# Journal of Nonlinear Mathematical Physics

Volume 26, Issue 1, December 2018, Pages 1 - 23

# A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Authors
Xianguo Geng, Yunyun Zhai*, Bo Xue, Jiao Wei
School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan 450001, People’s Republic of China,xggeng@zzu.edu.cn(Xianguo Geng);zhaiyy@zzu.edu.cn(Yunyun Zhai);xuebo@zzu.edu.cn(Bo Xue);weijiaozzu@163.com(Jiao Wei)
*Corresponding authors.
Corresponding Author
Yunyun Zhai
Received 24 January 2018, Accepted 24 May 2018, Available Online 6 January 2021.
DOI
10.1080/14029251.2019.1544785How to use a DOI?
Keywords
long wave-short wave type equations; Baker-Akhiezer function; meromorphic function; quasi-periodic solutions
Abstract

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties of the meromorphic functions, including asymptotic expansions at infinite points, we obtain the essential singularities and divisor of the Baker-Akhiezer function. Utilizing the theory of algebraic curves, quasi-periodic solutions for the entire hierarchy are finally derived in terms of the Riemann theta function.

Open Access

## 1. Introduction

Soliton equations have caught a great deal of attention for describing and explaining nonlinear phenomena in theoretical physics, fluid mechanics, nonlinear optics, plasma physics and other subjects  . Besides some famous equations such as the Korteweg-de Vries, sine-Gordon, Boussinesq and Kadomtsev Petviashvili equations [1, 31, 47, 50], a lot of new models possessing great significance have been proposed, for example, the Camassa-Holm, Degasperis-Procesi, Novikov, and Geng-Xue equations etc [6, 10, 22, 23, 30, 3537]. As the research moves along, more and more systematic methods have been developed to solve these soliton systems [1, 4, 14, 17, 29, 33, 44]. For examples, the inverse scattering transformation [1, 17], the bilinear transformation methods of Hirota  , the Bäcklund and Darboux transformations  , algebro-geometric method [4, 14, 32, 33] and others [19, 42].

Since 1970s, various methods in a series of papers [3 , 4, 7, 9, 14, 18, 24, 26, 3234, 41, 51] were developed on the basis of the theory of hyperelliptic curves to obtain quasi-periodic solutions of soliton equations associated with 2 × 2 matrix spectral problems such as the KdV, KP, nonlinear Schrödinger, Camassa-Holm, Toda lattice, Ablowitz-Ladik equations and so on. However, it is the trigonal curve [5, 11, 15, 49] rather than the hyperelliptic curve that is the theoretic foundation to obtain the quasi-periodic solutions to soliton equations related to the third order spectral problems. Although the reduction theory of Riemann theta functions has been applied to study quasi-periodic solutions of the Boussinesq equation in a few of literature [2, 43, 45, 46, 5255], the method is not a general scheme to construct quasi-periodic solutions of completely integrable systems. In 1999, Dickson and his partners proposed a unified framework which yields all algebro-geometric quasi-periodic solutions of the entire Boussinesq hierarchy [12, 13]. Shortly after that, this method was generalized to deal with the modified Boussinesq and the Kaup-Kupershmidt hierarchies [20, 21] based on the trigonal curve introduced by the characteristic polynomial of the Lax matrix. The trigonal curves in [12,13,20,21] have one collective feature, that is, one infinite point which is a branch point with the triple root. Recently, the research has developed to get the quasi-periodic solutions to the coupled modified Korteweg-de Vries hierarchy, the three wave resonant interaction hierarchy and the four-component AKNS soliton hierarchy associated with the trigonal curves having three different infinite points which are not branch points [25, 28, 39, 40].

In this paper, we first derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. The first nontrivial member in the hierarchy is the long wave-short wave type system

ut=2(vw)x,vt=vxxuvx,wt=wxxuwx.(1.1)

Equation (1.1) reduces to

ut=2(|v|2)x,vt=ivxxuvx,(1.2)
if x → −ix, t → −it, w = v*, which is different from the standard long wave-short wave resonance system [8, 38, 56]. Another principal subject of the present paper is to construct quasi-periodic solutions for the long wave-short wave type hierarchy on the basis of the theory of algebraic curves. To this end, one introduces the trigonal curve with the aid of the characteristic polynomial of the Lax matrix. A distinguishing feature for the trigonal curve associated with the hierarchy is that it has two infinite points, one of which is a double branch point and the other is not a branch point. Compared with references [12, 13 , 20, 21, 25, 28, 39, 40], the trigonal curve in this paper is more general. Therefore, we need to reinvestigate the local coordinates near infinite points, the Abelian differentials and other basic properties.

The outline of this paper is as follows. In section 2, we consider a 3 × 3 matrix spectral problem with three potentials and derive a hierarchy of nonlinear evolution equations with the aid of three sets of Lenard recursion equations and the stationary zero-curvature equation. In section 3, a trigonal curve is introduced by using the characteristic polynomial of the Lax matrix, on which the Baker-Akhiezer function together with two related meromorphic functions is given. Based on the asymptotic properties of the two meromorphic functions near the infinite points, the essential singularities of Baker-Akhiezer function are derived. Accordingly, the Abelian differentials of the second kind are given for the purpose of representation. In section 4, we investigate some properties of the two meromorphic functions and obtain the Dubrovin-type equations. Subsequently, we derive divisors of meromorphic functions and the Baker-Akhiezer function which are necessary for the construction of the quasi-periodic solutions. Section 5 finally constructs the Riemann theta function representations for the long wave-short wave type hierarchy according to the asymptotic properties and the quasi-periodic characters of the meromorphic function and the Baker-Akhiezer function.

## 2. A long wave-short wave type hierarchy

In this section, we shall derive a hierarchy of long wave-short wave type models. To this end, we consider a 3 × 3 matrix spectral problem with three potentials

ψx=Uψ,ψ=(ψ1ψ2ψ3),U=(uλv100w00),(2.1)
where u, v, w are three potentials, and λ a constant spectral parameter. We first introduce three sets of Lenard recursion equations:
Ksj=Jsj+1,j0,sj|(u,v,w)=0=0,j1,Ks^j=Js^j+1,j0,s^j|(u,v,w)=0=0,j1,Ks˜j=Js˜j+1,j0,s˜j|(u,v,w)=0=0,j1,(2.2)
where the two operators are defined as
K=(vwu2wv+v0uvvvv2uv2v0w+w02uvww2u2wv0vw0wv),(2.3)
J=(20000v0100w00102u2wv0vw0wv),(2.4)
and the starting points s0 = (0, 0, 0, 0, 1)T, ŝ0 = (0, 1, 0, 0, 0)T, s˜0=(1,0,v,w,vw)T. Then the sequences sj, ŝj, s˜j can be determined uniquely. For example, the first three members read as
s1=(00vwvw),s^1=(12u14ux18u212vwvx12uvwx12uwvxwvwx12uvw),s˜1=(vwvxwvxxuvxv2wwxx+uwxvw2s˜1(5)),(2.5)
where
s˜1(5)=vxxwvxwx+vwxxuvxw+uvwxv2w2.

In order to generate a hierarchy of nonlinear evolution equations associated with the spectral problem (2.1), we define a 3 × 3 matrix

V=(Vij)3×3=(V11λV12V13V21λV22V23V31λV32V33)(2.6)
with the elements
V11=(u)c+ve+λb,V12=c,V13=vbx+vc+dx,V21=cbx,V22=b,V23=d,V31=wcex,V32=e,V33=f.

