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

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; [email protected]

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 Bäcklund 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 [5–13], but allthe integrable couplings obtained are linear for the V =(V1, . . . , V

𝑚)

𝑇. As for how to generate nonlinear integrablecouplings, Ma proposed a general scheme [14]. Suppose thatan integrable system

𝑢𝑡= 𝐾 (𝑢) (1)

has a Lax pair 𝑈 and 𝑉, which belong to semisimple matrixLie algebras. Introduce an enlarged spectral matrix

𝑈 = 𝑈 (𝑢) = [

𝑈 (𝑢) 0

𝑈𝑎(V) 𝑈 (𝑢) + 𝑈

𝑎(V)] (2)

from a zero curvature representation

𝑈𝑡− 𝑉𝑥+ [𝑈,𝑉] = 0, (3)

where

𝑉 = 𝑉 (𝑢) = [

𝑉 (𝑢) 0

𝑉𝑎(𝑢) 𝑉 (𝑢) + 𝑉

𝑎(𝑢)

] , (4)

then we can give rise to

𝑈𝑡− 𝑉𝑥+ [𝑈,𝑉] = 0,

𝑈𝑎,𝑡− 𝑉𝑎,𝑥+ [𝑈,𝑉

𝑎] + [𝑈

𝑎, 𝑉] + [𝑈

𝑎, 𝑉𝑎] = 0.

(5)

This is an integrable coupling of (1), and it is a nonlinearintegrable coupling because the commutator [𝑈

𝑎, 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 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 super𝐺-𝐽 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:

𝐺1= span {𝑒

1, 𝑒2, 𝑒3} , (6)

where

𝑒1= (

1 0

0 −1

) , 𝑒2= (

0 1

1 0

) , 𝑒3= (

0 1

−1 0

) , (7)

which have the commutative relations

[𝑒1, 𝑒2] = 2𝑒

2, [𝑒

1, 𝑒3] = −2𝑒

3, [𝑒

2, 𝑒3] = 𝑒1. (8)

Let us introduce a Lie algebra with matrix blocks by using𝐺1in order to get nonlinear couplings of soliton hierarchy as

follows:

𝐺 = span {𝑔1, . . . , 𝑔

6} , (9)

where

𝑔1= (

𝑒10

0 𝑒1

) , 𝑔2= (

𝑒20

0 𝑒2

) ,

𝑔3= (

𝑒30

0 𝑒3

) , 𝑔4= (

0 0

𝑒1𝑒1

) ,

𝑔5= (

0 0

𝑒2𝑒2

) , 𝑔6= (

0 0

𝑒3𝑒3

) .

(10)

Define a commutator as follows:

[𝑎, 𝑏] = 𝑎𝑏 − 𝑏𝑎, 𝑎, 𝑏 ∈ 𝐺. (11)

A direct verification exhibits that

[𝑔1, 𝑔2] = 2𝑔

3, [𝑔

1, 𝑔3] = 2𝑔

2,

[𝑔2, 𝑔3] = −2𝑔

1, [𝑔

1, 𝑔5] = 2𝑔

6,

[𝑔1, 𝑔6] = 2𝑔

5, [𝑔

2, 𝑔4] = −2𝑔

6,

[𝑔2, 𝑔6] = −2𝑔

4,

[𝑔3, 𝑔4] = −2𝑔

5, [𝑔

3, 𝑔5] = 2𝑔

4,

[𝑔4, 𝑔5] = 2𝑔

6, [𝑔

4, 𝑔6] = 2𝑔

5,

[𝑔5, 𝑔6] = −2𝑔

4, [𝑔

1, 𝑔4] = [𝑔

3, 𝑔6] = 0.

(12)

Set

̃𝐺1= span {𝑔

1, 𝑔2, 𝑔3} ,

̃𝐺2= span {𝑔

4, 𝑔5, 𝑔6} , (13)

then we find that

𝐺 =̃𝐺1⊕̃𝐺2,

̃𝐺1≅ 𝐺1, [

̃𝐺1,̃𝐺2] ⊆̃𝐺2, (14)

and̃𝐺1and̃𝐺

2are all simple Lie subalgebras.

While we use Lie algebras to generate integrable hierar-chies of evolution equations, we actually employ their loopalgebras ̃𝐺 = 𝐺 ⊗ 𝐶(𝜆, 𝜆−1) to establish Lax pairs, where𝐶(𝜆, 𝜆

−1) 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:

̃𝐺 = span {𝑔

1(𝑛) , . . . , 𝑔

6(𝑛)} , (15)

where𝑔𝑖(𝑛) = 𝑔

𝑖𝜆

𝑛, [𝑔𝑖(𝑚), 𝑔

𝑗(𝑛)] = [𝑔

𝑖, 𝑔𝑗]𝜆

𝑚+𝑛, 1 ≤ 𝑖, 𝑗 ≤ 6,𝑚, 𝑛 ∈ 𝑍.

