# Algebro-Geometric Solutions for a Discrete Integrable Equation

Authors: Mengshuang Tao; Huanhe Dong
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.


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## Body

## 1. Introduction

As we all know, the generation of integrable system, determination of exact solution, and the properties of the conservation laws are becoming more and more rich [1–5]; in particular, the discrete integrable systems have many applications in statistical physics, quantum physics, and mathematical physics [6–11]. It is worth discussing the properties of discrete integrable systems, such as Darboux transformations [12, 13], Hamiltonian structures [14–16], exact solutions [17], and the transformed rational function method [18]. In the past decades, some methods have been proposed to gain explicit solutions of the continuous soliton equations, for instance, the algebro-geometric method [19, 20], the inverse scattering transformation [21], the Ba¨cklund transformation [22], and the sine-cosine method [23]. However, it is very hard to obtain algebro-geometric solutions for discrete soliton equations due to the treatment of discrete variables. In 1975, Its and Matveev first presented the algebro-geometric approaches [24], which permitted us to seek out a class of exact solutions to the soliton equations. The elliptic functions and multisoliton solutions may be acquired by these degenerated solutions [25]. Recently, Qiao et al. further improved the algebro-geometric methods by making use of the nonlinearization theory [26–29]. Trigonal curves are also systematically used to construct algebro-geometric solutions [30, 31]. But we note that there is few research to focus on the algebro-geometric solutions of discrete soliton equations.In this paper, we will generate the algebro-geometric solutions of the discrete integrable system by taking advantage of the Riemann-Jacobi inversion theorem and Abel coordinates. In Sections2 and 3, we will construct a new discrete integrable system by using Lie algebra and spectral problem. By introducing Abel-Jacobi coordinates, straightening out of the continuous and discrete flows will be given and placed in Section 4. Section 5 will be devoted to derive the algebro-geometric solutions of the abovementioned discrete integrable equation by utilizing the Riemann theta function.


