# Positive and Negative Integrable Hierarchies: Bi-Hamiltonian Structure and Darboux–Bäcklund Transformation

Authors: Ning Zhang; Xi-Xiang Xu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Two integrable hierarchies are derived from a novel discrete matrix spectral problem by discrete zero curvature equations. They correspond, respectively, to positive power and negative power expansions of Lax operators with respect to the spectral parameter. The bi-Hamiltonian structures of obtained hierarchies are established by a pair of Hamiltonian operators through discrete trace identity. The Liouville integrability of the obtained hierarchies is proved. Through a gauge transformation of the Lax pair, a Darboux–Bäcklund transformation is constructed for the first nonlinear different-difference equation in the negative hierarchy. Ultimately, applying the obtained Darboux–Bäcklund transformation, two exact solutions are given by means of mathematical software.


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## Body

## 1. Introduction

It is well known that the study of nonlinear integrable differential-difference equations (NIDDEs) has attracted much attention in recent decades [1–14]. Many problems in mathematical physics may be modeled by NIDDEs. Up to now, many important NIDDEs have been presented such as the Ablowitz–Ladik lattice [1], the Toda lattice [2], the relativistic Toda lattice [3], the modified Toda lattice [4, 5], the Merola–Ragnisco–Tu lattice [6], and the deformed reduced semidiscrete Kaup–Newell lattice [7–14]. Now, finding new NIDDEs, in lattice soliton theory, is still an important and complicated task. In general, we choose an appropriate discrete matrix spectral problem and a list of auxiliary spectral problems:(1)Eφn=φn+1=Unun,λφn,φntm=Vnmun,λφn,where for a lattice function fn=fn, the shift operator E and the inverse of E are defined by(2)Efn=fn+1,E−1fn=fn−1,n∈Z,Unun,λ is a square matrix, Vnmun,λ is a list of the same order square matrices of Unun,λ, un is a potential vector function, φn is the eigenfunction vector, λ is the spectral parameter, and λt=0. The integrability condition of (1) is Eφntm=Eφntm, and it is equivalent to(3)Untm=EVnmUn−UnVnm,m≥0.Here, (3) is called a discrete zero curvature equation. Usually, (3) determines a hierarchy of NIDDEs (or lattice soliton equations):(4)untm=Kun,un−1,un+1,….One of the important problems in the lattice soliton theory is to search for a Hamiltonian operatorJ1 and a hierarchy of conserved densities H˜mmm=0∞ so that (4) has the following Hamiltonian structures:(5)untm=J1δH˜nmδun,m≥0,where the Hamiltonian functionals H˜nm=∑n∈ZHnmm≥1 and the variational derivative δH˜nm/δun=∑m∈ZE−m∂Hnm/∂un+m. Furthermore, if there is another operator J2 so that J1 and J2 form a pair of Hamiltonian operators and(6)untm=J1δH˜nmδun=J2δH˜nm−1δun,m≥1,then the integrable hierarchy (4) possess a bi-Hamiltonian structure. According to the theory of Hamiltonian operators, if J1 is reversible, then the hierarchy (4) is integrable in Liouville sense ([4, 13] and their references). As is known to all, staring from a continuous matrix spectral problem, we only derive one integrable hierarchy, but for some suitable discrete matrix spectral problems, we can get two hierarchies of the NIDDEs [13]. In this paper, we are going to present two integrable hierarchies from a discrete matrix spectral problem. They, respectively, apply positive power and negative power expansions of Lax operators with respect to the spectral parameter. Theory is, respectively, called positive and negative integrable hierarchies. Moreover, as is known to all, Bäcklund transformation is a powerful method to obtain exact solutions of NIDDEs [15, 16]. This transformation is a relation between the new solution and the old solution of the NIDDEs. Based on a known solution, applying this transformation, a new solution may be derived. According to [15], Bäcklund transformation is usually divided into three types: the Wahlquist–Estabrook (WE) type [16, 17], the Hirota type [18], and the Darboux–Bäcklund type [19–22]. In the Darboux–Bäcklund type, the Lax pair plays a key role. A gauge transformation of the Lax pairs is called a Darboux–Bäcklund transformation if it transforms the Lax pair into another Lax pair of the same form.This paper is organized as follows. In Sections2 and 3, we introduce a novel discrete spectral problem:(7)Eφn=Unun,λφn,Unun,λ=pnλ1qnλ1,where φn=φn1,φn2 is the eigenfunction vector, un=pn,qnT is the potential vector, and pn=pn,t and qn=qn,t depend on integer n∈Z and real t∈R. Staring from spectral problem (7), positive and negative integrable hierarchies of NIDDEs are, respectively, presented by discrete zero curvature equations. Then, the Hamiltonian structure and bi-Hamiltonian structure of the obtained hierarchies are established by means of the discrete trace identity [11]. Afterwards, infinitely many common commuting conserved functionals of the obtained positive hierarchy are worked out. The Liouville integrability of the obtained positive hierarchy is proved. For the obtained negative integrable hierarchy, the same results can be similarly obtained. In Section 4, a Darboux–Bäcklund transformation is established though the gauge transformation of the Lax pair for the first NIDDE in the negative integrable hierarchy. In Section 5, using obtained Darboux–Bäcklund transformation, two exact solutions are given with the help of the mathematical software “Mathematica.” Finally, in Section 6, there will be some conclusions and remarks.


