A Hierarchy of Integrable Differential-Difference Equations and Darboux Transformation

Vol. 84 (2019)

REPORTS ON MATHEMATICAL PHYSICS

No. 3

A HIERARCHY OF INTEGRABLE DIFFERENTIAL-DIFFERENCE EQUATIONS AND DARBOUX TRANSFORMATION FANG -C HENG FAN , S HAO -Y UN S HI and Z HI -G UO X U School of Mathematics & State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, P. R. China (e-mail: [email protected]) (Received December 31, 2018 — Revised March 16, 2019) By embedding a free function into discrete zero curvature equation, we generalize the original hierarchy of integrable differential-difference equations into a new hierarchy which includes the famous relativistic Toda hierarchy. Infinitely many conservation laws and Darboux transformation for the first nontrivial system in the new hierarchy are constructed with the help of its Lax pair. The exact solutions of the system are generated by applying the obtained Darboux transformation. Finally, the figures of one- and two-soliton solutions with proper parameters are presented to illustrate the structures of soliton solutions. Keywords: integrable differential-difference equations, conservation law, Darboux transformation, soliton solutions.

1.

Introduction

Nonlinear integrable differential-difference equations (DDEs) have been the focus of many nonlinear studies since the original work of Fermi, Pasta and Ulam in the 1960s [1]. They have many applications in various scientific contexts ranging from mathematical physics, numerical analysis to quantum physics, as well as statistical physics [2, 3], so searching for new integrable DDEs and studying their associated properties is necessary and very meaningful. Up to now, many methods and theories of constructing integrable DDEs have been proposed and developed, such as the discrete Zakharov–Shabat eigenvalue problem [4, 5], the r-matrix approach [6, 7] and the Tu scheme [8], Among these existing methods of construction, the Tu scheme has turned out to be one of efficient methods to obtain integrable DDEs [9, 10], its main idea is looking for the proper spectral problem such that the discrete zero curvature equation holds. Various physically important integrable DDEs have been obtained and systematically investigated, for instance the Ablowitz–Ladik lattice [4, 5], the Toda lattice [11], the Volterra lattice [12, 13], the discrete nonlinear Schrödinger equation [14] and so on. Conservation laws is one of the most important tools in the study of DDEs. The existence of infinitely many conservation laws is a significant indicator of integrability [289]

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of the system [15]. From physical point of view, it is also very interesting to know whether there exist conservation laws for DDEs. Some conservation laws have a physical meaning, for instance conserved momentum and energy. Furthermore, conservation laws may facilitate the study of both quantitative and qualitative properties of solutions [16]. Meanwhile, the Darboux transformation method is a powerful scheme to explicitly derive exact solutions for DDEs. Its critical idea is to use the Lax pair and an initial seed solution of the system, then iterate the low-order wave solution to obtain high-order solution. In the past several decades, the Darboux transformation method has been widely applied to obtain exact solutions of various DDEs, such as discrete potential KdV equation [17], Kaup-Newell lattice equation [18] and so on. The discrete spectral problem, i.e. the Lax pair of system plays crucial role in the treatment of integrable DDEs. From the associated spectral problem, many new DDEs and the interrelated nice properties, such as the nonlinearization [19], the master symmetries [20], the Hamiltonian structures [21], the infinite conservation laws [22, 23], the Darboux transformation [9] and so on, can usually be investigated conveniently. As a key step, many spectral problems have been put forward and discussed. In [24–26], Ma, Xu, Wen and their co-works considered the following discrete (2 × 2)-matrix spectral problems ! 0 1 , Eφn = Un (un , λ)φn , Un (un , λ) = λqn λ + rn ! 0 1 , Eφn = Un (un , λ)φn , Un (un , λ) = (αλ + β)qn λ + rn ! 0 − q1n , Eφn = Un (un , λ)φn , Un (un , λ) = (αλ + β)qn λ + rn
T where E is shift operator defined by Ef (n, t) = f (n+1, t) = fn+1 , φn = φn,1 , φn,2 is the eigenfunction, un = (qn , rn )T are potential functions, λ is the spectral parameter and λt = 0, α, β are constants and α 2 + β 2 6= 0. Starting from these spectral problems, some new integrable equations were constructed and the Hamiltonian structures were derived to show their integrability in the Liouville sense. By using gauge transformation and Lax pair, Darboux transformation or N-fold Darboux transformation were established to present explicitly exact solutions or soliton solutions. As far as we know, the more research is focused on the discrete (2 × 2)-matrix spectral problems with two potential functions [27, 28], while for three or more potential functions, owing to their complexity, the relevant research is not satisfactory. Here, we point out that a lot of valuable and meaningful work has been done by Vakhnenko in [29–31]. Thus, it is necessary to make further research on the discrete (2 × 2)-matrix spectral problems with three or more potential functions.