## 2. The Discrete Integrable Hierarchy

We consider the algebra(1)h1=1000,h2=0001,e=0100,f=0010,which is the simple subalgebra of the Lie algebras A1, and corresponding loop algebras can be expressed as (2)hin=hiλn,en=eλn,fn=fλn,n∈Z.According to the loop algebras, we introduce the following discrete spectral problems(3)Eψn=Unψn,dψndt=Vnψn,where (4)Un=h1-1+h2-1+rnsnh21+rne1+snf1,Vn=∑n=0∞anh1-2n-h2-2n+bne-2n+1+cnf-2n+1.Thus(5)Un=1λrnsn1λ+λrnsn,Vn=ABC-A,where(6)A=∑n=0∞anλ-2n,B=∑n=0∞bnλ-2n+1,C=∑n=0∞cnλ-2n+1.According to the following stationary discrete zero curvature equation forVn,(7)EVnUn=UnVn,we get(8)1λEA-A+snEB-rnC=0,rnEA-A+1λ+λrnsnEB-1λB=0,1λEC-snEA+A-1λ+λrnsnC=0,rnEC-1λ+λrnsnEA-A-snB=0.Substituting (6) into (8) yields(9)Δan+snEbn+1-rncn+1=0,rnsnEbn+1+rnEan+an+Δbn=0,-snrncn+1-snEan+an+Δcn=0,-rnsnΔan+1+rnEcn+1-snbn+1-Δan=0,where Δ=E-1.We choose the initial valuesa0=-1/2, b0=0 and need to select zero constants for the inverse operation of the difference operator Δ in computing an,n≥1. On this condition, recursion relations (9) uniquely determine an,bn,cn,n≥1. Then, we obtain the first few quantities (10)a1=1sn-1rn,b1=1sn-1,c1=1rn,a2=-1sn-12rn2-1rn2snsn-1-1sn-12rn-1rn,b2=-1sn-12rn-1-1sn-12rn,c2=-1rn2sn-1-1rn2sn,….From(11)Eψn=Unψn,dψndt=Vnmψn,we have the discrete zero curvature equation (12)dUndtm-EVnmUn+UnVnm=0,where (13)Vnm=∑n=0manλ2m-2n-ambnλ2m-2n+1cnλ2m-2n+1-anλ2m-2n+am.Thus, we obtain the following integrable discrete hierarchy(14)rn,tm=Ebm-bm,sn,tm=Ecm-cm.And(15)rn,t=1sn-1sn-1,sn,t=1rn+1-1rn,with m=1. Equation (15) can be read as(16)∂tln⁡rn=1rn1sn-1sn-1,∂tln⁡sn=1sn1rn+1-1rn.It is easy to find that the Lax pair of (15) is given by(17)Eφn=Unφn,φtn=Vn1φn,where (18)Vn1=-12λ21sn-1λ1rnλ12λ2.In the following, we express Lenard's gradient sequencesSj0≤j∈Z, by the recursion equation(19)JnSj+1n=KnSjn,JnS-1n=0,j=0,1,…,with two operators (20)Kn=Δ0-snE+10ΔrnE+1rnΔsnΔ-rnsnΔ,Jn=rnsn000-rnsnE0rnΔsnΔ-rnsnΔ,where Sjn=sj1,sj2,sj3T.From the equationJnS-1(n)=0 and (19), respectively, (21)S-1n=00-12,S0n=1rn1sn-11sn-1rn,S1n=-1rn21sn-1+1sn-1sn-121rn-1+1rn-1sn-1rn1sn-1rn+1snrn+1sn-1rn-1.Equation (19) implies that(22)Δsj2+rnE+1sj3+rnsnEsj+12=0,Δsj1-snE+1sj3-rnsnsj+11=0,rnΔsj1+snΔsj2-rnsnΔsj3=0.The discrete integrable hierarchical (14) could be rewritten as generation of the following, so spectrum problem is(23)ψn+1=Unψn,ψntm=Vnmψn,where(24)Vnm=AnmBnmCnm-Anm,Anm=∑j=0msj-13λ2m-j-sm3,Bnm=∑j=0msj-12λ2m-j+1,Cnm=∑j=0msj-11λ2m-j+1.From the compatibility conditions of the discrete Lax pair (23), we can read that the hierarchical equation is(25)rn,tm=Esm-12-sm-12,sn,tm=Esm-11-sm-11.Thus, we also have (26)rnsntm=Xmn=Δsm-12Δsm-11.


