# Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

Authors: Chao Yue; Tiecheng Xia
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a3×3 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve Km−2 of arithmetic genus m−2, from which the corresponding Baker-Akhiezer function and meromorphic functions on Km−2 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.


---

## Body

## 1. Introduction

It is significantly important to search for solutions of nonlinear partial differential equations of mathematical physics. There are many methods to find the exact solutions [1, 2] and approximate solutions [1–3] of various nonlinear partial differential equations. Reaction diffusion equations are effective and important mathematical models, which contribute to explaining processes of the transition, diffusion, and fluidity of matter. Constructing exact solutions of such equations has been widely used in mathematics, physics, chemistry, biology, and other fields. Therefore, it is necessary for us to study algebro-geometric constructions of the coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3×3 matrix spectral problem. The third member in the hierarchy is
(1)u1,t2=12u1,xx+24u1−u12u2,u2,t2=−12u2,xx−24u2−u22u1,which is called the CCIRD equation compared with Equation (0.4) in Ref. [4].Algebro-geometric solution is closely associated with the inverse spectral theory [5, 6], and the solution of the KdV equation with an initial value problem was solved by the use of the method in Ref. [7]. Over the recent decades, integrable equations related to 2×2 matrix spectral problems have been extensively researched. Several systematic methods have been developed to construct algebro-geometric solutions for integrable equations such as KdV, Kadomtsev-Petviashvili equation, modified KdV, sine-Gordon, Ablowitz-Kaup-Newell-Segur, the Camassa-Holm equations, and Ablowitz-Ladik lattice [8–27]. But the study of algebro-geometric solutions of the whole hierarchy of 3×3 is still a challenging problem. Fortunately, in Ref. [28], a unified framework was proposed to yield algebro-geometric solutions of the whole Boussinesq hierarchy. Based on the work of that, a systematic method was proposed to define the trigonal curve and develop the framework to analyse soliton equations associated with the 3×3 matrix spectral problems, from which the algebro-geometric solutions of some entire hierarchies are obtained [29–34]. In Ref. [29], algebro-geometric quasi-periodic solutions to the three-wave resonant interaction hierarchy related to the trigonal curve with three infinite points were obtained. Wang and Geng constructed algebro-geometric solutions of a new hierarchy of soliton equations associated with a 3×3 matrix spectral problem [30] based on the methods used in [28, 29]. Later, Ma analysed the four-component AKNS soliton hierarchy, particularly asymptotics of the Baker-Akhiezer functions, in such a way that it proposes a general theory applicable to soliton hierarchies associated with 3×3 matrix spectral problems [31]. As a continuous study of [31], Ma constructed algebro-geometric solutions of the four-component AKNS soliton hierarchy in terms of a general theory of trigonal curves [32]. However, as far as we know, algebro-geometric solutions to the CCIRD hierarchy have not been investigated. The most important result of this paper is to give the explicit algebro-geometric solutions to the CCIRD hierarchy related to 3×3 matrix spectral problems by using the approaches used in [28–30], which complements the existing works in this area.The outline of this paper is as follows. In Section2, we obtain the CCIRD hierarchy related to a 3×3 matrix spectral problem based on the Lenard recursion equations. In Section 3, a trigonal curve Km−2 of arithmetic genus m−2 with three infinite points is introduced by the use of the characteristic polynomial of the Lax matrix for the stationary CCIRD equations, from which the stationary Baker-Akhiezer function and associated meromorphic functions are given on Km−2. Then, the stationary CCIRD equations are decomposed into the system of Dubrovin-type ordinary differential equations. In Section 4, we present the explicit theta function representations of the stationary Baker-Akhiezer function, of the meromorphic functions, and, in particular, of the potentials for the entire stationary CCIRD hierarchy. In Section 5, we extend all the Baker-Akhiezer functions, the meromorphic functions, the Dubrovin-type equations, and the theta function representations dealt with in Sections 3 and 4 to the time-dependent case.


## 2. The CCIRD Hierarchy

In the section, we shall derive the CCIRD hierarchy associated with a3×3 spectral problem:
(2)ψx=Uψ,U=iλu11u2−iλ111−2iλ,ψ=ψ1ψ2ψ3,where the potential u=u1,u2T and λ is a spectral parameter. Next, we introduce the Lenard gradient sequences
(3)Kgj−1=Jgj,gjui=0=0,j≥0,Kg^j−1=Jg^j,g^jui=0=0,j≥0,with two initial points
(4)g−1=0,0,0,0,0,0,1,0T,g^−1=0,0,0,0,0,0,0,1T,and two operators are defined as
(5)K=∂1000−1−2u1−u11∂0−u100−1−200∂1−102u2u20−u21∂001−100−10∂u212−1000u1∂−11−u20u110−1∂00−10−1110∂,J=2i00000000−i00000000−2i00000000−3i00000000i000000003i00−u20u110−1∂00−10−1110∂.Then, the sequencesgj and g^j,j≥0, can be uniquely determined and the first several members read as
(6)g0=iu1,−i,iu2,13i,−i,13i,0,0T,g^0=12iu1,−2i,12iu2,−13i,−2i,−13i,0,0T,g1=12u1,x−23,−23u1,−12u2,x−23,−23u2,−23u2,−23u1,12u1u2+1,89T,g^1=14u1,x+23,23u1,−14u2,x+23,23u2,23u2,23u1,14u1u2−1,199T.In order to obtain the CCIRD hierarchy, we solve the stationary zero-curvature equation(7)Vx−U,V=0,V=Vij3×3,which is equivalent to
(8)V11,x+u2V12+V13−u1V21−V31=0,(9)V22,x−u2V12+u1V21+V23−V32=0,(10)V33,x−V13−V23+V31+V32=0,(11)V12,x−2iλV12+u1V11−V22+V13−V32=0,(12)V13,x+iλV13+V11−V33+V12−u1V23=0,(13)V21,x+2iλV21+u2V22−V11+V23−V31=0,(14)V23,x+3iλV23+V22−V33−u2V13+V21=0,(15)V31,x−iλV31+V33−V11−V21+u2V32=0,(16)V32,x−3iλV32+V33−V22−V12+u1V31=0,where each entry Vij is a Laurent expansion in λ:
(17)Vij=∑k≥0Vij,k−1λ−k,1≤i≤3,1≤j≤3,and V11,V22,V33 satisfy V11+V22+V33=0. A direct calculation shows that (8)–(16) implies the Lenard equation
(18)KG=λJG,G=V12,V13,V21,V23,V31,V32,V22,V33T.Substituting Equation (17) into Equation (18) and collecting the same powers of λ, we get the following recursion relations:
(19)KGk−1=JGk,JG−1=0,where Gk=V12,k,V13,k,V21,k,V23,k,V31,k,V32,k,V22,k,V33,kT. Since equation JG−1=0 has the general solution
(20)G−1=α0g−1+β0g^−1,then Gj can be expressed as
(21)Gk=α0gk+β0g^k+⋯+αkg0+βkg^0+αk+1g−1+βk+1g^−1,k≥0,where αk and βk are arbitrary constants.Consider the auxiliary problem:(22)ψtr=V~rψ,V~r=V~ijr3×3,where each entry V~ijr has the form V~ijr=∑l=0rV~ij,l−1λr−l,1≤i≤3,1≤j≤3, and satisfies V~11r+V~22r+V~33r=0. Then, we introduce G~s=V~12,s,V~13,s,V~21,s,V~23,s,V~31,s,V~32,s,V~22,s,V~33,sT, which is determined by G~s=α~0gs+β~0g^s+⋯+α~sg0+β~sg^0+α~s+1g−1+β~s+1g^−1,−1≤s≤r−1,α~s and β~s are arbitrary constants. It is easy to find that G~s satisfies the Lenard equation KG~s=JG~s+1,JG~−1=0.Then, from the compatibility conditionUtr−V~xr+U,V~r=0 of Equations (2) and (22), we have
(23)utr=Xr,r≥0,where the vector fields Xl=Xlu;α¯l,β¯l=PKG~l−1=PJG~l, the constant vectors α¯l=α~0,⋯,α~l,β¯l=β~0,⋯,β~l, and the projective map γ=γ1,γ2,γ3,γ4,γ5,γ6,γ7,γ8T→γ1,γ2. The third member in the hierarchy (22) is (for convenience, we take α~1=β~1=0)
(24)u1,t2=α~012u1,xx−269u1−u12u2+β~014u1,xx−19u1−12u12u2,u2,t2=α~0−12u2,xx+269u2+u22u1+β~0−14u2,xx+19u2+12u22u1.Takingα~0=−11,α~0=70 in system (23), we have
(25)u1,t2=12u1,xx+24u1−u12u2,u2,t2=−12u2,xx−24u2−u22u1,which is called the CCIRD equation compared with Equation (0.4) in Ref. [4].


