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

Xianguo Geng; Yunyun Zhai; Bo Xue; Jiao Wei

doi:10.1080/14029251.2019.1544785

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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: 10.1080/14029251.2019.1544785 How 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.
Copyright: © 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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 [1] . 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, 35–37]. 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 [29] , the Bäcklund and Darboux transformations [44] , 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, 32–34, 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, 52–55], 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=vxx−uvx,wt=−wxx−uwx.(1.1)

Equation (1.1) reduces to

ut=2(|v|2)x, vt=−ivxx−uvx,(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,j≥0,sj|(u,v,w)=0=0,j≥1,Ks^j=Js^j+1,j≥0,s^j|(u,v,w)=0=0,j≥1,Ks˜j=Js˜j+1,j≥0,s˜j|(u,v,w)=0=0,j≥1,(2.2)

where the two operators are defined as

K=(−vw∂∂u−∂2w∂∂v+v∂0uv∂−∂v∂∂v−v∂∂2−u∂v2−v0∂w+w∂0−∂2−u∂−vww−∂2−u∂2∂w−v0vw∂0−w∂−v∂∂),(2.3)

J=(−2∂0000−v0100w00−10−∂2−u∂2∂w−v0vw∂0−w∂−v∂∂),(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 s_j, ŝ_j, s˜j can be determined uniquely. For example, the first three members read as

s1=(00−v−w−vw), s^1=(−12u−14ux−18u2−12vwvx−12uv−wx−12uwvxw−vwx−12uvw), s˜1=(−vw−vxwvxx−uvx−v2wwxx+uwx−vw2s˜1(5)),(2.5)

where

s˜1(5)=vxxw−vxwx+vwxx−uvxw+uvwx−v2w2.

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=c−bx,V22=b, V23=d, V31=wc−ex, V32=e, V33=f.

Then the stationary zero-curvature equation

Vx−[U,V]=0(2.7)

is equivalent to

−vw∂b+(∂u−∂2)c+w∂d+(∂v+v∂)e+2λ∂b=0,(uv∂−∂v∂)b+(∂v−v∂)c+(∂2−u∂)d+v2e−vf+λ(vb−d)=0,(∂w+w∂)c−(∂2+u∂+vw)e+wf+λ(−wb+e)=0,−(∂2+ub)b+2∂c+wd−ve=0,vw∂b−w∂d−v∂e+∂f=0.(2.8)

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

(b,c,d,e,f)=∑j≥0(bj,cj,dj,ej,fj)λ−j,(2.9)

equation (2.8) is equivalent to the following recursion equations

KSj=JSj+1, j≥0, JS0=0,(2.10)

where S_j = (b_j, c_j, d_j, e_j, f_j)^T. Since equation JS₀ = 0 has a general solution

S0=α0s0+β0s^0+δ0s˜0,(2.11)

then functions S_j given by

Sj=∑l=0j(αlsj−l+βls^j−l+δls˜j−l),(2.12)

satisfy the recursion equation (2.10), where α_j, β_j, δ_j 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))=∑j≥0r(b˜j,c˜j,d˜j,e˜j,f˜j)λr−j,(2.14)

with S˜j=(b˜j,c˜j,d˜j,e˜j,f˜j)T determined by

S˜j=∑l=0j(α˜lsj−l+β˜ls^j−l+δ˜ls˜j−l).(2.15)

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

(utr,vtr,wtr)T=Xr, r≥0,(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(vxx−uvx),wt0=α˜0w+β˜0wx+δ˜0(−wxx−uwx).(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+32vxxw−32vwxx−32(uvw)x−38u2ux)+2δ˜0(vxxw+vwxx+uvwx−uvxw−v2w2)x+β˜1ux+2δ˜1(vw)x,vt1=α˜0(−vxx+uvx)+β˜0(vxxx−34uxvx−32uvxx−32vvxw+38u2vx)+δ˜0(vxxxx−uxxvx−2uxvxx−2uvxxx−2vx2w−2vvxxw+uuxvx+u2vxx+2uvvxw)−α˜1v+β˜1vx+δ˜1(vxx−uvx),wt1=α˜0(wxx+uwx)+β˜0(wxxx−32vwwx+32uwxx+34uxwx+38u2wx)+δ˜0(−wxxxx−uxxwx−2uxwxx−2uwxxx−uuxwx−u2wxx+2vwx2+2vwwxx+2uvwwx)+α˜1w+β˜1wx+δ˜1(−wxx−uwx).(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=14uxxx−38u2ux+32vxxw−32vwxx−32(uvw)x,vt1=vxxx−34uxvx−32uvxx−32vvxw+38u2vx,wt1=wxxx+34uxwx+32uwxx−32vwwx+38u2wx,(2.19)

or

ut1=2(vxxw+vwxx+uvwx−uvxw−v2w2)x,vt1=vxxxx−uxxvx−2uxvxx−2uvxxx−2vx2w−2vvxxw +uuxvx+u2vxx+2uvvxw,wt1=−wxxxx−uxxwx−2uxwxx−2uwxxx−uuxwx−u2wxx +2vwx2+2vwwxx+2uvwwx.(2.20)

