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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 598570, 10 pageshttp://dx.doi.org/10.1155/2013/598570
Research ArticleSelf-Consistent Sources and Conservation Laws for NonlinearIntegrable Couplings of the Li Soliton Hierarchy
Han-yu Wei1,2 and Tie-cheng Xia1
1 Department of Mathematics, Shanghai University, Shanghai 200444, China2Department 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 Β© 2013 H.-y. Wei and T.-c. Xia. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
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 theinfinitely many conservation laws for the nonlinear integrable coupling of Li soliton hierarchy.
1. Introduction
Soliton theory has achieved great success during the lastdecades, it is being applied to many fields. The diversityand complexity of soliton theory enables investigators to doresearch fromdifferent views, such as binary nonlinearizationof soliton hierarchy [1] and BaΜcklund transformations ofsoliton systems from symmetry constraints [2].
Recently, with the development of integrable systems,integrable couplings have attracted much attention. Inte-grable couplings [3, 4] are coupled systems of integrableequations, which have been introduced when we study ofVirasoro symmetric algebras. It is an important topic tolook for integrable couplings because integrable couplingshavemuch richermathematical structures and better physicalmeanings. In recent years, many methods of searching forintegrable couplings have been developed [5β13], but allthe integrable couplings obtained are linear for the V =(V1, . . . , V
π)
π. As for how to generate nonlinear integrablecouplings, Ma proposed a general scheme [14]. Suppose thatan integrable system
π’π‘= πΎ (π’) (1)
has a Lax pair π and π, which belong to semisimple matrixLie algebras. Introduce an enlarged spectral matrix
π = π (π’) = [
π (π’) 0
ππ(V) π (π’) + π
π(V)] (2)
from a zero curvature representation
ππ‘β ππ₯+ [π,π] = 0, (3)
where
π = π (π’) = [
π (π’) 0
ππ(π’) π (π’) + π
π(π’)
] , (4)
then we can give rise to
ππ‘β ππ₯+ [π,π] = 0,
ππ,π‘β ππ,π₯+ [π,π
π] + [π
π, π] + [π
π, ππ] = 0.
(5)
This is an integrable coupling of (1), and it is a nonlinearintegrable coupling because the commutator [π
π, ππ] can
generate nonlinear terms.Soliton equation with self-consistent sources (SESCS)
[15β22] is an important part in soliton theory. Physically,
2 Abstract and Applied Analysis
the sources may result in solitary waves with a nonconstantvelocity and therefore lead to a variety of dynamics of physicalmodels. For applications, these kinds of systems are usuallyused to describe interactions between different solitary wavesand are relevant to some problems of hydrodynamics, solidstate physics, plasma physics, and so forth. How to obtain anintegrable coupling of the SESCS is an interesting topic; inthis paper, we will use new formula [23] presented by us togeneralize soliton hierarchy with self-consistent sources.
The conservation laws play an important role in dis-cussing the integrability for soliton hierarchy. An infinitenumber of conservation laws for KdV equation was firstdiscovered by Miura et al. [24]. The direct constructionmethod of multipliers for the conservation laws was pre-sented [25], the Lagrangian approach for evolution equa-tions was considered in [26], Wang and Xia established theinfinitely many conservation laws for the integrable superπΊ-π½ hierarchy [27], and the infinite conservation laws of thegeneralized quasilinear hyperbolic equations were derived in[28]. Comparatively, the less nonlinear integrable couplingsof the soliton equations have been considered for theirconservation laws.
This paper is organized as follows. In Section 2, a kind ofexplicit Lie algebras with the forms of blocks is introducedto generate nonlinear integrable couplings of Li solitonhierarchy. In Section 3, a new nonlinear integrable couplingof Li soliton hierarchy with self-consistent sources is derived.In Section 4, we obtain the conservation laws for the non-linear integrable couplings of Li hierarchy. Finally, someconclusions are given.