## 2. Positive Integrable Hierarchy and Its Bi-Hamiltonian Structure

Now, we want to deduce a hierarchy of NIDDEs associated with eigenvalue problem (7). For this purpose, we first solve the following stationary discrete zero curvature equation:(8)EMnUn−UnMn=Mn+1Un−UnMn=0.Let us set(9)Mn=AnBnCn−An.We find that equation (8) implies(10)pnAn+1−An−Cn+qnBn+1=0,pnBnλ−An+1+An−Bn+1=0,pnCn+1λ−qnAn+1+An−Cn=0,Cn+1−qnBnλ+An−An+1=0.Substituting expansions(11)An=∑m=0∞Anmλ−m,Bn=∑m=0∞Bnmλ−m,Cn=∑m=0∞Cnmλ−m,into (10) and comparing each power of λ in equation (10), we obtain the initial conditions:(12)pnAn+10−An0=Cn0−qnBn+10,Bn0=0,Cn+10=0,and the recursion relations:(13)pnAn+1m+1−Anm+1=Cnm+1−qnBn+1m+1,m≥0pnBnm+1=Anm+An+1m+Bn+1m,m≥0,pnCn+1m+1=qnAnm+An+1m+Cnm,m≥0.Proposition 1.
We take that(14)An0=12,Cn0=0,then Anm,Bnm,Cnmm≥0, which are solved by equation (13), are all local, and they are just rational functions in the two dependent variables pn and qn.Proof.
According to second and third equations in (13), we get that Bnm+1 and Cnm+1 can be shown locally by means of Anm,Bnm, and Cnmm≥0. In order to derived Anm+1m≥0 from first equation in (13), we require to apply operator D−1=E−1−1 to solve the related difference equation. Next, we will show that Anm+1m≥0 may be also deduced through an algebraic method rather than by solving the difference equation. Based on (8), we obtain that(15)E−1trMn2=2E−1An2+BnCnλ=0.
SoAn2+BnCnλ=γt, γt is an arbitrary function of time variable t only. Furthermore, we take that γt=0. Then, we obtain a recursion relation for Anm:(16)Anm+1=∑j=1mAnjAnm−j+1−∑j=1m+1BnjCnm−j+2,m≥0.
Therefore,Anm+1m≥0 can be derived locally by Anm,Bnm, and Cnmm≥0 and then Anm,Bnm,andCnmm≥0 are all local; they are just rational functions in the two dependent variables pn and qn.
The proof is completed.
Specially, we have(17)An1=−qn−1pn−1pn,Bn1=1pn,Cn1=qn−1pn−1,An2=qn−12pn−12pn2+qn−1qnpn+1pn2pn−1+qn−1qn−2pnpn−12pn−2−qn−1pnpn+1pn−1−qn−2pnpn−1pn−2,Bn2=−qnpn+1pn2−qn−1pn2pn−1+1pnpn+1,Cn2=−qn−12pn−12pn−qn−1qn−2pn−12pn−2+qn−2pn−1pn−2⋯.
Let us denote(18)Mnm=∑i=0mAniλm−i∑i=0mBniλm−i∑i=0mCniλm−i+1−∑i=0mAniλm−i,m≥0.
By means of (13), we get(19)EMnmUn−UnMnm=0pnBnm+1−pnCn+1m+1λAnm−An+1m.
Obviously, (19) is not compatible with Untm. So, we choose a correction term(20)Δnm=−Bnm−Anm00Anm,and set(21)M˜nm=Mnm+Δnm,m≥0.
We consider the following auxiliary spectral problems associated with the spectral problem (7):(22)φntm=M˜nmφn,m≥0.
