# A Hierarchy of Discrete Integrable Coupling System with Self-Consistent Sources

Authors: Yuqing Li; Huanhe Dong; Baoshu Yin
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

Integrable coupling system of a lattice soliton equation hierarchy is deduced. The Hamiltonian structure of the integrable coupling is constructed by using the discrete quadratic-form identity. The Liouville integrability of the integrable coupling is demonstrated. Finally, the discrete integrable coupling system with self-consistent sources is deduced.


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## Body

## 1. Introduction

Many physical problems may be modeled by soliton equation. The Hamiltonian structures of many systems have been obtained by the famous trace identity [1–6]. The study of integrable couplings of integrable systems has become the focus of common concern in recent years. It originates from the investigations on the symmetry problems and associated centerless Virasoro algebras [7]. Many integrable coupling systems have been constructed by using the methods of a direct method [8], perturbations [9], enlarging spectral problems [10, 11], creating new loop algebras [12, 13], and semidirect sums of Lie algebras [14, 15]. The Hamiltonian structures of the integrable couplings of lattice equations can be constructed by means of the discrete quadratic-form identity [16, 17].Since Mel’Nikov proposed a new kind of integrable model which was called soliton equations with self-consistent sources [18] in 1983, many soliton equations with self-consistent sources [19–23] have been presented in recent years. For applications, these kinds of systems are usually used to describe interactions between different solitary waves. In this paper, we deduce a hierarchy of discrete integrable coupling system with self-consistent sources which are few compared with the continuous ones.The paper will be organized as follows. We first get a hierarchy of integrable lattice soliton equation with self-consistent sources in Section2. In Section 3, a hierarchy of discrete integrable coupling system is derived by making use of the discrete zero curvature representation. By means of the discrete quadratic-form identity we establish the Hamiltonian structures of the hierarchy. Further, the resulting Hamiltonian equations are all proved to be integrable in Liouville sense. Finally, we give the integrable coupling systems with self-consistent sources.