2. Lie Algebras for Constructing NonlinearIntegrable Couplings of Li Soliton Hierarchy
Tu [29] presented a base of the Li algebra sl(2) as follows:
πΊ1= span {π
1, π2, π3} , (6)
where
π1= (
1 0
0 β1
) , π2= (
0 1
1 0
) , π3= (
0 1
β1 0
) , (7)
which have the commutative relations
[π1, π2] = 2π
2, [π
1, π3] = β2π
3, [π
2, π3] = π1. (8)
Let us introduce a Lie algebra with matrix blocks by usingπΊ1in order to get nonlinear couplings of soliton hierarchy as
follows:
πΊ = span {π1, . . . , π
6} , (9)
where
π1= (
π10
0 π1
) , π2= (
π20
0 π2
) ,
π3= (
π30
0 π3
) , π4= (
0 0
π1π1
) ,
π5= (
0 0
π2π2
) , π6= (
0 0
π3π3
) .
(10)
Define a commutator as follows:
[π, π] = ππ β ππ, π, π β πΊ. (11)
A direct verification exhibits that
[π1, π2] = 2π
3, [π
1, π3] = 2π
2,
[π2, π3] = β2π
1, [π
1, π5] = 2π
6,
[π1, π6] = 2π
5, [π
2, π4] = β2π
6,
[π2, π6] = β2π
4,
[π3, π4] = β2π
5, [π
3, π5] = 2π
4,
[π4, π5] = 2π
6, [π
4, π6] = 2π
5,
[π5, π6] = β2π
4, [π
1, π4] = [π
3, π6] = 0.
(12)
Set
ΜπΊ1= span {π
1, π2, π3} ,
ΜπΊ2= span {π
4, π5, π6} , (13)
then we find that
πΊ =ΜπΊ1βΜπΊ2,
ΜπΊ1β πΊ1, [
ΜπΊ1,ΜπΊ2] βΜπΊ2, (14)
andΜπΊ1andΜπΊ
2are all simple Lie subalgebras.
While we use Lie algebras to generate integrable hierar-chies of evolution equations, we actually employ their loopalgebras ΜπΊ = πΊ β πΆ(π, πβ1) to establish Lax pairs, whereπΆ(π, π
β1) represents a set of Laurent ploynomials in π and πΊ
is a Lie algebra. Based on this, we give the loop algebras of (9)as follows:
ΜπΊ = span {π
1(π) , . . . , π
6(π)} , (15)
whereππ(π) = π
ππ
π, [ππ(π), π
π(π)] = [π
π, ππ]π
π+π, 1 β€ π, π β€ 6,π, π β π.
We consider an auxiliary linear problem as follows:
(
π1
π2
π3
π4
)
π₯
= π (π’, π)(
π1
π2
π3
π4
),
π (π’, π) = π 1+
6
β
π=1
π’πππ(π) ,
(
π1
π2
π3
π4
)
π‘π
= ππ(π’, π)(
π1
π2
π3
π4
),
(16)
where π’ = (π’1, . . . , π’
π )
π, ππ= π 1+ π’1π1+ β β β + π’
6π6, π 1is a
pseudoregular element, π’π(π, π‘) = π’
π(π = 1, 2, . . . , 6), and π
π=
π(π₯, π‘) are field variables defined on π₯ β π , π‘ β π , ππ(π) β
ΜπΊ.
The compatibility of (16) gives rise to thewell-known zerocurvature equation
ππ‘β ππ₯+ [π,π] = 0, ππ‘
= 0. (17)
Abstract and Applied Analysis 3
The general scheme of searching for the consistent ππ,
and generating a hierarchy of zero curvature equations wasproposed in [30]. Solving the following equation:
ππ₯= [π,π] ,
π =
β
β
π=0
πππ
βπ
= (
π π + π 0 0
π β π βπ 0 0
π π + π π + π π + π + π + π
π β π βπ π β π + π β π β (π + π)
) ,
(18)
then we sesrch for πβΜπΊ, the new π
πcan be constructed by
ππ=
π
β
π=0
ππ(π’) π
πβπ+ π(π’, π) . (19)
Solving zero curvature (17), we could get evolution equationas follows:
π’π‘= πΎ(π’, π’
π₯, . . . ,
π
ππ’
ππ₯
π) . (20)
Now, we consider Li soliton hierarchy [31]. In order to setup nonlinear integrable couplings of the Li soliton hierarchywith self-consistent sources, we first consider the followingmatrix spectral problem:
ππ₯= π (π’, π) π,
π (π’, π) = β π1(1) + Vπ
1(0) + π’π
2(0) + Vπ
3(0)
β π4(1) + π
2π4(0) + π
1π5(0) + π
2π6(0) ,
(21)
that is,
π (π’, π)
= (
βπ + V π’ + V 0 0
π’ β V π β V 0 0
βπ + π2π1+ π2β2π + V + π
2π’ + V + π
1+ π2
π1β π2π β π2π’ β V + π
1β π2
2π β V β π2
)
= (
π1
0
π0π1+ π0
) ,
(22)
where π is a spectral parameter and π1satisfies π
π₯= π1π
which is matrix spectral problem of the Li soliton hierarchy[31].
