A Family of Integrable Different-Difference Equations, Its Hamiltonian

Hindawi Discrete Dynamics in Nature and Society Volume 2018, Article ID 4152917, 11 pages https://doi.org/10.1155/2018/4152917

Research Article A Family of Integrable Different-Difference Equations, Its Hamiltonian Structure, and Darboux-Bäcklund Transformation Xi-Xiang Xu

and Meng Xu

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China Correspondence should be addressed to Xi-Xiang Xu; xixiang [email protected] Received 26 December 2017; Accepted 5 May 2018; Published 7 June 2018 Academic Editor: Josef Diblik Copyright © 2018 Xi-Xiang Xu and Meng Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.

1. Introduction During the past decades, study of the integrable differentdifference 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 differentdifference 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 differentdifference 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 𝐸𝜑𝑛 = 𝑈𝑛 (𝑦𝑛 , 𝜆) 𝜑𝑛

(1)

𝐸 (𝑓𝑛 ) = 𝑓𝑛+1 = 𝑓 (𝑛 + 1, 𝑡) , 𝐸−1 (𝑓𝑛 ) = 𝑓𝑛−1 = 𝑓 (𝑛 − 1, 𝑡) ,

(2)

(3) 𝑛 ∈ 𝑍,

and (𝜑𝑛1 , 𝜑𝑛2 ) is eigenfunction vector, and 𝑦𝑛 = (𝑟𝑛 , 𝑠𝑛 )𝑇 is a potential vector. A family of evolution different-difference equations 𝑦𝑛𝑡 = 𝐹𝑚 (𝑦𝑛 ) , 𝑚 ≥ 1, 𝑚

(4)

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): 𝑈𝑛 (𝑦𝑛 , 𝜆)𝑡𝑚 − (𝐸𝑉𝑛(𝑚) ) 𝑈 (𝑦𝑛 , 𝜆) + 𝑈𝑛 (𝑦𝑛 , 𝜆) 𝑉𝑛(𝑚) = 0.

and a series of auxiliary spectral problems 𝜑𝑡𝑚 = 𝑉𝑛(𝑚) 𝜑𝑛 , 𝑚 ≥ 1,

where for a lattice function 𝑓𝑛 = 𝑓(𝑛, 𝑡), the shift operator 𝐸 and the inverse of 𝐸 are defined by

(5)

Moreover, in the lattice soliton theory, if we can seek out a Hamiltonian operator 𝐽 and a series of conserved functionals

2

Discrete Dynamics in Nature and Society

̃(𝑚) (𝑚 ≥ 1) so that (4) may be represented as the following 𝐻 𝑛 forms ̃(𝑚) 𝛿𝐻 (6) 𝑦𝑛,𝑡𝑚 = 𝐽 𝑛 , 𝛿𝑦𝑛

with

̃(𝑚) = ∑𝑛∈𝑍 𝐻(𝑚) , 𝛿𝐻 ̃𝑛 / where the Hamiltonian functionals 𝐻 𝑛 𝑛 −𝑚 (𝑚) 𝛿𝑦𝑛 = ∑𝑛∈𝑍 𝐸 (𝜕𝐻𝑛 /𝜕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 Section 2, we introduce a novel discrete spectral problem

Equation (9) implies

(7)

where 𝜆 is the spectral parameter and 𝜆 𝑡 = 0. It is easy to see that the matrix spectral problem (7) is equivalent to the following eigenvalue problem: 1 𝑠𝑛 𝐸 (𝑟𝑛 𝜉𝑛 ) + 𝑟𝑛 𝑠𝑛 𝜉𝑛 = 2 𝐸 (𝜉𝑛 ) , 𝜆 −1

(8)

the eigenfunction 𝜉𝑛 = 𝜑𝑛2 . 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.

2. An Integrable Different-Difference Family and Its Hamiltonian Structure We first solve the stationary discrete zero curvature equation (𝐸Γ𝑛 ) 𝑈𝑛 − 𝑈𝑛 Γ𝑛 = Γ𝑛+1 𝑈𝑛 − 𝑈𝑛 Γ𝑛 = 0,

(10)

𝑟𝑛 𝑠𝑛 (𝑎𝑛+1 − 𝑎𝑛 ) 𝜆 = 𝑟𝑛 𝑐𝑛+1 − 𝑠𝑛 𝑏𝑛 , 𝑟𝑛 𝑠𝑛 𝑏𝑛+1 𝜆 = −𝑟𝑛 (𝑎𝑛 + 𝑎𝑛+1 ) , 𝑟𝑛 𝑠𝑛 𝑐𝑛 𝜆 = −𝑠𝑛 (𝑎𝑛 + 𝑎𝑛+1 ) ,

(11)

𝑠𝑛 𝑏𝑛+1 − 𝑟𝑛 𝑐𝑛 = 0. Substituting Laurent series expansions ∞

𝑎𝑛 = ∑ 𝑎𝑛(𝑚) 𝜆−2𝑚 , 𝑚=0 ∞

𝑏𝑛 = ∑ 𝑏𝑛(𝑚) 𝜆−2𝑚+1 ,

(12)

𝑚=0 ∞

𝑐𝑛 = ∑ 𝑐𝑛(𝑚) 𝜆−2𝑚+1 𝑚=0

𝐸𝜑𝑛 = 𝑈𝑛 (𝑦𝑛 , 𝜆) 𝜑𝑛 , 0 𝑟𝑛 𝜆 ), 𝑈𝑛 (𝑦𝑛 , 𝜆) = ( 𝑠𝑛 𝜆 𝑟𝑛 𝑠𝑛 𝜆2

𝑎𝑛 𝑏𝑛 ). Γ𝑛 = ( 𝑐𝑛 −𝑎𝑛

(9)

into (11), we obtain the initial conditions: (0) (0) − 𝑎𝑛(0) ) = 𝑟𝑛 𝑐𝑛+1 − 𝑠𝑛 𝑏𝑛(0) , 𝑟𝑛 𝑠𝑛 (𝑎𝑛+1 (0) 𝑏𝑛+1 = 0,

(13)

𝑐𝑛(0) = 0 and the recursion relations: (𝑚+1) (𝑚+1) − 𝑎𝑛(𝑚+1) ) = 𝑟𝑛 𝑐𝑛+1 − 𝑠𝑛 𝑏𝑛(𝑚) , 𝑟𝑛 𝑠𝑛 (𝑎𝑛+1 (𝑚+1) (𝑚) 𝑟𝑛 𝑠𝑛 𝑏𝑛+1 = −𝑟𝑛 (𝑎𝑛(𝑚) + 𝑎𝑛+1 ),

(14)

(𝑚) ). 𝑟𝑛 𝑠𝑛 𝑐𝑛(𝑚+1) = −𝑠𝑛 (𝑎𝑛(𝑚) + 𝑎𝑛+1

We choose the initial values satisfying the above initial conditions, 1 𝑎𝑛(0) = − , 2

(15)

𝑏𝑛(0) = 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. {𝑎𝑛(𝑚) }𝑚≥1 may be deduced through an algebraic method rather than by solving the difference equation.