Then the stationary zero-curvature equation

Vx[U,V]=0(2.7)
is equivalent to
vwb+(u2)c+wd+(v+v)e+2λb=0,(uvv)b+(vv)c+(2u)d+v2evf+λ(vbd)=0,(w+w)c(2+u+vw)e+wf+λ(wb+e)=0,(2+ub)b+2c+wdve=0,vwbwdve+f=0.(2.8)

Expanding b, c, d, e, f into the Laurent polynomials in λ:

(b,c,d,e,f)=j0(bj,cj,dj,ej,fj)λj,(2.9)
equation (2.8) is equivalent to the following recursion equations
KSj=JSj+1,j0,JS0=0,(2.10)
where Sj = (bj, cj, dj, ej, fj)T. Since equation JS0 = 0 has a general solution
S0=α0s0+β0s^0+δ0s˜0,(2.11)
then functions Sj given by
Sj=l=0j(αlsjl+βls^jl+δls˜jl),(2.12)
satisfy the recursion equation (2.10), where αj, βj, δj are arbitrary constants.

Let ψ satisfy the spectral problem (2.1) and the auxiliary problem

ψtr=V˜(r)ψ,V˜(r)=(V˜ij(r))3×3,(2.13)
where V˜ij(r)=Vij(b˜(r),c˜(r),d˜(r),e˜(r),f˜(r)),
(b˜(r),c˜(r),d˜(r),e˜(r),f˜(r))=j0r(b˜j,c˜j,d˜j,e˜j,f˜j)λrj,(2.14)
with S˜j=(b˜j,c˜j,d˜j,e˜j,f˜j)T determined by
S˜j=l=0j(α˜lsjl+β˜ls^jl+δ˜ls˜jl).(2.15)

The constants α˜j, β˜j, δ˜j here are independent of the choice of αj, βj, δj. Then the compatibility condition of (2.1) and (2.13) yields the zero-curvature equation, UtrV˜x(r)+[U,V˜(r)]=0, which is equivalent to a hierarchy of nonlinear evolution equations

(utr,vtr,wtr)T=Xr,r0,(2.16)
where the vector fields Xr=P(KS˜r)=P(JS˜r+1), P is the projective map P(γ1, γ2, γ3, γ4, γ5)T → (γ1, γ2, γ3)T. The first member in the hierarchy for r = 0 is a long wave-short wave type system
ut0=β˜0ux+2δ˜0(vw)x,vt0=α˜0v+β˜0vx+δ˜0(vxxuvx),wt0=α˜0w+β˜0wx+δ˜0(wxxuwx).(2.17)

As α˜0=β˜0=0, δ˜0=1, t0 = t, equation (2.17) turns into (1.1). For r = 1, the second member in the hierarchy (2.16) reads as

ut1=2α˜0(vw)x+β˜0(14uxxx+32vxxw32vwxx32(uvw)x38u2ux)+2δ˜0(vxxw+vwxx+uvwxuvxwv2w2)x+β˜1ux+2δ˜1(vw)x,vt1=α˜0(vxx+uvx)+β˜0(vxxx34uxvx32uvxx32vvxw+38u2vx)+δ˜0(vxxxxuxxvx2uxvxx2uvxxx2vx2w2vvxxw+uuxvx+u2vxx+2uvvxw)α˜1v+β˜1vx+δ˜1(vxxuvx),wt1=α˜0(wxx+uwx)+β˜0(wxxx32vwwx+32uwxx+34uxwx+38u2wx)+δ˜0(wxxxxuxxwx2uxwxx2uwxxxuuxwxu2wxx+2vwx2+2vwwxx+2uvwwx)+α˜1w+β˜1wx+δ˜1(wxxuwx).(2.18)

If choosing β˜0=1, α˜0=α˜1=β˜1=δ˜0=δ˜1=0 or δ˜0=1, α˜0=β˜0=α˜1=β˜1=δ˜1=0, then (2.18) is respectively reduced to a new coupled mKdV equation

ut1=14uxxx38u2ux+32vxxw32vwxx32(uvw)x,vt1=vxxx34uxvx32uvxx32vvxw+38u2vx,wt1=wxxx+34uxwx+32uwxx32vwwx+38u2wx,(2.19)
or
ut1=2(vxxw+vwxx+uvwxuvxwv2w2)x,vt1=vxxxxuxxvx2uxvxx2uvxxx2vx2w2vvxxw+uuxvx+u2vxx+2uvvxw,wt1=wxxxxuxxwx2uxwxx2uwxxxuuxwxu2wxx+2vwx2+2vwwxx+2uvwwx.(2.20)

## 3. Meromorphic functions and Baker-Akhiezer function

In this section, we shall define a trigonal curve 𝒦m−1, the vector Baker-Akhiezer function and two meromorphic functions on 𝒦m−1. The Abelian differentials of the second kind are introduced on the basis of the analysis for Baker-Akhiezer function ψ2 at infinite points.

With the help of the n-th stationary flow, we introduce a Lax matrix V(n)=(Vij(n))3×3=((λnVij)+)3×3, which satisfies the Lax equation

Vx(n)[U,V(n)]=0,(3.1)
Vtr(n)[V˜(r),V(n)]=0.(3.2)

Then the characteristic polynomial m(λ, y) = det(yIV(n)) of the Lax matrix V(n) is a polynomial of λ independent of variables x and tr, which can be expressed in the following form

det(yIV(n))=y3y2Rm(λ)+ySm(λ)Tm(λ),(3.3)
where Rm(λ), Sm(λ) and Tm(λ) are polynomials with constant coefficients of λ
Rm(λ)=V11(n)+λV22(n)+V33(n)=2δ0λn+1+(2δ1+α0)λn+,Sm(λ)=|V11(n)λV12(n)V21(n)λV22(n)|+|V11(n)V13(n)V31(n)V33(n)|+|λV22(n)V23(n)λV32(n)V33(n)|=δ02λ2n+2+(2δ0δ1+2α0δ0β02)λ2n+1+,Tm(λ)=|V11(n)λV12(n)V13(n)V21(n)λV22(n)V23(n)V31(n)λV32(n)V33(n)|=λ[α0δ02λ3n+1+(2α0δ0δ1+α1δ02α0β02)λ3n+].(3.4)

This naturally leads to a trigonal curve 𝒦m−1 of degree m = 3n + 2 with respect to λ by

𝒦m1:m(λ,y)=y3y2Rm(λ)+ySm(λ)Tm(λ)=0.(3.5)

According to (3.4) and (3.5), the trigonal curve 𝒦m−1 can be compactified by adding two infinite points P1 and P2 for which we take P1 as a double point without loss of generality. 𝒦m−1 is nonsingular or smooth means that for every point Q0 = (λ0, y0) ∈ 𝒦m−1 \ {P1, P2}, (mλ,my)|(λ,y)=(λ0,y0)0. Adding the assumption of irreducibility, the trigonal curve 𝒦m−1 becomes connected. For the sake of convenience, we use the same symbol 𝒦m−1 in the following text to denote the three sheeted nonsingular compact Riemann surface. Obviously, the discriminant of (3.5) is Δ(λ)=Rm2Sm2+4Rm3Tm+4Sm318RmSmTm+27Tm2=4β02δ04λ6n+5+, which has at most 6n + 5 zeros. Therefore, the Riemann-Hurwitz formula shows that the arithmetic genus of 𝒦m−1 is 3n + 1 for β0δ0 ≠ 0 [16, 27, 48].

Equip the Riemann surface 𝒦m−1 with homology basis {𝕒j,𝕓j}j=1m1, which are independent and have intersection numbers as follows

𝕒j𝕓k=δjk,𝕒j𝕒k=0,𝕓j𝕓k=0,j,k=1,,m1.

