Theory and Modern Applications

# The application of trigonal curve theory to the second-order Benjamin-Ono hierarchy

## Abstract

By introducing two sets of Lenard recursion equations, the second-order Benjamin-Ono hierarchy is proposed. In view of the characteristic polynomial of Lax matrix, a trigonal curve of arithmetic genus $m-1$ is deduced. Then the trigonal curve theory is used to derive the explicit algebro-geometric solutions represented in theta functions to the second-order Benjamin-Ono hierarchy with the help of the properties of Baker-Akhiezer function, the meromorphic function and the three kinds of Abel differentials.

MSC:35Q51, 37K10, 14H70, 35C99.

## 1 Introduction

The principal aim of the present paper concerns the algebro-geometric solutions of the second-order Benjamin-Ono hierarchy with the aid of the theory of trigonal curves [13]. To the best of the authors’ knowledge, there have been no results about the algebro-geometric solutions of the second-order Benjamin-Ono equation [4, 5]

${u}_{tt}=\alpha {\left({u}^{2}\right)}_{xx}+\beta {u}_{xxxx},$
(1.1)

which is used in the analysis of long waves in shallow water and many other physical applications, where α is a constant controlling nonlinearity and the characteristic speed of the long waves, and β is the depth of the fluid, although there are some results about the exact solutions of (1.1), such as the pulse-type and kink-type solutions, periodic solitary wave and double periodic solutions, soliton solutions etc., by using the following methods: the Jacobi elliptic function expansion method, the bilinear method, the extended homoclinic test approach, the homogeneous balance method and the lattice Boltzmann method etc. [610].

Before turning to the contents of each section, it seems appropriate to review the existing literature on algebro-geometric solutions, which are of great importance for revealing inherent structure mechanism of solutions and describing the quasi-periodic behavior of nonlinear phenomena. During the last few years, there have been fairly mature techniques to construct algebro-geometric solutions of soliton equations associated with $2×2$ matrix spectral problems, such as the KdV, nonlinear Schrödinger, sine-Gordon, Toda equations and so on [1115]. Unfortunately, the situation is not so good for soliton equations associated with $3×3$ matrix spectral problems, which are more complicated and more difficult. In [16], a unified framework was proposed to yield all algebro-geometric solutions of the entire Boussinesq hierarchy. Recently, based on the characteristic polynomial of Lax matrix associated with the $3×3$ matrix spectral problems, we have developed the method in [16] to deal with some important soliton equations by introducing the trigonal curves of arithmetic genus $m-1$ and deriving the explicit Riemann theta function representations of the entire hierarchies, such as the modified Boussinesq, the Kaup-Kupershmidt hierarchies and others [1719].

The present paper is organized as follows. In Section 2, based on two kinds of different Lenard recursion equations, we derive the second-order Benjamin-Ono hierarchy, which relates to a $3×3$ matrix spectral problem. In Section 3, we introduce the Baker-Akhiezer function and the associated meromorphic function. Then the second-order Benjamin-Ono hierarchy is decomposed into the system of Dubrovin-type ordinary differential equations. In Section 4, the explicit Riemann theta function representations of the Baker-Akhiezer function and the meromorphic function, and especially of the solutions to the entire second-order Benjamin-Ono hierarchy are displayed by resorting to the Riemann theta functions, the holomorphic differentials, and the Abel map.

## 2 The zero-curvature representation to the second-order Benjamin-Ono hierarchy

In this section, we shall derive the second-order Benjamin-Ono hierarchy associated with the $3×3$ matrix spectral problem

${\psi }_{x}=U\psi ,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\psi =\left(\begin{array}{c}{\psi }_{1}\\ {\psi }_{2}\\ {\psi }_{3}\end{array}\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}U=\left(\begin{array}{ccc}0& 1& 0\\ u& 0& 1\\ v+\lambda & u& 0\end{array}\right),$
(2.1)

where u and v are two potentials, and λ is a constant spectral parameter. To this end, we introduce two sets of Lenard recursion equations

$K{g}_{j-1}=J{g}_{j},\phantom{\rule{1em}{0ex}}{g}_{j}{|}_{\left(u,v\right)=0}=0,j\ge 0,$
(2.2)
$K{\stackrel{ˆ}{g}}_{j-1}=J{\stackrel{ˆ}{g}}_{j},\phantom{\rule{1em}{0ex}}{\stackrel{ˆ}{g}}_{j}{|}_{\left(u,v\right)=0}=0,j\ge 0$
(2.3)

with two starting points

${g}_{-1}={\left(1,0\right)}^{T},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\stackrel{ˆ}{g}}_{-1}={\left(0,1\right)}^{T},$

where the initial conditions mean to identify constants of integration as zero, and two operators are defined as follows:

$\begin{array}{c}K=\left(\begin{array}{cc}\partial u+u\partial -{\partial }^{3}& \partial v+\frac{1}{2}v\partial \\ 2v\partial +\partial v& \frac{1}{6}{\partial }^{5}-\frac{1}{3}\left({\partial }^{3}u+u{\partial }^{3}\right)-\frac{1}{2}\left({\partial }^{2}u\partial +\partial u{\partial }^{2}\right)+{u}^{2}\partial +\partial {u}^{2}+\frac{2}{3}u\partial u\end{array}\right),\hfill \\ J=\left(\begin{array}{cc}0& -\frac{3}{2}\partial \\ -3\partial & 0\end{array}\right).\hfill \end{array}$

Hence ${g}_{j}$ and ${\stackrel{ˆ}{g}}_{j}$ are uniquely determined, for example, the first two members read as

${g}_{0}=-\frac{1}{3}\left(\begin{array}{c}v\\ 2u\end{array}\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\stackrel{ˆ}{g}}_{0}=\frac{1}{9}\left(\begin{array}{c}{u}_{xx}-4{u}^{2}\\ -6v\end{array}\right).$

In order to generate a hierarchy of evolution equations associated with the spectral problem (2.1), we solve the stationary zero-curvature equation

${V}_{x}-\left[U,V\right]=0,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}V={\left({V}_{ij}\right)}_{3×3},$
(2.4)

which is equivalent to

$\begin{array}{r}{V}_{11,x}+u{V}_{12}+\left(v+\lambda \right){V}_{13}-{V}_{21}=0,\\ {V}_{12,x}+u{V}_{13}+{V}_{11}-{V}_{22}=0,\\ {V}_{13,x}-{V}_{23}+{V}_{12}=0,\\ {V}_{21,x}+u\left({V}_{22}-{V}_{11}\right)+\left(v+\lambda \right){V}_{23}-{V}_{31}=0,\\ {V}_{22,x}+u\left({V}_{23}-{V}_{12}\right)+{V}_{21}-{V}_{32}=0,\\ {V}_{23,x}-u{V}_{13}+{V}_{22}-{V}_{33}=0,\\ {V}_{31,x}+u\left({V}_{32}-{V}_{21}\right)+\left(v+\lambda \right)\left({V}_{33}-{V}_{11}\right)=0,\\ {V}_{32,x}+u\left({V}_{33}-{V}_{22}\right)-\left(v+\lambda \right){V}_{12}+{V}_{31}=0,\\ {V}_{33,x}-u{V}_{23}-\left(v+\lambda \right){V}_{13}+{V}_{32}=0,\end{array}$
(2.5)

where each entry ${V}_{ij}={V}_{ij}\left(a,b\right)$ is a Laurent expansion in λ:

$\begin{array}{c}{V}_{11}=\frac{1}{3}\left(\frac{1}{2}{\partial }^{2}-u\right)b-\partial a,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{V}_{12}=a-\frac{1}{2}\partial b,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{V}_{13}=b,\hfill \\ {V}_{21}=\left(\frac{1}{6}{\partial }^{3}-\frac{1}{3}\partial u-\frac{1}{2}u\partial +v+\lambda \right)b+\left(u-{\partial }^{2}\right)a,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{V}_{22}=\frac{1}{3}\left(-{\partial }^{2}+2u\right)b,\hfill \\ {V}_{23}=a+\frac{1}{2}\partial b,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{V}_{31}=\left(\frac{1}{6}{\partial }^{4}-\frac{1}{3}{\partial }^{2}u-\frac{1}{2}\partial u\partial -\frac{1}{2}u{\partial }^{2}+{u}^{2}\right)b+\left(v+\lambda \right)a,\hfill \end{array}$
(2.6)
$\begin{array}{c}{V}_{32}=\left(-\frac{1}{6}{\partial }^{3}+\frac{1}{3}\partial u+\frac{1}{2}u\partial +v+\lambda \right)b+\left(u-{\partial }^{2}\right)a,\hfill \\ {V}_{33}=\frac{1}{3}\left(\frac{1}{2}{\partial }^{2}-u\right)b+\partial a,\hfill \\ a=\sum _{j\ge 0}{a}_{j-1}{\lambda }^{-j},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}b=\sum _{j\ge 0}{b}_{j-1}{\lambda }^{-j}.\hfill \end{array}$
(2.7)

A direct calculation shows that (2.5) and (2.6) imply the Lenard equation

$KG=\lambda JG,\phantom{\rule{1em}{0ex}}G={\left(a,b\right)}^{T}.$
(2.8)

Substituting (2.7) into (2.8) and collecting terms with the same powers of λ, we arrive at the following recursion relation:

$K{G}_{j-1}=J{G}_{j},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}J{G}_{-1}=0,\phantom{\rule{1em}{0ex}}j\ge 0,$
(2.9)

where ${G}_{j}={\left({a}_{j},{b}_{j}\right)}^{T}$. Since the equation $J{G}_{-1}=0$ has the general solution

${G}_{-1}={\alpha }_{0}{g}_{-1}+{\beta }_{0}{\stackrel{ˆ}{g}}_{-1},$
(2.10)

then ${G}_{j}$ can be expressed as

${G}_{j}={\alpha }_{0}{g}_{j}+{\beta }_{0}{\stackrel{ˆ}{g}}_{j}+\cdots +{\alpha }_{j}{g}_{0}+{\beta }_{j}{\stackrel{ˆ}{g}}_{0}+{\alpha }_{j+1}{g}_{-1}+{\beta }_{j+1}{\stackrel{ˆ}{g}}_{-1},\phantom{\rule{1em}{0ex}}j\ge 0,$
(2.11)

where ${\alpha }_{j}$ and ${\beta }_{j}$ are arbitrary constants.

