# Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Authors: Qianqian Yang; Qiulan Zhao; Xinyue Li
Journal: Complexity
 (2019)

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## Abstract

An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.


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## Body

## 1. Introduction

Investigation of nonlinear integrable lattice equations [1–6] is an active topic in the field of nonlinear science because its solutions can explain some physical phenomena [7–10]. Researchers in soliton field have proposed many valuable nonlinear integrable lattice equations [11–15] until now, but there are still many integrable lattice equations that have not been discovered. Generally speaking, nonlinear integrable lattice equations are related to their matrix spectral problems, respectively. Therefore, finding suitable matrix spectral problems is of great significance for us to derive integrable lattice equations and investigate their explicitly exact solutions.It is well known that it is not easy to get solutions for integrable lattice systems. Currently, there are several useful approaches which can help us obtain explicit solutions, for instance, the binary nonlinearization [16–18], Lie symmetry analysis [19, 20], and the Darboux transformation [21–29]. The Darboux matix method is one of important and effective techniques to obtain explicitly exact solutions of integrable lattice equations [30–34].In this paper, we will focus on this spectral problem:(1)Eφn=Unu,λφn,Un=unλ1+vnλun1,and its auxiliary problem,(2)φn,t=Vnu,λφn,where u=(un,vn)T, un and vn are potentials, and λ is a spectral parameter independent of t. According to the above spectral problem, an integrable lattice hierarchy in Liouville sense is presented firstly. Then, we investigate its bi-Hamiltonian structures using the discrete trace identity [35] proposed by G. Z. Tu. Its infinitely many conservation laws are also presented. Finally, the explicitly exact solutions of the integrable lattice equation derived from this matrix spectral problem are discussed by DT method.The structure of the article is as follows. In Section2, we will obtain a new integrable hierarchy in Liouville sense and present its bi-Hamiltonian structure in accordance with the matrix spectral problem. In Section 3, we will discuss conservation laws of the integrable hierarchy proposed in Section 2. In Section 4, its N-fold DT will be constructed and its exact solutions will be obtained. In Section 5, some conclusions are given.