For the present, we will choose as our basis the following set

ϖl(P)=13y2(P)2Rm(λ)y(P)+Sm(λ)[λl1dλ,1l2n+1,y(P)λl2n2dλ,2n+2l3n+1,(3.6)
which are 3n + 1 linearly independent holomorphic differentials on 𝒦m−1. By using the homology basis {𝕒j}j=1m1 and {𝕓j}j=1m1, the period matrices A = (Ajk) and B = (Bjk) can be constructed from
Ajk=𝕒kϖj,Bjk=𝕓kϖj.(3.7)

It is possible to show that the matrices A and B are invertible [16, 27, 48]. Now we define the matrices C and τ by C = A−1, τ = A−1B. The matrix τ can be shown to be symmetric (τjk = τkj) and has a positive-definite imaginary part (Imτ > 0). If we normalized ϖl(P) into new basis ωj=l=1m1Cjlϖl,,

𝕒kωj=δjk(3.8)
then we have 𝕓kωj=τjk, j, k = 1,...,m − 1.

The complex structure on 𝒦m−1 is defined in the usual way by introducing local coordinates ζQ0 : P → (λλ0) near points Q0 = (λ0, y(Q0)) ∈ 𝒦m−1 which are not branch points nor infinite points of 𝒦m−1, ζPj: Pλ−1/(3−j) near the points Pj𝒦m−1, j = 1, 2, and similar at others branch points of 𝒦m−1.

Given these preliminaries, let ψ(P, x, x0, tr, t0,r) denote the vector Baker-Akhiezer function by

ψx(P,x,x0,tr,t0,r)=U(u(x,tr),v(x,tr),w(x,tr);λ(P))ψ(P,x,x0,tr,t0,r),ψtr(P,x,x0,tr,t0,r)=V˜(r)(u(x,tr),v(x,tr),w(x,tr);λ(P))ψ(P,x,x0,tr,t0,r),V(n)(u(x,tr),v(x,tr),w(x,tr);λ(P))ψ(P,x,x0,tr,t0,r)=y(P)ψ(P,x,x0,tr,t0,r),ψ2(P,x,x0,t0,r,t0,r)=1,x,tr,P𝒦m1\{P1,P2}.(3.9)

Define two meromorphic functions ϕ2(P, x, tr) and ϕ3(P, x, tr) on 𝒦m−1 closely related to the Baker-Akhiezer function by

φ2(P,x,tr)=ψ1(P,x,x0,tr,t0,r)ψ2(P,x,x0,tr,t0,r),P𝒦m1,x,tr,(3.10)
φ3(P,x,tr)=ψ1(P,x,x0,tr,t0,r)ψ2(P,x,x0,tr,t0,r),P𝒦m1,x,tr.(3.11)

### Lemma 3.1.

Assume that (3.9), (3.10), (3.11) hold and let P = (λ, y(P)) ∈ 𝒦m−1 \ {P1, P2} and (λ, x, tr) ∈ ℂ3. Then meromorphic functions ϕ2(P, x, tr) and ϕ3(P, x, tr) have the following asymptotic expansions near Pj𝒦m−1, j = 1, 2, under the local coordinate ζ = λ−1/(3−j)

φ2(P,x,tr)=ζ0{ζ1+u2+18(2ux+u2+4vw)ζ+18(uux2vxw6vwx+uxx)ζ2+O(ζ3),asPP1,ζ=λ1/2,vxv+1v2[vvxxxvxvxx(uvx)xv+uvx2vxv2w]ζ+O(ζ2),asPP2,ζ=λ1,(3.12)
φ3(P,x,tr)=ζ0{1wζ1+O(1),asPP1,ζ=λ1/2,vxζ+O(ζ2),asPP2,ζ=λ1.(3.13)

### Proof.

Expressions (3.9) and (3.10) imply that meromorphic functions ϕ2(P, x, tr) satisfies the Riccati-type equation

φ2,xx+3φ2φ2,x+φ23(vxv+u)(φ2,x+φ22)=(uxuvxv+vw)φ2+λ(φ2vxv).(3.14)

We can insert the two following ansatzs into the above equation (3.14)

φ2(P,x,tr)=ζ0{κ1,1ζ1+κ1,0+κ1,1ζ+κ1,2ζ2+O(ζ3),asPP1,ζ=λ1/2,κ2,0+κ2,1ζ+O(ζ2),asPP2,ζ=λ1.(3.15)

A comparison of the same powers of ζ then proves the first expression (3.12) in this lemma. The first expression in (3.9) implies the relationship between ϕ2 and ϕ3

φ3=vφ2φ2,x+φ22uφ2λ.(3.16)

Utilizing the expansions of ϕ2 in (3.12), we can easily derive (3.13).

Taking advantage of (3.9), (3.12) and (3.13), we can calculate out the asymptotic behaviors of y(P) near P1, P2 as

y(P)=ζ0{ζ2n2(δ0+β0ζ+δ1ζ2+β1ζ3+O(ζ4)),asPP1,ζ=λ1/2,ζn(α0+α1ζ+O(ζ2)),asPP2,ζ=λ1.(3.17)

Subsequently, one infers that

ωj=ζ0{(Cj,2n+1δ0Cj,3n+1β0δ0+O(ζ))dζ,asPP1,ζ=λ1/2,(Cj,2n+1δ02+O(ζ))dζ,asPP2,ζ=λ1.(3.18)

Furthermore, we could write ωj in the following form:

ωk=l=0ρk,l(Pj)ζldζ,asPPj,ζ=λ1/(3j),(3.19)
where ρk,l(Pj) are constants, j = 1, 2; k = 1, 2,...,3n + 1.

From the first two expression of (3.9), we arrive at the formula of ψ2(P, x, x0, tr, t0,r) as follows

ψ2(P,x,x0,tr,t0,r)=exp(x0x[φ2(P,x,tr)]dx+t0,rtr[V˜21(r)(λ,x0,t)φ2(P,x0,t)+λV˜22(r)(λ,x0,t)+V˜23(r)(λ,x0,t)v(x0,t)(φ2,x(P,x0,t)+φ22(P,x0,t)u(x0,t)φ2(P,x0,t)λ)]dt),(3.20)
from which we can deduce the essential singularity of ψ2(P, x, x0, tr, t0,r) near Pj, j = 1, 2. For the sake of convenience, we define a function
Ir(P,x,tr)=V˜21(r)(λ,x,tr)φ2(P,x,tr)+λV˜22(r)(λ,x,tr)+V˜23(r)(λ,x,tr)v(x,tr)(φ2,r(P,x,tr)+φ22(P,x,tr)u(x,tr)φ2(P,x,tr)λ)(3.21)
whose three homogeneous cases is denoted by
I¯r(ε)(P,x,tr)=V˜21(r,ε)(λ,x,tr)φ2(P,x,tr)+λV˜¯22(r,ε)(λ,x,tr)+V˜23(r,ε)(λ,x,tr)v(x,tr)(φ2,x(P,x,tr)+φ22(P,x,tr)u(x,tr)φ2(P,x,tr)λ),ε=1,2,3,(3.22)
where
V˜¯ij(r,1)=V˜ij(r)|α˜0=1,α˜1==α˜r=β˜0==β˜r=δ˜0==δ˜r=0,V˜¯ij(r,2)=V˜ij(r)|β˜0=1,α˜1==α˜r=β˜1==β˜r=δ˜0==δ˜r=0,V˜¯ij(r,3)=V˜ij(r)|δ˜0=1,α˜0==α˜r=β˜0==β˜r=δ˜1==δ˜r=0.(3.23)

Homogeneous polynomials b˜¯(r,ε), c˜¯(r,ε), d˜¯(r,ε), e˜¯(r,ε), f˜¯(r,ε) and b˜¯j(ε), c˜¯j(ε), d˜¯j(ε), e˜¯j(ε), f˜¯j(ε) also have the similar stipulation.