We consider an auxiliary linear problem as follows:

(

𝜑1

𝜑2

𝜑3

𝜑4

)

𝑥

= 𝑈 (𝑢, 𝜆)(

𝜑1

𝜑2

𝜑3

𝜑4

),

𝑈 (𝑢, 𝜆) = 𝑅1+

6

∑

𝑖=1

𝑢𝑖𝑔𝑖(𝜆) ,

(

𝜑1

𝜑2

𝜑3

𝜑4

)

𝑡𝑛

= 𝑉𝑛(𝑢, 𝜆)(

𝜑1

𝜑2

𝜑3

𝜑4

),

(16)

where 𝑢 = (𝑢1, . . . , 𝑢

𝑠)

𝑇, 𝑈𝑛= 𝑅1+ 𝑢1𝑔1+ ⋅ ⋅ ⋅ + 𝑢

6𝑔6, 𝑅1is a

pseudoregular element, 𝑢𝑖(𝑛, 𝑡) = 𝑢

𝑖(𝑖 = 1, 2, . . . , 6), and 𝜑

𝑖=

𝜑(𝑥, 𝑡) are field variables defined on 𝑥 ∈ 𝑅, 𝑡 ∈ 𝑅, 𝑔𝑖(𝜆) ∈

̃𝐺.

The compatibility of (16) gives rise to thewell-known zerocurvature equation

𝑈𝑡− 𝑉𝑥+ [𝑈,𝑉] = 0, 𝜆𝑡

= 0. (17)

Abstract and Applied Analysis 3

The general scheme of searching for the consistent 𝑉𝑛,

and generating a hierarchy of zero curvature equations wasproposed in [30]. Solving the following equation:

𝑉𝑥= [𝑈,𝑉] ,

𝑉 =

∞

∑

𝑛=0

𝑉𝑛𝜆

−𝑛

= (

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)

then we sesrch for 𝑛∈̃𝐺, the new 𝑉

𝑛can be constructed by

𝑉𝑛=

𝑛

∑

𝑚=0

𝑉𝑚(𝑢) 𝜆

𝑛−𝑚+ 𝑛(𝑢, 𝜆) . (19)

Solving zero curvature (17), we could get evolution equationas follows:

𝑢𝑡= 𝐾(𝑢, 𝑢

𝑥, . . . ,

𝜕

𝑝𝑢

𝜕𝑥

𝑝) . (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:

𝜑𝑥= 𝑈 (𝑢, 𝜆) 𝜑,

𝑈 (𝑢, 𝜆) = − 𝑔1(1) + V𝑔

1(0) + 𝑢𝑔

2(0) + V𝑔

3(0)

− 𝑔4(1) + 𝑝

2𝑔4(0) + 𝑝

1𝑔5(0) + 𝑝

2𝑔6(0) ,

(21)

that is,

𝑈 (𝑢, 𝜆)

= (

−𝜆 + V 𝑢 + V 0 0

𝑢 − V 𝜆 − V 0 0

−𝜆 + 𝑝2𝑝1+ 𝑝2−2𝜆 + V + 𝑝

2𝑢 + V + 𝑝

1+ 𝑝2

𝑝1− 𝑝2𝜆 − 𝑝2𝑢 − V + 𝑝

1− 𝑝2

2𝜆 − V − 𝑝2

)

= (

𝑈1

0

𝑈0𝑈1+ 𝑈0

) ,

(22)

where 𝜆 is a spectral parameter and 𝑈1satisfies 𝜑

𝑥= 𝑈1𝜑

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 𝑉

𝑥= [𝑈,𝑉] of

the spectral problem (21) is firstly solved, we assume that asolution 𝑉 is given by the following:

𝑉 = (

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)

)

=

∞

∑

𝑛=0

𝑉𝑛𝜆

−𝑛

=

∞

∑

𝑛=0

×(

𝑎𝑛 𝑏𝑛 + 𝑐𝑛 0 0

𝑏𝑛 − 𝑐𝑛 −𝑎𝑛 0 0

𝑒𝑛 𝑓𝑛 + 𝑔𝑛 𝑎𝑛 + 𝑒𝑛 𝑏𝑛 + 𝑐𝑛 + 𝑓𝑛 + 𝑔𝑛

𝑓𝑛 − 𝑔𝑛 −𝑒𝑛 𝑏𝑛 − 𝑐𝑛 + 𝑓𝑛 − 𝑔𝑛 − (𝑎𝑛 + 𝑒𝑛)

)𝜆

−𝑛.

(23)Therefore, the condition (18) becomes the following recursionrelation:

𝑎𝑛,𝑥= 2V𝑏𝑛− 2𝑢𝑐𝑛,

𝑏𝑛,𝑥= −2𝑐

𝑛+1+ 2V𝑐𝑛− 2V𝑎𝑛,

𝑐𝑛,𝑥= −2𝑏

𝑛+1+ 2V𝑏𝑛− 2𝑢𝑎

𝑛,

𝑒𝑛,𝑥= − 2𝑢𝑔

𝑛+ 2V𝑓

𝑛− 2𝑝1𝑐𝑛

+ 2𝑝2𝑏𝑛− 2𝑝1𝑔𝑛+ 2𝑝2𝑓𝑛,

𝑓𝑛,𝑥= − 2𝑔

𝑛+1+ 2V𝑔

𝑛− 2V𝑒𝑛

+ 2𝑝2𝑐𝑛− 2𝑝2𝑎𝑛+ 2𝑝2𝑔𝑛− 2𝑝2𝑒𝑛,

𝑔𝑛,𝑥= − 2𝑓

𝑛+1+ 2V𝑓

𝑛− 2𝑢𝑒𝑛+ 2𝑝2𝑏𝑛

− 2𝑝1𝑎𝑛+ 2𝑝2𝑓𝑛− 2𝑝1𝑒𝑛.

