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

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.

Starting from a suitable matrix spectral problem   =   (  , )   (1) and a series of auxiliary spectral problems where for a lattice function   = (, ), the shift operator  and the inverse of  are defined by  (  ) =  +1 =  ( + 1, ) , and ( 1  ,  2  ) is eigenfunction vector, and   = (  ,   )  is a potential vector.
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 Section 2, we introduce a novel discrete spectral problem where  is the spectral parameter and   = 0.It is easy to see that the matrix spectral problem ( 7) is equivalent to the following eigenvalue problem: the eigenfunction   =  2  .Based on this discrete matrix spectral problem, we derive a family of integrable differentdifference 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.

An Integrable Different-Difference Family and Its Hamiltonian Structure
We first solve the stationary discrete zero curvature equation with Equation ( 9) implies Substituting Laurent series expansions into (11), we obtain the initial conditions: and the recursion relations: We choose the initial values satisfying the above initial conditions, and require selecting zero constant for the inverse operation of the difference operator ( − 1) in computing  ()  (  ≥ 1), then the recursion relation (14) uniquely determines  ()   ,  ()  ,  ()  (  ≥ 1).In addition, we have the following assertion.

Proposition 1. {𝑎 (𝑚)
} ≥1 may be deduced through an algebraic method rather than by solving the difference equation.
Proof.From (9), we know that Here () is an arbitrary function of time variable  only.Further, we choose () = 0.Then, we get a recursion relation for  ()  ( ≥ 1).Thus,  ()  ( ≥ 1) are all local, and they are just the rational function in two dependent variables   ,   .
The proof is completed.
The first few terms are given by Let us denote On the basis of the recursion relations ( 14), we have ) . ( It is obvious that ( 19) is not compatible with (  )   .Therefore, we choose the following modification term: Then we introduce auxiliary matrix spectral problem Let the time evolution of the eigenfunction of the spectral problem (7) obey the differential equation Then the compatibility conditions of ( 7) and ( 22) are It implies the family of integrable (in Lax sense) differentdifference equations.
When  = 0, (24) becomes a trivial linear system When  = 1 in ( 24), we obtain the first nontrivial integrable lattice equation 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 .(29) Following [18], we set and ⟨, ⟩ = tr(), where  and  are some order square matrices.It is easy to calculate that Hence By virtue of the discrete trace identity [18] Substituting expansions −2+11 into (33) and comparing the coefficients of  −2−1 in (33), we obtain When  = 0 in (34), through a direct calculation, we find that  = 0. Therefore, (34) can be written as We have Set We can obtain Evidently, the operator  is a skew-symmetric operator, i.e.,  = − * .In addition, through a direct calculation, we can prove that the operator  satisfies the Jacobi identity (28).Thus, we obtain the following assertion.
Proposition 2.  is a discrete Hamiltonian operator.Consequently, (22) have Hamiltonian structures ) ,  ≥ 0. (40) In particular, the different-difference equation ( 23) possesses the Hamiltonian structure ) . (42) Following ( 14) and ( 35), we have the following recursion relation: where Moreover, we have with It is easy to verify that  is a skew-symmetric operator.Namely, With the help of the operator Ψ, (40) may be written in the following form: (48) Furthermore, on the basis of [4,16], we may obtain a recursion operator ) . (49)

or (40). And they are in involution in pairs with respect to the Poisson bracket (29).
Proof.Due to (47), we know that  is a skew-symmetric operator, and a direct calculation shows Therefore Repeating the above argument, we can obtain Then combining (52) with (53) leads to and 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.

Darboux-Bäcklund Transformation and Exact Solution
Next we are going to establish a Darboux-Bäcklund transformation of (26).

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 differentdifference 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   ( ,   are required to be rapidly vanished at the infinity, and (  ,   )  2 denotes the standard inner product of   and   in the Euclidean space  2 .Operator  * is defined by ⟨  ,  *   ⟩ = ⟨  ,   ⟩; it is called the adjoint operator of .If an operator  has the property * = −, then  is called to be skewsymmetric.An operator  is called a Hamiltonian operator, if  is a skew-symmetric operator satisfying the Jacobi identity, i.e., ⟨  ,   ⟩ = − ⟨  ,   ⟩ , ⟨  (  ) [  ]   , ℎ  ⟩ +  (  ,   , ℎ  ) = 0.