Then, the compatibility condition of equations (7) and (22)(23)Eφntm=Eφntm,is equivalent to the discrete zero curvature equations(24)Untm=EM˜nmUn−UnM˜nm,m≥0,which give rise to the hierarchy of NLDDEs:(25)pntm=pn1−EAnm+Bnm,qntm=qnBnm−Cnm,m≥0.Remark 1.
Owing to the entries in matrixM˜nmm≥0 only has nonnegative power powers of the eigenvalue λ, then the integrable hierarchy (25) is called a positive integrable hierarchy associated with the discrete matrix spectral problem (7).
Whenm=0, (25) becomes a trivial linear system:(26)pnt0=0,qnt0=0.
Whenm=1 in (25), we obtain the first NIDDE in hierarchy (25):(27)pnt1=pnpn−1−qn−1pn−1pn−pnpn−qnpnpn+1,qnt1=qnpn−qn−1pn−1.
Furthermore, it is easy to derive the time part of the Lax pair of (27) is(28)φnt1=M˜n1φn=λ2−1pn1pnqn−1pn−1λ−λ2φn.
Next, we are going to establish the Hamiltonian structure for the hierarchy of NLDDE (25) by means of the discrete trace identity [11].
First, let us introduce some notions for further discussion. The Gateaux derivative and the inner product are defined, respectively, by(29)J′unvn=∂∂εJun+εvn|ε=0,fn,gn=∑n∈Zfn,gnR2,where fnandgn are demanded to be rapidly vanished at the infinity. The standard inner product of fn and gn in the Euclidean space R2 is given by fn,gnR2. Operator J∗ is defined by fn,J∗gn=Jfn,gn, and it is called the adjoint operator of J. If an operator J possesses the property J∗=−J, then J is called to be a skew-symmetric. A linear operator J is called a Hamiltonian operator if J is a skew-symmetric operator and fulfills the Jacobi identity, i.e.,(30)fn,Jgn=−Jfn,gn,J′unJfngn,hn+Cyclefn,gn,hn=0.
For a Hamiltonian operatorJ, we may define a corresponding Poisson bracket [4]:(31)fn,gnJ=δfnδun,Jδgnδun=∑n∈Zδfnδun,JδgnδunR2.
According to [11], we write(32)Sn=MnUn−1=An−qnBnλpn−qnλ−An−pnBnλpn−qnλAn+qnCnpn−qn−An+pnCnλpn−qn,and Y,Z=TrYZ, where Y and Z are the same order square matrices. We have(33)∂Un∂λ=pn0qn0,∂Un∂pn=λ000,∂Un∂qn=00λ0.
Hence,(34)Rn,∂Un∂λ=Anλ,Rn,∂Un∂pn=An−qnBnλpn−qn,Rn,∂Un∂qn=−An+pnBnλpn−qn.
By virtue of the discrete trace identity [11],(35)δδpn∑n∈ZSn,∂Un∂λ=λ−ε∂∂λλεSn,∂Un∂pn,δδqn∑n∈ZSn,∂Un∂λ=λ−ε∂∂λλεSn,∂Un∂qn.
Substituting expansionsAn=∑m=0∞Anmλ−m,Bn=∑m=0∞Bnmλ−m,andCn=∑m=0∞Cnmλ−m into (35) and comparing the coefficients of λ−m−1, we arrive at(36)δδpnδδqn∑n∈ZAnm=ε−mAnm−qnBnm+1pn−qn−Anm+pnBnm+1pn−qn.
Whenm=0 in equation (36), by means of a direct confirmation, we get that ε=0. Thus, equation (36) can be written as(37)δδpnδδqn∑n∈Z−Anmm=Anm−qnBnm+1pn−qn−Anm+pnBnm+1pn−qn,m>0.