## 2. Integrability and Bi-Hamiltonian Structures of a Lattice Hierarchy

Let us first introduce a shift operatorE, which obeys the following operations:(3)Efn=fn+1,E-1fn=fn-1,n∈Z,where f is a lattice function.In order to obtain a hierarchy of integrable lattice equations, we assume that(4)Vn=anbnλcn-an.By solving the stationary discrete zero curvature equation(5)EVnUn-UnVn=0,the following relationships will be derived:(6)an+1unλ+bn+1un-anunλ-cn1+vn=0,an+11+vnλ+bn+1-bnunλ+an1+vnλ=0,-an+1+anun+cn+1un-λcn=0,1+vncn+1-unbn-an+1-an=0.Substituting(7)an=∑m>=0anmλ-m,bn=∑m>=0bnmλ-m,cn=∑m>=0cnmλ-minto (6) follows the recursion relations (8)unan+1m-unanm-1+vncnm+1+unbn+1m+1=0,1+vnan+1m-unbnm+1+vnanm+bn+1m+1=0,uncn+1m-unan+1m-unanm-cnm+1=0,1+vncn+1m-an+1m-unbnm+anm=0.By calculation, we find bn(0)=0,cn(0)=0 and an(0) is a arbitrary constant. For simplicity of calculation, we take an(0)=0. Then, the recursion relations Equation (8) could uniquely determine an(m), bn(m), cn(m), and m>=1. The first few quantities are given by(9)an1=un1+vn-1,bn1=1+vn,cn1=un,an2=-un21+vn-12-unun-1vn11+vn-2-unun+1vn1+vn-1,bn+12=-un+11+vn2-unvn1+vn-1,cn2=-unun+1vn-un21+vn-1and so on. Now, for any integer m>=0, we define(10)
V
n
m
=
λ
m
V
n
=
∑
i
=
0
m
a
n
i
λ
m
-
i
∑
i
=
0
m
b
n
i
λ
m
-
i
∑
i
=
0
m
c
n
i
λ
m
-
i
-
∑
i
=
0
m
a
n
i
λ
m
-
i
.In order to derive the associated integrable lattice hierarchy, we give a modification Δn(m),(11)Δnm=-2anm+cnm000.Letting(12)Vnm=Vnm+Δnm,we can obtain that(13)EVnmUn-UnVnm=unλanm-an+1m+cn+1m-cnmvnλanm-an+1munanm-an+1m+cn+1m-cnm0.Then, the discrete zero-curvature equation(14)Un,t-EVnmUn+UnVn=0,m≥0is able to bring about the following integrable lattice hierarchy:(15)un,tm=unanm-an+1m+cn+1m-1-cnm-1,vn,tm=vnanm-an+1m.Eφn=Un(u,λ)φn and φn,t=Vn(u,λ)φn compose the Lax representation of the integrable lattice hierarchy Equation (15). When m=1, (15) is reduced to the following:(16)un,t=ununvn-1-un+1vn,vn,t=vnun-un+1+vnunvn-1-un+1vn,in which φn,t=Vn[1]φnVn[1] is(17)Vn1=-12λ-unvn-11+vn-1λun12λ-un-unvn-1.Next, we try to construct the bi-Hamiltonian structures of (15). In order to achieve this goal, we denote Γn=VnUn-1and M,N stands for the trace of the product of two square matrices of the same orders M and N. Then,(18)Γn,∂Un∂λ=-an+1,Γn,∂Un∂un=anun,Γn,∂Un∂vn=1vnan+1-cn+1.The discrete trace identity [35] plays an important role in constructing Hamiltonian structures of integrable lattice equations and studying their integrability. Here, we try to construct Hamiltonian structure of (15) by using (19)δδun∑n∈ZΓn,∂Un∂λ=λ-ε∂∂λλεΓn,∂Un∂uni,i=1,2.Substituting the expansions an=∑m>=0an(i)λ-i and cn=∑m>=0cn(i)λ-i into (19), we have(20)δδun∑n∈Z-anm+1=c-manmun1vnan+1m-cn+1m.To get the value of the constant c, we simply set m=0 in (20) and then find c=0. Therefore we have that(21)δHnmδun=anmun1vnan+1m-cn+1m,(22)Hnm=∑n∈zanm+1m.So we obtain the desired Hamiltonian form of (15),(23)
u
n
v
n
t
m
=
J
δ
H
n
m
δ
u
n
=
J
L
δ
H
n
m
-
1
δ
u
n
=
K
δ
H
n
m
-
1
δ
u
n
=
K
L
m
-
1
1
2
u
n
1
2
v
n
,where(24)J=0unvn-1+E-1unvn1-E0,(25)L=L11L12L21L22,(26)L11=-1unE-1-11+vnEpn2+1unE-1-1unE-1unvn+1un1-E-1un2,L12=-1unE-1-11+vnEunvn+1unE-1-1unE-1vn+vn2,L21=1vn1-E-1-1unE-1unvn-1vn1-E-1-1vnEun2,L22=1vn1-E-1-1unE-1vn+vn2-1vn1-E-1-1vnEunvn-1vn1-E-1-1unvn.In (23), the operator J is a Hamiltonian operator for the hierarchy Equation (15), and the operator L is a recursion operator for the hierarchy Equation (15). These two operators J and L satisfy the following relationships:(27)J∗=-J,JL=L∗J,where J∗ stands for the formal conjugation of J and(28)K=J=K11K12K21K22,(29)K11=unvnEun2-un2E-1unvn,K12=unvnEunvn-un2E-1vn+vn2+un2vn,K21=vn+vn2Eun2-unvnE-1unvn-un2vn,K22=vn+vn2Eunvn-unvnE-1vn+vn2.Remark. ( 1 ) The hierarchy of lattice equations (15) is an integrable Hamiltonian system in Liouville sense.( 2 ) The bi-Hamiltonian structure Equation (23) of the integrable lattice equation exists.


