# A Family of Integrable Different-Difference Equations, Its Hamiltonian Structure, and Darboux-Bäcklund Transformation

Authors: Xi-Xiang Xu; Meng Xu
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

An integrable family of the different-difference equations is derived from a discrete matrix spectral problem by the discrete zero curvature representation. Hamiltonian structure of obtained integrable family is established. Liouville integrability for the obtained family of discrete Hamiltonian systems is proved. Based on the gauge transformation between the Lax pair, a Darboux-Bäcklund transformation of the first nonlinear different-difference equation in obtained family is deduced. Using this Darboux-Bäcklund transformation, an exact solution is presented.


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## Body

## 1. Introduction

During the past decades, study of the integrable different-difference equations (or integrable lattice equations) has received considerable attention. They play the important roles in mathematical physics, lattice soliton theory, cellular automata, and so on. Many nonlinear integrable different-difference equations have been proposed and discussed, such as Ablowitz-Ladik lattice [1], Toda lattice [2], relativistic Toda lattices in polynomial form and rational form [3, 4], modified Toda lattice [5, 6], Volterra (or Langmuir) lattice [7], deformed reduced semi-discrete Kaup-Newell lattice [8], Merola-Ragnisco-Tu lattice [9], and so forth [10–18]. As we know, searching for novel integrable nonlinear different-difference equations is still an important and very difficult research topic. In the lattice soliton theory, the discrete zero curvature representation is a significant way to derive the integrable different-difference equations [18].Starting from a suitable matrix spectral problem(1)Eφn=Unyn,λφnand a series of auxiliary spectral problems(2)φtm=Vnmφn,m≥1,where for a lattice function fn=f(n,t), the shift operator E and the inverse of E are defined by (3)Efn=fn+1=fn+1,t,E-1fn=fn-1=fn-1,t,n∈Z,and (φn1,φn2) is eigenfunction vector, and yn=(rn,sn)T is a potential vector.A family of evolution different-difference equations(4)yntm=Fmyn,m≥1,is called to be integrable in Lax sense (or lattice soliton equations), if it can be written as a compatibility condition of equations (1) and (2): (5)Unyn,λtm-EVnmUyn,λ+Unyn,λVnm=0.Moreover, in the lattice soliton theory, if we can seek out a Hamiltonian operator J and a series of conserved functionals H~n(m)(m≥1) so that (4) may be represented as the following forms (6)yn,tm=JδH~nmδyn,where the Hamiltonian functionals H~n(m)=∑n∈Z‍Hn(m),δH~n/δyn=∑n∈Z‍E-m∂Hn(m)/∂yn+m,m≥1, then Hamiltonian structure of family (4) is established. In lattice soliton theory, (6) is called discrete Hamiltonian systems. Further, if we can deduce infinitely many involutive conserved functionals of family (6), then Liouville integrability of the integrable lattice family (4) is proved.In addition, it is well known that Darboux-Bäcklund transformation is an important and very effective method for solving integrable different-difference equations. This transformation is a formula between the new solution and the old solution of the different-difference equation. From a known solution, through this transformation, we can obtain another solution. According to [19], Bäcklund transformation are usually divided into three types: the Wahlquist-Estabrook (WE) type [20, 21], the Darboux-Bäcklund type [5, 6, 22, 23], and the Hirota type [24].This paper is organized as follows. In Section2, we introduce a novel discrete spectral problem (7)Eφn=Unyn,λφn,Unyn,λ=0rnλsnλrnsnλ2,where λ is the spectral parameter and λt=0. It is easy to see that the matrix spectral problem (7) is equivalent to the following eigenvalue problem: (8)snE-1rnξn+rnsnξn=1λ2Eξn, the eigenfunction ξn=φn2. Based on this discrete matrix spectral problem, we derive a family of integrable different-difference equations through the discrete zero curvature representation. Then, we establish the Hamiltonian structure of the obtained family by means of the discrete trace identity [18]. Afterwards, infinitely many common commuting conserved functionals of the obtained family are worked out. This guarantees Liouville integrability of the obtained family. In Section 3, we would like to derive a Bäcklund transformation of Darboux type (or Darboux-Bäcklund transformation) of the first nonlinear integrable different-difference equation in the obtained family; this transformation is constructed by means of the gauge transformation of Lax pair of the spectral problem, as application of Darboux-Bäcklund transformation, an exact solution is given. Some conclusions and remarks are given in the final section.