(24)

Choose the initial data𝑎0= 𝑒0= 𝛽, 𝑏

0= 𝑐0= 𝑓0= 𝑔0= 0, (25)

we see that all sets of functions 𝑎𝑛, 𝑏𝑛, 𝑐𝑛, 𝑒𝑛, 𝑓𝑛, and 𝑔

𝑛are

uniquely determined. In particular, the first few sets are asfollows:

𝑎1= 0, 𝑏

1= −𝑢𝛽, 𝑐

1= −V𝛽, 𝑒

1= 0,

𝑓1= −𝑝1𝛽, 𝑔

1= −𝑝2𝛽, 𝑎

2=

1

2

(V2− 𝑢

2) ,

𝑏2= (

1

2

V𝑥− 𝑢V)𝛽, 𝑐

2= (

1

2

𝑢𝑥− V2)𝛽,

𝑒2= (

1

2

V𝑝2−

1

2

𝑢𝑝1+

1

4

𝑢

2−

1

4

V2−

1

4

𝑝

2

1+

1

4

𝑝

2

2)𝛽,


𝑓2= (

1

4

𝑝2,𝑥−

1

4

V𝑥+

1

2

𝑢V −1

2

V𝑝1−

1

2

𝑢𝑝2−

1

2

𝑝1𝑝2)𝛽,

𝑔2= (

1

4

𝑝1,𝑥−

1

4

𝑢𝑥+

1

2

V2−

1

2

𝑝

2

2− V𝑝2)𝛽, . . . .

(26)

Considering

𝑉𝑛= 𝑉 +

𝑛,

𝑛

=(

− (𝑎𝑛 − 𝑐𝑛) 0 0 0

0 𝑎𝑛 − 𝑐𝑛 0 0

− (𝑒𝑛 − 𝑔𝑛) 0 − (𝑎𝑛 − 𝑐𝑛) − (𝑒𝑛 − 𝑔𝑛)

0 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

𝑢𝑡𝑛

= 𝐾𝑛= (

𝑢

V

𝑝1

𝑝2

)

𝑡𝑛

= (

𝑏𝑛,𝑥

−(𝑎𝑛− 𝑐𝑛)𝑥

𝑓𝑛,𝑥

−(𝑒𝑛− 𝑔𝑛)

𝑥

)

= 𝐽(

𝑏𝑛

𝑎𝑛− 𝑐𝑛

𝑓𝑛

𝑒𝑛− 𝑔𝑛

) = 𝐽𝐿

𝑛(

0

𝛽

0

𝛽

) , 𝑛 ≥ 0,

(28)

with theHamiltonian operator 𝐽 and the hereditary recursionoperator 𝐿, respectively, as follows:

𝐽 = (

𝜕 0 0 0

0 −𝜕 0 0

0 0 𝜕 0

0 0 0 −𝜕

) ,

𝐿 =(

0

1

2

𝜕 − 𝑢 0 0

𝜕

−1𝑢𝜕 +

1

2

𝜕 𝜕

−1V𝜕 + V 0 0

0 𝑀1

0 𝑀2

𝑀3

𝑀4

𝑀5𝑀6

),

(29)

where

𝑀1= −

1

4

𝜕 −

1

2

𝑝1+

1

2

𝑢,

𝑀2=

1

4

𝜕 −

1

2

𝑝1−

1

2

𝑢,

𝑀3= −

1

2

𝜕

−1𝑢 −

1

2

𝜕

−1𝑝1− 𝜕

−1𝑝1𝜕 −

1

4

𝜕,

𝑀4= −

1

2

𝜕

−1V𝜕 +

1

2

𝜕

−1𝑝2𝜕 + 𝜕

−1𝑝2𝑢 −

1

2

V +1

2

𝑝2,

𝑀5=

1

2

𝜕

−1𝑢𝜕 +

1

2

𝜕

−1𝑝1𝜕 +

1

4

𝜕,

𝑀6= − 2𝜕

−1𝑢V −

3

2

𝜕

−1𝑝1V +

1

2

𝜕

−1V𝜕

+

1

2

𝜕

−1𝑝2𝜕 −

1

2

V +1

2

𝑝2.

(30)

Obviously, when 𝑝1= 𝑝2= 0 in (28), the above results

become Li soliton hierarchy. So, we can say that (28) isintegrable coupling of the Li soliton hierarchy.

Taking 𝑛 = 2, we get that the nonlinear integrablecouplings of Li soliton hierarchy are as follows:

𝑢𝑡2

= (−

1

2

V𝑥𝑥− 𝑢𝑥V − 𝑢V

𝑥)𝛽,

V𝑡2

= (

1

2

𝑢𝑥𝑥− 3VV𝑥+ 𝑢𝑢𝑥)𝛽,

𝑝1,𝑡2

= (

1

4

𝑝2,𝑥−

1

4

V𝑥+

1

2

𝑢V −1

2

V𝑝1

−

1

2

𝑢𝑝2−

1

2

𝑝1𝑝2)

𝑥

𝛽,

𝑝2,𝑡2

= (

1

4

𝑝1,𝑥−

1

4

𝑢𝑥+

1

2

V2−

3

4

𝑝

2

2−

3

2

V𝑝2

+

1

2

𝑢𝑝1−

1

4

𝑢

2+

1

4

V2+

1

4

𝑝

2

1)

𝑥

𝛽.