## 3. Conservation Laws of (16)

Based on Lax representation of the hierarchy of integrable lattice equations (15), we can obtain its conversation laws. Substituting Φn=(ϕ1,n,ϕ2,n)T into the spectral problem (1), we have(30)ϕ1,n+1=unλϕ1,n+1+vnλϕ2,n,ϕ2,n+1=unϕ1,n+ϕ2,n,which demonstrates that(31)λunθnθn+1+λθn+1-unθn-1+vn=0,where we have θn=ϕ1,n/ϕ2,n.Then, if we suppose thatθn=∑j=0∞θn(j)λ-j, the following two recursion relationships will be found. One is(32)θn0=1un,θn1=-vnun+1un,θn2=vnvn+1un+2-vnun+1,θnj=-unθnj-1+un∑m=1j-1θnmθn+1j-m,and the other is(33)θn0=0,θn1=1+vn,θn2=-un-1vn-11+vn-2,θn+1j=unθnj-1-un∑m=1j-1θnmθnj-m.Based on (2) and (17), a direct calculation gives(34)ln⁡unθn+1t=E-1-λunθn-12λ+un+unvn-1.Then, we discuss conversation laws of the hierarchy of integrable lattice equations (15) according to the above two different recursion relationships, respectively. Substituting θn=∑j=0∞‍θn(j)λ-j into (34), equating the coefficients of λ on both sides of the equation, we can achieve an infinite number of conversation laws of (16) based on the (32). The first two of them are(35)12unt=E-1un+unvn-1+vnun+1,-2vnun+1t=E-1-unvnvn+1un+2+unvnun+1.Similarly, we also have another infinite conservation laws of (16) based on (33). The first two of them are(36)un+unvnt=E-1-unun-1vn-11+vn-1,-un1+vnvn+1+12un+12vnt=E-1-un2vn-un1+vn-12+un-1vn-11+vn-2.


## 4. The N-Fold DT and Exact Solutions of Eq. (16)

If a integrable lattice equation with Lax pairEΦn=UnΦn and Φn,t=VnΦn exists a gauge transformation(37)Φ~n=TnΦn,where Tn is a reversible matrix and(38)EΦ~n=U~nu,λΦ~n,Φ~n,t=V~nu,λΦ~n.The new potentials U~n(u,λ) and V~n(u,λ) possess the same form with the old potentials Un(u,λ) and Vn(u,λ), respectively. The gauge transformation in (37) composes Darboux transformation of an integrable lattice equation with the transformation between old potential functions Un(u,λ), Vn(u,λ) and new potential functions U~n(u,λ)  V~n(u,λ).Based on (37) and (38), we are also able to obtain the relations, i.e.,(39)U~n=Tn+1U~nTn-1,V~n=Tn,t+TnV~nTn-1.