3

Proof. From (9), we know that

Let the time evolution of the eigenfunction of the spectral problem (7) obey the differential equation

(𝐸 − 1) tr (Γ𝑛2 ) = (𝐸 − 1) (𝑎𝑛2 + 𝑏𝑛 𝑐𝑛 ) = 0. 𝑚

𝑚

𝑗=1

𝑗=1

𝑎𝑛(𝑚+1) = ∑𝑎𝑛(𝑗) 𝑎𝑛(𝑚+1−𝑗) + ∑ 𝑏𝑛(𝑗) 𝑐𝑛(𝑚+1−𝑗)

𝜑𝑛,𝑡𝑚 = Ω(𝑚) 𝑛 𝜑𝑛 , 𝑚 ≥ 0. (16)

Then the compatibility conditions of (7) and (22) are (𝑚) (𝑈𝑛 )𝑡𝑚 = (𝐸Ω(𝑚) 𝑛 ) 𝑈𝑛 − 𝑈𝑛 Ω𝑛 , 𝑚 ≥ 0

+ 𝛾 (𝑡) . 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.

𝑎𝑛(1) 𝑏𝑛(1)

1

=

𝑐𝑛(1) = 𝑎𝑛(2)

𝑠𝑛−1

(𝑚+1) − 𝑟𝑛 𝑎𝑛(𝑚) , 𝑟𝑛,𝑡𝑚 = −𝑟𝑛 𝑠𝑛 𝑏𝑛+1 (𝑚) 𝑠𝑛,𝑡𝑚 = 𝑟𝑛 𝑠𝑛 𝑐𝑛(𝑚+1) + 𝑠𝑛 𝑎𝑛+1 .

When 𝑚 = 0, (24) becomes a trivial linear system 𝑟𝑛,𝑡0 = −

,

V𝑛,𝑡0 =

1 1 1 =− 2 2 − − , 𝑟𝑛 𝑠𝑛−1 𝑟𝑛+1 𝑟𝑛 𝑠𝑛 𝑠𝑛−1 𝑟𝑛−1 𝑟𝑛 𝑠𝑛−1 𝑠𝑛−2

𝑐𝑛(2) = −

1 𝑟𝑛−1 𝑠𝑛−1 𝑠𝑛−2

−

(17)

1 , 2 𝑟𝑛 𝑠𝑛−1

𝑠𝑛,𝑡1

Let us denote 𝑚

∑𝑎𝑛(𝑖) 𝜆2𝑚−2𝑖 ∑𝑏𝑛(𝑖) 𝜆2𝑚−2𝑖+1

𝑖=0 ) , 𝑚 ≥ 0. (18) 𝑉𝑛(𝑚) = ( 𝑚𝑖=0 𝑚 (𝑖) 2𝑚−2𝑖+1 −∑𝑎𝑛(𝑖) 𝜆2𝑚−2𝑖 ∑𝑐𝑛 𝜆 𝑖=0

𝑖=0

(𝐸𝑉𝑛(𝑚) ) 𝑈𝑛 − 𝑈𝑛 𝑉𝑛(𝑚) 0 =( 𝑟𝑛 𝑠𝑛 𝑐𝑛(𝑚+1) 𝜆

(19)

It is obvious that (19) is not compatible with (𝑈𝑛 )𝑡𝑚 . Therefore, we choose the following modification term: Δ(𝑚) 𝑛 =(

0

0

0

𝑎𝑛(𝑚)

).

(20)

𝑟𝑛

𝑟𝑛+1 𝑠𝑛

(21)

(25)

,

𝑠 =− 𝑛 . 𝑟𝑛 𝑠𝑛−1

(26)

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 𝐽󸀠 (𝑦𝑛 ) [V𝑛 ] =

𝜕 󵄨 𝐽 (𝑦𝑛 + 𝜀V𝑛 )󵄨󵄨󵄨𝜀=0 , 𝜕𝜀 (27)

⟨𝑓𝑛 , 𝑔𝑛 ⟩ = ∑ (𝑓𝑛 , 𝑔𝑛 )𝑅2 . 𝑛∈𝑍

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 skewsymmetric. An operator 𝐽 is called a Hamiltonian operator, if 𝐽 is a skew-symmetric operator satisfying the Jacobi identity, i.e., ⟨𝑓𝑛 , 𝐽𝑔𝑛 ⟩ = − ⟨𝐽𝑓𝑛 , 𝑔𝑛 ⟩ ,

Then we introduce auxiliary matrix spectral problem (𝑚) + Δ(𝑚) Ω(𝑚) 𝑛 = 𝑉𝑛 𝑛 , 𝑚 ≥ 1.

𝑠𝑛 . 2

𝛿𝐻𝑛 𝜕𝐻𝑛 = ∑ 𝐸(−𝑚) ( ), 𝛿𝑦𝑛 𝑚∈𝑍 𝜕𝑦𝑛+𝑚

On the basis of the recursion relations (14), we have

(𝑚+1) −𝑟𝑛 𝑠𝑛 𝑏𝑛+1 𝜆 ). (𝑚) 𝑟𝑛 𝑠𝑛 (𝑎𝑛+1 − 𝑎𝑛(𝑚) )

𝑟𝑛 , 2

When 𝑚 = 1 in (24), we obtain the first nontrivial integrable lattice equation 𝑟𝑛,𝑡1 =

1 1 − ,⋅⋅⋅ 𝑟𝑛2 𝑠𝑛−1 𝑟𝑛 𝑟𝑛+1 𝑠𝑛

𝑚

(24) 𝑚 ≥ 0.

1 , 𝑟𝑛

𝑏𝑛(2) = −

(23)

It implies the family of integrable (in Lax sense) differentdifference equations.

The first few terms are given by 1 = , 𝑟𝑛 𝑠𝑛−1

(22)

⟨𝐽󸀠 (𝑢𝑛 ) [𝐽𝑓𝑛 ] 𝑔𝑛 , ℎ𝑛 ⟩ + 𝐶𝑦𝑐𝑙𝑒 (𝑓𝑛 , 𝑔𝑛 , ℎ𝑛 ) = 0.