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In this paper, motivated by the works mentioned above, we consider the following discrete (2 × 2)-matrix spectral problem with three potential functions, ! 0 pn Eφn = Un (un , λ)φn , Un (un , λ) = . (1) λqn λ + rn Here we are interested in generating the hierarchy of nonlinear integrable DDEs connected with (1) through the Tu scheme, and deriving an infinite number of conservation laws and Darboux transformation for the first nontrivial system in the hierarchy. To the best of our knowledge, the discrete (2 × 2)-matrix spectral problem (1) has not been investigated. The rest of the paper is organized as follows. In Section 2, starting from a discrete spectral problem, we derive an integrable DDEs hierarchy with a free function which includes the relativistic Toda hierarchy. Thanks to the Lax pair, infinite hierarchy of conservation laws and Darboux transformation for the first nontrivial system in the hierarchy are given in Section 3 and Section 4, respectively. As applications of the Darboux transformation, exact solutions of the system are given in Section 5, and the figures of one- and two-soliton solutions with proper parameters are shown graphically. Some remarks and summary are presented in the last section. 2.

A hierarchy of integrable DDEs

The main purpose of this section is to construct the hierarchy of integrable DDEs associated with the discrete spectral problem (1). We first solve the stationary discrete zero-curvature equation [8] (EVn )Un − Un Vn = 0, where Vn =

An Cn

(2)

! Bn . −An

A direct calculation from (2) leads to the following equations:   λqn Bn+1 − pn Cn = 0,    λB n+1 + pn (An + An+1 ) + rn Bn+1 = 0,  λCn + λqn (An + An+1 ) + rn Cn = 0,    λ△A + λq B − p C n n n n n+1 + rn △An = 0,

(3)

where △ is a difference operator defined by △fn = (E−1)fn = fn+1 −fn . Substituting expansions An =

∞ X j =0

an(j ) λ−j ,

Bn =

∞ X j =0

bn(j ) λ−j ,

Cn =

∞ X j =0

cn(j ) λ−j +1

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into (3), we obtain  (0) bn+1 = 0, cn(0) = 0, △an(0) = 0,    (j ) (j )    qn bn+1 − pn cn = 0, (j ) (j ) (j +1) (j ) bn+1 + pn (an + an+1 ) + rn bn+1 = 0,  (j ) (j ) (j ) (j +1)   + qn (an + an+1 ) + rn cn = 0, cn    (j +1) (j +1) (j ) (j +1) − pn cn+1 + rn △an = 0, + qn bn △an

(4) j ≥ 0.

Without loss of generality, we choose an(0) = − 21 , then (4) can be solved successively as an(0) = − 12 , bn(0) = 0, cn(0) = 0, an(1) = pn−1 qn , bn(1) = pn−1 , cn(1) = qn , (2) an = −pn−1 qn (pn−1 qn + pn qn+1 + pn−2 qn−1 + rn + rn−1 ) , bn(2) = −pn−1 (pn−2 qn−1 + pn−1 qn ) − rn−1 pn−1 , cn(2) = −qn (pn−1 qn + pn qn+1 ) − rn qn , .... Now, we set Vn(m) = λm Vn




+

=

m X j =0

(j )

an λm−j (j ) cn λm−j +1

(j )

bn λm−j (j ) −an λm−j

!