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

${\psi }_{{t}_{r}}={\stackrel{˜}{V}}^{\left(r\right)}\psi ,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\stackrel{˜}{V}}^{\left(r\right)}={\left({\stackrel{˜}{V}}_{ij}^{\left(r\right)}\right)}_{3×3},$
(2.12)

where each entry ${\stackrel{˜}{V}}_{ij}^{\left(r\right)}={\stackrel{˜}{V}}_{ij}\left({\stackrel{˜}{a}}^{\left(r\right)},{\stackrel{˜}{b}}^{\left(r\right)}\right)$,

${\stackrel{˜}{a}}^{\left(r\right)}=\sum _{j=0}^{r}{\stackrel{˜}{a}}_{j-1}{\lambda }^{r-j},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\stackrel{˜}{b}}^{\left(r\right)}=\sum _{j=0}^{r}{\stackrel{˜}{a}}_{j-1}{\lambda }^{r-j}$

with

${\stackrel{˜}{G}}_{j}={\left({\stackrel{˜}{a}}_{j},{\stackrel{˜}{b}}_{j}\right)}^{T}={\stackrel{˜}{\alpha }}_{0}{g}_{j}+{\stackrel{˜}{\beta }}_{0}{\stackrel{ˆ}{g}}_{j}+\cdots +{\stackrel{˜}{\alpha }}_{j}{g}_{0}+{\stackrel{˜}{\beta }}_{j}{\stackrel{ˆ}{g}}_{0}+{\stackrel{˜}{\alpha }}_{j+1}{g}_{-1}+{\stackrel{˜}{\beta }}_{j+1}{\stackrel{ˆ}{g}}_{-1},\phantom{\rule{1em}{0ex}}j\ge -1.$

Then the compatibility condition of (2.1) and (2.12) yields the zero-curvature equation, ${U}_{{t}_{r}}-{\stackrel{˜}{V}}_{x}^{\left(r\right)}+\left[U,{\stackrel{˜}{V}}^{\left(r\right)}\right]=0$, which is equivalent to the hierarchy of nonlinear evolution equations

${\left({u}_{{t}_{r}},{v}_{{t}_{r}}\right)}^{T}={\stackrel{˜}{X}}_{r},\phantom{\rule{1em}{0ex}}r\ge 0,$
(2.13)

where the vector fields ${\stackrel{˜}{X}}_{j}={\stackrel{˜}{X}}_{j}\left(u,v;{\stackrel{˜}{\underline{\alpha }}}^{\left(j\right)},{\stackrel{˜}{\underline{\beta }}}^{\left(j\right)}\right)=K{\stackrel{˜}{G}}_{j-1}=J{\stackrel{˜}{G}}_{j}$, and ${\stackrel{˜}{\underline{\alpha }}}^{\left(j\right)}=\left({\stackrel{˜}{\alpha }}_{0},\dots ,{\stackrel{˜}{\alpha }}_{j}\right)$, ${\stackrel{˜}{\underline{\beta }}}^{\left(j\right)}=\left({\stackrel{˜}{\beta }}_{0},\dots ,{\stackrel{˜}{\beta }}_{j}\right)$. The first nontrivial member in the hierarchy (2.13) is as follows:

$\begin{array}{r}{u}_{{t}_{0}}={\stackrel{˜}{\alpha }}_{0}{u}_{x}+{\stackrel{˜}{\beta }}_{0}{v}_{x},\\ {v}_{{t}_{0}}={\stackrel{˜}{\alpha }}_{0}{v}_{x}-\frac{1}{3}{\stackrel{˜}{\beta }}_{0}\left({u}_{xxx}-8u{u}_{x}\right).\end{array}$
(2.14)

For ${\stackrel{˜}{\alpha }}_{0}=0$, ${\stackrel{˜}{\beta }}_{0}=1$ (${t}_{0}=t$), equation (2.14) is reduced to the second-order Benjamin-Ono equation by canceling the variable v

${u}_{tt}=\frac{4}{3}{\left({u}^{2}\right)}_{xx}-\frac{1}{3}{u}_{xxxx}.$
(2.15)

The second one in the hierarchy (2.13) (as ${\stackrel{˜}{\alpha }}_{1}=0$, ${\stackrel{˜}{\beta }}_{1}=0$) can be written as

$\begin{array}{c}{u}_{{t}_{1}}=\frac{1}{3}{\stackrel{˜}{\alpha }}_{0}{\left({v}_{xx}-4uv\right)}_{x}-\frac{1}{54}{\stackrel{˜}{\beta }}_{0}{\left(6{u}_{xxxx}-60u{u}_{xx}-45{u}_{x}^{2}+40{u}^{3}+45{v}^{2}\right)}_{x},\hfill \\ {v}_{{t}_{1}}=-\frac{1}{27}{\stackrel{˜}{\alpha }}_{0}{\left(3{u}_{xxxx}-36u{u}_{xx}-18{u}_{x}^{2}+32{u}^{3}+18{v}^{2}\right)}_{x}\hfill \\ \phantom{{v}_{{t}_{1}}=}-\frac{1}{9}{\stackrel{˜}{\beta }}_{0}{\left({v}_{xxxx}-5{u}_{xx}v-10u{v}_{xx}-5{u}_{x}{v}_{x}+20{u}^{2}v\right)}_{x}.\hfill \end{array}$
(2.16)

For ${\stackrel{˜}{\alpha }}_{0}=0$, ${\stackrel{˜}{\beta }}_{0}=-9$ (${t}_{1}=t$), equation (2.16) is reduced to a 5-order coupled equation

$\begin{array}{r}{u}_{t}={u}_{xxxxx}-{\left(10u{u}_{xx}+9{u}_{x}^{2}-9{v}^{2}-\frac{20}{3}{u}^{3}\right)}_{x},\\ {v}_{t}={v}_{xxxxx}-{\left(5{u}_{xx}v+10u{v}_{xx}+5{u}_{x}{v}_{x}-20{u}^{2}v\right)}_{x}.\end{array}$
(2.17)

## 3 The meromorphic function and Dubrovin-type equations

In this section, we shall consider the Baker-Akhiezer function and the associated meromorphic function. By introducing the elliptic kind coordinates, we decompose the second-order Benjamin-Ono equation into the system of Dubrovin-type differential equations.

We first introduce the Baker-Akhiezer function $\psi \left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)$ by

$\begin{array}{r}{\psi }_{x}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)=U\left(u\left(x,{t}_{r}\right),v\left(x,{t}_{r}\right);\lambda \left(P\right)\right)\psi \left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right),\\ {\psi }_{{t}_{r}}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)={\stackrel{˜}{V}}^{\left(r\right)}\left(u\left(x,{t}_{r}\right),v\left(x,{t}_{r}\right);\lambda \left(P\right)\right)\psi \left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right),\\ {V}^{\left(n\right)}\left(u\left(x,{t}_{r}\right),v\left(x,{t}_{r}\right);\lambda \left(P\right)\right)\psi \left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)=y\left(P\right)\psi \left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right),\\ {\psi }_{1}\left(P,{x}_{0},{x}_{0},{t}_{0,r},{t}_{0,r}\right)=1,\end{array}$
(3.1)

where ${V}^{\left(n\right)}={\left({\lambda }^{n}V\right)}_{+}={\left({V}_{ij}^{\left(n\right)}\right)}_{3×3}$ and ${V}_{ij}^{\left(n\right)}={V}_{ij}\left({a}^{\left(n\right)},{b}^{\left(n\right)}\right)$,

${a}^{\left(n\right)}=\sum _{j=0}^{n}{a}_{j-1}{\lambda }^{n-j},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{b}^{\left(n\right)}=\sum _{j=0}^{n}{b}_{j-1}{\lambda }^{n-j}$

with ${a}_{j}$, ${b}_{j}$ determined by (2.11). The compatibility conditions of the first three expressions in (3.1) yield that

${U}_{{t}_{r}}-{\stackrel{˜}{V}}_{x}^{\left(r\right)}+\left[U,{\stackrel{˜}{V}}^{\left(r\right)}\right]=0,$
(3.2)
$-{V}_{x}^{\left(n\right)}+\left[U,{V}^{\left(n\right)}\right]=0,$
(3.3)
$-{V}_{{t}_{r}}^{\left(n\right)}+\left[{\stackrel{˜}{V}}^{\left(r\right)},{V}^{\left(n\right)}\right]=0.$
(3.4)

Through a direct calculation we can show that $yI-{V}^{\left(n\right)}$ satisfies equations (3.3) and (3.4). So is an independent constant of the variables x and ${t}_{r}$, from which we can define a trigonal curve ${\mathcal{K}}_{m-1}:{\mathcal{F}}_{m}\left(\lambda ,y\right)=0$ with the expansion

(3.5)

where

${S}_{m}=\sum _{1\le i

Immediately, from (2.10) if we choose ${\beta }_{0}=1$, ${\alpha }_{0}$ an arbitrary constant or ${\beta }_{0}=0$, ${\alpha }_{0}=1$, we shall know that the corresponding values of m in (3.5) are $3n+2$ or $3n+1$, respectively. For the convenience, the compactification of the curve ${\mathcal{K}}_{m-1}$ is denoted by the same symbol ${\mathcal{K}}_{m-1}$. Thus ${\mathcal{K}}_{m-1}$ becomes a three-sheeted Riemann surface of arithmetic genus $m-1$ when it is nonsingular or smooth.

