# Self-Consistent Sources and Conservation Laws for Nonlinear Integrable Couplings of the Li Soliton Hierarchy

Authors: Han-yu Wei; Tie-cheng Xia
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 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
(1)ut=K(u)
has a Lax pair U and V, which belong to semisimple matrix Lie algebras. Introduce an enlarged spectral matrix
(2)U-=U-(u-)=[U(u)0Ua(v)U(u)+Ua(v)]
from a zero curvature representation
(3)Ut--Vx-+[U-,V-]=0,
where
(4)V-=V-(u-)=[V(u)0Va(u-)V(u)+Va(u-)],
then we can give rise to
(5)Ut-Vx+[U,V]=0,Ua,t-Va,x+[U,Va]+[Ua,V]+[Ua,Va]=0.
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, 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 Section2, 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:
(6)G1=span{e1,e2,e3},
where
(7)e1=(100-1),e2=(0110),e3=(01-10),
which have the commutative relations
(8)[e1,e2]=2e2,[e1,e3]=-2e3,[e2,e3]=e1.
Let us introduce a Lie algebra with matrix blocks by using G1 in order to get nonlinear couplings of soliton hierarchy as follows:
(9)G=span{g1,…,g6},
where
(10)g1=(e100e1),g2=(e200e2),g3=(e300e3),g4=(00e1e1),g5=(00e2e2),g6=(00e3e3).
Define a commutator as follows:
(11)[a,b]=ab-ba,a,b∈G.
A direct verification exhibits that
(12)[g1,g2]=2g3,[g1,g3]=2g2,[g2,g3]=-2g1,[g1,g5]=2g6,[g1,g6]=2g5,[g2,g4]=-2g6,[g2,g6]=-2g4,[g3,g4]=-2g5,[g3,g5]=2g4,[g4,g5]=2g6,[g4,g6]=2g5,[g5,g6]=-2g4,[g1,g4]=[g3,g6]=0.
Set
(13)G1~=span{g1,g2,g3},G2~=span{g4,g5,g6},
then we find that
(14)G=G1~⊕G2~,G1~≅G1,[G1~,G2~]⊆G2~,
and G1~ and G2~ are all simple Lie subalgebras.While we use Lie algebras to generate integrable hierarchies of evolution equations, we actually employ their loop algebrasG~=G⊗C(λ,λ-1) to establish Lax pairs, where C(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:
(15)G~=span{g1(n),…,g6(n)},
where gi(n)=giλn, [gi(m),gj(n)]=[gi,gj]λm+n, 1≤i, j≤6, m,n∈Z.We consider an auxiliary linear problem as follows:(16)(φ1φ2φ3φ4)x=U(u,λ)(φ1φ2φ3φ4),U(u,λ)=R1+∑i=16‍uigi(λ),(φ1φ2φ3φ4)tn=Vn(u,λ)(φ1φ2φ3φ4),
where u=(u1,…,us)T, Un=R1+u1g1+⋯+u6g6,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
(17)Ut-Vx+[U,V]=0,λt=0.
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:
(18)Vx=[U,V],V=∑n=0∞‍Vnλ-n=(ab+c00b-c-a00ef+ga+eb+c+f+gf-g-eb-c+f-g-(a+e)),
then we sesrch for ▵n∈G~, the new Vn can be constructed by
(19)Vn=∑m=0n‍Vm(u)λn-m+▵n(u,λ).
Solving zero curvature (17), we could get evolution equation as follows:
(20)ut=K(u,ux,…,∂pu∂xp).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:
(21)φx=U(u,λ)φ,U(u,λ)=-g1(1)+vg1(0)+ug2(0)+vg3(0)-g4(1)+p2g4(0)+p1g5(0)+p2g6(0),
that is,
(22)U(u,λ)=(-λ+vu+v00u-vλ-v00-λ+p2p1+p2-2λ+v+p2u+v+p1+p2p1-p2λ-p2u-v+p1-p22λ-v-p2)=(U10U0U1+U0),
where λ is a spectral parameter and U1 satisfies φx=U1φ 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 equationVx=[U,V] of the spectral problem (21) is firstly solved, we assume that a solution V is given by the following:
(23)V=(ab+c00b-c-a00ef+ga+eb+c+f+gf-g-eb-c+f-g-(a+e))=∑n=0∞‍Vnλ-n=∑n=0∞‍×(anbn+cn00bn-cn-an00enfn+gnan+enbn+cn+fn+gnfn-gn-enbn-cn+fn-gn-(an+en))λ-n.