3. Meromorphic functions and Baker-Akhiezer function

In this section, we shall define a trigonal curve 𝒦_m−1, the vector Baker-Akhiezer function and two meromorphic functions on 𝒦_m−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 ℱ_m(λ, y) = det(yI −V⁽ⁿ⁾) of the Lax matrix V⁽ⁿ⁾ is a polynomial of λ independent of variables x and t_r, which can be expressed in the following form

det(yI−V(n))=y3−y2Rm(λ)+ySm(λ)−Tm(λ),(3.3)

where R_m(λ), S_m(λ) and T_m(λ) are polynomials with constant coefficients of λ

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

𝒦m−1:ℱm(λ,y)=y3−y2Rm(λ)+ySm(λ)−Tm(λ)=0.(3.5)

According to (3.4) and (3.5), the trigonal curve 𝒦_m−1 can be compactified by adding two infinite points P_∞₁ and P_∞₂ for which we take P_∞₁ as a double point without loss of generality. 𝒦_m−1 is nonsingular or smooth means that for every point Q₀ = (λ₀, y₀) ∈ 𝒦_m−1 \ {P_∞₁, P_∞₂}, (∂ℱm∂λ,∂ℱm∂y)|(λ,y)=(λ0,y0)≠0. Adding the assumption of irreducibility, the trigonal curve 𝒦_m−1 becomes connected. For the sake of convenience, we use the same symbol 𝒦_m−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+4Sm3−18RmSmTm+27Tm2=−4β02δ04λ6n+5+⋯, which has at most 6n + 5 zeros. Therefore, the Riemann-Hurwitz formula shows that the arithmetic genus of 𝒦_m−1 is 3n + 1 for β₀δ₀ ≠ 0 [16, 27, 48].

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

𝕒j∘𝕓k=δjk, 𝕒j∘𝕒k=0, 𝕓j∘𝕓k=0, j,k=1,…,m−1.

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

ϖl(P)=13y2(P)−2Rm(λ)y(P)+Sm(λ)[λl−1dλ,1≤l≤2n+1,y(P)λl−2n−2dλ,2n+2≤l≤3n+1,(3.6)

which are 3n + 1 linearly independent holomorphic differentials on 𝒦_m−1. By using the homology basis {𝕒j}j=1m−1 and {𝕓j}j=1m−1, the period matrices A = (A_jk) and B = (B_jk) 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 (τ_jk = τ_kj) and has a positive-definite imaginary part (Imτ > 0). If we normalized ϖ_l(P) into new basis ωj=∑l=1m−1Cjlϖl,,

∫𝕒kωj=δjk(3.8)

then we have ∫𝕓kωj=τjk, j, k = 1,...,m − 1.

The complex structure on 𝒦_m−1 is defined in the usual way by introducing local coordinates ζ_Q₀ : P → (λ − λ₀) near points Q₀ = (λ₀, y(Q₀)) ∈ 𝒦_m−1 which are not branch points nor infinite points of 𝒦_m−1, ζ_{P_{∞_j}}: P → λ^−1/(3−j) near the points P_{∞_j} ∈ 𝒦_m−1, j = 1, 2, and similar at others branch points of 𝒦_m−1.

Given these preliminaries, let ψ(P, x, x₀, t_r, t_0,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∈ℂ, P∈𝒦m−1\{P∞1,P∞2}.(3.9)

Define two meromorphic functions ϕ₂(P, x, t_r) and ϕ₃(P, x, t_r) on 𝒦_m−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), P∈𝒦m−1,x,tr∈ℂ,(3.10)

φ3(P,x,tr)=ψ1(P,x,x0,tr,t0,r)ψ2(P,x,x0,tr,t0,r), P∈𝒦m−1,x,tr∈ℂ.(3.11)

Lemma 3.1.

Assume that (3.9), (3.10), (3.11) hold and let P = (λ, y(P)) ∈ 𝒦_m−1 \ {P_∞₁, P_∞₂} and (λ, x, t_r) ∈ ℂ³. Then meromorphic functions ϕ₂(P, x, t_r) and ϕ₃(P, x, t_r) have the following asymptotic expansions near P_{∞_j} ∈ 𝒦_m−1, j = 1, 2, under the local coordinate ζ = λ^−1/(3−j)

φ2(P,x,tr)=ζ→0{ζ−1+u2+18(−2ux+u2+4vw)ζ+18(−uux−2vxw−6vwx+uxx)ζ2+O(ζ3),as P→P∞1, ζ=λ−1/2,vxv+1v2[vvxxx−vxvxx−(uvx)xv+uvx2−vxv2w]ζ+O(ζ2),as P→P∞2, ζ=λ−1,(3.12)

φ3(P,x,tr)=ζ→0{1wζ−1+O(1),as P→P∞1,ζ=λ−1/2,−vxζ+O(ζ2),as P→P∞2,ζ=λ−1.(3.13)

Proof.

