
𝜑3𝑗
𝜑4𝑗
𝑗 = 1, . . . , 𝑁.
(32)
Based on the result in [32], we show that the following
equation
𝛿𝐻𝑘 𝑁 𝛿𝜆 𝑗
+ ∑𝛼
=0
𝛿𝑢 𝑗=1 𝑗 𝛿𝑢
(33)
holds true, where 𝛼𝑗 is a constant. From (32), we may know
that
𝛿𝜆 𝑗
𝛿𝑢𝑖
= 𝛼𝑗 Tr (Ψ𝑗
𝜕𝑈 (𝑢, 𝜆 𝑗 )
𝜕𝑢𝑖
)
(34)
= 𝛼𝑗 Tr (Ψ𝑗 𝑔𝑖 (𝜆 𝑗 )) ,
2
2
𝜙1𝑗 𝜙2𝑗 −𝜙1𝑗
𝜙3𝑗 𝜙4𝑗 −𝜙3𝑗
2
2
𝜙
−𝜙1𝑗 𝜙2𝑗 𝜙4𝑗 −𝜙3𝑗 𝜙4𝑗
Ψ𝑗 = ( 2𝑗
2 ),
0
0
𝜙1𝑗 𝜙2𝑗 −𝜙1𝑗
2
0
0
𝜙2𝑗
−𝜙1𝑗 𝜙2𝑗
(35)
𝜕𝑈0 (𝑢, 𝜆 𝑗 )
𝜕𝑢𝑖
),
(36)
where
0
𝑈
𝑈=( 1
),
𝑈0 𝑈1 + 𝑈0
Ψ𝑗𝐴
2
−𝜙3𝑗
𝜙 𝜙
= ( 3𝑗2 4𝑗
),
𝜙4𝑗 −𝜙3𝑗 𝜙4𝑗
(37)
𝑁
𝛿𝜆 𝑗
𝛿𝐻𝑛+1
+ 𝐽 ∑ 𝛼𝑗
𝛿𝑢𝑖
𝛿𝑢
𝑗=1
(38)
= 𝐽𝐿𝑛
𝑁
𝛿𝜆 𝑗
𝛿𝐻1
+ 𝐽 ∑ 𝛼𝑗
,
𝛿𝑢𝑖
𝛿𝑢
𝑗=1
by taking 𝛼𝑗 = 1 and 𝛽𝑗 = 1 in formulas (34) and (36).
Therefore, we have nonlinear integrable coupling system of
the Li equations hierarchy with self-consistent sources as
follows:
𝑛 = 1, 2, . . . .
𝑢
V
=( )
𝑝1
𝑝2 𝑡
0
𝛽
= 𝐽𝐿𝑛 ( )
0
𝛽
2 (⟨Φ2 , Φ2 ⟩ − ⟨Φ1 , Φ1 ⟩)
2 (⟨Φ1 , Φ1 ⟩ + ⟨Φ2 , Φ2 ⟩ + 2 ⟨Φ1 , Φ2 ⟩)
)
+ 𝐽(
⟨Φ4 , Φ4 ⟩ − ⟨Φ3 , Φ3 ⟩
⟨Φ3 , Φ3 ⟩ + ⟨Φ4 , Φ4 ⟩ + 2 ⟨Φ3 , Φ4 ⟩
𝑁
and 𝛽𝑗 is a constant.
According to (34) and (36), we obtain a kind of nonlinear
integrable couplings with self-consistent sources as follows:
𝑢𝑡𝑛 = 𝐽
2 (⟨Φ2 , Φ2 ⟩ − ⟨Φ1 , Φ1 ⟩)
2 (⟨Φ1 , Φ1 ⟩ + ⟨Φ2 , Φ2 ⟩ + 2 ⟨Φ1 , Φ2 ⟩)
),
=(
⟨Φ4 , Φ4 ⟩ − ⟨Φ3 , Φ3 ⟩
⟨Φ3 , Φ3 ⟩ + ⟨Φ4 , Φ4 ⟩ + 2 ⟨Φ3 , Φ4 ⟩
𝑛
For 𝑖 = 3, 4 we define that
𝛿𝑢𝑖
( 𝛿𝑝1 )
𝑢𝑡𝑛 = 𝐾𝑛
𝑗 = 1, . . . , 𝑁.