To establish the nonlinear integrable coupling system ofthe Li soliton hierarchy, the adjoint equation π
π₯= [π,π] of
the spectral problem (21) is firstly solved, we assume that asolution π is given by the following:
π = (
π π + π 0 0
π β π βπ 0 0
π π + π π + π π + π + π + π
π β π βπ π β π + π β π β (π + π)
)
=
β
β
π=0
πππ
βπ
=
β
β
π=0
Γ(
ππ ππ + ππ 0 0
ππ β ππ βππ 0 0
ππ ππ + ππ ππ + ππ ππ + ππ + ππ + ππ
ππ β ππ βππ ππ β ππ + ππ β ππ β (ππ + ππ)
)π
βπ.
(23)Therefore, the condition (18) becomes the following recursionrelation:
ππ,π₯= 2Vππβ 2π’ππ,
ππ,π₯= β2π
π+1+ 2Vππβ 2Vππ,
ππ,π₯= β2π
π+1+ 2Vππβ 2π’π
π,
ππ,π₯= β 2π’π
π+ 2Vπ
πβ 2π1ππ
+ 2π2ππβ 2π1ππ+ 2π2ππ,
ππ,π₯= β 2π
π+1+ 2Vπ
πβ 2Vππ
+ 2π2ππβ 2π2ππ+ 2π2ππβ 2π2ππ,
ππ,π₯= β 2π
π+1+ 2Vπ
πβ 2π’ππ+ 2π2ππ
β 2π1ππ+ 2π2ππβ 2π1ππ.
(24)
Choose the initial dataπ0= π0= π½, π
0= π0= π0= π0= 0, (25)
we see that all sets of functions ππ, ππ, ππ, ππ, ππ, and π
πare
uniquely determined. In particular, the first few sets are asfollows:
π1= 0, π
1= βπ’π½, π
1= βVπ½, π
1= 0,
π1= βπ1π½, π
1= βπ2π½, π
2=
1
2
(V2β π’
2) ,
π2= (
1
2
Vπ₯β π’V)π½, π
2= (
1
2
π’π₯β V2)π½,
π2= (
1
2
Vπ2β
1
2
π’π1+
1
4
π’
2β
1
4
V2β
1
4
π
2
1+
1
4
π
2
2)π½,
4 Abstract and Applied Analysis
π2= (
1
4
π2,π₯β
1
4
Vπ₯+
1
2
π’V β1
2
Vπ1β
1
2
π’π2β
1
2
π1π2)π½,
π2= (
1
4
π1,π₯β
1
4
π’π₯+
1
2
V2β
1
2
π
2
2β Vπ2)π½, . . . .
(26)
Considering
ππ= π +
π,
π
=(
β (ππ β ππ) 0 0 0
0 ππ β ππ 0 0
β (ππ β ππ) 0 β (ππ β ππ) β (ππ β ππ)
0 ππ β ππ 0 ππ β ππ + ππ β ππ
).