Therefore, the condition (18) becomes the following recursion relation:
(24)an,x=2vbn-2ucn,bn,x=-2cn+1+2vcn-2van,cn,x=-2bn+1+2vbn-2uan,en,x=-2ugn+2vfn-2p1cn+2p2bn-2p1gn+2p2fn,fn,x=-2gn+1+2vgn-2ven+2p2cn-2p2an+2p2gn-2p2en,gn,x=-2fn+1+2vfn-2uen+2p2bn-2p1an+2p2fn-2p1en.
Choose the initial data
(25)a0=e0=β,b0=c0=f0=g0=0,
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:
(26)a1=0,b1=-uβ,c1=-vβ,e1=0,f1=-p1β,g1=-p2β,a2=12(v2-u2),b2=(12vx-uv)β,c2=(12ux-v2)β,e2=(12vp2-12up1+14u2-14v2-14p12+14p22)β,f2=(14p2,x-14vx+12uv-12vp1-12up2-12p1p2)β,g2=(14p1,x-14ux+12v2-12p22-vp2)β,….
Considering
(27)Vn=V+▵n,▵n=(-(an-cn)0000an-cn00-(en-gn)0-(an-cn)-(en-gn)0en-gn0an-cn+en-gn).
From the zero curvature equation Ut-Vx+[U,V]=0, we obtain the nonlinear integrable coupling system
(28)utn=Kn=(uvp1p2)tn=(bn,x-(an-cn)xfn,x-(en-gn)x)=J(bnan-cnfnen-gn)=JLn(0β0β),n≥0,
with the Hamiltonian operator J and the hereditary recursion operator L, respectively, as follows:
(29)J=(∂0000-∂0000∂0000-∂),L=(012∂-u00∂-1u∂+12∂∂-1v∂+v000M10M2M3M4M5M6),
where
(30)M1=-14∂-12p1+12u,M2=14∂-12p1-12u,M3=-12∂-1u-12∂-1p1-∂-1p1∂-14∂,M4=-12∂-1v∂+12∂-1p2∂+∂-1p2u-12v+12p2,M5=12∂-1u∂+12∂-1p1∂+14∂,M6=-2∂-1uv-32∂-1p1v+12∂-1v∂+12∂-1p2∂-12v+12p2.
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.Takingn=2, we get that the nonlinear integrable couplings of Li soliton hierarchy are as follows:
(31)ut2=(-12vxx-uxv-uvx)β,vt2=(12uxx-3vvx+uux)β,p1,t2=(14p2,x-14vx+12uv-12vp1-12up2-12p1p2)xβ,p2,t2=(14p1,x-14ux+12v2-34p22-32vp2+12up1-14u2+14v2+14p12)xβ.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.


## 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 λj, j=1,2,…,N and the systems of (16) become in the following form:
(32)(φ1jφ2jφ3jφ4j)x=U(u,λj)(φ1jφ2jφ3jφ4j)=∑i=16‍uigi(λ)(φ1jφ2jφ3jφ4j),j=1,…,N,(φ1jφ2jφ3jφ4j)tn=Vn(u,λj)(φ1jφ2jφ3jφ4j)=[∑m=0n‍Vm(u)λjn-m+Δn(u,λj)]×(φ1jφ2jφ3jφ4j),j=1,…,N.Based on the result in [32], we show that the following equation
(33)δHkδu+∑j=1N‍αjδλjδu=0
holds true, where αj is a constant. From (32), we may know that
(34)δλjδui=αjTr(Ψj∂U(u,λj)∂ui)=αjTr(Ψjgi(λj)),i=1,2,
where Tr denotes the trace of a matrix and
(35)Ψj=(ϕ1jϕ2j-ϕ1j2ϕ3jϕ4j-ϕ3j2ϕ2j2-ϕ1jϕ2jϕ4j2-ϕ3jϕ4j00ϕ1jϕ2j-ϕ1j200ϕ2j2-ϕ1jϕ2j),j=1,…,N.Fori=3,4 we define that
(36)δλjδui=βjTr(ΨjA∂U0(u,λj)∂ui),
where
(37)U=(U10U0U1+U0),ΨjA=(ϕ3jϕ4j-ϕ3j2ϕ4j2-ϕ3jϕ4j),
and βj is a constant.According to (34) and (36), we obtain a kind of nonlinear integrable couplings with self-consistent sources as follows:
(38)utn=JδHn+1δui+J∑j=1N‍αjδλjδu=JLnδH1δui+J∑j=1N‍αjδλjδu,n=1,2,….