## 3. The Stationary Baker-Akhiezer Function

In the section, we are devoted to detailed study of the stationary Baker-Akhiezer function and the associated meromorphic functions. Then, the system of Dubrovin-type differential equations is derived. Let us consider the stationary CCIRD hierarchy:Xn=0,n≥0, which is equivalent to the stationary zero-curvature equation, Vxn=U,Vn,Vn=λnV+=Vi,jn3×3,Vi,jn=∑k=0nVij,k−1λn−k, and Vij,k−1 determined by (21). It is easy to verify that the matrix yI−Vn also satisfies the stationary zero-curvature equation. Then, the characteristic polynomial of Lax matrix Vn,Fmλ,y=detyI−Vn, is independent of variable x with the expansion
(26)detyI−Vn=y3+Smλy−Tmλ,where Smλ and Tmλ are polynomials with constant coefficients of λ:
(27)Smλ=∑1≤i<j≤3ViinVijnVjinVjjn=−α02+β02+α0β0λ2n−2α0α1+2β0β1+α0β1+α1β0λ2n−1+⋯,(28)Tmλ=V11nV12nV13nV21nV22nV23nV31nV32nV33n=−α0β0α0+β0λ3n−α0+β0α0β1+α1β0+α0β0α1+β1λ3n−1+⋯.It is easy to find thatTmλ is a polynomial of degree 3n with respect to λ as α0β0α0+β0≠0. Then, Fmλ,y=0 naturally leads to a trigonal curve
(29)Km−2:Fmλ,y=y3+Smλy−Tmλ=0,with m=3n. For convenience, we denote the compactification of the curve Km−2 by the same symbol Km−2. Hence, Km−2 becomes a three-sheeted Riemann surface of arithmetic genus m−2 if it is nonsingular or smooth. Here, the meaning of nonsingular is that at each point P′=λ′,y′∈Km−2,∂Fmλ,y/∂λ,∂Fm(λ,y)/∂yλ,y=λ′,y′≠0 holds. For m≥4, these curves are typically nonhyperelliptic. Point P on Km−2 is represented as pairs P=λ,yP satisfying (29) along with P∞j=∞,∞j,j=1,2,3, the three different points at infinity, which can be computed from the curve y3+Smλy−Tmλ=0 by choosing λ=ζ−1. The complex structure on Km−2 is defined in the usual way by introducing local coordinate ζQ=λ−λQ near point Q=λQ,yQ∈Km−2 which is neither branch nor singular point of Km−2 except the three points P∞1,P∞2,P∞3, at infinity with local coordinate λ=ζ−1 and local coordinate ζQ0=λ−λQ0−1/3 near branch or singular point Q0=λQ0,yQ0∈Km−2.Next, we shall define the meromorphic functionsϕ2P,x and ϕ3P,x on Km−2 as follows:
(30)ϕ2P,x=ψ2P,x,x0ψ1P,x,x0,P∈Km−2,x∈ℂ,ϕ3P,x=ψ3P,x,x0ψ1P,x,x0,P∈Km−2,x∈ℂ,with the stationary Baker-Akhiezer function ψP,x,x0 defined by
(31)ψxP,x,x0=Uux;λPψP,x,x0,(32)Vnux;λPψP,x,x0=yPψP,x,x0,(33)ψ1P,x0,x0=1,P=λ,y∈Km−2\P∞1,P∞2,P∞3,x∈ℂ.By using (31)–(33), a direct calculation gives that
(34)ϕ2=Fm−2y2V23n−yCm+Dm=y2V13n−yAm+BmEm−2=yV23n+CmyV13n+Am,(35)ϕ3=Fm−2y2V32n−yCm+Dm=y2V12n−yAm+Bm−Em−2=yV32n+CmyV12n+Am,where
(36)Am=V12nV23n−V13nV22n,Bm=V13nV11nV33n−V13nV31n+V12nV11nV23n−V13nV21n,Cm=V13nV21n−V11nV23n,Dm=V23nV22nV33n−V23nV32n+V21nV13nV22n−V12nV23n,(37)Am=V13nV32n−V12nV33n,Bm=V13nV11nV22n−V12nV21n+V13nV11nV32n−V12nV31n,Cm=V12nV31n−V11nV32n,Dm=V32nV22nV33n−V23nV32n+V31nV12nV33n−V13nV32n,(38)Em−2=V13n2V32n+V12nV13nV22n−V33n−V12n2V23n,Em−2=V23n2V31n+V21nV33nV11n−V33n−V21n2V13n,Fm−2=V12n2V32n+V31nV32nV11n−V22n−V31n2V12n.Through straightforward calculations, we obtain some main interrelationships among polynomialsAm,⋯,Dm,Am,⋯,Dm,Em−2,Fm−2,Fm−2,Sm,Tm, some of which are summarized below:
(39)V13nFm−2=V23nDm−SmV23n2−Cm2,AmFm−2=TmV23n2+CmDm,(40)V23nEm−2=V13nBm−SmV13n2−Am2,CmEm−2=TmV13n2+AmBm,(41)V23nBm+V13nDm−V13nV23nSm+AmCm=0,V13nV23nTm+V23nAmSm+V13nCmSm−BmCm−AmDm=0,V23nAmTm+V13nCmTm+Em−2Fm−2−BmDm=0,(42)V12nFm−2=V32nDm−SmV32n2−Cm2,AmFm−2=TmV32n2+CmDm,(43)−V32nEm−2=V12nBm−SmV12n2−Am2,−CmEm−2=TmV12n2+AmBm,(44)V32nBm+V12nDm−V12nV32nSm+AmCm=0,V12nV32nTm+V32nAmSm+V13nCmSm−BmCm−AmDm=0,V32nAmTm+V12nCmTm+Em−2Fm−2−BmDm=0,(45)Em−2,x=iλEm−2−u12V13nSm−3Bm+2V12nSm−3Bm,V23nFm−2,x=V21−u2V23n2V23nSm−3Dm−5iλV23n+3V22nFm−2,V32nFm−2,x=V31−V32n2V23nSm−3Dm+4iλV32n−3V33nFm−2.By observing Equations (21) and (38), one infers that Em−2, Fm−2, and Fm−2 are polynomials with respect to λ of degree m−2. Therefore, we may write them in the following forms:
(46)Em−2λ,x=α0−β0α0+2β02α0+β02u1∏j=1m−2λ−μjx,(47)Fm−2λ,x=α0−β0α0+2β02α0+β06u2∏j=1m−2λ−νjx,(48)Fm−2λ,x=β0−α0α0+2β02α0+β03∏j=1m−2λ−ξjx,where α0−β0α0+2β02α0+β0≠0.Define(49)μ^ix=μjx,yμ^jx=μjx,−Amμjx,xV13nμjx,x=μjx,−Amμjx,xV12nμjx,x∈Km−2,ν^jx=νjx,yν^jx=νjx,−Cmνjx,xV23nνjx,x∈Km−2,ξ^jx=ξjx,yξ^jx=ξjx,−Cmξjx,xV32nξjx,x∈Km−2,1≤j≤m−2,x∈ℂ.In order to more distinctly put forward the properties ofϕ2P,x,ϕ3P,x, and ψ1P,x,x0, we introduce the holomorphic map ∗, changing sheets, which is defined by
(50)∗:Km−1→Km−1,P=λ,yiλ→P∗=λ,yi+1mod3λ,i=0,1,2,P∗∗≔P∗∗,etc.,where yjλ,j=0,1,2, satisfy Fmλ,y=0, that is,
(51)y−y0λy−y1λy−y2λ=y3+Smλy−Tmλ=0.Furthermore, the positive divisors onKm−2 of degree m−2 are defined as
(52)DP1,⋯,Pm−2=Km−2→ℕ0,P→DP1,⋯,Pm−2=0,ifP∉P1,⋯,Pm−2,k,ifPoccursktimes∈P1,⋯,Pm−2,with ℕ0=ℕ∪0.Further properties ofϕ2P,x,ϕ3P,x, and ψ1P,x,x0 are summarized as follows:
(53)ϕ2,xP,x+u1ϕ22P,x+ϕ2P,xϕ3P,x+2iλϕ2P,x−ϕ3P,x−u2=0,(54)ϕ3,xP,x+ϕ32P,x+u1ϕ2P,xϕ3P,x−iλϕ3P,x−ϕ2P,x−1=0,(55)ϕ2P,xϕ2P∗,xϕ2P∗∗,x=−Fm−2λ,xEm−2λ,x,(56)ϕ3P,xϕ3P∗,xϕ3P∗∗,x=Fm−2λ,xEm−2λ,x,(57)ϕ2P,x+ϕ2P∗,x+ϕ2P∗∗,x=−2V13nλ,xSmλ−3Bmλ,xEm−2λ,x=Em−2,xλ,xEm−2λ,x−iλ+3V11nλ,xV13nλ,x×V13nλ,xu1xV13nλ,x−V12nλ,x,(58)ϕ3P,x+ϕ3P∗,x+ϕ3P∗∗,x=2V12nλ,xSmλ−3Bmλ,xEm−2λ,x=−Em−2,xλ,xEm−2λ,x−iλ+3u1xV11nλ,xV12nλ,x×V12nλ,xu1xV13nλ,x−V12nλ,x,(59)u1ϕ2P,x+ϕ2P∗,x+ϕ2P∗∗,x+ϕ3P,x+ϕ3P∗,x+ϕ3P∗∗,x=Em−2,xλ,xEm−2λ,x−iλ,(60)1ϕ2P,x+1ϕ2P∗,x+1ϕ2P∗∗,x=−5iλV23nλ,x−3V22nλ,xFm−2λ,x−V23nλ,xFm−2,xλ,xV21nλ,x−u2V23nλ,xFm−2λ,x,(61)1ϕ3P,x+1ϕ3P∗,x+1ϕ3P∗∗,x=4iλV32nλ,x−3V33nλ,xFm−2λ,x−V32nλ,xFm−2,xλ,xV31nλ,x−V32nλ,xFm−2λ,x,(62)ϕ2P,xϕ3P,x+ϕ2P∗,xϕ3P∗,x+ϕ2P∗∗,xϕ3P∗∗,x=1V32nλ,xV31nλ,x−V32nλ,xFm−2λ,x×V31nλ,xV32nλ,xFm−2,xλ,x+V31nλ,x3V33nλ,x−4iλV32nλ,xFm−2λ,x−3V33nλ,xV32nλ,x,(63)ϕ3P,xϕ2P,x+ϕ3P∗,xϕ2P∗,x+ϕ3P∗∗,xϕ2P∗∗,x=1V23nλ,xV21nλ,x−u2V23nλ,xFm−2λ,x×V21nλ,xV23nλ,xFm−2,xλ,x+V21nλ,x3V22nλ,x−4iλV23nλ,xFm−2λ,x−3V22nλ,xV23nλ,x,(64)ψ1P,x,x0=exp∫x0xiλ+u1x′ϕ2P,x′+ϕ3P,x′dx′,(65)ψ1P,x,x0ψ1P∗,x,x0ψ1P∗∗,x,x0=Em−2λ,xEm−2λ,x0exp2iλx−x0.The dynamics of the zerosμjxj=1m−2, νjxj=1m−2, and ξjxj=1m−2 of Em−2λ,x, Fm−2λ,x, and Fmλ,x are described according to Dubrovin-type equations as follows.Lemma 1.
Assume the zerosμjxj=1m−2, νjxj=1m−2, and ξjxj=1m−2 of Em−2λ,x, Fm−2λ,x, and Fmλ,x remain distinct, respectively, and let x∈ℂ. Then, μjxj=1m−2, νjxj=1m−2, and ξjxj=1m−2 satisfy the system of differential equations
(66)μj,xx=2V12nμjx,x−u1V13nμjx,x3y2μ^jx+Smμjxα0−β0α0+2β02α0+β0u1∏k=1k≠jm−2μjx−μkx,1≤j≤m−2,(67)νj,xx=6V21nνjx,x−u2V23nνjx,x3y2ν^jx+Smνjxα0−β0α0+2β02α0+β0u2∏k=1k≠jm−2νjx−νkx,1≤j≤m−2,(68)ξj,xx=−3V31nξjx,x−V32nξjx,x3y2ξ^jx+Smξjxα0−β0α0+2β02α0+β0∏k=1k≠jm−2ξjx−ξkx,1≤j≤m−2.Proof.
Substitutingλ=μjx into (40) and (43), we have
(69)V13nμjx,x2Smμjx−V13nμjx,xBmμjx+Am2μjx=0,V12nμjx,x2Smμjx−V12nμjx,xBmμjx+Am2μjx=0.
Then, we get(70)Bmμjx,x=SmμjxV13nμjx,x+Am2μjx,xV13nμjx,x=Smμjx+y2μjxV13nμjx,x,(71)Bmμjx,x=SmμjxV12nμjx,x+Am2μjx,xV12nμjx,x=Smμjx+y2μjxV12nμjx,x.
Then, insertingλ=μjx into the first equation of (45) and in view of (70) and (71), we arrive at
(72)Em−2,xμjx,x=u1V13nμjx,x−V12nμjx,x3y2μ^jx+Smμjx.
On the other hand, differentiating (46) with respect to x, we find
(73)Em−2,xμjx,x=β0−α0α0+2β02α0+β0u12μj,xx∏k=1k≠jm−2μjx−μkx.
Comparing (72) and (73), we derive (66). Similarly, we can prove that Equations (67) and (68) hold.☐