= 𝛽𝑗 Tr (Ψ𝑗𝐴
(39)
𝑖 = 1, 2,
where Tr denotes the trace of a matrix and
𝛿𝜆 𝑗
𝛿𝑢
( 𝛿𝜆 𝑗 )
(
)
𝑁 𝛿𝜆
𝑁 (
)
𝑗
( 𝛿V )
= ∑(
∑
)
( 𝛿𝜆 𝑗 )
𝑗=1 𝛿𝑢
𝑗=1 (
)
( 𝛿𝑝 )
1
𝛿𝜆 𝑗
2
2
− 𝜑1𝑗
)
2 ∑ (𝜑2𝑗
𝑗=1
)
( 𝑁
)
(
2
2
0
(2 ∑ (𝜑1𝑗
+
𝜑
+
2𝜑
𝜑
)
1𝑗 2𝑗 )
2𝑗
)
(
𝛽
)
( 𝑗=1
= 𝐽𝐿𝑛 ( ) + 𝐽 (
),
𝑁
0
)
(
2
2
)
(
∑ (𝜑4𝑗 − 𝜑3𝑗 )
𝛽
)
(
)
(
𝑗=1
𝑁
2
2
+ 𝜑4𝑗
+ 2𝜑3𝑗 𝜑4𝑗 )
∑ (𝜑3𝑗
( 𝑗=1
)
(40)
6
Abstract and Applied Analysis
𝜑4𝑗,𝑥 = (𝑝1 − 𝑝2 ) 𝜑1𝑗 + (𝜆 − 𝑝2 ) 𝜑2𝑗
with
+ (𝑢 − V + 𝑝1 − 𝑝2 ) 𝜑3𝑗
𝜑1𝑗,𝑥 = (−𝜆 + V) 𝜑1𝑗 + (𝑢 + V) 𝜑2𝑗 ,
+ (2𝜆 − V − 𝑝2 ) 𝜑4𝑗 ,
𝜑2𝑗,𝑥 = (𝑢 − V) 𝜑1𝑗 + (𝜆 − V) 𝜑2𝑗 ,
𝑗 = 1, . . . , 𝑁.
(43)
𝜑3𝑗,𝑥 = (−𝜆 + 𝑝2 ) 𝜑1𝑗 + (𝑝1 + 𝑝2 ) 𝜑2𝑗
+ (−2𝜆 + V + 𝑝2 ) 𝜑3𝑗 + (𝑢 + V + 𝑝1 + 𝑝2 ) 𝜑4𝑗 , (41)
𝜑4𝑗,𝑥 = (𝑝1 − 𝑝2 ) 𝜑1𝑗 + (𝜆 − 𝑝2 ) 𝜑2𝑗
In what follows, we will construct conservation laws for the
nonlinear integrable couplings of the Li hierarchy. For the
coupled spectral problem of Li hierarchy
+ (𝑢 − V + 𝑝1 − 𝑝2 ) 𝜑3𝑗
+ (2𝜆 − V − 𝑝2 ) 𝜑4𝑗 ,
4. Conservation Laws for the Nonlinear
Integrable Couplings of Li Soliton Hierarchy
𝑗 = 1, . . . , 𝑁,
where Φ𝑖 = (𝜑𝑖1 , . . . , 𝜑𝑖𝑁), 𝑖 = 1, 2, 3, 4, and ⟨⋅, ⋅⟩ is the
standard inner product in 𝑅𝑁.