## 2. A Hierarchy of Integrable Lattice Soliton Equations with Self-Consistent Sources

We first briefly describe our notations. Assumefn=f(n) is a lattice function; the shift operator E and the inverse of E are defined by
(1)Efn=f(n+1),E-1fn=f(n-1),n∈Z,Ekfn=f(n+k),n,k∈Z.A system of discrete equations(2)∂tmun=Km(un)
is said to have a discrete Lax pair
(3)Eφn=Un(un,λ)φn,∂tmφn=Vn[m](un,λ)φn,
if it is equivalent to the compatibility condition
(4)∂tmUn(un,λ)=(EVn[m](un,λ))Un(un,λ)-Un(un,λ)Vn[m](un,λ).
In [16], a Lie algebra is presented as
(5)G=span⁡{ω1,ω2,ω3,ω4,ω5,ω6,ω7,ω8},
where
(6)ω1=(1000000000100000),ω2=(0000010000000001),ω3=(0100000000010000),ω4=(0000100000000010),ω5=(0000000010000000),ω6=(0000000000000100),ω7=(0000000000001000),ω8=(0000000001000000).
Set G1=span⁡{ω1,ω2,ω3,ω4} and G2=span⁡{ω5,ω6,ω7,ω8}; it is easy to see that G, G1, and G2 construct three Lie algebra, and
(7)G=G1⊕G2,[G1,G2]≡G1G2-G2G1⊆G2.
So G2 is an Abelian ideal of the Lie algebra G. The corresponding loop algebra G~ is defined by
(8)G~=span⁡{ωi(m),i=1,2,…,8},ωi(m)=ωiλm.In [15], a new discrete matrix spectral problem has been proposed:
(9)Eϕn=U~n(rn,λ)ϕn,U~n(rn,λ)=rnω2(1)+ω3(0),
by solving the stationary discrete zero curvature equation
(10)(EΓn)Un-UnΓn=0,
where
(11)Γn=anω1(0)-anω2(0)+bnω3(0)+cnω4(1),an=∑m=0∞an(m)λ-m,bn=∑m=0∞bn(m)λ-m,cn=∑m=0∞cn(m)λ-m,
and introducing the auxiliary spectral problems associated with the spectral problem (9)
(12)∂tmϕn=V~n[m]ϕn,m≥0,V~n[m]=∑i=0m[an(i)ω1(m-i)-an(i)ω2(m-i)1111ii+bn(i)ω3(m-i)+cn(i)ω4(m-i+1)]-an(m)ω1(0)+an(m)ω2(0),
a hierarchy of integrable lattice soliton equations with a potential rn has been presented:
(13)∂tmrn=rn(an+1(m)-an(m)),m≥0,
where
(14)an(0)=-12,an(1)=1rnrn-1,an(2)=-1rnrn-1(1rnrn-1+1rn-2rn-1+1rn+1rn),….
Equation (13) possesses the following Hamiltonian forms [15]:
(15)∂tmrn=J~δF~n(m)δrn=M~δF~n(m-1)δrn,m≥1,
where
(16)J~=rn(1-E)(1+E)-1rn,M~=E-E-1,F~n(m)=∑n∈zFn(m),Fn(m)=-an(m)m,m≥1.Next, we will construct a hierarchy of integrable lattice soliton equations (13) with self-consistent sources. For n distinct real λj, consider the auxiliary linear problem
(17)E(ϕ1jϕ2jϕ3jϕ4j)=U~n(rn,λj)(ϕ1jϕ2jϕ3jϕ4j),(ϕ1jϕ2jϕ3jϕ4j)tm=V~n(rn,λj)(ϕ1jϕ2jϕ3jϕ4j).
Based on the results in [24], we show the following equation:
(18)δF~n(m)δrn+∑j=1Nδλjδrn=0,
where
(19)δλjδrn=12Tr⁡(ψj∂U~(rn,λj)∂rn),ψj=(ϕ1jϕ2j-ϕ1j2ϕ3jϕ4j-ϕ3j2ϕ2j2-ϕ1jϕ2jϕ4j2-ϕ3jϕ4j00ϕ1jϕ2j-ϕ1j200ϕ2j2-ϕ1jϕ2j),kkkkkkkkkkkkkkkkikkkj=1,2,…,N.
According to the approach proposed in [24–26], through a direct computation, we obtain the discrete integrable hierarchy with self-consistent sources as follows:
(20)∂tmrn=J~(δF~n(m)δrn+∑j=1Nδλjδrn)=J~(δF~n(m)δrn-∑j=1Nλjϕ1jϕ2j),m≥1.Takingm=1 in the above system, under t1→t, we can obtain the following equation with self-consistent sources:
(21)∂trn=1rn+1-1rn-1-rn(1-E)(1+E)-1rn∑j=1Nλjϕ1jϕ2j.