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

φ2,xx+3φ2φ2,x+φ23−(vxv+u)(φ2,x+φ22)=(ux−uvxv+vw)φ2+λ(φ2−vxv).(3.14)

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

φ2(P,x,tr)=ζ→0{κ1,−1ζ−1+κ1,0+κ1,1ζ+κ1,2ζ2+O(ζ3),as P→P∞1,ζ=λ−1/2,κ2,0+κ2,1ζ+O(ζ2),as P→P∞2,ζ=λ−1.(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+φ22−uφ2−λ.(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 P_∞₁, P_∞₂ as

y(P)=ζ→0{ζ−2n−2(δ0+β0ζ+δ1ζ2+β1ζ3+O(ζ4)), as P→P∞1, ζ=λ−1/2,ζ−n(α0+α1ζ+O(ζ2)), as P→P∞2, ζ=λ−1.(3.17)

Subsequently, one infers that

ωj=ζ→0{(−Cj,2n+1−δ0Cj,3n+1β0δ0+O(ζ))dζ, as P→P∞1, ζ=λ−1/2,(−Cj,2n+1δ02+O(ζ))dζ, as P→P∞2, ζ=λ−1.(3.18)

Furthermore, we could write ω_j in the following form:

ωk=∑l=0∞ρk,l(P∞j)ζldζ, as P→P∞j, ζ=λ−1/(3−j),(3.19)

where ρ_k,l(P_{∞_j}) 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, t_r, t_0,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, t_r, t_0,r) near P_{∞_j}, 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, t_r), v(x, t_r) and w(x, t_r) satisfy the r-th nonlinear evolution equations (2.16). Moreover, let P ∈ 𝒦_m−1 \ {P_∞₁, P_∞₂}, (x, x₀, t_r, t_0,r) ∈ ℂ⁴. Then

ψ2(P,x,x0,tr,t0,r)=ζ→0{exp(ζ−1(x−x0)+∑l=0r(δ˜r−l+β˜r−lζ)ζ−2l−2(tr−t0,r) +12∂−1u(x,tr)−12∂−1u(x0,t0,r)+O(ζ)), as P→P∞1, ζ=λ−1/2,v(x,tr)v(x0,t0,r)exp(∑l=0rα˜r−lζ−l(tr−t0,r)+O(ζ)), as P→P∞2, ζ=λ−1.(3.24)

Proof.

To investigate the property of ψ₂(P, x, x₀, t_r, t_0,r) near P_∞₁, 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)+(∂u−∂2)c˜¯r(1)+wd˜¯r,x(1)+(∂v+v∂)e˜¯r(1)]2+O(ζ),I¯r(2)(P,x,tr)=ζ−2r−1+∂−1[−vwb˜¯r,x(2)+(∂u−∂2)c˜¯r(2)+wd˜¯r,x(2)+(∂v+v∂)e˜¯r(2)]2+O(ζ),I¯r(3)(P,x,tr)=ζ−2r−2+∂−1[−vwb˜¯r,x(3)+(∂u−∂2)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)+(∂u−∂2)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)=ζ→0∑j=0∞σj(1)(x,tr)ζj, P→P∞1,(3.27)

for some coefficients {σj(1)(x,tr)}j∈ℕ0 to be determined. Observing

φ2,tr=[V˜¯21(r,1)φ2+λV˜¯22(r,1)+V˜¯23(r,1)v(φ2,x+φ22−uφ2−λ)]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, t_r) near P_∞₁ 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)=ζ−2I¯r(1)+(c˜¯r+1(1)−b˜¯r+1,x(1))φ2+b˜¯r+1(1)ζ−2+d˜¯r+1(1)v(φ2,x+φ22−uφ2−ζ−2)=∂−1[−vwb˜¯r+1,x(1)+(∂u−∂2)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 P_∞₁. Similarly, one can prove the other two expressions in (3.25), which yield the expansion of I_r(P, x, t_r) near P_∞₁ as follows

Ir(P,x,tr)=∑l=0r(δ˜r−l+β˜r−lζ)ζ−2l−2+∂−1[−vwb˜r,x+(∂u−∂2)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 P_∞₂, we can similarly prove the second expression in lemma 3.2.

Let ωP∞s,j(2)(P), j ≥ 2, s = 1, 2, denote the normalized Abelian differential of the second kind holomorphic on 𝒦_m−1 \ {P_{∞_s}} satisfying

∫𝕒kωP∞s,j(2)(P)=0, k=1,…,3n+1,(3.32)

ωP∞s,j(2)(P)=ζ→0(ζ−j+O(1))dζ, as P→P∞,ζ=λ1/(s−3).(3.33)

According to the asymptotic behaviors of ψ₂(P, x, x₀, t_r, t_0,r) in (3.24), we introduce the corresponding Abelian differential of the second kind

Ω2(2)(P)=ωP∞1,2(2)(P)(3.34)

and

Ω˜2r+3(2)(P)=∑l=0r(2l+2)δ˜r−lωP∞1,2l+3(2)(P)+∑l=0r(2l+1)β˜r−lωP∞1,2l+2(2)(P)+∑l=1rlα˜r−lωP∞2,l+1(2)(P).(3.35)

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

∫Q0PΩ2(2)(P)={−ζ−1+e1(2)(Q0)+O(ζ), as P→P∞1, ζ=λ−1/2,e2(2)(Q0)+O(ζ), as P→P∞2, ζ=λ−1,(3.36)