## 2. An Integrable Different-Difference Family and Its Hamiltonian Structure

We first solve the stationary discrete zero curvature equation(9)EΓnUn-UnΓn=Γn+1Un-UnΓn=0,with(10)Γn=anbncn-an.Equation (9) implies(11)rnsnan+1-anλ=rncn+1-snbn,rnsnbn+1λ=-rnan+an+1,rnsncnλ=-snan+an+1,snbn+1-rncn=0.Substituting Laurent series expansions (12)an=∑m=0∞anmλ-2m,bn=∑m=0∞bnmλ-2m+1,cn=∑m=0∞cnmλ-2m+1into (11), we obtain the initial conditions: (13)rnsnan+10-an0=rncn+10-snbn0,bn+10=0,cn0=0and the recursion relations: (14)rnsnan+1m+1-anm+1=rncn+1m+1-snbnm,rnsnbn+1m+1=-rnanm+an+1m,rnsncnm+1=-snanm+an+1m.We choose the initial values satisfying the above initial conditions, (15)an0=-12,bn0=0,and require selecting zero constant for the inverse operation of the difference operator (E-1) in computing an(m)(m≥1), then the recursion relation (14) uniquely determines an(m),bn(m),cn(m)(m≥1). In addition, we have the following assertion.Proposition 1.
{ a n ( m ) } m ≥ 1 may be deduced through an algebraic method rather than by solving the difference equation.Proof.
From (9), we know that (16)E-1trΓn2=E-1an2+bncn=0.anm+1=∑j=1manjanm+1-j+∑j=1mbnjcnm+1-j+γt. Here γ(t) is an arbitrary function of time variable t only. Further, we choose γ(t)=0. Then, we get a recursion relation for an(m)(m≥1). Thus, an(m)(m≥1) are all local, and they are just the rational function in two dependent variables rn,sn.
The proof is completed.The first few terms are given by(17)an1=1rnsn-1,bn1=1sn-1,cn1=1rn,an2=-1rn2sn-12-1rn+1rnsnsn-1-1rn-1rnsn-1sn-2,bn2=-1rn-1sn-1sn-2-1rnsn-12,cn2=-1rn2sn-1-1rnrn+1sn,⋯ Let us denote (18)Vnm=∑i=0maniλ2m-2i∑i=0mbniλ2m-2i+1∑i=0mcniλ2m-2i+1-∑i=0maniλ2m-2i,m≥0.On the basis of the recursion relations (14), we have (19)EVnmUn-UnVnm=0-rnsnbn+1m+1λrnsncnm+1λrnsnan+1m-anm.It is obvious that (19) is not compatible with (Un)tm. Therefore, we choose the following modification term: (20)Δnm=000anm.Then we introduce auxiliary matrix spectral problem (21)Ωnm=Vnm+Δnm,m≥1.Let the time evolution of the eigenfunction of the spectral problem (7) obey the differential equation (22)φn,tm=Ωnmφn,m≥0.Then the compatibility conditions of (7) and (22) are (23)Untm=EΩnmUn-UnΩnm,m≥0It implies the family of integrable (in Lax sense) different-difference equations. (24)rn,tm=-rnsnbn+1m+1-rnanm,sn,tm=rnsncnm+1+snan+1m.m≥0.When m=0, (24) becomes a trivial linear system (25)rn,t0=-rn2,vn,t0=sn2.When m=1 in (24), we obtain the first nontrivial integrable lattice equation (26)rn,t1=rnrn+1sn,sn,t1=-snrnsn-1.In Section 3, we are going to construct its Darboux-Bäcklund transformation.Now let us introduce some concepts for further discussion. The Gateaux derivative, the variational derivative, and the inner product are defined, respectively, by(27)J′ynvn=∂∂εJyn+εvnε=0,δHnδyn=∑m∈ZE-m∂Hn∂yn+m,fn,gn=∑n∈Zfn,gnR2.fn,gn are required to be rapidly vanished at the infinity, and (fn,gn)R2 denotes the standard inner product of fn and gn in the Euclidean space R2. Operator J∗ is defined by 〈fn,J∗gn〉=〈Jfn,gn〉; it is called the adjoint operator of J. If an operator J has the property J∗=-J, then J is called to be skew-symmetric. An operator J is called a Hamiltonian operator, if J is a skew-symmetric operator satisfying the Jacobi identity, i.e., (28)fn,Jgn=-Jfn,gn,J′unJfngn,hn+Cyclefn,gn,hn=0.Based on a given Hamiltonian operator J, we can define the corresponding Poisson bracket [10, 11, 18] (29)fn,gnJ=δfnδyn,Jδgnδyn=∑n∈Zδfnδyn,JδgnδynR2.Following [18], we set (30)
R
n
=
Γ
n
U
n
-
1
=
-
a
n
+
b
n
r
n
λ
a
n