## 3. A Hierarchy of Discrete Integrable Coupling System with Self-Consistent Sources

First, we will give out the integrable couplings of the hierarchy (13). Consider the discrete isospectral problem
(22)Eϕn=Un(un,λ)ϕn,Un(un,λ)=rnω2(1)+ω3(0)+ω4(1)+snω6(1),
in which un=(rn,sn)T is the potential, rn=r(n,t) and sn=s(n,t) are real functions defined over Z×R, λ is a spectral parameter, λt=0, and ϕn=(ϕ1(n),ϕ2(n),ϕ3(n),ϕ4(n))T is the eigenfunction vector.We solve the stationary discrete zero curvature equation(23)(EΓn)Un-UnΓn=0,
where
(24)Γn=anω1(0)-anω2(0)+bnω3(0)+cnω4(1)+enω5(0)-enω6(0)+gnω7(1)+fnω8(0).
Equation (23) gives
(25)bn+1=cn,an+λrnbn+1+an+1=0,an+λrncn+an+1=0,cn+1-bn+rn(an-an+1)=0,fn+1=gn,en+1+en+λrnfn+1+λsnbn+1=0,en+1+en+λrngn+λsncn=0,-fn+gn+1+rn(en-en+1)+sn(an-an+1)=0.
Substituting the expansions
(26)an=∑m=0∞an(m)λ-m,bn=∑m=0∞bn(m)λ-m,cn=∑m=0∞cn(m)λ-m,en=∑m=0∞en(m)λ-m,fn=∑m=0∞fn(m)λ-m,gn=∑m=0∞gn(m)λ-m
into (25), we can get the recursion relation
(27)an+1(m)+an(m)=-rnbn+1(m+1),an+1(m)+an(m)=-rncn(m+1),rn(anm-an+1(m))+cn+1(m)-bnm=0,en+1(m)+en(m)=-snbn+1(m+1)-rnfn+1(m+1),en+1(m)+en(m)=-sncn(m+1)-rngn(m+1),sn(an(m)-an+1(m))+rn(en(m)-en+1(m))-fn(m)+gn+1(m)=0.
The initial values are taken as
(28)an(0)=-12,bn(0)=0,cn(0)=0,en(0)=-12,fn(0)=0,gn(0)=0.
Note that the definition of the inverse operator of D=(E-1) does not yield any arbitrary constant in computing an(m) and en(m), m≥1. Thus, the recursion relation (27) uniquely determines
(29)an(m),bn(m),cn(m),en(m),fn(m),gn(m),m≥1,
and the first few quantities are given by
(30)an(1)=1rnrn-1,bn(1)=1rn-1,cn(1)=1rn,en(1)=1rn-1rn-sn-1rn-12rn-snrn2rn-1,fn(1)=1rn-1-sn-1rn-12,gn(1)=1rn-snrn2,an(2)=-1rnrn-1(1rnrn-1+1rn-2rn-1+1rn+1rn),bn(2)=-1rn-12(1rn-2+1rn),cn(2)=-1rn2(1rn-1+1rn+1),en(2)=-1rnrn-1(1rnrn-1+1rnrn+1+1rn-2rn-1)+2snrn-1rn3(1rn+1+1rn-1)+2sn-1rnrn-13(1rn-2+1rn)+1rn2rn+12(sn-1rn+2+sn+1rn-1)+1rn-12rnrn-2(sn-2rn-2+snrn),fn(2)=2sn-1rn-12(1rn+1rn-2)+1rn-12rn-2(sn-2rn-2-1)+1rn-12rn(snrn-1),gn(2)=2snrn2(1rn+1+1rn-1)+1rn2rn-1(sn-1rn-1-1)+1rn2rn+1(sn+1rn+1-1).
Set
(31)Vn(m)=∑i=0m[an(i)ω1(m-i)-an(i)ω2(m-i)+bn(i)ω3(m-i)kkkkk+cn(i)ω4(m-i+1)+en(i)ω5(m-i)-en(i)ω6kkkkk×(m-i)+gn(i)ω7(m-i+1)+fn(i)ω8(m-i)],
so
(32)E(Vn(m))Un-UnVn(m)=-rncn(m+1)ω3(0)+rncn(m+1)ω4(1)-(en(m)+en+1(m))ω7(1)+(en(m)+en+1(m))ω8(0).
Take ηn(m)=-an(m)ω1(0)+an(m)ω2(0)-en(m)ω5(0)+en(m)ω6(0), m≥0, and let
(33)Vn[m]=Vn(m)+ηn(m).
We introduce the auxiliary spectral problems associated with the spectral problem (22):
(34)∂tmϕn=Vn[m]ϕn,m≥0.
The compatibility conditions of (22) and (34) are
(35)∂tmUn=(EVn[m])Un-UnVn[m],m≥0,
which give rise to the following hierarchy of integrable lattice equations:
(36)∂tmrn=rn(an+1(m)-an(m)),m≥0,∂tmsn=sn(an+1(m)-an(m))-rn(en(m)-en+1(m)),m≥0.So (35) is the discrete zero curvature representation of (36); the discrete spectral problems (22) and (34) constitute the Lax pairs of (36), and (36) are a hierarchy of Lax integrable nonlinear lattice equations. It is easy to verify that the first nonlinear lattice equation in (36), when m=1, under t1→t, is
(37)∂trn=(E-E-1)1rn,∂tsn=(E-1-E)snrn2+(E-E-1)1rn.In (36) the first lattice equations
(38)∂tmrn=rn(an+1(m)-an(m)),m≥0,
constitute a hierarchy of integrable lattice soliton equations with a potential rn; in the view of integrable coupling theory [7, 13, 17], (36) are integrable coupling systems of (13) or (15).In what follows, we would like to establish the Hamiltonian structures for the integrable coupling systems (36).Seta=∑i=18aiωi, b=∑i=18biωi, and c=∑i=18ciωi∈G. We define a map
(39)σ:G⟶R8,a⟼(a1,a2,…,a8)T,a∈G.
Following [16], we introduce the matrix
(40)F=(1000100001000100000100100010000110000000010000000010000000010000).It is easy to verify thatF meets FT=F. Under the definition of the quadratic-form function
(41){a,b}=aTFb,
we have {ab,c}={a,bc} and a,b,c∈G. Set Rn=ΓnUn-1; through a direct calculation, we get
(42){Rn,∂Un∂λ}=en+anλ+rn(cn+gn)+sncn,{Rn,∂Un∂rn}=λ(cn+gn),{Rn,∂Un∂sn}=λcn.
By the discrete quadratic-form identity [16]
(43)δδrn∑n∈Z{Rn,∂Un∂λ}=(λ-γ(∂∂λ)λγ){Rn,∂Un∂rn},δδsn∑n∈Z{Rn,∂Un∂λ}=(λ-γ(∂∂λ)λγ){Rn,∂Un∂sn},
with γ being a constant to be determined, we have
(44)δδrn∑n∈Z‍[en+anλ+rn(cn+gn)+sncn]=λ-λ(∂∂λ)λγ[λ(cn+gn)],δδsn∑n∈Z‍[en+anλ+rn(cn+gn)+sncn]=λ-λ(∂∂λ)λγ(λcn).
By the substitution of
(45)an=∑m=0∞an(m)λ-m,bn=∑m=0∞bn(m)λ-m,cn=∑m=0∞cn(m)λ-m,en=∑m=0∞en(m)λ-m,fn=∑m=0∞fn(m)λ-m,gn=∑m=0∞gn(m)λ-m
into (44) and comparing the coefficients of λ-m-1 in (44), we get
(46)(δδrnδδsn)∑n∈Z‍[en(m)+an(m)+rn(cn(m+1)+gn(m+1))+sncn(m+1)]=(-m+γ)(cn(m+1)+gn(m+1)cn(m+1)).
When m=0 in (46), a direct calculation shows that γ=0. So we have
(47)(δδrnδδsn)×∑n∈Z‍([en(m)+an(m)+rn(cn(m+1)+gn(m+1))iiiiiiiiiiiiiiiii+sncn(m+1)](-m)-1)=(cn(m+1)+gn(m+1)cn(m+1)).
Set
(48)H~n(m)=∑n∈Z‍[-en(m)-an(m)-rn(cn(m+1)+gn(m+1))-sncn(m+1)]m,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikm≥1.Now we can rewrite those lattice equations in (36) as
(49)(rnsn)tm=J(δH~n(m)δrnδH~n(m)δsn),
where J is a local difference operator defined by
(50)J=(J11J12J21J22),
where
(51)J11=0,J12=J21=rn(1+E)-1(1-E)rn,J22=sn(1+E)-1(1-E)rn+rn(1+E)-1(1-E)sn+rn(1+E)-1(E-1)rn.
Obviously, the operator J is a skew-symmetric operator; that is, J*=-J. Moreover, we can prove that the operator J satisfies the Jacobi identity
