A New Discrete Integrable System Derived from a Generalized Ablowitz-Ladik Hierarchy and Its Darboux Transformation

Abstract

We find an interesting phenomenon that the discrete system appearing in a reference can be reduced to the old integrable system given by Merola, Ragnisco, and Tu in another reference. Differing from the works appearing in the above two references, a new discrete integrable system is obtained by the generalized Ablowitz-Ladik hierarchy; the Darboux transformation of this new discrete integrable system is established further. As applications of this Darboux transformation, different kinds of exact solutions of this new system are explicitly given. Investigatingthe properties of these exact solutions, we find that these exact solutions are not pure soliton solutions, but their dynamic characteristics are very interesting.

1. Introduction

It is well known that the nonlinear integrable lattice systems have been intensively investigated in many scientific fields such as physics, chemistry, and biology. The scientific researchers are not only interested in their rich mathematical structures, but also their applications in science, such as mathematical physics, numerical analysis, computer science, statistical physics, and quantum physics. Various integrable lattice systems have been studied extensively, such as the Toda lattice and the relativistic Toda lattice, the Ablowitz-Ladik lattice, the Volterra lattice, the Suris lattice, and some discrete classic equation such as discrete KdV equation, discrete Schrödinger equation, [1–18].

Recently, some new lattice hierarchies and their integrability, Darboux transformation, conversation law, exact solution, and dynamic characteristics have been holding more and more attention.In addition, the scheme for constructing nonlinear discrete integrable couplings [19] and new transformation methods which are used to directly solve discrete integrable systems [20] are also worth paying attention to. Among the multitudinous new integrable hierarchies, we will discuss the so-called new discrete lattice system which is given by [21, equation (2.12)]𝑞_𝑛,𝑡=𝑞²_𝑛𝑟_𝑛+1−𝑞_𝑛−1,𝑟_𝑛,𝑡=−𝑟²_𝑛𝑞_𝑛−1+𝑟_𝑛+1, (1.1)

which is derived by Qin. It is very interesting that this equation can be reduced to the old integrable lattice system under a series of local coordinate transformation, which is given by Merola et al. [22]. To see this,we give the whole transformation procedure as follows.

Taking the local coordinate transformation 𝑟_𝑛+1→𝑟_𝑛̃, (1.1) can be rewritten as𝑞_𝑛,𝑡=𝑞²_𝑛𝑟_𝑛̃−𝑞_𝑛−1,𝑟̃_𝑛,𝑡=−𝑟²_𝑛̃𝑞_𝑛+𝑟̃_𝑛+1. (1.2)

Taking another local coordinate transformation 𝑛→−𝑛, (1.2) can be rewritten as𝑞_−𝑛,𝑡=𝑞²_−𝑛𝑟̃_−𝑛−𝑞_−𝑛−1,𝑟̃_−𝑛,𝑡=−𝑟̃²_−𝑛𝑞_−𝑛+𝑟̃_−𝑛+1. (1.3)

Taking the third coordinate transformation −𝑛→𝑛 and −(𝑛+1)→𝑛+1,−(𝑛−1)→𝑛−1, (1.3) can be rewritten as𝑞𝑛_,𝑡=𝑞𝑛²𝑟𝑛̃−𝑞𝑛₊₁,𝑟𝑛̃_,𝑡=−𝑟𝑛²̃𝑞𝑛+𝑟𝑛̃₋₁. (1.4)

To simplify notation, we still use 𝑛 and 𝑟_𝑛 to take the place of 𝑛 and 𝑟_𝑛̃. Thus, (1.4) can be rewritten as𝑞_𝑛,𝑡=𝑞²_𝑛𝑟_𝑛−𝑞_𝑛+1,𝑟_𝑛,𝑡=−𝑞_𝑛𝑟²_𝑛+𝑟_𝑛−1. (1.5)

Equation (1.5)is equivalent to the following Equation:𝑞|𝑟⟩_𝑡1=𝑞𝐸𝑞∣𝑟−𝐸₋₁−𝑟⟩,𝑞∣𝑟| (1.6)

where 𝐸 is a shift operator and |𝑞⟩↔𝑞_𝑛,|𝑟⟩↔𝑟_𝑛,𝑡₁↔𝑡. Equation (1.6) was given in [22] as the 44th equation. The results investigated by Qin in [21] are different from those given in [22]; however, these two systems are equivalent under the above coordinate transformations, noting that the coordinate scales of these two systems are different, and both work are interesting and very useful.

The rest of this paper is organized as follows. In Section 2, based on Qin’s work, we will derive an indeed new lattice system from the generalized Ablowitz-Ladik hierarchy and establish its Darboux transformation. In Section 3, under the Darboux transformation, using nonzero or zero constant as seeds, we will obtain different kinds of exact solutions of this new system and discuss their dynamic characteristics. It isworthy of mentioning that the exact solutions of this new lattice system which we will obtain have many particular phenomena; they are not pure soliton solutions and their waveforms are very interesting. In Section 4, brief conclusions are given.

2. New Discrete Lattice System and Its Darboux Transformation

As in [21], we introduce the following discrete spectral problem:𝐸𝜓_𝑛=𝑈_𝑛𝑞_𝑛,𝑟_𝑛,𝜆𝜓_𝑛, (2.1)

where𝑈_𝑛=1+𝜃𝑞_𝑛𝑟_𝑛𝜆𝑞_𝑛𝜆𝑟_𝑛𝜆²,𝜃=0,1. (2.2)

This is a generalization of the Ablowitz-Ladik spectral problem; its corresponding auxiliary problem is shown as follows:𝜓_𝑛,𝑡=𝑉_𝑛𝑞_𝑛,𝑟_𝑛,𝜆𝜓_𝑛. (2.3)

First, we consider the stationary discrete zero curvature equation𝐸𝑉_𝑛𝑈_𝑛−𝑈_𝑛𝑉_𝑛=𝑂, (2.4)

where𝑉_𝑛=^∞𝑗=0𝑎_𝑛^(𝑗)𝜆^∞−2𝑗_𝑗=0𝑏_𝑛^(𝑗)𝜆_∞^−2𝑗+1_𝑗=0𝑐_𝑛^(𝑗)𝜆^−2𝑗+1−^∞_𝑗=0𝑎_𝑛^(𝑗)𝜆^−2𝑗,𝑂=0000. (2.5)