Moreover, we have(38)−pn1−E−1pnBnm+1−Anm−pnCn+1m+1+qnE−1pnBnm+1−Anm+qnAn+1m=J1Anm−qnBnm+1pn−qn−Anm+pnBnm+1pn−qn=J2Anm−1−qnBnmpn−qn,−An+1m−1+pnCnmpn−qn.,m≥1,where(39)J1=0−pn2+pnqn+pnE−1pn−qn−pnqn+pn2−pn−qnEpn−pn−qnEqn+qnE−1pn−qn,J2=E−1pn−qn−pn−qnEpn−qn−pn−qnE−pn+qn+E−1pn−qn0.
Furthermore, we can get(40)J1−1=Q1−pn2+pnqn+pnEpn−qn−pnqn+pn2−pn−qnEpn−pn−qnEqn+qnE−1pn−qn,where Q1=−1/pn1−E−1−1Eqn/pn−qn1−E−11/pn−1/pn1−E−1−1qn/pn−qnE−11−E−11/pn and(41)J2−1=0−1pn−qn1−E−1−11−E−11pn−qnQ2.
In above matrix,Q2=−1−E−11/pn−qn1−E−1+1−E−1−11/pn−qn1−E−1−1.Proposition 2.
For all values of two arbitrary constantsα and β,(42)Jα,β=αJ1+βJ2,is a Hamiltonian operator.
Expressly,J1=J1,0andJ2=J0,1 constitute a pair of Hamiltonian operators.Proof.
Obviously, the operatorJα,β is a skew-symmetric operator, i.e., Jα,β=−Jα,β∗. Furthermore, by a direct and tedious calculation, we can prove that the operator Jα,β fulfills the Jacobi identity (30).
The proposition is proved.
So we obtain the following proposition.Proposition 3.
Equation (25) possesses the following Hamiltonian structure:(43)untm=pnqntm=J1Anm−qnBnm+1pn−qn,An+1m+Cnm+1pn−qn.=J1δH˜nmδun,m≥1,where H˜nm=∑n∈ZHnmandHnm=Anm/mm≥1.According to equation (13), we get the recursion relation:(44)δHn˜m+1δun=ΦδH˜nmδun,m≥0,Φ=J1−1J2.Furthermore, we have(45)un,tm=pnqntm=J1δH˜nm+1δun=J2δH˜nm−1δun,m>0.That is, (25) is a hierarchy of discrete bi-Hamiltonian systems.Using the operatorΦ, the positive integrable hierarchy (25) can be written as follows:(46)untm=pnqntm=J1δH˜nmδun=J1ΦδH˜nm−1δun⋯=J1Φm−1δH˜n1δun,m>0.Obviously, the integrable NLDDE (27) possesses Hamiltonian structure:(47)un,t1=pnqnt=J1δH˜n1δun.Next, we prove the Liouville integrability of the discrete bi-Hamiltonian systems (25). It is crucial to make known the existence of infinite involutive conserved functionals.Proposition 4.
H˜nmm=0∞ are conserved functionals of the whole family (25) or (45). And they are in involution in pairs with respect to the Poisson bracket (31).Proof.
Though a direct calculation, we have(48)J1Φ∗=J2∗=−J2=−J1Φ,namely,(49)Φ∗J1=J1Φ.
Therefore,(50)H˜nm,H˜nlJ1=δH˜nmδun,J1δH˜nlδun=Φm−1δH˜n1δun,J1Φl−1δH˜n1δun=Φm−1δH˜n1δun,Φ∗J1Φl−2δH˜n1δun=ΦmδH˜n1δun,J1Φl−2δH˜n1δun=H˜nm+1,H˜nl−1J1=⋯=H˜nm,H˜nlJ1=H˜nm+l−1,H˜n1J1.
Repeating the above argument, we can obtain(51)H˜nl,H˜nmJ1=H˜nm+l−1,H˜n1J1.
By equations (51) and (52), we have(52)H˜nm,H˜nlJ1=0,m,l≥1,(53)H˜nmtl=δH˜nmδun,utl=δH˜nmδun,J1δH˜nlδun=H˜nm,H˜nlJ1=0,m,l≥1.
The proposition is proved.
Based on (45) and the Propositions 3 and 4, we can obtain the following theorem.Theorem 1.
The integrable NLDDE in hierarchy (25) is all Liouville integrable discrete bi-Hamiltonian systems.