### Lemma 3.2.

Suppose that u(x, tr), v(x, tr) and w(x, tr) satisfy the r-th nonlinear evolution equations (2.16). Moreover, let P𝒦m−1 \ {P1, P2}, (x, x0, tr, t0,r) ∈ ℂ4. Then

ψ2(P,x,x0,tr,t0,r)=ζ0{exp(ζ1(xx0)+l=0r(δ˜rl+β˜rlζ)ζ2l2(trt0,r)+121u(x,tr)121u(x0,t0,r)+O(ζ)),asPP1,ζ=λ1/2,v(x,tr)v(x0,t0,r)exp(l=0rα˜rlζl(trt0,r)+O(ζ)),asPP2,ζ=λ1.(3.24)

### Proof.

To investigate the property of ψ2(P, x, x0, tr, t0,r) near P1, one shall take the local coordinate as λ = ζ−2. We use the inductive method to prove the subsequent expression

I¯r(1)(P,x,tr)=1[vwb˜¯r,x(1)+(u2)c˜¯r(1)+wd˜¯r,x(1)+(v+v)e˜¯r(1)]2+O(ζ),I¯r(2)(P,x,tr)=ζ2r1+1[vwb˜¯r,x(2)+(u2)c˜¯r(2)+wd˜¯r,x(2)+(v+v)e˜¯r(2)]2+O(ζ),I¯r(3)(P,x,tr)=ζ2r2+1[vwb˜¯r,x(3)+(u2)c˜¯r(3)+wd˜¯r,x(3)+(v+v)e˜¯r(3)]2+O(ζ).(3.25)

In fact, for r = 1, a direct calculation shows that

I¯1(1)(P,x,tr)=1[vwb˜¯1,x(1)+(u2)c˜¯1(1)+wd˜¯1,x(1)+(v+v)e˜¯1(1)]2+O(ζ).(3.26)

Suppose that I¯r(1)(P,x,tr) has the following expansion

I¯r(1)(P,x,tr)=ζ0j=0σj(1)(x,tr)ζj,PP1,(3.27)
for some coefficients {σj(1)(x,tr)}j0 to be determined. Observing
φ2,tr=[V˜¯21(r,1)φ2+λV˜¯22(r,1)+V˜¯23(r,1)v(φ2,x+φ22uφ2λ)]x,(3.28)
we arrive at
σj,x(1)=κ1,j,tr,j=0,1,2,3,(3.29)

Taking use of (2.2), (2.14) and lemma 3.1, we get three expressions

σ0(1)=1κ1,0,tr=b˜¯r+1(1)σ1(1)=1κ1,1,tr=b˜¯r+1,x(1)c˜¯r+1(1),σ2(1)=1κ1,2,tr=14b˜¯r+1,xx(1)+14ub˜¯r+1,x(1)+14ve˜¯r+1(1)34wd˜¯r+1(1)+12f˜r+1(1),(3.30)
where the integration constants are taken as zero because there is no arbitrary constants in the expansions of ϕ2(P, x, tr) near P1 nor in the coefficients of the homogeneous polynomials V˜¯i,j(r,1) with the condition ∂∂−1 = −1 = 1. It is easy to see that
I¯r+1(1)(P,x,tr)=ζ2I¯r(1)+(c˜¯r+1(1)b˜¯r+1,x(1))φ2+b˜¯r+1(1)ζ2+d˜¯r+1(1)v(φ2,x+φ22uφ2ζ2)=1[vwb˜¯r+1,x(1)+(u2)c˜¯r+1(1)+wd˜¯r+1,x(1)+(v+v)e˜¯r+1(1)]2+O(ζ).

Thus I¯r(1)(P,x,tr) is proved to have the expansion as seen in (3.25) near P1. Similarly, one can prove the other two expressions in (3.25), which yield the expansion of Ir(P, x, tr) near P1 as follows

Ir(P,x,tr)=l=0r(δ˜rl+β˜rlζ)ζ2l2+1[vwb˜r,x+(u2)c˜r+wd˜r,x+(v+v)e˜r]2+O(ζ).(3.31)

Substituting (3.12) and (3.31) into (3.20), we arrive at the first expression in (3.24) right now. Under the local coordinate ζ = λ−1 near P2, we can similarly prove the second expression in lemma 3.2.

Let ωPs,j(2)(P), j ≥ 2, s = 1, 2, denote the normalized Abelian differential of the second kind holomorphic on 𝒦m−1 \ {Ps} satisfying

𝕒kωPs,j(2)(P)=0,k=1,,3n+1,(3.32)
ωPs,j(2)(P)=ζ0(ζj+O(1))dζ,asPP,ζ=λ1/(s3).(3.33)

According to the asymptotic behaviors of ψ2(P, x, x0, tr, t0,r) in (3.24), we introduce the corresponding Abelian differential of the second kind

Ω2(2)(P)=ωP1,2(2)(P)(3.34)
and
Ω˜2r+3(2)(P)=l=0r(2l+2)δ˜rlωP1,2l+3(2)(P)+l=0r(2l+1)β˜rlωP1,2l+2(2)(P)+l=1rlα˜rlωP2,l+1(2)(P).(3.35)

From (3.33), (3.34) and (3.35), we conclude that

Q0PΩ2(2)(P)={ζ1+e1(2)(Q0)+O(ζ),asPP1,ζ=λ1/2,e2(2)(Q0)+O(ζ),asPP2,ζ=λ1,(3.36)
Q0PΩ˜2r+3(2)(P)={l=0rδ˜rlζ2l2l=0rβ˜rlζ2l2+e˜1(2)(Q0)+O(ζ),asPP1,ζ=λ1/2,l=1rα˜rlζ1+e˜1(2)(Q0)+O(ζ),asPP2,ζ=λ1,(3.37)
where e1(2)(Q0), e2(2)(Q0), e˜1(2)(Q0) and e˜2(2)(Q0) are integration constants with Q0 an appropriately chosen base point on 𝒦m−1 \ {P1, P2}. The 𝕓-periods of the differential Ω2(2)(P) and Ω˜2r+3(2)(P) are denoted by
U_2(2)=(U2,1(2),,U2,m1(2)),U2,k(2)=12πi𝕓kΩ2(2)(P),k=1,,m1,(3.38)
U_˜2r+3(2)=(U˜2r+3,1(2),,U˜2r+3,m1(2)),U˜2r+3,j(2)=12πi𝕓kΩ˜2r+3(2)(P),j=1,,m1.(3.39)

By the relationship between the normalized Abelian differential of the second kind and the normalized holomorphic differential ω_, we can derive that

U2,k(2)=ρk,0(P1),k=1,2,,m1,(3.40)
U˜2r+3,k(2)=l=0rδ˜rlρk,2l+1(P1)+l=0rβ˜rlρk,2l(P1)+l=1rα˜rlρk,l1(P2),k=1,2,,m1.(3.41)

## 4. Divisors of meromorphic functions and Baker-Akhiezer function

In this section, we shall investigate the properties of meromorphic functions and Baker-Akhiezer function on the finite part of the Riemann surface 𝒦m−1 including the divisors, which are necessary for construction of the Riemann theta function representations.