## 3. Decomposition of the Differential-Difference Equations

In this section, we shall resolve the discrete systems (16) into solvable ordinary differential equations. We assume that (23) has two basic solutions ψ(n)=ψ1n,ψ1nT, ϕ(n)=ϕ1n,ϕ1nT, and we define a Lax matrix Wn as follows: (27)Wn=12ϕnψnT+ψnϕnTσ=fngnhn-fn,σ=0-110,and Wn should meet the following equations:(28)Wn+1Un-UnWn=0,Wn,tm=Vnm,Wn.It is easy to see that (28) can be written as(29)1λΔfn+snEgn-rnhn=0,rnEfn+1λΔgn+λrnsnEgn+rnfn=0,1λΔhn-snEfn-snfn-λrnsnhn=0,rnEhn-1λΔfn-λrnsnΔfn-sngn=0,fntm=Bnmhn-Bnmgn,gntm=2Anmgn-2Bnmfn,hntm=2Cnmfn-2Anmhn,where(30)fn=∑j=0N+1fj-1nλ2N+1-j,gn=∑j=1N+1gj-1nλ2N+1-j+1,hn=∑j=1N+1hj-1nλ2N+1-j+1.Substituting (30) into (29) yields(31)JnGj+1=KnGjn,JnG-1n=0,KnGN-1n=0,j≥0,where Gj(n)=hjn,gjn,fjnT.It is evident that(32)G-1=α0S-1n,where α0 is a constant.Acting withJn-1Kn and Kn-1Jn, respectively, on (32) yields (33)G0n=α0S0n+α1S-1n.Thus(34)Gkn=α0Skn+α1Sk-1n+⋯+αk+1S-1n,where α0,α1,…,αk+1 are constants.Substituting (34) into the KnGN-1n=0 gives the following discrete stationary equation: (35)α0XNn+α1XN-1n+⋯+αNX0n=0.According to (31), we have (α0=1)(36)h0n=1rn,g0n=1sn-1,f0n=1rnsn-1-12α1,h1n=-1rn21sn-1+1sn+α11rn,g1n=-1sn-121rn-1+1rn+α11sn-1.Define the elliptic coordinatesμj and νj by expressing g(n) and h(n):(37)gn=1sn-1λ∏j=1Nλ2-μjn2≡1sn-1λ∏j=1Nλ~-μjn~,hn=1rnλ∏j=1Nλ2-νjn2≡1rnλ∏j=1Nλ~-νjn~,where we denote λ2, μj(n)2, and νj(n)2 with λ~, μjn~, and νjn~, respectively.By comparing coefficients of the same power forλ, we get(38)g1n=-1sn-1∑j=1Nμj~n,h1n=-1rn∑j=1Nνj~n,g2n=1sn-1∑i<jNμi~nμj~n,h2n=1rn∑i<jNνi~nνj~n.Equation (38) can be rewritten as(39)1sn-11rn-1+1rn-α1=∑j=1Nμj~n,1rn1sn-1+1sn-α1=∑j=1Nνj~n,by making use of (37).Thus, (16) can be written as(40)∂tln⁡rn=∑j=1Nνj~n+α1,∂tln⁡sn=∑j=1Nμj~n+1+α1.Consider the function detWn which is a 4N+2th-order polynomial in λ:(41)-det⁡Wn=f2n+gnhn=14λ2∏j=12Nλ2-λj2=14λ~∏j=12Nλ~-λj~=14Rλ~.Substituting (30) into (41) and comparing coefficients of the same powers of λ read (42)α1=-12∑j=12Nλj~and deduce that (43)fnλ~=μ~kn=12Rμ~kn,fnλ~=ν~kn=12Rν~kn,gnt1=2An1gn-2Bn1fn,hnt1=2Cn1fn-2An1hn,by taking m=1.Hence, it follows that(44)gnt1λ~=μ~kn=-1sn-1μknRμ~kn=-1sn-1μkn∂t1μ~kn∏i=1,i≠kNμk~n-μi~n,hnt1λ~=ν~kn=1rnνknRν~kn=1rnνkn∂t1ν~kn∏i=1,i≠kNνk~n-νi~n.Again from (37) and (44), we have (45)∂t1μ~knRμ~kn=-1∏i=1,i≠kNμ~kn-μ~in,∂t1ν~knRν~kn=1∏i=1,i≠kNν~kn-ν~in.Similarly, whenm=2(46)gnt2λ~=μ~kn=-1sn-1μkn3-1sn-121rn-1+1rnμknRμ~kn=-1sn-1μkn∂t2μ~kn∏i=1,i≠kNμk~n-μi~n,hnt2λ~=ν~kn=1rnνkn3-1rn21sn-1+1snνknRν~kn=1rnνkn∂t2ν~kn∏i=1,i≠kNνk~n-νi~n.Thus(47)∂t2μ~knRμ~kn=-μ~kn-∑j=1Nμj~n-α1∏i=1,i≠kNμ~kn-μ~in,∂t2ν~knRν~kn=ν~kn-∑j=1Nνj~n-α1∏i=1,i≠kNν~kn-ν~in.