∫Q0PΩ˜2r+3(2)(P)={−∑l=0rδ˜r−lζ−2l−2−∑l=0rβ˜r−lζ−2l−2+e˜1(2)(Q0)+O(ζ), as P→P∞1, ζ=λ−1/2,−∑l=1rα˜r−lζ−1+e˜1(2)(Q0)+O(ζ), as P→P∞2, ζ=λ−1,(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 𝒦_m−1 \ {P_∞₁, P_∞₂}. 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,m−1(2)), U2,k(2)=12πi∫𝕓kΩ2(2)(P), k=1,…,m−1,(3.38)

U_˜2r+3(2)=(U˜2r+3,1(2),…,U˜2r+3,m−1(2)), U˜2r+3,j(2)=12πi∫𝕓kΩ˜2r+3(2)(P), j=1,…,m−1.(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(P∞1), k=1,2,…,m−1,(3.40)

U˜2r+3,k(2)=∑l=0rδ˜r−lρk,2l+1(P∞1)+∑l=0rβ˜r−lρk,2l(P∞1)+∑l=1rα˜r−lρk,l−1(P∞2), k=1,2,…,m−1.(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 𝒦_m−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 𝒦_m−2. For a fixed λ, let y_i(λ), i = 0, 1, 2, denote the three roots of polynomial ℱ_m(λ, y) = 0, that is

(y−y0(λ))(y−y1(λ))(y−y2(λ))=y3−y2Rm+ySm−Tm=0.(4.1)

Then points (λ, y₀(λ)), (λ, y₁(λ)) and (λ, y₂(λ)) are on the three different sheets of Riemann surface 𝒦_m−1, respectively. Let P = (λ, y_i(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 y_i(λ), i = 0, 1, 2 and the coefficients R_m, S_m, T_m

y0+y1+y2=Rm,y0y1+y0y2+y1y2=Sm,y0y1y2=Tm,y02+y12+y22=Rm2−2Sm,y03+y13+y23=Rm3−3RmSm+3Tm,y02y12+y02y22+y12y22=Sm2−2RmTm,∏j=02(3yj2−2yjRm+Sm)=Δ(λ).(4.2)

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

φ2=yV13(n)+CmyV23(n)+Am=λFm−1y2V13(n)−y(RmV13(n)+Cm)+Dm=y2V23(n)−y(RmV23(n)+Am)+BmEm−1,(4.3)

φ3=yV12(n)+𝒞myV32(n)+𝒜m=−Fm−1y2V12(n)−y(RmV12(n)+𝒞m)+𝒟m=y2V32(n)−y(RmV32(n)+𝒜m)+ℬmℰm−1,(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),ℬm=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)

Em−1=V21(n)(V11(n)V23(n)−V13(n)V21(n))+V23(n)(V23(n)V31(n)−V21(n)V33(n)),Fm−1=V13(n)(V13(n)V32(n)−V12(n)V33(n))+λV12(n)(V13(n)V22(n)−V12(n)V23(n)),ℰm−1=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 E_m−1, F_m−1 and ℰ_m−1 are polynomials with respect to λ of degree 3n + 1 for β₀δ₀ ≠ 0. Therefore, we can rewrite them in the following form:

Em−1(λ,x,tr)=β0δ02v∏j=13n+1(λ−μj(x,tr)),(4.8)

Fm−1(λ,x,tr)=β0δ02v∏j=13n+1(λ−νj(x,tr)),(4.9)

ℰm−1(λ,x,tr)=β0δ02w2∏j=13n+1(λ−ξj(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 E_m−1(λ, x, t_r), F_m−1(λ, x, t_r), ℰ_m−1(λ, x, t_r) 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

Fm−1|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))∈𝒦m−1,(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))∈𝒦m−1,(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))∈𝒦m−1(4.13)

with 1 ≤ j ≤ 3n + 1, (x, t_r) ∈ ℂ².

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

(φ2(P,x,tr))=𝒟P0,ν^1(x,tr),…,ν^3n+1(x,tr)(P)−𝒟P∞1,μ^1(x,tr),…,μ^3n+1(x,tr)(P),(4.14)

(φ3(P,x,tr))=𝒟P∞2,ν^1(x,tr),…,ν^3n+1(x,tr)(P)−𝒟P∞1,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, t_r, t_0,r) on 𝒦_m−1 \ {P_∞₁, P_∞₂}. From (4.3) and (4.4), we can easily obtain the interrelationships among the polynomials A_m, B_m, C_m, D_m, 𝒜_m, ℬ_m, 𝒞_m, 𝒟_m, E_m−1, F_m−1, ℰ_m−1, R_m, S_m, T_m, which we list below:

λV23(n)Fm−1=V13(n)Dm−(V13(n))2Sm−V13(n)RmCm−Cm2,λAmFm−1=(V13(n))2Tm+CmDm,(4.16)

V13(n)Em−1=V23(n)Bm−(V23(n))2Sm−V23(n)RmAm−Am2,CmEm−1=(V23(n))2Tm+AmBm,(4.17)

0=V13(n)Bm+V23(n)Dm−V13(n)V23(n)Sm+AmCm,0=V13(n)V23(n)RmSm+V13(n)V23(n)Tm+V13(n)AmSm+V23(n)CmSm−V13(n)RmBm−V23(n)RmDm−BmCm−AmDm,0=V13(n)V23(n)RmTm+V13(n)AmTm+V23(n)CmTm+λEm−1Fm−1−BmDm,(4.18)