(31)

So, we can say that the system in (28) with 𝑛 ≥ 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 𝑁 distinct 𝜆

𝑗, 𝑗 = 1, 2, . . . , 𝑁 and the systems

of (16) become in the following form:

(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

𝜑4𝑗

)

𝑥

= 𝑈 (𝑢, 𝜆𝑗)(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

𝜑4𝑗

)

=

6

∑

𝑖=1

𝑢𝑖𝑔𝑖(𝜆)(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

𝜑4𝑗

), 𝑗 = 1, . . . , 𝑁,


(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

𝜑4𝑗

)

𝑡𝑛

= 𝑉𝑛(𝑢, 𝜆𝑗)(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

𝜑4𝑗

)

= [

𝑛

∑

𝑚=0

𝑉𝑚(𝑢) 𝜆

𝑛−𝑚

𝑗+ Δ𝑛(𝑢, 𝜆𝑗)]

×(

𝜑1𝑗

𝜑2𝑗

𝜑3𝑗

𝜑4𝑗

), 𝑗 = 1, . . . , 𝑁.

(32)

Based on the result in [32], we show that the followingequation

𝛿𝐻𝑘

𝛿𝑢

+

𝑁

∑

𝑗=1

𝛼𝑗

𝛿𝜆𝑗

𝛿𝑢

= 0 (33)

holds true, where 𝛼𝑗is a constant. From (32), we may know

that

𝛿𝜆𝑗

𝛿𝑢𝑖

= 𝛼𝑗Tr(Ψ

𝑗

𝜕𝑈 (𝑢, 𝜆𝑗)

𝜕𝑢𝑖

)

= 𝛼𝑗Tr (Ψ𝑗𝑔𝑖(𝜆𝑗)) , 𝑖 = 1, 2,

(34)

where Tr denotes the trace of a matrix and

Ψ𝑗= (

𝜙1𝑗𝜙2𝑗

−𝜙

2

1𝑗𝜙3𝑗𝜙4𝑗

−𝜙

2

3𝑗

𝜙

2

2𝑗−𝜙1𝑗𝜙2𝑗

𝜙

2


0 0 𝜙1𝑗𝜙2𝑗

−𝜙

2

1𝑗

0 0 𝜙

2


),

𝑗 = 1, . . . , 𝑁.

(35)

For 𝑖 = 3, 4 we define that

𝛿𝜆𝑗

𝛿𝑢𝑖

= 𝛽𝑗Tr(Ψ

𝑗𝐴

𝜕𝑈0(𝑢, 𝜆𝑗)

𝜕𝑢𝑖

) , (36)

where

𝑈 = (

𝑈1

0

𝑈0𝑈1+ 𝑈0

) ,

Ψ𝑗𝐴= (

𝜙3𝑗𝜙4𝑗

−𝜙

2

3𝑗

𝜙

2


) ,

(37)

and 𝛽𝑗is a constant.

According to (34) and (36), we obtain a kind of nonlinearintegrable couplings with self-consistent sources as follows:

𝑢𝑡𝑛

= 𝐽

𝛿𝐻𝑛+1

𝛿𝑢𝑖

+ 𝐽

𝑁

∑

𝑗=1

𝛼𝑗

𝛿𝜆𝑗

𝛿𝑢

= 𝐽𝐿

𝑛𝛿𝐻1

𝛿𝑢𝑖

+ 𝐽

𝑁

∑

𝑗=1

𝛼𝑗

𝛿𝜆𝑗

𝛿𝑢

, 𝑛 = 1, 2, . . . .

(38)

Therefore, according to formulas (34) and (36), we have thefollowing results by direct computations:

𝑁

∑

𝑗=1

𝛿𝜆𝑗

𝛿𝑢

=

𝑁

∑

𝑗=1

(

(

(

(

(

(

(

(

(

𝛿𝜆𝑗

𝛿𝑢

𝛿𝜆𝑗

𝛿V

𝛿𝜆𝑗

𝛿𝑝1

𝛿𝜆𝑗

𝛿𝑝1

)

)

)

)

)

)

)

)

)

=(

2(⟨Φ2, Φ2⟩ − ⟨Φ

1, Φ1⟩)

2 (⟨Φ1, Φ1⟩ + ⟨Φ

2, Φ2⟩ + 2 ⟨Φ

1, Φ2⟩)

⟨Φ4, Φ4⟩ − ⟨Φ

3, Φ3⟩

⟨Φ3, Φ3⟩ + ⟨Φ

4, Φ4⟩ + 2 ⟨Φ

3, Φ4⟩

) ,

(39)

by taking 𝛼𝑗= 1 and 𝛽

𝑗= 1 in formulas (34) and (36).

