Self-Consistent Sources and Conservation Laws for Nonlinear ......Self-Consistent Sources and Conservation Laws for Nonlinear Integrable Couplings of the Li Soliton Hierarchy Han-yuWei1,2
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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 © 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 BaÌcklund 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 [5â13], but allthe integrable couplings obtained are linear for the V =(V1, . . . , V
ð)
ð. As for how to generate nonlinear integrablecouplings, Ma proposed a general scheme [14]. Suppose thatan integrable system
ð¢ð¡= ðŸ (ð¢) (1)
has a Lax pair ð and ð, which belong to semisimple matrixLie algebras. Introduce an enlarged spectral matrix
ð = ð (ð¢) = [
ð (ð¢) 0
ðð(V) ð (ð¢) + ð
ð(V)] (2)
from a zero curvature representation
ðð¡â ðð¥+ [ð,ð] = 0, (3)
where
ð = ð (ð¢) = [
ð (ð¢) 0
ðð(ð¢) ð (ð¢) + ð
ð(ð¢)
] , (4)
then we can give rise to
ðð¡â ðð¥+ [ð,ð] = 0,
ðð,ð¡â ðð,ð¥+ [ð,ð
ð] + [ð
ð, ð] + [ð
ð, ðð] = 0.
(5)
This is an integrable coupling of (1), and it is a nonlinearintegrable coupling because the commutator [ð
ð, ðð] can
generate nonlinear terms.Soliton equation with self-consistent sources (SESCS)
[15â22] 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 superðº-ðœ 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:
ðº1= span {ð
1, ð2, ð3} , (6)
where
ð1= (
1 0
0 â1
) , ð2= (
0 1
1 0
) , ð3= (
0 1
â1 0
) , (7)
which have the commutative relations
[ð1, ð2] = 2ð
2, [ð
1, ð3] = â2ð
3, [ð
2, ð3] = ð1. (8)
Let us introduce a Lie algebra with matrix blocks by usingðº1in order to get nonlinear couplings of soliton hierarchy as
follows:
ðº = span {ð1, . . . , ð
6} , (9)
where
ð1= (
ð10
0 ð1
) , ð2= (
ð20
0 ð2
) ,
ð3= (
ð30
0 ð3
) , ð4= (
0 0
ð1ð1
) ,
ð5= (
0 0
ð2ð2
) , ð6= (
0 0
ð3ð3
) .
(10)
Define a commutator as follows:
[ð, ð] = ðð â ðð, ð, ð â ðº. (11)
A direct verification exhibits that
[ð1, ð2] = 2ð
3, [ð
1, ð3] = 2ð
2,
[ð2, ð3] = â2ð
1, [ð
1, ð5] = 2ð
6,
[ð1, ð6] = 2ð
5, [ð
2, ð4] = â2ð
6,
[ð2, ð6] = â2ð
4,
[ð3, ð4] = â2ð
5, [ð
3, ð5] = 2ð
4,
[ð4, ð5] = 2ð
6, [ð
4, ð6] = 2ð
5,
[ð5, ð6] = â2ð
4, [ð
1, ð4] = [ð
3, ð6] = 0.
(12)
Set
Ìðº1= span {ð
1, ð2, ð3} ,
Ìðº2= span {ð
4, ð5, ð6} , (13)
then we find that
ðº =Ìðº1âÌðº2,
Ìðº1â ðº1, [
Ìðº1,Ìðº2] âÌðº2, (14)
andÌðº1andÌðº
2are all simple Lie subalgebras.
While we use Lie algebras to generate integrable hierar-chies of evolution equations, we actually employ their loopalgebras Ìðº = ðº â ð¶(ð, ðâ1) to establish Lax pairs, whereð¶(ð, ð
â1) represents a set of Laurent ploynomials in ð and ðº
is a Lie algebra. Based on this, we give the loop algebras of (9)as follows:
Ìðº = span {ð
1(ð) , . . . , ð
6(ð)} , (15)
whereðð(ð) = ð
ðð
ð, [ðð(ð), ð
ð(ð)] = [ð
ð, ðð]ð
ð+ð, 1 †ð, ð †6,ð, ð â ð.