Substituting (2.2) and (2.5) into (2.4), we obtain a series of recursive formulas as follows:1+𝜃𝑞_𝑛𝑟_𝑛(𝐸−1)𝑎_𝑛^(𝑗)+𝑟_𝑛𝑏^(𝑗+1)_𝑛+1−𝑞_𝑛𝑐_𝑛^(𝑗+1)𝑏=0,_{(𝑗+1)𝑛+1}+𝑞_𝑛(𝐸+1)𝑎_𝑛^(𝑗)−1+𝜃𝑞_𝑛𝑟_𝑛𝑏_𝑛^(𝑗)=0,−𝑐_𝑛^(𝑗+1)−𝑟_𝑛(𝐸+1)𝑎_𝑛^(𝑗)+1+𝜃𝑞_𝑛𝑟_𝑛𝑐^(𝑗)_𝑛+1𝑞=0,_𝑛𝑐^(𝑗)_𝑛+1−𝑟_𝑛𝑏_𝑛^(𝑗)−(𝐸−1)𝑎_𝑛^(𝑗)𝑏=0,_(0)𝑛+1=𝑐_𝑛⁽⁰⁾=0. (2.6)

Second, we consider the discrete zero curvature equation𝑈_𝑛,𝑡=𝐸𝑉_𝑛^(𝑚)𝑈_𝑛−𝑈_𝑛𝑉_𝑛^(𝑚) (2.7) with𝑉_𝑛^(𝑚)=𝜆^2𝑚𝑉⁺_𝑛+Δ_𝑛^(𝑚), (2.8)

where the positive power term 𝑉⁺_𝑛 and modification term Δ_𝑛^(𝑚) are, respectively, given by𝑉⁺_𝑛=^𝑚𝑗=0𝑎_𝑛^(𝑗)𝜆^𝑚−2𝑗_𝑗=0𝑏_𝑛^(𝑗)𝜆_𝑚^−2𝑗+1_𝑗=0𝑐_𝑛^(𝑗)𝜆^−2𝑗+1−^𝑚_𝑗=0𝑎_𝑛^(𝑗)𝜆^−2𝑗,Δ_𝑛^(𝑚)=−(1−𝜃)𝑎_𝑛^(𝑚)000. (2.9)

Substituting 𝑈_𝑛,𝑉_𝑛^(𝑚) into the discrete zero curvature equation (2.7) and recurring to the stationary discrete zero curvature equation (2.3) yield𝑈_𝑛,𝑡=𝐸Δ_𝑛^(𝑚)𝑈_𝑛−𝑈_𝑛Δ_𝑛^(𝑚). (2.10)

Thus, we obtain𝑞_{𝑛,𝑡𝑚}=−𝑏^(𝑚+1)_𝑛+1−(1−𝜃)𝑞_𝑛𝑎^(𝑚)_𝑛+1,𝑟_{𝑛,𝑡𝑚}=𝑐_𝑛^(𝑚+1)+(1−𝜃)𝑟_𝑛𝑎_𝑛^(𝑚),𝜃𝑞_𝑛𝑟_𝑛𝑡𝑚=𝜃1+𝜃𝑞_𝑛𝑟_𝑛𝑎^(𝑚)_𝑛+1−𝑎_𝑛^(𝑚),𝑚=0,1,2,…. (2.11) Obviously, when 𝜃=0 and 𝜃=1, all equations in (2.11) satisfy compatibility condition between (2.1) and the following auxiliary spectral problem:𝜓_𝑛,𝑡=𝑉_𝑛^(𝑚)𝜓_𝑛. (2.12)

When 𝑗=0,1 and 𝜃=1,𝑚=1, letting𝑎_𝑛⁽⁰⁾=−1/2 and considering the integral constant as zero, by using (2.6) and (2.11), Qin derived (1.1). Moreover, when 𝑗=0,1,2 and 𝜃=1,𝑚=2,   letting 𝑎_𝑛⁽⁰⁾=−1/2 and considering the integral constant as zero, by using (2.6) and (2.11) again,we successfully obtain an indeed new lattice system as follows:𝑞_𝑛,𝑡=𝑞_𝑛−1𝑞_𝑛−1𝑟_𝑛+2𝑞_𝑛𝑟_𝑛+1−𝑞²_𝑛𝑞_𝑛𝑟²_𝑛+1−𝑟_𝑛+2−𝑞_𝑛−2,𝑟_𝑛,𝑡=−𝑟_𝑛+1𝑞_𝑛𝑟_𝑛+1+2𝑞_𝑛−1𝑟_𝑛+𝑟²_𝑛𝑞²_𝑛−1𝑟_𝑛−𝑞_𝑛−2+𝑟_𝑛+2. (2.13)

Obviously, (2.13) is different from equation (2.12) appearing in [21], and their dynamic properties are very different, see the below discussion.

The Lax pairs of (2.13) are given by 𝑈_𝑛 in (2.1) and 𝑉_𝑛⁽²⁾ in (2.12) when 𝜃=1,𝑚=2. The 𝑉_𝑛⁽²⁾ is given by𝑉_𝑛⁽²⁾=𝑎_𝑛⁽⁰⁾𝜆⁴+𝑎_𝑛⁽¹⁾𝜆²+𝑎_𝑛⁽²⁾𝑏_𝑛⁽⁰⁾𝜆⁵+𝑏_𝑛⁽¹⁾𝜆³+𝑏_𝑛⁽²⁾𝜆𝑐_𝑛⁽⁰⁾𝜆⁵+𝑐_𝑛⁽¹⁾𝜆³+𝑐_𝑛⁽²⁾𝜆−𝑎_𝑛⁽⁰⁾𝜆⁴−𝑎_𝑛⁽¹⁾𝜆²−𝑎_𝑛⁽²⁾. (2.14)

By using (2.6), when 𝑗=0,1,2, substituting the values 𝑎_𝑛^(𝑗),𝑏_𝑛^(𝑗),𝑐_𝑛^(𝑗) into the above matrix and comparing the coefficients of the different power of 𝜆, we rewrite the matrix 𝑉_𝑛⁽²⁾ as𝑉_𝑛⁽²⁾=𝑉_4𝑛𝜆⁴+𝑉_3𝑛𝜆³+𝑉_2𝑛𝜆²+𝑉_1𝑛𝜆+𝑉_0𝑛, (2.15)

where 𝑉_4𝑛=−120012,𝑉_3𝑛=0𝑞_𝑛−1𝑟_𝑛0,𝑉_2𝑛=𝑞_𝑛−1𝑟_𝑛00−𝑞_𝑛−1𝑟_𝑛,𝑉_1𝑛=0𝑞_𝑛−21+𝑞_𝑛−1𝑟_𝑛−1−𝑞_𝑛−1𝑞_𝑛−1𝑟_𝑛+𝑞_𝑛−2𝑟_𝑛−1𝑟_𝑛+11+𝑞_𝑛𝑟_𝑛−𝑟_𝑛𝑞_𝑛𝑟_𝑛+1+𝑞_𝑛−1𝑟_𝑛0,𝑉_0𝑛=−𝑞²_𝑛−1𝑟²_𝑛+𝑞_𝑛−1𝑟_𝑛+1+𝑞_𝑛−2𝑟_𝑛00𝑞²_𝑛−1𝑟²_𝑛−𝑞_𝑛−1𝑟_𝑛+1−𝑞_𝑛−2𝑟_𝑛. (2.16)