### 4.1. The N-Fold DT of Eq. (16)

The selection ofTn is very important in constructing the Darboux transformation. The solution of the soliton equation can be obtained much more easily by an appropriate Darboux matrix. For this integrable lattice equation (16), we take (40)Tn=anbncndn=1-bnN-1λN+∑i=0N-1dniλi∑i=0N-1bniλi0λN+∑i=0N-1dniλiwith bn(i) and dn(i) being all functions with respect to n and t and N being a positive integer number. It is easy to find that det⁡(Tn) is a (2N)-th order polynomial with respect to λ. If we assume that the 2N parameters λj(λj≠λi,j≠i;λj≠0,j=1,2,…,2N) are the roots of the det⁡(Tn), det⁡(Tn) can be expressed as(41)det⁡Tn=1-bnN-1∏i=12Nλ-λj.Setting φn=(φn,1,φn,2)T and ψn=(ψn,1,ψn,2)T being the column vectors of Φn, we find that the column vectors of Φ~n=TnΦn are linear dependent because of det⁡Tn(λi)=0. Then for λ=λj,j=1,2,…,2N, according to (37), we have(42)∑i=0N-1dni+δj,nbniλji=-1-bnN-1λjN,∑i=0N-1δj,ndniλji=-δj,nλjN,in which(43)δj,n=φ2,n+γjψ2,nφ1,n+γjψ1,n,δj,n+1=λjun+δj,nun+1+vnδj,n,i=1,…,2N.and the parameters λj and γi are so chosen that the denominators in (43) are nonzero.Theorem 1.
The matrixU~n defined by (38) has the same form as Un, i.e.,(44)U~n=u~nλ1+v~nλu~n1.where the transformation from the old potentials un and vn into the new potentials u~n and v~n is given by(45)u~n=un1-bnN-1,v~n=1-bn+1N-1vn.Proof.
LetTn-1=Tn∗/det⁡Tn and(46)Tn+1UnTn∗=f11λ,nf12λ,nf21λ,nf22λ,n.It can be see that f11 and f12 are (2N-1)-th order polynomials with respect to λ, and f21 and f22 are (2N)-th order polynomials with respect to λ. From (42), we have(47)anλj=-δj,nbnλj.Moreover, (46) can be described as(48)Tn+1UnTn∗=det⁡TnPn,together with(49)Pn=λ-1P11λ,nλ-1P12λ,nP21λ,nP22λ,n,where λi(i=1,2,…,2N) are roots of fst(λ,n)(s,t=1,2), Pn is a 2-order square matrix, and Pjk(j,k=1,2) are the functions with respect to n and t. Thus, (48) can be written as(50)Tn+1Un=PnTn.Equating the coefficients of λN and λN-1 on both sides of (50), we find(51)P11=un1-bnN-1=u~n,P12=1+vn1-bn+1N-1=1+v~n,P21=un1-bnN-1=u~n,P22=1.From (51), we can see that Pn=U~n. The proof is completed.Theorem 2.
Under the transformation (37) and (45), the matrix V~n defined by (38) has the same form as Vn, i.e.,(52)V~n=-12-u~nv~n-11+v~n-1λu~n12-u~n1+v~n-1.Proof.
LetTn-1=Tn∗/det⁡Tn and(53)Tn,t+TnVnTn∗=g11λ,ng12λ,ng21λ,ng22λ,n,in which all gjk(j,k=1,2) are the functions of n and t. Through a series of calculations, we can figure out that g11(λ,n), g21(λ,n), and g22(λ,n) are (2N+1)-th order polynomials with respect to λ, and g12(λ,n) is (2N)-th order polynomial with respect to λ. We can verify from (47) that λi(i=1,2,…,2N) are the roots of gjk(j,k=1,2). So, we know(54)Tn,t+TnVnTn∗=det⁡TnRn,together with(55)Rn=λR111λ,n+R110λ,nR120λ,nλR211λ,n+R210λ,nλR221λ,n+R220λ,n.where all the Rjk(j,k=1,2) are the functions with respect to n and t. Thus, (54) can be rewritten as(56)Tn,t+TnVn=RnTn,Equating the coefficients of λN+1 and λN on both sides of (56), we find that(57)R111=-12,R110=-un1-bnN-11-bn-1N-1vn-1=-u~nv~n-1,R120=1+1-bnN-1=1+v~n-1,R211=un1-bnN-1=u~n,R210=0,R221=12,R220=-u~n1+v~n-1.From (38) and (57), we are able to see that Rn=V~n. The proof is completed.