We consider an auxiliary linear problem as follows:
(
ð1
ð2
ð3
ð4
)
ð¥
= ð (ð¢, ð)(
ð1
ð2
ð3
ð4
),
ð (ð¢, ð) = ð 1+
6
â
ð=1
ð¢ððð(ð) ,
(
ð1
ð2
ð3
ð4
)
ð¡ð
= ðð(ð¢, ð)(
ð1
ð2
ð3
ð4
),
(16)
where ð¢ = (ð¢1, . . . , ð¢
ð )
ð, ðð= ð 1+ ð¢1ð1+ â â â + ð¢
6ð6, ð 1is a
pseudoregular element, ð¢ð(ð, ð¡) = ð¢
ð(ð = 1, 2, . . . , 6), and ð
ð=
ð(ð¥, ð¡) are field variables defined on ð¥ â ð , ð¡ â ð , ðð(ð) â
Ìðº.
The compatibility of (16) gives rise to thewell-known zerocurvature equation
ðð¡â ðð¥+ [ð,ð] = 0, ðð¡
= 0. (17)
Abstract and Applied Analysis 3
The general scheme of searching for the consistent ðð,
and generating a hierarchy of zero curvature equations wasproposed in [30]. Solving the following equation:
ðð¥= [ð,ð] ,
ð =
â
â
ð=0
ððð
âð
= (
ð ð + ð 0 0
ð â ð âð 0 0
ð ð + ð ð + ð ð + ð + ð + ð
ð â ð âð ð â ð + ð â ð â (ð + ð)
) ,
(18)
then we sesrch for ðâÌðº, the new ð
ðcan be constructed by
ðð=
ð
â
ð=0
ðð(ð¢) ð
ðâð+ ð(ð¢, ð) . (19)
Solving zero curvature (17), we could get evolution equationas follows:
ð¢ð¡= ðŸ(ð¢, ð¢
ð¥, . . . ,
ð
ðð¢
ðð¥
ð) . (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:
ðð¥= ð (ð¢, ð) ð,
ð (ð¢, ð) = â ð1(1) + Vð
1(0) + ð¢ð
2(0) + Vð
3(0)
â ð4(1) + ð
2ð4(0) + ð
1ð5(0) + ð
2ð6(0) ,
(21)
that is,
ð (ð¢, ð)
= (
âð + V ð¢ + V 0 0
ð¢ â V ð â V 0 0
âð + ð2ð1+ ð2â2ð + V + ð
2ð¢ + V + ð
1+ ð2
ð1â ð2ð â ð2ð¢ â V + ð
1â ð2
2ð â V â ð2
)
= (
ð1
0
ð0ð1+ ð0
) ,
(22)
where ð is a spectral parameter and ð1satisfies ð
ð¥= ð1ð
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 ð
ð¥= [ð,ð] of
the spectral problem (21) is firstly solved, we assume that asolution ð is given by the following:
ð = (
ð ð + ð 0 0
ð â ð âð 0 0
ð ð + ð ð + ð ð + ð + ð + ð
ð â ð âð ð â ð + ð â ð â (ð + ð)
)
=
â
â
ð=0
ððð
âð
=
â
â
ð=0
Ã(
ðð ðð + ðð 0 0
ðð â ðð âðð 0 0
ðð ðð + ðð ðð + ðð ðð + ðð + ðð + ðð
ðð â ðð âðð ðð â ðð + ðð â ðð â (ðð + ðð)
)ð
âð.
(23)Therefore, the condition (18) becomes the following recursionrelation:
ðð,ð¥= 2Vððâ 2ð¢ðð,
ðð,ð¥= â2ð
ð+1+ 2Vððâ 2Vðð,
ðð,ð¥= â2ð
ð+1+ 2Vððâ 2ð¢ð
ð,
ðð,ð¥= â 2ð¢ð
ð+ 2Vð
ðâ 2ð1ðð
+ 2ð2ððâ 2ð1ðð+ 2ð2ðð,
ðð,ð¥= â 2ð
ð+1+ 2Vð
ðâ 2Vðð
+ 2ð2ððâ 2ð2ðð+ 2ð2ððâ 2ð2ðð,
ðð,ð¥= â 2ð
ð+1+ 2Vð
ðâ 2ð¢ðð+ 2ð2ðð
â 2ð1ðð+ 2ð2ððâ 2ð1ðð.