## 4. Straightening out of the Continuous and Discrete Flows

In order to acquire the algebro-geometric solutions of systems (16), we first introduce the Riemann surface Γ of the hyperelliptic curve with genus N: (48)Γ:ξ2=Rλ~,Rλ~=∏j=12Nλ~-λ~j,which has two infinite points ∞1 and ∞2, not branch point of Γ. We fix a set of regular cycle paths: a1,…,aN; b1,…,bN, which are independent and have the intersection numbers: (49)ak∘aj=bk∘bj=0,ak∘bj=δkj,1≤k,j≤N.We choose the holomorphic differentials, onΓ(50)ω~l=λl-1dλRλ,1≤l≤N,and define (51)Akj=∫ajω~i,Bkj=∫bjω~i,where A=(Aij)N×N,B=(Bij)N×N.Thus, we denote the matricesC and τ by (52)C=ckj=A-1,τ=A-1Band verify that τ is symmetric and has positive defined imaginary part.By normalizingω~j into the new basis ωj, (53)ωj=∑l=1N-1cjlω~l,j=1,2,…,N,which meets (54)∫akωj=∑l=1Ncjl∫akω~l=∑l=1NcjlAlk=δjk,∫bkωj=∑l=1Ncjl∫bkω~l=∑l=1NcjlBlk=τjk.The Abel map A(p) is introduced as (55)Ap=∫p0pω,and the Able-Jacobi coordinates are defined as(56)ρ1n=A∑k=1Npμ~kn=∑k=1N∫p0pμ~knω,ρ2n=A∑k=1Npν~kn=∑k=1N∫p0pν~knω,where (57)pμ~k=λ~=μ~kn,ξ=Rμ~kn∈Γ,pν~k=λ~=ν~kn,ξ=Rν~kn∈Γ,and p0 is a chosen base point on Γ.The components of the Abel-Jacobi coordinates in (56) are(58)ρj1n=∑k=1N∫p0pμ~kωj=∑k=1N∑l=1N∫λ~p0μ~kncjlλ~l-1dλ~Rλ~,ρj2n=∑k=1N∫p0pν~kωj=∑k=1N∑l=1N∫λ~p0ν~kncjlλ~l-1dλ~Rλ~,where λ~(p0) is the local coordinate of p0.We infer that(59)∂tρj1n=∑k=1N∑l=1Ncjlμ~knl-1∂tμ~knRν~kn=-∑k=1N∑l=1Ncjlμ~knl-1∏i=1,i≠kNμ~kn-μ~in=-cjN≡Ωj1,∂tρj2n=∑k=1N∑l=1Ncjlν~knl-1∂tν~knRν~kn=∑k=1N∑l=1Ncjlν~knl-1∏i=1,i≠kNν~kn-ν~in=cjN≡-Ωj1.Similarly, we have (60)∂t2ρj1n=-∑l=1N∑k=1Ncjlμ~knl-1μ~kn-∑j=1Nμj~n-α1∏i=1,i≠kNμ~kn-μ~in=Ωj2,∂t2ρj2n=∑l=1N∑k=1Ncjlν~knl-1ν~kn-∑j=1Nνj~n-α1∏i=1,i≠kNν~kn-ν~in=-Ωj2.Let the fundamental solution matrix of (3) be of the form (61)Qn=ϕn,ϕ^n=ϕ1nϕ^1nϕ2nϕ^2n,Q0=I.It is easy to obtain that (62)Qn+1=UnUn-1⋯U0,from which we have (63)ϕ11=1λ,ϕ21=s0,ϕ^11=r0,ϕ^21=1λ+λr0s0,ϕ12=1λ2+r1s0,ϕ22=s11λ+s01λ+λr1s1,ϕ^12=r01λ+1λ+λr0s0r1,ϕ^22=s1r0+1λ+λr1s11λ+λr0s0,⋮Suppose thatδ is eigenvalue of the Lax matrix Wn in the solution space of equation ψ(n+1)=Unψ(n), which is invariant under the action of Wn due to (EWn)Un=UnWn. The corresponding eigenfunction is ψ(n) that can be called the Baker function which satisfies that (64)ψn+1=Unψn,Wnψn=δψn.It is easy to check that(65)det⁡δI-Wn=δ2-f2n-gnhn=0,which has two eigenvalues δ±=±δ, where(66)δ=f2n-gnhn=12Rλ~.The corresponding Baker function can be taken as(67)ϕ±n=ϕn+b±ϕ^n,ϕ^±n=ϕ^n+c±ϕn,where (68)b±=±δ-f0g0orb±=h0±δ+f0,c±=±δ+f0h0orc±=g0±δ-f0.Letp±(n,λ), q±(n,λ) be the components of the Baker functions ϕ±(n) and ϕ^±(n), respectively. Actually, starting from (69)Qn+1=UnQn,WnQn=WnUn-1Qn-1=⋯=Un-1Un-2⋯U0W0Q0=QnW0Q0=QnW0,we can infer that(70)p+n,λp-n,λ=s-1sn-1∏j=1Nλ~-μ~jnλ~-μ~j0,p+n,λp-n,λ=r0rn∏j=1Nλ~-ν~jnλ~-ν~j0.Similarly, we have(71)Δρ1n=ρ1n+1-ρ1n=Ω0modJ,Δρ2n=ρ2n+1-ρ2n=Ω0modJ,where Ω(0)=∫∞1∞2ω.