### 4.2. Exact Solutions of (16)

In this section, the explicit solutions of the integrable lattice equation (16) will be discussed under its Darboux transformation. We can find un=1,vn=1 is a trivial solution of (16) easily. Starting from this trivial solution, the solutions of the Lax Pair Equations (1) and (17) can be obtained,(58)φn=φn,1φn,2=τ1neρ1tτ1n+1eρ1t,ψn=ψn,1ψn,2=τ2neρ2tτ2n+1eρ2t,in which(59)τ1=-λi2+6λi+1+λi-1,τ2=λi2+6λi+1+λi-1(60)ρ1=123+λi2+6λi+1,ρ2=-12-3+λi2+6λi+1.According to (43), we have(61)δi,n=τ1n+1eρ1t+γiτ2n+1eρ2tτ1neρ1t+γiτ2neρ2tBy using of the Cramer rules, solving (42) follows(62)bnN-1=ΔbnN-1Δtogether with(63)Δ=1λ1…λ1N-1δ1δ1λ1…δ1λ1N-2δ1-λ1λ1N-11λ2…λ2N-1δ2δ2λ2…δ2λ2N-2δ2-λ2λ2N-1⋮⋮⋱⋮⋮⋮⋱⋮⋮1λ2N…λ2NN-1δ2Nδ2Nλ2N…δ2Nλ2NN-2δ2N-λ2Nλ2NN-1,and Δbn(N-1) is obtained from Δ by replacing its (2N)-th column with (-λ1N,-λ2N,…,-λ2NN)T, i.e.,(64)ΔbnN-1=1λ1…λ1N-1δ1δ1λ1…δ1λ1N-2-λ1N1λ2…λ2N-1δ2δ2λ2…δ2λ2N-2-λ2N⋮⋮⋱⋮⋮⋮⋱⋮⋮1λ2N…λ2NN-1δ2Nδ2Nλ2N…δ2Nλ2NN-2-λ2NN.By using (41), (45), and (62), the solution of (16) can be derived, and it is expressed as(65)u~n=11-bnN-1,v~n=1-bn+1N-1.In order to investigate the structure of the solutions (65) fully, we plot their structure figures as shown in Figures 1–4 when N=1 and N=2.Figure 1
A structure of soliton solution forun in (66) with λ1=0.5,λ2=2,γ1=1,γ2=-1. Figures (a) and (b) are three-dimensional graphs of un. Figure (c) is the density plot of un and Figures (d), (e), and (f) are structures of the solution un at different time.
(a)
u n 1 (b)
u n 2 (c)
density ofun (d)
t=-5 (e)
t=0 (f)
t=5Figure 2
A structure of soliton solution forvn in (66) with λ1=0.5,λ2=2,γ1=1,γ2=-1. Figures (a) and (b) are three-dimensional graphs of vn. Figure (c) is the density plot of vn and Figures (d), (e), and (f) are structures of the solution vn at different time.
(a)
v n 1 (b)
v n 2 (c)
density ofvn (d)
t=-5 (e)
t=0 (f)
t=5Figure 3
A structure of soliton solution forun in (68) with λ1=0.5,λ2=2,λ3=1,λ4=4,γ1=1,γ2=-1,γ3=2,γ4=-2. Figures (a) and (b) are three-dimensional graphs of un, Figure (c) is the density plot of un, and Figures (d), (e), and (f) are structures of the solution un at different time.
(a)
u n 1 (b)
u n 2 (c)
density ofun (d)
t=-5 (e)
t=0 (f)
t=5Figure 4
A structure of soliton solution forvn in (68) with λ1=0.5,λ2=2,λ3=1,λ4=4,γ1=1,γ2=-1,γ3=2,γ4=-2. Figures (a) and (b) are three-dimensional graphs of vn, Figure (c) is the density plot of vn, and Figures (d), (e), and (f) are structures of the solution vn at different time.
(a)
1 (b)
2 (c)
density ofvn (d)
t=-5 (e)
t=0 (f)
t=5(I) In particular, whenN=1, solving (42) leads to(66)u~n=11-bn0,v~n=1-bn+10.where(67)Δ=1δ1,n-λ11δ2,n-λ2,Δbn0=1-λ11-λ2.Figures 1 and 2 show the structures of the soliton solutions u~n and v~n in (65).(II) When N=2, solving (42) leads to(68)u~n=11-bn1,v~n=1-bn+11.where(69)Δ=1λ1δ1δ1,n-λ1λ11λ2δ2δ2,n-λ2λ21λ3δ3δ3,n-λ3λ31λ4δ4δ4,n-λ4λ4,Δbn1=1λ1δ1λ121λ2δ2λ221λ3δ3λ321λ4δ4λ42.Figures 3 and 4 show the structures of the soliton solutions u~n and v~n in (68).