(52)〈J′(un)[Jfn]gn,hn〉+Cycle(fn,gn,hn)=0.So we have the following facts.Proposition 1.
Jis a discrete Hamiltonian operator.Set(53)δH~n(m)δun=ζnδH~n(m-1)δun.
From the recursion relation (27) we can get the recursion operator ζn in (53).Therefore, we have(54)(rnsn)tm=JδH~n(m)δun=JζnδH~n(m-1)δun=JζnmδH~n(0)δun,m≥0.
So (49) are a family of Hamiltonian systems. The hierarchy of lattice equations (36) possesses Hamiltonian structures (54). Furthermore, a direct calculation shows that
(55)M=Jζn=(0E-E-1E-E-1E-1-E).
It is easy to verify that the operator M is a skew-symmetric operator; that is, M*=-M. So we have the following.Proposition 2.
{
H
~
n
(
m
)
}
m
≥
1 defined by (48) forms an infinite set of conserved functionals of the hierarchy (36), and H~n(m), m≥1, are involution in pairs with respect to the Poisson bracket.Proof.
We can find thatM*=-M. Namely, (Jζn)*=-Jζn, and then ζn*J=Jζn. Hence
(56){H~n(m),H~n(l)}J=〈δH~n(m)δun,JδH~n(l)δun〉=〈ζnm-1δH~n(1)δun,Jζnl-1δH~n(1)δun〉=〈ζnm-1δH~n(1)δun,ζn*Jζnl-2δH~n(1)δun〉=〈ζnmδH~n(1)δun,Jζnl-2δH~n(1)δun〉={H~n(m+1),H~n(l-1)}J=⋯={H~n(m+l-1),H~n(1)}J.
Similarly, we get
(57){H~n(l),H~n(m)}J={H~n(m+l-1),H~n(1)}J.
This implies that
(58){H~n(l),H~n(m)}J=-{H~n(m),H~n(l)}J.
Thus
(59){H~n(m),H~n(l)}J=0,m,l≥1,(H~n(m))tl=〈δH~n(m)δun,untl〉=〈δH~n(m)δun,JδH~n(l)δun〉={H~n(m),H~n(l)}J=0,m,l≥1.In summary, we obtain the following theorem.Theorem 3.
The lattice equations in (36) or the discrete Hamiltonian equations in (49) are all discrete Liouville integrable Hamiltonian systems.Now we search for the integrable coupling systems with self-consistent sources. Forn distinct real λj, consider the auxiliary linear problem
(60)E(ϕ1jϕ2jϕ3jϕ4j)=Un(un,λj)(ϕ1jϕ2jϕ3jϕ4j),(ϕ1jϕ2jϕ3jϕ4j)tm=Vn[m](un,λj)(ϕ1jϕ2jϕ3jϕ4j).
Based on the results in [24], we show the following equation:
(61)δHn(m)δun+∑j=1Nδλjδun=0,
where
(62)δλjδun=12Tr⁡(ψj∂U(un,λj)∂un),ψj=(ϕ1jϕ2j-ϕ1j2ϕ3jϕ4j-ϕ3j2ϕ2j2-ϕ1jϕ2jϕ4j2-ϕ3jϕ4j00ϕ1jϕ2j-ϕ1j200ϕ2j2-ϕ1jϕ2j),kkkkkkkkkkkkkkkkkkkkkj=1,2,…,N.
According to the approach proposed in [24–26], through a direct computation, we get the discrete integrable hierarchy with self-consistent sources as follows:
(63)(rnsn)tm=J(δH~n(m)δrn+∑j=1NδλjδrnδH~n(m)δsn+∑j=1Nδλjδsn)=J(δH~n(m)δrn-∑j=1Nλjϕ1jϕ2jδH~n(m)δsn-∑j=1Nλjϕ3jϕ4j),m≥0.
When m=1 in the above system, under t1→t, we can obtain the following coupling equations with self-consistent sources:
(64)∂trn=(E-E-1)1rn-rn(1-E)(1+E)-1rn∑j=1Nλjϕ1jϕ2j,∂tsn=(E-1-E)snrn2+(E-E-1)1rn-rn(1-E)×(1+E)-1rn(∑j=1Nλjϕ1jϕ2j+∑j=1Nλjϕ3jϕ4j)-[sn(1+E)-1(1-E)rn+rn(1+E)-1(1-E)sn]×∑j=1Nλjϕ3jϕ4j.



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