(24)
Choose the initial datað0= ð0= ðœ, ð
0= ð0= ð0= ð0= 0, (25)
we see that all sets of functions ðð, ðð, ðð, ðð, ðð, and ð
ðare
uniquely determined. In particular, the first few sets are asfollows:
ð1= 0, ð
1= âð¢ðœ, ð
1= âVðœ, ð
1= 0,
ð1= âð1ðœ, ð
1= âð2ðœ, ð
2=
1
2
(V2â ð¢
2) ,
ð2= (
1
2
Vð¥â ð¢V)ðœ, ð
2= (
1
2
ð¢ð¥â V2)ðœ,
ð2= (
1
2
Vð2â
1
2
ð¢ð1+
1
4
ð¢
2â
1
4
V2â
1
4
ð
2
1+
1
4
ð
2
2)ðœ,
4 Abstract and Applied Analysis
ð2= (
1
4
ð2,ð¥â
1
4
Vð¥+
1
2
ð¢V â1
2
Vð1â
1
2
ð¢ð2â
1
2
ð1ð2)ðœ,
ð2= (
1
4
ð1,ð¥â
1
4
ð¢ð¥+
1
2
V2â
1
2
ð
2
2â Vð2)ðœ, . . . .
(26)
Considering
ðð= ð +
ð,
ð
=(
â (ðð â ðð) 0 0 0
0 ðð â ðð 0 0
â (ðð â ðð) 0 â (ðð â ðð) â (ðð â ðð)
0 ðð â ðð 0 ðð â ðð + ðð â ðð
).
(27)
From the zero curvature equation ðð¡â ðð¥+ [ð,ð] = 0, we
obtain the nonlinear integrable coupling system
ð¢ð¡ð
= ðŸð= (
ð¢
V
ð1
ð2
)
ð¡ð
= (
ðð,ð¥
â(ððâ ðð)ð¥
ðð,ð¥
â(ððâ ðð)
ð¥
)
= ðœ(
ðð
ððâ ðð
ðð
ððâ ðð
) = ðœð¿
ð(
0
ðœ
0
ðœ
) , ð ⥠0,
(28)
with theHamiltonian operator ðœ and the hereditary recursionoperator ð¿, respectively, as follows:
ðœ = (
ð 0 0 0
0 âð 0 0
0 0 ð 0
0 0 0 âð
) ,
ð¿ =(
0
1
2
ð â ð¢ 0 0
ð
â1ð¢ð +
1
2
ð ð
â1Vð + V 0 0
0 ð1
0 ð2
ð3
ð4
ð5ð6
),
(29)
where
ð1= â
1
4
ð â
1
2
ð1+
1
2
ð¢,
ð2=
1
4
ð â
1
2
ð1â
1
2
ð¢,
ð3= â
1
2
ð
â1ð¢ â
1
2
ð
â1ð1â ð
â1ð1ð â
1
4
ð,
ð4= â
1
2
ð
â1Vð +
1
2
ð
â1ð2ð + ð
â1ð2ð¢ â
1
2
V +1
2
ð2,
ð5=
1
2
ð
â1ð¢ð +
1
2
ð
â1ð1ð +
1
4
ð,
ð6= â 2ð
â1ð¢V â
3
2
ð
â1ð1V +
1
2
ð
â1Vð
+
1
2
ð
â1ð2ð â
1
2
V +1
2
ð2.
(30)
Obviously, when ð1= ð2= 0 in (28), the above results
become Li soliton hierarchy. So, we can say that (28) isintegrable coupling of the Li soliton hierarchy.