Next, we establish the Darboux transformation of this new discrete lattice system (2.13). Actually, the Darboux transformation is a special gauge transformation𝜓_𝑛=𝑇_𝑛𝜓_𝑛 (2.17)

of the solutions of the Lax pairs (2.1) and (2.12) when 𝑚=2, where 𝜓_𝑛 must satisfy Lax pairs (2.1) and (2.12) with some 𝑈_𝑛 and 𝑉_𝑛⁽²⁾, that is,𝐸𝜓_𝑛=𝑈_𝑛𝜓_𝑛,𝑇_𝑛+1𝑈_𝑛=𝑈_𝑛𝑇_𝑛, (2.18)𝜓_𝑛,𝑡=𝑉_𝑛⁽²⁾𝜓_𝑛,𝑇_𝑛,𝑡+𝑇_𝑛𝑉_𝑛⁽²⁾=𝑉_𝑛⁽²⁾𝑇_𝑛. (2.19)

Meanwhile, it is required that 𝑈_𝑛 and 𝑉_𝑛⁽²⁾ have the same forms as 𝑈_𝑛 and 𝑉_𝑛⁽²⁾, respectively. Thus, the old potentials 𝑞_𝑛,𝑟_𝑛 in 𝑈_𝑛 and 𝑉_𝑛⁽²⁾ are mapped into new potentials 𝑞_𝑛,𝑟_𝑛̃ in 𝑈_𝑛 and 𝑉_𝑛⁽²⁾. We assume 𝑇_𝑛 has the following form:𝑇_𝑛=𝜆²+𝐴_𝑛𝐵_𝑛𝜆𝐶_𝑛𝜆𝜆²+𝐷_𝑛, (2.20)

where 𝐴_𝑛,𝐵_𝑛,𝐶_𝑛,𝐷_𝑛 are unknown functions which will be determined later.

Let 𝑦_𝑛=(𝑦¹_𝑛,𝑦²_𝑛)^𝑇 and 𝑧_𝑛=(𝑧¹_𝑛,𝑧²_𝑛)^𝑇 be two linear independent solutions of Lax pairs (2.1) and (2.12), and suppose that𝜓_𝑛=𝑦¹_𝑛𝑧¹_𝑛𝑦²_𝑛𝑧²_𝑛. (2.21)

Substituting (2.21) into (2.17), we obtain𝜓_𝑛=𝜆²+𝐴_𝑛𝑦¹_𝑛+𝐵_𝑛𝜆𝑦²_𝑛𝜆²+𝐴_𝑛𝑧¹_𝑛+𝐵_𝑛𝜆𝑧²_𝑛𝐶_𝑛𝜆𝑦¹_𝑛+𝜆²+𝐷_𝑛𝑦²_𝑛𝐶_𝑛𝜆𝑧¹_𝑛+𝜆²+𝐷_𝑛𝑧²_𝑛. (2.22)

We assume that 𝜆₁ and 𝜆₂ are two roots of det𝑇_𝑛(𝜆)=0. Therefore, when 𝜆=𝜆_𝑖(𝑖=1,2), the two columns of 𝑇_𝑛 and 𝜓_𝑛 are linear dependent, respectively. Thus, there exist two group of nonzero parameters 𝛼_𝑖=𝛼_𝑖(𝑛) and 𝛾_𝑖=𝛾_𝑖(𝑛) (𝑖=1,2) which satisfy the following equations:𝜆²_𝑖+𝐴_𝑛+𝛼_𝑖𝐵_𝑛𝜆_𝑖𝐶=0,_𝑛𝜆_𝑖+𝛼_𝑖𝜆²_𝑖+𝐷_𝑛𝜆=0,_2𝑖+𝐴_𝑛𝑦¹_𝑛+𝐵_𝑛𝜆_𝑖𝑦²_𝑛=𝛾_𝑖𝜆²_𝑖+𝐴_𝑛𝑧¹_𝑛+𝐵_𝑛𝜆_𝑖𝑧²_𝑛,𝐶_𝑛𝜆_𝑖𝑦¹_𝑛+𝜆²_𝑖+𝐷_𝑛𝑦²_𝑛=𝛾_𝑖𝐶_𝑛𝜆_𝑖𝑧¹_𝑛+𝜆²_𝑖+𝐷_𝑛𝑧²_𝑛. (2.23)

Solving the above equations, it follows that𝐴_𝑛=𝜆₁𝜆₂𝜆₁𝛼₂−𝜆₂𝛼₁𝜆₁𝛼₁−𝜆₂𝛼₂,𝐵_𝑛=𝜆²₂−𝜆²₁𝜆₁𝛼₁−𝜆₂𝛼₂,𝐶_𝑛=𝛼₁𝛼₂𝜆²₂−𝜆²₁𝜆₁𝛼₂−𝜆₂𝛼₁,𝐷_𝑛=𝜆₁𝜆₂𝜆₁𝛼₁−𝜆₂𝛼₂𝜆₁𝛼₂−𝜆₂𝛼₁,𝛼_𝑖=𝛼_𝑖(𝑛)=𝛾_𝑖𝑧²_𝑛−𝑦²_𝑛𝛾_𝑖𝑧¹_𝑛−𝑦¹_𝑛,𝑖=1,2, (2.24)(2.25)

where parameters 𝜆_𝑖 and 𝛾_𝑖 (note: 𝜆₁≠𝜆₂,𝛾₁≠𝛾₂) are suitably chosen such that all the denominators in (2.24) and (2.25) are nonzero.

When 𝜃=1, substituting (2.21) into (2.1), it follows,𝑦¹_𝑛+1=1+𝑞_𝑛𝑟_𝑛𝑦¹_𝑛+𝑞_𝑛𝑦²_𝑛𝜆_𝑖,𝑧¹_𝑛+1=1+𝑞_𝑛𝑟_𝑛𝑧¹_𝑛+𝑞_𝑛𝑧²_𝑛𝜆_𝑖,𝑦²_𝑛+1=𝑟_𝑛𝑦¹_𝑛𝜆_𝑖+𝑦²_𝑛𝜆²_𝑖,𝑧²_𝑛+1=𝑟_𝑛𝑧¹_𝑛𝜆_𝑖+𝑧²_𝑛𝜆²_𝑖. (2.26)

