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

Authors: Xianbin Wu; Weiguo Rui; Xiaochun Hong
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## 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.


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## Body

## 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)](1.1)qn,t=qn2rn+1-qn-1,rn,t=-rn2qn-1+rn+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 transformationrn+1→r̃n,  (1.1) can be rewritten as(1.2)qn,t=qn2r̃n-qn-1,r̃n,t=-r̃n2qn+r̃n+1.Taking another local coordinate transformationn→-n¯,  (1.2) can be rewritten as(1.3)q-n¯,t=q-n¯2r̃-n¯-q-n¯-1,r̃-n¯,t=-r̃-n¯2q-n¯+r̃-n¯+1.Taking the third coordinate transformation-n¯→n̂ and -(n¯+1)→n̂+1,-(n¯-1)→n̂-1, (1.3) can be rewritten as(1.4)qn̂,t=qn̂2r̃n̂-qn̂+1,r̃n̂,t=-r̃n̂2qn̂+r̃n̂-1.To simplify notation, we still usen and rn to take the place of n̂ and r̃n. Thus, (1.4) can be rewritten as(1.5)qn,t=qn2rn-qn+1,rn,t=-qnrn2+rn-1.Equation (1.5)is equivalent to the following Equation:(1.6)(|q〉|r〉)t1=((〈q∣r〉-E)|q〉(E-1-〈q∣r〉)|r〉),whereE is a shift operator and |q〉↔qn,|r〉↔rn,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 Section2, 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:(2.1)Eψn=Un(qn,rn,λ)ψn,where(2.2)Un=(1+θqnrnλqnλrnλ2),θ=0,1.This is a generalization of the Ablowitz-Ladik spectral problem; its corresponding auxiliary problem is shown as follows:(2.3)ψn,t=Vn(qn,rn,λ)ψn.First, we consider the stationary discrete zero curvature equation(2.4)(EVn)Un-UnVn=O,where(2.5)Vn=(∑j=0∞an(j)λ-2j∑j=0∞bn(j)λ-2j+1∑j=0∞cn(j)λ-2j+1-∑j=0∞an(j)λ-2j),O=(0000).Substituting (2.2) and (2.5) into (2.4), we obtain a series of recursive formulas as follows:(2.6)(1+θqnrn)[(E-1)an(j)]+rnbn+1(j+1)-qncn(j+1)=0,bn+1(j+1)+qn[(E+1)an(j)]-(1+θqnrn)bn(j)=0,-cn(j+1)-rn[(E+1)an(j)]+(1+θqnrn)cn+1(j)=0,qncn+1(j)-rnbn(j)-(E-1)an(j)=0,bn+1(0)=cn(0)=0.Second, we consider the discrete zero curvature equation(2.7)Un,t=(EVn(m))Un-UnVn(m)
with(2.8)Vn(m)=(λ2mVn+)+Δn(m),where the positive power termVn+ and modification term Δn(m) are, respectively, given by(2.9)Vn+=(∑j=0man(j)λ-2j∑j=0mbn(j)λ-2j+1∑j=0mcn(j)λ-2j+1-∑j=0man(j)λ-2j),Δn(m)=(-(1-θ)an(m)000).SubstitutingUn,Vn(m) into the discrete zero curvature equation (2.7) and recurring to the stationary discrete zero curvature equation (2.3) yield(2.10)Un,t=(EΔn(m))Un-UnΔn(m).Thus, we obtain(2.11)qn,tm=-bn+1(m+1)-(1-θ)qnan+1(m),rn,tm=cn(m+1)+(1-θ)rnan(m),θ(qnrn)tm=θ(1+θqnrn)(an+1(m)-an(m)),m=0,1,2,….