## 3. Negative Integrable Lattice Hierarchy and Its Bi-Hamiltonian Structure

In this section, we would like to derive the negative integrable hierarchy associated with matrix spectral problem (7). To this end, we first consider the following stationary discrete zero curvature equation:(54)ENnUn−UnNn=Nn+1Un−UnNn=0,with(55)Nn=anbncn−an.On the basis of equation (54), we arrive at(56)pnan+1−an−cn+qnbn+1=0,bn+1+an+1+an−bnλpn=0,cn+qnan+1+an−pncn+1λ=0,cn+1−qnbnλ+an−an+1=0.Here, we expandan,bn,cn by the nonnegative power power of λ:(57)an=∑m=0∞anmλm,bn=∑m=0∞bnmλm,cn=∑m=0∞cnmλm.Substituting the above expansions into (56), we get the following initial conditions:(58)pnan+10−an0−cn0+qnbn+10=0,bn+10=−an0+an+10,cn=−qnan0+an+10,and recursion relations(59)an+1m+1−anm+1=cn+1m−qnbnm,m≥0,bn+1m+1=−an+1m+1−anm+1+pnbnm,m≥0,cnm+1=−qnanm+1+an+1m+1+pncn+1m,m≥0.If the initial values are chosen as(60)an0=12,then we get that(61)bn0=−1,cn0=−qn.Similar to Proposition1, we can obtain that anm,bnm,cnmm≥1 are all local, and they are just polynomial functions in the two dependent variables pn and qn, and anmm≥1 may be deduced through an algebraic method rather than by solving the difference equation.The first few terms are given by(62)an1=−qn,bn1=qn+qn−1−pn−1,cn1=qnqn+1+qn2−pnqn+1,….Set(63)Nnm=∑i=0maniλ−m+i−1∑i=0mbniλ−m+i−1∑i=0mcniλ−m+i−∑i=0maniλ−m+i−1,m≥0,At this point,(64)ENnmUn−UnNnm=0−pnbnmpncn+1mλcn+1m−pnbnm.It is obvious that (64) is not also compatible with Untm. So, we choose the following correction term:(65)ηnm=pn−1bn−1m−an−1m+100−anm+1.Then, we introduce auxiliary matrix spectral problem:(66)N˜nm=Nnm+ηnm,m≥1.Through a direct calculation, we obtain that(67)Untm=EN˜nUn−UnN˜n,n≥0,it is equivalent to(68)pn,tm=pn1−E−1pnbnm−anm+1,qn,tm=pncn+1m−qnpn−1bn−1m−an−1m+1−qnan+1m+1,m≥0.Remark 2.
Because the entries in matrixN˜nmm≥1 only have negative powers of the eigenvalue λ, then the integrable hierarchy (68) is called a negative integrable hierarchy associated with the discrete matrix spectral problem (7).
Whenm=0, (68) becomes(69)pn,t0=pnpn−1−pn−pnqn−1−qn,qn,t0=qnqn+1−qn−1+pn−1qn−pnqn+1.
If we setpn=0, (69) is reduced to the well-known Volterra lattice qnt=qnqn+1−qn−1; namely, (69) is a generalized Volterra lattice.
In next section, we are going to establish a Darboux–Ba¨cklund transformation of (69). It is easy to get the time part of the Lax pair of (69) is(70)φnt1=N˜n1φn=12λ−pn−1+qn−1−1λ−qn−12λ+qnφn.
In the discrete variational identity (35), we replace Mn with Nn; at this point, the following equations hold:(71)δδpn∑n∈ZS˜n,∂Un∂λ=λ−ε∂∂λλεS˜n,∂Un∂pn,δδqn∑n∈ZS˜n,∂Un∂λ=λ−ε∂∂λλεS˜n,∂Un∂qn,where S˜n=NnUn−1.
Substituting expansions(72)an=∑m=0∞anmλm,bn=∑m=0∞bnmλm,cn=∑m=0∞cnmλm,into (71), we get(73)δK˜nmδpnδK˜nmδqn=anm+1−qnbnmpn−qn−anm+1+pnbnmpn−qn,m≥0,where K˜nm=∑n∈Z−anm+1/mm≥0. There is the recursion relation as follows:(74)δK˜nmδpnδK˜nmδqn=Φ−1δK˜nm−1δpnδK˜nm−1δqn,m≥0,where Φ−1=J2−1J1.
Based on (56), we get(75)pn1−E−1pnbnm−anm+1,pncn+1m−qnpn−1bn−1man−1m+1+an+1m+1=−J2anm+2−qnbn+1m+1pn−qn,−anm+2+pnbnm+1pn−qn=−J1anm+1−qnbn+1mpn−qn,−anm+1+pnbn+1mpn−qn,m≥0.
Then, we have(76)untm=pnqntm=−J2δK˜nmδun=−J1δK˜nm−1δun,m>0.
Thus, the integrable hierarchy (68) has a bi-Hamiltonian structure (76). Furthermore, similar to integrable hierarchy (25), we can prove that integrable hierarchy (68) is also Liouville integrable.
With the help of the operatorΦ−1, the negative integrable hierarchy (68) can be written as follows:(77)untm=pnqntm=−J2δK˜nmδun=−J2Φ−1δK˜nm−1δun⋯=−J2Φ−1m−1δK˜n1δun,m>0.