(28)

4


Based on a given Hamiltonian operator 𝐽, we can define the corresponding Poisson bracket [10, 11, 18] {𝑓𝑛 , 𝑔𝑛 }𝐽 = ⟨

𝛿𝑓𝑛 𝛿𝑔𝑛 𝛿𝑓 𝛿𝑔 ⟩ = ∑ ⟨ 𝑛,𝐽 𝑛⟩ . ,𝐽 𝛿𝑦𝑛 𝛿𝑦𝑛 𝛿𝑦 𝛿𝑦𝑛 𝑅2 𝑛 𝑛∈𝑍

̃(𝑚) = ∑ (− 𝐻 𝑛 𝑛∈𝑍

(29)

𝑟𝑛 𝑐𝑛(𝑚+1) ). 2𝑚

(36)

We have

Following [18], we set

𝑅𝑛 = Γ𝑛 (𝑈𝑛 )

Set

(𝑚)

−1

𝑎𝑛 𝑏 −𝑎𝑛 + 𝑛 𝑟𝑛 𝜆 𝑠𝑛 𝜆 =( ) 𝑎𝑛 𝑐𝑛 −𝑐𝑛 − 𝑟𝑛 𝜆 𝑠𝑛 𝜆

(30)

(31)

0 0 𝜕𝑈𝑛 =( ). 𝜕𝑠𝑛 𝜆 𝑟𝑛 𝜆2

(𝑚+1) −𝑟𝑛 𝑠𝑛 𝑏𝑛+1

− 𝑟𝑛 𝑎𝑛(𝑚) ) (𝑚) 𝑟𝑛 𝑠𝑛 𝑐𝑛(𝑚+1) + 𝑠𝑛 𝑎𝑛+1

𝑎𝑛(𝑚) 𝑟𝑛 = 𝐽 ( (𝑚) ) 𝑎𝑛+1 − 𝑠𝑛 −

𝑚 ≥ 1.

(38)

We can obtain 0 −𝑟𝑛 𝑠𝑛 𝐽=( ). 𝑟𝑛 𝑠𝑛 0

(39)

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.

Hence 𝜕𝑈𝑛 ⟩ = 𝑟𝑛 𝑐𝑛 , 𝜕𝜆

𝜕𝑈 𝑎 ⟨𝑅𝑛 , 𝑛 ⟩ = − 𝑛 , 𝜕𝑟𝑛 𝑟𝑛

(32)

𝑎 𝜕𝑈 ⟨𝑅𝑛 , 𝑛 ⟩ = − 𝑛+1 . 𝜕𝑠𝑛 𝑠𝑛

Proposition 2. 𝐽 is a discrete Hamiltonian operator. Consequently, (22) have Hamiltonian structures 𝑟𝑛

𝑦𝑛𝑡 = ( ) 𝑚 𝑠𝑛 𝑡

𝑚

By virtue of the discrete trace identity [18] 𝜕𝑈 𝜕𝑈 𝜕 𝛿 ∑ ⟨𝑅𝑛 , 𝑛 ⟩ = 𝜆−𝜀 𝜆𝜀 ⟨𝑅𝑛 , 𝑛 ⟩ , 𝛿𝑟𝑛 𝑛∈𝑍 𝜕𝜆 𝜕𝜆 𝜕𝑟𝑛 𝜕𝑈 𝜕𝑈 𝜕 𝛿 ∑ ⟨𝑅𝑛 , 𝑛 ⟩ = 𝜆−𝜀 𝜆𝜀 ⟨𝑅𝑛 , 𝑛 ⟩ . 𝛿𝑠𝑛 𝑛∈𝑍 𝜕𝜆 𝜕𝜆 𝜕𝑠𝑛

𝑎(𝑚) − 𝑛 (𝑚) ̃ 𝛿𝐻 𝑟𝑛 = 𝐽 𝑛 = 𝐽 ( (𝑚) ), 𝑎𝑛+1 𝛿𝑦𝑛 − 𝑠𝑛

(33)

̃(𝑚) = ∑ (𝐻(𝑚) ) , 𝐻 𝑛 𝑛 𝑛∈𝑍

𝐻𝑛(𝑚) = (

𝑟𝑛 𝑐𝑛(𝑚+1) ), −2𝑚

In particular, the different-difference equation (23) possesses the Hamiltonian structure (34)

When 𝑚 = 0 in (34), through a direct calculation, we find that 𝜀 = 0. Therefore, (34) can be written as (𝑚)

𝑎 𝛿 − 𝑛 𝑟𝑛 𝑐𝑛(𝑚+1) 𝑟𝑛 𝛿𝑟𝑛 ) ( 𝛿 ) ∑ (− ) = ( (𝑚) 𝑎 2𝑚 𝑛+1 𝑛∈𝑍 − 𝛿𝑠𝑛 𝑠 𝑛

𝑚 ≥ 1.

(41) 𝑚 ≥ 1.

in (33), we obtain

𝑎(𝑚) 𝛿 − 𝑛 𝑟𝑛 𝛿𝑟 ). ( 𝛿𝑛 ) ∑ (𝑟𝑛 𝑐𝑛(𝑚+1) ) = (𝜀 − 2𝑚) ( (𝑚) −𝑎𝑛+1 𝑛∈𝑍 𝛿𝑠𝑛 𝑠𝑛

(40) 𝑚 ≥ 0.

(𝑚) −2𝑚 Substituting expansions 𝑎𝑛 = ∑∞ , 𝑏𝑛 = 𝑚=0 𝑎𝑛 𝜆 ∞ ∞ (𝑚) −2𝑚+1 (𝑚) −2𝑚+11 ∑𝑚=0 𝑏𝑛 𝜆 , 𝑐𝑛 = ∑𝑚=0 𝑐𝑛 𝜆 into (33) and com−2𝑚−1

paring the coefficients of 𝜆

(37)

Set

(

0 𝑟𝑛 𝜕𝑈𝑛 ), =( 𝜕𝜆 𝑠𝑛 2𝑟𝑛 𝑠𝑛 𝜆 0 𝜆 𝜕𝑈𝑛 =( ), 𝜕𝑟𝑛 0 𝑠𝑛 𝜆2

𝑚 ≥ 1.

𝑛

and ⟨𝑀, 𝑁⟩ = tr(𝑀𝑁), where 𝑀 and 𝑁 are some order square matrices. It is easy to calculate that

⟨𝑅𝑛 ,

𝑎 𝛿 − 𝑛 𝛿𝑟𝑛 (𝑚) 𝑛 ) ̃ ) = ( 𝑟(𝑚) ( 𝛿 ) ∑ (𝐻 𝑛 𝑎𝑛+1 𝑛∈𝑍 − 𝛿𝑠𝑛 𝑠

̃(1) 𝑟𝑛 𝛿𝐻 𝑦𝑛,𝑡 = ( ) = 𝐽 𝑛 , 𝛿𝑢𝑛 𝑠𝑛 𝑡 ̃(1) 𝐻 𝑛

1 1 = ∑( + ). 2𝑟𝑛+1 𝑠𝑛 𝑛∈𝑍 2𝑟𝑛 𝑠𝑛−1

(42)

Following (14) and (35), we have the following recursion relation: (35)

̃(𝑚+1) ̃(𝑚) 𝛿𝐻 𝛿𝐻 𝑛 = Ψ 𝑛 , 𝑚 ≥ 1, 𝛿𝑢𝑛 𝛿𝑢𝑛

(43)


5 With the help of the operator Ψ, (40) may be written in the following form:

where Ψ11 Ψ12 ). Ψ=( Ψ21 Ψ12

̃(𝑚) ̃(𝑚−1) 𝑟𝑛 𝛿𝐻 𝛿𝐻 𝑦𝑛,𝑡𝑚 = ( ) = 𝐽 𝑛 = 𝑀 𝑛 𝛿𝑢𝑛 𝛿𝑢𝑛 𝑠𝑛 𝑡

(44)