,

where “ + " denotes the choice of positive power of λ. By (4), we get ! (m+1) 0 −bn+1   (m) (m) (EVn )Un − Un Vn = . (m) λcn(m+1) rn an(m) − an+1 Let

φn,tm = Vn(m) φn ,

m ≥ 0,

(5)

then the compatibility conditions between (1) and (5), i.e. the discrete zero curvature equation Un,t = (EVn(m) )Un − Un Vn(m) , gives rise to the following integrable hierarchy  (m+1)   pn,tm = −bn+1 , (m+1) qn,tm = cn  ,   r = r a (m) − a (m) , n,tm

n

n

n+1

(6) m ≥ 0,

which admits a Lax pair constituted by the spectral problem (1) and (5). In what follows, we derive some more general hierarchies associated with (1) by embedding a free function δn(m) into Vn(m) . Indeed, set (m) V(m) + 1(m) n = Vn n

A HIERARCHY OF INTEGRABLE DIFFERENTIAL-DIFFERENCE EQUATIONS AND. . .

with 1(m) n =

δn(m) 0

293

! 0 , 0

then through a direct calculation, we obtain (m) (EV(m) n )Un − Un Vn =

0   λ cn(m+1) − qn δn(m)

(m) ! (m+1) −b n+1 + pn δn+1  , (m) rn an(m) − an+1

from which we get the following integrable hierarchy  (m+1) (m)   pn,tm = −bn+1 + pn δn+1 , (m) qn,tm = cn(m+1)  − qn δn ,  (m) r (m) m ≥ 0. n,tm = rn an − an+1 ,

(7)

We remark that when pn = 1 and if we choose choose δn(m) = bn(m+1) , (7) is the relativistic Toda hierarchy [24]. Choose δn(m) = cn(m) and let m = 1, we obtain the following first nontrivial system from hierarchy (7)    pn,t = pn (pn−1 qn + pn qn+1 ) + rn pn + pn qn+1 , (8) qn,t = −qn (pn−1 qn + pn qn+1 ) − rn qn − qn2 ,   r = r (p q − p q ) , n,t n n−1 n n n+1

and the corresponding Lax pair Eφn = Un φn , φn,t = Vn φn ,

Un = Vn =

0 λqn

! pn φn , λ + rn

− 12 λ + pn−1 qn + qn λqn

(9) ! pn−1 φn . 1 λ − pn−1 qn 2

(10)

The rest of paper will focus on studying of the infinite number of conservation laws and the Darboux transformation for integrable differential-difference equations (8). 3.

Infinitely many conservation laws The existence of infinite number of conservation laws acts as one of important characteristics for integrable systems. Some known successful methods for finding infinitely many conservation laws of DDEs have been proposed. In [32], using the direct methods, several of conservation laws for a system can be obtained. However, this method has a shortcoming. On the one hand, if we want to obtain more conservation laws, the calculation is very tedious and complex. On the other hand, the method cannot provide the justification whether there exist infinitely many conservation laws for a system. In [33, 34], by means of the recursive approach,

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infinitely many conservation laws for the semi-discrete matrix NLS equation were derived. In [22, 23], Zhang and Chen proposed a simple approach to constructing the conservation laws for Lax integrable systems, the essential idea of this approach was a simple equality f (n + 1) = (E − 1) ln f (n), ln f (n) by which the conservation law can easily be obtained from the concerned Lax pair. In [35, 36], Vakhnenko modified procedure of the recursive approach, the technique is based on a collection of several distinct generating equations and a set of auxiliary Riccati equations, instead of a single generating equation and a single auxiliary Riccati equation typical for the traditional considerations [22, 23, 33, 34]. In addition, we remark that this method is applicable for any semi-discrete integrable system associated with the auxiliary square spectral and evolution matrices of an arbitrary order. In what follows, we will derive the infinite number of conservation laws for (8). In order to obtain conservation laws from Lax pair dierctly, we choose the method from [22, 23]. By (9) and (10), we have φ2,n+1 = λpn−1 qn φ2,n−1 + (λ + rn ) φ2,n ,   1 λ − pn−1 qn φ2,n . φ2,n,t = λpn−1 qn φ2,n−1 + 2 Let Ŵn =