Taking ð = 2, we get that the nonlinear integrablecouplings of Li soliton hierarchy are as follows:
ð¢ð¡2
= (â
1
2
Vð¥ð¥â ð¢ð¥V â ð¢V
ð¥)ðœ,
Vð¡2
= (
1
2
ð¢ð¥ð¥â 3VVð¥+ ð¢ð¢ð¥)ðœ,
ð1,ð¡2
= (
1
4
ð2,ð¥â
1
4
Vð¥+
1
2
ð¢V â1
2
Vð1
â
1
2
ð¢ð2â
1
2
ð1ð2)
ð¥
ðœ,
ð2,ð¡2
= (
1
4
ð1,ð¥â
1
4
ð¢ð¥+
1
2
V2â
3
4
ð
2
2â
3
2
Vð2
+
1
2
ð¢ð1â
1
4
ð¢
2+
1
4
V2+
1
4
ð
2
1)
ð¥
ðœ.
(31)
So, we can say that the system in (28) with ð ⥠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 ð distinct ð
ð, ð = 1, 2, . . . , ð and the systems
of (16) become in the following form:
(
ð1ð
ð2ð
ð3ð
ð4ð
)
ð¥
= ð (ð¢, ðð)(
ð1ð
ð2ð
ð3ð
ð4ð
)
=
6
â
ð=1
ð¢ððð(ð)(
ð1ð
ð2ð
ð3ð
ð4ð
), ð = 1, . . . , ð,
Abstract and Applied Analysis 5
(
ð1ð
ð2ð
ð3ð
ð4ð
)
ð¡ð
= ðð(ð¢, ðð)(
ð1ð
ð2ð
ð3ð
ð4ð
)
= [
ð
â
ð=0
ðð(ð¢) ð
ðâð
ð+ Îð(ð¢, ðð)]
Ã(
ð1ð
ð2ð
ð3ð
ð4ð
), ð = 1, . . . , ð.
(32)
Based on the result in [32], we show that the followingequation
ð¿ð»ð
ð¿ð¢
+
ð
â
ð=1
ðŒð
ð¿ðð
ð¿ð¢
= 0 (33)
holds true, where ðŒðis a constant. From (32), we may know
that
ð¿ðð
ð¿ð¢ð
= ðŒðTr(Κ
ð
ðð (ð¢, ðð)
ðð¢ð
)
= ðŒðTr (Κððð(ðð)) , ð = 1, 2,
(34)
where Tr denotes the trace of a matrix and
Κð= (
ð1ðð2ð
âð
2
1ðð3ðð4ð
âð
2
3ð
ð
2
2ðâð1ðð2ð
ð
2
4ðâð3ðð4ð
0 0 ð1ðð2ð
âð
2
1ð
0 0 ð
2
2ðâð1ðð2ð
),
ð = 1, . . . , ð.
(35)
For ð = 3, 4 we define that
ð¿ðð
ð¿ð¢ð
= ðœðTr(Κ
ððŽ
ðð0(ð¢, ðð)
ðð¢ð
) , (36)
where
ð = (
ð1
0
ð0ð1+ ð0
) ,
ΚððŽ= (
ð3ðð4ð
âð
2
3ð
ð
2
4ðâð3ðð4ð
) ,
(37)
and ðœðis a constant.
According to (34) and (36), we obtain a kind of nonlinearintegrable couplings with self-consistent sources as follows:
ð¢ð¡ð
= ðœ
ð¿ð»ð+1
ð¿ð¢ð
+ ðœ
ð
â
ð=1
ðŒð
ð¿ðð
ð¿ð¢
= ðœð¿
ðð¿ð»1
ð¿ð¢ð
+ ðœ
ð
â
ð=1
ðŒð
ð¿ðð
ð¿ð¢
, ð = 1, 2, . . . .
(38)
Therefore, according to formulas (34) and (36), we have thefollowing results by direct computations:
ð
â
ð=1
ð¿ðð
ð¿ð¢
=
ð
â
ð=1
(
(
(
(
(
(
(
(
(
ð¿ðð
ð¿ð¢
ð¿ðð
ð¿V
ð¿ðð
ð¿ð1
ð¿ðð
ð¿ð1
)
)
)
)
)
)
)
)
)
=(
2(âšÎŠ2, Ί2â© â âšÎŠ
1, Ί1â©)
2 (âšÎŠ1, Ί1â© + âšÎŠ
2, Ί2â© + 2 âšÎŠ
1, Ί2â©)
âšÎŠ4, Ί4â© â âšÎŠ
3, Ί3â©
âšÎŠ3, Ί3â© + âšÎŠ
4, Ί4â© + 2 âšÎŠ
3, Ί4â©
) ,
(39)
by taking ðŒð= 1 and ðœ
ð= 1 in formulas (34) and (36).