## 4. Algebro-Geometric Solutions of the Stationary CCIRD Hierarchy

In the section, we continue our study of the stationary CCIRD hierarchy and will obtain explicit Riemann theta function representations for the two meromorphic functionsϕ2P,x, ϕ3P,x, the Baker-Akhiezer function ψ1P,x,x0, and the algebro-geometric solutions u1 and u2 for the CCIRD hierarchy. By introducing the local coordinate ζ=λ−1 near P∞1,P∞2,P∞3∈Km−2, we have the following lemma.Lemma 2.
Suppose thatu satisfies the nth stationary CCIRD system Xn=0. Moreover, let P∈Km−2\P∞1,P∞2,P∞3,x,x0∈ℂ2. Then,
(74)ϕ2P,x=ζ→0−2iu1−1xζ−1+u1,xxu12x−23u1−2x+Oζ,asP→P∞1,13+Oζ,asP→P∞2,−i2u2xζ+12+14u2,xxζ2+Oζ3,asP→P∞3,(75)ϕ3P,x=ζ→023u1−1x+Oζ,asP→P∞1,iζ−1−13u1x+Oζ,as→P∞2,iζ+12u2xζ2+Oζ3,asP→P∞3,(76)ψ1P,x=ζ→0u1xu1x0exp−ix−x0ζ−1+Oζ,asP→P∞1,exp2ix−x0ζ−1+Oζ,asP→P∞2,expix−x0ζ−1+Oζ,asP→P∞3.Proof.
Substituting the three sets of ansatz(77)ϕ2P,x=ζ→0κ1,−1ζ−1+κ1,0+Oζ,ϕ3P,x=ζ→0χ1,0+Oζ,nearP∞1,ϕ2P,x=ζ→0κ2,0+Oζ,ϕ3P,x=ζ→0κ2,−1ζ−1+χ2,0+Oζ,nearP∞2,ϕ2P,x=ζ→0κ3,1ζ+κ3,2ζ2+Oζ3,ϕ3P,x=ζ→0χ3,1ζ+χ3,2ζ2+Oζ3,nearP∞3,into Riccati-type Equations (53) and (54), and comparing the coefficients of the same powers of ζ, we derive (74) and (75). Equation (76) then follows from inserting (74) and (75) into (64).
One infers from Equations (34), (35), (74), and (75) that the divisors ϕ2P,x and ϕ3P,x of ϕ2P,x and ϕ3P,x are as follows:
(78)ϕ2P,x=DP∞3,ν^1x,⋯,ν^m−2xP−DP∞1,μ^1x,⋯,μ^m−2xP,(79)ϕ3P,x=DP∞3,ξ^1x,⋯,ξ^m−2xP−DP∞2,μ^1x,⋯,μ^m−2xP,which means that P∞3,ν^1x,⋯,ν^m−2x are m−1 zeros of ϕ2P,x and P∞1,μ^1x,⋯,μ^m−2x its m−1 poles, P∞3,ξ^1x,⋯,ξ^m−2x are m−1 zeros of ϕ3P,x and P∞2,μ^1x,⋯,μ^m−2x its m−1 poles.
A tedious calculation reveals that the asymptotic behaviors ofyP and Sm near P∞1,P∞2,P∞3 are given as
(80)yP=ζ→0ζ−nα0+α1ζ+Oζ2,asP→P∞1,ζ−nβ0+β1ζ+Oζ2,asP→P∞2,−ζ−nα0+β0+α1+β1ζ+Oζ2,asP→P∞3,Sm=ζ→0−ζ−2nα02+β02+α0β0+2α0+β0α1+β1−α0β1−α1β0ζ+Oζ2,asP→P∞j,j=1,2,3.
Equip the Riemann surfaceKm−2 with an appropriate fixed homology basis aj,bjj=1m−2, in such a way that the intersection matrix of cycles satisfies aj∘bk=δj,k,aj∘ak=0,bj∘bk=0,j,k=1,⋯,m−2. For the present, we introduce the holomorphic differentials ϖlP on Km−2 defined by
(81)ϖlP=13y2P+Smλl−1dλ,1≤l≤2n−1,yPλl+n−mdλ,2n≤l≤m−2.
By using the basisaj and bj, the matrices A and B can be constructed from
(82)Ajk=∫akϖj,Bjk=∫bkϖj,and it is possible to show that matrices A and B are invertible. Now, we define the matrices C and τ by C=A−1,τ=A−1B. One can see that matrix τ=τijm−1×m−1 is symmetric, and it has a positive definite imaginary part. If we normalize ϖl into the new basis ωj,(83)ωj=∑l=1m−2Cjlϖl,then we have ∫akωj=δjk,∫bkωj=τjk,j,k=1,⋯,m−2.
Then, the Laurent expansion of (83) near P∞1,P∞2,P∞3 yields the following results:
(84)ω=ω1,⋯,ωj,⋯,ωm−2,(85)ωj=ζ→0−Cj,2n−1+α0Cj,m−2α0−β02α0+β0+Oζdζ,P→P∞1,Cj,2n−1+β0Cj,m−2α0−β0α0+2β0+Oζdζ,P→P∞2,−Cj,2n−1−α0+β0Cj,m−2α0+2β02α0+β0+Oζdζ,P→P∞3.
LetωP∞s,22P,s=1,2,3, denote the normalized Abelian differential of the second kind satisfying
(86)∫akωP∞s,22P=0,k=1,⋯,m−2,ωP∞s,22P=ζ→0ζ−2+O1dζasP→P∞s,and introduce
(87)Ω2P=−iωP∞1,22P+2iωP∞2,22P+iωP∞3,22P,then we have
(88)∫Q0PΩ2P=iζ−1+e12Q0+Oζ,asP→P∞1,−2iζ−1+e22Q0+Oζ,asP→P∞2,−iζ−1+e32Q0+Oζ,asP→P∞3,where e12Q0,e22Q0,e32Q0 are integral constants and Q0 is an appropriately chosen base point on Km−2\P∞1,P∞2,P∞3. The b-periods of the differential Ω2P are denoted by
(89)U22=U2,12,⋯,U2,m−22,U2,k2=12πi∫bkΩ2P,k=1,⋯,m−2.
Then, from (85) and (87), we have
(90)U2,k2=12πi−i∫bkωP∞1,22P+2i∫bkωP∞2,22P+i∫bkωP∞3,22P=−iCk,2n−1+α0Ck,3n−2α0−β02α0+β0+2iCk,2n−1+β0Ck,3n−2α0−β0α0+2β0−iCk,2n−1−α0+β0Ck,3n−2α0+2β02α0+β0=1α0+2β02α0+β0α0−β04iα0+5iβ0Ck,2n−1+2iα02+6iα0β0+iβ02Ck,3n−2,in which we used 1/2πi∫bkωQ1,22P=ρk,0,ifwk=∑l=0∞ρk,lζQ1dζQ1.
Furthermore, the normalized Abelian differential of the third kindωP∞3,P∞j3P,j=1,2, is holomorphic on Km−2\P∞3,P∞j with simple poles at P∞3 and P∞j with residues 1 and -1, respectively, that is,
(91)ωP∞3,P∞13P=ζ→0−ζ−1+O1dζ,asP→P∞1,O1dζ,asP→P∞2,ζ−1+O1dζ,asP→P∞3,(92)ωP∞3,P∞23P=ζ→0O1dζ,asP→P∞1,−ζ−1+O1dζ,asP→P∞2,ζ−1+O1dζ,asP→P∞3,(93)∫Q0PωP∞3,P∞13P=−lnζ+e1,∞13Q0+Oζ,asP⟶P∞1,e1,∞23Q0+Oζ,asP⟶P∞2,lnζ+e1,∞33Q0+Oζ,asP⟶P∞3,(94)∫Q0PωP∞3,P∞23P=e2,∞13Q0+Oζ,asP⟶P∞1,−lnζ+e2,∞23Q0+Oζ,asP⟶P∞2,lnζ+e2,∞33Q0+Oζ,asP⟶P∞3,where ek,∞j3Q0,k=1,2,j=1,2,3, are integration constants. Let Tm−2 be the period lattice Z¯∈ℂm−2∣Z¯=N¯+τL¯;N¯,L¯∈ℤm−2. The complex torus Jm−2=ℂm−2/Tm−2 is called the Jacobian variety of Km−2. An Abel map A¯:Km−2⟶Jm−2 is defined by
(95)AP=A1P,⋯,Am−2P=∫Q0Pω1,⋯,∫Q0Pωm−2modTm−2,with the natural linear extension to the factor group DivKm−2(96)A∑nkPk=∑nkAPk.