(11) (12)

φ2,n−1 , then by (11), φ2,n rn Ŵn+1 + λŴn+1 + λpn−1 qn Ŵn Ŵn+1 − 1 = 0,

(13)

which is a discrete Riccati equation. Suppose that Ŵn =

∞ X

λj gn(j )

(14)

j =0

(j )

is a solution of (13), where gn are the functions to be determined. Substituting (14) into (13), comparing the coefficients of the same power of λ, we get gn(0) = gn(j +1)

1 , rn−1 =−

gn(1) = − 1

rn−1

fn(j )

1 2 rn−1

−

pn−2 qn−1 , 2 rn−2 rn−1

j pn−2 qn−1 X (k) (j −k) − g g , rn−1 k=0 n−1 n

j = 1, 2, . . . .

A direct calculation from (11) and (12) gives φ2,n+1 = λpn−1 qn Ŵn + λ + rn , φ2,n

φ2,n,t λ = λpn−1 qn Ŵn + − pn−1 qn . φ2,n 2

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Note that from

  φ2,n+1 φ2,n,t ln = (E − 1) , φ2,n t φ2,n

we get



where 9n = with

ln rn + ln (1 + λ9n )

∞ X 1 pn−1 qn zj λj , Ŵn + = rn rn j =0

z0 =

pn−1 qn 1 + , rn rn−1 rn

zj =



t

˜ n, = (E − 1) 9

˜ n = λpn−1 qn Ŵn − pn−1 qn = 9

∞ X

λj wj ,

j =0

pn−1 qn (j ) gn , rn

wj = pn−1 qn gn(j −1) ,

w0 = −pn−1 qn ,

(15)

j = 1, 2, . . . .

˜ n into (15), and making a comparison of the same Now substituting 9n and 9 powers of λ on both sides, we obtain the following set of infinitely many conservation laws for (8) (j ) ρn,t = (E − 1)Jn(j ) , where

ρn(0) = ln rn , ρn(j ) = zj −1 − Jn(j )

= wj ,

Jn(0) = −pn−1 qn ,

ρn(1) =

1 pn−1 qn + , rn rn−1 rn

Jn(1) =

pn−1 qn , rn−1

1 X (−1)j −1 j j −2 z0 , zl1 zl2 + · · · + (−1)j −2 z0 z1 + 2 l +l =j −2 j 1

2

j = 2, 3 . . . .

At the end of this section, we remark that by using the similar method, one can obtain infinitely many conservation laws for the other system in hierarchy (7). 4.

Darboux transformation In this section, we proceed to construct the Darboux transformation for (8). First, we review the notation of Darboux transformation. In essence, the Darboux transformation is a special gauge transformation φ˜ n = Tn φn ,

which can transform the spectral problem Eφn = Un φn ,

φn,t = Vn φn

E φ˜ n = U˜ n φ˜ n ,

φ˜ n,t = V˜n φ˜ n ,

into the following spectral problem

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where

U˜ n = Tn+1 Un Tn−1 ,

V˜n = (Tn,t + Tn Vn )Tn−1 ,

and the matrixes U˜ n and V˜n have the same structures as Un , Vn , respectively. Therefore, we can obtain the same DDEs from the discrete zero curvature equation. What’s more, the relationship between the two solutions of the system can be established in the process of constructing Darboux transformation. By using this relationship, the other new solutions can be obtained from initial ones. Let ϕn (λ) = (ϕ1,n (λ), ϕ2,n (λ))T and ψn (λ) = (ψ1,n (λ), ψ2,n (λ))T be two basic solutions of (9) and (10) related to the solution (pn , qn , rn ) of (8). We define ϕ2,n (λi ) + γi ψ2,n (λi ) , i = 1, 2, (16) σi,n = ϕ1,n (λi ) + γi ψ1,n (λi ) and