## 4.1. The N-Fold DT of Eq. (16)

The selection ofTn is very important in constructing the Darboux transformation. The solution of the soliton equation can be obtained much more easily by an appropriate Darboux matrix. For this integrable lattice equation (16), we take (40)Tn=anbncndn=1-bnN-1λN+∑i=0N-1dniλi∑i=0N-1bniλi0λN+∑i=0N-1dniλiwith bn(i) and dn(i) being all functions with respect to n and t and N being a positive integer number. It is easy to find that det⁡(Tn) is a (2N)-th order polynomial with respect to λ. If we assume that the 2N parameters λj(λj≠λi,j≠i;λj≠0,j=1,2,…,2N) are the roots of the det⁡(Tn), det⁡(Tn) can be expressed as(41)det⁡Tn=1-bnN-1∏i=12Nλ-λj.Setting φn=(φn,1,φn,2)T and ψn=(ψn,1,ψn,2)T being the column vectors of Φn, we find that the column vectors of Φ~n=TnΦn are linear dependent because of det⁡Tn(λi)=0. Then for λ=λj,j=1,2,…,2N, according to (37), we have(42)∑i=0N-1dni+δj,nbniλji=-1-bnN-1λjN,∑i=0N-1δj,ndniλji=-δj,nλjN,in which(43)δj,n=φ2,n+γjψ2,nφ1,n+γjψ1,n,δj,n+1=λjun+δj,nun+1+vnδj,n,i=1,…,2N.and the parameters λj and γi are so chosen that the denominators in (43) are nonzero.Theorem 1.
The matrixU~n defined by (38) has the same form as Un, i.e.,(44)U~n=u~nλ1+v~nλu~n1.where the transformation from the old potentials un and vn into the new potentials u~n and v~n is given by(45)u~n=un1-bnN-1,v~n=1-bn+1N-1vn.Proof.
LetTn-1=Tn∗/det⁡Tn and(46)Tn+1UnTn∗=f11λ,nf12λ,nf21λ,nf22λ,n.It can be see that f11 and f12 are (2N-1)-th order polynomials with respect to λ, and f21 and f22 are (2N)-th order polynomials with respect to λ. From (42), we have(47)anλj=-δj,nbnλj.Moreover, (46) can be described as(48)Tn+1UnTn∗=det⁡TnPn,together with(49)Pn=λ-1P11λ,nλ-1P12λ,nP21λ,nP22λ,n,where λi(i=1,2,…,2N) are roots of fst(λ,n)(s,t=1,2), Pn is a 2-order square matrix, and Pjk(j,k=1,2) are the functions with respect to n and t. Thus, (48) can be written as(50)Tn+1Un=PnTn.Equating the coefficients of λN and λN-1 on both sides of (50), we find(51)P11=un1-bnN-1=u~n,P12=1+vn1-bn+1N-1=1+v~n,P21=un1-bnN-1=u~n,P22=1.From (51), we can see that Pn=U~n. The proof is completed.Theorem 2.
Under the transformation (37) and (45), the matrix V~n defined by (38) has the same form as Vn, i.e.,(52)V~n=-12-u~nv~n-11+v~n-1λu~n12-u~n1+v~n-1.Proof.
LetTn-1=Tn∗/det⁡Tn and(53)Tn,t+TnVnTn∗=g11λ,ng12λ,ng21λ,ng22λ,n,in which all gjk(j,k=1,2) are the functions of n and t. Through a series of calculations, we can figure out that g11(λ,n), g21(λ,n), and g22(λ,n) are (2N+1)-th order polynomials with respect to λ, and g12(λ,n) is (2N)-th order polynomial with respect to λ. We can verify from (47) that λi(i=1,2,…,2N) are the roots of gjk(j,k=1,2). So, we know(54)Tn,t+TnVnTn∗=det⁡TnRn,together with(55)Rn=λR111λ,n+R110λ,nR120λ,nλR211λ,n+R210λ,nλR221λ,n+R220λ,n.where all the Rjk(j,k=1,2) are the functions with respect to n and t. Thus, (54) can be rewritten as(56)Tn,t+TnVn=RnTn,Equating the coefficients of λN+1 and λN on both sides of (56), we find that(57)R111=-12,R110=-un1-bnN-11-bn-1N-1vn-1=-u~nv~n-1,R120=1+1-bnN-1=1+v~n-1,R211=un1-bnN-1=u~n,R210=0,R221=12,R220=-u~n1+v~n-1.From (38) and (57), we are able to see that Rn=V~n. The proof is completed.