When 𝑛 = 2 and 𝛽 = 2, we obtain nonlinear integrable
couplings of Li hierarchy with self-consistent sources
𝑈 (𝑢, 𝜆)
=(
𝑢+V
0
0
𝑢−V
𝜆−V
0
0
−2𝜆 + V + 𝑝2
𝑢 + V + 𝑝1 + 𝑝2
−𝜆 + 𝑝2 𝑝1 + 𝑝2
𝑁
1
2
2
− 𝜑1𝑗
),
𝑢𝑡2 = − V𝑥𝑥 − 𝑢𝑥 V − 𝑢V𝑥 + 2𝜕∑ (𝜑2𝑗
2
𝑗=1
𝑝1 − 𝑝2 𝜆 + −𝑝2 𝑢 − V + 𝑝1 − 𝑝2
),
2𝜆 − V − 𝑝2
(44)
1
𝑢 − 3VV𝑥 + 𝑢𝑢𝑥
2 𝑥𝑥
V𝑡2 =
−𝜆 + V
we introduce the variables
𝑁
2
2
− 2𝜕∑ (𝜑1𝑗
+ 𝜑2𝑗
+ 2𝜑1𝑗 𝜑2𝑗 ) ,
𝑗=1
𝑝1𝑡2
𝑀=
1
1
1
1
1
= ( 𝑝2,𝑥 − V𝑥 + 𝑢V − V𝑝1 − 𝑢𝑝2
4
4
2
2
2
𝑁
1
2
2
− 𝑝1 𝑝2 ) + 𝜕∑ (𝜑4𝑗
− 𝜑3𝑗
),
2
𝑥
𝑗=1
𝜑2
,
𝜑1
𝑁=
𝜑3
,
𝜑1
𝐾=
𝜑4
.
𝜑1
(45)
From (44), we have
𝑀𝑥 = 𝑢 − V + 2𝜆𝑀 − 2V𝑀 − (𝑢 + V) 𝑀2 ,
1
1
1
3
3
1
𝑝2𝑡2 = ( 𝑝1,𝑥 − 𝑢𝑥 + V2 − 𝑝22 − V𝑝2 + 𝑢𝑝1
4
4
2
4
2
2
𝑁𝑥 = − 𝜆 + 𝑝2 − 𝜆𝑁 + (𝑝1 + 𝑝2 ) 𝑀
1
1
1
− 𝑢2 + V2 + 𝑝12 )
4
4
4
𝑥
+ 𝑝2 𝑁 + (𝑢 + V + 𝑝1 + 𝑝2 ) 𝐾 − (𝑢 + V) 𝑁𝑀,
𝑁
𝐾𝑥 = 𝑝1 − 𝑝2 + 3𝜆𝐾 + 𝜆𝑀 − 𝑝2 𝑀
2
2
− 𝜕∑ (𝜑3𝑗
+ 𝜑4𝑗
+ 2𝜑3𝑗 𝜑4𝑗 ) ,
𝑗=1
(42)
with
− (2V + 𝑝2 ) 𝐾 + (𝑢 − V + 𝑝1 − 𝑝2 ) 𝑁 − (𝑢 + V) 𝐾𝑀.
(46)
We expand 𝑀, 𝑁, and 𝐾 in powers of 𝜆 as follows:
𝜑1𝑗,𝑥 = (−𝜆 + V) 𝜑1𝑗 + (𝑢 + V) 𝜑2𝑗 ,
𝜑2𝑗,𝑥 = (𝑢 − V) 𝜑1𝑗 + (𝜆 − V) 𝜑2𝑗 ,
𝜑3𝑗,𝑥 = (−𝜆 + 𝑝2 ) 𝜑1𝑗 + (𝑝1 + 𝑝2 ) 𝜑2𝑗
+ (−2𝜆 + V + 𝑝2 ) 𝜑3𝑗 + (𝑢 + V + 𝑝1 + 𝑝2 ) 𝜑4𝑗 ,
∞
∞
𝑀 = ∑𝑚𝑗 𝜆−𝑗 ,
𝑁 = ∑𝑛𝑗 𝜆−𝑗 ,
𝑗=1
𝑗=1
∞
(47)
−𝑗
𝐾 = ∑ 𝑘𝑗 𝜆 .