𝑚

Moreover, we have 𝑀11 𝑀12 ) 𝐽Ψ = 𝑀 = ( 𝑀21 𝑀22

̃(0) 𝛿𝐻 = 𝐽Ψ𝑚 𝑛 , 𝑚 ≥ 1. 𝛿𝑢𝑛

(45)

Furthermore, on the basis of [4, 16], we may obtain a recursion operator

with −1

𝑀11 = −𝑟𝑛 (1 − 𝐸−1 )

−1

− 𝐸−1

+ 𝑟𝑛 (1 − 𝐸)

+ 𝑟𝑛 (1 − 𝐸)−1 − 𝑟𝑛 (1 − 𝐸)−1

1 1 𝐸 (1 − 𝐸)−1 𝑠𝑛 , 𝑠𝑛 𝑟𝑛

−

−1 1 −1 𝑟𝑛 𝑠𝑛 −1 𝑟𝑛 𝐸 − 𝑠𝑛 (1 − 𝐸−1 ) 𝐸 𝑟𝑛 𝑠𝑛 𝑟𝑛 𝑠𝑛

(𝐽Ψ)∗ = 𝑀∗ = −𝑀 = −𝐽Ψ,

(50)

Ψ∗ 𝐽 = 𝐽Ψ

(51)

namely,

Therefore (46) ̃(𝑙) } = ⟨ ̃(𝑚) , 𝐻 {𝐻 𝑛 𝑛 𝐽

−1 −1 1 1 (1 − 𝐸−1 ) 𝑟𝑛 + 𝑠𝑛 (1 − 𝐸−1 ) 𝐸 𝑟𝑛 𝑠𝑛

̃(𝑚) 𝛿𝐻 ̃(𝑙) 𝛿𝐻 𝑛 ,𝐽 𝑛 ⟩ 𝛿𝑢𝑛 𝛿𝑢𝑛

𝑠𝑛 −1 1 𝐸 (1 − 𝐸 ) 𝑟𝑛 𝑟𝑛 𝑠𝑛

= ⟨Ψ𝑚−1

̃(1) ̃(1) 𝛿𝐻 𝛿𝐻 𝑛 , 𝐽Ψ𝑙−1 𝑛 ⟩ 𝛿𝑢𝑛 𝛿𝑢𝑛

1 −1 𝑟𝑛 𝐸 𝑟𝑛 𝑠𝑛

= ⟨Ψ𝑚−1

̃(1) ∗ 𝑙−2 𝛿𝐻 ̃(1) 𝛿𝐻 𝑛 𝑛 , Ψ 𝐽Ψ ⟩ 𝛿𝑢𝑛 𝛿𝑢𝑛

−1 −1

−1 −1

− 𝑠𝑛 (1 − 𝐸 ) − 𝑠𝑛 (1 − 𝐸−1 ) −

Proof. Due to (47), we know that 𝑀 is a skew-symmetric operator, and a direct calculation shows

1 𝑟𝑛 𝐸 𝑠𝑛 𝑠𝑛 1 −1 1 𝐸 (1 − 𝐸)−1 𝑟𝑛 𝑠𝑛

−𝐸

̃(𝑚) }∞ are conserved functionals of the Proposition 3. {𝐻 𝑚=0 𝑛 whole family (24) or (40). And they are in involution in pairs with respect to the Poisson bracket (29).

𝑟 1 1 (1 − 𝐸)−1 𝑠𝑛 + 𝑛 𝐸 (1 − 𝐸)−1 𝑠𝑛 𝑠𝑛 𝑠𝑛 𝑟𝑛 −1

−1

−1 1 −1 1 𝐸 (1 − 𝐸−1 ) 𝑟𝑛 𝑠𝑛 𝑟𝑛

= ⟨Ψ𝑚

−1 𝑠𝑛 −1 1 𝐸 (1 − 𝐸−1 ) 𝑟𝑛 , 𝑟𝑛 𝑠𝑛

𝑀22 = 𝑠𝑛 (1 − 𝐸)

−1

− 𝑠𝑛 (1 − 𝐸)

̃(1) ̃(1) 𝛿𝐻 𝛿𝐻 𝑛 , 𝐽Ψ𝑙−2 𝑛 ⟩ 𝛿𝑢𝑛 𝛿𝑢𝑛

̃(𝑙) } = {𝐻 ̃(𝑚+𝑙−1) , 𝐻 ̃(1) } . ̃(𝑚) , 𝐻 = {𝐻 𝑛 𝑛 𝐽 𝑛 𝑛 𝐽 Repeating the above argument, we can obtain

1 1 𝐸 (1 − 𝐸)−1 𝑟𝑛 . 𝑠𝑛 𝑟𝑛

̃(𝑙) , 𝐻 ̃(𝑚) } = {𝐻 ̃(𝑚+𝑙−1) , 𝐻 ̃(1) } . {𝐻 𝑛 𝑛 𝑛 𝑛 𝐽 𝐽

It is easy to verify that 𝑀 is a skew-symmetric operator. Namely, 𝑀∗ = −𝑀.

(52)

̃(𝑚+1) , 𝐻 ̃(𝑙−1) } = ⋅ ⋅ ⋅ = {𝐻 𝑛 𝑛 𝐽

−1 1 −1 1 𝐸 (1 − 𝐸−1 ) 𝑠𝑛 𝑟𝑛 𝑠𝑛 −1

(49)

Next, we prove Liouville integrability of the discrete Hamiltonian systems (40). It is crucial to show the existence of infinite involutive conserved functionals.

1 −1 1 𝐸 (1 − 𝐸)−1 𝑟𝑛 , 𝑟𝑛 𝑠𝑛

𝑟𝑛 𝑠𝑛 1 𝐸 − 𝑟𝑛 (1 − 𝐸)−1 𝐸−1 𝑠𝑛 𝑟𝑛 𝑟𝑛

𝑀12 = 𝐸−1 −

𝑀21 = −𝐸 −

∗ ∗ Ψ21 Ψ11 ). Φ = Ψ∗ = ( ∗ ∗ Ψ12 Ψ22

1 1 𝐸 (1 − 𝐸)−1 𝑟𝑛 𝑠𝑛 𝑟𝑛

+ 𝑟𝑛 (1 − 𝐸−1 )

(48)

(47)

(53)

Then combining (52) with (53) leads to ̃(𝑚) , 𝐻 ̃(𝑙) } = 0, {𝐻 𝑛 𝑛 𝐽

𝑚, 𝑙 ≥ 1,

(54)

6


and ̃(𝑚) ) = ⟨ (𝐻 𝑛 𝑡 𝑙

̃(𝑚) ̃(𝑚) 𝛿𝐻 ̃(𝑙) 𝛿𝐻 𝛿𝐻 𝑛 , 𝑢𝑡𝑙 ⟩ = ⟨ 𝑛 , 𝐽 𝑛 ⟩ 𝛿𝑢𝑛 𝛿𝑢𝑛 𝛿𝑢𝑛

̃(𝑙) } = 0, ̃(𝑚) , 𝐻 = {𝐻 𝑛 𝑛 𝐽

problem into another spectral problem of the same form [5, 6, 8, 23]. We introduce the following gauge transformation: ̃𝑛 = 𝑇𝑛 𝜑𝑛 , 𝜑

(55)

which can transform two spectral problems (7) and (56) into

𝑚, 𝑙 ≥ 1.