## 4.2. Exact Solutions of (16)

In this section, the explicit solutions of the integrable lattice equation (16) will be discussed under its Darboux transformation. We can find un=1,vn=1 is a trivial solution of (16) easily. Starting from this trivial solution, the solutions of the Lax Pair Equations (1) and (17) can be obtained,(58)φn=φn,1φn,2=τ1neρ1tτ1n+1eρ1t,ψn=ψn,1ψn,2=τ2neρ2tτ2n+1eρ2t,in which(59)τ1=-λi2+6λi+1+λi-1,τ2=λi2+6λi+1+λi-1(60)ρ1=123+λi2+6λi+1,ρ2=-12-3+λi2+6λi+1.According to (43), we have(61)δi,n=τ1n+1eρ1t+γiτ2n+1eρ2tτ1neρ1t+γiτ2neρ2tBy using of the Cramer rules, solving (42) follows(62)bnN-1=ΔbnN-1Δtogether with(63)Δ=1λ1…λ1N-1δ1δ1λ1…δ1λ1N-2δ1-λ1λ1N-11λ2…λ2N-1δ2δ2λ2…δ2λ2N-2δ2-λ2λ2N-1⋮⋮⋱⋮⋮⋮⋱⋮⋮1λ2N…λ2NN-1δ2Nδ2Nλ2N…δ2Nλ2NN-2δ2N-λ2Nλ2NN-1,and Δbn(N-1) is obtained from Δ by replacing its (2N)-th column with (-λ1N,-λ2N,…,-λ2NN)T, i.e.,(64)ΔbnN-1=1λ1…λ1N-1δ1δ1λ1…δ1λ1N-2-λ1N1λ2…λ2N-1δ2δ2λ2…δ2λ2N-2-λ2N⋮⋮⋱⋮⋮⋮⋱⋮⋮1λ2N…λ2NN-1δ2Nδ2Nλ2N…δ2Nλ2NN-2-λ2NN.By using (41), (45), and (62), the solution of (16) can be derived, and it is expressed as(65)u~n=11-bnN-1,v~n=1-bn+1N-1.In order to investigate the structure of the solutions (65) fully, we plot their structure figures as shown in Figures 1–4 when N=1 and N=2.Figure 1
A structure of soliton solution forun in (66) with λ1=0.5,λ2=2,γ1=1,γ2=-1. Figures (a) and (b) are three-dimensional graphs of un. Figure (c) is the density plot of un and Figures (d), (e), and (f) are structures of the solution un at different time.
(a)
u n 1 (b)
u n 2 (c)
density ofun (d)
t=-5 (e)
t=0 (f)
t=5Figure 2
A structure of soliton solution forvn in (66) with λ1=0.5,λ2=2,γ1=1,γ2=-1. Figures (a) and (b) are three-dimensional graphs of vn. Figure (c) is the density plot of vn and Figures (d), (e), and (f) are structures of the solution vn at different time.
(a)
v n 1 (b)
v n 2 (c)
density ofvn (d)
t=-5 (e)
t=0 (f)
t=5Figure 3
A structure of soliton solution forun in (68) with λ1=0.5,λ2=2,λ3=1,λ4=4,γ1=1,γ2=-1,γ3=2,γ4=-2. Figures (a) and (b) are three-dimensional graphs of un, Figure (c) is the density plot of un, and Figures (d), (e), and (f) are structures of the solution un at different time.
(a)
u n 1 (b)
u n 2 (c)
density ofun (d)
t=-5 (e)
t=0 (f)
t=5Figure 4
A structure of soliton solution forvn in (68) with λ1=0.5,λ2=2,λ3=1,λ4=4,γ1=1,γ2=-1,γ3=2,γ4=-2. Figures (a) and (b) are three-dimensional graphs of vn, Figure (c) is the density plot of vn, and Figures (d), (e), and (f) are structures of the solution vn at different time.
(a)
1 (b)
2 (c)
density ofvn (d)
t=-5 (e)
t=0 (f)
t=5(I) In particular, whenN=1, solving (42) leads to(66)u~n=11-bn0,v~n=1-bn+10.where(67)Δ=1δ1,n-λ11δ2,n-λ2,Δbn0=1-λ11-λ2.Figures 1 and 2 show the structures of the soliton solutions u~n and v~n in (65).(II) When N=2, solving (42) leads to(68)u~n=11-bn1,v~n=1-bn+11.where(69)Δ=1λ1δ1δ1,n-λ1λ11λ2δ2δ2,n-λ2λ21λ3δ3δ3,n-λ3λ31λ4δ4δ4,n-λ4λ4,Δbn1=1λ1δ1λ121λ2δ2λ221λ3δ3λ321λ4δ4λ42.Figures 3 and 4 show the structures of the soliton solutions u~n and v~n in (68).


## 5. Conclusions

In this article, we have derived an integrable lattice hierachy on the basis of a new matrix spectral problem. Then, some properties of this hierarchy have been shown, such as the Liouville integrability, the bi-Hamiltonian structures, and infinitely many conservation laws. Eventually, the Darboux transformation of the first integrable equation in this hierarchy has been constructed and the exact solutions of the integrable equation have been investigated via graphs. The binary nonlinearization [36–38] is also a useful method to investigate explicit solutions. We will investigate the binary nonlinearization with regard to (16) associated with spectral problems (1) and (2) in subsequent papers.



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