an =

λ2 σ1,n − λ1 σ2,n , σ2,n − σ1,n

σ1,n σ2,n (λ2 − λ1 ) cn = , λ1 σ2,n − λ2 σ1,n

bn =

λ1 − λ2 , σ2,n − σ1,n


λ1 λ2 σ1,n − σ2,n dn = , λ1 σ2,n − λ2 σ1,n

(17)

where nonzero parameters λ1 , λ2 with λ1 6= λ2 and γ1 , γ2 are suitably chosen such that the denominators in (16) and (17) are nonzero. It is obvious that the matrix Tn is a key step for constructing Darboux transformation, a proper Tn will ensure the correctness of Darboux transformation. Hereby, we take Tn as ! λ + an bn Tn = . λcn λ + dn It is easy to verify that λ1 and λ2 are roots of det Tn , which means det Tn = (λ − λ1 )(λ − λ2 ).

In order to construct the Darboux transformation, we need to prove that U˜ n and ˜ Vn have the same forms as Un and Vn , respectively. We first show that the structure of Un is the same as U˜ n . Let ! 1 2 F (λ) F (λ) n n Fn (λ) = Tn+1 Un Tn∗ = , Fn3 (λ) Fn4 (λ) j

where Tn∗ denotes the joint matrix of Tn , and the details of Fn (λ) (j = 1, 2, 3, 4) are omitted. We find that Fn1 (λ) and Fn2 (λ) are polynomial in λ of order 2, Fn3 (λ) and Fn4 (λ) are polynomials in λ of order 3. From (16) and (17) we obtain λi + an + bn σi,n = 0,

By (9) and (16),

σi,n+1 =

λi cn + (λi + dn ) σi,n = 0, λi + rn λi qn + , pn σi,n pn

i = 1, 2.

i = 1, 2.

(18) (19)

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297

Moreover, with (18) and (19), it can be verified that λ1 and λ2 are the roots of j Fn (λ) (j = 1, 2, 3, 4). Taking Fn1 (λ) as an example, fn1 (λi ) = λ2i bn+1 qn + λi σi,n (λi + dn ) (bn+1 + pn ) + λi bn+1 dn qn
−pn σi,n (λi + dn ) λi + σi,n+1 bn+1 + rn bn+1 σi,n (λi + dn ) = 0,

i = 1, 2.

! Pn1 , Pn3 λ + Pn4

(20)

Therefore, there exists a matrix Pn such that

0 Tn+1 Un Tn∗ = det Tn · Pn = (det Tn ) Pn2 λ

with Pnk (k = 1, 2, 3, 4) are independent of λ. To determine Pnk , we rewrite (20) as (21)

Tn+1 Un = Pn Tn . Comparing the coefficients of λ in (21), we get Pn1 = pn + bn+1 , Pn2 = qn − cn , Pn3 = 1, Pn4 = d1+n − dn + cn+1 pn − bn qn + bn cn + rn . Let p˜ n = pn + bn+1 , q˜n = qn − cn , r˜n = d1+n − dn + cn+1 pn − bn qn + bn cn + rn , then

Pn1 = p˜ n ,

Pn2 = q˜n ,

(22)

Pn4 = r˜n ,

which imply U˜ n = Tn+1 Un Tn−1 = Pn =

0 q˜n

! p˜ n . λ + r˜n

Thus, the matrix U˜ n has the same form as Un . Under the transformation (22), we can verify the invariance form for Vn , i.e. the matrix V˜n has the structure as ! 1 − λ + p ˜ q ˜ p ˜ n−1 n n−1 2 . V˜n = 1 λq˜n λ − p˜ n−1 q˜n 2 Obviously, we get the relationship (22) between the two solutions of (8) in this process. Based on the above facts, we finally conclude the following theorem. T HEOREM 1. The gauge transformation (φn , pn , qn , rn ) → (φ˜ n , p˜ n , q˜n , r˜n ) is a Darboux transformation of the integrable DDEs (8). Furthermore, the solutions (pn , qn , rn ) of (8) are mapped into new solutions (p˜ n , q˜n , r˜n ) under the Bäcklund transformation p˜ n = pn + bn+1 ,

q˜n = qn − cn ,

r˜n = dn+1 − dn + cn+1 pn − bn qn + bn cn + rn .