̃𝑛 𝜑̃𝑛 , 𝐸̃ 𝜑𝑛 = 𝜑̃𝑛+1 = 𝑈

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.

(59)

with ̃𝑛 = 𝑇𝑛+1 𝑈𝑛 (𝑇𝑛 )−1 , 𝑈

(60)

̃ (1) = (Ω(1) + 𝑇𝑛 Ω(1) ) (𝑇𝑛 )−1 . Ω 𝑛 𝑛𝑡 𝑛 We suppose that 𝑇𝑛 has the following form: 𝑡11 [𝑛] 𝑡12 [𝑛] 𝜆 ( 1). 𝑡21 [𝑛] 𝑡22 [𝑛] 𝜆 + 𝜆 𝜆+

Next we are going to establish a Darboux-Bäcklund transformation of (26). When 𝑚 = 1 in (23), we obtain the time part of the Lax pair of (26) 1 𝜆2 1 𝜆 + 2 𝑟𝑛 𝑠𝑛−1 𝑠𝑛−1 ) 𝜑𝑛 . = Ω(1) 𝜑 = ( 𝑛 𝑛 1 𝜆2 𝜆 𝑟𝑛 2

(58)

̃ (1) 𝜑̃𝑛 , 𝜑̃𝑛𝑡 = Ω 𝑛

3. Darboux-Bäcklund Transformation and Exact Solution

−

𝜑𝑛,𝑡

(57)

(56)

It is crucial to look for a suitable gauge transformation of a matrix spectral problem; it can transform the matrix spectral

(61)

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 ̃𝑛 and Ω ̃ (1) in (60) have the same form with 𝑈𝑛 𝑇𝑛 such that 𝑈 𝑛 (1) and Ω𝑛 , respectively. Let 𝜙𝑛 = (𝜙𝑛1 , 𝜙𝑛2 )𝑇 , 𝜓𝑛 = (𝜓𝑛1 , 𝜓𝑛2 )𝑇 be two real linear independent solutions of (7) and (56). Let 𝜙𝑛1 𝜓𝑛1 𝐾𝑛 = ( 2 2 ) . 𝜙𝑛 𝜓𝑛

(62)

̃𝑛 = 𝑇𝑛 𝐾𝑛 , we obtain that From equation 𝐾

𝑡11 [𝑛] 1 𝑡 [𝑛] ) 𝜙𝑛 + 𝑡12 [𝑛] 𝜙𝑛2 (𝜆 + 11 ) 𝜓𝑛1 + 𝑡12 [𝑛] 𝜓𝑛2 𝜆 𝜆 ̃ 𝐾𝑛 = ( ). 1 1 𝑡21 [𝑛] 𝜙𝑛1 + (𝑡22 [𝑛] 𝜆 + ) 𝜙𝑛2 𝑡21 [𝑛] 𝜓𝑛1 + (𝑡22 [𝑛] 𝜆 + ) 𝜓𝑛2 𝜆 𝜆 (𝜆 +

We assume that ±𝜆 1 and ±𝜆 2 are four roots of det(𝑇𝑛 ). When 𝜆 = ±𝜆 𝑖 (𝑖 = 1, 2), two columns of (63) are linear dependent. Without loss of generality, there exist two nonzero constants 𝜅1 and 𝜅2 , which satisfy (𝜆 +

𝑡11 [𝑛] 1 ) 𝜙𝑛 + 𝑡12 [𝑛] 𝜙𝑛2 𝜆

= 𝜅𝑖 ((𝜆 + 𝑡21 [𝑛] 𝜙𝑛1

1 + (𝑡22 [𝑛] 𝜆 + ) 𝜙𝑛2 𝜆

where 𝜆 = ±𝜆 𝑖 (𝑖 = 1, 2).

Set 𝛼𝑖 [𝑛] =

𝜙𝑛2 − 𝜅𝑖 𝜓𝑛2 , 𝑖 = 1, 2. 𝜙𝑛1 − 𝜅𝑖 𝜓𝑛1

(65)

Solving (64), we have

𝑡11 [𝑛] ) 𝜓𝑛1 + 𝑡12 [𝑛] 𝜓𝑛2 ) , 𝜆

= 𝜅𝑖 (𝑡21 [𝑛] 𝜓𝑛1 + (𝑡22 [𝑛] 𝜆 +

(63)

1 ) 𝜓2 ) 𝜆 𝑛

(64)

𝑡11 [𝑛] =

𝜆 1 𝜆 2 (𝜆 1 𝛼2 [𝑛] − 𝜆 2 𝛼1 [𝑛]) , 𝜆 1 𝛼1 [𝑛] − 𝜆 2 𝛼2 [𝑛]

𝑡12 [𝑛] =

𝜆22 − 𝜆21 , 𝜆 1 𝛼1 [𝑛] − 𝜆 2 𝛼2 [𝑛]

𝑡21 [𝑛] =

(𝜆22 − 𝜆21 ) 𝛼1 [𝑛] 𝛼2 [𝑛] 𝜆 1 𝜆 2 (𝜆 1 𝛼1 [𝑛] − 𝜆 2 𝛼2 [𝑛])

,

Discrete Dynamics in Nature and Society 𝑡22 [𝑛] =

7

𝜆 1 𝛼2 [𝑛] − 𝜆 2 𝛼1 [𝑛] . 𝜆 1 𝜆 2 (𝜆 1 𝜎1 [𝑛] − 𝜆 2 𝜎2 [𝑛])

with (66)

In (66), 𝜆 𝑖 , 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 ±𝜆 𝑖 (𝑖 = 1, 2) are four roots of det(𝑇𝑛 ) = 0, and then det (𝑇𝑛 ) = 𝑡22 [𝑛]

(𝜆2 − 𝜆21 ) (𝜆2 − 𝜆22 ) 𝜆2

𝑡22 [𝑛 + 1] =

ℎ12 (𝜆, 𝑛) = (𝑟𝑛 + 𝑟𝑛 𝑠𝑛 𝑡12 [𝑛 + 1]) 𝜆3 + (𝑟𝑛 𝑡11 [𝑛]

(67)

.

𝜉𝑖 (𝑛) 𝜒𝑖 (𝑛)

(𝑖 = 1, 2) ,

(68)

where 𝜉𝑖 (𝑛) = 𝜆 𝑖 𝑠𝑛 + (𝑟𝑛 𝑠𝑛 𝜆2𝑖 )𝛼𝑖 [𝑛], 𝜒𝑖 (𝑛) = 𝑟𝑛 𝜆 𝑖 𝛼𝑖 [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].