Considering the nonspecial divisorDμ^¯x=∑k−1m−2μ^kx,Dν^¯x=∑k−1m−2ν^kx, and Dξ^¯x=∑k−1m−2ξ^kx, we define
(97)A∑k=1m−2Pkjx=∑k=1m−2APkjx=∑k=1m−2∫Q0Pkjxω=ρjx,j=1,2,3,where Pk1x=μ^kx,Pk2x=ν^kx,Pk3x=ξ^kx,μ^¯x=μ^1x,⋯,μ^m−2x∈σm−2Km−2,ν^¯x=ν^1x,⋯,ν^m−2x∈σm−2Km−2,ξ^¯x=ξ^1x,⋯,ξ^m−2x∈σm−2Km−2, and
σm−2Km−2=Q1,⋯,Qn∣Qj∈Km−2,j=1,⋯,n denotes the nth symmetric power of Km−2.☐Theorem 3.
Assume that the curveKm−2 is nonsingular, and let x,x0∈ℂ. Then,
(98)ρ1x=ρ1x0+U22x−x0,ρ2x=ρ2x0+U22x−x0,ρ3x=ρ3x0+U22x−x0.Proof.
We prove only the first linearity of the Abel map with respect tox in (98). Assume that μjx≠μj′x for j≠j′; then, one computes
(99)ddxρl1x=ddx∑j=1m−2∫Q0μ^jωl=∑j=1m−2μj,xωlμ^j=∑j=1m−2μj,x∑k=1m−2Clkω¯k=∑j=1m−22V12nμj,x−u1V13nμj,x3y2μ^j+Smμjα0−β0α0+2β02α0+β0u1∏k=1r≠jm−2μjx−μrx∑k=1m−n−1Clkμjk−13y2μ^j+Smμj+∑k=m−nm−2Clkyμ^jμjk+n−m3y2μ^j+Smμj=−2α0−β0α0+2β02α0+β0∑j=1m−2∑k=1m−n−1Clku1V13nμj,x−V12nμj,xμjk−1u1∏r=1r≠jm−2μj−μr+∑j=1m−2∑k=m−nm−2Clku1V13nμj,x−V12nμj,xyμ^jμjk+n−mu1∏r=1r≠jm−2μj−μr=−2α0−β0α0+2β02α0+β0∑k=1m−n−1Clk∑j=1m−2−2iα0−5/2iβ0μjk+n−2∏r=1r≠jm−2μj−μr+∑k=m−nm−2Clk∑j=1m−2Amμj,x−u1Amμj,xμjk+n−mu1∏r=1r≠jm−2μj−μr,which yields by the use of the standard Lagrange interpolation argument that
(100)ddxρl1x=U2,l2,which implies the first representation of (98). The second and third equalities in (98) follow from the same calculation.
Denote byθz¯ the Riemann theta function associated with Km−2 equipped with a fixed homology basis. For convenience, the function z¯:Km−2×σm−2Km−2→ℂ is defined as
(101)z¯P,Q¯=M−AP+∑Q′∈Q¯DQ′AQ′,P∈Km−2,Q¯=Q1,⋯,Qm−2∈σm−2Km−2,j=1,2,3,where M is the vector of Riemann constants. Then, we get
(102)z¯P,μ^¯x=M−AP+ρ1x,P∈Km−2,z¯P,ν^¯x=M−AP+ρ2x,P∈Km−2,z¯P,ξ^¯x=M−AP+ρ3x,P∈Km−2.
In view of (98), we could rewrite them as
(103)z¯P∞j,μ^¯x=M1j+U22x,z¯P∞j,ν^¯x=M2j+U22x,z¯P∞j,ξ^¯x=M3j+U22x,where Msj=M−AP∞j+ρsx0−U22x0,j=1,2,3,s=1,2,3.
Combined with the above results, the theta function representations ofϕ2P,x,ϕ3P,x,ψ1P,x,x0 and the algebro-geometric solutions of the stationary CCIRD hierarchy are presented in the next theorem.☐Theorem 4.
Assume that the curveKm−2 is nonsingular. Let P=λ,y∈Km−2\P∞1,P∞2,P∞3 and let x,x0∈ℂ2. Suppose that Dμ^¯x or Dν^¯x or Dξ^¯x is nonspecial. Then,
(104)ϕ2P,x=N1x0θz¯P,ν^¯xθz¯P∞3,μ¯^xθz¯P∞1,ν^¯xθz¯P,μ^¯xexp∫Q0PωP∞3,P∞13+e12Q0−e32Q0x,(105)ϕ3P,x=N2x0θz¯P,ξ^¯xθz¯P∞3,μ^¯xθz¯P∞2,ξ^¯xθz¯P,μ^¯xexp∫Q0PωP∞3,P∞23+e22Q0−e32Q0x,(106)ψ1P,x,x0=θz¯P,μ^¯xθz¯P∞3,μ^¯x0θz¯P∞3,μ^¯xθz¯P,μ^¯x0expe32Q0−∫Q0PΩ2x−x0,(107)N1x0=−2iu1x0θz¯P∞1,μ^¯x0θz¯P∞3,μ^¯x0expe32Q0−e12Q0x0−e1,∞13Q0,(108)N2x0=iθz¯P∞2,μ^¯x0θz¯P∞3,μ^¯x0expe32Q0−e22Q0x0−e2,∞23Q0.
Finally, the theta representations ofu1x and u2x are of the form
(109)u1x=u1x0θz¯P∞1,μ^¯xθz¯P∞3,μ^¯x0θz¯P∞3,μ^¯xθz¯P∞1,μ^¯x0expe32Q0−e12Q0x−x0,(110)u2x=4u1x0θz¯P∞3,ν^¯xθz¯P∞1,μ^¯x0θz¯P∞1,ν^¯xθz¯P∞3,μ^¯x0×expe1,∞33Q0−e1,∞13Q0−e32Q0−e12Q0x−x0.Proof.
LetΨ1 be defined by the right-hand side of (106). We intend to prove that ψ1=Ψ1 with ψ1 given by (64). For that purpose, we first inspect the zeros and poles of ψ1. Since they can only come from zeros of ϕ2 and ϕ3, one can compute by using (34) and (35) that
(111)iλ+u1xϕ2P,x+ϕ3P,x=iλ+u1y2V13n−yAm+BmEm−2−y2V12n−yAm+BmEm−2=1Em−2u1V13n−V12ny2−u1Am−Amy+13Em−2,x−iλEm−2+2u1V13n−V12nSm+iλ=13Em−2,xEm−2+23u1V13n−V12n3y2+SmEm−2−u1V13nyy+Am/V13n−V12nyy+Am/V12nEm−2+23iλ=λ→μjx−μj,xxλ−μjx+O1=λ→μjx∂xlnλ−μjx+O1.
Then,(112)ψ1P,x,x0=exp∫x0xiλ+u1x′ϕ2P,x′+ϕ3P,x′dx′=λ−μjxλ−μjx0O1=λ−μjxO1,forPnearμ^jx≠μ^jx0,O1,forPnearμ^jx=μ^jx0,λ−μjx0−1O1,forPnearμ^jx0≠μ^jx,where O1≠0. Consequently, all zeros and poles of ψ1 and Ψ1 on Km−2\P∞1,P∞2,P∞3 are simple and coincident. It remains to identify the essential singularities of ψ1 and Ψ1 at P∞1,P∞2,P∞3. Considering (64), (76), (87), and the expression for Ψ1 in (106), we deduce that ψ1 and Ψ1 share the same singularities and zeros. The Riemann-Roch uniqueness results in the holomorphic function Ψ1/ψ1=γ, where γ is a constant. By (76), (87), and the right-hand side of (106), we have
(113)Ψ1P,x,x0ψ1P,x,x0=ζ→0expiζ−1x−x0+Oζ1+Oζexpiζ−1x−x0+Oζ=ζ→01+Oζ,P→P∞3.
Then, we concludeγ=1, with which the proof of (106) is completed. By using the asymptotic properties of ψ1 near P∞1, we get (109). Equations (78), (79), and (93) immediately yield that ϕ2 and ϕ3 have the following forms:
(114)ϕ2P,x=N^1xθz¯P,ν^¯xθz¯P,μ^¯xexp∫Q0PωP∞3,P∞13,ϕ3P,x=N^2xθz¯P,ξ^¯xθz¯P,μ^¯xexp∫Q0PωP∞3,P∞23.
Taking into account the asymptotic expansions ofϕ2 and ϕ3 near P∞1, P∞3, we have
(115)u1x=−2iN^1xθz¯P∞1,μ^¯xθz¯P∞1,ν^¯xexp−e1,∞13Q0,u2x=2iN^1xθz¯P∞3,ν^¯xθz¯P∞3,μ^¯xexpe1,∞33Q0,1u1x=32N^2xθz¯P∞1,ξ^¯xθz¯P∞1,μ^¯xexpe2,∞13Q0,which together with (109) show the expressions (107), (108), and (110).☐