For convenience, we define three points P, P*, P** on three different sheets of the same Riemann surface 𝒦m−2. For a fixed λ, let yi(λ), i = 0, 1, 2, denote the three roots of polynomial m(λ, y) = 0, that is

(yy0(λ))(yy1(λ))(yy2(λ))=y3y2Rm+ySmTm=0.(4.1)

Then points (λ, y0(λ)), (λ, y1(λ)) and (λ, y2(λ)) are on the three different sheets of Riemann surface 𝒦m−1, respectively. Let P = (λ, yi(λ)), i = 0, 1, 2, be an arbitrary point in the three points, then the other two points are defined as P* and P**, respectively. From (4.1), we can derive the relationships between the roots yi(λ), i = 0, 1, 2 and the coefficients Rm, Sm, Tm

y0+y1+y2=Rm,y0y1+y0y2+y1y2=Sm,y0y1y2=Tm,y02+y12+y22=Rm22Sm,y03+y13+y23=Rm33RmSm+3Tm,y02y12+y02y22+y12y22=Sm22RmTm,j=02(3yj22yjRm+Sm)=Δ(λ).(4.2)

Using (3.9), (3.10) and (3.11), a direct calculation shows that

φ2=yV13(n)+CmyV23(n)+Am=λFm1y2V13(n)y(RmV13(n)+Cm)+Dm=y2V23(n)y(RmV23(n)+Am)+BmEm1,(4.3)
φ3=yV12(n)+𝒞myV32(n)+𝒜m=Fm1y2V12(n)y(RmV12(n)+𝒞m)+𝒟m=y2V32(n)y(RmV32(n)+𝒜m)+mm1,(4.4)
where
Am=V13(n)V21(n)V11(n)V23(n),Bm=λ(V22(n)V23(n)V33(n)(V23(n))2V32(n)+V13(n)V21(n)V22(n)V12(n)V21(n)V23(n)),Cm=λ(V12(n)V23(n)V13(n)V22(n)),Dm=V11(n)V13(n)V33(n)(V13(n))2V31(n)+λV11(n)V12(n)V23(n)λV12(n)V13(n)V21(n),(4.5)
𝒜m=V12(n)V31(n)V11(n)V32(n),m=λV22(n)V32(n)V33(n)λV23(n)(V32(n))2V12(n)+V31(n)V33(n)V13(n)V31(n)V32(n),𝒞m=V13(n)V32(n)V12(n)V3(n),𝒟m=λV11(n)V12(n)V22(n)λ(V12(n))2V21(n)+λV11(n)V13(n)V32(n)V12(n)V13(n)V31(n),(4.6)
Em1=V21(n)(V11(n)V23(n)V13(n)V21(n))+V23(n)(V23(n)V31(n)V21(n)V33(n)),Fm1=V13(n)(V13(n)V32(n)V12(n)V33(n))+λV12(n)(V13(n)V22(n)V12(n)V23(n)),m1=V31(n)(V11(n)V32(n)V12(n)V31(n))+λV31(n)(V21(n)V32(n)V22(n)V31(n)).(4.7)

It can be inferred from (4.7) that Em−1, Fm−1 and m−1 are polynomials with respect to λ of degree 3n + 1 for β0δ0 ≠ 0. Therefore, we can rewrite them in the following form:

Em1(λ,x,tr)=β0δ02vj=13n+1(λμj(x,tr)),(4.8)
Fm1(λ,x,tr)=β0δ02vj=13n+1(λνj(x,tr)),(4.9)
m1(λ,x,tr)=β0δ02w2j=13n+1(λξj(x,tr)),(4.10)
where {μj(x,tr)}j=13n+1, {νj(x,tr)}j=13n+1, {ξj(x,tr)}j=13n+1 are zeros of Em−1(λ, x, tr), Fm−1(λ, x, tr), m−1(λ, x, tr) respectively. Define P0 = (0, 0). Since
𝒞m(νj(x,tr),x,tr)V12(n)(νj(x,tr),x,tr)=Cm(νj(x,tr),x,tr)V13(n)(νj(x,tr),x,tr)
which can be deduced from
Fm1|λ=νj(x,tr)=[V13(n)(V13(n)V32(n)V12(n)V33(n))λV12(n)(V12(n)V23(n)V13(n)V22(n))]|λ=νj(x,tr)=[V13(n)(νj(x,tr),x,tr)𝒞m(νj(x,tr),x,tr)V12(n)(νj(x,tr),x,tr)Cm(νj(x,tr),x,tr)]=0,
we can define
μ^j(x,tr)=(μj(x,tr),y(μj(x,tr))=(μj(x,tr),Am(μj(x,tr),x,tr)V23(n)(μj(x,tr),x,tr))𝒦m1,(4.11)
ν^j(x,tr)=(νj(x,tr),y(νj(x,tr))=(νj(x,tr),Cm(μj(x,tr),x,tr)V13(n)(μj(x,tr),x,tr)),=(νj(x,tr),𝒞m(νj(x,tr),x,tr)V12(n)(νj(x,tr),x,tr))𝒦m1,(4.12)
ξ^j(x,tr)=(ξj(x,tr),y(ξj(x,tr)))=(ξj(x,tr),𝒜m(ξj(x,tr),x,tr)V32(n)(ξj(x,tr),x,tr))𝒦m1(4.13)
with 1 ≤ j ≤ 3n + 1, (x, tr) ∈ ℂ2.

Observing (4.3), (4.4) and lemma 3.1, we obtain the divisors (ϕ2(P, x, tr)) and (ϕ3(P, x, tr)) of ϕ2(P, x, tr) and ϕ3(P, x, tr) as follows

(φ2(P,x,tr))=𝒟P0,ν^1(x,tr),,ν^3n+1(x,tr)(P)𝒟P1,μ^1(x,tr),,μ^3n+1(x,tr)(P),(4.14)
(φ3(P,x,tr))=𝒟P2,ν^1(x,tr),,ν^3n+1(x,tr)(P)𝒟P1,ξ^1(x,tr),,ξ^3n+1(x,tr)(P).(4.15)

Now we are in a position to discuss zeros and poles of ψ2(P, x, x0, tr, t0,r) on 𝒦m−1 \ {P1, P2}. From (4.3) and (4.4), we can easily obtain the interrelationships among the polynomials Am, Bm, Cm, Dm, 𝒜m, m, 𝒞m, 𝒟m, Em−1, Fm−1, m−1, Rm, Sm, Tm, which we list below:

λV23(n)Fm1=V13(n)Dm(V13(n))2SmV13(n)RmCmCm2,λAmFm1=(V13(n))2Tm+CmDm,(4.16)
V13(n)Em1=V23(n)Bm(V23(n))2SmV23(n)RmAmAm2,CmEm1=(V23(n))2Tm+AmBm,(4.17)
0=V13(n)Bm+V23(n)DmV13(n)V23(n)Sm+AmCm,0=V13(n)V23(n)RmSm+V13(n)V23(n)Tm+V13(n)AmSm+V23(n)CmSmV13(n)RmBmV23(n)RmDmBmCmAmDm,0=V13(n)V23(n)RmTm+V13(n)AmTm+V23(n)CmTm+λEm1Fm1BmDm,(4.18)
V32(n)Fm1=V12(n)𝒟m(V12(n))2SmV12(n)Rm𝒞m𝒞m2,𝒜mFm1=(V12(n))2Tm+𝒞m𝒟m,(4.19)
V12(n)m1=V32(n)m(V32(n))2SmV32(n)Rm𝒜m𝒜m2,𝒞mm1=(V32(n))2Tm+𝒜mm,(4.20)
0=V12(n)m+V32(n)𝒟mV12(n)V32(n)Sm+𝒜m𝒞m,0=V12(n)V32(n)RmSm+V12(n)V32(n)Tm+V12(n)𝒜mSm+V32(n)𝒞mSmV12(n)RmmV32(n)Rm𝒟mm𝒞m𝒜m𝒟m,0=V12(n)V32(n)RmTm+V12(n)𝒜mTm+V32(n)𝒞mTmm1Fm1m𝒟m.(4.21)

Taking use of above relationships, we arrive at the evolution of Em−1(λ, x, tr), Fm−1(λ, x, tr), m−1(λ, x, tr) with respect to x and tr respectively in the subsequent lemma.