𝑡21 [𝑛 + 1] =

+ 𝑠𝑛 𝑡12 [𝑛 + 1] 𝑡22 [𝑛]) 𝜆2 ,

− 𝑠𝑛 𝑡12 [𝑛] 𝑡12 [𝑛 + 1]) 𝜆, +

𝛼𝑖 [𝑛 + 1] =

𝑡12 [𝑛 + 1] =

+ (−𝑟𝑛 𝑡21 [𝑛] − 𝑟𝑛 𝑠𝑛 𝑡12 [𝑛 + 1] 𝑡2 1 [𝑛]

+ 𝑟𝑛 𝑡11 [𝑛 + 1] + 𝑟𝑛 𝑠𝑛 𝑡11 [𝑛] 𝑡12 [𝑛 + 1]

From (65), we get

𝑡11 [𝑛 + 1] =

ℎ11 (𝜆, 𝑛) = 𝑠𝑛 𝑡12 [𝑛 + 1] − 𝑟𝑛 𝑡11 [𝑛 + 1] 𝑡21 [𝑛]

𝜆 1 𝜉1 (𝑛) 𝜒2 (𝑛) − 𝜆 2 𝜉2 (𝑛) 𝜒1 (𝑛)

+ 𝑠𝑛 𝑡22 [𝑛] 𝑡22 [𝑛 + 1]) 𝜆3 + (−𝑟𝑛 𝑠𝑛 𝑡21 [𝑛] − 𝑟𝑛 𝑡21 [𝑛] 𝑡21 [𝑛 + 1] + 𝑠𝑛 𝑡22 [𝑛] + 𝑠𝑛 𝑡22 [𝑛 + 1]) 𝜆 +

𝜆 1 𝜆 2 (𝜆 1 𝜉1 (𝑛) 𝜒2 (𝑛) − 𝜆 2 𝜉2 (𝑛) 𝜒1 (𝑛))

ℎ22 (𝜆, 𝑛) = 𝑟𝑛 𝑠𝑛 𝑡22 [𝑛 + 1] 𝜆4 + (𝑟𝑛 𝑠𝑛 + 𝑟𝑛 𝑡21 [𝑛 + 1]

+ 𝑟𝑛 𝑠𝑛 𝑡11 [𝑛] − 𝑠𝑛 𝑡12 [𝑛] + 𝑟𝑛 𝑡11 [𝑛] 𝑡21 [𝑛] . According to (66) and (69), we can find that (69)

𝑠𝑛 𝑡12 [𝑛 + 1] − 𝑟𝑛 𝑡11 [𝑛 + 1] 𝑡21 [𝑛] = 0,

,

− 𝑟𝑛 𝑡21 [𝑛] − 𝑟𝑛 𝑠𝑛 𝑡12 [𝑛 + 1] 𝑡2 1 [𝑛]

𝜆 1 𝜉2 (𝑛) 𝜒1 (𝑛) − 𝜆 2 𝜉1 (𝑛) 𝜒2 (𝑛) . 𝜆 1 𝜆 2 (𝜆 1 𝜉1 (𝑛) 𝜒2 (𝑛) − 𝜆 2 𝜉2 (𝑛) 𝜒1 (𝑛))

(70)

Thus, we have ℎ11 (𝜆, 𝑛) = 0. Moreover, it is easy to see that 𝜆2 ℎ12 (𝜆, 𝑛) and 𝜆2 ℎ21 (𝜆, 𝑛) are five-order polynomial in 𝜆, and 𝜆2 ℎ22 (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 ℎ12 (±𝜆 𝑖 ) , ℎ21 (±𝜆 𝑖 ) ,

where the relations between old potentials 𝑟𝑛 and 𝑠𝑛 and new potentials 𝑟̃𝑛 and 𝑠̃𝑛 are given by 𝑟̃𝑛 =

𝑟𝑛 , 𝑡22 [𝑛] − 𝑟𝑛 𝑡21 [𝑛]

𝑠̃𝑛 =

𝑠𝑛 . 𝑡11 [𝑛]

(75)

ℎ22 (±𝜆 𝑖 ) (𝑖 = 1, 2)

(71)

are all equal zero. Therefore, we have 𝑇𝑛+1 𝑈𝑛 𝑇𝑛∗ = det (𝑇𝑛 ) 𝜏𝑛 ,

Proof. One has 𝑇𝑛+1 𝑈𝑛 𝑇𝑛∗

(74)

+ 𝑠𝑛 𝑡12 [𝑛 + 1] 𝑡22 [𝑛] = 0.

̃𝑛 defined by (60) has the same Proposition 6. The matrix 𝑈 form as 𝑈𝑛 , that is, 0 𝑟̃𝑛 𝜆 ̃𝑛 = ( 𝑈 ), 𝑠̃𝑛 𝜆 𝑟̃𝑛 𝑠̃𝑛 𝜆2

𝑠𝑛 , 𝜆

+ 𝑟𝑛 𝑠𝑛 𝑡11 [𝑛] 𝑡22 [𝑛 + 1] − 𝑠𝑛 𝑡12 [𝑛] 𝑡22 [𝑛 + 1]) 𝜆2

,

(𝜆21 − 𝜆22 ) 𝜉1 (𝑛) 𝜉2 (𝑛)

(73)

ℎ21 (𝜆, 𝑛) = (−𝑟𝑛 𝑠𝑛 𝑡21 [𝑛] 𝑡22 [𝑛 + 1]

𝜆 1 𝜆 2 (𝜆 1 𝜉2 (𝑛) 𝜒1 (𝑛) − 𝜆 2 𝜉1 (𝑛) 𝜒2 (𝑛)) , 𝜆 1 𝜉 (𝑛) 𝜒2 (𝑛) − 𝜆 2 𝜉2 (𝑛) 𝜒1 (𝑛) (𝜆22 − 𝜆21 ) 𝜒1 (𝑛) 𝑐ℎ𝑖2 (𝑛)

𝑟𝑛 𝑡11 [𝑛] 𝑡11 [𝑛 + 1] , 𝜆

(76)

with ℎ11 (𝜆, 𝑛) ℎ12 (𝜆, 𝑛) ), =( ℎ21 (𝜆, 𝑛) ℎ22 (𝜆, 𝑛)

(72)

𝜏𝑛 = (

0

(1) 𝜆 𝜏12

(1) (2) 2 (0) 𝜏21 𝜆 𝜏22 𝜆 + 𝜏22

)

(77)

8


where 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

𝑇𝑛+1 𝑈𝑛 = 𝜏𝑛 𝑇𝑛 .