𝑗=1
Abstract and Applied Analysis
7
Substituting (47) into (46) and comparing the coefficients of
the same power of 𝜆, we obtain the following:
𝑚1 =
1
(V − 𝑢) ,
2
𝑘1 =
1
1
(𝑝 − 𝑝1 ) + (𝑢 − V) ,
3 2
6
and a recursion formula for 𝑚𝑗 , 𝑛𝑗 , and 𝑘𝑗 as follows:
𝑗−1
1
1
𝑚𝑗+1 = 𝑚𝑗,𝑥 + V𝑚𝑗 + (𝑢 + V) ∑ 𝑚𝑙 𝑚𝑗−𝑙 ,
2
2
𝑙=1
𝑛1 = 𝑝2 ,
𝑛𝑗+1 = − 𝑛𝑗,𝑥 + (𝑝1 + 𝑝2 ) 𝑚𝑗 + 𝑝2 𝑛𝑗
1
1
𝑚2 = (V − 𝑢)𝑥 + (V2 − 𝑢V) ,
4
2
𝑗−1
+ (𝑢 + V + 𝑝1 + 𝑝2 ) 𝑘𝑗 − (𝑢 + V) ∑ 𝑚𝑙 𝑛𝑗−𝑙 ,
1
2
2
4
1
1
𝑛2 = −𝑝2,𝑥 − 𝑝12 − 𝑢𝑝1 + V𝑝2 + 𝑝22 + 𝑢2 − V2 ,
3
3
3
3
6
6
5
1
5
5
𝑘2 =
(𝑢 − V)𝑥 + (𝑝2 − 𝑝1 )𝑥 − V2 + 𝑢V
36
9
18
18
𝑙=1
𝑘𝑗+1
1
1
1
1
= 𝑘𝑗,𝑥 − 𝑚𝑗,𝑥 − V𝑚𝑗 + 𝑝2 𝑚𝑗
3
6
3
3
4
4
4
2
2
+ V𝑝2 − 𝑢𝑝2 + 𝑝22 − 𝑝1 𝑝2 − V𝑝1 ,
3
9
9
9
9
𝑚3 =
1
1
1
3
(V − 𝑢)𝑥𝑥 + 𝑢3 − 𝑢2 V + VV𝑥
8
8
8
4
+ (𝑢 + V + 𝑝1 + 𝑝2 ) 𝐾]
=
4
11
14
16
− 𝑝1 𝑢V + 𝑝2 V2 − 𝑢𝑝1 𝑝2 + V𝑝22
9
9
9
9
16
7
7
5
5
2
+ 𝑝23 − 𝑝2 𝑢2 − 𝑝2 𝑝12 + V𝑢2 − V3 − V𝑝12 ,
9
9
9
18
18
9
19
1
5
(𝑢 − V)𝑥𝑥 + (𝑝2 − 𝑝1 )𝑥𝑥 + (𝑢 − V)𝑥
216
27
54
+
7
5
5
5
𝑝2 V𝑥 + 𝑢𝑝2,𝑥 − 𝑝2 𝑢𝑥 − 𝑝2 𝑝1,𝑥
27
27
27
27
+
5
2
4
103 3
𝑝1 𝑝2,𝑥 − 𝑝1 V𝑥 − V𝑝1,𝑥 −
V
27
27
27
216
+
101 2 1 2 7
16
𝑢V + V𝑢 + 𝑝2 V2 − 𝑝2 𝑢V
216
8
18
27
−
32
16
4
19
𝑝 V2 + V𝑝22 − 𝑝1 𝑝2 V − 𝑝1 V2