Therefore, we have nonlinear integrable coupling system ofthe Li equations hierarchy with self-consistent sources asfollows:
ð¢ð¡ð
= ðŸð
= (
ð¢
V
ð1
ð2
)
ð¡ð
= ðœð¿
ð(
0
ðœ
0
ðœ
)
+ ðœ(
2 (âšÎŠ2, Ί2â© â âšÎŠ
1, Ί1â©)
2 (âšÎŠ1, Ί1â© + âšÎŠ
2, Ί2â© + 2 âšÎŠ
1, Ί2â©)
âšÎŠ4, Ί4â© â âšÎŠ
3, Ί3â©
âšÎŠ3, Ί3â© + âšÎŠ
4, Ί4â© + 2 âšÎŠ
3, Ί4â©
)
= ðœð¿
ð(
0
ðœ
0
ðœ
) + ðœ
(
(
(
(
(
(
(
(
(
(
(
2
ð
â
ð=1
(ð
2
2ðâ ð
2
1ð)
2
ð
â
ð=1
(ð
2
1ð+ ð
2
2ð+ 2ð1ðð2ð)
ð
â
ð=1
(ð
2
4ðâ ð
2
3ð)
ð
â
ð=1
(ð
2
3ð+ ð
2
4ð+ 2ð3ðð4ð)
)
)
)
)
)
)
)
)
)
)
)
,
(40)
6 Abstract and Applied Analysis
with
ð1ð,ð¥= (âð + V) ð
1ð+ (ð¢ + V) ð
2ð,
ð2ð,ð¥= (ð¢ â V) ð
1ð+ (ð â V) ð
2ð,
ð3ð,ð¥= (âð + ð
2) ð1ð+ (ð1+ ð2) ð2ð
+ (â2ð + V + ð2) ð3ð+ (ð¢ + V + ð
1+ ð2) ð4ð,
ð4ð,ð¥= (ð1â ð2) ð1ð+ (ð â ð
2) ð2ð
+ (ð¢ â V + ð1â ð2) ð3ð
+ (2ð â V â ð2) ð4ð, ð = 1, . . . , ð,
(41)
where Ίð= (ðð1, . . . , ð
ðð), ð = 1, 2, 3, 4, and âšâ , â â© is the
standard inner product in ð ð.When ð = 2 and ðœ = 2, we obtain nonlinear integrable
couplings of Li hierarchy with self-consistent sources
ð¢ð¡2
= â
1
2
Vð¥ð¥â ð¢ð¥V â ð¢V
ð¥+ 2ð
ð
â
ð=1
(ð
2
2ðâ ð
2
1ð) ,
Vð¡2
=
1
2
ð¢ð¥ð¥â 3VVð¥+ ð¢ð¢ð¥
â 2ð
ð
â
ð=1
(ð
2
1ð+ ð
2
2ð+ 2ð1ðð2ð) ,
ð1ð¡2
= (
1
4
ð2,ð¥â
1
4
Vð¥+
1
2
ð¢V â1
2
Vð1â
1
2
ð¢ð2
â
1
2
ð1ð2)
ð¥
+ ð
ð
â
ð=1
(ð
2
4ðâ ð
2
3ð) ,
ð2ð¡2
= (
1
4
ð1,ð¥â
1
4
ð¢ð¥+
1
2
V2â
3
4
ð
2
2â
3
2
Vð2+
1
2
ð¢ð1
â
1
4
ð¢
2+
1
4
V2+
1
4
ð
2
1)
ð¥
â ð
ð
â
ð=1
(ð
2
3ð+ ð
2
4ð+ 2ð3ðð4ð) ,
(42)
with
ð1ð,ð¥= (âð + V) ð
1ð+ (ð¢ + V) ð
2ð,
ð2ð,ð¥= (ð¢ â V) ð
1ð+ (ð â V) ð
2ð,
ð3ð,ð¥= (âð + ð
2) ð1ð+ (ð1+ ð2) ð2ð
+ (â2ð + V + ð2) ð3ð+ (ð¢ + V + ð
1+ ð2) ð4ð,
ð4ð,ð¥= (ð1â ð2) ð1ð+ (ð â ð
2) ð2ð
+ (ð¢ â V + ð1â ð2) ð3ð
+ (2ð â V â ð2) ð4ð, ð = 1, . . . , ð.