## 5. Algebro-Geometric Solutions of the CCIRD Hierarchy

In this section, we extend the results of Sections3 and 4 to the time-dependent CCIRD hierarchy. In particular, we obtain Riemann theta function representations for the time-dependent Baker-Akhiezer function, the meromorphic function, and algebro-geometric solutions of the CCIRD hierarchy.Similar to (31)–(33), we consider the following time-dependent Baker-Akhiezer function:
(116)ψxP,x,x0,tr,t0,r=Uux,tr;λPψP,x,x0,tr,t0,r,ψtrP,x,x0,tr,t0,r=V~rux,tr;λPψP,x,x0,tr,t0,r,Vnux,tr;λPψP,x,x0,tr,t0,r=yPψP,x,x0,tr,t0,r,ψ1P,x0,x0,t0,r,t0,r=1,P=λ,y∈Km−2\P∞1,P∞2,P∞3,x,tr,x0,t0,r∈ℂ.The compatibility conditions of the first three equations in (116) show that
(117)Utr−V~xr+U,Vr=0,(118)−Vxn+U,Vn=0,(119)−Vtrn+V~r,Vn=0.It is easy to find thatyI−Vn satisfies (118) and (119). Then, the characteristic polynomial of Lax matrix Vn for the CCIRD hierarchy is a constant independent of variables x and tr with the expansion
(120)detyI−Vn=y3+Smλy−Tmλ,where Smλ and Tmλ are defined as in (27) and (28). Then, the CCIRD curve Km−2 is determined by
(121)Km−2:Fmλ,y=y3+Smλy−Tmλ=0.Closely related toψP,x,x0,tr,t0,r are the following two meromorphic functions ϕ2P,x,tr and ϕ3P,x,tr on Km−2 defined by
(122)ϕ2P,x,tr=ψ2P,x,x0,tr,t0,rψ1P,x,x0,tr,t0,r,P∈Km−2,x,tr,x0,t0,r∈ℂ,(123)ϕ3P,x,tr=ψ3P,x,x0,tr,t0,rψ1P,x,x0,tr,t0,r,P∈Km−2,x,tr,x0,t0,r∈ℂ,which imply from (116) that
(124)ϕ2P,x,tr=Fm−2λ,x,try2V23nλ,x,tr−yCmλ,x,tr+Dmλ,x,tr=y2V13nλ,x,tr−yAmλ,x,tr+Bmλ,x,trEm−2λ,x,tr=yV23nλ,x,tr+Cmλ,x,tryV13nλ,x,tr+Amλ,x,tr,(125)ϕ3P,x,tr=Fm−2λ,x,try2V32nλ,x,tr−yCmλ,x,tr+Dmλ,x,tr=y2V12nλ,x,tr−yAmλ,x,tr+Bmλ,x,tr−Em−2λ,x,tr=yV32nλ,x,tr+Cmλ,x,tryV12nλ,x,tr+Amλ,x,tr,where P=λ,y∈Km−2,x,tr∈ℂ2 and Amλ,x,tr,⋯,Dmλ,x,tr,Amλ,x,tr,⋯,Dmλ,x,tr,Em−2,Fm−2λ,x,tr,Fm−2λ,x,tr are defined as in (36)–(38). Hence, (39)–(45) also hold in the present context. Similarly, we have
(126)Em−2λ,x,tr=α0−β0α0+2β02α0+β02u1x,tr∏j=1m−2λ−μjx,tr,(127)Fm−2λ,x,tr=α0−β0α0+2β02α0+β06u2x,tr∏j=1m−2λ−νjx,tr,(128)Fm−2λ,x,tr=β0−α0α0+2β02α0+β03∏j=1m−2λ−ξjx,tr.After definingμ^jx,tr,ν^jx,tr,ξ^jx,tr as (46)–(48) by replacing x with x,tr, one infers from (74), (75), (124), and (128) that the divisors ϕ2P,x,tr and ϕ3P,x,tr of ϕ2P,x,tr and ϕ3P,x,tr are as follows:
(129)ϕ2P,x,tr=DP∞3,ν^1x,tr,⋯,ν^m−2x,trP−DP∞1,μ^1x,tr,⋯,μ^m−2x,trP,ϕ3P,x,tr=DP∞3,ξ^1x,tr,⋯,ξ^m−2x,trP−DP∞2,μ^1x,tr,⋯,μ^m−2x,trP.Differentiating (122) and (123) with respect to tr and using (116), we get
(130)ϕ2,tr=ψ2ψ1tr=V~21rψ1+V~22rψ2+V~23rψ3ψ1−ϕ2V~11rψ1+V~12rψ2+V~13rψ3ψ1=V~21r+V~22r−V~11rϕ2+V~23rϕ3−V~12rϕ22−V~13rϕ2ϕ3,ϕ3,tr=ψ3ψ1tr=V~31rψ1+V~32rψ2+V~33rψ3ψ1−ϕ3V~11rψ1+V~12rψ2+V~13rψ3ψ1=V~31r+V~33r−V~11rϕ3+V~32rϕ2−V~13rϕ32−V~12rϕ2ϕ3.Further properties ofϕ2P,x,tr and ϕ3P,x,tr can be presented, similar to (55)–(63), replacing x with x,tr,P,x with P,x,tr, etc. The four important ones of that are given as follows:
(131)ϕ2,xP,x,tr+u1x,trϕ22P,x,tr+ϕ2P,x,trϕ3P,x,tr+2iλϕ2P,x,tr−ϕ3P,x,tr−u2x,tr=0,ϕ3,xP,x,tr+ϕ32P,x,tr+u1x,trϕ2P,x,trϕ3P,x,tr−iλϕ3P,x,tr−ϕ2P,x,tr−1=0,ϕ2,trP,x,tr=V~21rx,tr+V~22rx,tr−V~11rx,trϕ2P,x,tr+V~23rx,trϕ3P,x,tr−V~12rx,trϕ22P,x,tr−V~13rx,trϕ2P,x,trϕ3P,x,tr,ϕ3,trP,x,tr=V~31rx,tr+V~33rx,tr−V~11rx,trϕ3P,x,tr+V~32rx,trϕ2P,x,tr−V~13rx,trϕ32P,x,tr−V~12rx,trϕ2P,x,trϕ3P,x,tr.Lemma 5.
Assume (116) and (117) and let λ,x,tr∈ℂ3. Then,
(132)Em−2,trλ,x,tr=Em−2,xλ,x,trV13nV~12r−V12nV~13ru1V13n−V12n+Em−23V~11r−u1V~13r−V~12ru1V13n−V12nV11n−iλV13nV~12r−V12nV~13ru1V13n−V12n,(133)Fm−2,trλ,x,tr=Fm−2,xλ,x,trV21nV~23r−V23nV~21rV21n−u2V23n+Fm−2λ,x,tr3V~22r−V~21r−u2V~23rV21n−u2V23nV22n+5iλV21nV~23r−V23nV~21rV21n−u2V23n,(134)Fm−2,trλ,x,tr=Fm−2,xλ,x,trV31nV~32r−V32nV~31rV31n−V32n+Fm−2λ,x,tr3V~33r−V~31r−V~32rV31n−V32nV33n−4iV31nV~32r−V32nV~31rV31n−V32n.Proof.
Differentiating (59) (by replacing x with x,tr) with respect to tr, we get
(135)Em−2,xEm−2tr=∂x∂trlnEm−2=u1ϕ2+ϕ2∗+ϕ2∗∗+ϕ3+ϕ3∗+ϕ3∗∗tr=u1,tr+u1V~22r−V~11r+V~32rϕ2+ϕ2∗+ϕ2∗∗+V~33r−V~11r+u1V~23rϕ3+ϕ3∗+ϕ3∗∗−u1V~12rϕ22+ϕ2∗2+ϕ2∗∗2−V~13rϕ32+ϕ3∗2+ϕ3∗∗2−u1V~13r+V~12rϕ2ϕ3+ϕ2∗ϕ3∗+ϕ2∗∗ϕ3∗∗+3u1V~21r+V~31r=V~12rϕ2+ϕ2∗+ϕ2∗∗+V~13rϕ3+ϕ3∗+ϕ3∗∗+3V~11rx.
Without loss of generality, we take the integration constant as zero, then get(136)∂trlnEm−2λ,x,tr=V~12rϕ2+ϕ2∗+ϕ2∗∗+V~13rϕ3+ϕ3∗+ϕ3∗∗+3V~11r=Em−2,xEm−2λ,x,tr−iλV13nV~12r−V12nV~13ru1V13n−V12n+3V~11r−u1V~13r−V~12ru1V13n−V12nV11n, which implies equation (132). Differentiating (55) (by replacing x with x,tr) with respect to tr and using (55), (60), and (63), we can deduce