Obviously, when θ=0 and θ=1, all equations in (2.11) satisfy compatibility condition between (2.1) and the following auxiliary spectral problem:(2.12)ψn,t=Vn(m)ψn.Whenj=0,1 and θ=1,m=1, lettingan(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 an(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:(2.13)qn,t=qn-1(qn-1rn+2qnrn+1)-qn2(qnrn+12-rn+2)-qn-2,rn,t=-rn+1(qnrn+1+2qn-1rn)+rn2(qn-12rn-qn-2)+rn+2.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 Un in (2.1) and Vn(2) in (2.12) when θ=1,m=2. The Vn(2) is given by(2.14)Vn(2)=(an(0)λ4+an(1)λ2+an(2)bn(0)λ5+bn(1)λ3+bn(2)λcn(0)λ5+cn(1)λ3+cn(2)λ-an(0)λ4-an(1)λ2-an(2)).By using (2.6), when j=0,1,2, substituting the values an(j),bn(j),cn(j) into the above matrix and comparing the coefficients of the different power of λ, we rewrite the matrix Vn(2) as(2.15)Vn(2)=V4nλ4+V3nλ3+V2nλ2+V1nλ+V0n,where(2.16)V4n=(-120012),V3n=(0qn-1rn0),V2n=(qn-1rn00-qn-1rn),V1n=(0qn-2(1+qn-1rn-1)-qn-1(qn-1rn+qn-2rn-1)rn+1(1+qnrn)-rn(qnrn+1+qn-1rn)0),V0n=(-qn-12rn2+qn-1rn+1+qn-2rn00qn-12rn2-qn-1rn+1-qn-2rn).Next, we establish the Darboux transformation of this new discrete lattice system (2.13). Actually, the Darboux transformation is a special gauge transformation(2.17)ψ̃n=Tnψnof the solutions of the Lax pairs (2.1) and (2.12) when m=2, where ψ̃n must satisfy Lax pairs (2.1) and (2.12) with some Ũn and Ṽn(2), that is,(2.18)Eψ̃n=Ũnψ̃n,Tn+1Un=ŨnTn,(2.19)ψ̃n,t=Ṽn(2)ψ̃n,Tn,t+TnVn(2)=Ṽn(2)Tn.Meanwhile, it is required thatŨn and Ṽn(2) have the same forms as Un and Vn(2), respectively. Thus, the old potentials qn,rn in Un and Vn(2) are mapped into new potentials q̃n,r̃n in Ũn and Ṽn(2). We assume Tn has the following form:(2.20)Tn=(λ2+AnBnλCnλλ2+Dn),whereAn,Bn,Cn,Dn are unknown functions which will be determined later.Letyn=(yn1,yn2)T and zn=(zn1,zn2)T be two linear independent solutions of Lax pairs (2.1) and (2.12), and suppose that(2.21)ψn=(yn1zn1yn2zn2).Substituting  (2.21) into (2.17), we obtain(2.22)ψ̃n=((λ2+An)yn1+Bnλyn2(λ2+An)zn1+Bnλzn2Cnλyn1+(λ2+Dn)yn2Cnλzn1+(λ2+Dn)zn2).We assume thatλ1 and λ2 are two roots of detTn(λ)=0. Therefore, when λ=λi(i=1,2), the two columns of Tn and ψ̃n are linear dependent, respectively. Thus, there exist two group of nonzero parameters αi=αi(n) and γi=γi(n) (i=1,2) which satisfy the following equations:(2.23)λi2+An+αiBnλi=0,Cnλi+αi(λi2+Dn)=0,(λi2+An)yn1+Bnλiyn2=γi[(λi2+An)zn1+Bnλizn2],Cnλiyn1+(λi2+Dn)yn2=γi[Cnλizn1+(λi2+Dn)zn2].Solving