Next we shall introduce the meromorphic function ${\varphi }_{1}\left(P,x,{t}_{r}\right)$, which is closely related to $\psi \left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)$, by

${\varphi }_{1}\left(P,x,{t}_{r}\right)=\frac{{\partial }_{x}{\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)}{{\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)},\phantom{\rule{1em}{0ex}}P\in {\mathcal{K}}_{m-1},x\in \mathbb{C},$
(3.6)

which implies from (3.1) that

$\begin{array}{rcl}{\varphi }_{1}\left(P,x,{t}_{r}\right)& =& \frac{\epsilon \left(m\right){F}_{m}\left(\lambda ,x,{t}_{r}\right)}{{y}^{2}{V}_{23}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)-y{C}_{m}\left(\lambda ,x,{t}_{r}\right)+{D}_{m}\left(\lambda ,x,{t}_{r}\right)}\\ =& \frac{{y}^{2}{V}_{13}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)-y{A}_{m}\left(\lambda ,x,{t}_{r}\right)+{B}_{m}\left(\lambda ,x,{t}_{r}\right)}{-\epsilon \left(m\right){E}_{m-1}\left(\lambda ,x,{t}_{r}\right)}\\ =& \frac{y{V}_{23}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)+{C}_{m}\left(\lambda ,x,{t}_{r}\right)}{y{V}_{13}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)+{A}_{m}\left(\lambda ,x,{t}_{r}\right)},\end{array}$
(3.7)

where $P=\left(\lambda ,y\right)\in {\mathcal{K}}_{m-1}$, $\left(x,{t}_{r}\right)\in {\mathbb{C}}^{2}$,

$\begin{array}{r}{A}_{m}={V}_{12}^{\left(n\right)}{V}_{23}^{\left(n\right)}-{V}_{13}^{\left(n\right)}{V}_{22}^{\left(n\right)},\\ {B}_{m}={V}_{13}^{\left(n\right)}\left({V}_{11}^{\left(n\right)}{V}_{33}^{\left(n\right)}-{V}_{13}^{\left(n\right)}{V}_{31}^{\left(n\right)}\right)+{V}_{12}^{\left(n\right)}\left({V}_{11}^{\left(n\right)}{V}_{23}^{\left(n\right)}-{V}_{13}^{\left(n\right)}{V}_{21}^{\left(n\right)}\right),\\ {C}_{m}={V}_{13}^{\left(n\right)}{V}_{21}^{\left(n\right)}-{V}_{11}^{\left(n\right)}{V}_{23}^{\left(n\right)},\\ {D}_{m}={V}_{23}^{\left(n\right)}\left({V}_{22}^{\left(n\right)}{V}_{33}^{\left(n\right)}-{V}_{23}^{\left(n\right)}{V}_{32}^{\left(n\right)}\right)+{V}_{21}^{\left(n\right)}\left({V}_{13}^{\left(n\right)}{V}_{22}^{\left(n\right)}-{V}_{12}^{\left(n\right)}{V}_{23}^{\left(n\right)}\right),\end{array}$
(3.8)
$\begin{array}{r}{E}_{m-1}=-\epsilon \left(m\right)\left[{V}_{13}^{\left(n\right)}\left({V}_{13}^{\left(n\right)}{V}_{32}^{\left(n\right)}-{V}_{12}^{\left(n\right)}{V}_{33}^{\left(n\right)}\right)+{V}_{12}^{\left(n\right)}\left({V}_{13}^{\left(n\right)}{V}_{22}^{\left(n\right)}-{V}_{12}^{\left(n\right)}{V}_{23}^{\left(n\right)}\right)\right],\\ {F}_{m}=\epsilon \left(m\right)\left[{V}_{23}^{\left(n\right)}\left({V}_{23}^{\left(n\right)}{V}_{31}^{\left(n\right)}-{V}_{21}^{\left(n\right)}{V}_{33}^{\left(n\right)}\right)+{V}_{21}^{\left(n\right)}\left({V}_{11}^{\left(n\right)}{V}_{23}^{\left(n\right)}-{V}_{13}^{\left(n\right)}{V}_{21}^{\left(n\right)}\right)\right],\end{array}$
(3.9)

and

which is introduced to ensure that ${E}_{m-1}$, ${F}_{m}$ are both monic polynomials. It is easy to see that there exist various interrelationships between polynomials ${A}_{m}$, ${B}_{m}$, ${C}_{m}$, ${D}_{m}$, ${E}_{m-1}$, ${F}_{m}$ and ${S}_{m}$, ${T}_{m}$, some of which are summarized as follows:

$\begin{array}{r}\epsilon \left(m\right){V}_{13}^{\left(n\right)}{F}_{m}={V}_{23}^{\left(n\right)}{D}_{m}-{S}_{m}{\left({V}_{23}^{\left(n\right)}\right)}^{2}-{C}_{m}^{2},\\ \epsilon \left(m\right){A}_{m}{F}_{m}={T}_{m}{\left({V}_{23}^{\left(n\right)}\right)}^{2}+{C}_{m}{D}_{m},\\ \epsilon \left(m\right){V}_{23}^{\left(n\right)}{E}_{m-1}={S}_{m}{\left({V}_{13}^{\left(n\right)}\right)}^{2}-{V}_{13}^{\left(n\right)}{B}_{m}+{A}_{m}^{2},\\ -\epsilon \left(m\right){C}_{m}{E}_{m-1}={T}_{m}{\left({V}_{13}^{\left(n\right)}\right)}^{2}+{A}_{m}{B}_{m},\end{array}$
(3.10)
$\begin{array}{r}{V}_{23}^{\left(n\right)}{B}_{m}+{V}_{13}^{\left(n\right)}{D}_{m}-{V}_{13}^{\left(n\right)}{V}_{23}^{\left(n\right)}{S}_{m}+{A}_{m}{C}_{m}=0,\\ {V}_{13}^{\left(n\right)}{V}_{23}^{\left(n\right)}{T}_{m}+{V}_{23}^{\left(n\right)}{A}_{m}{S}_{m}+{V}_{13}^{\left(n\right)}{C}_{m}{S}_{m}-{B}_{m}{C}_{m}-{A}_{m}{D}_{m}=0,\\ {V}_{23}^{\left(n\right)}{A}_{m}{T}_{m}+{V}_{13}^{\left(n\right)}{C}_{m}{T}_{m}-{E}_{m-1}{F}_{m}-{B}_{m}{D}_{m}=0,\end{array}$
(3.11)
$\begin{array}{r}\epsilon \left(m\right){E}_{m-1,x}=2{S}_{m}{V}_{13}^{\left(n\right)}-3{B}_{m},\\ {V}_{23}^{\left(n\right)}{F}_{m,x}=-3{V}_{22}^{\left(n\right)}{F}_{m}+\epsilon \left(m\right)\left({V}_{21}^{\left(n\right)}-u{V}_{23}^{\left(n\right)}\right)\left(2{V}_{23}^{\left(n\right)}{S}_{m}-3{D}_{m}\right).\end{array}$
(3.12)

For displaying the properties of ${\varphi }_{1}\left(P,x,{t}_{r}\right)$ exactly, we introduce the holomorphic map , changing sheets, as

$\begin{array}{c}\ast :\left\{\begin{array}{c}{\mathcal{K}}_{m-1}\to {\mathcal{K}}_{m-1},\hfill \\ P=\left(\lambda ,{y}_{i}\left(\lambda \right)\right)\to {P}^{\ast }=\left(\lambda ,{y}_{i+1\left(mod3\right)}\left(\lambda \right)\right),\phantom{\rule{1em}{0ex}}i=0,1,2,\hfill \end{array}\hfill \\ {P}^{\ast \ast }:={\left({P}^{\ast }\right)}^{\ast },\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{etc.,}\hfill \end{array}$

where ${y}_{i}\left(\lambda \right)$, $i=0,1,2$, denote the three branches of $y\left(P\right)$ satisfying ${\mathcal{F}}_{m}\left(\lambda ,y\right)=0$. Then it is easy to show the properties of ${\varphi }_{1}\left(P,x,{t}_{r}\right)$ immediately:

$\begin{array}{c}{\varphi }_{1,xx}\left(P,x,{t}_{r}\right)+3{\varphi }_{1}\left(P,x,{t}_{r}\right){\varphi }_{1,x}\left(P,x,{t}_{r}\right)+{\varphi }_{1}^{3}\left(P,x,{t}_{r}\right)-2u\left(x,{t}_{r}\right){\varphi }_{1}\left(P,x,{t}_{r}\right)\hfill \\ \phantom{\rule{1em}{0ex}}={u}_{x}\left(x,{t}_{r}\right)+v\left(x,{t}_{r}\right)+\lambda ,\hfill \end{array}$
(3.13)
$\begin{array}{c}{\varphi }_{1,{t}_{r}}\left(P,x,{t}_{r}\right)={\partial }_{x}\left[{\stackrel{˜}{V}}_{11}^{\left(r\right)}\left(\lambda ,x,{t}_{r}\right)+{\stackrel{˜}{V}}_{12}^{\left(r\right)}\left(\lambda ,x,{t}_{r}\right){\varphi }_{1}\left(P,x,{t}_{r}\right)\hfill \\ \phantom{{\varphi }_{1,{t}_{r}}\left(P,x,{t}_{r}\right)=}+{\stackrel{˜}{V}}_{13}^{\left(r\right)}\left(\lambda ,x,{t}_{r}\right)\left({\varphi }_{1,x}\left(P,x,{t}_{r}\right)+{\varphi }_{1}^{2}\left(P,x,{t}_{r}\right)-u\left(x,{t}_{r}\right)\right)\right],\hfill \end{array}$
(3.14)
${\varphi }_{1}\left(P,x,{t}_{r}\right){\varphi }_{1}\left({P}^{\ast },x,{t}_{r}\right){\varphi }_{1}\left({P}^{\ast \ast },x,{t}_{r}\right)=\frac{{F}_{m}\left(\lambda ,x,{t}_{r}\right)}{{E}_{m-1}\left(\lambda ,x,{t}_{r}\right)},$
(3.15)
${\varphi }_{1}\left(P,x,{t}_{r}\right)+{\varphi }_{1}\left({P}^{\ast },x,{t}_{r}\right)+{\varphi }_{1}\left({P}^{\ast \ast },x,{t}_{r}\right)=\frac{{E}_{m-1,x}\left(\lambda ,x,{t}_{r}\right)}{{E}_{m-1}\left(\lambda ,x,{t}_{r}\right)},$
(3.16)
$\begin{array}{c}y\left(P\right){\varphi }_{1}\left(P,x,{t}_{r}\right)+y\left({P}^{\ast }\right){\varphi }_{1}\left({P}^{\ast },x,{t}_{r}\right)+y\left({P}^{\ast \ast }\right){\varphi }_{1}\left({P}^{\ast \ast },x,{t}_{r}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{3{T}_{m}\left(\lambda \right){V}_{32}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)+2{S}_{m}\left(\lambda \right){A}_{m}\left(\lambda ,x,{t}_{r}\right)}{-\epsilon \left(m\right){E}_{m-1}\left(\lambda ,x,{t}_{r}\right)},\hfill \end{array}$
(3.17)
$\begin{array}{c}\frac{1}{{\varphi }_{1}\left(P,x,{t}_{r}\right)}+\frac{1}{{\varphi }_{1}\left({P}^{\ast },x,{t}_{r}\right)}+\frac{1}{{\varphi }_{1}\left({P}^{\ast \ast },x,{t}_{r}\right)}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{-3{V}_{22}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)}{{V}_{21}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)-u\left(x,{t}_{r}\right){V}_{23}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\frac{{V}_{23}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)}{{V}_{21}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)-u\left(x,{t}_{r}\right){V}_{23}^{\left(n\right)}\left(\lambda ,x,{t}_{r}\right)}\frac{{F}_{m,x}\left(\lambda ,x,{t}_{r}\right)}{{F}_{m}\left(\lambda ,x,{t}_{r}\right)}.\hfill \end{array}$
(3.18)

After tedious calculations, we have the following lemma.

Lemma 3.1 Assume (3.1), (3.2), and let $\left(\lambda ,x,{x}_{0},{t}_{r},{t}_{0,r}\right)\in {\mathbb{C}}^{5}$. Then

$\begin{array}{c}{E}_{m-1,{t}_{r}}\left(\lambda ,x,{t}_{r}\right)={E}_{m-1,x}\left({\stackrel{˜}{V}}_{12}^{\left(r\right)}-\frac{{\stackrel{˜}{V}}_{13}^{\left(r\right)}}{{V}_{13}^{\left(n\right)}}{V}_{12}^{\left(n\right)}\right)+3{E}_{m-1}\left({\stackrel{˜}{V}}_{11}^{\left(r\right)}-\frac{{\stackrel{˜}{V}}_{13}^{\left(r\right)}}{{V}_{13}^{\left(n\right)}}{V}_{11}^{\left(n\right)}\right),\hfill \\ {F}_{m,{t}_{r}}\left(\lambda ,x,{t}_{r}\right)={F}_{m,x}\left({\stackrel{˜}{V}}_{23}^{\left(r\right)}-\frac{{\stackrel{˜}{V}}_{21}^{\left(r\right)}-u{\stackrel{˜}{V}}_{23}^{\left(r\right)}}{{V}_{21}^{\left(n\right)}-u{V}_{23}^{\left(n\right)}}{V}_{23}^{\left(n\right)}\right)\hfill \\ \phantom{{F}_{m,{t}_{r}}\left(\lambda ,x,{t}_{r}\right)=}+3{F}_{m}\left({\stackrel{˜}{V}}_{22}^{\left(r\right)}-\frac{{\stackrel{˜}{V}}_{21}^{\left(r\right)}-u{\stackrel{˜}{V}}_{23}^{\left(r\right)}}{{V}_{21}^{\left(n\right)}-u{V}_{23}^{\left(n\right)}}{V}_{22}^{\left(n\right)}\right).\hfill \end{array}$
(3.19)

Moreover, by institute of (3.2), (3.6), (3.16), and (3.19), we arrive at the properties of ${\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)$ immediately.

Lemma 3.2 Assume (3.1), (3.6), $P=\left(\lambda ,y\left(P\right)\right)\in {\mathcal{K}}_{m-1}\setminus \left\{{P}_{\mathrm{\infty }}\right\}$, and let $\left(\lambda ,x,{x}_{0},{t}_{r},{t}_{0,r}\right)\in {\mathbb{C}}^{5}$. Then

$\begin{array}{c}\frac{{\psi }_{1,{t}_{r}}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)}{{\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)}={\stackrel{˜}{V}}_{13}^{\left(r\right)}\left(\lambda ,x,{t}_{r}\right)\left[{\varphi }_{1,x}\left(P,x,{t}_{r}\right)+{\varphi }_{1}^{2}\left(P,x,{t}_{r}\right)-u\left(x,{t}_{r}\right)\right]\hfill \\ \phantom{\frac{{\psi }_{1,{t}_{r}}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)}{{\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)}=}+{\stackrel{˜}{V}}_{12}^{\left(r\right)}\left(\lambda ,x,{t}_{r}\right){\varphi }_{1}\left(P,x,{t}_{r}\right)+{\stackrel{˜}{V}}_{11}^{\left(r\right)}\left(\lambda ,x,{t}_{r}\right),\hfill \end{array}$
(3.20)
${\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right){\psi }_{1}\left({P}^{\ast },x,{x}_{0},{t}_{r},{t}_{0,r}\right){\psi }_{1}\left({P}^{\ast \ast },x,{x}_{0},{t}_{r},{t}_{0,r}\right)=\frac{{E}_{m-1}\left(\lambda ,x,{t}_{r}\right)}{{E}_{m-1}\left(\lambda ,{x}_{0},{t}_{0,r}\right)},$
(3.21)
${\psi }_{1,x}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right){\psi }_{1,x}\left({P}^{\ast },x,{x}_{0},{t}_{r},{t}_{0,r}\right){\psi }_{1,x}\left({P}^{\ast \ast },x,{x}_{0},{t}_{r},{t}_{0,r}\right)=\frac{{F}_{m}\left(\lambda ,x,{t}_{r}\right)}{{E}_{m-1}\left(\lambda ,{x}_{0},{t}_{0,r}\right)},$
(3.22)
$\begin{array}{c}{\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=exp\left({\int }_{{x}_{0}}^{x}{\varphi }_{1}\left(P,{x}^{\prime },{t}_{r}\right)\phantom{\rule{0.2em}{0ex}}d{x}^{\prime }\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\int }_{{t}_{0,r}}^{{t}_{r}}\left[{\stackrel{˜}{V}}_{13}^{\left(r\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)\left(\frac{y\left(P\right)-{V}_{11}^{\left(n\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}{{V}_{13}^{\left(n\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}-\frac{{V}_{12}^{\left(n\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}{{V}_{13}^{\left(n\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}{\varphi }_{1}\left(P,{x}_{0},{t}^{\prime }\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\stackrel{˜}{V}}_{12}^{\left(r\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right){\varphi }_{1}\left(P,{x}_{0},{t}^{\prime }\right)+{\stackrel{˜}{V}}_{11}^{\left(r\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)\right]\phantom{\rule{0.2em}{0ex}}d{t}^{\prime }\right),\hfill \end{array}$
(3.23)
$\begin{array}{c}{\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)\hfill \\ \phantom{\rule{1em}{0ex}}={\left[\frac{{E}_{m-1}\left(\lambda ,x,{t}_{r}\right)}{{E}_{m-1}\left(\lambda ,{x}_{0},{t}_{0,r}\right)}\right]}^{1/3}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×exp\left({\int }_{{x}_{0}}^{x}\frac{y{\left(P\right)}^{2}{V}_{13}^{\left(n\right)}\left(\lambda ,{x}^{\prime },{t}_{r}\right)-y\left(P\right){A}_{m}\left(\lambda ,{x}^{\prime },{t}_{r}\right)+\frac{2}{3}{S}_{m}\left(\lambda \right){V}_{13}^{\left(n\right)}\left(\lambda ,{x}^{\prime },{t}_{r}\right)}{-\epsilon \left(m\right){E}_{m-1}\left(\lambda ,{x}^{\prime },{t}_{r}\right)}\phantom{\rule{0.2em}{0ex}}d{x}^{\prime }\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\int }_{{t}_{0,r}}^{{t}_{r}}\left[\frac{y{\left(P\right)}^{2}{V}_{13}^{\left(n\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)-y\left(P\right){A}_{m}\left(\lambda ,{x}_{0},{t}^{\prime }\right)+\frac{2}{3}{S}_{m}\left(\lambda \right){V}_{13}^{\left(n\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}{-\epsilon \left(m\right){E}_{m-1}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×\left({\stackrel{˜}{V}}_{12}^{\left(r\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)-\frac{{\stackrel{˜}{V}}_{13}^{\left(r\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}{{V}_{13}^{\left(n\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}{V}_{12}^{\left(n\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+y\left(P\right)\frac{{\stackrel{˜}{V}}_{13}^{\left(r\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}{{V}_{13}^{\left(n\right)}\left(\lambda ,{x}_{0},{t}^{\prime }\right)}\right]\phantom{\rule{0.2em}{0ex}}d{t}^{\prime }\right).\hfill \end{array}$
(3.24)