(27)
From the zero curvature equation ππ‘β ππ₯+ [π,π] = 0, we
obtain the nonlinear integrable coupling system
π’π‘π
= πΎπ= (
π’
V
π1
π2
)
π‘π
= (
ππ,π₯
β(ππβ ππ)π₯
ππ,π₯
β(ππβ ππ)
π₯
)
= π½(
ππ
ππβ ππ
ππ
ππβ ππ
) = π½πΏ
π(
0
π½
0
π½
) , π β₯ 0,
(28)
with theHamiltonian operator π½ and the hereditary recursionoperator πΏ, respectively, as follows:
π½ = (
π 0 0 0
0 βπ 0 0
0 0 π 0
0 0 0 βπ
) ,
πΏ =(
0
1
2
π β π’ 0 0
π
β1π’π +
1
2
π π
β1Vπ + V 0 0
0 π1
0 π2
π3
π4
π5π6
),
(29)
where
π1= β
1
4
π β
1
2
π1+
1
2
π’,
π2=
1
4
π β
1
2
π1β
1
2
π’,
π3= β
1
2
π
β1π’ β
1
2
π
β1π1β π
β1π1π β
1
4
π,
π4= β
1
2
π
β1Vπ +
1
2
π
β1π2π + π
β1π2π’ β
1
2
V +1
2
π2,
π5=
1
2
π
β1π’π +
1
2
π
β1π1π +
1
4
π,
π6= β 2π
β1π’V β
3
2
π
β1π1V +
1
2
π
β1Vπ
+
1
2
π
β1π2π β
1
2
V +1
2
π2.
(30)
Obviously, when π1= π2= 0 in (28), the above results
become Li soliton hierarchy. So, we can say that (28) isintegrable coupling of the Li soliton hierarchy.
Taking π = 2, we get that the nonlinear integrablecouplings of Li soliton hierarchy are as follows:
π’π‘2
= (β
1
2
Vπ₯π₯β π’π₯V β π’V
π₯)π½,
Vπ‘2
= (
1
2
π’π₯π₯β 3VVπ₯+ π’π’π₯)π½,
π1,π‘2
= (
1
4
π2,π₯β
1
4
Vπ₯+
1
2
π’V β1
2
Vπ1
β
1
2
π’π2β
1
2
π1π2)
π₯
π½,
π2,π‘2
= (
1
4
π1,π₯β
1
4
π’π₯+
1
2
V2β
3
4
π
2
2β
3
2
Vπ2
+
1
2
π’π1β
1
4
π’
2+
1
4
V2+
1
4
π
2
1)
π₯
π½.
(31)
So, we can say that the system in (28) with π β₯ 2provides a hierarchy of nonlinear integrable couplings for theLi hierarchy of the soliton equation.
3. Self-Consistent Sources for the NonlinearIntegrable Couplings of Li Soliton Hierarchy
According to (16), now we consider a new auxiliary linearproblem. For π distinct π
π, π = 1, 2, . . . , π and the systems
of (16) become in the following form:
(
π1π
π2π
π3π
π4π
)
π₯
= π (π’, ππ)(
π1π
π2π
π3π
π4π
)
=
6
β
π=1
π’πππ(π)(
π1π
π2π
π3π
π4π
), π = 1, . . . , π,
Abstract and Applied Analysis 5
(
π1π
π2π
π3π
π4π
)
π‘π
= ππ(π’, ππ)(
π1π
π2π
π3π
π4π
)
= [
π
β
π=0
ππ(π’) π
πβπ
π+ Ξπ(π’, ππ)]
Γ(
π1π
π2π
π3π
π4π
), π = 1, . . . , π.
(32)
Based on the result in [32], we show that the followingequation
πΏπ»π
πΏπ’
+
π
β
π=1
πΌπ
πΏππ
πΏπ’
= 0 (33)
holds true, where πΌπis a constant. From (32), we may know
that
πΏππ
πΏπ’π
= πΌπTr(Ξ¨
π
ππ (π’, ππ)
ππ’π
)
= πΌπTr (Ξ¨πππ(ππ)) , π = 1, 2,
(34)
where Tr denotes the trace of a matrix and
Ξ¨π= (
π1ππ2π
βπ
2
1ππ3ππ4π
βπ
2
3π
π
2
2πβπ1ππ2π
π
2
4πβπ3ππ4π
0 0 π1ππ2π
βπ
2
1π
0 0 π
2
2πβπ1ππ2π
),
π = 1, . . . , π.
(35)
For π = 3, 4 we define that
πΏππ
πΏπ’π
= π½πTr(Ξ¨
ππ΄
ππ0(π’, ππ)
ππ’π
) , (36)
where
π = (
π1
0
π0π1+ π0
) ,
Ξ¨ππ΄= (
π3ππ4π
βπ
2
3π
π
2
4πβπ3ππ4π
) ,
(37)
and π½πis a constant.