Therefore, we have nonlinear integrable coupling system ofthe Li equations hierarchy with self-consistent sources asfollows:

𝑢𝑡𝑛

= 𝐾𝑛

= (

𝑢

V

𝑝1

𝑝2

)

𝑡𝑛

= 𝐽𝐿

𝑛(

0

𝛽

0

𝛽

)

+ 𝐽(

2 (⟨Φ2, Φ2⟩ − ⟨Φ

1, Φ1⟩)

2 (⟨Φ1, Φ1⟩ + ⟨Φ

2, Φ2⟩ + 2 ⟨Φ

1, Φ2⟩)

⟨Φ4, Φ4⟩ − ⟨Φ

3, Φ3⟩

⟨Φ3, Φ3⟩ + ⟨Φ

4, Φ4⟩ + 2 ⟨Φ

3, Φ4⟩

)

= 𝐽𝐿

𝑛(

0

𝛽

0

𝛽

) + 𝐽

(

(

(

(

(

(

(

(

(

(

(

2

𝑁

∑

𝑗=1

(𝜑

2

2𝑗− 𝜑

2

1𝑗)

2

𝑁

∑

𝑗=1

(𝜑

2

1𝑗+ 𝜑

2

2𝑗+ 2𝜑1𝑗𝜑2𝑗)

𝑁

∑

𝑗=1

(𝜑

2

4𝑗− 𝜑

2

3𝑗)

𝑁

∑

𝑗=1

(𝜑

2

3𝑗+ 𝜑

2

4𝑗+ 2𝜑3𝑗𝜑4𝑗)

)

)

)

)

)

)

)

)

)

)

)

,

(40)


with

𝜑1𝑗,𝑥= (−𝜆 + V) 𝜑

1𝑗+ (𝑢 + V) 𝜑

2𝑗,

𝜑2𝑗,𝑥= (𝑢 − V) 𝜑

1𝑗+ (𝜆 − V) 𝜑

2𝑗,

𝜑3𝑗,𝑥= (−𝜆 + 𝑝

2) 𝜑1𝑗+ (𝑝1+ 𝑝2) 𝜑2𝑗

+ (−2𝜆 + V + 𝑝2) 𝜑3𝑗+ (𝑢 + V + 𝑝

1+ 𝑝2) 𝜑4𝑗,

𝜑4𝑗,𝑥= (𝑝1− 𝑝2) 𝜑1𝑗+ (𝜆 − 𝑝

2) 𝜑2𝑗

+ (𝑢 − V + 𝑝1− 𝑝2) 𝜑3𝑗

+ (2𝜆 − V − 𝑝2) 𝜑4𝑗, 𝑗 = 1, . . . , 𝑁,

(41)

where Φ𝑖= (𝜑𝑖1, . . . , 𝜑

𝑖𝑁), 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

𝑢𝑡2

= −

1

2

V𝑥𝑥− 𝑢𝑥V − 𝑢V

𝑥+ 2𝜕

𝑁

∑

𝑗=1

(𝜑

2

2𝑗− 𝜑

2

1𝑗) ,

V𝑡2

=

1

2

𝑢𝑥𝑥− 3VV𝑥+ 𝑢𝑢𝑥

− 2𝜕

𝑁

∑

𝑗=1

(𝜑

2

1𝑗+ 𝜑

2

2𝑗+ 2𝜑1𝑗𝜑2𝑗) ,

𝑝1𝑡2

= (

1

4

𝑝2,𝑥−

1

4

V𝑥+

1

2

𝑢V −1

2

V𝑝1−

1

2

𝑢𝑝2

−

1

2

𝑝1𝑝2)

𝑥

+ 𝜕

𝑁

∑

𝑗=1

(𝜑

2

4𝑗− 𝜑

2

3𝑗) ,

𝑝2𝑡2

= (

1

4

𝑝1,𝑥−

1

4

𝑢𝑥+

1

2

V2−

3

4

𝑝

2

2−

3

2

V𝑝2+

1

2

𝑢𝑝1

−

1

4

𝑢

2+

1

4

V2+

1

4

𝑝

2

1)

𝑥

− 𝜕

𝑁

∑

𝑗=1

(𝜑

2

3𝑗+ 𝜑

2

4𝑗+ 2𝜑3𝑗𝜑4𝑗) ,

(42)

with

𝜑1𝑗,𝑥= (−𝜆 + V) 𝜑

1𝑗+ (𝑢 + V) 𝜑

2𝑗,

𝜑2𝑗,𝑥= (𝑢 − V) 𝜑

1𝑗+ (𝜆 − V) 𝜑

2𝑗,

𝜑3𝑗,𝑥= (−𝜆 + 𝑝

2) 𝜑1𝑗+ (𝑝1+ 𝑝2) 𝜑2𝑗

+ (−2𝜆 + V + 𝑝2) 𝜑3𝑗+ (𝑢 + V + 𝑝

1+ 𝑝2) 𝜑4𝑗,

𝜑4𝑗,𝑥= (𝑝1− 𝑝2) 𝜑1𝑗+ (𝜆 − 𝑝

2) 𝜑2𝑗

+ (𝑢 − V + 𝑝1− 𝑝2) 𝜑3𝑗

+ (2𝜆 − V − 𝑝2) 𝜑4𝑗, 𝑗 = 1, . . . , 𝑁.