By inspection of (3.9), one shall know that ${E}_{m-1}$ and ${F}_{m}$ are both monic polynomials with respect to λ of degree $m-1$ and m, respectively. Hence we may decompose them into

${E}_{m-1}\left(\lambda ,x,{t}_{r}\right)=\prod _{j=1}^{m-1}\left(\lambda -{\mu }_{j}\left(x,{t}_{r}\right)\right),$
(3.25)
${F}_{m}\left(\lambda ,x,{t}_{r}\right)=\prod _{l=0}^{m-1}\left(\lambda -{\nu }_{l}\left(x,{t}_{r}\right)\right).$
(3.26)

Define

$\begin{array}{c}{\stackrel{ˆ}{\mu }}_{j}\left(x,{t}_{r}\right)=\left({\mu }_{j}\left(x,{t}_{r}\right),y\left({\stackrel{ˆ}{\mu }}_{j}\left(x,{t}_{r}\right)\right)\right)=\left({\mu }_{j}\left(x,{t}_{r}\right),-\frac{{A}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)}{{V}_{13}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)}\right)\in {\mathcal{K}}_{m-1},\hfill \\ \phantom{\rule{1em}{0ex}}1\le j\le m-1,\left(x,{t}_{r}\right)\in {\mathbb{C}}^{2},\hfill \end{array}$
(3.27)
$\begin{array}{c}{\stackrel{ˆ}{\nu }}_{l}\left(x,{t}_{r}\right)=\left({\nu }_{l}\left(x,{t}_{r}\right),y\left({\stackrel{ˆ}{\nu }}_{l}\left(x,{t}_{r}\right)\right)\right)=\left({\nu }_{l}\left(x,{t}_{r}\right),-\frac{{C}_{m}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)}{{V}_{23}^{\left(n\right)}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)}\right)\in {\mathcal{K}}_{m-1},\hfill \\ \phantom{\rule{1em}{0ex}}0\le l\le m-1,\left(x,{t}_{r}\right)\in {\mathbb{C}}^{2}.\hfill \end{array}$
(3.28)

The dynamics of the zeros ${\mu }_{j}\left(x,{t}_{r}\right)$ and ${\nu }_{l}\left(x,{t}_{r}\right)$ of ${E}_{m-1}\left(\lambda ,x,{t}_{r}\right)$ and ${F}_{m}\left(\lambda ,x,{t}_{r}\right)$ are then described in terms of Dubrovin-type equations as follows.

Lemma 3.3 (i) Suppose that the zeros ${\mu }_{j}{\left(x,{t}_{r}\right)}_{j=1,\dots ,m-1}$ of ${E}_{m-1}\left(P,x,{t}_{r}\right)$ remain distinct for $\left(x,{t}_{r}\right)\in {\mathrm{\Omega }}_{\mu }$, where ${\mathrm{\Omega }}_{\mu }\subseteq {\mathbb{C}}^{2}$ is open and connected. Then ${\mu }_{j}{\left(x,{t}_{r}\right)}_{j=1,\dots ,m-1}$ satisfy the system of differential equations

$\begin{array}{c}{\mu }_{j,x}\left(x,{t}_{r}\right)=\frac{\epsilon \left(m\right){V}_{13}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)\left[3{y}^{2}\left({\stackrel{ˆ}{\mu }}_{j}\left(x,{t}_{r}\right)\right)+{S}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right)\right)\right]}{\underset{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.2em}{0ex}}k\ne j}{{\prod }_{k=1}^{m-1}}\left({\mu }_{j}\left(x,{t}_{r}\right)-{\mu }_{k}\left(x,{t}_{r}\right)\right)},\hfill \\ \phantom{\rule{1em}{0ex}}1\le j\le m-1,\hfill \end{array}$
(3.29)
$\begin{array}{c}{\mu }_{j,{t}_{r}}\left(x,{t}_{r}\right)=\left[{V}_{13}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right){\stackrel{˜}{V}}_{12}^{\left(r\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)\hfill \\ \phantom{{\mu }_{j,{t}_{r}}\left(x,{t}_{r}\right)=}-{\stackrel{˜}{V}}_{13}^{\left(r\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right){V}_{12}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)\right]\hfill \\ \phantom{{\mu }_{j,{t}_{r}}\left(x,{t}_{r}\right)=}×\frac{\epsilon \left(m\right)\left[3{y}^{2}\left({\stackrel{ˆ}{\mu }}_{j}\left(x,{t}_{r}\right)\right)+{S}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right)\right)\right]}{\underset{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.2em}{0ex}}k\ne j}{{\prod }_{k=1}^{m-1}}\left({\mu }_{j}\left(x,{t}_{r}\right)-{\mu }_{k}\left(x,{t}_{r}\right)\right)},\phantom{\rule{1em}{0ex}}1\le j\le m-1.\hfill \end{array}$
(3.30)
1. (ii)

Suppose that the zeros ${\nu }_{l}{\left(x,{t}_{r}\right)}_{l=0,\dots ,m-1}$ of ${F}_{m}\left(P,x,{t}_{r}\right)$ remain distinct for $\left(x,{t}_{r}\right)\in {\mathrm{\Omega }}_{\nu }$, where ${\mathrm{\Omega }}_{\nu }\subseteq {\mathbb{C}}^{2}$ is open and connected. Then ${\nu }_{l}{\left(x,{t}_{r}\right)}_{l=0,\dots ,m-1}$ satisfy the system of differential equations

$\begin{array}{c}{\nu }_{l,x}\left(x,{t}_{r}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{\epsilon \left(m\right)\left[{V}_{21}^{\left(n\right)}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)-u{V}_{23}^{\left(n\right)}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)\right]\left[3{y}^{2}\left({\stackrel{ˆ}{\nu }}_{l}\left(x,{t}_{r}\right)\right)+{S}_{m}\left({\nu }_{l}\left(x,{t}_{r}\right)\right)\right]}{\underset{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.2em}{0ex}}k\ne l}{{\prod }_{k=0}^{m-1}}\left({\nu }_{l}\left(x,{t}_{r}\right)-{\nu }_{k}\left(x,{t}_{r}\right)\right)},\hfill \\ \phantom{\rule{1em}{0ex}}0\le l\le m-1,\hfill \end{array}$
(3.31)
$\begin{array}{c}{\nu }_{l,{t}_{r}}\left(x,{t}_{r}\right)=\left[\left({V}_{21}^{\left(n\right)}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)-u{V}_{23}^{\left(n\right)}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)\right){\stackrel{˜}{V}}_{23}^{\left(r\right)}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)\hfill \\ \phantom{{\nu }_{l,{t}_{r}}\left(x,{t}_{r}\right)=}-\left({\stackrel{˜}{V}}_{21}^{\left(r\right)}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)-u{\stackrel{˜}{V}}_{23}^{\left(r\right)}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)\right){V}_{23}^{\left(n\right)}\left({\nu }_{l}\left(x,{t}_{r}\right),x,{t}_{r}\right)\right]\hfill \\ \phantom{{\nu }_{l,{t}_{r}}\left(x,{t}_{r}\right)=}×\frac{\epsilon \left(m\right)\left[3{y}^{2}\left({\stackrel{ˆ}{\nu }}_{l}\left(x,{t}_{r}\right)\right)+{S}_{m}\left({\nu }_{l}\left(x,{t}_{r}\right)\right)\right]}{\underset{\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.2em}{0ex}}k\ne l}{{\prod }_{k=0}^{m-1}}\left({\nu }_{l}\left(x,{t}_{r}\right)-{\nu }_{k}\left(x,{t}_{r}\right)\right)},\phantom{\rule{1em}{0ex}}0\le l\le m-1.\hfill \end{array}$
(3.32)

Proof Using (3.10), we have ($\lambda ={\mu }_{j}\left(x,{t}_{r}\right)$)

$\begin{array}{c}{S}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right)\right){\left({V}_{13}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)\right)}^{2}-{B}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right){V}_{13}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)\hfill \\ \phantom{\rule{1em}{0ex}}+{A}_{m}^{2}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)=0,\hfill \end{array}$
(3.33)

that is,

$\begin{array}{rcl}{B}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)& =& {S}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right)\right){V}_{13}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)+\frac{{A}_{m}^{2}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)}{{V}_{13}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)}\\ =& \left[{S}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right)\right)+{y}^{2}\left({\stackrel{ˆ}{\mu }}_{j}\left(x,{t}_{r}\right)\right)\right]{V}_{13}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right).\end{array}$

After substituting ${B}_{m}$ into (3.12), we get

$\begin{array}{c}\epsilon \left(m\right){E}_{m-1,x}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=-{V}_{13}^{\left(n\right)}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)\left[3{y}^{2}\left({\stackrel{ˆ}{\mu }}_{j}\left(x,{t}_{r}\right)\right)+{S}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right)\right)\right].\hfill \end{array}$
(3.34)

On the other hand, derivatives of the expression in (3.25) with respect to x and ${t}_{r}$ respectively, are

${E}_{m-1,x}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)=-{\mu }_{j,x}\left(x,{t}_{r}\right)\underset{k\ne j}{\overset{m-1}{\prod _{k=1}}}\left({\mu }_{j}\left(x,{t}_{r}\right)-{\mu }_{k}\left(x,{t}_{r}\right)\right),$
(3.35)
${E}_{m-1,{t}_{r}}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)=-{\mu }_{j,{t}_{r}}\left(x,{t}_{r}\right)\underset{k\ne j}{\overset{m-1}{\prod _{k=1}}}\left({\mu }_{j}\left(x,{t}_{r}\right)-{\mu }_{k}\left(x,{t}_{r}\right)\right).$
(3.36)