s
n
λ
-
c
n
-
a
n
r
n
λ
c
n
s
n
λand M,N=tr(MN), where M and N are some order square matrices. It is easy to calculate that (31)∂Un∂λ=0rnsn2rnsnλ,∂Un∂rn=0λ0snλ2,∂Un∂sn=00λrnλ2.Hence (32)Rn,∂Un∂λ=rncn,Rn,∂Un∂rn=-anrn,Rn,∂Un∂sn=-an+1sn.By virtue of the discrete trace identity [18] (33)δδrn∑n∈ZRn,∂Un∂λ=λ-ε∂∂λλεRn,∂Un∂rn,δδsn∑n∈ZRn,∂Un∂λ=λ-ε∂∂λλεRn,∂Un∂sn.Substituting expansions an=∑m=0∞‍an(m)λ-2m,bn=∑m=0∞‍bn(m)λ-2m+1,cn=∑m=0∞‍cn(m)λ-2m+11 into (33) and comparing the coefficients of λ-2m-1 in (33), we obtain (34)
δ
δ
r
n
δ
δ
s
n
∑
n
∈
Z
r
n
c
n
m
+
1
=
ε
-
2
m
-
a
n
m
r
n
-
a
n
+
1
m
s
n
.When m=0 in (34), through a direct calculation, we find that ε=0. Therefore, (34) can be written as (35)δδrnδδsn∑n∈Z-rncnm+12m=-anmrn-an+1msnm≥1.Set(36)H~nm=∑n∈Z-rncnm+12m.We have (37)δδrnδδsn∑n∈ZH~nm=-anmrn-an+1msnm≥1.Set (38)-rnsnbn+1m+1-rnanmrnsncnm+1+snan+1m=J-anmrn-an+1msnm≥1.We can obtain (39)J=0-rnsnrnsn0.Evidently, the operator J is a skew-symmetric operator, i.e., J=-J∗. In addition, through a direct calculation, we can prove that the operator J satisfies the Jacobi identity (28). Thus, we obtain the following assertion.Proposition 2.
J is a discrete Hamiltonian operator.
Consequently, (22) have Hamiltonian structures (40)yntm=rnsntm=JδH~nmδyn=J-anmrn-an+1msn,m≥0.(41)H~nm=∑n∈ZHnm,Hnm=rncnm+1-2m,m≥1.In particular, the different-difference equation (23) possesses the Hamiltonian structure (42)yn,t=rnsnt=JδH~n1δun,H~n1=∑n∈Z12rnsn-1+12rn+1sn. Following (14) and (35), we have the following recursion relation: (43)δH~nm+1δun=ΨδH~nmδun,m≥1, where (44)Ψ=Ψ11Ψ12Ψ21Ψ12.Moreover, we have (45)JΨ=M=M11M12M21M22with (46)M11=-rn1-E-1-11snE1rn1-E-1rn+rn1-E-1-11rnE-11sn1-E-1rn,M12=E-1-rnsnEsnrn-rn1-E-11rnE-1-E-11sn1-E-1sn+rnsnE1rn1-E-1sn+rn1-E-11snErnsn+rn1-E-11rnE-11sn1-E-1-rn1-E-11snE1rn1-E-1sn,M21=-E-snrnE-1rnsn-sn1-E-1-11rnE-1rnsn-E1rn1-E-1-1rn+sn1-E-1-11snE-snrnE-11sn1-E-1-1rn-sn1-E-1-11rnE-1rnsn-sn1-E-1-11snE-11rn1-E-1-1rn-snrnE-11sn1-E-1-1rn,M22=sn1-E-11rnE-11sn1-E-1-1sn-sn1-E-11snE1rn1-E-1rn.It is easy to verify that M is a skew-symmetric operator. Namely, (47)M∗=-M.With the help of the operator Ψ, (40) may be written in the following form: (48)yn,tm=rnsntm=JδH~nmδun=MδH~nm-1δun=JΨmδH~n0δun,m≥1.Furthermore, on the basis of [4, 16], we may obtain a recursion operator (49)Φ=Ψ∗=Ψ11∗Ψ21∗Ψ12∗Ψ22∗.Next, we prove Liouville integrability of the discrete Hamiltonian systems (40). It is crucial to show the existence of infinite involutive conserved functionals.Proposition 3.
{ H ~ n ( m ) } m = 0 ∞ are conserved functionals of the whole family (24) or (40). And they are in involution in pairs with respect to the Poisson bracket (29).Proof.
Due to (47), we know that M is a skew-symmetric operator, and a direct calculation shows (50)JΨ∗=M∗=-M=-JΨ,namely, (51)Ψ∗J=JΨTherefore(52)H~nm,H~nlJ=δH~nmδun,JδH~nlδun=Ψm-1δH~n1δun,JΨl-1δH~n1δun=Ψm-1δH~n1δun,Ψ∗JΨl-2δH~n1δun=ΨmδH~n1δun,JΨl-2δH~n1δun=H~nm+1,H~nl-1J=⋯=H~nm,H~nlJ=H~nm+l-1,H~n1J.Repeating the above argument, we can obtain (53)H~nl,H~nmJ=H~nm+l-1,H~n1J.Then combining (52) with (53) leads to (54)H~nm,H~nlJ=0,m,l≥1,and(55)H~nmtl=δH~nmδun,utl=δH~nmδun,JδH~nlδun=H~nm,H~nlJ=0,m,l≥1.
The proof is completed.Remark 4.
According to the above proposition, we can get that (24) is not only Lax integrable but also Liouville integrable. Based on (40) and Proposition 2, we can obtain the following theorem.Theorem 5.
The integrable different-difference equations in family (24) are all Liouville integrable discrete Hamiltonian systems.