54 2
27
27
27
−
16 2 16 3 16
2
𝑢𝑝2 + 𝑝2 − 𝑝1 𝑝22 + 𝑝1 𝑢2
27
27
27
9
+
1 2 1 2 1 3 2
2
𝑢V + 𝑢𝑝1 + 𝑝1 − 𝑢V𝑝1 − 𝑢𝑝1 𝑝2
18
3
9
9
9
𝑗−1
𝜕
[−𝜆 + 𝑝2 + (𝑝1 + 𝑝2 ) 𝑀 + (−2𝜆 + V + 𝑝2 ) 𝑁
𝜕𝑡
1
5
1
1
1
+ 𝑢𝑝2,𝑥 + V𝑢𝑥 − V𝑝1,𝑥 + 𝑝1 𝑝2,𝑥 − 𝑝2 𝑝1,𝑥
9
36
9
9
9
47
7
19
4
VV + V𝑢 +
𝑢V − V𝑝
108 𝑥 27 𝑥 108 𝑥 27 2,𝑥
1
1
(𝑢 + V) ∑ 𝑚𝑙 𝑚𝑗−𝑙 + (𝑢 + V) ∑ 𝑚𝑙 𝑘𝑗−𝑙 .
6
3
𝑙=1
𝑙=1
𝜕
𝜕
[−𝜆 + V + (𝑢 + V) 𝑀] =
[𝑎 + (𝑏 + 𝑐) 𝑀] ,
𝜕𝑡
𝜕𝑥
7
5
1
1
5
VV + 𝑝 𝑝 + 𝑝 V − 𝑝 𝑢 − 𝑢V
36 𝑥 9 1 1,𝑥 9 1 𝑥 9 2 𝑥 36 𝑥
−
−
Because of
5
5
32
7
𝑛3 = 𝑝2,𝑥𝑥 + (𝑝1 𝑢)𝑥 − (𝑝2 V)𝑥 − 𝑝2 𝑝2,𝑥 − 𝑢𝑢𝑥
9
9
9
36
𝑘3 =
1
1
(2V + 𝑝2 ) 𝑘𝑗 − (𝑢 − V + 𝑝1 − 𝑝2 ) 𝑛𝑗
3
3
𝑗−1
1
1
5
5
− V𝑢𝑥 − 𝑢V𝑥 + V3 − 𝑢V2 ,
2
4
8
8
+
+
(49)
(50)
𝜕
[𝑒 + (𝑓 + 𝑔) 𝑀 + (𝑎 + 𝑒) 𝑁
𝜕𝑥
+ (𝑏 + 𝑐 + 𝑓 + 𝑔) 𝐾] ,
where
1
𝑎 = 𝜉0 𝜆2 + 𝜉1 𝜆 + 𝜉0 (V2 − 𝑢2 ) ,
2
1
𝑏 = −𝜉0 𝑢𝜆 + 𝜉0 (−𝑢V + V𝑥 ) − 𝜉1 𝑢,
2
1
𝑐 = −𝜉0 V𝜆 + 𝜉0 (−V2 + 𝑢𝑥 ) − 𝜉1 V,
2
𝑒 = 𝜉0 𝜆2 + 𝜉1 𝜆
1
1
1
1
1
1
+ 𝜉0 ( V𝑝2 − 𝑢𝑝1 + 𝑢2 − V2 − 𝑝12 + 𝑝22 ) ,
2
2
4
4
4
4
1
1
1
1
𝑓 = − 𝜉0 𝑝1 𝜆 + 𝜉0 ( 𝑝2,𝑥 − V𝑥 + 𝑢V − V𝑝1
4
4
2
2
1
1
− 𝑢𝑝2 − 𝑝1 𝑝2 ) − 𝜉1 𝑝1 ,
2
2
1
1
1
𝑔 = − 𝜉0 𝑝2 𝜆 + 𝜉0 ( 𝑝1,𝑥 − 𝑢𝑥 + V2
4
4
2
1
1
1
− V𝑝12 − 𝑝2 𝑝12 − 𝑢3 , . . . ,
9
9
8
(48)
1
− 𝑝22 − V𝑝2 ) − 𝜉1 𝑝2 .