### Lemma 4.1.

Assume that (3.9) holds and let (λ, x, tr) ∈ ℂ3. Then

Em1,x=uEm1(RmAm+2V23(n)Sm3Bm),Fm1,x=2uFm1+(RmCm2V13(n)Sm+3Dm)+v(Rm𝒞m+2V12(n)Sm3𝒟m),m1,x=um1+w(Rm𝒜m2V32(n)Sm+3m).(4.22)
Em1,tr(λ,x,tr)=Em1[1utr+3λV˜22(r)+V˜23(r)(Rm3λV22(n))V23(n)]+V˜23(r)V21(n)V˜21(r)V23(n)V23(n)(RmAm+2V23(n)Sm3Bm)=Em1[1utr+3λV˜22(r)+uV˜21(r)+V˜23(r)(RmuV21(n)3λV22(n))V23(n)]+V˜21(r)V23(n)V˜23(r)V21(n)V23(n)Em1,x,(4.23)
Fm1,tr(λ,x,tr)=(3V˜11(r)1utr)Fm1+V˜21(r)(RmCm2V13(r)Sm+3Dm)+V˜13(r)(Rm𝒞m+2V12(b)Sm3𝒟m)=Fm1[3V˜11(r)1utr+V˜12(r)(vRm3vV11(n)+2uV13(n))vV12(n)V13(n)]V˜13(r)(Rm3V11(n)+2uV12(n))vV12(n)V13(n)V˜12(r)V˜13(n)V˜12(n)V˜13(r)vV12(n)V13(n)Fm1,x,(4.24)
m1,tr(λ,x,tr)=m1[1utr+3V˜33(r)+uV˜31(r)w+V˜32(r)wV˜33(n)(wRmuV31(n)3wV33(n))]+V˜31(r)V32(n)V31(n)V˜32(r)wV32(n)m1,x=m1[1utr+3V˜33(r)+V˜32(r)(Rm3V33(n))V32(n)]+V31(n)V˜32(r)V˜31(r)V32(n)V32(n)(Rm𝒜m+2V32(n)Sm3m).(4.25)

### Proof.

Considering that V(n) satisfies the nth stationary equation, we can prove equations (4.22) directly. In order to prove (4.23), (4.24) and (4.25), we first show several expressions about the meromorphic functions ϕ2(P, x, tr) and ϕ3(P, x, tr)

φ2(P,x,tr)+φ2(P*,x,tr)+φ2(P**,x,tr)=RmAm2V23(n)Sm+3BmEm1(λ,x,tr),(4.26)
φ3(P,x,tr)+φ3(P*,x,tr)+φ3(P**,x,tr)=RmAm2V32(n)Sm+3mm1(λ,x,tr),(4.27)
φ2(P,x,tr)+φ2(P*,x,tr)+φ2(P**,x,tr)=λFm1(λ,x,tr)Em1(λ,x,tr),(4.28)
φ3(P,x,tr)+φ3(P*,x,tr)+φ3(P**,x,tr)=Fm1(λ,x,tr)m1(λ,x,tr),(4.29)
1φ2(P,x,tr)+1φ2(P*,x,tr)+1φ2(P**,x,tr)=RmCm2V13(n)Sm+3DmλFm1(λ,x,tr)=(vRm3vV11(n)+2vV13(n))Fm1(λ,x,tr)V13(n)Fm1,x(λ,x,tr)λ(vV12(n)V13(n))Fm1(λ,x,tr),(4.30)
1φ3(P,x,tr)+1φ3(P*,x,tr)+1φ3(P**,x,tr)=Rm𝒞m+2V12(n)Sm3𝒟mFm1(λ,x,tr)=(Rm3V11(n)+2uV12(n))Fm1(λ,x,tr)V12(n)Fm1,x(λ,x,tr)(V13(n)vV12(n))Fm1(λ,x,tr),(4.31)
φ2(P,x,tr)φ3(P,x,tr)+φ2(P*,x,tr)φ3(P*,x,tr)+φ2(P**,x,tr)φ3(P**,x,tr)=V21(n)Em1,x(λ,x,tr)+(RmuV21(n)3λV22(n))Em1(λ,x,tr)V23(n)Em1(λ,x,tr),(4.32)
φ3(P,x,tr)φ2(P,x,tr)+φ3(P*,x,tr)φ2(P*,x,tr)+φ3(P**,x,tr)φ2(P**,x,tr)=V31(n)m1,x(λ,x,tr)+(wRmuV31(n)3wV33(n))m1(λ,x,tr)λwV32(n)m1(λ,x,tr).(4.33)

Expression (4.26) implies that

Em1,xEm1=u+φ2(P,x,tr)++φ2(P*,x,tr)++φ2(P**,x,tr).(4.34)

Differentiating (4.34) with respect to tr, we can derive

(Em1,xEm1)tr=xtr(lnEm1)=[u+φ2(P,x,tr)+φ2(P*,x,tr)+φ2(P**,x,tr)]tr=utr+x(ψ2,tr(P,x,x0,tr,t0,r)ψ2(P,x,x0,tr,t0,r)+ψ2,tr(P*,x,x0,tr,t0,r)ψ2(P*,x,x0,tr,t0,r)+ψ2,tr(P**,x,x0,tr,t0,r)ψ2(P**,x,x0,tr,t0,r))=utr+x(V˜21(r)(φ2(P,x,tr)+φ2(P*,x,tr)+φ2(P**,x,tr))+3λV˜22(r)+V˜23(r)(φ2(P,x,tr)φ3(P,x,tr)+φ2(P*,x,tr)φ3(P*,x,tr)+φ2(P**,x,tr)φ3(P**,x,tr))).(4.35)

Without loss of generality, taking the integration constant of (4.35) to be zero and substituting (4.26) and (4.32) into (4.35) can indicate (4.23). Expressions (4.24) and (4.25) can be proved similarly.

Lemma 4.1 naturally yields the dynamics of the zeros μj(x, tr), νj(x, tr) and ξj(x, tr) of Em−1(λ, x, tr), Fm−1(λ, x, tr) and m−1(λ, x, tr) in terms of Dubrovin-type equations in the subsequent lemma.

### Lemma 4.2.

Suppose that the zeros {μj(x, tr)}j=1,...,3n+1, {νj(x, tr)}j=1,...,3n+1 and {ξj(x, tr)}j=1,...,3n+1 of Em−1(λ, x, tr), Fm−1(λ, x, tr) and ℰm−1(λ, x, tr) remain distinct for (x, tr) ∈ Ωμ, (x, tr) ∈ Ων and (x, tr) ∈ Ωξ, respectively, where Ωμ, Ων, Ωξ ⊆ ℂ2 are open and connected. Then {μj(x, tr)}j=1,...,3n+1, {νj(x, tr)}j=1,...,3n+1 and {ξj(x, tr)}j=1,...,3n+1 satisfy the system of differential equations

μj,x(x,tr)=V23(n)(3y22Rmy+Sm)|λ=μj(x,tr)β0δ02vk=1kj3n+1(μj(x,tr)μk(x,tr)),(4.36)
μj,tr(x,tr)=(V21(n)V˜23(r))V˜21(r)V23(n)(3y22Rmy+Sm)|λ=μj(x,tr)β0δ02vk=1kj3n+1(μj(x,tr)μk(x,tr)),(4.37)
νj,x(x,tr)=(vV12(n)V13(n))(3y22Rmy+Sm)|λ=νj(x,tr)β0δ02vk=1kj3n+1(νj(x,tr)νk(x,tr)),(4.38)
νj,tr(x,tr)=(V12(n)V˜13(r)V˜12(r)V13(n))(3y22Rmy+Sm)|λ=νj(x,tr)β0δ02vxk=1kj3n+1(νj(x,tr)νk(x,tr)),(4.39)
ξj,x(x,tr)=V32(n)(3y22Rmy+Sm)|λ=ξj(x,tr)β0δ02wk=1kj3n+1(ξj(x,tr)ξk(x,tr)),(4.40)
ξj,tr(x,tr)=(V31(n)V˜32(r)V˜31(r)V33(n)(3y22Rmy+Sm)|λ=ξj(x,tr)β0δ02w2k=1kj3n+1(ξj(x,tr)ξk(x,tr))(4.41)
with 1 ≤ j ≤ 3n + 1.