According to (34) and (36), we obtain a kind of nonlinearintegrable couplings with self-consistent sources as follows:
π’π‘π
= π½
πΏπ»π+1
πΏπ’π
+ π½
π
β
π=1
πΌπ
πΏππ
πΏπ’
= π½πΏ
ππΏπ»1
πΏπ’π
+ π½
π
β
π=1
πΌπ
πΏππ
πΏπ’
, π = 1, 2, . . . .
(38)
Therefore, according to formulas (34) and (36), we have thefollowing results by direct computations:
π
β
π=1
πΏππ
πΏπ’
=
π
β
π=1
(
(
(
(
(
(
(
(
(
πΏππ
πΏπ’
πΏππ
πΏV
πΏππ
πΏπ1
πΏππ
πΏπ1
)
)
)
)
)
)
)
)
)
=(
2(β¨Ξ¦2, Ξ¦2β© β β¨Ξ¦
1, Ξ¦1β©)
2 (β¨Ξ¦1, Ξ¦1β© + β¨Ξ¦
2, Ξ¦2β© + 2 β¨Ξ¦
1, Ξ¦2β©)
β¨Ξ¦4, Ξ¦4β© β β¨Ξ¦
3, Ξ¦3β©
β¨Ξ¦3, Ξ¦3β© + β¨Ξ¦
4, Ξ¦4β© + 2 β¨Ξ¦
3, Ξ¦4β©
) ,
(39)
by taking πΌπ= 1 and π½
π= 1 in formulas (34) and (36).
Therefore, we have nonlinear integrable coupling system ofthe Li equations hierarchy with self-consistent sources asfollows:
π’π‘π
= πΎπ
= (
π’
V
π1
π2
)
π‘π
= π½πΏ
π(
0
π½
0
π½
)
+ π½(
2 (β¨Ξ¦2, Ξ¦2β© β β¨Ξ¦
1, Ξ¦1β©)
2 (β¨Ξ¦1, Ξ¦1β© + β¨Ξ¦
2, Ξ¦2β© + 2 β¨Ξ¦
1, Ξ¦2β©)
β¨Ξ¦4, Ξ¦4β© β β¨Ξ¦
3, Ξ¦3β©
β¨Ξ¦3, Ξ¦3β© + β¨Ξ¦
4, Ξ¦4β© + 2 β¨Ξ¦
3, Ξ¦4β©
)
= π½πΏ
π(
0
π½
0
π½
) + π½
(
(
(
(
(
(
(
(
(
(
(
2
π
β
π=1
(π
2
2πβ π
2
1π)
2
π
β
π=1
(π
2
1π+ π
2
2π+ 2π1ππ2π)
π
β
π=1
(π
2
4πβ π
2
3π)
π
β
π=1
(π
2
3π+ π
2
4π+ 2π3ππ4π)
)
)
)
)
)
)
)
)
)
)
)
,
(40)
6 Abstract and Applied Analysis
with
π1π,π₯= (βπ + V) π
1π+ (π’ + V) π
2π,
π2π,π₯= (π’ β V) π
1π+ (π β V) π
2π,
π3π,π₯= (βπ + π
2) π1π+ (π1+ π2) π2π
+ (β2π + V + π2) π3π+ (π’ + V + π
1+ π2) π4π,
π4π,π₯= (π1β π2) π1π+ (π β π
2) π2π
+ (π’ β V + π1β π2) π3π
+ (2π β V β π2) π4π, π = 1, . . . , π,
(41)
where Ξ¦π= (ππ1, . . . , π
ππ), π = 1, 2, 3, 4, and β¨β , β β© is the
standard inner product in π π.When π = 2 and π½ = 2, we obtain nonlinear integrable
couplings of Li hierarchy with self-consistent sources
π’π‘2
= β
1
2
Vπ₯π₯β π’π₯V β π’V
π₯+ 2π
π
β
π=1
(π
2
2πβ π
2
1π) ,
Vπ‘2
=
1
2
π’π₯π₯β 3VVπ₯+ π’π’π₯
β 2π
π
β
π=1
(π
2
1π+ π
2
2π+ 2π1ππ2π) ,
π1π‘2
= (
1
4
π2,π₯β
1
4
Vπ₯+
1
2
π’V β1
2
Vπ1β
1
2
π’π2
β
1
2
π1π2)
π₯
+ π
π
β
π=1
(π
2
4πβ π
2
3π) ,
π2π‘2
= (
1
4
π1,π₯β
1
4
π’π₯+
1
2
V2β
3
4
π
2
2β
3
2
Vπ2+
1
2
π’π1
β
1
4
π’
2+
1
4
V2+
1
4
π
2
1)
π₯
β π
π
β
π=1
(π
2
3π+ π
2
4π+ 2π3ππ4π) ,
(42)
with
π1π,π₯= (βπ + V) π
1π+ (π’ + V) π
2π,
π2π,π₯= (π’ β V) π
1π+ (π β V) π
2π,
π3π,π₯= (βπ + π
2) π1π+ (π1+ π2) π2π
+ (β2π + V + π2) π3π+ (π’ + V + π
1+ π2) π4π,
π4π,π₯= (π1β π2) π1π+ (π β π
2) π2π
+ (π’ β V + π1β π2) π3π
+ (2π β V β π2) π4π, π = 1, . . . , π.