Comparing (3.34) and (3.35), we can obtain (3.29). From (3.19), one can know

$\begin{array}{c}{E}_{m-1,{t}_{r}}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)\hfill \\ \phantom{\rule{1em}{0ex}}={E}_{m-1,x}\left({\mu }_{j}\left(x,{t}_{r}\right),x,{t}_{r}\right)\frac{{V}_{13}^{\left(n\right)}{\stackrel{˜}{V}}_{12}^{\left(r\right)}-{\stackrel{˜}{V}}_{13}^{\left(r\right)}{V}_{12}^{\left(n\right)}}{{V}_{13}^{\left(n\right)}}\hfill \\ \phantom{\rule{1em}{0ex}}=-{\mu }_{j,x}\left(x,{t}_{r}\right)\underset{k\ne j}{\overset{m-1}{\prod _{k=1}}}\left({\mu }_{j}\left(x,{t}_{r}\right)-{\mu }_{k}\left(x,{t}_{r}\right)\right)\frac{{V}_{13}^{\left(n\right)}{\stackrel{˜}{V}}_{12}^{\left(r\right)}-{\stackrel{˜}{V}}_{13}^{\left(r\right)}{V}_{12}^{\left(n\right)}}{{V}_{13}^{\left(n\right)}}\hfill \\ \phantom{\rule{1em}{0ex}}=-\epsilon \left(m\right)\left[3{y}^{2}\left({\stackrel{ˆ}{\mu }}_{j}\left(x,{t}_{r}\right)\right)+{S}_{m}\left({\mu }_{j}\left(x,{t}_{r}\right)\right)\right]\left({V}_{13}^{\left(n\right)}{\stackrel{˜}{V}}_{12}^{\left(r\right)}-{\stackrel{˜}{V}}_{13}^{\left(r\right)}{V}_{12}^{\left(n\right)}\right),\hfill \end{array}$
(3.37)

then we have (3.30). Similarly, we can prove (3.31) and (3.32). □

## 4 Algebro-geometric solutions to the second-order Benjamin-Ono hierarchy

In our final and principal section, we obtain Riemann theta function representations for the Baker-Akhiezer function and the meromorphic function; especially, the theta function representations for general algebro-geometric solutions u, v of the second-order Benjamin-Ono hierarchy. For the convenience, we assume that the curve ${\mathcal{K}}_{m-1}$ is nonsingular.

For investigating the asymptotic expansion of ${\varphi }_{1}\left(P,x,{t}_{r}\right)$ near ${P}_{\mathrm{\infty }}$, we choose the local coordinate $\zeta ={\lambda }^{-\frac{1}{3}}$, then we get the following lemma.

Lemma 4.1 Let $\left(x,{t}_{r}\right)\in {\mathbb{C}}^{2}$, near ${P}_{\mathrm{\infty }}\in {\mathcal{K}}_{m-1}$, we have

(4.1)

where

$\begin{array}{c}{\kappa }_{0}=1,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\kappa }_{1}=0,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\kappa }_{2}=\frac{2}{3}u,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\kappa }_{3}=\frac{1}{3}\left(v-{u}_{x}\right),\hfill \\ {\kappa }_{4}=\frac{1}{9}{u}_{xx}-\frac{1}{3}{v}_{x},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\kappa }_{5}=\frac{2}{9}\left({v}_{xx}-u{u}_{x}-uv\right),\hfill \\ {\kappa }_{j}=-\frac{1}{3}\left[{\kappa }_{j-2,xx}+3\sum _{i=2}^{j-1}{\kappa }_{j-1-i}{\kappa }_{i,x}+\sum _{i=2}^{j-1}{\kappa }_{i}{\kappa }_{j-i}+\sum _{i=2}^{j-1}\sum _{l=0}^{j-i}{\kappa }_{i}{\kappa }_{l}{\kappa }_{j-i-l}-2u{\kappa }_{j-2}\right]\phantom{\rule{1em}{0ex}}\left(j\ge 4\right).\hfill \end{array}$
(4.2)

Proof In terms of the local coordinate $\zeta ={\lambda }^{-\frac{1}{3}}$, (3.13) reads

${\varphi }_{1,xx}+3{\varphi }_{1}{\varphi }_{1,x}+{\varphi }_{1}^{3}-2u{\varphi }_{1}={u}_{x}+v+{\zeta }^{-3}.$
(4.3)

Then, by inserting the power series ansatz of ${\varphi }_{1}\left(P,x,{t}_{r}\right)$ in ζ as follows:

${\varphi }_{1}\left(P,x,{t}_{r}\right)\underset{\zeta \to 0}{=}\frac{1}{\zeta }\sum _{j=0}^{\mathrm{\infty }}{\kappa }_{j}\left(x,{t}_{r}\right){\zeta }^{j}$
(4.4)

into (4.3)

$\begin{array}{c}{\zeta }^{-1}\sum _{j=0}^{\mathrm{\infty }}{\kappa }_{j,xx}{\zeta }^{j}+3{\zeta }^{-2}\sum _{j=0}^{\mathrm{\infty }}\sum _{i=0}^{\mathrm{\infty }}{\kappa }_{j}{\kappa }_{i,x}{\zeta }^{\left(j+i\right)}+{\zeta }^{-3}\sum _{j=0}^{\mathrm{\infty }}\sum _{i=0}^{\mathrm{\infty }}\sum _{l=0}^{\mathrm{\infty }}{\kappa }_{j}{\kappa }_{i}{\kappa }_{l}{\zeta }^{\left(j+i+l\right)}-2u{\zeta }^{-1}\sum _{j=0}^{\mathrm{\infty }}{\kappa }_{j}{\zeta }^{j}\hfill \\ \phantom{\rule{1em}{0ex}}={u}_{x}+v+{\zeta }^{-3},\hfill \end{array}$
(4.5)

and comparing the same powers of ζ in (4.5), we arrive at (4.2). □

One infers, from (3.7), (3.25), (3.26), and (4.1), that the divisor (${\varphi }_{1}\left(P,x,{t}_{r}\right)$) of ${\varphi }_{1}\left(P,x,{t}_{r}\right)$ is given by

$\left({\varphi }_{1}\left(P,x,{t}_{r}\right)\right)={\mathcal{D}}_{{\stackrel{ˆ}{\nu }}_{0}\left(x,{t}_{r}\right),\dots ,{\stackrel{ˆ}{\nu }}_{m-1}\left(x,{t}_{r}\right)}\left(P\right)-{\mathcal{D}}_{{P}_{\mathrm{\infty }},{\stackrel{ˆ}{\mu }}_{1}\left(x,{t}_{r}\right),\dots ,{\stackrel{ˆ}{\mu }}_{m-1}\left(x,{t}_{r}\right)}\left(P\right).$
(4.6)

That is, ${\stackrel{ˆ}{\nu }}_{0}\left(x,{t}_{r}\right),\dots ,{\stackrel{ˆ}{\nu }}_{m-1}\left(x,{t}_{r}\right)$ are the m zeros of ${\varphi }_{1}\left(P,x,{t}_{r}\right)$ and ${P}_{\mathrm{\infty }}$, ${\stackrel{ˆ}{\mu }}_{1}\left(x,{t}_{r}\right),\dots ,{\stackrel{ˆ}{\mu }}_{m-1}\left(x,{t}_{r}\right)$ are its m poles.

A straightforward calculation reveals that the asymptotic behaviors of $y\left(P\right)$ and ${S}_{m}\left(\lambda \right)$ near ${P}_{\mathrm{\infty }}$ are

(4.7)
(4.8)

Next we will introduce the three kinds of holomorphic differentials and show some properties of them. The holomorphic differentials ${\eta }_{l}\left(P\right)$ on ${\mathcal{K}}_{m-1}$ are defined by

${\eta }_{l}\left(P\right)=\frac{1}{3y{\left(P\right)}^{2}+{S}_{m}}\left\{\begin{array}{cc}{\lambda }^{l-1}\phantom{\rule{0.2em}{0ex}}d\lambda ,\hfill & 1\le l\le m-n-1,\hfill \\ y\left(P\right){\lambda }^{l+n-m}\phantom{\rule{0.2em}{0ex}}d\lambda ,\hfill & m-n\le l\le m-1.\hfill \end{array}$
(4.9)

To construct the theta function and normalize the holomorphic differentials, we choose a homology basis on ${\mathcal{K}}_{m-1}$ so that they satisfy

Introducing an invertible matrix $E={\left({E}_{j,k}\right)}_{\left(m-1\right)×\left(m-1\right)}$ and $\underline{e}\left(k\right)=\left({e}_{1}\left(k\right),\dots ,{e}_{m-1}\left(k\right)\right)$, where

and the normalized holomorphic differentials ${\omega }_{j}$ for $j=1,\dots ,m-1$,

(4.10)

Let ${\omega }_{{P}_{\mathrm{\infty }},2}^{\left(2\right)}\left(P\right)$ denote the normalized second Abel differential defined by

${\omega }_{{P}_{\mathrm{\infty }},2}^{\left(2\right)}\left(P\right)=-\sum _{j=1}^{m-1}{z}_{j}{\eta }_{j}\left(P\right)-\frac{1}{3y{\left(P\right)}^{2}+{S}_{m}}\left\{\begin{array}{cc}{\lambda }^{2n}\phantom{\rule{0.2em}{0ex}}d\lambda ,\hfill & m=3n+1,\hfill \\ y\left(P\right){\lambda }^{n}\phantom{\rule{0.2em}{0ex}}d\lambda ,\hfill & m=3n+2,\hfill \end{array}$
(4.11)

which is holomorphic on ${\mathcal{K}}_{m-1}\setminus \left\{{P}_{\mathrm{\infty }}\right\}$ with a pole of order 2 at ${P}_{\mathrm{\infty }}$, and the constants ${\left\{{z}_{j}\right\}}_{j=1,\dots ,m-1}$ are determined by the normalization condition