(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

𝑈 (𝑢, 𝜆)

= (

−𝜆 + V 𝑢 + V 0 0

𝑢 − V 𝜆 − V 0 0

−𝜆 + 𝑝2𝑝1+ 𝑝2−2𝜆 + V + 𝑝

2𝑢 + V + 𝑝

1+ 𝑝2

𝑝1− 𝑝2𝜆 + −𝑝

2𝑢 − V + 𝑝

1− 𝑝2

2𝜆 − V − 𝑝2

),

(44)

we introduce the variables

𝑀 =

𝜑2

𝜑1

, 𝑁 =

𝜑3

𝜑1

, 𝐾 =

𝜑4

𝜑1

. (45)

From (44), we have

𝑀𝑥= 𝑢 − V + 2𝜆𝑀 − 2V𝑀− (𝑢 + V)𝑀

2,

𝑁𝑥= − 𝜆 + 𝑝

2− 𝜆𝑁 + (𝑝

1+ 𝑝2)𝑀

+ 𝑝2𝑁 + (𝑢 + V + 𝑝

1+ 𝑝2)𝐾 − (𝑢 + V)𝑁𝑀,

𝐾𝑥= 𝑝1− 𝑝2+ 3𝜆𝐾 + 𝜆𝑀 − 𝑝

2𝑀

− (2V + 𝑝2)𝐾 + (𝑢 − V + 𝑝

1− 𝑝2)𝑁 − (𝑢 + V) 𝐾𝑀.

(46)

We expand𝑀,𝑁, and𝐾 in powers of 𝜆 as follows:

𝑀 =

∞

∑

𝑗=1

𝑚𝑗𝜆

−𝑗, 𝑁 =

∞

∑

𝑗=1

𝑛𝑗𝜆

−𝑗,

𝐾 =

∞

∑

𝑗=1

𝑘𝑗𝜆

−𝑗.

(47)


Substituting (47) into (46) and comparing the coefficients ofthe same power of 𝜆, we obtain the following:

𝑚1=

1

2

(V − 𝑢) , 𝑛1= 𝑝2,

𝑘1=

1

3

(𝑝2− 𝑝1) +

1

6

(𝑢 − V) ,

𝑚2=

1

4

(V − 𝑢)𝑥+

1

2

(V2− 𝑢V) ,

𝑛2= −𝑝2,𝑥−

1

3

𝑝

2

1−

2

3

𝑢𝑝1+

2

3

V𝑝2+

4

3

𝑝

2

2+

1

6

𝑢

2−

1

6

V2,

𝑘2=

5

36

(𝑢 − V)𝑥+

1

9

(𝑝2− 𝑝1)

𝑥−

5

18

V2+

5

18

𝑢V

+

2

3

V𝑝2−

4

9

𝑢𝑝2+

4

9

𝑝

2

2−

4

9

𝑝1𝑝2−

2

9

V𝑝1,

𝑚3=

1

8

(V − 𝑢)𝑥𝑥+

1

8

𝑢

3−

1

8

𝑢

2V +

3

4

VV𝑥

−

1

2

V𝑢𝑥−

1

4

𝑢V𝑥+

5

8

V3−

5

8

𝑢V2,

𝑛3= 𝑝2,𝑥𝑥+

5

9

(𝑝1𝑢)

𝑥−

5

9

(𝑝2V)𝑥−

32

9

𝑝2𝑝2,𝑥−

7

36

𝑢𝑢𝑥

+

7

36

VV𝑥+

5

9

𝑝1𝑝1,𝑥+

1

9

𝑝1V𝑥−

1

9

𝑝2𝑢𝑥−

5

36

𝑢V𝑥

+

1

9

𝑢𝑝2,𝑥+

5

36

V𝑢𝑥−

1

9

V𝑝1,𝑥+

1

9

𝑝1𝑝2,𝑥−

1

9

𝑝2𝑝1,𝑥

−

4

9

𝑝1𝑢V +

11

9

𝑝2V2−

14

9

𝑢𝑝1𝑝2+

16

9

V𝑝2

2

+

16

9

𝑝

3

2−

7

9

𝑝2𝑢

2−

7

9

𝑝2𝑝

2

1+

5

18

V𝑢2−

5

18

V3−

2

9

V𝑝2

1,

𝑘3=

19

216

(𝑢 − V)𝑥𝑥+

1

27

(𝑝2− 𝑝1)

𝑥𝑥+

5

54

(𝑢 − V)𝑥

−

47

108

VV𝑥+

7

27

V𝑢𝑥+

19

108

𝑢V𝑥−

4

27

V𝑝2,𝑥

+

7

27

𝑝2V𝑥+

5

27

𝑢𝑝2,𝑥−

5

27

𝑝2𝑢𝑥−

5

27

𝑝2𝑝1,𝑥

+

5

27

𝑝1𝑝2,𝑥−

2

27

𝑝1V𝑥−

4

27

V𝑝1,𝑥−

103

216

V3

+

101

216

𝑢V2+

1

8

V𝑢2+

7

18

𝑝2V2−

16

27

𝑝2𝑢V

−

19

54

𝑝2V2+

32

27

V𝑝2

2−

16

27

𝑝1𝑝2V −

4

27

𝑝1V2

−

16

27

𝑢𝑝

2

2+

16

27

𝑝

3

2−

16

27

𝑝1𝑝

2

2+

2

9

𝑝1𝑢

2

+

1

18

𝑢V2+

1

3

𝑢𝑝

2

1+

1

9

𝑝

3

1−

2

9

𝑢V𝑝1−

2

9

𝑢𝑝1𝑝2

−

1

9

V𝑝2

1−

1

9

𝑝2𝑝

2

1−

1

8

𝑢

3, . . . ,

(48)

and a recursion formula for𝑚𝑗, 𝑛𝑗, and 𝑘

𝑗as follows:

𝑚𝑗+1=

1

2

𝑚𝑗,𝑥+ V𝑚𝑗+

1

2

(𝑢 + V)

𝑗−1

∑

𝑙=1

𝑚𝑙𝑚𝑗−𝑙,

𝑛𝑗+1= − 𝑛

𝑗,𝑥+ (𝑝1+ 𝑝2)𝑚𝑗+ 𝑝2𝑛𝑗

+ (𝑢 + V + 𝑝1+ 𝑝2) 𝑘𝑗− (𝑢 + V)

𝑗−1

∑

𝑙=1

𝑚𝑙𝑛𝑗−𝑙,

𝑘𝑗+1=

1

3

𝑘𝑗,𝑥−

1

6

𝑚𝑗,𝑥−

1

3

V𝑚𝑗+

1

3

𝑝2𝑚𝑗

+

1

3

(2V + 𝑝2) 𝑘𝑗−

1

3

(𝑢 − V + 𝑝1− 𝑝2) 𝑛𝑗

−

1

6

(𝑢 + V)

𝑗−1

∑

𝑙=1

𝑚𝑙𝑚𝑗−𝑙+

1

3

(𝑢 + V)

𝑗−1

∑

𝑙=1

𝑚𝑙𝑘𝑗−𝑙.

(49)

Because of

𝜕

𝜕𝑡

[−𝜆 + V + (𝑢 + V)𝑀] =𝜕

𝜕𝑥

[𝑎 + (𝑏 + 𝑐)𝑀] ,

𝜕

𝜕𝑡

[−𝜆 + 𝑝2+ (𝑝1+ 𝑝2)𝑀 + (−2𝜆 + V + 𝑝

2)𝑁

+ (𝑢 + V + 𝑝1+ 𝑝2)𝐾]

=

𝜕

𝜕𝑥

[𝑒 + (𝑓 + 𝑔)𝑀 + (𝑎 + 𝑒)𝑁

+ (𝑏 + 𝑐 + 𝑓 + 𝑔)𝐾] ,

(50)

where

𝑎 = 𝜉0𝜆

2+ 𝜉1𝜆 +

1

2

𝜉0(V2− 𝑢

2) ,

𝑏 = −𝜉0𝑢𝜆 + 𝜉

0(−𝑢V +

1

2

V𝑥) − 𝜉1𝑢,

𝑐 = −𝜉0V𝜆 + 𝜉

0(−V2+

1

2

𝑢𝑥) − 𝜉1V,

𝑒 = 𝜉0𝜆

2+ 𝜉1𝜆

+ 𝜉0(

1

2

V𝑝2−

1

2

𝑢𝑝1+

1

4

𝑢

2−

1

4

V2−

1

4

𝑝

2

1+

1

4

𝑝

2

2) ,

𝑓 = − 𝜉0𝑝1𝜆 + 𝜉0(

1

4

𝑝2,𝑥−

1

4

V𝑥+

1

2

𝑢V −1

2

V𝑝1

−

1

2

𝑢𝑝2−

1

2

𝑝1𝑝2) − 𝜉1𝑝1,

𝑔 = − 𝜉0𝑝2𝜆 + 𝜉0(

1

4

𝑝1,𝑥−

1

4

𝑢𝑥+

1

2

V2

−

1

2

𝑝

2

2− V𝑝2) − 𝜉1𝑝2.

(51)


Assume that

𝜎 = −𝜆 + V + (𝑢 + V)𝑀,

𝜃 = 𝑎 + (𝑏 + 𝑐)𝑀,

𝜌 = − 𝜆 + 𝑝2+ (𝑝1+ 𝑝2)𝑀 + (−2𝜆 + V + 𝑝

2)𝑁

+ (𝑢 + V + 𝑝1+ 𝑝2)𝐾,

𝛿 = 𝑒 + (𝑓 + 𝑔)𝑀 + (𝑎 + 𝑒)𝑁 + (𝑏 + 𝑐 + 𝑓 + 𝑔)𝐾.