2
(51)
8
Abstract and Applied Analysis
Assume that
𝜎 = −𝜆 + V + (𝑢 + V) 𝑀,
𝜃 = 𝑎 + (𝑏 + 𝑐) 𝑀,
𝜌 = − 𝜆 + 𝑝2 + (𝑝1 + 𝑝2 ) 𝑀 + (−2𝜆 + V + 𝑝2 ) 𝑁
(52)
+ (𝑢 + V + 𝑝1 + 𝑝2 ) 𝐾,
+
11
55
43
V𝑝 𝑝 + V𝑝2 − 𝑝 𝑢2
36 1 2 18 2 36 2
−
53
93
𝑢𝑝 𝑝 + 𝑝3
18 1 2 36 2
3
1
4
− 𝑝2 𝑝12 − V𝑝12 − V𝑝22
4
18
9
1
4
− 𝑢V𝑝1 − 𝑝1 𝑝22
9
9
𝛿 = 𝑒 + (𝑓 + 𝑔) 𝑀 + (𝑎 + 𝑒) 𝑁 + (𝑏 + 𝑐 + 𝑓 + 𝑔) 𝐾.
Then, (50) can be written as 𝜎𝑡 = 𝜃𝑥 and 𝜌𝑡 = 𝛿𝑥 , which are
the right form of conservation laws. We expand 𝜎, 𝜃, 𝜌, and 𝛿
as series in powers of 𝜆 with the coefficients, which are called
conserved densities and currents, respectively:
+
1 3 1
V + 𝑢V𝑥 )
18
36
5
5
+ 𝜉1 (−2𝑝2,𝑥 + V𝑝1 − 𝑢𝑝1 + V𝑝2 − 𝑢𝑝2
3
3
7
1
1
1
+ 𝑝22 + 𝑢2 − V2 − 𝑝12 ) , . . . .
3
6
6
3
∞
𝜎 = −𝜆 + V + (𝑢 + V) ∑𝜎𝑗 𝜆−𝑗 ,
(54)
𝑗=1
∞
𝜃 = 𝜉0 𝜆2 + 𝜉1 𝜆 + ∑𝜃𝑗 𝜆−𝑗 ,
The recursion relations for 𝜎𝑗 , 𝜃𝑗 , 𝜌𝑗 , and 𝛿𝑗 are as follows:
𝑗=1
∞
(53)
−𝑗
𝜌 = −𝜆 + 𝑝2 + ∑𝜌𝑗 𝜆 ,
𝑗=1
∞
𝛿 = 𝜉0 𝜆2 + 𝜉1 𝜆 + ∑𝛿𝑗 𝜆−𝑗 ,
𝑗=1
where 𝜉0 and 𝜉1 are constants of integration. The first
conserved densities and currents are read as follows:
𝜎1 =
1 2
(V − 𝑢2 ) ,
2
1
3
3
1
𝜃1 = 𝜉0 ( 𝑢𝑢𝑥 − 𝑢V𝑥 − VV𝑥 ) − 𝜉1 (V2 − 𝑢2 ) ,
2
4
4
2
𝜌1 =
1
1
4
1
𝑢𝑝1 + V𝑝2 − 𝑝22 − 𝑢2
2
3
3
6
1
1
1
+ V2 + 𝑝12 + 𝑢𝑝1 + 2𝑝2,𝑥 ,
6
3
6
𝛿1 = 𝜉0 (2𝑝2,𝑥𝑥 +
1
41
41
𝑝V + 𝑝𝑢 − 𝑝V
36 1 𝑥 36 1 𝑥 36 2 𝑥
−
1
41
1
𝑝2 𝑢𝑥 − V𝑝2,𝑥 − V𝑝1,𝑥
36
36
36
+
13
1
1
VV𝑥 − V𝑢𝑥 + 𝑢𝑝2,𝑥
36
36
36
+
41
13
257
𝑢𝑝 − 𝑢𝑢 −
𝑝𝑝
36 1,𝑥 36 𝑥 36 2 2,𝑥
+
41
1
𝑝𝑝 + 𝑝𝑝
36 1 1,𝑥 36 1 2,𝑥
−
1
47
1
𝑝2 𝑝1,𝑥 + 𝑝2 V2 − V𝑢2
36
36
18
𝜎𝑗 = (𝑢 + V) 𝑚𝑗 ,
1
1
𝜃𝑗 = − 𝜉0 (𝑢 + V) 𝑚𝑗+1 + 𝜉0 ( 𝑢𝑥 + V𝑥 − 𝑢V − V2 ) 𝑚𝑗
2
2
− 𝜉1 (𝑢 + V) 𝑚𝑗 ,