(137)−Fm−2Em−2tr=ϕ2ϕ2∗ϕ2∗∗ϕ2,trϕ2+ϕ2,tr∗ϕ2∗+ϕ2,tr∗∗ϕ2∗∗=ϕ2ϕ2∗ϕ2∗∗V~21r1ϕ2+1ϕ2∗+1ϕ2∗∗+V~23rϕ3ϕ2+ϕ3∗ϕ2∗+ϕ3∗∗ϕ2∗∗+3V~22r=−V~12rϕ2+ϕ2∗+ϕ2∗∗−V~13rϕ3+ϕ3∗+ϕ3∗∗−3V~11r=−Fm−2Em−2V21nV~23r−V23nV~21rV21n−u2V23nFm−2,xFm−2+5iλ+3V~22r−V~21r−u2V~23rV21n−u2V23nV22n−Em−2,trEm−2.
Thus, we prove the expression (133). The last one can be proved in the same way.☐We present some properties ofψ1P,x,x0,tr,t0,r as follows.Lemma 6.
Assume (116) and (122), P=λ,y∈Km−2\P∞, and let λ,x,x0,tr,t0,r∈ℂ5. Then,
(138)ψ1P,x,x0,tr,t0,r=exp∫x0xiλ+u1x′,trϕ2P,x′,tr+ϕ3P,x′,trdx′+∫t0,rtrV~11rλ,x0,t′+V~12rλ,x0,t′ϕ2P,x0,t′+V~13rλ,x0,t′ϕ3P,x0,t′dt′,(139)ψ1P,x,x0,tr,t0,rψ1P∗,x,x0,tr,t0,rψ1P∗∗,x,x0,tr,t0,r==Em−2λ,x,trEm−2λ,x0,t0,rexp2iλx−x0,(140)ψ1P,x,x0,tr,t0,r=Em−2λ,x,trEm−2λ,x0,t0,r1/3×exp∫x0x23iλ+y2u1V13n−V12n−yu1Am−Am+2/3u1V13n−V12nSmEm−2dx′+∫t0,rtry2u1V13n−V12n−yu1Am−Am+2/3u1V13n−V12nSmEm−2dt′.
Similar to Lemma1, the zeros μjxj=1m−2, νjxj=1m−2, and ξjxj=1m−2 of Em−2λ,x, Fm−2λ,x, and Fmλ,x are described in terms of Dubrovin-type equations as follows.Lemma 7.
(i)
Assume the zerosμjx,trj=1m−2 of Em−2λ,x,tr remain distinct for x,tr∈Ωμ, where Ωμ∈ℂ2 is open and connected. Then, μjx,trj=1m−2 satisfy the system of differential equations(141)μj,xx,tr=2V12nμjx,tr,x,tr−u1x,trV13nμjx,tr,x,tr3y2μ^jx,tr+Smμjx,trα0−β0α0+2β02α0+β0u1x,tr∏k=1k≠jm−2μjx,tr−μkx,tr,1≤j≤m−2,μj,trx,tr=2V~13rV12n−V~12rV13nλ=μjx,tr3y2μ^jx,tr+Smμjx,trα0−β0α0+2β02α0+β0u1x,tr∏k=1k≠jm−2μjx,tr−μkx,tr,1≤j≤m−2(ii)
Assume the zerosνjx,trj=1m−2 of Fm−2λ,x,tr remain distinct for x,tr∈Ων, where Ων∈ℂ2 is open and connected. Then, νjx,trj=1m−2 satisfy the system of differential equations(142)νj,xx,tr=6V21nνjx,tr,x,tr−u2x,trV23nνjx,tr,x,tr3y2ν^jx,tr+Smνjx,trα0−β0α0+2β02α0+β0u2x,tr∏k=1k≠jm−2νjx,tr−νkx,tr,1≤j≤m−2,νj,trx,tr=6V~23rV21n−V~21rV23nλ=νjx,tr3y2ν^jx,tr+Smνjx,trα0−β0α0+2β02α0+β0u2x,tr∏k=1k≠jm−2νjx,tr−νkx,tr,1≤j≤m−2(iii)
Assume the zerosξjx,trj=1m−2 of Fm−2λ,x,tr remain distinct for x,tr∈Ωξ, where Ωξ∈ℂ2 is open and connected. Then, ξjx,trj=1m−2 satisfy the system of differential equations(143)ξj,xx,tr=−3V31nξjx,tr,x,tr−V32nξjx,tr,x,tr3y2ξ^jx,tr+Smξjx,trα0−β0α0+2β02α0+β0∏k=1k≠jm−2ξjx,tr−ξkx,tr,1≤j≤m−2,ξj,trx,tr=−3V~32rV31n−V~31rV32nλ=ξjx,tr3y2ξ^jx,tr+Smξjx,trα0−β0α0+2β02α0+β0∏k=1k≠jm−2ξjx,tr−ξkx,tr,1≤j≤m−2.
For convenience, we introduce the notation(144)V~1jr=V~1jr,1,0=∑k=0rV~1j,k−11,0λr−k,j=1,2,3,V~1jr,0,1=∑k=0rV~1j,k−10,1λr−k,j=1,2,3,where
(145)V~1j,k−11,0=V~1j,k−1α~0=1,β~0=0,α~1,⋯,α~r,β~1,⋯,β~r∈ℝ,j=1,2,3,V~1j,k−10,1=V~1j,k−1α~0=0,β~0=1,α~1,⋯,α~r,β~1,⋯,β~r∈ℝ,j=1,2,3,and the corresponding homogeneous cases
(146)V~¯1jr=V~¯1jr,1,0=∑k=0rV~¯1j,k−11,0λr−k,j=1,2,3,V~¯1jr,0,1=∑k=0rV~¯1j,k−10,1λr−k,j=1,2,3,with
(147)V~¯1j,k−11,0=V~1j,k−1α~0=1,β~0=0,α~1=⋯=α~r=β~1=⋯=β~r=0,j=1,2,3,V~¯1j,k−10,1=V~1j,k−1α~0=0,β~0=1,α~1=⋯=α~r=β~1=⋯=β~r=0,j=1,2,3.
In view of (138), we denote the function Irp,x,tr by
(148)IrP,x,tr=V~11rλ,x,tr+V~12rλ,x,trϕ2P,x,tr+V~13rλ,x,trϕ3P,x,tr,and I¯rp,x,tr is the associated homogeneous quantity replacing V~11r,V~12r,V~13r by the corresponding homogeneous polynomials V~¯11r,V~¯12r,V~¯13r, that is,
(149)I¯rP,x,tr=V~¯11rλ,x,tr+V~¯12rλ,x,trϕ2P,x,tr+V~¯13rλ,x,trϕ3P,x,tr,especially
(150)I¯r1,0=V~11r+V~12rϕ2+V~13rϕ3α~0=1,β~0=0,α~1=⋯=α~r=β~1=⋯=β~r=0,I¯r0,1=V~11r+V~12rϕ2+V~13rϕ3α~0=0,β~0=1,α~1=⋯=α~r=β~1=⋯=β~r=0.Lemma 8.
Assumex,tr∈ℂ2,λ=ζ−1 denotes the local coordinate near P∞j,j=1,2,3. Then,
(151)I¯r1,0=ζ−r+2iu1V~¯12,r1,0+Oζ,P⟶P∞1,−iV~¯13,r1,0+Oζ,P⟶P∞2,ζ−r+Oζ,P⟶P∞3,(152)I¯r0,1=2iu1V~¯12,r0,1+Oζ,P⟶P∞1,ζ−r−iV~¯13,r0,1+Oζ,P⟶P∞2,ζ−r+Oζ,P⟶P∞3.Proof.
We only prove (151) and accordingly obtain (152). From (149), it is easy to see that
(153)I¯r1,0=−ζ−1+iu1ϕ2−iϕ3=ζ−1+2iu1V~¯12,11,0+Oζ,P⟶P∞1,−iV~¯13,11,0+Oζ,P⟶P∞2,ζ−1+Oζ,P⟶P∞3;thus, (151) is right for r=1. By using (116), (117), and (149), we have
(154)∂xI¯r1,0P,x,tr=∂xV~¯11r,1,0λ,x,tr+V~¯12r,1,0λ,x,trϕ2P,x,tr+V~¯13r,1,0λ,x,trϕ3P,x,tr=∂tiλ+u1x,trϕ2P,x,tr+ϕ3P,x,tr.(i)
WhenP→P∞1, by investigating (153), one can assume that I¯r1,0P,x,tr has the following expansion(155)I¯r1,0P,x,tr=ζ→0ζ−r+∑j=0∞ς1jx,trζj,asP→P∞1,for some coefficients ς1jx,trj∈ℕ0. Suppose that asP⟶P∞1(156)ϕ2=∑j=0∞κ1,j−1ζj−1,ϕ3=∑j=0∞χ1,jζj,where κ1,j−1 and χ1,j are defined in (74) and (75). Substituting (155) and (156) into (154) and comparing the same powers of ζ yield
(157)ς1j,x=u1κ1,j+χ1,jtr,j=0,1,2,⋯,ς10,x=u1κ1,0+χ1,0tr=2iu1V~¯12,r1,0x,ς11,x=u1κ1,1+χ1,1tr=−V~¯11,r1,0−u1,xu12V~¯12,r1,0−23u1V~¯13,r1,0−1u1V~¯12,r1,0+2iu1V~¯12,r+11,0x,from which it can be inferred