(52)

Then, (50) can be written as 𝜎𝑡= 𝜃𝑥and 𝜌

𝑡= 𝛿𝑥, which are

the right form of conservation laws. We expand 𝜎, 𝜃, 𝜌, and 𝛿as series in powers of 𝜆 with the coefficients, which are calledconserved densities and currents, respectively:

𝜎 = −𝜆 + V + (𝑢 + V)

∞

∑

𝑗=1

𝜎𝑗𝜆

−𝑗,

𝜃 = 𝜉0𝜆

2+ 𝜉1𝜆 +

∞

∑

𝑗=1

𝜃𝑗𝜆

−𝑗,

𝜌 = −𝜆 + 𝑝2+

∞

∑

𝑗=1

𝜌𝑗𝜆

−𝑗,

𝛿 = 𝜉0𝜆

2+ 𝜉1𝜆 +

∞

∑

𝑗=1

𝛿𝑗𝜆

−𝑗,

(53)

where 𝜉0and 𝜉

1are constants of integration. The first

conserved densities and currents are read as follows:

𝜎1=

1

2

(V2− 𝑢

2) ,

𝜃1= 𝜉0(

1

2

𝑢𝑢𝑥−

3

4

𝑢V𝑥−

3

4

VV𝑥) −

1

2

𝜉1(V2− 𝑢

2) ,

𝜌1=

1

2

𝑢𝑝1+

1

3

V𝑝2−

4

3

𝑝

2

2−

1

6

𝑢

2

+

1

6

V2+

1

3

𝑝

2

1+

1

6

𝑢𝑝1+ 2𝑝2,𝑥,

𝛿1= 𝜉0(2𝑝2,𝑥𝑥+

1

36

𝑝1V𝑥+

41

36

𝑝1𝑢𝑥−

41

36

𝑝2V𝑥

−

1

36

𝑝2𝑢𝑥−

41

36

V𝑝2,𝑥−

1

36

V𝑝1,𝑥

+

13

36

VV𝑥−

1

36

V𝑢𝑥+

1

36

𝑢𝑝2,𝑥

+

41

36

𝑢𝑝1,𝑥−

13

36

𝑢𝑢𝑥−

257

36

𝑝2𝑝2,𝑥

+

41

36

𝑝1𝑝1,𝑥+

1

36

𝑝1𝑝2,𝑥

−

1

36

𝑝2𝑝1,𝑥+

47

36

𝑝2V2−

1

18

V𝑢2

+

11

36

V𝑝1𝑝2+

55

18

V𝑝2

2−

43

36

𝑝2𝑢

2

−

53

18

𝑢𝑝1𝑝2+

93

36

𝑝

3

2

−

3

4

𝑝2𝑝

2

1−

1

18

V𝑝2

1−

4

9

V𝑝2

2

−

1

9

𝑢V𝑝1−

4

9

𝑝1𝑝

2

2

+

1

18

V3+

1

36

𝑢V𝑥)

+ 𝜉1(−2𝑝2,𝑥+ V𝑝1−

5

3

𝑢𝑝1+

5

3

V𝑝2− 𝑢𝑝2

+

7

3

𝑝

2

2+

1

6

𝑢

2−

1

6

V2−

1

3

𝑝

2

1) , . . . .

(54)

The recursion relations for 𝜎𝑗, 𝜃𝑗, 𝜌𝑗, and 𝛿

𝑗are as follows:

𝜎𝑗= (𝑢 + V)𝑚

𝑗,

𝜃𝑗= − 𝜉

0(𝑢 + V)𝑚

𝑗+1+ 𝜉0(

1

2

𝑢𝑥+

1

2

V𝑥− 𝑢V − V

2)𝑚𝑗

− 𝜉1(𝑢 + V)𝑚

𝑗,

𝜌𝑗= (𝑝1+ 𝑝2)𝑚𝑗− 2𝑛𝑗+1+ (V + 𝑝

2) 𝑛𝑗

+ (𝑢 + V + 𝑝1+ 𝑝2) 𝑘𝑗,

𝛿𝑗= 𝜉0[ − (𝑝

1+ 𝑝2)𝑚𝑗+1

+ (

1

4

𝑝2,𝑥+

1

4

𝑝1,𝑥−

1

4

V𝑥

−

1

4

𝑢𝑥+

1

2

𝑢V −1

2

V𝑝1

−

1

2

𝑢𝑝2−

1

2

𝑝1𝑝2+

1

2

V2

−

1

2

𝑝

2

2− 𝑝2V)𝑚𝑗

+ 2𝑛𝑗+2+ (

1

2

V2−

1

2

𝑢

2+

1

2

V𝑝2

−

1

2

𝑢𝑝1+

1

4

𝑢

2

+

1

4

𝑝

2

2−

1

4

V2−

1

4

𝑝

2

1) 𝑛𝑗

− (𝑢 + V + 𝑝1+ 𝑝2) 𝑘𝑗+1


+ (

1

4

𝑢𝑥+

1

4

V𝑥+

1

4

𝑝1,𝑥+

1

4

𝑝2,𝑥−

1

2

𝑢V

−

1

2

V2−

1

2

V𝑝1−

1

2

𝑢𝑝2

−

1

2

𝑝1𝑝2−

1

2

𝑝

2

2− V𝑝2)]

+ 𝜉1[2𝑛𝑗+1− (𝑝1+ 𝑝2)𝑚𝑗

− (𝑢 + V + 𝑝1+ 𝑝2) 𝑘𝑗] ,

(55)

where𝑚𝑗, 𝑛𝑗, and 𝑘

𝑗can be calculated from (49).The infinite

conservation laws of nonlinear integrable couplings (37) canbe easily obtained in (45)–(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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