Now we turn to consider expression (3.20) from which one can obtain the subsequent proposition.

### Proposition 4.1.

Let P = (λ, y) ∈ 𝒦m−1 \ {P1, P2}, (x, x0, tr, t0,r) ∈ ℂ4. Then ψ2(P, x, x0, tr, t0,r) on 𝒦m−1 \ {P1, P2} has 3n + 1 zeros and 3n + 1 poles which are μ^1(x,tr),,μ^3n+1(x,tr) and μ^1(x0,t0,r),,μ^3n+1(x0,t0,r), respectively.

### Proof.

By using (4.3), (4.11) and (4.23), we can compute that

φ2(P,x,tr)=y2V23(n)y(RmV23(n)+Am)+BmEm1=1Em1{y2V23(n)y(RmV23(n)+Am)+13(Em1,x+uEm1+RmAm+2V23(n)Sm)}=13Em1,xEm1+1Em1{23V23(n)(3y22Rmy+Sm)+V23(n)(Rm3y)(y+AmV23(n))}+13u=μj,x(x,tr)λμj(x,tr)+O(1)=xln(λμj(x,tr))+O(1),λμj(x,tr).(4.42)

On the other hand, since

1v(x,tr)(φ2,x(P,x,tr)+φ22(P,x,tr)u(x,tr)φ2(P,x,tr)λ)=φ2(P,x,tr)φ3(P,x,tr),(4.43)
we can similarly derive that
V˜21(r)(λ,x,tr)φ2(P,x,tr)+λV˜22(r)(λ,x,tr)+V˜23(r)φ2(P,x,tr)φ3(P,x,tr)=μj,tr(x,tr)λμj(x,tr)+O(1)=trln(λμj(x,tr))+O(1),λμj(x,tr).(4.44)

Substituting expressions (4.42) and (4.44) into (3.20) yields the proposition.

## 5. Quasi-periodic solutions

In this section, we shall construct the Riemann theta function representations for the Baker-Akhiezer function ψ2(P, x, x0, tr, t0,r) and two meromorphic functions ϕ2(P, x, tr), ϕ3(P, x, tr), and in particular, that of solutions for the entire long wave-short wave type hierarchy.

We denote the period lattice 𝒯m1={z_𝒦m1|z_=N_+L_τ,N_,L_𝕑m1}. The complex torus 𝒥m−1 = ℂm−1/𝒯m−1 is called the Jacobian variety of 𝒦m−1. An Abel map 𝒜_:𝒦m1𝒯m1 is defined as

𝒜_(P)=(Q0Pω1,,Q0Pωm1)(mod𝒯m1)(5.1)
with the natural linear extension to the factor group Div(𝒦m−1)
𝒜_(nkPk)=nk𝒜_(Pk).(5.2)

Define

ρ_(1)(x,tr)=𝒜_(k=13n+1μ^k(x,tr))=k=13n+1Q0μ^k(x,tr)ω_,ρ_(2)(x,tr)=𝒜_(k=13n+1ν^k(x,tr))=k=13n+1Q0ν^k(x,tr)ω_,ρ_(3)(x,tr)=𝒜_(k=13n+1ξ^k(x,tr))=k=13n+1Q0ξ^k(x,tr)ω_,(5.3)
where ρ_(1)(x,tr), ρ_(2)(x,tr), ρ_(3)(x,tr) can be linearized on 𝒥m−1 in the following text.

Let θ(z_) denote the Riemann theta function associated with 𝒦m−1 equipped with homology basis and holomorphic differentials as before:

θ(z_)=N_𝕑m1exp{πiN_τ,N_>+2πi<N_,z_>},(5.4)
where z_=(z1,,zm1)m1 is a complex vector, the diamond brackets denote the Euclidean scalar product:
<N_,z_>=i=1m1Nizi,<N_τ,N_>=i,j=1m1τijNiNj.(5.5)

Expression (5.4) implies that

θ(z_+N_+M_τ)=exp{πi<M_τ,M_>2πi<M_,z_>}θ(z_).(5.6)

For brevity, define the function z_:𝒦m1×σm1𝒦m1m1 by

z_(P,Q_)=M_𝒜_(P)+QQ_𝒟(Q)𝒜_(Q),P𝒦m1,Q_=(Q1,Qm1)σm1𝒦m1,(5.7)
where σm−1𝒦m−1 denotes the (m − 1)-th symmetric power of 𝒦m−1 and M_=(M1,,Mm1) is the vector of Riemann constant depending on the base point Q0 by the following expression
Mj=12(1+τjj)l=1ljm1𝕒lωl(P)Q0Pωj,j=1,,m1.(5.8)

Then we have

θ(z_(P,μ^_(x,tr)))=θ(M_𝒜_(P)+ρ_(1)(x,tr)),P𝒦m1,θ(z_(P,ν^_(x,tr)))=θ(M_𝒜_(P)+ρ_(2)(x,tr)),P𝒦m1,θ(z_(P,ξ^_(x,tr)))=θ(M_𝒜_(P)+ρ_(3)(x,tr)),P𝒦m1.(5.9)

According to divisors as seen in (4.14), (4.15) of the meromorphic functions ϕ2(x, tr) and ϕ3(x, tr), we need to introduce Abelian differentials of the third kind for their representations in terms of Riemann theta function.

Let ωQ1,Q2(3)(P) denote the normalized Abelian differential of the third kind holomorphic on 𝒦m−1 \ {Q1, Q2} and having simple poles at Ql with residues (−1)l+1, l = 1,2, then

𝕒kωQ1,Q2(3)(P)=0,𝕓kωQ1,Q2(3)(P)=2πiQ2Q1ωk,k=1,,3n+1.(5.10)

Especially, we introduce ωP0,P1(3)(P) and ωP2,P1(3)(P) as follows

ωP2,P1(3)(P)=δ0(2y(P)Rm(λ))λndλ2(3y2(P)2Rm(λ)y(P)+Sm(λ))+j=13n+1γjϖj,(5.11)
ωP2,P1(3)(P)=δ0(2y(P)Rm(λ))λndλ2(3y2(P)2Rm(λ)y(P)+Sm(λ))+(y2(P)Rm(λ)y(P)+Sm(λ))dλλ(3y2(P)2Rm(λ)y(P)+Sm(λ))+j=13n+1ηjϖj,(5.12)
where the γj, ηj, j = 1,...,3n + 1, are uniquely determined by the requirement of normalized condition, that is vanishing 𝕒–periods
𝕒kωP2,P1(3)(P)=0,𝕒kωP0,P1(3)(P)=0.(5.13)

From (5.11) and (5.12), we can directly calculate that

ωP0,P1(3)(P)=ζ0{(ζ1+O(1))dζ,asPP0,ζ=λ,(ζ1+Λ+O(ζ))dζ,asPP1,ζ=λ1/2,O(1)dζ,asPP2,ζ=λ1,(5.14)
ωP2,P1(3)(P)=ζ0{(ζ1+O(1))dζ,asPP1,ζ=λ1/2,(ζ1+O(1))dζ,asPP2,ζ=λ1,(5.15)
where Λ=12β0δ0(α0δ0+2β022η2n+12δ0η3n+1). Then we have
Q0PωP0,P1(3)(P)=ζ0{lnζ+e1,0(3)(Q0)+O(ζ),asPP0,ζ=λ,lnζ+e1,1(3)(Q0)+Λζ+O(ζ2),asPP1,ζ=λ1/2,e1,2(3)Q0+O(ζ),asPP2,ζ=λ1,(5.16)
and
Q0PωP2,P1(3)(P)=ζ0{lnζ+e2,1(3)(Q0)+O(ζ),asPP1,ζ=λ1/2,lnζ+e2,2(3)Q0+O(ζ),asPP2,ζ=λ1,(5.17)
where e1,0(3)(Q0), e1,1(3)(Q0), e1,2(3)(Q0), e2,1(3)(Q0), e2,2(3)(Q0) are integration constants.