(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
ð (ð¢, ð)
= (
âð + V ð¢ + V 0 0
ð¢ â V ð â V 0 0
âð + ð2ð1+ ð2â2ð + V + ð
2ð¢ + V + ð
1+ ð2
ð1â ð2ð + âð
2ð¢ â V + ð
1â ð2
2ð â V â ð2
),
(44)
we introduce the variables
ð =
ð2
ð1
, ð =
ð3
ð1
, ðŸ =
ð4
ð1
. (45)
From (44), we have
ðð¥= ð¢ â V + 2ðð â 2Vðâ (ð¢ + V)ð
2,
ðð¥= â ð + ð
2â ðð + (ð
1+ ð2)ð
+ ð2ð + (ð¢ + V + ð
1+ ð2)ðŸ â (ð¢ + V)ðð,
ðŸð¥= ð1â ð2+ 3ððŸ + ðð â ð
2ð
â (2V + ð2)ðŸ + (ð¢ â V + ð
1â ð2)ð â (ð¢ + V) ðŸð.
(46)
We expandð,ð, andðŸ in powers of ð as follows:
ð =
â
â
ð=1
ððð
âð, ð =
â
â
ð=1
ððð
âð,
ðŸ =
â
â
ð=1
ððð
âð.
(47)
Abstract and Applied Analysis 7
Substituting (47) into (46) and comparing the coefficients ofthe same power of ð, we obtain the following:
ð1=
1
2
(V â ð¢) , ð1= ð2,
ð1=
1
3
(ð2â ð1) +
1
6
(ð¢ â V) ,
ð2=
1
4
(V â ð¢)ð¥+
1
2
(V2â ð¢V) ,
ð2= âð2,ð¥â
1
3
ð
2
1â
2
3
ð¢ð1+
2
3
Vð2+
4
3
ð
2
2+
1
6
ð¢
2â
1
6
V2,
ð2=
5
36
(ð¢ â V)ð¥+
1
9
(ð2â ð1)
ð¥â
5
18
V2+
5
18
ð¢V
+
2
3
Vð2â
4
9
ð¢ð2+
4
9
ð
2
2â
4
9
ð1ð2â
2
9
Vð1,
ð3=
1
8
(V â ð¢)ð¥ð¥+
1
8
ð¢
3â
1
8
ð¢
2V +
3
4
VVð¥
â
1
2
Vð¢ð¥â
1
4
ð¢Vð¥+
5
8
V3â
5
8
ð¢V2,
ð3= ð2,ð¥ð¥+
5
9
(ð1ð¢)
ð¥â
5
9
(ð2V)ð¥â
32
9
ð2ð2,ð¥â
7
36
ð¢ð¢ð¥
+
7
36
VVð¥+
5
9
ð1ð1,ð¥+
1
9
ð1Vð¥â
1
9
ð2ð¢ð¥â
5
36
ð¢Vð¥
+
1
9
ð¢ð2,ð¥+
5
36
Vð¢ð¥â
1
9
Vð1,ð¥+
1
9
ð1ð2,ð¥â
1
9
ð2ð1,ð¥
â
4
9
ð1ð¢V +
11
9
ð2V2â
14
9
ð¢ð1ð2+
16
9
Vð2
2
+
16
9
ð
3
2â
7
9
ð2ð¢
2â
7
9
ð2ð
2
1+
5
18
Vð¢2â
5
18
V3â
2
9
Vð2
1,
ð3=
19
216
(ð¢ â V)ð¥ð¥+
1
27
(ð2â ð1)
ð¥ð¥+
5
54
(ð¢ â V)ð¥
â
47
108
VVð¥+
7
27
Vð¢ð¥+