(158)ς10x,tr=ε1,0tr+2iu1V~¯12,r1,0,ς11x,tr=ε1,1tr−V~¯11,r1,0−u1,xu12V~¯12,r1,0−23u1V~¯13,r1,0−1u1V~¯12,r1,0+2iu1V~¯12,r+11,0,where ε1,0tr and ε1,1tr are integration constants. We find that the coefficients of the power series for ϕ2P,x,tr,ϕ3P,x,tr near P∞1 and the coefficients of the homogeneous polynomials V~¯1j,r1,0 are differential polynomials in u, with no arbitrary integration constants in their construction, and the definition of I¯r1,0, it follows that it also can have no arbitrary integration constants and must consist purely of differential polynomials in u, and ε1,0tr=ε1,1tr=0. Therefore, we have
(159)I¯r1,0P,x,tr=ζ−r+2iu1V~¯12,r1,0−V~¯11,r1,0+u1,xu12V~¯12,r1,0+23u1V~¯13,r1,0−1u1V~¯12,r1,0−2iu1V~¯12,r+11,0ζ+Oζ2,asP⟶P∞1.
On the other hand, we find(160)I¯r+11,0P,x,tr=V~¯11r+1,1,0+V~¯12r+1,1,0ϕ2+V~¯13r+1,1,0ϕ3=ζ−1I¯r1,0+V~¯11,r1,0+V~¯12,r1,0ϕ2+V~¯13,r1,0ϕ3=ζ−r−1+2iu1V~¯12,r+11,0+Oζ,asP⟶P∞1.(ii)
WhenP→P∞2, by analysing (153), one can assume that I¯r1,0P,x,tr has the following form:(161)I¯r1,0P,x,tr=ζ→0∑j=0∞ς2jx,trζj,asP→P∞2,for some coefficients ς2jx,trj∈ℕ0. Suppose that asP→P∞2(162)ϕ2=∑j=0∞κ2,jζj,ϕ3=∑j=0∞χ2,j−1ζj−1,where κ2,j and χ2,j−1 are defined in (74) and (75). Inserting (161) and (162) into (154) and comparing the same powers of ζ imply
(163)ς2j,x=u1κ2,j+χ2,jtr,j=0,1,2,⋯,ς20,x=u1κ2,0+χ2,0tr=−iV~¯13,r1,0x,ς21,x=u1κ2,1+χ2,1tr=−V~¯11,r1,0−13V~¯12,r1,0−u1V~¯13,r1,0−iV~¯13,r+11,0x, from which it can be inferred
(164)ς20x,tr=ε2,0tr−iV~¯13,r1,0,ς21x,tr=ε2,1tr−V~¯11,r1,0−13V~¯12,r1,0−u1V~¯13,r1,0−iV~¯13,r+11,0,where ε2,0tr and ε2,1tr are integration constants. Managed together, we find that ε2,0tr=ε2,1tr=0. Then,
(165)I¯r1,0P,x,tr=−iV~¯13,r1,0−V~¯11,r1,0+13V~¯12,r1,0−u1V~¯12,r1,0+iV~¯13,r+11,0ζ+Oζ2,asP⟶P∞2.
On the other hand, we get(166)I¯r+11,0P,x,tr=V~¯11r+1,1,0+V~¯12r+1,1,0ϕ2+V~¯13r+1,1,0ϕ3=ζ−1I¯r1,0+V~¯11,r1,0+V~¯12,r1,0ϕ2+V~¯13,r1,0ϕ3=−iV~¯13,r+11,0+Oζ,asP⟶P∞2.(iii)
WhenP→P∞3, in terms of (153), one can assume that I¯r1,0P,x,tr has the following expansion:(167)I¯r1,0P,x,tr=ζ→0−ζ−r+∑j=1∞ς3jx,trζj,asP→P∞3,for some coefficients ς3jx,trj∈ℕ. Suppose that asP→P∞3(168)ϕ2=∑j=1∞κ3,jζj,ϕ3=∑j=1∞χ3,jζj,where κ3,j and χ3,j are defined in (74) and (75). Substituting (167) and (168) into (154) and comparing the same powers of ζ yield
(169)ς3j,x=u1κ3,j+χ3,jtr,j=0,1,2,⋯,ς31,x=u1κ3,1+χ3,1tr=−V~¯11,r1,0x,from which one can infer
(170)ς31x,tr=ε3,1tr−V~¯11,r1,0,where ε3,1tr is an integration constant. Similarly, one can conclude ε3,1tr=0. Therefore,
(171)I¯r1,0P,x,tr=−ζ−r−V~¯11,r1,0ζ+Oζ2,asP⟶P∞3.
On the other hand,(172)I¯r+11,0P,x,tr=V~¯11r+1,1,0+V~¯12r+1,1,0ϕ2+V~¯13r+1,1,0ϕ3=ζ−1I¯r1,0+V~¯11,r1,0+V~¯12,r1,0ϕ2+V~¯13,r1,0ϕ3=−ζ−r−1+Oζ,asP⟶P∞3.
Thus, we complete the proof of (151). Similarly, we can prove that (152) is right.
From (116), one infers that
(173)IrP,x,tr=∑l=0rα~r−lI¯l1,0P,x,tr+β~r−lI¯l1,0P,x,tr.
Therefore,(174)∫t0,rtrIrP,x,t′dt′=ζ→0tr−t0,r∑l=0rα~r−lζ−l+lnu1x,tru1x,t0,r+Oζ,asP→P∞1,tr−t0,r∑l=0rβ~r−lζ−l+Oζ,asP→P∞2,t0,r−tr∑l=0rα~r−l+β~r−lζ−l+Oζ,asP→P∞3.
LetωP∞l,j2,j∈ℕ,l=1,2,3, be the normalized differential of the second kind holomorphic on Km−2\P∞l with a pole of order j at P∞l,
(175)ωP∞l,j2P=ζ→0ζ−j+O1dζ,asP→P∞l.
Furthermore, we define the normalized differential of the second kind by(176)Ω~r+12P=∑l=0rα~r−llωP∞1,l+12+∑l=0rβ~r−llωP∞2,l+12−∑l=0rα~r−l+β~r−llωP∞3,l+12.
In addition, we define the vector ofb-periods of the differential of the second kind Ω~r+12,(177)U~r+12=U~r+1,12,⋯,U~r+1,m−22,U~r+1,k2=12πi∫bkΩ~r+12,k=1,⋯,m−2.
Integrating Equation (176) gives rise to
(178)∫Q0PΩ~r+12=−∑l=0rα~r−lζ−l+e12Q0+Oζ,asP→P∞1,−∑l=0rβ~r−lζ−l+e22Q0+Oζ,asP→P∞2,∑l=0rα~r−l+β~r−lζ−l+e32Q0+Oζ,asP→P∞3,where e12Q0,e22Q0,e32Q0 are constants.☐Given these results, the theta function representations ofϕ2P,x,tr,ϕ3P,x,tr and ψ1P,x,x0,tr,t0,r, particularly, the algebro-geometric solutions of CCIRD hierarchy, are shown as follows.Theorem 9.
Assume that the curveKm−2 is nonsingular. Let P=λ,y∈Km−2\P∞1,P∞2,P∞3 and let x,x0,tr,t0,r∈ℂ4. Suppose that Dμ^¯x,tr or Dν^¯x,tr or Dξ^¯x,tr is nonspecial. Then,
(179)ϕ2P,x,tr=N1x0,t0,rθz¯P,ν^¯x,trθz¯P∞3,μ^¯x,trθz¯P∞1,ν^¯x,trθz¯P,μ^¯x,tr×exp∫Q0PωP∞3,P∞13+e12Q0−e32Q0x+e12Q0−e32Q0tr,(180)ϕ3P,x,tr=N2x0,t0,rθz¯P,ξ^¯x,trθz¯P∞3,μ^¯x,trθz¯P∞2,ξ^¯x,trθz¯P,μ^¯x,tr×exp∫Q0PωP∞3,P∞23+e22Q0−e32Q0x+e22Q0−e32Q0tr,(181)ψ1P,x,x0,tr,t0,r=θz¯P,μ^¯x,trθz¯P∞3,μ^¯x0,t0,rθz¯P∞3,μ^¯x,trθz¯P,μ^¯x0,t0,r×expe32Q0−∫Q0PΩ2x−x0+e32Q0−∫Q0PΩ~r+12Ptr−t0,r,(182)N1x0,t0,r=−2iu1x0,t0,rθz¯P∞1,μ^¯x0,t0,rθz¯P∞3,μ^¯x0,t0,r×expe32Q0−e12Q0x0+e32Q0−e12Q0t0,r−e1,∞13Q0,(183)N2x0,t0,r=iθz¯P∞2,μ^¯x0,t0,rθz¯P∞3,μ^¯x0,t0,r×expe32Q0−e22Q0x0+e32Q0−e22Q0t0,r−e2,∞23Q0.
Finally, the theta representations ofu1x,tr and u2x,tr read