−V32(n)Fm−1=V12(n)𝒟m−(V12(n))2Sm−V12(n)Rm𝒞m−𝒞m2,−𝒜mFm−1=(V12(n))2Tm+𝒞m𝒟m,(4.19)

V12(n)ℰm−1=V32(n)ℬm−(V32(n))2Sm−V32(n)Rm𝒜m−𝒜m2,𝒞mℰm−1=(V32(n))2Tm+𝒜mℬm,(4.20)

0=V12(n)ℬm+V32(n)𝒟m−V12(n)V32(n)Sm+𝒜m𝒞m,0=V12(n)V32(n)RmSm+V12(n)V32(n)Tm+V12(n)𝒜mSm+V32(n)𝒞mSm−V12(n)Rmℬm−V32(n)Rm𝒟m−ℬm𝒞m−𝒜m𝒟m,0=V12(n)V32(n)RmTm+V12(n)𝒜mTm+V32(n)𝒞mTm−ℰm−1Fm−1−ℬm𝒟m.(4.21)

Taking use of above relationships, we arrive at the evolution of E_m−1(λ, x, t_r), F_m−1(λ, x, t_r), ℰ_m−1(λ, x, t_r) with respect to x and t_r respectively in the subsequent lemma.

Lemma 4.1.

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

Em−1,x=−uEm−1−(RmAm+2V23(n)Sm−3Bm),Fm−1,x=2uFm−1+(−RmCm−2V13(n)Sm+3Dm)+v(Rm𝒞m+2V12(n)Sm−3𝒟m),ℰm−1,x=−uℰm−1+w(−Rm𝒜m−2V32(n)Sm+3ℬm).(4.22)

Em−1,tr(λ,x,tr)=Em−1[−∂−1utr+3λV˜22(r)+V˜23(r)(Rm−3λV22(n))V23(n)]+V˜23(r)V21(n)−V˜21(r)V23(n)V23(n)(RmAm+2V23(n)Sm−3Bm)=Em−1[−∂−1utr+3λV˜22(r)+uV˜21(r)+V˜23(r)(Rm−uV21(n)−3λV22(n))V23(n)]+V˜21(r)V23(n)−V˜23(r)V21(n)V23(n)Em−1,x,(4.23)

Fm−1,tr(λ,x,tr)=(3V˜11(r)−∂−1utr)Fm−1+V˜21(r)(−RmCm−2V13(r)Sm+3Dm)+V˜13(r)(Rm𝒞m+2V12(b)Sm−3𝒟m)=Fm−1[3V˜11(r)−∂−1utr+V˜12(r)(vRm−3vV11(n)+2uV13(n))vV12(n)−V13(n)]−V˜13(r)(Rm−3V11(n)+2uV12(n))vV12(n)−V13(n)−V˜12(r)V˜13(n)−V˜12(n)V˜13(r)vV12(n)−V13(n)Fm−1,x,(4.24)

ℰm−1,tr(λ,x,tr)=ℰm−1[−∂−1utr+3V˜33(r)+uV˜31(r)w+V˜32(r)wV˜33(n)(wRm−uV31(n)−3wV33(n))]+V˜31(r)V32(n)−V31(n)V˜32(r)wV32(n)ℰm−1,x=ℰm−1[−∂−1utr+3V˜33(r)+V˜32(r)(Rm−3V33(n))V32(n)]+V31(n)V˜32(r)−V˜31(r)V32(n)V32(n)(Rm𝒜m+2V32(n)Sm−3ℬm).(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, t_r) and ϕ₃(P, x, t_r)

φ2(P,x,tr)+φ2(P*,x,tr)+φ2(P**,x,tr)=−RmAm−2V23(n)Sm+3BmEm−1(λ,x,tr),(4.26)

φ3(P,x,tr)+φ3(P*,x,tr)+φ3(P**,x,tr)=−RmAm−2V32(n)Sm+3ℬmℰm−1(λ,x,tr),(4.27)

φ2(P,x,tr)+φ2(P*,x,tr)+φ2(P**,x,tr)=−λFm−1(λ,x,tr)Em−1(λ,x,tr),(4.28)

φ3(P,x,tr)+φ3(P*,x,tr)+φ3(P**,x,tr)=Fm−1(λ,x,tr)ℰm−1(λ,x,tr),(4.29)

1φ2(P,x,tr)+1φ2(P*,x,tr)+1φ2(P**,x,tr)=−RmCm−2V13(n)Sm+3DmλFm−1(λ,x,tr)=(vRm−3vV11(n)+2vV13(n))Fm−1(λ,x,tr)−V13(n)Fm−1,x(λ,x,tr)λ(vV12(n)−V13(n))Fm−1(λ,x,tr),(4.30)