19
108
ð¢Vð¥â
4
27
Vð2,ð¥
+
7
27
ð2Vð¥+
5
27
ð¢ð2,ð¥â
5
27
ð2ð¢ð¥â
5
27
ð2ð1,ð¥
+
5
27
ð1ð2,ð¥â
2
27
ð1Vð¥â
4
27
Vð1,ð¥â
103
216
V3
+
101
216
ð¢V2+
1
8
Vð¢2+
7
18
ð2V2â
16
27
ð2ð¢V
â
19
54
ð2V2+
32
27
Vð2
2â
16
27
ð1ð2V â
4
27
ð1V2
â
16
27
ð¢ð
2
2+
16
27
ð
3
2â
16
27
ð1ð
2
2+
2
9
ð1ð¢
2
+
1
18
ð¢V2+
1
3
ð¢ð
2
1+
1
9
ð
3
1â
2
9
ð¢Vð1â
2
9
ð¢ð1ð2
â
1
9
Vð2
1â
1
9
ð2ð
2
1â
1
8
ð¢
3, . . . ,
(48)
and a recursion formula forðð, ðð, and ð
ðas follows:
ðð+1=
1
2
ðð,ð¥+ Vðð+
1
2
(ð¢ + V)
ðâ1
â
ð=1
ððððâð,
ðð+1= â ð
ð,ð¥+ (ð1+ ð2)ðð+ ð2ðð
+ (ð¢ + V + ð1+ ð2) ððâ (ð¢ + V)
ðâ1
â
ð=1
ððððâð,
ðð+1=
1
3
ðð,ð¥â
1
6
ðð,ð¥â
1
3
Vðð+
1
3
ð2ðð
+
1
3
(2V + ð2) ððâ
1
3
(ð¢ â V + ð1â ð2) ðð
â
1
6
(ð¢ + V)
ðâ1
â
ð=1
ððððâð+
1
3
(ð¢ + V)
ðâ1
â
ð=1
ððððâð.
(49)
Because of
ð
ðð¡
[âð + V + (ð¢ + V)ð] =ð
ðð¥
[ð + (ð + ð)ð] ,
ð
ðð¡
[âð + ð2+ (ð1+ ð2)ð + (â2ð + V + ð
2)ð
+ (ð¢ + V + ð1+ ð2)ðŸ]
=
ð
ðð¥
[ð + (ð + ð)ð + (ð + ð)ð
+ (ð + ð + ð + ð)ðŸ] ,
(50)
where
ð = ð0ð
2+ ð1ð +
1
2
ð0(V2â ð¢
2) ,
ð = âð0ð¢ð + ð
0(âð¢V +
1
2
Vð¥) â ð1ð¢,
ð = âð0Vð + ð
0(âV2+
1
2
ð¢ð¥) â ð1V,
ð = ð0ð
2+ ð1ð
+ ð0(
1
2
Vð2â
1
2
ð¢ð1+
1
4
ð¢
2â
1
4
V2â
1
4
ð
2
1+
1
4
ð
2
2) ,
ð = â ð0ð1ð + ð0(
1
4
ð2,ð¥â
1
4
Vð¥+
1
2
ð¢V â1
2
Vð1
â
1
2
ð¢ð2â
1
2
ð1ð2) â ð1ð1,
ð = â ð0ð2ð + ð0(
1
4
ð1,ð¥â
1
4
ð¢ð¥+
1
2
V2
â
1
2
ð
2
2â Vð2) â ð1ð2.
(51)
8 Abstract and Applied Analysis
Assume that
ð = âð + V + (ð¢ + V)ð,
ð = ð + (ð + ð)ð,
ð = â ð + ð2+ (ð1+ ð2)ð + (â2ð + V + ð
2)ð
+ (ð¢ + V + ð1+ ð2)ðŸ,
ð¿ = ð + (ð + ð)ð + (ð + ð)ð + (ð + ð + ð + ð)ðŸ.