## 3. Darboux-Bäcklund Transformation and Exact Solution

Next we are going to establish a Darboux-Bäcklund transformation of (26).Whenm=1 in (23), we obtain the time part of the Lax pair of (26) (56)
φ
n
,
t
=
Ω
n
1
φ
n
=
-
λ
2
2
+
1
r
n
s
n
-
1
1
s
n
-
1
λ
1
r
n
λ
λ
2
2
φ
n
.It is crucial to look for a suitable gauge transformation of a matrix spectral problem; it can transform the matrix spectral problem into another spectral problem of the same form [5, 6, 8, 23]. We introduce the following gauge transformation: (57)φn~=Tnφn,which can transform two spectral problems (7) and (56) into (58)Eφn~=φ~n+1=U~nφ~n,(59)φ~nt=Ω~n1φ~n,with (60)U~n=Tn+1UnTn-1,Ω~n1=Ωnt1+TnΩn1Tn-1.We suppose that Tn has the following form: (61)λ+t11nλt12nt21nt22nλ+1λ.In (61), t11[n], t12[n], t21[n], and t22[n] are undetermined functions of variables n and t. Now we would like to construct Tn such that U~n and Ω~n(1) in (60) have the same form with Un and Ωn(1), respectively.Letϕn=(ϕn1,ϕn2)T,ψn=(ψn1,ψn2)T be two real linear independent solutions of (7) and (56). Let (62)Kn=ϕn1ψn1ϕn2ψn2. From equation K~n=TnKn, we obtain that(63)K~n=λ+t11nλϕn1+t12nϕn2λ+t11nλψn1+t12nψn2t21nϕn1+t22nλ+1λϕn2t21nψn1+t22nλ+1λψn2.We assume that ±λ1 and ±λ2 are four roots of det(Tn). When λ=±λi(i=1,2), two columns of (63) are linear dependent. Without loss of generality, there exist two nonzero constants κ1 and κ2, which satisfy(64)λ+t11nλϕn1+t12nϕn2=κiλ+t11nλψn1+t12nψn2,t21nϕn1+t22nλ+1λϕn2=κit21nψn1+t22nλ+1λψn2where λ=±λi(i=1,2).Set(65)αin=ϕn2-κiψn2ϕn1-κiψn1,i=1,2.Solving (64), we have (66)t11n=λ1λ2λ1α2n-λ2α1nλ1α1n-λ2α2n,t12n=λ22-λ12λ1α1n-λ2α2n,t21n=λ22-λ12α1nα2nλ1λ2λ1α1n-λ2α2n,t22n=λ1α2n-λ2α1nλ1λ2λ1σ1n-λ2σ2n.In (66), λi,κi(i=1,2) are suitably chosen such that all the denominations in (65) and (66) are nonzero. Through a direct but tedious calculation, we can get that ±λi(i=1,2) are four roots of det(Tn)=0, and then (67)detTn=t22nλ2-λ12λ2-λ22λ2.From (65), we get (68)αin+1=ξinχini=1,2,where ξi(n)=λisn+(rnsnλi2)αi[n],χi(n)=rnλiαi[n](i=1,2).Based on (65) and (66), we can get the expressions of t11[n+1],t12[n+1],t21[n+1],t22[n+1].(69)t11n+1=λ1λ2λ1ξ2nχ1n-λ2ξ1nχ2nλ1ξnχ2n-λ2ξ2nχ1n,t12n+1=λ22-λ12χ1nchi2nλ1ξ1nχ2n-λ2ξ2nχ1n,t21n+1=λ12-λ22ξ1nξ2nλ1λ2λ1ξ1nχ2n-λ2ξ2nχ1n,t22n+1=λ1ξ2nχ1n-λ2ξ1nχ2nλ1λ2λ1ξ1nχ2n-λ2ξ2nχ1n.Proposition 6.
The matrixU~n defined by (60) has the same form as Un, that is, (70)U~n=0r~nλs~nλr~ns~nλ2,where the relations between old potentials rn and sn and new potentials r~n and s~n are given by (71)r~n=rnt22n-rnt21n,s~n=snt11n.Proof.
One has(72)Tn+1UnTn∗=h11λ,nh12λ,nh21λ,nh22λ,n, with (73)h11λ,n=snt12n+1-rnt11n+1t21n+-rnt21n-rnsnt12n+1t21n+snt12n+1t22nλ2,h12λ,n=rn+rnsnt12n+1λ3+rnt11n+rnt11n+1+rnsnt11nt12n+1-snt12nt12n+1λ,+rnt11nt11n+1λ,h21λ,n=-rnsnt21nt22n+1+snt22nt22n+1λ3+-rnsnt21n-rnt21nt21n+1+snt22n+snt22n+1λ+snλ,h22λ,n=rnsnt22n+1λ4+rnsn+rnt21n+1+rnsnt11nt22n+1-snt12nt22n+1λ2+rnsnt11n-snt12n+rnt11nt21n.According to (66) and (69), we can find that (74)snt12n+1-rnt11n+1t21n=0,-rnt21n-rnsnt12n+1t21n+snt12n+1t22n=0.Thus, we have h11(λ,n)=0. Moreover, it is easy to see that λ2h12(λ,n) and λ2h21(λ,n) are five-order polynomial in λ, and λ2h22(λ,n) is