the above equations, it follows that(2.24)An=λ1λ2(λ1α2-λ2α1)λ1α1-λ2α2,Bn=λ22-λ12λ1α1-λ2α2,Cn=α1α2(λ22-λ12)λ1α2-λ2α1,Dn=λ1λ2(λ1α1-λ2α2)λ1α2-λ2α1,(2.25)αi=αi(n)=γizn2-yn2γizn1-yn1,i=1,2,where parametersλi and  γi (note: λ1≠λ2,γ1≠γ2) are suitably chosen such that all the denominators in (2.24) and (2.25) are nonzero.Whenθ=1, substituting (2.21) into (2.1), it follows,(2.26)yn+11=(1+qnrn)yn1+qnyn2λi,zn+11=(1+qnrn)zn1+qnzn2λi,yn+12=rnyn1λi+yn2λi2,zn+12=rnzn1λi+zn2λi2.Applying the shift operatorE to all expressions in (2.26) once, it follows(2.27)yn1=(1+qn-1rn-1)yn-11+qn-1yn-12λi,zn1=(1+qn-1rn-1)zn-11+qn-1zn-12λiyn2=rn-1yn-11λi+yn-12λi2,zn2=rn-1zn-11λi+zn-12λi2.Solving (2.27) yields(2.28)yn-11=λi-1(yn1λi-qn-1yn2),zn-11=λi-1(zn1λi-qn-1zn2),yn-12=λi-2[(1+qn-1rn-1)yn2-rn-1yn1λi],zn-12=λi-2[(1+qn-1rn-1)zn2-rn-1zn1λi].By using (2.26)–(2.28), from (2.25), we get(2.29)αi(n+1)=μiνi,αi(n-1)=βiωi,i=1,2,where(2.30)μi=μi(n)=rnλi+αiλi2,νi=νi(n)=1+qnrn+αiqnλi,βi=βi(n)=(1+qn-1rn-1)αi-rn-1λi,ωi=ωi(n)=λi2-αiqn-1λi.According to the above analysis, we derive some useful functions in forthcoming computation process as follows:(2.31)An+1=λ1λ2(λ1μ2ν1-λ2μ1ν2)λ1μ1ν2-λ2μ2ν1,Bn+1=ν1ν2(λ22-λ12)λ1μ1ν2-λ2μ2ν1,Cn+1=μ1μ2(λ22-λ12)λ1μ2ν1-λ2μ1ν2,Dn+1=λ1λ2(λ1μ1ν2-λ2μ2ν1)λ1μ2ν1-λ2μ1ν2,(2.32)An-1=λ1λ2(λ1β2ω1-λ2β1ω2)λ1β1ω2-λ2β2ω1,Bn-1=ω1ω2(λ22-λ12)λ1β1ω2-λ2β2ω1,Cn-1=β1β2(λ22-λ12)λ1β2ω1-λ2β1ω2,Dn-1=λ1λ2(λ1β1ω2-λ2β2ω1)λ1β2ω1-λ2β1ω2.We know that the matrixŨn keeps the same form as Un in the procedure of mapping the old potentials qn,rn in Un into new potentials q̃n,r̃n in Ũn. We can suppose that(2.33)Ũn=(1+q̃nr̃nλq̃nλr̃nλ2).SubstitutingTn+1,Un,Tn,Ũn into the second formula in (2.18) and comparing the coefficients of the same power of λ, it follows that(2.34)1+q̃nr̃n+q̃nCn=1+qnrn+rnBn+1,(2.35)An(1+q̃nr̃n)=An+1(1+qnrn),(2.36)q̃n=qn+Bn+1,(2.37)(1+q̃nr̃n)Bn+q̃nDn=qnAn+1,(2.38)r̃n+Cn=rn,(2.39)r̃nAn=(1+qnrn)Cn+1+rnDn+1,(2.40)r̃nBn+Dn=qnCn+1+Dn+1.From (2.36) and (2.38), it is easy to find that there exists a transformation which maps the old potentials qn,rn into new ones q̃n,r̃n as follows:(2.41)q̃n=qn+Bn+1,r̃n=rn-Cn.Respectively, substituting (2.41) into (2.34), (2.35), (2.37), (2.39), and (2.40), we obtain five conditional equations as follows:(2.42)(qn+Bn+1)(rn-Cn)+(qn+Bn+1)Cn-qnrn-rnBn+1=0,An[1+(qn+Bn+1)(rn-Cn)]-An+1(1+qnrn)=0,Bn[1+(qn+Bn+1)(rn-Cn)]+(qn+Bn+1)Dn-qnAn=0,(rn-Cn)An-Cn+1(1+qnrn)-rnDn+1=0,(rn-Cn)Bn-qnCn+1+Dn-Dn+1=0.