# A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Authors: Ning Zhang; Xi-Xiang Xu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.


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## Body

## 1. Introduction

Over the past three decades, the integrable nonlinear differential-difference systems (INDDEs) have received considerable attention. Many INDDEs have been proposed and studied [1–15]. Finding new INDDEs is still an important and difficult work. The discrete zero curvature representation is one of the most significant tools to generate the INDDEs. Furthermore, the Hamiltonian structure of the INDDEs can be established by discrete trace identity or discrete variational identity [4, 5]. For a family of INDDEs(1)utm=Knmun,m≥0,one of the interesting problems in the theory of lattice soliton and integrable systems is to look for a Hamiltonian operator J and a sequence-conserved functional H˜nm,m≥0 so that equation (1) may be represented as the following Hamiltonian form:(2)un,tm=δH˜nmδun,where the Hamiltonian functional is H˜nm=∑n∈ZHnmm≥0. The variational derivative is δH˜nm/δun=∑n∈ZE−mδHnm/δun+m,m≥0. If we can discover infinitely many involutive conserved functionals for a family of discrete Hamiltonian system, the Liouville integrability of the the discrete Hamiltonian system (2) is proved [9–16].For a lattice functionfn=fn, the shift operator E and the inverse of E are defined by(3)Efn=fn+1,E−1fn=fn−1,n∈Z.In this paper, we introduce the following spectral problem:(4)Eφn=Unun,λφn,Unun,λ=0rnλ−snλrnsnλ2−1,where φn=φn1,φn2 is the eigenfunction vector, λ is the spectral parameter and λt=0, rn,snTis the potential vector, and rn=rn,t,sn=sn,t depend on integer n∈Z and real t∈R. Starting from spectral problem (4), a novel family of INDDEs are deduced. We are going to prove that the obtained family has a triple Hamiltonian structure (tri-Hamiltonian structure). Furthermore, its Louville integrability is presented. It should be pointed out that, in the theory of lattice soliton and integrable systems, the integrable families, which possess concise tri-Hamiltonian structure, are very rare [12–18]. Also, the symmetrical algebraic structure of equations is an important research direction for INDDEs [19–24]. Furthermore, the research of nonisospectral INDDEs has been widespread concern, the algebraic structure of isospectral and nonisospectral vector fields is established in [16–18], and the key of the theory is to derive the corresponding nonisospectral family of INDDEs.This paper is organized as follows. In Section2, starting from the matrix spectral problem (4), by means of the discrete zero curvature representation, we derive a family of INDDEs. In Section 3, we establish a triple Hamiltonian structure (tri-Hamiltonian structure) for the obtained integrable family through the discrete trace identity [4]. Infinitely many commuting symmetries and infinitely many commuting conserved functionals for the obtained family are given. The Louville integrability of the obtained family is demonstrated. In Section 4, a nonisospectral integrable family associated with the obtained family is deduced by solving an initial nonisospectral discrete zero curvature equation and the corresponding characteristic operator equation. Lie algebra of isospectral and nonisospectral vector fields is presented. Finally, in Section 5, there will be some conclusions and remarks.