a six-order polynomial in λ. Through a tedious but direct computation or by computer algebra system (for instance, Mathematica, Maple), we can obtain that (75)h12±λi,h21±λi,h22±λii=1,2are all equal zero. Therefore, we have (76)Tn+1UnTn∗=detTnτn,with (77)τn=0τ121λτ211λτ222λ2+τ220where τjl(i)(i=0,1,2,j,l=1,2) are all independent of λ. We know that Tn-1=Tn∗/det(Tn). Hence we obtain (78)Tn+1Un=τnTn.Equating the coefficients of λi(i=0,1,2) in (78), we have (79)τ121=rnt22n-rnt21n,τ211=snt11n,τ222=rnt22n-rnt21nsnt11n,τ220=0.
The proof is completed.Proposition 7.
The matrixΩ~n(1) defined by (60) has the same form as Ωn(1) in (56) under the transformation (71), i.e., (80)
Ω
~
n
1
=
-
λ
2
2
+
1
r
~
n
s
~
n
-
1
λ
s
~
n
-
1
λ
r
~
n
λ
2
2
.Proof.
Let(81)Tn,t+TnΩn1Tn∗=η11λ,nη12λ,nη21λ,nη22λ,n.Here (82)η11λ,n=-12λ4t22n+-12-t21nsn-1-12t12nt21n+t11nt22n+t12nt22nrnλ2+1+t11nt22nrnsn-1-t11n2+t12nrn-t12nt21nsn-1+t11ntt22n-t21nt12nt+t11n+rnsn-1t11ntrnsn-1λ2,η12λ,n=1sn-1+t12nλ3+2t11nsn-1-t12nrnsn-1-t12n2rn+t11nt12n+t12ntλ+rnt11n2-t11nt12n-rnsn-1t12nt11nt-t11nt12ntrnsn-1λ,η21λ,n=-t21nt22n+t22n2rnλ3+-t21n-t21n2sn-1+2t22nrn+t21nt22nrnsn-1+t22nt21nt-t21nt22ntλ+sn-1+t21n+rnsn-1t22ntrnsn-1λ,η22λ,n=12t22nλ4+12+t21nsn-1+t12nt21+t11nt22n2-t12nt22nrn+t22ntλ2+t11n2-t12nrn+t11nt21nsn-1-t12nt21nrnsn-1-t12nt21nt+t11nt22nt. It is easy to get that λ2η11(λ,n) and λ2η22(λ,n) are six-order polynomials in λ;λ2η12(λ,n) and λ2η21(λ,n) are five-order polynomials in λ. According to (56) and (65), we have(83)αi,tn=λirn+λi2-1rnsn-1αin-λisn-1αin2,i=1,2.Through lengthy but direct calculation, we can verify that (84)ηjk±λi,n=0,i,j,k=1,2. Therefore, the following equation is derived: (85)Tnt+TnΩn1Tn∗=detTnμn,with (86)μn=μ112λ2+μ110μ121λμ211λμ222λ2+μ220,where μ11(2)μ11(0),μ12(1),μ21(1),μ22(2),μ22(0)are all independent of λ. From (85), we get (87)Tnt+TnΩn1=μnTn.Comparing the coefficients of λ in (87), we have (88)μ112=-12,μ110=1r~ns~n-1,μ121=1s~n-1,μ211=1r~n,μ222=12,μ220=0.The proof is completed.Theorem 8.
Equation (71) is a Darboux-Bäcklund transformation of (26). That is, each solution (rn,sn)T of the integrable lattice system (26) is mapped into its new solution (r~n,s~n)T under transformation (71).Next, using obtained Darboux-Bäcklund transformation (71), we derive an exact solution of (26).First, it is easy to find thatrn=e-t,sn=-et constitute a trivial solution of (71). Substituting this solution into the corresponding Lax pair, we get (89)Eφn=0e-tλ-etλ-λ2φn,φnt=-λ22-1-e-tλetλλ22φn.Solving the above two equations, we get two real linear independent solutions: (90)ϕn1ϕn2=C1λ,n22+5exp-t2-3+5expt2,ψn1ψn2=C2λ,n22+5exp-t23+5expt2,where (91)C1λ,n=-72+5-352+5λ-λ+λ2-4n--λλ+λ2-4n2913+5λ2-4λ-λ+λ2-4n+-λλ+λ2-4n,C2λ,n=λ-λ+λ2-4n--λλ+λ2-4n.From (65), we have (92)αin=ϕn2-κiψn2ϕn1-κiψn1=-3+5expt/2C1λi,n-κiC2λi,n22+5exp-t/2C1λi,n+κiC2λi,n,i=1,2.In (92), λ1=(2+5), λ2=-(2+5).By means of the Darboux-Bäcklund transformation (71), we deduce a new exact solution of (26). (93)r~n=λ1λ2λ1σ1n-λ2σ2nexp-tλ1α2n-λ2α1n-λ22-λ12α1nα2nexp-t,s~n=-λ1α1n-λ2α2nexptλ1λ2λ1α2n-λ2α1n.This transformation can be done continually. Thus, we can obtain a series of exact solutions of (26).