Given the asymptotic expansions at P1, P2 as in lemma 3.1, lemma 3.2 and the divisors (ϕ2(P, x, tr)), (ϕ3(P, x, tr)), 𝒟=μ^1(x0,t0,r)++μ^3n+1(x0,t0,r) as simple poles on 𝒦m−1 \ {P1, P2} of ψ2(P, x, x0, tr, t0,r), the representations of ϕ2(P, x, tr), ϕ3(P, x, tr) and ψ2(P, x, x0, tr, t0,r) can be uniquely determined in the following theorem, as well as those of potentials u(x, tr), v(x, tr), w(x, tr).

### Theorem 5.1.

Let P = (λ, y) ∈ 𝒦m−1 \ {P1, P2} and let (x0, t0,r) ∈ ℂ2, (x, tr) ∈ Ωμ ⊆ ℂ2, where Ωμ is open and connected. Suppose that 𝒟μ^_(x,tr) or 𝒟ν^_(x,tr) or 𝒟ξ^_(x,tr) is nonspecial for (x, tr) ∈ Ωμ. Then ϕ2(P, x, tr), ϕ3(P, x, tr) and ψ2(P, x, x0, tr, t0,r) have the following representations

φ2(P,x,tr)=θ(z_(P,ν^_(x,tr)))θ(z_(P1,μ^_(x,tr)))θ(z_(P1,ν^_(x,tr)))θ(z_(P,μ^_(x,tr)))exp(Q0PωP0,P1(3)(P)e1,1(3)(Q0)),(5.18)
φ3(P,x,tr)=vx(x,tr)θ(z_(P,ν^_(x,tr)))θ(z_(P2,ξ^_(x,tr)))θ(z_(P2,ν^_(x,tr)))θ(z_(P,ξ^_(x,tr)))exp(Q0PωP2,P1(3)(P)e2,1(3)(Q0)),(5.19)
ψ2(P,x,x0,tr,t0,r)=exp(121u(x,tr)121u(x0,t0,r))θ(z_(P,μ^_(x,tr)))θ(z_(P1,μ^_(x,t0,r)))θ(z_(P1,μ^_(x,tr)))θ(z_(P,μ^_(x,t0,r)))×exp((e1(2)(Q0)Q0PΩ2(2)(P))(xx0)+(e˜1(2)(Q0)Q0PΩ˜2r+3(2)(P))(trt0,r)),(5.20)
and potentials u(x, tr), v(x, tr), w(x, tr) are of the form
u(x,tr)=2xlnθ(z_(P1,μ^_(x,tr)))θ(z_(P1,ν^_(x,tr)))+2Λ,(5.21)
v(x,tr)=v(x0,t0,r)exp(121u(x,tr)121u(x0,t0,r))θ(z_(P2,μ^_(x,tr)))θ(z_(P1,μ^_(x,t0,r)))θ(z_(P1,μ^_(x,tr)))θ(z_(P2,μ^_(x,t0,r)))×exp((e1(2)(Q0)e2(2)(Q0))(xx0)+(e˜1(2)(Q0)e˜2(2)(Q0)α˜r)(trt0,r)),(5.22)
w(x,tr)=1vx(x,tr)θ(z_(P2,ν^_(x,tr)))θ(z_(P1,ξ^_(x,tr)))θ(z_(P1,ν^_(x,tr)))θ(z_(P2,ξ^_(x,tr)))exp(e2,2(3)e2,1(3)),(5.23)
where the paths of integration in the integrals and in the Abel mapping are the same.

### Proof.

Assume temporarily that μj(x, tr) ≠ μj′ (x, tr) for jj′ and (x,tr)Ω˜μΩμ, where Ω˜μ is open and connected. Let the right hand side of (5.18) be denoted by Φ2. Noting (4.15) and (5.14), we can see that ϕ2 and Φ2 have the identical 3n + 2 simple poles P1, μ^1(x,tr),,μ^3n+1(x,tr) and simple zeros P0, ν^1(x,tr),,ν^3n+1(x,tr). Since the arithmetic genus of the Riemann surface is 3n + 1, utilizing the Riemann-Roch theorem, we conclude that the holomorphic function Φ2φ2=γ, a constant with respect to P. Using (3.12) and (5.14), we can derive that

Φ2φ2=ζ0(1+O(ζ))(ζ1+O(1))ζ1+O(1)=ζ01+O(ζ),asPP1,ζ=λ1/2,(5.24)
which yields that γ = 1. Similarly, we can prove expression (5.19). The asymptotic expansions of the Baker-Akhiezer function ψ2(P, x, x0, tr, t0,r) in lemma 3.2 and divisor in proposition 4.1 reveal that ψ2(P, x, x0, tr, t0,r) has the Riemann theta representation (5.20), where the function u(x, tr) can be determined later. With help of a meromorphic differential
Ω(x,x0,tr,t0,r)=λln(ψ2(P,x,x0,tr,t0,r))dλ,(5.25)
the Abel map can be linearized in the following form 
ρ_(j)(x,tr)=ρ_(j)(x0,t0,r)+U_2(2)(xx0)+U˜_2r+3(2)(trt0,r)(mod𝒯m1),j=1,2,3.(5.26)

Therefore, θ(z_(P1,μ^_(x,tr))) and θ(z_(P1,ν^_(x,tr))) could be written briefly in the following form

θ(z_(P1,μ^_(x,tr)))=θ(M_(1)+U_2(2)x+U˜_2r+3(2)tr),θ(z_(P1,ν^_(x,tr)))=θ(M_(2)+U_2(2)x+U˜_2r+3(2)tr)(5.27)
where
M_(j)=M_𝒜(P1)+ρ_(j)(x0,t0,r)U_2(2)(x0)U_˜2r+3(2)t0,r,j=1,2.

In order to derive (5.21), we expand ϕ2 of (5.18) near P1 under the local coordinate ζ = λ−1/2

φ2=ζ0ζ1+xlnθ(z_(P1,μ^_(x,tr)))θ(z_(P1,μ^_(x,tr)))+Λ+O(ζ),(5.28)
from which we can derive expression (5.21) by comparing asymptotic expansion (5.28) of ϕ2 with (3.12) in lemma 3.1. Representations for v and w as seen in (5.22) and (5.23) can be deduced similarly by expanding ψ2 and ϕ3 respectively near P2 and P1. Observing the continuity of the Abel map 𝒜_, one can extend the result from (x,tr)Ω˜μ to (x, tr) ∈ Ωμ, which completes the proof of the theorem.

## Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant Nos. 11871440, 11501520, 11522112) and Outstanding Young Talent Research Fund of Zhengzhou University (Grant No. 1521315001).