1φ3(P,x,tr)+1φ3(P*,x,tr)+1φ3(P**,x,tr)=Rm𝒞m+2V12(n)Sm−3𝒟mFm−1(λ,x,tr)=(Rm−3V11(n)+2uV12(n))Fm−1(λ,x,tr)−V12(n)Fm−1,x(λ,x,tr)(V13(n)−vV12(n))Fm−1(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)Em−1,x(λ,x,tr)+(Rm−uV21(n)−3λV22(n))Em−1(λ,x,tr)V23(n)Em−1(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)ℰm−1,x(λ,x,tr)+(wRm−uV31(n)−3wV33(n))ℰm−1(λ,x,tr)λwV32(n)ℰm−1(λ,x,tr).(4.33)

Expression (4.26) implies that

Em−1,xEm−1=−u+φ2(P,x,tr)++φ2(P*,x,tr)++φ2(P**,x,tr).(4.34)

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

(Em−1,xEm−1)tr=∂x∂tr(lnEm−1)=[−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 μ_j(x, t_r), ν_j(x, t_r) and ξ_j(x, t_r) of E_m−1(λ, x, t_r), F_m−1(λ, x, t_r) and ℰ_m−1(λ, x, t_r) in terms of Dubrovin-type equations in the subsequent lemma.

Lemma 4.2.

Suppose that the zeros {μ_j(x, t_r)}_j=1,...,3n+1, {ν_j(x, t_r)}_j=1,...,3n+1 and {ξ_j(x, t_r)}_j=1,...,3n+1 of E_m−1(λ, x, t_r), F_m−1(λ, x, t_r) and ℰ_m−1(λ, x, t_r) remain distinct for (x, t_r) ∈ Ω_μ, (x, t_r) ∈ Ω_ν and (x, t_r) ∈ Ω_ξ, respectively, where Ω_μ, Ω_ν, Ω_ξ ⊆ ℂ² are open and connected. Then {μ_j(x, t_r)}_j=1,...,3n+1, {ν_j(x, t_r)}_j=1,...,3n+1 and {ξ_j(x, t_r)}_j=1,...,3n+1 satisfy the system of differential equations

μj,x(x,tr)=−V23(n)(3y2−2Rmy+Sm)|λ=μj(x,tr)β0δ02v∏k=1k≠j3n+1(μj(x,tr)−μk(x,tr)),(4.36)

μj,tr(x,tr)=(V21(n)V˜23(r))−V˜21(r)V23(n)(3y2−2Rmy+Sm)|λ=μj(x,tr)β0δ02v∏k=1k≠j3n+1(μj(x,tr)−μk(x,tr)),(4.37)

νj,x(x,tr)=(vV12(n)−V13(n))(3y2−2Rmy+Sm)|λ=νj(x,tr)β0δ02v∏k=1k≠j3n+1(νj(x,tr)−νk(x,tr)),(4.38)

νj,tr(x,tr)=(V12(n)V˜13(r)−V˜12(r)V13(n))(3y2−2Rmy+Sm)|λ=νj(x,tr)β0δ02vx∏k=1k≠j3n+1(νj(x,tr)−νk(x,tr)),(4.39)

ξj,x(x,tr)=−V32(n)(3y2−2Rmy+Sm)|λ=ξj(x,tr)β0δ02w∏k=1k≠j3n+1(ξj(x,tr)−ξk(x,tr)),(4.40)

ξj,tr(x,tr)=(V31(n)V˜32(r)−V˜31(r)V33(n)(3y2−2Rmy+Sm)|λ=ξj(x,tr)β0δ02w2∏k=1k≠j3n+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) ∈ 𝒦_m−1 \ {P_∞₁, P_∞₂}, (x, x₀, t_r, t_0,r) ∈ ℂ⁴. Then ψ₂(P, x, x₀, t_r, t_0,r) on 𝒦_m−1 \ {P_∞₁, P_∞₂} 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)+BmEm−1=1Em−1{y2V23(n)−y(RmV23(n)+Am)+13(Em−1,x+uEm−1+RmAm+2V23(n)Sm)}=13Em−1,xEm−1+1Em−1{23V23(n)(3y2−2Rmy+Sm)+V23(n)(Rm3−y)(y+AmV23(n))}+13u=−μj,x(x,tr)λ−μj(x,tr)+O(1)=∂xln(λ−μj(x,tr))+O(1), λ→μj(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)λ−μj(x,tr)+O(1)=∂trln(λ−μj(x,tr))+O(1), λ→μj(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, t_r, t_0,r) and two meromorphic functions ϕ₂(P, x, t_r), ϕ₃(P, x, t_r), and in particular, that of solutions for the entire long wave-short wave type hierarchy.

We denote the period lattice 𝒯m−1={z_∈𝒦m−1|z_=N_+L_τ, N_,L_∈𝕑m−1}. The complex torus 𝒥_m−1 = ℂ^m−1/𝒯_m−1 is called the Jacobian variety of 𝒦_m−1. An Abel map 𝒜_:𝒦m−1→𝒯m−1 is defined as

𝒜_(P)=(∫Q0Pω1,…,∫Q0Pωm−1) (mod𝒯m−1)(5.1)

with the natural linear extension to the factor group Div(𝒦_m−1)

𝒜_(∑nkPk)=∑nk𝒜_(Pk).(5.2)

Define

ρ_(1)(x,tr)=𝒜_(∑k=13n+1μ^k(x,tr))=∑k=13n+1∫Q0μ^k(x,tr)ω_,ρ_(2)(x,tr)=𝒜_(∑k=13n+1ν^k(x,tr))=∑k=13n+1∫Q0ν^k(x,tr)ω_,ρ_(3)(x,tr)=𝒜_(∑k=13n+1ξ^k(x,tr))=∑k=13n+1∫Q0ξ^k(x,tr)ω_,(5.3)

where ρ_(1)(x,tr), ρ_(2)(x,tr), ρ_(3)(x,tr) can be linearized on 𝒥_m−1 in the following text.