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 Ṽn(2) defined by (2.19) has the same form as Vn(2), that is,(2.43)Ṽn(2)=Ṽ4nλ4+Ṽ3nλ3+Ṽ2nλ2+Ṽ1nλ+Ṽ0n,where all the forms ofṼ4n, Ṽ3n, Ṽ2n, Ṽ1n, Ṽ0n are the same as the forms of V4n, V3n, V2n, V1n, V0n. The matrices Ṽ4n, Ṽ3n, Ṽ2n, Ṽ1n, Ṽ0n can be obtained as long as we use the q̃ and r̃ to take the place of q and r in the matrices V4n, V3n, V2n, V1n, V0n, respectively.To simplify computation, according to the coefficients of the different power ofλ, we also rewrite the matrices Tn,Tn,t as(2.44)Tn=(λ2+AnBnλCnλλ2+Dn)=Iλ2+T1nλ+T0n,
where(2.45)I=(1001),T1n=(0BnCn0),T0n=(An00Dn),Tn,t=(An,tBn,tλCn,tλDn,t)=T1n,tλ+T0n,t,(2.46)T1n,t=(0Bn,tCn,t0),T0n,t=(An,t00Dn,t).SubstitutingTn,t,Vn(2),Ṽn(2) into the second formula in (2.19) yields(2.47)(T1n,tλ+T0n,t)+(Iλ2+T1nλ+T0n)(V4nλ4+V3nλ3+V2nλ2+V1nλ+V0n)-(Ṽ4nλ4+Ṽ3nλ3+Ṽ2nλ2+Ṽ1nλ+Ṽ0n)(Iλ2+T1nλ+T0n)=O.Letting the coefficients of theλ6,λ5,λ4,λ3,λ2,λ1,λ0 be zero, it follows that(2.48)V4n-Ṽ4n=O,(2.49)(T1nV4n-Ṽ4nT1n)+(V3n-Ṽ3n)=O,(2.50)(T0nV4n-Ṽ4nT0n)+(T1nV3n-Ṽ3nT1n)+(V2n-Ṽ2n)=O,(2.51)(T0nV3n-Ṽ3nT0n)+(T1nV2n-Ṽ2nT1n)+(Vn-Ṽ1n)=O,(2.52)(T0nV2n-Ṽ2nT0n)+(T1nV1n-Ṽ1nT1n)+(V0n-Ṽ0n)=O,(2.53)T1n,t+(T0nV1n-Ṽ1nT0n)+(T1nV0n-Ṽ0nT1n)=O,(2.54)T0n,t+(T0nV0n-Ṽ0nT0n)=O.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:(2.55)An,t=λ1λ2(λ12-λ22)(α2,tα1-α2α1,t)(λ1α1-λ2α2)2,Bn,t=(λ12-λ22)(λ1α1,t-λ2α2,t)(λ1α1-λ2α2)2,Cn,t=(λ12-λ22)(-λ1α22α1,t+λ2α12α2,t)(-λ1α2+λ2α1)2,Dn,t=λ1λ2(λ12-λ22)(α2α1,t-α1α2,t)(-λ1α2+λ2α1)2,(2.56)αi,t=v21(λi)-2v11(λi)αi+v12(λi)αi2,where(2.57)v11(λi)=-12λi4+qn-1rnλi2-qn-12rn2+qn-1rn+1+qn-2rn,v12(λi)=qn-1λi3+[qn-2(1+qn-1rn-1)-qn-1(qn-1rn+qn-2rn-1)]λi,v21(λi)=rnλi3+[rn+1(1+qnrn)-rn(qnrn+1+qn-1rn)]λi,i=1,2.On the other hand,(2.58)det⁡Tn=(λ2)2+(An+Dn-BnCn)λ2+AnDn.Thus, we obtain(2.59)An+Dn-BnCn=-(λ12+λ22),AnDn=λ12λ22,and their differentiation with respect to timet(2.60)An,t+Dn,t=Bn,tCn+BnCn,t,An,tDn+AnDn,t=0.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(qn,rn) of the discrete lattice system (2.13) are mapped into their new solutions (q̃n,r̃n) 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  qn=rn=0  are a couple of seed solutions of system (2.13). In addition, all the constants that satisfy conditional equation qnrn=2±3 are seed solutions of system (2.13).Without loss of generality, we only consider two kinds of seed solutions, that is, qn=rn=0 andqn=1,rn=2+3.