Applying the shift operator 𝐸 to all expressions in (2.26) once, it follows𝑦¹_𝑛=1+𝑞_𝑛−1𝑟_𝑛−1𝑦¹_𝑛−1+𝑞_𝑛−1𝑦²_𝑛−1𝜆_𝑖,𝑧¹_𝑛=1+𝑞_𝑛−1𝑟_𝑛−1𝑧¹_𝑛−1+𝑞_𝑛−1𝑧²_𝑛−1𝜆_𝑖𝑦²_𝑛=𝑟_𝑛−1𝑦¹_𝑛−1𝜆_𝑖+𝑦²_𝑛−1𝜆²_𝑖,𝑧²_𝑛=𝑟_𝑛−1𝑧¹_𝑛−1𝜆_𝑖+𝑧²_𝑛−1𝜆²_𝑖. (2.27)

Solving (2.27) yields𝑦¹_𝑛−1=𝜆_𝑖⁻¹𝑦¹_𝑛𝜆_𝑖−𝑞_𝑛−1𝑦²_𝑛,𝑧¹_𝑛−1=𝜆_𝑖⁻¹𝑧¹_𝑛𝜆_𝑖−𝑞_𝑛−1𝑧²_𝑛,𝑦²_𝑛−1=𝜆_𝑖⁻²1+𝑞_𝑛−1𝑟_𝑛−1𝑦²_𝑛−𝑟_𝑛−1𝑦¹_𝑛𝜆_𝑖,𝑧²_𝑛−1=𝜆_𝑖⁻²1+𝑞_𝑛−1𝑟_𝑛−1𝑧²_𝑛−𝑟_𝑛−1𝑧¹_𝑛𝜆_𝑖. (2.28)

By using (2.26)–(2.28), from (2.25), we get𝛼_𝑖()=𝜇𝑛+1^𝑖𝜈_𝑖,𝛼_𝑖()=𝛽𝑛−1^𝑖𝜔_𝑖,𝑖=1,2, (2.29)

where𝜇_𝑖=𝜇_𝑖(𝑛)=𝑟_𝑛𝜆_𝑖+𝛼_𝑖𝜆²_𝑖,𝜈_𝑖=𝜈_𝑖(𝑛)=1+𝑞_𝑛𝑟_𝑛+𝛼_𝑖𝑞_𝑛𝜆_𝑖,𝛽_𝑖=𝛽_𝑖(𝑛)=1+𝑞_𝑛−1𝑟_𝑛−1𝛼_𝑖−𝑟_𝑛−1𝜆_𝑖,𝜔_𝑖=𝜔_𝑖(𝑛)=𝜆²_𝑖−𝛼_𝑖𝑞_𝑛−1𝜆_𝑖. (2.30)

According to the above analysis, we derive some useful functions in forthcoming computation process as follows:𝐴_𝑛+1=𝜆₁𝜆₂𝜆₁𝜇₂𝜈₁−𝜆₂𝜇₁𝜈₂𝜆₁𝜇₁𝜈₂−𝜆₂𝜇₂𝜈₁,𝐵_𝑛+1=𝜈₁𝜈₂𝜆²₂−𝜆²₁𝜆₁𝜇₁𝜈₂−𝜆₂𝜇₂𝜈₁,𝐶_𝑛+1=𝜇₁𝜇₂𝜆²₂−𝜆²₁𝜆₁𝜇₂𝜈₁−𝜆₂𝜇₁𝜈₂,𝐷_𝑛+1=𝜆₁𝜆₂𝜆₁𝜇₁𝜈₂−𝜆₂𝜇₂𝜈₁𝜆₁𝜇₂𝜈₁−𝜆₂𝜇₁𝜈₂,𝐴_𝑛−1=𝜆₁𝜆₂𝜆₁𝛽₂𝜔₁−𝜆₂𝛽₁𝜔₂𝜆₁𝛽₁𝜔₂−𝜆₂𝛽₂𝜔₁,𝐵_𝑛−1=𝜔₁𝜔₂𝜆²₂−𝜆²₁𝜆₁𝛽₁𝜔₂−𝜆₂𝛽₂𝜔₁,𝐶_𝑛−1=𝛽₁𝛽₂𝜆²₂−𝜆²₁𝜆₁𝛽₂𝜔₁−𝜆₂𝛽₁𝜔₂,𝐷_𝑛−1=𝜆₁𝜆₂𝜆₁𝛽₁𝜔₂−𝜆₂𝛽₂𝜔₁𝜆₁𝛽₂𝜔₁−𝜆₂𝛽₁𝜔₂. (2.31)(2.32)

We know that the matrix 𝑈_𝑛 keeps the same form as 𝑈_𝑛 in the procedure of mapping the old potentials 𝑞_𝑛,𝑟_𝑛 in 𝑈_𝑛 into new potentials 𝑞_𝑛,𝑟_𝑛̃ in 𝑈_𝑛. We can suppose that𝑈_𝑛=1+𝑞_𝑛𝑟_𝑛̃𝜆𝑞_𝑛𝜆𝑟_𝑛̃𝜆². (2.33)

Substituting 𝑇_𝑛+1,𝑈_𝑛,𝑇_𝑛,𝑈_𝑛 into the second formula in (2.18) and comparing the coefficients of the same power of 𝜆, it follows that1+𝑞_𝑛𝑟_𝑛̃+𝑞_𝑛𝐶_𝑛=1+𝑞_𝑛𝑟_𝑛+𝑟_𝑛𝐵_𝑛+1,𝐴_𝑛1+𝑞_𝑛𝑟_𝑛̃=𝐴_𝑛+11+𝑞_𝑛𝑟_𝑛,𝑞_𝑛=𝑞_𝑛+𝐵_𝑛+1,1+𝑞_𝑛𝑟_𝑛̃𝐵_𝑛+𝑞_𝑛𝐷_𝑛=𝑞_𝑛𝐴_𝑛+1,𝑟_𝑛̃+𝐶_𝑛=𝑟_𝑛,𝑟_𝑛̃𝐴_𝑛=1+𝑞_𝑛𝑟_𝑛𝐶_𝑛+1+𝑟_𝑛𝐷_𝑛+1,𝑟_𝑛̃𝐵_𝑛+𝐷_𝑛=𝑞_𝑛𝐶_𝑛+1+𝐷_𝑛+1. (2.34)(2.35)(2.36)(2.37)(2.38)(2.39)(2.40)

From (2.36) and (2.38), it is easy to find that there exists a transformation which maps the old potentials 𝑞_𝑛,𝑟_𝑛 into new ones 𝑞_𝑛,𝑟_𝑛̃ as follows:𝑞_𝑛=𝑞_𝑛+𝐵_𝑛+1,𝑟_𝑛̃=𝑟_𝑛−𝐶_𝑛. (2.41)