(184)u1x,tr=u1x0,t0,rθz¯P∞1,μ^¯x,trθz¯P∞3,μ^¯x0,t0,rθz¯P∞3,μ^¯x,trθz¯P∞1,μ^¯x0,t0,r×expe32Q0−e12Q0x−x0+e32Q0−e12Q0tr−t0,r,(185)u2x,tr=4u1x0,t0,rθz¯P∞3,ν^¯x,trθz¯P∞1,μ^¯x0,t0,rθz¯P∞1,ν^¯x,trθz¯P∞3,μ^¯x0,t0,r×expe1,∞33Q0−e1,∞13Q0−e32Q0−e12Q0x−x0−e32Q0−e12Q0tr−t0,r.Proof.
The proof of (179), (180), and (182)–(185) is similar to that of Theorem 4, so we only need to prove (181). Let Ψ1 be defined by the right-hand side of (181). We want to prove that ψ1=Ψ1 with ψ1 given by (138). Then, we compute by using (124) and (125) that
(186)iλ+u1x,trϕ2P,x,tr+ϕ3P,x,tr=iλ+u1y2V13n−yAm+BmEm−2−y2V12n−yAm+BmEm−2=1Em−2u1V13n−V12ny2−u1Am−Amy+13Em−2,x−iλEm−2+2u1V13n−V12nSm+iλ=13Em−2,xEm−2+23u1V13n−V12n3y2+SmEm−2−u1V13nyy+Am/V13n−V12nyy+Am/V12nEm−2+23iλ=λ→μjx,tr−μj,xx,trλ−μjx,tr+O1=λ→μjx,tr∂xlnλ−μjx,tr+O1,V~11rλ,x,tr+V~12rλ,x,trϕ2P,x,tr+V~13rλ,x,trϕ3P,x,tr=V~11r+V~12ry2V13n−yAm+BmEm−2−V~13ry2V12n−yAm+BmEm−2=1Em−2V~12rV13n−V~13rV12ny2−V~12rAm−V~13rAmy+13Em−2,tr−23V~13rV12n−V~12rV13nSm=13Em−2,trEm−2+23V~12rV13n−V~13rV12n3y2+SmEm−2−V~12rV13nyy+Am/V13n−V~13rV12nyy+Am/V12nEm−2=λ→μjx,trμj,,trx,trλ−μjx,tr+O1=λ→μjx,tr∂trlnλ−μjx,tr+O1.
Then,(187)ψ1P,x,x0,tr,t0,r=exp∫x0xiλ+u1x′,trϕ2P,x′,tr+ϕ3P,x′,trdx′+∫t0,rtrV~11rλ,x0,t′+V~12rλ,x0,t′ϕ2P,x0,t′+V~13rλ,x0,t′ϕ3P,x0,t′dt′=λ−μjx,trλ−μjx0,trλ−μjx0,trλ−μjx0,t0,rO1=λ−μjx,trλ−μjx0,t0,rO1=λ−μjx,trO1,forPnearμ^jx,tr≠μ^jx0,t0,r,O1,forPnearμ^jx,tr=μ^jx0,t0,r,λ−μjx0,t0,r−1O1,forPnearμ^jx0,t0,r≠μ^jx,tr,where O1≠0. Consequently, all zeros and poles of ψ1 and Ψ1 on Km−2\P∞1,P∞2,P∞3 are simple and coincident. Similar to Theorem 4, one can find that ψ1 and Ψ1 have the same essential singularities at P∞1,P∞2,P∞3. Then, the Riemann-Roch uniqueness results in Ψ1=ψ1.
We can explicitly rewriteθz¯P∞j,μ^¯x,tr,θz¯P∞j,ν^¯x,tr, and θz¯P∞j,ξ^¯x,tr as
(188)θz¯P∞j,μ¯x,tr=θM1j+U22x+U~r+12tr,θz¯P∞j,ν¯x,tr=θM2j+U22x+U~r+12tr,θz¯P∞j,ξ¯x,tr=θM3j+U22x+U~r+12tr,whereMsj=M−AP∞j+ρ¯sx0,t0,r−U22x0−U~r+12t0,r,j=1,2,3,s=1,2,3, with the aid of the following theorem.☐Theorem 10.
Assume that the curveKm−2 is nonsingular and let x,tr,x0,t0,r∈ℂ2. Then,
(189)ρ1x,tr=ρ1x0,t0,r+U22x−x0+U~r+12tr−t0,r,ρ2x,tr=ρ2x0,t0,r+U22x−x0+U~r+12tr−t0,r,ρ3x,tr=ρ3x0,t0,r+U22x−x0+U~r+12tr−t0,r.Proof.
First, we consider a meromorphic differential(190)Ωx,x0,tr,t0,r=∂∂λlnψ1P,x,x0,tr,t0,rdλ.
From the representation (180), we have
(191)Ωx,x0,tr,t0,r=−x−x0Ω2−tr−t0,rΩ~r+12−∑j=1m−2ωμ^x0,t0,r,μ^x,tr3+ω~,where ω~ denotes a holomorphic differential on Km−2, that is, ω~=∑j=1m−2ωjej for some ej∈ℂ,j=1,⋯,m−2. Because ψ1P,x,x0,tr,t0,r is single-valued on Km−2, all a- and b-periods of Ω are integer multiples of 2πi, and hence,
(192)2πimk=∫akΩx,x0,tr,t0,r=∫akω~=ek,k=1,⋯,m−2,for some mk∈ℤ. Similarly, for some nk∈ℤ,
(193)2πink=∫bkΩx,x0,tr,t0,r=−x−x0∫bkΩ2−tr−t0,r∫bkΩ~r+12+∑j=1m−2∫bkωμ^jx,tr,μ^jx0,t0,r3=∫bkω~−x−x0∫bkΩ2−tr−t0,r∫bkΩ~r+12+2πi∑j=1m−2∫μ^jx0,t0,rμ^jx,trωk=2πi∑j=1m−2mj∫bkωj−2πix−x0U2,k2−2πitr−t0,rU~r+1,k2+2πi∑j=1m−2mjτj,k+2πi∑j=1m−2∫Q0μ^jx,trωk−∑j=1m−2∫Q0μ^jx0,t0,rωk,in terms of
(194)∫bkωQ1,Q23=2πi∫Q1Q2ωk,k=1,⋯,m−1.
By symmetry ofτ, this is equivalent to x,tr∈Ωμ(195)ρ1x,tr=ρ1x0,t0,r+U22x−x0+U~r+12tr−t0,r.
Using this equality and the linear equivalenceDP∞1,μ^¯x,tr~DP∞2,ν^¯x,tr, and DP∞1,μ^¯x,tr~DP∞3,ξ^¯x,tr, that is,
(196)AP∞1+ρ1x,tr=AP∞2+ρ2x,tr,AP∞1+ρ1x,tr=AP∞3+ρ3x,tr,we can present the other two equalities
(197)ρ2x,tr=ρ2x0,t0,r+U22x−x0+U~r+12tr−t0,r,ρ3x,tr=ρ3x0,t0,r+U22x−x0+U~r+12tr−t0,r.


## 6. Conclusions

In this paper, firstly we obtain the CCIRD hierarchy related to a3×3 matrix spectral problem based on the Lenard recursion equations, and a trigonal curve Km−2 of arithmetic genus m−2 with three infinite points is introduced by using the characteristic polynomial of Lax matrix for the stationary CCIRD equations, from which the stationary Baker-Akhiezer function and associated meromorphic functions are given on Km−2. Then, the stationary CCIRD equations are decomposed into the system of Dubrovin-type ordinary differential equations. Furthermore, we present the explicit theta function representations of the stationary Baker-Akhiezer function, of the meromorphic functions, and, in particular, of the potentials for the entire stationary CCIRD hierarchy. Finally, we extend all the Baker-Akhiezer function, the meromorphic functions, the Dubrovin-type equations, and the theta function representations dealt with in Sections 3 and 4 to the time-dependent case. The technology presented in this paper can be applied to other hierarchies related to 3×3 matrix spectral problems, to get more various algebro-geometric solutions, which will enrich and supplement the known results.



---

*Source: 6618932-2021-07-22.xml*