## 2. The Family of Integrable Differential-Difference Equations

In this section, we shall derive a family of integrable differential-difference equations associated with eigenvalue problem (4). To this end, we first solve the following stationary discrete zero curvature equation:(5)EχnUn−Unχn=χn+1Un−Unχn=0.Upon setting(6)χn=AnBnCn−An,we find that equation (5) becomes(7)An+1−Anλ2=rnCn+1+snBnrnsnλ+An+1−Anrnsn,Bn+1λ2=−An+1+Ansnλ+Bn+1rnsn,Cnλ2=An+1+Anrnλ+Cnrnsn.Substituting expansions(8)An=∑m=0∞Anmλ−2m,Bn=∑m=0∞Bnmλ−2m+1,Cn=∑m=0∞Cnmλ−2m+1,into (7) and comparing each power of λ in the equations of (7), we obtain the initial conditions:(9)An+10−An0=0,Bn+10=0,Cn0=0,and the recursion relations:(10)An+1m+1−Anm+1=−Bnm+1rn+Cn+1m+1sn+An+1m−Anmrnsn,m≥0,Bn+1m+1=−Anm+An+1msn+Bn+1mrnsn,m≥0,Cnm+1=Anm+An+1mrn+Cnmrnsn,m≥0.Proposition 1.
If the initial values are chosen by(11)An0=−12,Bn0=0,then Anm,Bnm,andCnmm≥0, which are solved by equation (10), are all local, and they are just rational functions in the two dependent variables rn and sn.Proof.
On the basis of second and third equations in equation (10), we see that Bnm+1 and Cnm+1can be solved locally by Anm,Bnm, and Cnmm≥0. In order to obtain Anm+1m≥0 from the first equation in the equation (10), we need to use operator D−1=E−1−1 to solve the corresponding difference equation. In the following, we are going to show that Anm+1m≥1 may be deduced through an algebraic method rather than by solving the difference equations. From (5), we know that(12)E−1trχn2=2E−1An2+BnCn=0.
This tells usAn2+BnCn=γt, where γt is an arbitrary function of time variable t only [10, 14]. Furthermore, we select γt=0. Then, we obtain a recursion relation for Anm:(13)Anm+1=∑j=1mAnjAnm−j+1−∑j=1mBnjCnm−j+1,m≥1.
Therefore,Anm+1m≥1 can be determined locally by Anm,Bnm, and Cnmm≥0, and then Anm,Bnm,andCnmm≥0 are all local and they are just rational functions in the two dependent variables rn and sn.
The proof is completed.
In particular, we have(14)An1=−1rnsn−1,Bn1=1sn−1,Cn1=−1rn,….
Set(15)χnm=∑i=0mAniλ2m−2i∑i=0mBniλ2m−2i+1∑i=0mCniλ2m−2i+1−∑i=0mAniλ2m−2i+Anm.
Let us introduce the following auxiliary spectral problems associated with the spectral problem (4):(16)φntm=χnmφn,m≥0.
Then, the compatibility condition of (4) and (16)(17)Eφntm=Eφntm,is equivalent to the discrete zero curvature equations(18)Untm=EχnmUn−Unχnm,m≥0,which give rise to the family of integrable differential-difference equations:(19)rntm=−rnsnBn+1m+1−rnAnm,sntm=−rnsnCnm+1+snAn+1m,m≥0.
Whenm=0, (19) becomes a trivial linear system:(20)rnt0=−rn2,snt0=sn2.
And, whenm=2, we obtain the first INDDE in family (19) as follows:(21)rnt1=−rnrn+1sn−1sn,snt1=1rn+snrnsn−1.