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5. Exact solutions In this section, we apply the Darboux transformation obtained in Section 4 to construct some explicit exact solutions for integrable DDEs (8). Obviously, (8) has a seed solution pn = 1, qn = 1, rn = −3 and the Lax pair related to the seed solution is ! ! 1 0 1 − 12 λ + 2 φn . (23) Eφn = φn , φn,t = 1 λ−1 λ 1 λ−3 2 We can obtain two linearly independent solutions of (23) as ! ! ! ! ϕ1,n (λ) −τ n−1 eρt ψ1,n (λ) −µn−1 e−ρt = , = , ϕ2,n (λ) τ n eρt ψ2,n (λ) µn e−ρt where λ−3 τ = τ (λ) = + ρ, 2

λ−3 µ = µ(λ) = − ρ, 2

√ λ2 − 2λ + 9 ρ = ρ(λ) = . 2 (24)

Based on (16), we have σi,n =

τin eρi t + γi µni e−ρi t

e−ρi t τin−1 eρi t + γi µn−1 i

,

τi = τ (λi ),

µi = µ(λi ),

ρi = ρ(λi ),

i = 1, 2.

By Theorem 1, the integrable DDEs (8) admits the following explicit exact solutions  (λ1 − λ2 ) σ1,n σ2,n   , p˜ n = 1 +    λ2 σ1,n − λ1 σ2,n − σ1,n σ2,n (λ1 − λ2 )     σ1,n σ2,n (λ1 − λ2 ) q˜n = 1 + , (25) λ1 σ2,n − λ2 σ1,n  

   3 λ2 σ1,n − λ1 σ2,n λ2 σ1,n − λ1 σ2,n − (λ1 − λ2 ) σ1,n σ2,n   


. r˜n = σ1,n − σ2,n 3σ1,n σ2,n (λ1 − λ2 ) − λ1 λ2 σ1,n − σ2,n

Next, we choose suitable parameters λi and γi (i = 1, 2) to illustrate the correctness of our results. When λ1 = −1.5, λ2 = −1.51, γ1 = −2, γ2 = −1, then all the denominators in (17) are meaningful, and we can show that the solutions (25) have the form of sinh and cosh by symbolic computation with Mathematica. Here we plot the figures of the solutions to illustrate the structures of soliton solutions. Fig. 1(a) and Fig. 1(b) show the anti-bell-shaped soliton. We remark that if the resulting solution is taken as the new starting point, and if we make the derived Darboux transformation once again, then another new solutions of (8) are obtained. This process can be done continuously. Therefore, we can obtain a series of explicit solutions for (8).

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(b)

Fig. 1. Plot of the solutions. (a) One-soliton solution pn with the parameters λ1 = −1.5, λ2 = −1.51, γ1 = −2, γ2 = −1. (b) One-soliton solution qn with the same parameters as those in (a).

6.

Discussion of the unsolved problems In Section 5, we get the exact solutions (25) for integrable DDEs (8) by using the Darboux transformation. When λ1 = −1.5, λ2 = −1.51, γ1 = −2, γ2 = −1, we obtain one-soliton solutions. However, the soliton solutions we get are not enough for physical applications, because the spectral parameters λi and γi are the free parameters and the ranges of their definitions are determinable via the symmetry properties of field functions with the boundary conditions for the field functions to be taken into account. Therefore, we need to rearrange the formulae for p˜ n , q˜n , r˜n in (25) into the form with strictly defined free physical parameters such as the velocity of soliton, the mean coordinate of the soliton wave packet, kinetic energy and amplitude of soliton. For now, it is an unsolved work. We have tried the following method. First, introducing some appropriate parametrization of the spectral parameter λ, i.e. √ λ = + 2(z − z−1 ) + 1, or alternatively

√ λ = − 2(z − z−1 ) + 1,

the square roots in (24) and (25) can be eliminated. Next, denoting z = exp(+ξ ) and introducing the proper parametrization for γi , we try to rearrange the formulae for p˜ n , q˜n , r˜n to obtain the analytical or soliton solutions with strictly defined free physical parameters. Unfortunately, we did not get the expected results. 7.