Therefore, according to formulas (34) and (36), we have the following results by direct computations:
(39)∑j=1N‍δλjδu=∑j=1N‍(δλjδuδλjδvδλjδp1δλjδp1)=(2(〈Φ2,Φ2〉-〈Φ1,Φ1〉)2(〈Φ1,Φ1〉+〈Φ2,Φ2〉+2〈Φ1,Φ2〉)〈Φ4,Φ4〉-〈Φ3,Φ3〉〈Φ3,Φ3〉+〈Φ4,Φ4〉+2〈Φ3,Φ4〉),
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:
(40)utn=Kn=(uvp1p2)tn=JLn(0β0β)+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〉)=JLn(0β0β)+J(2∑j=1N‍(φ2j2-φ1j2)2∑j=1N‍(φ1j2+φ2j2+2φ1jφ2j)∑j=1N‍(φ4j2-φ3j2)∑j=1N‍(φ3j2+φ4j2+2φ3jφ4j)),
with
(41)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,φ4j,x=(p1-p2)φ1j+(λ-p2)φ2j+(u-v+p1-p2)φ3j+(2λ-v-p2)φ4j,j=1,…,N,
where Φi=(φi1,…,φiN), i=1,2,3,4, and 〈·,·〉 is the standard inner product in RN.Whenn=2 andβ=2, we obtain nonlinear integrable couplings of Li hierarchy with self-consistent sources
(42)ut2=-12vxx-uxv-uvx+2∂∑j=1N‍(φ2j2-φ1j2),vt2=12uxx-3vvx+uux-2∂∑j=1N‍(φ1j2+φ2j2+2φ1jφ2j),p1t2=(14p2,x-14vx+12uv-12vp1-12up2-12p1p2)x+∂∑j=1N‍(φ4j2-φ3j2),p2t2=(14p1,x-14ux+12v2-34p22-32vp2+12up1-14u2+14v2+14p12)x-∂∑j=1N‍(φ3j2+φ4j2+2φ3jφ4j),
with
(43)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,φ4j,x=(p1-p2)φ1j+(λ-p2)φ2j+(u-v+p1-p2)φ3j+(2λ-v-p2)φ4j,j=1,…,N.


## 4. Conservation Laws for the Nonlinear Integrable Couplings of Li Soliton Hierarchy

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(44)U(u,λ)=(-λ+vu+v00u-vλ-v00-λ+p2p1+p2-2λ+v+p2u+v+p1+p2p1-p2λ+-p2u-v+p1-p22λ-v-p2),
we introduce the variables
(45)M=φ2φ1,N=φ3φ1,K=φ4φ1.
From (44), we have
(46)Mx=u-v+2λM-2vM-(u+v)M2,Nx=-λ+p2-λN+(p1+p2)M+p2N+(u+v+p1+p2)K-(u+v)NM,Kx=p1-p2+3λK+λM-p2M-(2v+p2)K+(u-v+p1-p2)N-(u+v)KM.
We expand M, N, and K in powers of λ as follows:
(47)M=∑j=1∞‍mjλ-j,N=∑j=1∞‍njλ-j,K=∑j=1∞‍kjλ-j.