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

θ(z_)=∑N_∈𝕑m−1exp{πi≤N_τ,N_>+2πi<N_,z_>},(5.4)

where z_=(z1,…,zm−1)∈ℂm−1 is a complex vector, the diamond brackets denote the Euclidean scalar product:

<N_,z_>=∑i=1m−1Nizi, <N_τ,N_>=∑i,j=1m−1τijNiNj.(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_:𝒦m−1×σm−1𝒦m−1→ℂm−1 by

z_(P,Q_)=M_−𝒜_(P)+∑Q′∈Q_𝒟(Q′)𝒜_(Q′), P∈𝒦m−1,Q_=(Q1,…Qm−1)∈σm−1𝒦m−1,(5.7)

where σ^m−1𝒦_m−1 denotes the (m − 1)-th symmetric power of 𝒦_m−1 and M_=(M1,…,Mm−1) is the vector of Riemann constant depending on the base point Q₀ by the following expression

Mj=12(1+τjj)−∑l=1l≠jm−1∫𝕒lωl(P)∫Q0Pωj, j=1,…,m−1.(5.8)

Then we have

θ(z_(P,μ^_(x,tr)))=θ(M_−𝒜_(P)+ρ_(1)(x,tr)), P∈𝒦m−1,θ(z_(P,ν^_(x,tr)))=θ(M_−𝒜_(P)+ρ_(2)(x,tr)), P∈𝒦m−1,θ(z_(P,ξ^_(x,tr)))=θ(M_−𝒜_(P)+ρ_(3)(x,tr)), P∈𝒦m−1.(5.9)

According to divisors as seen in (4.14), (4.15) of the meromorphic functions ϕ₂(x, t_r) and ϕ₃(x, t_r), 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 𝒦_m−1 \ {Q₁, Q₂} and having simple poles at Q_l with residues (−1)^l+1, l = 1,2, then

∫𝕒kωQ1,Q2(3)(P)=0, ∫𝕓kωQ1,Q2(3)(P)=2πi∫Q2Q1ωk, k=1,…,3n+1.(5.10)

Especially, we introduce ωP0,P∞1(3)(P) and ωP∞2,P∞1(3)(P) as follows

ωP∞2,P∞1(3)(P)=δ0(2y(P)−Rm(λ))λndλ2(3y2(P)−2Rm(λ)y(P)+Sm(λ))+∑j=13n+1γjϖj,(5.11)

ωP∞2,P∞1(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 γ_j, η_j, j = 1,...,3n + 1, are uniquely determined by the requirement of normalized condition, that is vanishing 𝕒–periods

∫𝕒kωP∞2,P∞1(3)(P)=0, ∫𝕒kωP0,P∞1(3)(P)=0.(5.13)

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

ωP0,P∞1(3)(P)=ζ→0{(ζ−1+O(1))dζ, as P→P0, ζ=λ,(−ζ−1+Λ+O(ζ))dζ, as P→P∞1, ζ=λ−1/2,O(1)dζ, as P→P∞2, ζ=λ−1,(5.14)

ωP∞2,P∞1(3)(P)=ζ→0{(−ζ−1+O(1))dζ, as P→P∞1, ζ=λ−1/2,(ζ−1+O(1))dζ, as P→P∞2, ζ=λ−1,(5.15)

where Λ=12β0δ0(α0δ0+2β02−2η2n+1−2δ0η3n+1). Then we have

∫Q0PωP0,P∞1(3)(P)=ζ→0{lnζ+e1,0(3)(Q0)+O(ζ), as P→P0, ζ=λ,−lnζ+e1,∞1(3)(Q0)+Λζ+O(ζ2), as P→P∞1, ζ=λ−1/2,e1,∞2(3)Q0+O(ζ), as P→P∞2, ζ=λ−1,(5.16)

and

∫Q0PωP∞2,P∞1(3)(P)=ζ→0{−lnζ+e2,∞1(3)(Q0)+O(ζ), as P→P∞1, ζ=λ−1/2,lnζ+e2,∞2(3)Q0+O(ζ), as P→P∞2, ζ=λ−1,(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 P_∞₁, P_∞₂ as in lemma 3.1, lemma 3.2 and the divisors (ϕ₂(P, x, t_r)), (ϕ₃(P, x, t_r)), 𝒟=μ^1(x0,t0,r)+⋯+μ^3n+1(x0,t0,r) as simple poles on 𝒦_m−1 \ {P_∞₁, P_∞₂} of ψ₂(P, x, x₀, t_r, t_0,r), the representations of ϕ₂(P, x, t_r), ϕ₃(P, x, t_r) and ψ₂(P, x, x₀, t_r, t_0,r) can be uniquely determined in the following theorem, as well as those of potentials u(x, t_r), v(x, t_r), w(x, t_r).