## 4. Darboux–Bäcklund Transformation

We introduce a gauge transformation of spectral problem (7):(78)φ˜n=Tnφn.Under this transformation, two spectral problems (7) and (70) become(79)Eφn˜=φ˜n+1=U˜nφ˜n,φ˜nt=N˜n1φ˜n,with(80)U˜n=Tn+1UnTn−1,N˜n1=Tnt+TnNn1Tn−1.Here, we suppose(81)Tn=Tnaλ+11−TnbTnbTncλTndλ+1,where Tna,Tnb,Tnc,andTnd are undetermined functions of variables n and t and n∈Z,t∈R. Next, we are going to solve Tna,Tnb,Tnc,andTnd such that U˜n and N˜n1 in equation (80) are provided with the same form with Un and Nn1, i.e.,(82)U˜n=p˜nλ1q˜nλ1,N˜1=12λ−p˜n−1+q˜n−1−1λ−q˜n−12λ+q˜n.For two different realsλ1andλ2, we can get that yn=yn1,yn2Tandzn=zn1,zn2T are two real linear independent solutions of equations (7) and (70):(83)ωn=yn1zn1y22zn2.Letω˜n=Tnωn, and we get(84)ω˜n=Tnaλ+11−Tnbyn1+Tnbyn2Tnaλ+11−Tnbzn1+Tnbzn2Tncλyn1+Tndλ+1yn2Tncλzn1+Tndλ+1zn2.We consider(85)Tnaλi+11−Tnbyn1+Tnbyn2=κiTnaλi+11−Tnbzn1+Tnbzn2,Tncλyn1+Tndλi+1yn2=κiTncλizn1+Tndλi+1zn2.In the above equation, seti=1,2, κjj=1,2, are nonzero constants.Solving equation (85) for Tna,Tnb,Tnc,andTnd, we obtain that(86)Tna=σ1n−σ2nλ1σ2n−λ2σ1n,Tnb=−λ1−λ2λ1−λ2+λ2σ1n−λ1σ2n,Tnc=−σ1nσ2nλ1−λ2λ1λ2σ1n−σ2n,Tnd=λ1σ2n−λ2σ1nλ1λ2σ1n−σ2n.In equation (86),(87)σin=zn2λi−κiyn2λizn1λi−κiyn1λi,i=1,2.Through direct calculation, we have(88)DetTn=1−Tnbλ−λ1λ−λ2λ1λ2.Proposition 5.
The matrixU˜n defined by (80) has the same form as Un in equation (7), and the original potentials pnandqn are changed into new potentials p˜nandq˜n by means of(89)p˜n=−TndpnpnTnb+TnaTnb−Tna,q˜n=qn−Tnc1−Tnb.Proof.
We know thatTn∗ is the adjoint matrix of Tn; from equations (7) and (87), we obtain that(90)σin+1=ζinτin,i=1,2,with(91)ζin=λiqn+σin,τin=λipn+σin,i=1,2.
From equations (90) and (91), we arrive at(92)Tn+1a=ζ1nτ2n−ζ2nτ1nλ1ζ2nτ1n−λ2ζ1nτ2n,Tn+1b=λ1−λ2τ1nτ2nλ1−λ2τ1nτ1nτ2n+λ1ζ2nτ1n−λ2ζ1nτ2n,Tn+1c=λ1−λ2ζ1nζ2nλ1λ2ζ1nτ2n−ζ2nτ1n,Tn+1d=λ1λ2λ1ζ2τ1n−λ2ζ1nτ2nλ1λ1λ2ζ1nτ2n−ζ2nτ1n.
As a result, we have(93)Tn+1UnTn∗=f11λ,nf12λ,nf21λ,nf22λ,n,where(94)f11λ,n=Tn+1a1−Tn+1bTndpnλ3+−Tn+1a1−Tn+1bTnc+Tn+1a1−Tn+1bpn++1−Tn+1bTndpn+Tn+1bDnqnλ2+−Tnc+1−Tn+1bpn+qnTn+1bλ,f12λ,n=Tn+1aTn+1b−1TnaTnb−1+pnTnbλ2+Tna1−Tnb−Tn+1a1−TnbTn+1b−1+TnbTn+1b−1pn−Tn+1bqnλ+1−Tnb,f21λ,n=pnTndTn+1c+qnTndTn+1dλ3+TncTn+1d−Tn+1cTnc+pnTn+1c+qnTn+1d+qnTndλ2+qn−Tncλ,f22λ,n=TnaTn+1c−qnTnbTn+1d−TnaTnbTn+1c+TnaTn+1d−TnaTnbTn+1d−pnTnbTn+1cλ2+Tna−TnaTnb+Tn+1c−TnbTn+1c+Tn+1d−TnbTn+1d−qnTnbλ+1−Tnb.