Respectively, substituting (2.41) into (2.34), (2.35), (2.37), (2.39), and (2.40), we obtain five conditional equations as follows:𝑞_𝑛+𝐵_𝑛+1𝑟_𝑛−𝐶_𝑛+𝑞_𝑛+𝐵_𝑛+1𝐶_𝑛−𝑞_𝑛𝑟_𝑛−𝑟_𝑛𝐵_𝑛+1𝐴=0,_𝑛1+𝑞_𝑛+𝐵_𝑛+1𝑟_𝑛−𝐶_𝑛−𝐴_𝑛+11+𝑞_𝑛𝑟_𝑛𝐵=0,_𝑛1+𝑞_𝑛+𝐵_𝑛+1𝑟_𝑛−𝐶_𝑛+𝑞_𝑛+𝐵_𝑛+1𝐷_𝑛−𝑞_𝑛𝐴_𝑛𝑟=0,_𝑛−𝐶_𝑛𝐴_𝑛−𝐶_𝑛+11+𝑞_𝑛𝑟_𝑛−𝑟_𝑛𝐷_𝑛+1𝑟=0,_𝑛−𝐶_𝑛𝐵_𝑛−𝑞_𝑛𝐶_𝑛+1+𝐷_𝑛−𝐷_𝑛+1=0. (2.42)

Substituting (2.24)-(2.25), (2.29)-(2.30), and (2.31)-(2.32) into the above five conditional equations, taking a tedious direct computation or using the software Maple, we can verify all the above five equations are identities.

In fact, we have directly verified that the (2.41) is the Darboux transformation of the spectral problem (2.1) under the gauge transformation (2.17). Thus, the proof process such as those in [21] can be simplified. Next, we need to verify that the matrix 𝑉_𝑛⁽²⁾ defined by (2.19) has the same form as 𝑉_𝑛⁽²⁾, that is,𝑉_𝑛⁽²⁾=𝑉_4𝑛𝜆⁴+𝑉_3𝑛𝜆³+𝑉_2𝑛𝜆²+𝑉_1𝑛𝜆+𝑉_0𝑛, (2.43)

where all the forms of 𝑉_4𝑛, 𝑉_3𝑛, 𝑉_2𝑛, 𝑉_1𝑛, 𝑉_0𝑛 are the same as the forms of 𝑉_4𝑛, 𝑉_3𝑛, 𝑉_2𝑛, 𝑉_1𝑛, 𝑉_0𝑛. The matrices 𝑉_4𝑛, 𝑉_3𝑛, 𝑉_2𝑛, 𝑉_1𝑛, 𝑉_0𝑛 can be obtained as long as we use the 𝑞 and 𝑟̃ to take the place of 𝑞 and 𝑟 in the matrices 𝑉_4𝑛, 𝑉_3𝑛, 𝑉_2𝑛, 𝑉_1𝑛, 𝑉_0𝑛, respectively.

To simplify computation, according to the coefficients of the different power of 𝜆, we also rewrite the matrices 𝑇_𝑛,𝑇_𝑛,𝑡 as𝑇_𝑛=𝜆²+𝐴_𝑛𝐵_𝑛𝜆C_𝑛𝜆𝜆²+𝐷_𝑛=𝐼𝜆²+𝑇_1𝑛𝜆+𝑇_0𝑛, (2.44) where𝐼=1001,𝑇_1𝑛=0𝐵_𝑛𝐶_𝑛0,𝑇_0𝑛=𝐴_𝑛00𝐷_𝑛,𝑇_𝑛,𝑡=𝐴_𝑛,𝑡𝐵_𝑛,𝑡𝜆𝐶_𝑛,𝑡𝜆𝐷_𝑛,𝑡=𝑇_1𝑛,𝑡𝜆+𝑇_0𝑛,𝑡,𝑇_1𝑛,𝑡=0𝐵_𝑛,𝑡𝐶_𝑛,𝑡0,𝑇_0𝑛,𝑡=𝐴_𝑛,𝑡00𝐷_𝑛,𝑡. (2.45)(2.46)

Substituting 𝑇_𝑛,𝑡,𝑉_𝑛⁽²⁾,𝑉_𝑛⁽²⁾ into the second formula in (2.19) yields𝑇_1𝑛,𝑡𝜆+𝑇_0𝑛,𝑡+𝐼𝜆²+𝑇_1𝑛𝜆+𝑇_0𝑛𝑉_4𝑛𝜆⁴+𝑉_3𝑛𝜆³+𝑉_2𝑛𝜆²+𝑉_1𝑛𝜆+𝑉_0𝑛−𝑉_4𝑛𝜆⁴+𝑉_3𝑛𝜆³+𝑉_2𝑛𝜆²+𝑉_1𝑛𝜆+𝑉_0𝑛𝐼𝜆²+𝑇_1𝑛𝜆+𝑇_0𝑛=𝑂. (2.47)

Letting the coefficients of the 𝜆⁶,𝜆⁵,𝜆⁴,𝜆³,𝜆²,𝜆¹,𝜆⁰ be zero, it follows that𝑉_4𝑛−𝑉_4𝑛𝑇=𝑂,_1𝑛𝑉_4𝑛−𝑉_4𝑛𝑇_1𝑛+𝑉_3𝑛−𝑉_3𝑛𝑇=𝑂,_0𝑛𝑉_4𝑛−𝑉_4𝑛𝑇_0𝑛+𝑇_1𝑛𝑉_3𝑛−𝑉_3𝑛𝑇_1𝑛+𝑉_2𝑛−𝑉_2𝑛𝑇=𝑂,_0𝑛𝑉_3𝑛−𝑉_3𝑛𝑇_0𝑛+𝑇_1𝑛𝑉_2𝑛−𝑉_2𝑛𝑇_1𝑛+𝑉_𝑛−𝑉_1𝑛𝑇=𝑂,_0𝑛𝑉_2𝑛−𝑉_2𝑛𝑇_0𝑛+𝑇_1𝑛𝑉_1𝑛−𝑉_1𝑛𝑇_1𝑛+𝑉_0𝑛−𝑉_0𝑛𝑇=𝑂,_1𝑛,𝑡+𝑇_0𝑛𝑉_1𝑛−𝑉_1𝑛𝑇_0𝑛+𝑇_1𝑛𝑉_0𝑛−𝑉_0𝑛𝑇_1𝑛𝑇=𝑂,_0𝑛,𝑡+𝑇_0𝑛𝑉_0𝑛−𝑉_0𝑛𝑇_0𝑛=𝑂. (2.48)(2.49)(2.50)(2.51)(2.52)(2.53)(2.54)