The -periods of the differential ${\omega }_{{P}_{\mathrm{\infty }},2}^{\left(2\right)}$ are denoted by

(4.12)

On the other hand, ${\omega }_{{P}_{\mathrm{\infty }},3}^{\left(2\right)}\left(P\right)$ denotes the normalized third Abel differential which is holomorphic on ${\mathcal{K}}_{m-1}\setminus \left\{{P}_{\mathrm{\infty }}\right\}$ with a pole of order 3 at ${P}_{\mathrm{\infty }}$

(4.13)

and the -periods of it are defined by

Furthermore, the normalized third Abel differential ${\omega }_{{P}_{\mathrm{\infty }},{\stackrel{ˆ}{\nu }}_{0}\left(x\right)}^{\left(3\right)}\left(P\right)$ is holomorphic on ${\mathcal{K}}_{m-1}\setminus \left\{{P}_{\mathrm{\infty }},{\stackrel{ˆ}{\nu }}_{0}\left(x\right)\right\}$ with simple poles at ${P}_{\mathrm{\infty }}$ and ${\stackrel{ˆ}{\nu }}_{0}\left(x\right)$ with residues ±1, respectively, that is,

(4.14)

Then

(4.15)

with ${e}^{\left(3\right)}\left({P}_{0}\right)$ being an integration constant.

A straightforward Laurent expansion of (4.9), (4.10), and (4.11) near ${P}_{\mathrm{\infty }}$ yields the following results.

Lemma 4.2 Near ${P}_{\mathrm{\infty }}$ in the local coordinate $\zeta ={\lambda }^{-\frac{1}{3}}$, the differentials $\underline{\omega }$ and ${\omega }_{{P}_{\mathrm{\infty }},2}^{\left(2\right)}$ have the Laurent series

$\underline{\omega }=\left({\omega }_{1},\dots ,{\omega }_{m-1}\right)\underset{\zeta \to 0}{=}\left({\underline{\rho }}_{0}+{\underline{\rho }}_{1}\zeta +{\underline{\rho }}_{2}{\zeta }^{3}+O\left({\zeta }^{4}\right)\right)\phantom{\rule{0.2em}{0ex}}d\zeta ,$
(4.16)

with

$\begin{array}{r}{\underline{\rho }}_{0}=\left\{\begin{array}{cc}-\underline{e}\left(m-n-1\right),\hfill & m=3n+2,\hfill \\ -\underline{e}\left(m-1\right),\hfill & m=3n+1,\hfill \end{array}\\ {\underline{\rho }}_{1}=\left\{\begin{array}{cc}-\underline{e}\left(m-1\right)+{\alpha }_{0}\underline{e}\left(m-n-1\right),\hfill & m=3n+2,\hfill \\ -\underline{e}\left(m-n-1\right),\hfill & m=3n+1,\hfill \end{array}\\ {\underline{\rho }}_{2}=\left\{\begin{array}{cc}\left(2{\beta }_{1}-{\alpha }_{0}^{3}\right)\underline{e}\left(m-n-1\right)+{\alpha }_{0}^{2}\underline{e}\left(m-1\right)-\underline{e}\left(m-n-2\right),\hfill & m=3n+2,\hfill \\ {\alpha }_{1}\underline{e}\left(m-1\right)+{\beta }_{1}\underline{e}\left(m-n-1\right)-\underline{e}\left(m-2\right),\hfill & m=3n+1,\hfill \end{array}\\ {\omega }_{{P}_{\mathrm{\infty }},2}^{\left(2\right)}\left(P\right)\underset{\zeta \to 0}{=}\left\{\begin{array}{c}\left({\zeta }^{-2}+{z}_{m-n-1}-{\alpha }_{0}^{2}+\left(-{\beta }_{1}+{\alpha }_{0}^{3}-{\alpha }_{0}{z}_{m-n-1}+{z}_{m-1}\right)\zeta +O\left({\zeta }^{2}\right)\right)\phantom{\rule{0.2em}{0ex}}d\zeta ,\hfill \\ \phantom{\rule{1em}{0ex}}m=3n+2,\hfill \\ \left({\zeta }^{-2}+{z}_{m-1}-{\beta }_{1}+\left({z}_{m-n-1}-2{\alpha }_{1}\right)\zeta +O\left({\zeta }^{2}\right)\right)\phantom{\rule{0.2em}{0ex}}d\zeta ,\hfill \\ \phantom{\rule{1em}{0ex}}m=3n+1.\hfill \end{array}\end{array}$
(4.17)

From Lemma 4.2 we infer

(4.18)

where ${e}_{2}^{\left(2\right)}\left({P}_{0}\right)$ is an appropriate constant, and

$\begin{array}{r}{q}_{1}=\left\{\begin{array}{cc}-{z}_{m-n-1}+{\alpha }_{0}^{2},\hfill & m=3n+2,\hfill \\ -{z}_{m-1}+{\beta }_{1},\hfill & m=3n+1,\hfill \end{array}\\ {q}_{2}=\left\{\begin{array}{cc}\frac{1}{2}\left(-{\beta }_{1}+{\alpha }_{0}^{3}-{\alpha }_{0}{z}_{m-n-1}+{z}_{m-1}\right),\hfill & m=3n+2,\hfill \\ \frac{1}{2}{z}_{m-n-1}-{\alpha }_{1},\hfill & m=3n+1.\hfill \end{array}\end{array}$
(4.19)

Let $\theta \left(\underline{\lambda }\right)$ denote the Riemann theta function [2022] associated with ${\mathcal{K}}_{m-1}$ and the appropriately fixed homology basis . Next we choose a convenient base point ${P}_{0}\in {\mathcal{K}}_{m-1}\setminus \left\{{P}_{\mathrm{\infty }}\right\}$. For brevity, define the function $\underline{\lambda }:{\mathcal{K}}_{m-1}×{\sigma }^{m-1}{\mathcal{K}}_{m-1}\to \mathbb{C}$ by

$\begin{array}{c}\underline{\lambda }\left(P,\underline{Q}\right)={\underline{\mathrm{\Xi }}}_{{P}_{0}}-{\underline{A}}_{{P}_{0}}\left(P\right)+{\underline{\alpha }}_{{P}_{0}}\left({\mathcal{D}}_{\underline{Q}}\right),\phantom{\rule{1em}{0ex}}P\in {\mathcal{K}}_{m-1},\hfill \\ \underline{Q}=\left({Q}_{1},\dots ,{Q}_{m-1}\right)\in {\sigma }^{m-1}{\mathcal{K}}_{m-1},\hfill \end{array}$

where ${\underline{\mathrm{\Xi }}}_{{P}_{0}}$ is the vector of Riemann constants, and the Abel maps ${\underline{A}}_{{P}_{0}}\left(P\right)$ and ${\underline{\alpha }}_{{P}_{0}}\left(P\right)$ are defined by (period lattice ${L}_{m-1}=\left\{\underline{z}\in {\mathbb{C}}^{m-1}|\underline{z}=\underline{N}+\tau \underline{M},\underline{N},\underline{M}\in {\mathbb{Z}}^{m-1}\right\}$)

$\begin{array}{c}{\underline{A}}_{{P}_{0}}:{\mathcal{K}}_{m-1}\to \mathcal{J}\left({\mathcal{K}}_{m-1}\right)={\mathbb{C}}^{m-1}/{L}_{m-1},\hfill \\ P↦{\underline{A}}_{{P}_{0}}\left(P\right)=\left({A}_{{P}_{0},1}\left(P\right),\dots ,{A}_{{P}_{0},m-1}\left(P\right)\right)=\left({\int }_{{P}_{0}}^{P}{\omega }_{1},\dots ,{\int }_{{P}_{0}}^{P}{\omega }_{m-1}\right)\left(mod{L}_{m-1}\right),\hfill \end{array}$

and

$\begin{array}{c}{\underline{\alpha }}_{{P}_{0}}:\text{Div}\left({\mathcal{K}}_{m-1}\right)\to \mathcal{J}\left({\mathcal{K}}_{m-1}\right),\hfill \\ \mathcal{D}↦{\underline{\alpha }}_{{P}_{0}}\left(\mathcal{D}\right)=\sum _{P\in {\mathcal{K}}_{m-1}}\mathcal{D}\left(P\right){\underline{A}}_{{P}_{0}}\left(P\right).\hfill \end{array}$

In view of these preparations, we give the theta function representation of our fundamental object ${\varphi }_{1}\left(P,x,{t}_{r}\right)$.