Then, we find thatf11λ,n and f21λ,n are two 3th-order polynomial in λ and f12λ,n and f22λ,n are two 2th-order polynomial in λ. By a tedious but direct computation or by a mathematical software, we get that λ1,λ2 are two roots of fijλ,n1≤i,j≤2. Thus, we may assume(95)Tn+1UnTn∗=DetTnαn,with(96)αn=α111λ+α110α120α211λ+α210α220,where αi11,αij01≤i,j≤2 are all independent of λ. Thus, we obtain(97)Tn+1Un=αnTn.
By comparing the coefficients ofλii=0,1, in both sides of equation (97), we get that(98)α111=−TndpnpnTnb+TnaTnb−Tna,α110=0,α12=1,α211=qn−Tnc1−Tnb,α210=0,α220=1.
Hence, we obtain thatαn=U˜n.
The proof is finished.Proposition 6.
The matrixN˜n1 given in (80) has the same form as Nn1 in (70) by means of the transformation (89).Proof.
Let us denote(99)Tnt+TnNn1Tn∗=g11λ,ng12λ,ng21λ,ng22λ,n,where(100)g11λ,n=Tn,ta1−TnbTnd−TnaTn,tbTndpn−1−qn−1λ2+Tn,ta1−Tnb−TnaTn,tb+Tna1−TnbTnc−Tn,tbTnc+12Tna1−TnbTnd−Tn,tbTnd−Tna1−Tnbpn−1−1−TnbTndpn−1+Tna1−Tnbqn−1+1−TnbTndqn−1−TnbTncqn−TnbTndqnλ+12Tna1−Tnb−TnbTnc+Tnd1−Tnb+Tnc−Tn,tb−1−Tnbpn−1+1−Tnbqn−1−Tnbqn+1−Tnb2λ,g12λ,n=−Tna2+2Tna2Tnb−Tn,taTnb−Tna2Tnb2+Tn,taTnb2+TnaTn,tb+TnaTnbpn−1−qn−1−TnaTnb2pn−1−qn−1+TnaTnb1−Tnbqnλ+Tna3Tnb−2−Tnb2+Tn,tb+Tnb1−Tnbpn−1−qn−1+Tnbqn−1−Tnbλ,g21λ,n=Tn,tcTnd−TncTn,td−TncTndpn−1−qn−1−TncTndqn−Tnd2qnλ2+Tnc2+Tn,tc+TncTnd−Tncpn−1−qn−1−Tncqn−2Tndqnλ+Tnc−qn.g22λ,n=TnaTn,td−TnaTnbTn,td+TnaTnd1−Tnbqnλ2+TnaTndTnb−12+TnaTncTnb−1−TnbTn,tc1−TnbTn,td+TnbTncpn−1−qn−1+Tna1−Tnbqn+Tndqnλ+12TnaTnb−1+Tnb−2Tnc+TndTnb−1+2qn+Tnb−12λ.
Based on (70) and (87), we get(101)σi,tn=−qn+qn−1λi+pn−1−qn−1σin+1λiσin2,i=1,2.
Through tediously long calculation, we can find thatgijλi,n=0i,j=1,2. So, we have(102)Tn,t+TnNn1Tn∗=DetTnβn,where(103)βn=β110+β11−1λβ12−1λβ210β220+β22−1λ,and β110,β11−1,β12−1,β210,β220,andβ22−1 are all independent of λ.
By means of equation (102), we obtain(104)Tn,t+TnNn1=βnTn.
Comparing the coefficients ofλii=−1,0 in (104), we have(105)β110=q˜n−1−p˜n−1,β11−1=12,β12−1=−1,β211=−qn˜,β220=q˜n,β22−1=−12.
Thus,(106)Tn,t+T.Nn1Tn−1=N˜n1.
The proof is finished.
The transformations (78) and (89), namely, from ψn;pn,qn to ψ˜n;p˜n,q˜n constitute a Darboux–Bäcklund transformation of the NIDDE (69).
In conclusion, according to the Propositions5 and 6, we have the theorem.Theorem 2.
Every solutionpn,qnT of the NLDDE (69) is changed into a new solution p˜n,q˜nT under the Darboux–Bäcklund transformation (89).