(43)
4. Conservation Laws for the NonlinearIntegrable Couplings of Li Soliton Hierarchy
In what follows, we will construct conservation laws for thenonlinear integrable couplings of the Li hierarchy. For thecoupled spectral problem of Li hierarchy
π (π’, π)
= (
βπ + V π’ + V 0 0
π’ β V π β V 0 0
βπ + π2π1+ π2β2π + V + π
2π’ + V + π
1+ π2
π1β π2π + βπ
2π’ β V + π
1β π2
2π β V β π2
),
(44)
we introduce the variables
π =
π2
π1
, π =
π3
π1
, πΎ =
π4
π1
. (45)
From (44), we have
ππ₯= π’ β V + 2ππ β 2Vπβ (π’ + V)π
2,
ππ₯= β π + π
2β ππ + (π
1+ π2)π
+ π2π + (π’ + V + π
1+ π2)πΎ β (π’ + V)ππ,
πΎπ₯= π1β π2+ 3ππΎ + ππ β π
2π
β (2V + π2)πΎ + (π’ β V + π
1β π2)π β (π’ + V) πΎπ.
(46)
We expandπ,π, andπΎ in powers of π as follows:
π =
β
β
π=1
πππ
βπ, π =
β
β
π=1
πππ
βπ,
πΎ =
β
β
π=1
πππ
βπ.
(47)
Abstract and Applied Analysis 7
Substituting (47) into (46) and comparing the coefficients ofthe same power of π, we obtain the following:
π1=
1
2
(V β π’) , π1= π2,
π1=
1
3
(π2β π1) +
1
6
(π’ β V) ,
π2=
1
4
(V β π’)π₯+
1
2
(V2β π’V) ,
π2= βπ2,π₯β
1
3
π
2
1β
2
3
π’π1+
2
3
Vπ2+
4
3
π
2
2+
1
6
π’
2β
1
6
V2,
π2=
5
36
(π’ β V)π₯+
1
9
(π2β π1)
π₯β
5
18
V2+
5
18
π’V
+
2
3
Vπ2β
4
9
π’π2+
4
9
π
2
2β
4
9
π1π2β
2
9
Vπ1,
π3=
1
8
(V β π’)π₯π₯+
1
8
π’
3β
1
8
π’
2V +
3
4
VVπ₯
β
1
2
Vπ’π₯β
1
4
π’Vπ₯+
5
8
V3β
5
8
π’V2,
π3= π2,π₯π₯+
5
9
(π1π’)
π₯β
5
9
(π2V)π₯β
32
9
π2π2,π₯β
7
36
π’π’π₯
+
7
36
VVπ₯+
5
9
π1π1,π₯+
1
9
π1Vπ₯β
1
9
π2π’π₯β
5
36
π’Vπ₯
+
1
9
π’π2,π₯+
5
36
Vπ’π₯β
1
9
Vπ1,π₯+
1
9
π1π2,π₯β
1
9
π2π1,π₯
β
4
9
π1π’V +
11
9
π2V2β
14
9
π’π1π2+
16
9
Vπ2
2
+
16
9
π
3
2β
7
9
π2π’
2β
7
9
π2π
2
1+
5
18
Vπ’2β
5
18
V3β
2