So (2.48) holds. By dint of (2.41), we can easily verify that (2.49) and (2.50) hold. Under the transformation (2.41), substituting (2.24), (2.30), (2.31), and (2.32) into (2.51) and (2.52), taking a tedious direct computation or using the software Maple, we can verify that (2.51) and (2.52) hold. To simplify computation to verify (2.53) and (2.54), we need some other auxiliary conditions:𝐴_𝑛,𝑡=𝜆₁𝜆₂𝜆²₁−𝜆²₂𝛼_2,𝑡𝛼₁−𝛼₂𝛼_1,𝑡𝜆₁𝛼₁−𝜆₂𝛼₂²,𝐵_𝑛,𝑡=𝜆²₁−𝜆²₂𝜆₁𝛼_1,𝑡−𝜆₂𝛼_2,𝑡𝜆₁𝛼₁−𝜆₂𝛼₂²,𝐶_𝑛,𝑡=𝜆²₁−𝜆²₂−𝜆₁𝛼²₂𝛼_1,𝑡+𝜆₂𝛼²₁𝛼_2,𝑡−𝜆₁𝛼₂+𝜆₂𝛼₁²,𝐷_𝑛,𝑡=𝜆₁𝜆₂𝜆²₁−𝜆²₂𝛼₂𝛼_1,𝑡−𝛼₁𝛼_2,𝑡−𝜆₁𝛼₂+𝜆₂𝛼₁², (2.55)𝛼_𝑖,𝑡=𝑣₂₁𝜆_𝑖−2𝑣₁₁𝜆_𝑖𝛼_𝑖+𝑣₁₂𝜆_𝑖𝛼²_𝑖, (2.56)

where𝑣₁₁𝜆_𝑖1=−2𝜆⁴_𝑖+𝑞_𝑛−1𝑟_𝑛𝜆²_𝑖−𝑞²_𝑛−1𝑟²_𝑛+𝑞_𝑛−1𝑟_𝑛+1+𝑞_𝑛−2𝑟_𝑛,𝑣₁₂𝜆_𝑖=𝑞_𝑛−1𝜆³_𝑖+𝑞_𝑛−21+𝑞_𝑛−1𝑟_𝑛−1−𝑞_𝑛−1𝑞_𝑛−1𝑟_𝑛+𝑞_𝑛−2𝑟_𝑛−1𝜆_𝑖,𝑣₂₁𝜆_𝑖=𝑟_𝑛𝜆³_𝑖+𝑟_𝑛+11+𝑞_𝑛𝑟_𝑛−𝑟_𝑛𝑞_𝑛𝑟_𝑛+1+𝑞_𝑛−1𝑟_𝑛𝜆_𝑖,𝑖=1,2. (2.57)

On the other hand,det𝑇_𝑛=𝜆²²+𝐴_𝑛+𝐷_𝑛−𝐵_𝑛𝐶_𝑛𝜆²+𝐴_𝑛𝐷_𝑛. (2.58)

Thus, we obtain𝐴_𝑛+𝐷_𝑛−𝐵_𝑛𝐶_𝑛=−𝜆²₁+𝜆²₂,𝐴_𝑛𝐷_𝑛=𝜆²₁𝜆²₂, (2.59)

and their differentiation with respect to time 𝑡𝐴_𝑛,𝑡+𝐷_𝑛,𝑡=𝐵_𝑛,𝑡𝐶_𝑛+𝐵_𝑛𝐶_𝑛,𝑡,𝐴_𝑛,𝑡𝐷_𝑛+𝐴_𝑛𝐷_𝑛,𝑡=0. (2.60)

Applying (2.24), (2.30), (2.31), (2.32), (2.41), and (2.55)–(2.60), we can verify that (2.53) and (2.54) hold. Hence, the following theorem holds.

Theorem 2.1. The solutions (𝑞_𝑛,𝑟_𝑛) of the discrete lattice system (2.13) are mapped into their new solutions (𝑞_𝑛,𝑟_𝑛̃) under the Darboux transformation (2.41).

3. New Exact Solutions and Their Properties

In this section, we will construct exact solutions of discrete system (2.13) by using the Darboux transformation (2.41). Obviously, the 𝑞_𝑛=𝑟_𝑛=0 are a couple of seed solutions of system (2.13). In addition, all the constants that satisfy conditional equation 𝑞_𝑛𝑟_𝑛=2±√3 are seed solutions of system (2.13).Without loss of generality, we only consider two kinds of seed solutions, that is, 𝑞_𝑛=𝑟_𝑛=0 and𝑞_𝑛=1,𝑟_𝑛=2+√3.

3.1. The Exact Solution Obtained by Nonzero Seed and Its Properties

When 𝜃=1 and 𝑚=2, substituting the seed solutions 𝑞_𝑛=1,𝑟_𝑛=2+√3 into the Lax pairs (2.1) and (2.12), it follows that𝐸𝜓_𝑛=3+√√3𝜆2+3𝜆𝜆²𝜓_𝑛,𝜓_𝑛,𝑡=𝑎₁₁𝑎₁₂𝑎₂₁𝑎₂₂𝜓_𝑛, (3.1)

where𝑎₁₁=𝑎₁₁(𝜆)1=−2𝜆⁴+2+√3𝜆²−3+2√3,𝑎₁₂=𝑎₁₂(𝜆)=𝜆³−1+√3𝑎𝜆,₂₁=𝑎₂₁(𝜆)=2+√3𝜆³−5+3√3𝜆,𝑎₂₂=𝑎₂₂(𝜆)=12𝜆⁴−2+√3𝜆²+3+2√3. (3.2)

 Equation (3.1) has two real linear independent basic solutions as follows:𝑦_𝑛=𝜏^𝑛₁𝑎exp²²+𝑎₁₁+√Δ2𝑡𝑎₂₂−𝑎₁₁+√Δ2𝑎₁₂,𝑧_𝑛=𝜏^𝑛₂𝑎exp²²+𝑎₁₁−√Δ2𝑡𝑎₂₂−𝑎₁₁−√Δ2𝑎₁₂, (3.3)

where𝜏₁=2+√3𝑎₂₂−𝑎₁₁+√Δ𝜆+2𝑎₁₂𝜆²2𝑎₁₂,𝜏₂=2+√3𝑎₂₂−𝑎₁₁−√Δ𝜆+2𝑎₁₂𝜆²2𝑎₁₂,Δ=𝑎₂₂−𝑎₁₁²+4𝑎₁₂𝑎₂₁=1+√3−𝜆²_𝑖²𝜆_𝑖−1²+2+√3𝜆_𝑖+1²+2+√3>0. (3.4)(3.5)

By using (2.25), we obtain𝛼_𝑖=2𝜆³_𝑖−1+√3𝜆_𝑖𝛾_𝑖−𝛿^𝑛_𝑖𝑒^𝜉𝑖𝑡𝛾_𝑖1+√3−𝜆²_𝑖3+√3−𝜆²_𝑖−𝜉_𝑖−𝛿^𝑛_𝑖1+√3−𝜆²_𝑖3+√3−𝜆²_𝑖+𝜉_𝑖𝑒^𝜉𝑖𝑡, (3.6)

where𝛿_𝑖=𝜏₁𝜆_𝑖𝜏₂𝜆_𝑖=2+√31+√3−𝜆²_𝑖3+√3−𝜆²_𝑖+𝜉_i𝜆_𝑖+2𝜆³_𝑖−1+√3𝜆_𝑖𝜆²_𝑖2+√31+√3−𝜆²_𝑖3+√3−𝜆²_𝑖−𝜉_𝑖𝜆_𝑖+2𝜆³_𝑖−1+√3𝜆_𝑖𝜆²_𝑖,𝜉_𝑖=1+√3−𝜆²_𝑖²𝜆_𝑖−1²+2+√3𝜆_𝑖+1²+2+√3,𝑖=1,2. (3.7)