## 3. Tri-Hamiltonian Structure

Next, we shall establish a tri-Hamiltonian structure for integrable family (19). First, we introduce some concepts. The variational derivative, the Gateaux derivative, and the inner product are defined, respectively, by(22)δHnδun=∑m∈ZE−m∂Hn∂un+m,J′unvn=∂∂εJun+εvn|ε=0,fn,gn=∑n∈Zfn,gnR2,where fnandgn are required to be rapidly vanished at infinity and fn,gnR2 denotes the standard inner product of fn and gn in the Euclidean space R2. The adjoint operator J∗ of J is defined by fn,J∗gn=Jfn,gn. If an operator J has the property J∗=−J, then J is said to be skew-symmetric. If a skew-symmetric operator J meets the Jacobi identity, i.e.,(23)J′unJfngn,hn+Cyclefn,gn,hn=0,then operator J is called a Hamiltonian operator. Based on a given Hamiltonian operator J, we can define a Poisson bracket [4]:(24)fn,gnJ=δfnδun,Jδgnδun=∑n∈Zδfnδun,JδgnδunR2,(25)1−E−1=12∑k=0∞Ek−∑k=−∞−1Ek,1−E−1−1=12∑k=−∞−1Ek+1−∑k=0∞Ek+1.Following [4], we set(26)Sn=χnUn−1=snBnλ+Anrnsnλ2−1rnsnλ2−Ansnλ−snAnλ+Cnrnsnλ2−1rnsnλ2−Cnsnλ,Y,Z is defined as TrYZ, where Y and Z are the some order square matrices. Hence,(27)Rn,∂Un∂λ=−Cn1+rnsnλ2snλ2,Rn,∂Un∂rn=−snAnλ+Cnrnsnλ,Rn,∂Un∂sn=An−rnλCnsn.Then, the discrete trace identity becomes(28)δδrn∑n∈ZSn,∂Un∂λ=λ−ε∂∂λλεSn,∂Un∂rn,δδsn∑n∈ZSn,∂Un∂λ=λ−ε∂∂λλεSn,∂Un∂sn.Substituting expansionsAn=∑m=0∞Anmλ−2m,Bn=∑m=0∞Bnmλ−2m+1,andCn=∑m=0∞Cnmλ−2m+1 into (28) and comparing the coefficients of λ−2m−1, we arrive at(29)δδrnδδsn∑n∈ZHnm=ε−2m−snAnm+Cnmrnsn−snAn+1m+Cnmsn2.Whenm=0 in equation (29), through a direct calculation, we find that ε=0. Thus, equation (29) can be written as(30)δδrnδδsnH˜nm=−snAnm+Cnmrnsn−snAn+1m+Cnmsn2,m≥1,where(31)H˜nm=∑j=0m−rnsnCnm+Cnm+1sn,m≥1.Moreover, we have(32)untm=rntmsntm=−rnsnBn+1m+1−rnAnm+An+1m,rnsnCnm+1+snAnm+An+1m=Θ−snAnm+1+Cnm+1rnsn−snAn+1m+1+Cnm+1sn2=J−snAnm+Cnmrnsn−snAn+1m+Cnmsn2=K−snAnm−1+Cnm−1rnsn−snAn+1m−1+Cnm−1sn2,m≥1,where(33)Θ=Θ11Θ12Θ21Θ22,where(34)Θ11=rn1−E−1rnEsn1−E−1−1rn−rn1−E−1⋅snE−1rn1−E−1−1rn+rn1−E−1rn2sn−rn2sn1−E−1rn−1,Θ12=rn1−E−1rnEsn1−E−1sn−rn1−E−1⋅snE−1rn1−E−1sn−rn1−E−1rnsn2−rnsn21−E−1sn,Θ21=sn1−E−1−1rnEsn1−E−1−1rn−sn1−E−1−1⋅snE−1rn1−E−1rn−sn1−E−1−1rn2sn−rnsn21−E−1−1rn,Θ22=sn1−E−1−1rnEsn1−E−1sn−sn1−E−1−1snE−1rn⋅1−E−1−1rn−sn1−E−1−1rn2sn+rn2sn1−E−1sn,J=0−rnsnrnsn0,K=K11K12K21K22,with(35)K11=−rn1−E−1−11snE1rn1−E−1rn+rn1−E−1−1⋅1rnE−11sn1−E−1rn−1sn1−E−1rn+rn1−E−1−11sn,K12=−rn1−E−1−11snE1rn1−E−1−1sn+rn1−E−1−11rnE−1⋅1sn1−E−1−1sn−1sn1−E−1−1sn−rn1−E−1−11rn,K21=−sn1−E−11snE1rn1−E−1rn+sn1−E−1⋅1rnE−11sn1−E−1rn+1rn1−E−1rn+sn1−E−11sn,K22=−sn1−E−11snE1rn1−E−1−1sn+sn1−E−11rnE−1⋅1sn1−E−1−1sn+1rn1−E−1−1sn−sn1−E−11rn.For