9
Vπ2
1,
π3=
19
216
(π’ β V)π₯π₯+
1
27
(π2β π1)
π₯π₯+
5
54
(π’ β V)π₯
β
47
108
VVπ₯+
7
27
Vπ’π₯+
19
108
π’Vπ₯β
4
27
Vπ2,π₯
+
7
27
π2Vπ₯+
5
27
π’π2,π₯β
5
27
π2π’π₯β
5
27
π2π1,π₯
+
5
27
π1π2,π₯β
2
27
π1Vπ₯β
4
27
Vπ1,π₯β
103
216
V3
+
101
216
π’V2+
1
8
Vπ’2+
7
18
π2V2β
16
27
π2π’V
β
19
54
π2V2+
32
27
Vπ2
2β
16
27
π1π2V β
4
27
π1V2
β
16
27
π’π
2
2+
16
27
π
3
2β
16
27
π1π
2
2+
2
9
π1π’
2
+
1
18
π’V2+
1
3
π’π
2
1+
1
9
π
3
1β
2
9
π’Vπ1β
2
9
π’π1π2
β
1
9
Vπ2
1β
1
9
π2π
2
1β
1
8
π’
3, . . . ,
(48)
and a recursion formula forππ, ππ, and π
πas follows:
ππ+1=
1
2
ππ,π₯+ Vππ+
1
2
(π’ + V)
πβ1
β
π=1
ππππβπ,
ππ+1= β π
π,π₯+ (π1+ π2)ππ+ π2ππ
+ (π’ + V + π1+ π2) ππβ (π’ + V)
πβ1
β
π=1
ππππβπ,
ππ+1=
1
3
ππ,π₯β
1
6
ππ,π₯β
1
3
Vππ+
1
3
π2ππ
+
1
3
(2V + π2) ππβ
1
3
(π’ β V + π1β π2) ππ
β
1
6
(π’ + V)
πβ1
β
π=1
ππππβπ+
1
3
(π’ + V)
πβ1
β
π=1
ππππβπ.
(49)
Because of
π
ππ‘
[βπ + V + (π’ + V)π] =π
ππ₯
[π + (π + π)π] ,
π
ππ‘
[βπ + π2+ (π1+ π2)π + (β2π + V + π
2)π
+ (π’ + V + π1+ π2)πΎ]
=
π
ππ₯
[π + (π + π)π + (π + π)π
+ (π + π + π + π)πΎ] ,
(50)
where
π = π0π
2+ π1π +
1
2
π0(V2β π’
2) ,
π = βπ0π’π + π
0(βπ’V +
1
2
Vπ₯) β π1π’,
π = βπ0Vπ + π
0(βV2+
1
2
π’π₯) β π1V,
π = π0π
2+ π1π
+ π0(
1
2
Vπ2β
1
2
π’π1+
1
4
π’
2β
1
4
V2β
1
4
π
2
1+
1
4
π
2
2) ,
π = β π0π1π + π0(
1
4
π2,π₯β
1
4
Vπ₯+
1
2
π’V β1
2
Vπ1
β
1
2
π’π2β
1
2
π1π2) β π1π1,
π = β π0π2π + π0(
1
4
π1,π₯β
1
4
π’π₯+
1
2
V2
β
1
2
π
2
2β Vπ2) β π1π2.
(51)
8 Abstract and Applied Analysis
Assume that
π = βπ + V + (π’ + V)π,
π = π + (π + π)π,
π = β π + π2+ (π1+ π2)π + (β2π + V + π
2)π
+ (π’ + V + π1+ π2)πΎ,
πΏ = π + (π + π)π + (π + π)π + (π + π + π + π)πΎ.