(78)

The proof is completed. ̃ (1) defined by (60) has the same Proposition 7. The matrix Ω 𝑛 form as Ω(1) 𝑛 in (56) under the transformation (71), i.e.,

Equating the coefficients of 𝜆𝑖 (𝑖 = 0, 1, 2) in (78), we have (1) = 𝜏12

𝑟𝑛 , 𝑡22 [𝑛] − 𝑟𝑛 𝑡21 [𝑛]

(1) 𝜏21 =

𝑠𝑛 , 𝑡11 [𝑛]

(2) 𝜏22 =

̃ (1) = ( Ω 𝑛

𝑠 𝑟𝑛 ( 𝑛 ), 𝑡22 [𝑛] − 𝑟𝑛 𝑡21 [𝑛] 𝑡11 [𝑛]

(79)

−

𝜆 𝜆2 1 + 𝑟̃𝑛 𝑠̃𝑛−1 𝑠̃𝑛−1 2 𝜆2 2

𝜆 𝑟̃𝑛

).

Proof. Let 𝜂11 (𝜆, 𝑛) 𝜂12 (𝜆, 𝑛) ∗ ). (𝑇𝑛,𝑡 + 𝑇𝑛 Ω(1) 𝑛 ) 𝑇𝑛 = ( 𝜂21 (𝜆, 𝑛) 𝜂22 (𝜆, 𝑛)

(0) 𝜏22 = 0.

(80)

(81)

Here

𝑡 [𝑛] 𝑡22 [𝑛] 2 1 + 𝑡11 [𝑛] 𝑡22 [𝑛] 1 1 𝑡 [𝑛] 1 𝜂11 (𝜆, 𝑛) = − 𝜆4 𝑡22 [𝑛] + (− − 21 − (𝑡12 [𝑛] 𝑡21 [𝑛] + 𝑡11 [𝑛] 𝑡22 [𝑛]) + 12 )𝜆 + 2 2 𝑠𝑛−1 2 𝑟𝑛 𝑟𝑛 𝑠𝑛−1 − 𝜂12 (𝜆, 𝑛) = (

𝑡11 [𝑛] + 𝑟𝑛 𝑠𝑛−1 (𝑡11 [𝑛])𝑡 𝑡11 [𝑛] 𝑡12 [𝑛] 𝑡12 [𝑛] 𝑡21 [𝑛] − + (𝑡11 [𝑛])𝑡 𝑡22 [𝑛] − 𝑡21 [𝑛] (𝑡12 [𝑛])𝑡 + , + 2 𝑟𝑛 𝑠𝑛−1 𝑟𝑛 𝑠𝑛−1 𝜆2 1

𝑠𝑛−1

2

2𝑡 [𝑛] 𝑡12 [𝑛] (𝑡12 [𝑛]) + 𝑡12 [𝑛]) 𝜆 + ( 11 − − + 𝑡11 [𝑛] 𝑡12 [𝑛] + (𝑡12 [𝑛])𝑡 ) 𝜆 𝑠𝑛−1 𝑟𝑛 𝑠𝑛−1 𝑟𝑛 3

2

+

𝑟𝑛 (𝑡11 [𝑛]) − 𝑡11 [𝑛] 𝑡12 [𝑛] − 𝑟𝑛 𝑠𝑛−1 (𝑡12 [𝑛] (𝑡11 [𝑛])𝑡 − 𝑡11 [𝑛] (𝑡12 [𝑛])𝑡 ) , 𝑟𝑛 𝑠𝑛−1 𝜆 2

𝜂21 (𝜆, 𝑛) = (−𝑡21 [𝑛] 𝑡22 [𝑛] +

(𝑡22 [𝑛]) ) 𝜆3 𝑟𝑛

(82)

2

(𝑡 [𝑛]) 2𝑡 [𝑛] 𝑡21 [𝑛] 𝑡22 [𝑛] + (−𝑡21 [𝑛] − 21 + 22 + + 𝑡22 [𝑛] (𝑡21 [𝑛])𝑡 − 𝑡21 [𝑛] (𝑡22 [𝑛])𝑡 ) 𝜆 𝑠𝑛−1 𝑟𝑛 𝑟𝑛 𝑠𝑛−1 +

𝑠𝑛−1 + 𝑡21 [𝑛] + 𝑟𝑛 𝑠𝑛−1 (𝑡22 [𝑛])𝑡 , 𝑟𝑛 𝑠𝑛−1 𝜆

𝑡 [𝑛] 𝑡12 [𝑛] 1 1 𝑡 [𝑛] 𝑡12 [𝑛] 𝑡21 + 𝑡11 [𝑛] 𝑡22 [𝑛] 𝑡12 [𝑛] 𝑡22 [𝑛] + + (𝑡22 [𝑛])𝑡 ) 𝜆2 + 11 𝜂22 (𝜆, 𝑛) = 𝑡22 [𝑛] 𝜆4 + ( + 21 − − 2 2 𝑠𝑛−1 2 𝑟𝑛 2 𝑟𝑛 +

𝑡11 [𝑛] 𝑡21 [𝑛] 𝑡12 [𝑛] 𝑡21 [𝑛] − − 𝑡12 [𝑛] (𝑡21 [𝑛])𝑡 + 𝑡11 [𝑛] (𝑡22 [𝑛])𝑡 . 𝑠𝑛−1 𝑟𝑛 𝑠𝑛−1

It is easy to get that 𝜆2 𝜂11 (𝜆, 𝑛) and 𝜆2 𝜂22 (𝜆, 𝑛) are six-order polynomials in 𝜆; 𝜆2 𝜂12 (𝜆, 𝑛) and 𝜆2 𝜂21 (𝜆, 𝑛) are five-order polynomials in 𝜆. According to (56) and (65), we have 𝜆 𝜆 1 𝛼𝑖,𝑡 [𝑛] = 𝑖 + (𝜆2𝑖 − ) 𝛼 [𝑛] − 𝑖 𝛼𝑖 [𝑛]2 , 𝑟𝑛 𝑟𝑛 𝑠𝑛−1 𝑖 𝑠𝑛−1

(83)

Therefore, the following equation is derived: ∗ (𝑇𝑛𝑡 + 𝑇𝑛 Ω(1) 𝑛 ) 𝑇𝑛 = det (𝑇𝑛 ) 𝜇𝑛 ,

with

𝑖 = 1, 2. Through lengthy but direct calculation, we can verify that 𝜂𝑗𝑘 (±𝜆 𝑖 , 𝑛) = 0, 𝑖, 𝑗, 𝑘 = 1, 2.

(84)

(85)

𝜇𝑛 = (

(2) 2 (0) 𝜇11 𝜆 + 𝜇11

(1) 𝜇12 𝜆

(1) 𝜆 𝜇21

(2) 2 (0) 𝜇22 𝜆 + 𝜇22

),

(86)


9

(2) (0) (1) (1) (2) (0) where 𝜇11 𝜇11 , 𝜇12 , 𝜇21 , 𝜇22 , 𝜇22 are all independent of 𝜆. From (85), we get

(𝑇𝑛𝑡 + 𝑇𝑛 Ω(1) 𝑛 ) = 𝜇𝑛 𝑇𝑛 .

where 𝐶1 (𝜆, 𝑛) = (−7√2 + √5 − 3√5 (2 + √5))

(87)

𝑛

Comparing the coefficients of 𝜆 in (87), we have

𝑛

⋅ (29 (13 + √5) √𝜆2 − 4) (𝜆 (−𝜆 + √𝜆2 − 4))

1 (2) 𝜇11 =− , 2 (0) = 𝜇11 (1) = 𝜇12 (1) 𝜇21

𝑠̃𝑛−1

𝑛

𝐶2 (𝜆, 𝑛) = (𝜆 (−𝜆 + √𝜆2 − 4)) 𝑛

− ((−𝜆 (𝜆 + √𝜆2 − 4)) ) .