𝜌𝑗 = (𝑝1 + 𝑝2 ) 𝑚𝑗 − 2𝑛𝑗+1 + (V + 𝑝2 ) 𝑛𝑗
+ (𝑢 + V + 𝑝1 + 𝑝2 ) 𝑘𝑗 ,
𝛿𝑗 = 𝜉0 [ − (𝑝1 + 𝑝2 ) 𝑚𝑗+1
1
1
1
+ ( 𝑝2,𝑥 + 𝑝1,𝑥 − V𝑥
4
4
4
1
1
1
− 𝑢𝑥 + 𝑢V − V𝑝1
4
2
2
1
1
1
− 𝑢𝑝2 − 𝑝1 𝑝2 + V2
2
2
2
1
− 𝑝22 − 𝑝2 V) 𝑚𝑗
2
1
1
1
+ 2𝑛𝑗+2 + ( V2 − 𝑢2 + V𝑝2
2
2
2
1
1
− 𝑢𝑝1 + 𝑢2
2
4
1
1
1
+ 𝑝22 − V2 − 𝑝12 ) 𝑛𝑗
4
4
4
− (𝑢 + V + 𝑝1 + 𝑝2 ) 𝑘𝑗+1
Abstract and Applied Analysis
9
1
1
1
1
1
+ ( 𝑢𝑥 + V𝑥 + 𝑝1,𝑥 + 𝑝2,𝑥 − 𝑢V
4
4
4
4
2
1
1
1
− V2 − V𝑝1 − 𝑢𝑝2
2
2
2
1
1
− 𝑝1 𝑝2 − 𝑝22 − V𝑝2 )]
2
2
+ 𝜉1 [2𝑛𝑗+1 − (𝑝1 + 𝑝2 ) 𝑚𝑗
− (𝑢 + V + 𝑝1 + 𝑝2 ) 𝑘𝑗 ] ,
(55)
where 𝑚𝑗 , 𝑛𝑗 , and 𝑘𝑗 can be calculated from (49). The infinite
conservation laws of nonlinear integrable couplings (37) can
be easily obtained in (45)–(55), respectively.
5. Conclusions
In this paper, a new explicit Lie algebra was introduced, and
a new nonlinear integrable couplings of Li soliton hierarchy
with self-consistent sources was worked out. Then, the
conservation laws of Li soliton hierarchy were also obtained.
The method can be used to other soliton hierarchy with selfconsistent sources. In the near future, we will investigate
exact solutions of nonlinear integrable couplings of soliton
equations with self-consistent sources which are derived by
using our method.
Acknowledgments
The study is supported by the National Natural Science
Foundation of China (Grant nos. 11271008, 61072147, and
1071159 ), the Shanghai Leading Academic Discipline Project
(Grant no. J50101), and the Shanghai University Leading
Academic Discipline Project (A. 13-0101-12-004).
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