(52)
Then, (50) can be written as ππ‘= ππ₯and π
π‘= πΏπ₯, which are
the right form of conservation laws. We expand π, π, π, and πΏas series in powers of π with the coefficients, which are calledconserved densities and currents, respectively:
π = βπ + V + (π’ + V)
β
β
π=1
πππ
βπ,
π = π0π
2+ π1π +
β
β
π=1
πππ
βπ,
π = βπ + π2+
β
β
π=1
πππ
βπ,
πΏ = π0π
2+ π1π +
β
β
π=1
πΏππ
βπ,
(53)
where π0and π
1are constants of integration. The first
conserved densities and currents are read as follows:
π1=
1
2
(V2β π’
2) ,
π1= π0(
1
2
π’π’π₯β
3
4
π’Vπ₯β
3
4
VVπ₯) β
1
2
π1(V2β π’
2) ,
π1=
1
2
π’π1+
1
3
Vπ2β
4
3
π
2
2β
1
6
π’
2
+
1
6
V2+
1
3
π
2
1+
1
6
π’π1+ 2π2,π₯,
πΏ1= π0(2π2,π₯π₯+
1
36
π1Vπ₯+
41
36
π1π’π₯β
41
36
π2Vπ₯
β
1
36
π2π’π₯β
41
36
Vπ2,π₯β
1
36
Vπ1,π₯
+
13
36
VVπ₯β
1
36
Vπ’π₯+
1
36
π’π2,π₯
+
41
36
π’π1,π₯β
13
36
π’π’π₯β
257
36
π2π2,π₯
+
41
36
π1π1,π₯+
1
36
π1π2,π₯
β
1
36
π2π1,π₯+
47
36
π2V2β
1
18
Vπ’2
+
11
36
Vπ1π2+
55
18
Vπ2
2β
43
36
π2π’
2
β
53
18
π’π1π2+
93
36
π
3
2
β
3
4
π2π
2
1β
1
18
Vπ2
1β
4
9
Vπ2
2
β
1
9
π’Vπ1β
4
9
π1π
2
2
+
1
18
V3+
1
36
π’Vπ₯)
+ π1(β2π2,π₯+ Vπ1β
5
3
π’π1+
5
3
Vπ2β π’π2
+
7
3
π
2
2+
1
6
π’
2β
1
6
V2β
1
3
π
2
1) , . . . .
(54)
The recursion relations for ππ, ππ, ππ, and πΏ
πare as follows:
ππ= (π’ + V)π
π,
ππ= β π
0(π’ + V)π
π+1+ π0(
1
2
π’π₯+
1
2
Vπ₯β π’V β V
2)ππ
β π1(π’ + V)π
π,
ππ= (π1+ π2)ππβ 2ππ+1+ (V + π
2) ππ
+ (π’ + V + π1+ π2) ππ,
πΏπ= π0[ β (π
1+ π2)ππ+1
+ (
1
4
π2,π₯+
1
4
π1,π₯β
1
4
Vπ₯
β
1
4
π’π₯+
1
2
π’V β1
2
Vπ1
β
1
2
π’π2β
1
2
π1π2+
1
2
V2
β
1
2
π
2
2β π2V)ππ
+ 2ππ+2+ (
1
2
V2β
1
2
π’
2+
1
2
Vπ2
β
1
2
π’π1+
1
4
π’
2
+
1
4
π
2
2β
1
4
V2β
1
4
π
2
1) ππ
β (π’ + V + π1+ π2) ππ+1
Abstract and Applied Analysis 9
+ (
1
4
π’π₯+
1
4
Vπ₯+
1
4
π1,π₯+
1
4
π2,π₯β
1
2
π’V
β
1
2
V2β
1
2
Vπ1β
1
2
π’π2
β
1
2
π1π2β
1
2
π
2
2β Vπ2)]
+ π1[2ππ+1β (π1+ π2)ππ
β (π’ + V + π1+ π2) ππ] ,
(55)
whereππ, ππ, and π
πcan be calculated from (49).The infinite
conservation laws of nonlinear integrable couplings (37) canbe easily obtained in (45)β(55), respectively.
5. Conclusions
In this paper, a new explicit Lie algebra was introduced, anda new nonlinear integrable couplings of Li soliton hierarchywith self-consistent sources was worked out. Then, theconservation laws of Li soliton hierarchy were also obtained.The method can be used to other soliton hierarchy with self-consistent sources. In the near future, we will investigateexact solutions of nonlinear integrable couplings of solitonequations with self-consistent sources which are derived byusing our method.
Acknowledgments
The study is supported by the National Natural ScienceFoundation of China (Grant nos. 11271008, 61072147, and1071159 ), the Shanghai Leading Academic Discipline Project(Grant no. J50101), and the Shanghai University LeadingAcademic Discipline Project (A. 13-0101-12-004).
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