## 5. Algebro-Geometric Solutions

The well-known Riemann theta function ofΓ is defined by (72)θξ∣τ=∑z∈ZNexp⁡πiτz,z+2πiξ,z,ξ∈CN,where ξ=ξ1,…,ξNT, ξ,z=∑j=1Nξjzj.According to the Riemann theorem, there exists a constantM(i)∈CN so that(i)
F1=θ(Ap-ρ1n-M1) has exactly N zeros at λ~=μ~1(n),…,μ~Nn;(ii)
F2=θ(Ap-ρ2n-M2) has exactly N zeros at λ~=ν~1(n),…,ν~Nn.We have the inversion formula(73)∑j=1Nμ~jnk=IkΓ-∑s=12Resλ~=∞sλ~kdln⁡F1λ~,∑j=1Nν~jnk=IkΓ-∑s=12Resλ~=∞sλ~kdln⁡F2λ~,with the constant Ik(Γ)=∑j=1N∫ajλ~kωj. Through a standard treatment, we arrive at(74)∑j=1Nμ~jnk=IkΓ+∂tln⁡θρ1n+M1+π2θρ1n+M1+π1,∑j=1Nν~jnk=IkΓ+∂tln⁡θρ2n+M2+π1θρ2n+M2+π2,where π(s)=∫∞sp0ω.Substituting (74) into (40) yields (75)rn=exp⁡-lnθρ2n+M2+π1θρ2n+M2+π2-12t∑j=12Nλ~j+I1Γt,sn=exp⁡-ln⁡θρ1n+1+M1+π2θρ1n+1+M1+π1-12t∑j=12Nλ~j+I1Γt.Thus(76)rn=exp⁡-lnθΩ0n-Ω1t+Υ^1θΩ0n-Ω1t+Υ^2-12t∑j=12Nλ~j+I1Γt,sn=exp⁡-lnθΩ0n+1+Ω1t+Υ2θΩ0n+1+Ω1t+Υ1-12t∑j=12Nλ~j+I1Γt,where (77)Υ1=ρ01+M1+π1,Υ2=ρ01+M1+π2,Υ^1=ρ02+M2+π1,Υ^2=ρ02+M2+π2.which is the algebro-geometric solution to (16).Remark 1.
We have concluded the algebro-geometric solutions of the discrete system (16). It is significance of a major work for investigating numerical solutions of the discrete integrable system (16) like the way presented in [32]. Comparing the numerical solutions and algebro-geometric solutions about the discrete integrable system, we can get lots of useful properties. These problems will be studied in the future.



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