Therefore, by using the Darboux transformation (2.41), (2.24), and (2.31), we obtain the new explicit solution of (2.13) as follows:𝑞_𝑛=1+𝐵_𝑛+1𝜈=1+¹𝜈₂𝜆²₂−𝜆²₁𝜆₁𝜇₁𝜈₂−𝜆₂𝜇₂𝜈₁,𝑟_𝑛̃=2+√3−𝐶_𝑛=2+√𝛼3−¹𝛼₂𝜆²₂−𝜆²₁𝜆₁𝛼₂−𝜆₂𝛼₁, (3.8)where𝜇_𝑖=2+√3𝜆_𝑖+𝛼_𝑖𝜆²_𝑖,𝜈_𝑖=3+√3+𝛼_𝑖𝜆_𝑖 (3.9)

which are reduced from (2.30); the parameters 𝛼_𝑖,𝑖=1,2 are defined by (3.6) and 𝜆_𝑖≠±1+√3,𝜆₁≠±𝜆₂.

Equation (3.8) is a very complicated solution due to (3.6), and it is hard to analyze its dynamical behavior. Same as Xu, we did not know whether the26th solution given in [23] was a soliton solution. We also do not know whether the solution (3.8) is a soliton solution if we only estimate the type of solution from its expression. By the further detailed analysis, we know that the solution (3.8) is not a pure soliton solution, its properties and profiles vary accordingly while the parameters vary. In other words, under the different values of parameters, it contains different waveform. However, in [21], treating 𝑛 as continuous variable, Qin showed that solution (4.4) possesses properties of one-soliton solutions and two-soliton solutions when 𝛾₁,𝛾₂,𝜆₁, and 𝜆₂ are taken as different parameters, respectively. In fact, the 𝑛 is a discrete variable, we cannot regard it as a continuous variable. The dynamic properties and profiles of solution would be distortion if we regard the discrete variable 𝑛 as a continuous variable. Therefore, regarding 𝑛 as a discrete variable, we will plot graphs of solution (3.8), some new properties and interesting phenomena will be produced. This shows that our analytic work on dynamic properties of exact solution obtained by the Darboux transformation is very different from the Qin’s work.

As an example, when 𝛾_𝑖(𝑖=1,2) are restricted in the region 0<𝛾_𝑖<1, we investigate the dynamical behavior of solution (3.8). The term 𝛿^𝑛_𝑖𝑒^𝜉𝑖𝑡 in (3.6) can be rewritten as 𝛿^𝑛_𝑖𝑒^𝜉𝑖𝑡=𝑒^{𝑛ln(𝛿𝑖)+𝜉𝑖𝑡} when 𝛿_𝑖>0, but it cannot be rewritten as this form when 𝛿_𝑖<0. It is for this reason that solution (3.8) has variety of properties. For example, when 𝛿₁>0 and 𝛿₂>0 are both held, solution (3.8) has no oscillation behavior. This can be seen in Figure 1.

(a) t = −0.003

(b) t = 0

(c) t = 0.002

(d) t = −0.003

(e) t = 0

(f) t = 0.002

Figure 1 

The profiles of 𝑝𝑛,𝑟𝑛̃ in (3.8) for 𝜆1=5,𝜆2=12,𝛾1=0.5,𝛾2=0.6,𝛿1>0,𝛿2>0.

Figure 1 shows the properties of double-kink soliton and single-kink soliton with singularity for solutions 𝑝_𝑛,𝑟_𝑛̃, respectively. Figures 1(a)–1(c) show a procedure of leftwards moving for discrete double kink soliton. The waveform gradually changes from Figure 1(a) to Figure 1(c). Particulary, when 𝑡=0, the waveform becomes a static singular kink soliton, which can be seen in Figure 1(b). Figures 1(d)–1(f) also show a procedure of waveform gradually changes for discrete kink wave. The waveform is a discrete single kink soliton when the time 𝑡<0, which can be seen in Figure 1(d). Particulary, when the time 𝑡=0, the waveform also becomes a static singular kink soliton, which can be seen in Figure 1(e). When the time 𝑡>0, the waveform becomes a moving singular kink wave, which can be seen in Figure 1(f).

When both 𝛿₁<0 and 𝛿₂<0 hold, the solution (3.8) is discrete breather solution; it has breather oscillation behavior shown in Figure 2.In fact, Figure 2 shows the properties of breather soliton with double-kink characteristic for solutions 𝑝_𝑛,𝑟_𝑛̃.

(a) t = −2

(b) t = 0

(c) t = 2

(d) t = −2

(e) t = 0

(f) t = 2

Figure 2 

The profiles of 𝑝𝑛,𝑟𝑛̃ in (3.8) for 𝜆1=1.8,𝜆2=2,𝛾1=0.5,𝛾2=0.6,𝛿1<0,𝛿2<0.

Figures 2(a)–2(c) show a procedure of breather oscillation which moves from the right to the left as the time 𝑡 increases from −2 to 2. The waveform and amplitude gradually change from Figure 2(a) to Figure 2(c). Figures 2(d)–2(f) also show a procedure of another breather oscillation which moves from the right to the left; the properties and profiles are very similar to those shown in Figures 2(a)–2(c).

However, when 𝛿₁>0,𝛿₂<0 (or 𝛿₁<0,𝛿₂>0), solution (3.8) has no breather oscillation behavior anymore. In fact, Figure 3 shows some shapes of singular kink and anti-kink soliton for solutions 𝑝_𝑛,𝑟_𝑛̃.

(a) t = −2

(b) t = 0

(c) t = 3

(d) t = −2

(e) t = 0

(f) t = 3

Figure 3 

The profiles of 𝑝𝑛,𝑟𝑛̃ in (3.8) for 𝜆1=5,𝜆2=1,𝛾1=0.5,𝛾2=0.6,𝛿1>0,𝛿2<0.

Figures 3(a)–3(c) show a procedure of singular antikink soliton which moves from the right to the left as the time 𝑡 increases from −2 to 3. The waveform and amplitude gradually change from the form in Figure 3(a) to that in Figure 3(c). Figures 3(d)–3(f) also show a procedure of singular kink soliton which moves from the right to the left; the other properties and profiles are very similar to those shown in Figures 3(a)–3(c) except for the static case at the time 𝑡=0. The waveform in Figure 3(b) has singularity; however, this is not the case in Figure 3(e).