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

Whenθ=1 and m=2, substituting the seed solutions qn=1,rn=2+3 into the Lax pairs (2.1) and (2.12), it follows that(3.1)Eψn=(3+3λ(2+3)λλ2)ψn,ψn,t=(a11a12a21a22)ψn,where(3.2)a11=a11(λ)=-12λ4+(2+3)λ2-(3+23),a12=a12(λ)=λ3-(1+3)λ,a21=a21(λ)=(2+3)λ3-(5+33)λ,a22=a22(λ)=12λ4-(2+3)λ2+(3+23).Equation (3.1) has two real linear independent basic solutions as follows:(3.3)yn=τ1nexp⁡(a22+a11+Δ2t)(a22-a11+Δ2a12),zn=τ2nexp⁡(a22+a11-Δ2t)(a22-a11-Δ2a12),where(3.4)τ1=(2+3)(a22-a11+Δ)λ+2a12λ22a12,τ2=(2+3)(a22-a11-Δ)λ+2a12λ22a12,(3.5)Δ=(a22-a11)2+4a12a21=(1+3-λi2)2[(λi-1)2+2+3][(λi+1)2+2+3]>0.By using (2.25), we obtain(3.6)αi=2[λi3-(1+3)λi](γi-δineξit)γi[(1+3-λi2)(3+3-λi2)-ξi]-δin[(1+3-λi2)(3+3-λi2)+ξi]eξit,where(3.7)δi=τ1(λi)τ2(λi)=(2+3)[(1+3-λi2)(3+3-λi2)+ξi]λi+2[λi3-(1+3)λi]λi2(2+3)[(1+3-λi2)(3+3-λi2)-ξi]λi+2[λi3-(1+3)λi]λi2,ξi=(1+3-λi2)2[(λi-1)2+2+3][(λi+1)2+2+3],i=1,2.Therefore, by using the Darboux transformation (2.41), (2.24), and (2.31), we obtain the new explicit solution of (2.13) as follows:(3.8)q̃n=1+Bn+1=1+ν1ν2(λ22-λ12)λ1μ1ν2-λ2μ2ν1,r̃n=2+3-Cn=2+3-α1α2(λ22-λ12)λ1α2-λ2α1,where(3.9)μi=(2+3)λi+αiλi2,νi=3+3+αiλiwhich are reduced from (2.30); the parameters αi,i=1,2 are defined by (3.6) and λi≠±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, whenγi(i=1,2) are restricted in the region 0<γi<1, we investigate the dynamical behavior of solution (3.8). The term δineξit in (3.6) can be rewritten as δineξit=enln(δi)+ξit when δi>0, but it cannot be rewritten as this form when δi<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.The profiles ofp̃n,r̃n in (3.8) for λ1=5,λ2=12,γ1=0.5,γ2=0.6,δ1>0,δ2>0.