(52)
Then, (50) can be written as ðð¡= ðð¥and ð
ð¡= ð¿ð¥, which are
the right form of conservation laws. We expand ð, ð, ð, and ð¿as series in powers of ð with the coefficients, which are calledconserved densities and currents, respectively:
ð = âð + V + (ð¢ + V)
â
â
ð=1
ððð
âð,
ð = ð0ð
2+ ð1ð +
â
â
ð=1
ððð
âð,
ð = âð + ð2+
â
â
ð=1
ððð
âð,
ð¿ = ð0ð
2+ ð1ð +
â
â
ð=1
ð¿ðð
âð,
(53)
where ð0and ð
1are constants of integration. The first
conserved densities and currents are read as follows:
ð1=
1
2
(V2â ð¢
2) ,
ð1= ð0(
1
2
ð¢ð¢ð¥â
3
4
ð¢Vð¥â
3
4
VVð¥) â
1
2
ð1(V2â ð¢
2) ,
ð1=
1
2
ð¢ð1+
1
3
Vð2â
4
3
ð
2
2â
1
6
ð¢
2
+
1
6
V2+
1
3
ð
2
1+
1
6
ð¢ð1+ 2ð2,ð¥,
ð¿1= ð0(2ð2,ð¥ð¥+
1
36
ð1Vð¥+
41
36
ð1ð¢ð¥â
41
36
ð2Vð¥
â
1
36
ð2ð¢ð¥â
41
36
Vð2,ð¥â
1
36
Vð1,ð¥
+
13
36
VVð¥â
1
36
Vð¢ð¥+
1
36
ð¢ð2,ð¥
+
41
36
ð¢ð1,ð¥â
13
36
ð¢ð¢ð¥â
257
36
ð2ð2,ð¥
+
41
36
ð1ð1,ð¥+
1
36
ð1ð2,ð¥
â
1
36
ð2ð1,ð¥+
47
36
ð2V2â
1
18
Vð¢2
+
11
36
Vð1ð2+
55
18
Vð2
2â
43
36
ð2ð¢
2
â
53
18
ð¢ð1ð2+
93
36
ð
3
2
â
3
4
ð2ð
2
1â
1
18
Vð2
1â
4
9
Vð2
2
â
1
9
ð¢Vð1â
4
9
ð1ð
2
2
+
1
18
V3+
1
36
ð¢Vð¥)
+ ð1(â2ð2,ð¥+ Vð1â
5
3
ð¢ð1+
5
3
Vð2â ð¢ð2
+
7
3
ð
2
2+
1
6
ð¢
2â
1
6
V2â
1
3
ð
2
1) , . . . .
(54)
The recursion relations for ðð, ðð, ðð, and ð¿
ðare as follows:
ðð= (ð¢ + V)ð
ð,
ðð= â ð
0(ð¢ + V)ð
ð+1+ ð0(
1
2
ð¢ð¥+
1
2
Vð¥â ð¢V â V
2)ðð
â ð1(ð¢ + V)ð
ð,
ðð= (ð1+ ð2)ððâ 2ðð+1+ (V + ð
2) ðð
+ (ð¢ + V + ð1+ ð2) ðð,
ð¿ð= ð0[ â (ð
1+ ð2)ðð+1
+ (
1
4
ð2,ð¥+
1
4
ð1,ð¥â
1
4
Vð¥
â
1
4
ð¢ð¥+
1
2
ð¢V â1
2
Vð1
â
1
2
ð¢ð2â
1
2
ð1ð2+
1
2
V2
â
1
2
ð
2
2â ð2V)ðð
+ 2ðð+2+ (
1
2
V2â
1
2
ð¢
2+
1
2
Vð2
â
1
2
ð¢ð1+
1
4
ð¢
2
+
1
4
ð
2
2â
1
4
V2â
1
4
ð
2
1) ðð
â (ð¢ + V + ð1+ ð2) ðð+1
Abstract and Applied Analysis 9
+ (
1
4
ð¢ð¥+
1
4
Vð¥+
1
4
ð1,ð¥+
1
4
ð2,ð¥â
1
2
ð¢V
â
1
2
V2â
1
2
Vð1â
1
2
ð¢ð2
â
1
2
ð1ð2â
1
2
ð
2
2â Vð2)]
+ ð1[2ðð+1â (ð1+ ð2)ðð
â (ð¢ + V + ð1+ ð2) ðð] ,
(55)
whereðð, ðð, and ð
ðcan be calculated from (49).The infinite
conservation laws of nonlinear integrable couplings (37) canbe easily obtained in (45)â(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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