3.2. The Exact Solution Obtained by Zero Seed and Its Properties

When 𝜃=1 and 𝑚=2, substituting the seed solutions 𝑞_𝑛=𝑟_𝑛=0 into the Lax pairs (2.1) and (2.12), it follows that𝐸𝜓_𝑛=100𝜆²𝜓_𝑛,𝜓_𝑛,𝑡=−12𝜆⁴0012𝜆⁴𝜓_𝑛. (3.10)

Equation (3.10) has two real linear independent basic solutions as follows:𝑦_𝑛−1=exp2𝜆⁴𝑡10,𝑧_𝑛=𝜆^2𝑛1exp2𝜆⁴𝑡01. (3.11)

From (3.11), let 𝑦¹_𝑛=exp((−1/2)𝜆⁴𝑡),𝑦²_𝑛=0,𝑧¹_𝑛=0,𝑧²_𝑛=exp((1/2)𝜆⁴𝑡); by using (2.25) and (2.30),we obtain𝛼_𝑖=−𝛾_𝑖𝜆_𝑖^2𝑛𝑒^{𝜆4𝑖𝑡},𝜇_𝑖=𝛼_𝑖𝜆²_𝑖,𝜈_𝑖=1. (3.12)

Similarly, by using the Darboux transformation (2.41) and (2.24), (2.31), and (3.12), we obtain another explicit solution of (2.13) as follows:𝑞_𝑛=𝐵_𝑛+1=𝜈₁𝜈₂𝜆²₂−𝜆²₁𝜆₁𝜇₁𝜈₂−𝜆₂𝜇₂𝜈₁=𝜆²₁−𝜆²₂𝛾₁𝜆₁^2𝑛+3𝑒^𝜆41𝑡−𝛾₂𝜆₂^2𝑛+3𝑒^𝜆42𝑡,𝑟_𝑛̃=−𝐶_𝑛𝛼=−¹𝛼₂𝜆²₂−𝜆²₁𝜆₁𝛼₂−𝜆₂𝛼₁=𝛾₁𝛾₂𝜆₁𝜆₂^2𝑛𝜆²₂−𝜆²₁𝑒^{(𝜆41+𝜆42)𝑡}𝜆₁𝛾₂𝜆₂^2𝑛𝑒^𝜆42𝑡−𝜆₂𝛾₁𝜆₁^2𝑛𝑒^𝜆41𝑡. (3.13)

In fact, solution (3.13) is not a pure soliton solution for arbitrary parametric values. By the further detailed analysis, we know that solution (3.13) becomes a soliton solution if one of the following parametric conditions holds:(i)𝜆₁>1,0<𝜆₂<1or0<𝜆₁<1,𝜆₂>1,and𝛾₁𝛾₂<0;(ii)𝜆₁<−1,0<𝜆₂<1or0<𝜆₁<1,𝜆₂<−1and𝛾₁𝛾₂>0;(iii)|𝜆₁|=1,|𝜆₂|≠1,𝜆₁𝜆₂<0,and𝛾₁𝛾₂>0;(iv)|𝜆₁|≠1,|𝜆₂|=1,𝜆₁𝜆₂>0,and𝛾₁𝛾₂<0.

Under the above parametric conditions (i) or (ii), the solutions 𝑝_𝑛,𝑟_𝑛̃ in (3.13) show discrete bright or dark soliton, which are shown in Figure 4. Under the above parametric conditions (iii) or (iv), the solutions 𝑝_𝑛,𝑟_𝑛̃ in (3.13) show discrete kink or anti-kink soliton, which are shown in Figure 5.

(a) 𝑡=0.2, 𝑛∈[−20,20]

(b) 𝑡=0.2, 𝑛[−40,40]

(c) 𝑡=0.2, 𝑛∈[−20,20]

(d) 𝑡=0.2, 𝑛∈[−40,40]

Figure 4 

The solutions of 𝑝𝑛,𝑟𝑛̃ in (3.13) show a shape of discrete bright or dark soliton for fixed parametric values. Take (a) and (c) parameters as 𝜆1=2,𝜆2=0.9,𝛾1=0.5,𝛾2=0.6, and take (b) and (d) parameters as 𝜆1=0.8,𝜆2=−1.2,𝛾1=0.5,𝛾2=1.6.

(a) 𝑡=0.2,𝑛∈[−20,20]

(b) 𝑡=0.2,𝑛∈[−40,40]

(c) 𝑡=0.2,𝑛∈[−20,20]

(d) 𝑡=0.2,𝑛∈[−40,40]

Figure 5 

The solutions of 𝑝𝑛,𝑟𝑛̃ in (3.13) show a shape of discrete kink or antikink soliton for fixed parametric values. Take (a) and (c) parameters as 𝜆1=1,𝜆2=2,𝛾1=−0.5,𝛾2=0.6, and take (a) and (c) parameters as 𝜆1=−1,𝜆2=0.9,𝛾1=−0.5,𝛾2=−0.6.

Under the other parametric conditions, solution (3.13) is not soliton, therefore, we omit those parts of graphs and discussions on dynamic properties.

4. Conclusions

In this paper, we show that the discrete system (1.1) which is given by Qin in [21] can be reduced to the old integrable lattice system which is given by Merola et al. [22]. Based on the Qin’s work, we present an indeed new discrete integrable system (2.13). Its Darboux transformation is established, and two complex exact solutions (3.8) and (3.13) are obtained by this transformation. By the further detailed analysis, we know that neither of these two exact solutions is pure soliton solutions. Their waveforms vary accordingly while the parameters vary. In other words, under different parametric conditions, these kinds of exact solutions show different dynamic behavior and profiles. In addition, by using different seeds, the obtained solutions are different, so are their properties and waveforms. The solutions obtained by nonzero constant seed mainly show the breather oscillation or singular behavior, but the solutions obtained by zero seed mainly show the profiles of discrete bright or dark soliton and kink or antikink soliton. These show that some exact solutions of discrete integrable systems obtained by the Darboux transformation are it is hard to know whether soliton solutions if we estimate only the types of solution from their expression. It is just because this reason that Xu did not know whether the solution obtained in [23] was a soliton solution. Perhaps, as in [24, 25], we call these kinds of solutions N-wave solutions or complexitons which is more accurate.Now, under what kind of parametric conditions, can the exact solutions of discrete integrable systems obtained by Darboux transformation show properties of soliton? This is a very interesting problem. We wish more and more researchers pay attention to this enthralling problem in the future works.

Acknowledgments

The authors thank three reviewers very much for their useful comments and helpful suggestions. This work was financially supported by the Natural Science Foundation of China (Grant no. 11161038). It was also supported by the Natural Science Foundations of Yunnan Province (Grant no. 2011FZ193) and Zhejiang Province (Grant no. Y2111160).