(a)
t = −0.003(b)
t = 0(c)
t = 0.002(d)
t = −0.003(e)
t = 0(f)
t = 0.002Figure1 shows the properties of double-kink soliton and single-kink soliton with singularity for solutions p̃n,r̃n, 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 bothδ1<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 p̃n,r̃n.The profiles ofp̃n,r̃n in (3.8) for λ1=1.8,λ2=2,γ1=0.5,γ2=0.6,δ1<0,δ2<0.
(a)
t = −2(b)
t = 0(c)
t = 2(d)
t = −2(e)
t = 0(f)
t = 2Figures2(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, whenδ1>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 p̃n,r̃n.The profiles ofp̃n,r̃n in (3.8) for   λ1=5,λ2=1,γ1=0.5,γ2=0.6,δ1>0,δ2<0.
(a)
t = −2(b)
t = 0(c)
t = 3(d)
t = −2(e)
t = 0(f)
t = 3Figures3(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

Whenθ=1 and m=2, substituting the seed solutions qn=rn=0 into the Lax pairs (2.1) and (2.12), it follows that(3.10)Eψn=(100λ2)ψn,ψn,t=(-12λ40012λ4)ψn.Equation (3.10) has two real linear independent basic solutions as follows:(3.11)yn=exp⁡(-12λ4t)(10),zn=λ2nexp⁡(12λ4t)(01).From (3.11), let yn1=exp((-1/2)λ4t),yn2=0,zn1=0,zn2=exp((1/2)λ4t); by using (2.25) and (2.30),we obtain(3.12)αi=-γiλi2neλi4t,μi=αiλi2,νi=1.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:(3.13)q̃n=Bn+1=ν1ν2(λ22-λ12)λ1μ1ν2-λ2μ2ν1=λ12-λ22γ1λ12n+3eλ14t-γ2λ22n+3eλ24t,r̃n=-Cn=-α1α2(λ22-λ12)λ1α2-λ2α1=γ1γ2(λ1λ2)2n(λ22-λ12)e(λ14+λ24)tλ1γ2λ22neλ24t-λ2γ1λ12neλ14t.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 solutionsp̃n,r̃n 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 p̃n,r̃n in (3.13) show discrete kink or anti-kink soliton, which are shown in Figure 5.The solutions ofp̃n,r̃n 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]The solutions ofp̃n,r̃n 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.
(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]Under the other parametric conditions, solution (3.13) is not soliton, therefore, we omit those parts of graphs and discussions on dynamic properties.


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

Whenθ=1 and m=2, substituting the seed solutions qn=1,rn=2+3 into the Lax pairs (2.1) and (2.12), it follows that(3.1)Eψn=(3+3λ(2+3)λλ2)ψn,ψn,t=(a11a12a21a22)ψn,where(3.2)a11=a11(λ)=-12λ4+(2+3)λ2-(3+23),a12=a12(λ)=λ3-(1+3)λ,a21=a21(λ)=(2+3)λ3-(5+33)λ,a22=a22(λ)=12λ4-(2+3)λ2+(3+23).Equation (3.1) has two real linear independent basic solutions as follows:(3.3)yn=τ1nexp⁡(a22+a11+Δ2t)(a22-a11+Δ2a12),zn=τ2nexp⁡(a22+a11-Δ2t)(a22-a11-Δ2a12),where(3.4)τ1=(2+3)(a22-a11+Δ)λ+2a12λ22a12,τ2=(2+3)(a22-a11-Δ)λ+2a12λ22a12,(3.5)Δ=(a22-a11)2+4a12a21=(1+3-λi2)2[(λi-1)2+2+3][(λi+1)2+2+3]>0.By using (2.25), we obtain(3.6)αi=2[λi3-(1+3)λi](γi-δineξit)γi[(1+3-λi2)(3+3-λi2)-ξi]-δin[(1+3-λi2)(3+3-λi2)+ξi]eξit,where(3.7)δi=τ1(λi)τ2(λi)=(2+3)[(1+3-λi2)(3+3-λi2)+ξi]λi+2[λi3-(1+3)λi]λi2(2+3)[(1+3-λi2)(3+3-λi2)-ξi]λi+2[λi3-(1+3)λi]λi2,ξi=(1+3-λi2)2[(λi-1)2+2+3][(λi+1)2+2+3],i=1,2.Therefore, by