three arbitrary constantsα,β,andγ, it is easy to verify that the operator Mα,β,γ=αΘ+βJ+γK is a skew-symmetric operator, i.e.,Mα,β,γ=−Mα,β,γ∗. Furthermore, by a straightforward and lengthy calculation, we can prove that the operator Mα,β,γ fulfills the Jacobian identity (22). So, we can get the following proposition.Proposition 2.
For all values of three arbitrary constantsα,β,andγ, Mα,β,γ=αΘ+βJ+γK is a Hamiltonian operator.
Furthermore, we can obtain that integrable family (19) possesses tri-Hamiltonian structure(36)ΘδH˜nm+1δun=JδH˜nmδun=KδH˜nmδun,m≥1.
Moreover, from (7), we find the recursion relation(37)δH˜nm+1δun=GδH˜nmδun,where(38)G=G11G12G21G22,G11=−1rn1−E−11snE1rn1−E−1rn+1rn1−E−1⋅1rnE1sn1−E−1rn+1rn1−E−11sn+1rn1sn1−E−1rn,G12=−1rn1−E−11snE1rn1−E−1−1sn+1rn1−E−11rnE−1⋅1sn1−E−1−1sn−1rn1−E−11rn+1rn2sn1−E−1rn,G21=1sn1−E−1−11snE1rn1−E−1rn−1sn1−E−1−11rn⋅E−11sn1−E−1rn−1sn1−E−1−11sn+1rn2sn1−E−1rn,G22=1sn1−E−1−11snE1rn1−E−1−1sn−1sn1−E−1−11rn⋅E−11sn1−E−1−1sn+1sn1−E−1−11rn+1rnsn21−E−1−1sn,where Φ=G∗ is a recursion operator. By means of the operator G, we have(39)untm=JδH˜nmδun=JGδH˜nm−1δun=JG2δH˜nm−2δun=⋯=JGmδH˜n0δun=ΦJδH˜nm−1δun=⋯=ΦmJδH˜n0δun.Now, we would like to prove the Liouville integrability of the discrete Hamiltonian systems (36). It is crucial to show the existence of infinite involutive conserved functionals.Proposition 3.
H˜nmm=0∞ are conserved functionals of the whole family (19). And, they are in involution in pairs with respect to the Poisson bracket (24).Proof.
A direct calculation shows(40)JG∗=K∗=−K=−JG,that is,(41)G∗J=JG.
So,(42)H˜nm,H˜nlJ=δH˜nmδun,JδH˜nlδun=Gm−1δH˜n1δun,JGl−1δH˜n1δun=Gm−1δH˜n1δun,G∗JGl−2δH˜n1δun=GmδH˜n1δun,JGl−2δH˜n1δun=H˜nm+1,H˜nl−1J=⋯=H˜nm+l−1,H˜n1J.
Repeating the above argumentation, we can get that(43)H˜nl,H˜nmJ=H˜nm+l−1,H˜n1J.
Then, combining the above two equations, we have(44)H˜nm,H˜nlJ=0,m,l≥1,H˜nmtl=δH˜nmδun,utl=δH˜nmδun,JδH˜nlδun=H˜nm,H˜nlJ=0,m,l≥1.
The proof is finished.
Based on system (39) and Proposition 3, we can obtain the following theorem.Theorem 1.
Each INDDE in family (19) is Liouville-integrable discrete Hamiltonian system and has tri-Hamitonian structure (36).