## 4. Conclusions and Remarks

In this work, based on a Lax pair, we have deduced a novel family of integrable different-difference equations using the discrete zero curvature equation and established the Hamiltonian structure of the obtained integrable family of different-difference equations in virtue of the discrete trace identity. And then the Liouville integrability of the obtained family is demonstrated. With the help of a gauge transformation of the Lax pair, a Darboux-Bäcklund transformation for the first nonlinear different-difference equation in the obtained family is presented, and using the obtained Darboux-Bäcklund transformation, an exact solution is derived. It is worth noting that in (61) we can also study the generalized form of Tn(94)Tn2N+1=Tn11Tn12Tn21Tn22, with (95)Tn11=λ2N+1+t112N-1nλ2N-1+t112N-3nλ2N-3+⋯+t11-2N+1nλ-2N+1+t11-2N-1nλ2N+1,N≥1,Tn12=t122Nnλ2N+t122N-2nλ2N-2+⋯+t12-2Nnλ2N,N≥1,Tn21=t212Nnλ2N+t212N-2nλ2N-2+⋯+t21-2Nnλ2N,N≥1,Tn22=λ2N+1t222N+1n+t222N-1nλ2N-3+t222N-5nλ2N-5+⋯+t11-2N+1nλ2N-1+1λ2N+1,N≥1.In addition, many interesting problems deserve further investigation for obtained integrable family, for example, symmetry constraint [7], symmetries and master symmetries [13], and integrable coupling systems [25, 26].



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*Source: 4152917-2018-06-07.xml*