using the Darboux transformation (2.41), (2.24), and (2.31), we obtain the new explicit solution of (2.13) as follows:(3.8)q̃n=1+Bn+1=1+ν1ν2(λ22-λ12)λ1μ1ν2-λ2μ2ν1,r̃n=2+3-Cn=2+3-α1α2(λ22-λ12)λ1α2-λ2α1,where(3.9)μi=(2+3)λi+αiλi2,νi=3+3+αiλiwhich are reduced from (2.30); the parameters αi,i=1,2 are defined by (3.6) and λi≠±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, whenγi(i=1,2) are restricted in the region 0<γi<1, we investigate the dynamical behavior of solution (3.8). The term δineξit in (3.6) can be rewritten as δineξit=enln(δi)+ξit when δi>0, but it cannot be rewritten as this form when δi<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.The profiles ofp̃n,r̃n in (3.8) for λ1=5,λ2=12,γ1=0.5,γ2=0.6,δ1>0,δ2>0.
(a)
t = −0.003(b)
t = 0(c)
t = 0.002(d)
t = −0.003(e)
t = 0(f)
t = 0.002Figure1 shows the properties of double-kink soliton and single-kink soliton with singularity for solutions p̃n,r̃n, 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 bothδ1<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 p̃n,r̃n.The profiles ofp̃n,r̃n in (3.8) for λ1=1.8,λ2=2,γ1=0.5,γ2=0.6,δ1<0,δ2<0.
(a)
t = −2(b)
t = 0(c)
t = 2(d)
t = −2(e)
t = 0(f)
t = 2Figures2(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, whenδ1>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 p̃n,r̃n.The profiles ofp̃n,r̃n in (3.8) for   λ1=5,λ2=1,γ1=0.5,γ2=0.6,δ1>0,δ2<0.
(a)
t = −2(b)
t = 0(c)
t = 3(d)
t = −2(e)
t = 0(f)
t = 3Figures3(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

Whenθ=1 and m=2, substituting the seed solutions qn=rn=0 into the Lax pairs (2.1) and (2.12), it follows that(3.10)Eψn=(100λ2)ψn,ψn,t=(-12λ40012λ4)ψn.Equation (3.10) has two real linear independent basic solutions as follows:(3.11)yn=exp⁡(-12λ4t)(10),zn=λ2nexp⁡(12λ4t)(01).From (3.11), let yn1=exp((-1/2)λ4t),yn2=0,zn1=0,zn2=exp((1/2)λ4t); by using (2.25) and (2.30),we obtain(3.12)αi=-γiλi2neλi4t,μi=αiλi2,νi=1.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:(3.13)q̃n=Bn+1=ν1ν2(λ22-λ12)λ1μ1ν2-λ2μ2ν1=λ12-λ22γ1λ12n+3eλ14t-γ2λ22n+3eλ24t,r̃n=-Cn=-α1α2(λ22-λ12)λ1α2-λ2α1=γ1γ2(λ1λ2)2n(λ22-λ12)e(λ14+λ24)tλ1γ2λ22neλ24t-λ2γ1λ12neλ14t.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 solutionsp̃n,r̃n 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 p̃n,r̃n in (3.13) show discrete kink or anti-kink soliton, which are shown in Figure 5.The solutions ofp̃n,r̃n 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]The solutions ofp̃n,r̃n 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.
(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]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.



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*Source: 652076-2012-03-12.xml*