Conclusions In this paper, by embedding a free function into zero curvature equation, we enlarge the original integrable DDE hierarchy (6) into a new hierarchy (7), which can be reduced to the well-known relativistic Toda hierarchy. Further, infinitely many conservation laws and Darboux transformation for the first nontrivial system (8) are derived by means of the Lax pair. As an application of the Darboux transformation,

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the explicit exact solutions is given. Soliton structures of those solutions are shown graphically. In fact, with the help of Darboux transformation, the multi-soliton solutions for (8) can be obtained successfully. Acknowledgements This work was completed with the support by NSFC grant (No. 11771177, 11301210, 11501242), China Automobile Industry Innovation and Development Joint Fund (No. U1664257), Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 2017TD-20), Science and Technology Development Project of Jilin Province (20170520055JH) and the scientific research project of The Education Department of Jilin Province (JJKH20160398KJ). We thank for the reviewer’s valuable advice and hard work, the good idea and skillful method in Section 6 is given by the reviewer. REFERENCES [1] E. Fermi, J. Pasta and S. Ulam: Studies of nonlinear problems I, Los Alamos Report LA-1940, Los Alamos, New Mexico 1955. [2] D. J. Kaup: Variational solutions for the discrete nonlinear Schrödinger equation, Math. Comput. Simul. 69 (2005), 322–333. [3] M. Toda: Waves in nonlinear lattice, Prog. Theor. Phys. Suppl. 45 (1970), 174–200. [4] M. J. Ablowitz and J. F. Ladik: Nonlinear differential-difference equations, J. Math. Phys. 16 (1975), 598–603. [5] M. J. Ablowitz and J. F. Ladik: A nonlinear difference scheme and inverse scattering, Stud. Appi. Math. 55 (1976), 213–229. [6] A. G. Reyman and M. A. Semenev-Tian-Shansky: Compatible Poisson structures for Lax equations: an r-matrix approach, Phys. Lett. A 130 (1988), 456–460. [7] M. Błaszak and K. Marciniak: R-matrix approach to lattice integrable systems, J. Math. Phys. 35 (1994), 4661–4682. [8] G. Z. Tu: A trace identity and its applications to theory of discrete integrable systems, J. Phys. A 23 (1990), 3903–3922. [9] Y. T. Wu and X. G. Geng: A new hierarchy integrable differential-difference equations and Darboux transformation, J. Phys. A 31 (1998), L677–L684. [10] A. Pickering and Z. N. Zhu: New integrable lattice hierarchies, Phys. Lett. A 349 (2002), 439–445. [11] M. Toda: Theory of Nonlinear Lattices, Springer, Berlin 1989. [12] M. Wadati: Transformation theories for nonlinear discrete systems, Prog. Theor. Phys. Suppl. 59 (1977), 36–63. [13] M. J. Ablowitz and P. A. Clarkson: Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge 1991. [14] M. J. Albowitz and H. Segur: Solitons and Inverse Scattering Transformation, SIAM, Philadephia 1981. [15] A. S. Fokas: Symmetries and integrability, Stud. Appl. Math. 77 (1987), 253–299. [16] E. G. B. Hohler and K. Olaussen: Using conservation laws to solve Toda field theories, Int. J. Mod. Phys. A 11 (1996), 1831–1853. [17] Y. Shi, J. J. C. Nimmo and D. J. Zhang: Darboux and binary Darboux transformations for discrete integrable systems I. Discrete potential KdV equation, J. Phys. A: Math. Theor. 47 (2014), 025205, 11 pages. [18] N. Liu and X. Y. Wen: Dynamics and elastic interactions of the discrete multi-dark soliton solutions for the Kaup-Newell lattice equation, Mod. Phys. Lett. B 32 (2018), 1850085, 19 pages.

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