## 5. Exact Solutions

Next, we will use the Darboux–Bäcklund transformation (89) to find two solutions of equation (69).First, we consider a seed solution of (69) (a simple special solution) pn,qnT=1,0T. Substituting this solution into the corresponding Lax pair (7) and (70), we have(107)Eψn=λ101ψn,ψnt=12λ−1−1λ0−12λψn.Solving above two equations, we arrive at the solutions(108)ψn=ψn1λ,tψn2λ,t=λnexpλ−2λ2t/2λ2−exp−t/2λλ−1exp−t2λ,ϕn=ϕn1λ,tϕn2λ,t=−λnexpλ−2λ2t/2λ2−exp−t/2λλ−1−exp−t2λ.Then, we obtain(109)σin=ψn2λi,t−κiϕn2λi,tψn1λi,t−κiϕn1λi,t=1+κiλinexp−t/2λi1−λi1−κiλinexpλi−2λi2/2λi2+1+κiexp−t/2λi,i=1,2.Using the Darboux–Ba¨cklund transformation (89) and with the help of mathematical software ”Mathematica,” we obtain an exact solution of (69):(110)p˜n=−TndTnb+TnaTnb−Tna=−λ1σ2n−λ2σ1n2λ1−λ2−λ1σ2n−λ2σ1nλ1λ2σ1n−σ2nλ1−λ2λ1σ2n−λ2σ1n+σ1n−σ2nλ1−λ2−1,q˜n=Tnc1−Tnb=λ1−λ2−λ1σ2n−λ2σ1nλ1−λ2σ1n−σ2nλ2σ1n−λ1σ2nλ1λ2σ1n−σ2n.Then, it easy to verify thatpn,qnT=1,−1T is another seed solution of (69). Substituting it into the corresponding Lax pair, it is found that(111)Eψn=11−λ1ψn,ψnt=12λ−2−1λ1−12λψn.Set(112)χ1λ,t=expt−3λ−λ2−6λ+12λ,χ2λ,t=expt−3λ+λ2−6λ+12λ,h1λ,n=2−n−11−λ+λ2−6λ+11+λ−λ2−6λ+1nλ2−6λ+1+2−n−1−1+λ+λ2−6λ+11+λ+λ2−6λ+1nλ2−6λ+1,h2λ,n=2−nλλ2−6λ+11+λ−λ2−6λ+1n−1+λ+λ2−6λ+1n,l1λ,n=2nλ2−6λ+11+λ−λ2−6λ+1n−1+λ+λ2−6λ+1n,l2λ,n=2−n−1−1+λ+λ2−6λ+11+λ−λ2−6λ+1nλ2−6λ+1+2−n−11−λ+λ2−6λ+11+λ+λ2−6λ+1nλ2−6λ+1.By means of mathematical software “Mathematica,” we may get that two real linear independent solutions as follows:(113)ψ˜n=ψ˜n1λ,tψ˜n2λ,t,ϕ˜n=ϕ˜n1λ,tϕ˜n2λ,t,where(114)ψ˜n1λ,t=h1λ,nλ−1χ1λ,n+λ+1χ2λ,n2λ2−6λ+1+χ1λ,n+χ2λ,n2+h2λ,nχ1λ,n−χ2λ,n2λ2−6λ+1,ψ˜n2λ,t=h1λ,nλχ1λ,n−λχ2λ,n2λ2−6λ+1+h2λ,n1−λχ1λ,n−λ−1χ2λ,n2λ2−6λ+1+χ1λ,n+χ2λ,n2,ϕ˜n1λ,t=l1λ,nλ−1χ1λ,n+λ+1χ2λ,n2λ2−6λ+1+χ1λ,n+χ2λ,n2+l2λ,nχ1λ,n−χ2λ,n2λ2−6λ+1,ϕ˜n2λ,t=l1λ,nλχ1λ,n−λχ2λ,n2λ2−6λ+1+l2λ,n1−λχ1λ,n−λ−1χ2λ,n2λ2−6λ+1+χ1λ,n+χ2λ,n2.Based on (87), we have(115)σ˜in=ψ˜n2λi,t−γiϕ˜n2λi,tψ˜n1λi,t−γiϕ˜n1λi,t,i=1,2.Though the Darboux–Bäcklund transformation (89), we arrive at another exact solution of (69):(116)p˜n=−TndTnb+TnaTnb−Tna=λ1σ˜2n−λ2σ˜1n−λ1+λ2+λ1σ˜2n−λ2σ˜1nλ1λ2σ˜1n−σ˜2nλ1−λ2+σ˜1n−σ˜2n,q˜n=−1+Tnc1−Tnb=λ1−λ2+λ2σ˜1n−λ1σ˜2nλ1λ2σ˜1n−σ˜2n−λ1−λ2σ˜1nσ˜2nλ1λ2σ˜1n−σ˜2n.


## 6. Conclusions and Remarks

In this paper, we have deduced two hierarchies of NIDDEs from a discrete matrix spectral problem by the discrete zero curvature equation. The obtained hierarchies, respectively, work in concert with positive power and negative power expansions of Lax operators with respect to the spectral parameter. Two bi-Hamiltonian forms for the obtained integrable hierarchies are given by the discrete trace identity. And then, the Liouville integrability of the obtained hierarchies is demonstrated. Furthermore, by the aid of a gauge transformation of the Lax pair, a Darboux–Bäcklund transformation for the first NIDDE in the negative integrable hierarchy was presented. Applying the obtained Darboux–Bäcklund transformation and “Mathematica,” we get two exact solutions. These solutions also are called one-fold solutions. This Darboux–Bäcklund transformation is continuously doneN times, then N-fold solution of (69) can be derived. Besides, we can get the Darboux–Bäcklund transformation of the first NIDDE (27) in the positive hierarchy in a similar way. Recently, in the soliton theory, some new types of explicit solutions for the continuous soliton equations have been found, for instance, abundant lump solutions and interaction solutions [23–26]. For the NIDDE (69), explicit solutions of these types can also be researched. These results will appear in later papers.In addition, many interesting problems deserve further investigation for the NIDDEs in the obtained hierarchies (25) and (68), such as symmetries constraint, integrable coupling systems by semidirect sums of Lie algebra, symmetries, and master symmetries.



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