Theorem 5.1.

Let P = (λ, y) ∈ 𝒦_m−1 \ {P_∞₁, P_∞₂} and let (x₀, t_0,r) ∈ ℂ², (x, t_r) ∈ Ω_μ ⊆ ℂ², where Ω_μ is open and connected. Suppose that 𝒟μ^_(x,tr) or 𝒟ν^_(x,tr) or 𝒟ξ^_(x,tr) is nonspecial for (x, t_r) ∈ Ω_μ. Then ϕ₂(P, x, t_r), ϕ₃(P, x, t_r) and ψ₂(P, x, x₀, t_r, t_0,r) have the following representations

φ2(P,x,tr)=θ(z_(P,ν^_(x,tr)))θ(z_(P∞1,μ^_(x,tr)))θ(z_(P∞1,ν^_(x,tr)))θ(z_(P,μ^_(x,tr)))exp(∫Q0PωP0,P∞1(3)(P)−e1,∞1(3)(Q0)),(5.18)

φ3(P,x,tr)=−vx(x,tr)θ(z_(P,ν^_(x,tr)))θ(z_(P∞2,ξ^_(x,tr)))θ(z_(P∞2,ν^_(x,tr)))θ(z_(P,ξ^_(x,tr)))exp(∫Q0PωP∞2,P∞1(3)(P)−e2,∞1(3)(Q0)),(5.19)

ψ2(P,x,x0,tr,t0,r)=exp(12∂−1u(x,tr)−12∂−1u(x0,t0,r))θ(z_(P,μ^_(x,tr)))θ(z_(P∞1,μ^_(x,t0,r)))θ(z_(P∞1,μ^_(x,tr)))θ(z_(P,μ^_(x,t0,r)))×exp((e1(2)(Q0)−∫Q0PΩ2(2)(P))(x−x0)+(e˜1(2)(Q0)−−∫Q0PΩ˜2r+3(2)(P))(tr−t0,r)),(5.20)

and potentials u(x, t_r), v(x, t_r), w(x, t_r) are of the form

u(x,tr)=2∂xlnθ(z_(P∞1,μ^_(x,tr)))θ(z_(P∞1,ν^_(x,tr)))+2Λ,(5.21)

v(x,tr)=v(x0,t0,r)exp(12∂−1u(x,tr)−12∂−1u(x0,t0,r))θ(z_(P∞2,μ^_(x,tr)))θ(z_(P∞1,μ^_(x,t0,r)))θ(z_(P∞1,μ^_(x,tr)))θ(z_(P∞2,μ^_(x,t0,r)))×exp((e1(2)(Q0)−e2(2)(Q0))(x−x0)+(e˜1(2)(Q0)−e˜2(2)(Q0)−α˜r)(tr−t0,r)),(5.22)

w(x,tr)=−1vx(x,tr)θ(z_(P∞2,ν^_(x,tr)))θ(z_(P∞1,ξ^_(x,tr)))θ(z_(P∞1,ν^_(x,tr)))θ(z_(P∞2,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 μ_j(x, t_r) ≠ μ_j′ (x, t_r) for j ≠ j′ and (x,tr)∈Ω˜μ⊆Ωμ, 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 P_∞₁, 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=ζ→0(1+O(ζ))(ζ−1+O(1))ζ−1+O(1)=ζ→01+O(ζ), as P→P∞1, ζ=λ−1/2,(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, t_r, t_0,r) in lemma 3.2 and divisor in proposition 4.1 reveal that ψ₂(P, x, x₀, t_r, t_0,r) has the Riemann theta representation (5.20), where the function u(x, t_r) 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 [25]

ρ_(j)(x,tr)=ρ_(j)(x0,t0,r)+U_2(2)(x−x0)+U˜_2r+3(2)(tr−t0,r) (mod𝒯m−1), j=1,2,3.(5.26)

Therefore, θ(z_(P∞1,μ^_(x,tr))) and θ(z_(P∞1,ν^_(x,tr))) could be written briefly in the following form

θ(z_(P∞1,μ^_(x,tr)))=θ(M_(1)+U_2(2)x+U˜_2r+3(2)tr),θ(z_(P∞1,ν^_(x,tr)))=θ(M_(2)+U_2(2)x+U˜_2r+3(2)tr)(5.27)

where

M_(j)=M_−𝒜(P∞1)+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 P_∞₁ under the local coordinate ζ = λ^−1/2

φ2=ζ→0ζ−1+∂xlnθ(z_(P∞1,μ^_(x,tr)))θ(z_(P∞1,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 P_∞₂ and P_∞₁. Observing the continuity of the Abel map 𝒜_, one can extend the result from (x,tr)∈Ω˜μ to (x, t_r) ∈ Ω_μ, 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).

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 1
Pages: 1 - 23
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2019.1544785 How to use a DOI?
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TY  - JOUR
AU  - Xianguo Geng
AU  - Yunyun Zhai
AU  - Bo Xue
AU  - Jiao Wei
PY  - 2021
DA  - 2021/01/06
TI  - A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 1
EP  - 23
VL  - 26
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1544785
DO  - 10.1080/14029251.2019.1544785
ID  - Geng2021
ER  -

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