, (88)

1 = , 𝑟̃𝑛

From (65), we have 𝛼𝑖 [𝑛] =

1 (2) = , 𝜇22 2

=

(0) = 0. 𝜇22

𝜙𝑛2 − 𝜅𝑖 𝜓𝑛2 𝜙𝑛1 − 𝜅𝑖 𝜓𝑛1

− (3 + √5) 𝑒𝑥𝑝 (𝑡/2) (𝐶1 (𝜆 𝑖 , 𝑛) − 𝜅𝑖 𝐶2 (𝜆 𝑖 , 𝑛)) 2√2 + √5𝑒𝑥𝑝 (−𝑡/2) (𝐶1 (𝜆 𝑖 , 𝑛) + 𝜅𝑖 𝐶2 (𝜆 𝑖 , 𝑛))

The proof is completed.

Next, using obtained Darboux-Bäcklund transformation (71), we derive an exact solution of (26). First, it is easy to find that 𝑟𝑛 = 𝑒−𝑡 , 𝑠𝑛 = −𝑒𝑡 constitute a trivial solution of (71). Substituting this solution into the corresponding Lax pair, we get 𝑒−𝑡 𝜆 ) 𝜑𝑛 , −𝑒𝑡 𝜆 −𝜆2

(92)

(89)

Solving the above two equations, we get two real linear independent solutions: 𝑡 2√2 + √5𝑒𝑥𝑝 (− ) 2 ), ( 2 ) = 𝐶1 (𝜆, 𝑛) ( 𝑡 𝜙𝑛 − (3 + √5) 𝑒𝑥𝑝 ( ) 2 𝜙𝑛1

𝑡 2√2 + √5𝑒𝑥𝑝 (− ) 2 = 𝐶2 (𝜆, 𝑛) ( 𝑡 ), (3 + √5) 𝑒𝑥𝑝 ( ) 2

In (92), 𝜆 1 = (2 + √5), 𝜆 2 = −(2 + √5). By means of the Darboux-Bäcklund transformation (71), we deduce a new exact solution of (26). 𝑟̃𝑛 =

𝜆 1 𝜆 2 (𝜆 1 𝜎1 [𝑛] − 𝜆 2 𝜎2 [𝑛]) 𝑒𝑥𝑝 (−𝑡) , (𝜆 1 𝛼2 [𝑛] − 𝜆 2 𝛼1 [𝑛]) − ((𝜆22 − 𝜆21 ) 𝛼1 [𝑛] 𝛼2 [𝑛]) 𝑒𝑥𝑝 (−𝑡) (93)

𝑠̃𝑛 = −

(𝜆 1 𝛼1 [𝑛] − 𝜆 2 𝛼2 [𝑛]) 𝑒𝑥𝑝 (𝑡) . 𝜆 1 𝜆 2 (𝜆 1 𝛼2 [𝑛] − 𝜆 2 𝛼1 [𝑛])

This transformation can be done continually. Thus, we can obtain a series of exact solutions of (26).

0

𝜆2 − − 1 −𝑒−𝑡 𝜆 𝜑𝑛𝑡 = ( 2 ) 𝜑𝑛 . 𝜆2 𝑡 𝑒𝜆 2

𝜓𝑛1 ( 2) 𝜓𝑛

,

𝑖 = 1, 2.

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 (̃𝑟𝑛 , 𝑠̃𝑛 )𝑇 under transformation (71).

𝐸𝜑𝑛 = (

(91)

𝑛

+ ((−𝜆 (𝜆 + √𝜆2 − 4)) ) ,

1 , 𝑟̃𝑛 𝑠̃𝑛−1 1

𝑛

⋅ (𝜆 (−𝜆 + √𝜆2 − 4)) − ((−𝜆 (𝜆 + √𝜆2 − 4)) )

(90)

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 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 𝑇𝑛 𝑇𝑛(2𝑁+1)

𝑇𝑛11 𝑇𝑛12 = ( 21 22 ) , 𝑇𝑛 𝑇𝑛

(94)

10


with 𝑇𝑛11

=𝜆

2𝑁+1

+

+ ⋅⋅⋅ + 𝑇𝑛12

=

(2𝑁) 𝑡12

+

(2𝑁−1) 𝑡11

[𝑛] 𝜆

2𝑁−1

+

(2𝑁−3) 𝑡11

−2𝑁+1 −2𝑁−1 𝑡11 [𝑛] 𝑡11 [𝑛] + , 𝜆−2𝑁+1 𝜆2𝑁+1

[𝑛] 𝜆

2𝑁

+

−2𝑁 𝑡12 [𝑛] , 𝜆2𝑁

(2𝑁−2) 𝑡12

[𝑛] 𝜆

2𝑁−2

[𝑛] 𝜆

𝑇𝑛22

=

+ ⋅⋅⋅ [9]

(95)

𝑁 ≥ 1, [𝑛] +

(2𝑁−1) 𝑡22

[10]

[11]

[𝑛] 𝜆

(2𝑁−5) + 𝑡22 [𝑛] 𝜆2𝑁−5 + ⋅ ⋅ ⋅ +

+

[8]

𝑁 ≥ 1,

−2𝑁 𝑡21 [𝑛] , 2𝑁 𝜆

(2𝑁+1) 𝜆2𝑁+1 𝑡22

[7]

𝑁 ≥ 1,

(2𝑁) (2𝑁−2) 𝑇𝑛21 = 𝑡21 [𝑛] 𝜆2𝑁 + 𝑡21 [𝑛] 𝜆2𝑁−2 + ⋅ ⋅ ⋅

+

2𝑁−3

2𝑁−3

−2𝑁+1 𝑡11 [𝑛] 𝜆2𝑁−1

1 , 𝑁 ≥ 1. 𝜆2𝑁+1

[12]

[13]

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].

[14]

Conflicts of Interest

[15]

The authors declare that there are no conflicts of interest regarding the publication of this paper.

[16]

Acknowledgments This work was supported by the Science and Technology Plan Projects of the Educational Department of Shandong Province of China (Grant no. J08LI08).

[17]

[18]

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Discrete Dynamics in Nature and Society [25] X.-X. Xu and Y.-P. Sun, “An integrable coupling hierarchy of Dirac integrable hierarchy, its Liouville integrability and Darboux transformation,” Journal of Nonlinear Sciences and Applications (JNSA), vol. 10, no. 6, pp. 3328–3343, 2017. [26] X.-X. Xu, “An integrable coupling hierarchy of the MKdVintegrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 344–353, 2010.

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