## 4. Lie Algebraic Structure of Vector Fields

In Section2, we have discussed the spectral problem (4) in the isospectral case, namely, dλ/dtm=0. Next, we shall investigate the spectral problem (4) in the nonisospectral case. Now, let us suppose that dλ/dtm=λ2m+1 in (4). First, we shall derive a nonisospectral integrable family using the method proposed in [23, 24]. The nonisospectral initial zero curvature equation (45)EVn0Un−UnVn=Un′σn0+λ∂Un∂λ,is presented. In this equation, the corresponding solutions are easily solved as follows:(46)σn0=−rn−sn,Vn0=1001.Now, we introduce the operator equation:(47)EΩnXUn−UnΩnX=Un′ΦX+λ2∂Un∂λ,X=Xn1,Xn2T,In the above equation(48)ΩnX=Ωn11Ωn12Ωn21Ωn22,here, Ωnij1≤i,j≤2 is the undetermined function. This (47) is called a characteristic operator equation [23, 24]. Through a straightforward computation, the following results are obtained:(49)Ωn11X=−1−E−1X1rn2sn−1−E−1X2rnsn2−1−E−11sn⋅E1rn1−E−1−1X1rn+1rn1−E−1−1X2sn−X2rnsn−1−E−1⋅1rnE−11sn1−E−1X1rn+1sn1−E−1X2sn+X1rnsn,Ωn12X=−E−1−1sn1−E−1X1rn−1sn1−E−1X2sn+X1rnsnλ,Ωn21X=1rn1−E−1−1X1rn+1rn1−E−1−1X2sn−X2rnsnλ,Ωn22X=−λ21−E−1X1rn+1−E−1X2sn.Let us set(50)σnm=Φmσn0,m≥1,Vnm=λ2mVn0+∑j=1mλ2m−2jΩnσnj−1,m≥1.Then, we obtain that(51)EVnmUn−UnVnm=λ2mEVn0Un−UnVn0+∑j=1mλ2m−2jEΩσnj−1Un−UnΩσnj−1=λ2mUn′σn0+λ∂Un∂λ+∑j=1mλ2m−2jUn′Φσnj−1−λ2Un′Φσnj−1=λ2mUn′σn0+λ∂Un∂λ+∑j=1mλ2m−2jUn′σnj−λ2Un′Φσnj−1=Un′σnm+λ2m+1∂Un∂λ.Thus, we get a integrable family of nonisospectralλt=λm+1 discrete evolution equations(52)untm=σnm,m≥0,and based on (51), the integrable family has the following nonisospectral λt=λ2m+1 zero curvature representation(53)Untm=EVnmUn−UnVnm,m≥0.Following [23, 24], let KandS be vector fields; XandY be matrix operators; and fandg be smooth functions, then the following Lie bracket can be defined by(54)K,X,f,S,Y,g=K,S,X,Y,f,g,where(55)K,S=K′S−S′K,K′S=ddϵ|ϵ=0Ku+ϵS,V,W=V′S−W′K+V,W+gVλ−fWλ,V,W=VW−WV,f,gλ=f′λgλ−fλg′λ.We start our calculation as follows:(56)χnm|rn,sn=∞,∞=λ2m−120012,Vnm|rn,sn=∞,∞=λ2m1001,Knm|rn,sn=∞,∞=0,ρnl|rn,sn=∞,∞=Φl,ρn0|rn,sn=∞,∞=0,l,m≥1,χn,λm|rn,sn=∞,∞=2mλ2m−1−120012,Vn,λm=2mλ2m−11001.By the above results, we can get(57)χnm,χnl|rn,sn=∞,∞=0,χnm,Vnl|rn,sn=∞,∞=2mχnm+l|rn,sn=∞,∞,Vnm,Vnl|rn,sn=∞,∞=2m−lVnm+l|rn,sn=∞,∞.Moreover, it is easy to find thatUn′ is an injective map. Using the above Lie algebra relations, we can arrive at the Lie algebra of vector fields for the isospectal and nonisospectal families (19) and (52) as follows:(58)Knm,Knl=0,(59)Knm,ρnl=2mKnm+l,(60)ρnm,ρnl=2m−lρnm+1.From equations (58) and (59), we obtain(61)Knm,Knm,ρnl=0,m,l≥1.Hence,ρnll≥0 are all the master symmetries of un,tm=Knmm≥0. Furthermore, the vector field τml=tKnm,ρnl+ρnlm,l≥0 is the symmetries of utl=Kmm. Here,(62)Knm,τml,m,l≥0,constitute a symmetry algebra of Witt type.


## 5. Conclusions and Remarks

In this paper, we have deduced a family of integrable differential-difference equations through the discrete zero curvature equation. With the help of discrete zero curvature equations, we derive a family of INDDEs. A triple Hamiltonian structure (tri-Hamiltonian structure) for the obtained integrable family is established by the discrete trace identity. Louville integrability of the obtained family is proved. Then, a nonisospectral integrable family associated with the obtained family is derived by solving an initial nonisospectral discrete zero curvature equation and the corresponding characteristic operator equation. Ultimately, Lie algebra of isospectral and nonisospectral vector fields is obtained. In addition, there are other problems worth further investigation for the integrable family (9). For example, the inverse scattering transformation, Hamiltonian structures, binary nonlinearization of Lax pairs, and integrable coupling systems by semidirect sums of Lie algebra.



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