Substituting (47) into (46) and comparing the coefficients of the same power of λ, we obtain the following:
(48)m1=12(v-u),n1=p2,k1=13(p2-p1)+16(u-v),m2=14(v-u)x+12(v2-uv),n2=-p2,x-13p12-23up1+23vp2+43p22+16u2-16v2,k2=536(u-v)x+19(p2-p1)x-518v2+518uv+23vp2-49up2+49p22-49p1p2-29vp1,m3=18(v-u)xx+18u3-18u2v+34vvx-12vux-14uvx+58v3-58uv2,n3=p2,xx+59(p1u)x-59(p2v)x-329p2p2,x-736uux+736vvx+59p1p1,x+19p1vx-19p2ux-536uvx+19up2,x+536vux-19vp1,x+19p1p2,x-19p2p1,x-49p1uv+119p2v2-149up1p2+169vp22+169p23-79p2u2-79p2p12+518vu2-518v3-29vp12,k3=19216(u-v)xx+127(p2-p1)xx+554(u-v)x-47108vvx+727vux+19108uvx-427vp2,x+727p2vx+527up2,x-527p2ux-527p2p1,x+527p1p2,x-227p1vx-427vp1,x-103216v3+101216uv2+18vu2+718p2v2-1627p2uv-1954p2v2+3227vp22-1627p1p2v-427p1v2-1627up22+1627p23-1627p1p22+29p1u2+118uv2+13up12+19p13-29uvp1-29up1p2-19vp12-19p2p12-18u3,…,
and a recursion formula for mj, nj, and kj as follows:
(49)mj+1=12mj,x+vmj+12(u+v)∑l=1j-1‍mlmj-l,nj+1=-nj,x+(p1+p2)mj+p2nj+(u+v+p1+p2)kj-(u+v)∑l=1j-1‍mlnj-l,kj+1=13kj,x-16mj,x-13vmj+13p2mj+13(2v+p2)kj-13(u-v+p1-p2)nj-16(u+v)∑l=1j-1‍mlmj-l+13(u+v)∑l=1j-1‍mlkj-l.
Because of
(50)∂∂t[-λ+v+(u+v)M]=∂∂x[a+(b+c)M],∂∂t[-λ+p2+(p1+p2)M+(-2λ+v+p2)N+(u+v+p1+p2)K]=∂∂x[e+(f+g)M+(a+e)N+(b+c+f+g)K],
where
(51)a=ξ0λ2+ξ1λ+12ξ0(v2-u2),b=-ξ0uλ+ξ0(-uv+12vx)-ξ1u,c=-ξ0vλ+ξ0(-v2+12ux)-ξ1v,e=ξ0λ2+ξ1λ+ξ0(12vp2-12up1+14u2-14v2-14p12+14p22),f=-ξ0p1λ+ξ0(14p2,x-14vx+12uv-12vp1-12up2-12p1p2)-ξ1p1,g=-ξ0p2λ+ξ0(14p1,x-14ux+12v2-12p22-vp2)-ξ1p2.
Assume that
(52)σ=-λ+v+(u+v)M,θ=a+(b+c)M,ρ=-λ+p2+(p1+p2)M+(-2λ+v+p2)N+(u+v+p1+p2)K,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:
(53)σ=-λ+v+(u+v)∑j=1∞‍σjλ-j,θ=ξ0λ2+ξ1λ+∑j=1∞‍θjλ-j,ρ=-λ+p2+∑j=1∞‍ρjλ-j,δ=ξ0λ2+ξ1λ+∑j=1∞‍δjλ-j,
where ξ0 and ξ1 are constants of integration. The first conserved densities and currents are read as follows:
(54)σ1=12(v2-u2),θ1=ξ0(12uux-34uvx-34vvx)-12ξ1(v2-u2),ρ1=12up1+13vp2-43p22-16u2+16v2+13p12+16up1+2p2,x,δ1=ξ0(2p2,xx+136p1vx+4136p1ux-4136p2vx-136p2ux-4136vp2,x-136vp1,x+1336vvx-136vux+136up2,x+4136up1,x-1336uux-25736p2p2,x+4136p1p1,x+136p1p2,x-136p2p1,x+4736p2v2-118vu2+1136vp1p2+5518vp22-4336p2u2-5318up1p2+9336p23-34p2p12-118vp12-49vp22-19uvp1-49p1p22+118v3+136uvx)+ξ1(-2p2,x+vp1-53up1+53vp2-up2+73p22+16u2-16v2-13p12),….
The recursion relations for σj,θj,ρj, and δj are as follows:
(55)σj=(u+v)mj,θj=-ξ0(u+v)mj+1+ξ0(12ux+12vx-uv-v2)mj-ξ1(u+v)mj,ρj=(p1+p2)mj-2nj+1+(v+p2)nj+(u+v+p1+p2)kj,δj=ξ0[12p22-(p1+p2)mj+1+(14p2,x+14p1,x-14vx-14ux+12uv-12vp1-12up2-12p1p2+12v2-12p22-p2v)mj+2nj+2+(12v2-12u2+12vp2-12up1+14u2+14p22-14v2-14p12)nj-(u+v+p1+p2)kj+1+(14ux+14vx+14p1,x+14p2,x-12uv-12v2-12vp1-12up2-12p1p2-12p22-vp2)]+ξ1[2nj+1-(p1+p2)mj-(u+v+p1+p2)kj],
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 self-consistent 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.



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