Theorem 4.3 Let $P=\left(\lambda ,y\right)\in {\mathcal{K}}_{m-1}\setminus \left\{{P}_{\mathrm{\infty }}\right\}$, and let $\left(x,{t}_{r}\right),\left({x}_{0},{t}_{0,r}\right)\in {\mathrm{\Omega }}_{\mu }$, where ${\mathrm{\Omega }}_{\mu }\subseteq {\mathbb{C}}^{2}$ is open and connected. Suppose also that ${\mathcal{D}}_{\stackrel{ˆ}{\underline{\mu }}\left(x,{t}_{r}\right)}$, or equivalently, ${\mathcal{D}}_{\stackrel{ˆ}{\underline{\nu }}\left(x,{t}_{r}\right)}$ is nonspecial for $\left(x,{t}_{r}\right)\in {\mathrm{\Omega }}_{\mu }$. Then

${\varphi }_{1}\left(P,x,{t}_{r}\right)=\frac{\theta \left(\underline{\lambda }\left(P,\stackrel{ˆ}{\underline{\nu }}\left(x,{t}_{r}\right)\right)\right)\theta \left(\underline{\lambda }\left({P}_{\mathrm{\infty }},\stackrel{ˆ}{\underline{\mu }}\left(x,{t}_{r}\right)\right)\right)}{\theta \left(\underline{\lambda }\left({P}_{\mathrm{\infty }},\stackrel{ˆ}{\underline{\nu }}\left(x,{t}_{r}\right)\right)\right)\theta \left(\underline{\lambda }\left(P,\stackrel{ˆ}{\underline{\mu }}\left(x,{t}_{r}\right)\right)\right)}exp\left({e}^{\left(3\right)}\left({P}_{0}\right)-{\int }_{{P}_{0}}^{P}{\omega }_{{P}_{\mathrm{\infty }},{\stackrel{ˆ}{\nu }}_{0}\left(x,{t}_{r}\right)}^{\left(3\right)}\right).$
(4.20)

Proof Let Φ denote the right-hand side of (4.20). From (4.15) it follows that

$exp\left({e}^{\left(3\right)}\left({P}_{0}\right)-{\int }_{{P}_{0}}^{P}{\omega }_{{P}_{\mathrm{\infty }},{\stackrel{ˆ}{\nu }}_{0}\left(x,{t}_{r}\right)}^{\left(3\right)}\right)\underset{\zeta \to 0}{=}{\zeta }^{-1}+O\left(1\right).$
(4.21)

Using (4.6) we immediately know that ${\varphi }_{1}$ has simple poles at $\underline{\stackrel{ˆ}{\mu }}\left(x,{t}_{r}\right)$ and ${P}_{\mathrm{\infty }}$, and simple zeros at ${\stackrel{ˆ}{\nu }}_{0}\left(x,{t}_{r}\right)$, $\underline{\stackrel{ˆ}{\nu }}\left(x,{t}_{r}\right)$. By (4.20) and the Riemann vanishing theorem, we see that Φ has the same properties. Using the Riemann-Roch theorem [21, 22], we conclude that the holomorphic function $\frac{\mathrm{\Phi }}{{\varphi }_{1}}=\gamma$, where γ is a constant. Using (4.21) and Lemma 4.1, we have

(4.22)

from which we conclude $\gamma =1$. □

Let ${\omega }_{{P}_{\mathrm{\infty }},s}^{\left(2\right)}$, $s=3r+2$ (or $3r+1$), $r\in {\mathbb{N}}_{0}$, be the normalized differential of the second kind holomorphic on ${\mathcal{K}}_{m-1}\setminus \left\{{P}_{\mathrm{\infty }}\right\}$, with a pole of order s at ${P}_{\mathrm{\infty }}$,

Then we define the normalized differentials as

(4.23)

where

$\left({\stackrel{˜}{\alpha }}_{0},{\stackrel{˜}{\beta }}_{0}\right)=\left\{\begin{array}{cc}\left({\stackrel{˜}{\alpha }}_{0},1\right),\hfill & s=3r+2,\hfill \\ \left(1,0\right),\hfill & s=3r+1,\hfill \end{array}\phantom{\rule{1em}{0ex}}{\stackrel{˜}{\alpha }}_{0}\in \mathbb{C}.$

In addition, we define the vector of -periods of them as

(4.24)

Motivated by the second integration in (3.23), one defines the function ${I}_{s}\left(P,x,{t}_{r}\right)$, meromorphic on ${\mathcal{K}}_{m-1}×{\mathbb{C}}^{2}$, by

$\begin{array}{rcl}{I}_{s}\left(P,x,{t}_{r}\right)& =& {\stackrel{˜}{V}}_{11}^{\left(r\right)}\left(\lambda ,x,{t}_{r}\right)+{\stackrel{˜}{V}}_{12}^{\left(r\right)}\left(\lambda ,x,{t}_{r}\right){\varphi }_{1}\left(P,x,{t}_{r}\right)+{\stackrel{˜}{V}}_{13}^{\left(r\right)}\left(\lambda ,x,{t}_{r}\right)\left({\varphi }_{1,x}\left(P,x,{t}_{r}\right)\\ +{\varphi }_{1}^{2}\left(P,x,{t}_{r}\right)-u\left(x,{t}_{r}\right)\right).\end{array}$
(4.25)

Denote by ${\overline{I}}_{s}\left(P,x,{t}_{r}\right)$ the associated homogeneous one replacing ${\stackrel{˜}{V}}_{1j}^{\left(r\right)}$ by ${\overline{\stackrel{˜}{V}}}_{1j}^{\left(r\right)}$, where

${\overline{\stackrel{˜}{V}}}_{1j}^{\left(r\right)}=\left\{\begin{array}{cc}{\stackrel{˜}{V}}_{1j}^{\left(r\right)}{|}_{{\stackrel{˜}{\alpha }}_{0}=1,{\stackrel{˜}{\alpha }}_{1}=\cdots ={\stackrel{˜}{\alpha }}_{r}={\stackrel{˜}{\beta }}_{0}={\stackrel{˜}{\beta }}_{1}=\cdots ={\stackrel{˜}{\beta }}_{r}=0},\hfill & s=3r+1,\hfill \\ {\stackrel{˜}{V}}_{1j}^{\left(r\right)}{|}_{{\stackrel{˜}{\beta }}_{0}=1,{\stackrel{˜}{\alpha }}_{0}={\stackrel{˜}{\alpha }}_{1}=\cdots ={\stackrel{˜}{\alpha }}_{r}={\stackrel{˜}{\beta }}_{1}=\cdots ={\stackrel{˜}{\beta }}_{r}=0},\hfill & s=3r+2,\hfill \end{array}\phantom{\rule{1em}{0ex}}j=1,2,3.$

Lemma 4.4 Let $s=3r+2$ (or $3r+1$), $r\in {\mathbb{N}}_{0}$, $\left(x,{t}_{r}\right)\in {\mathbb{C}}^{2}$, and $\lambda ={\zeta }^{-3}$ be the local coordinate near ${P}_{\mathrm{\infty }}$. Then

(4.26)

Proof For the sake of convenience, we introduce the notation ${\stackrel{˜}{V}}_{1j}^{\left(r,s\right)}={\stackrel{˜}{V}}_{1j}^{\left(r\right)}$, $j=1,2,3$. From (2.12) and (4.25), one easily gets

$\begin{array}{rcl}{\overline{I}}_{s}\left(P,x,{t}_{r}\right)& =& {\overline{\stackrel{˜}{V}}}_{11}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right)+{\overline{\stackrel{˜}{V}}}_{12}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right){\varphi }_{1}\left(P,x,{t}_{r}\right)\\ +{\overline{\stackrel{˜}{V}}}_{13}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right)\left({\varphi }_{1,x}\left(P,x,{t}_{r}\right)+{\varphi }_{1}^{2}\left(P,x,{t}_{r}\right)-u\right)\\ =& \frac{1}{6}{\overline{\stackrel{˜}{b}}}_{xx}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right)-\frac{1}{3}u{\overline{\stackrel{˜}{b}}}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right)-{\overline{\stackrel{˜}{a}}}_{x}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right)\\ -\left[{\overline{\stackrel{˜}{a}}}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right)-\frac{1}{2}{\overline{\stackrel{˜}{b}}}_{x}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right)\right]{\varphi }_{1}\left(P,x,{t}_{r}\right)\\ +{\overline{\stackrel{˜}{b}}}^{\left(r,s\right)}\left[{\varphi }_{1,x}\left(P,x,{t}_{r}\right)+{\varphi }_{1}^{2}\left(P,x,{t}_{r}\right)-u\left(x,{t}_{r}\right)\right].\end{array}$

From (4.1), we can see

$\begin{array}{c}{\overline{I}}_{1}={\varphi }_{3}\left(P,x,{t}_{r}\right)={\zeta }^{-1}+O\left(\zeta \right),\hfill \\ {\overline{I}}_{2}=-\frac{1}{3}u\left(x,{t}_{r}\right)+{\varphi }_{1,x}\left(P,x,{t}_{r}\right)-{\varphi }_{1}^{2}\left(P,x,{t}_{r}\right)-u\left(x,{t}_{r}\right)={\zeta }^{-2}+O\left(\zeta \right).\hfill \end{array}$

So (4.26) is correct for $s=1$ and $s=2$. Then one may rewrite (4.26) as

(4.27)

for some coefficients ${\left\{{\delta }_{j}\left(x,{t}_{r}\right)\right\}}_{j\in \mathbb{N}}$. From (3.20) and (4.25), we can see

$\begin{array}{c}{\partial }_{x}{\overline{I}}_{s}\left(P,x,{t}_{r}\right)\hfill \\ \phantom{\rule{1em}{0ex}}={\partial }_{x}\left({\overline{\stackrel{˜}{V}}}_{12}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right){\varphi }_{1}\left(P,x,{t}_{r}\right)+{\overline{\stackrel{˜}{V}}}_{13}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right)\left({\varphi }_{1,x}\left(P,x,{t}_{r}\right)+{\varphi }_{1}^{2}\left(P,x,{t}_{r}\right)-u\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\overline{\stackrel{˜}{V}}}_{11}^{\left(r,s\right)}\left(\lambda ,x,{t}_{r}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}={\varphi }_{1,{t}_{r}}\left(P,x,{t}_{r}\right),\hfill \end{array}$

that is,

${\partial }_{x}\left(-{\zeta }^{-s}+\sum _{j=1}^{\mathrm{\infty }}{\delta }_{j}\left(x,{t}_{r}\right){\zeta }^{j}\right)={\left({\zeta }^{-1}+\sum _{j=1}^{\mathrm{\infty }}{\kappa }_{j}\left(x,{t}_{r}\right){\zeta }^{j-1}\right)}_{{t}_{r}}={\left(\sum _{j=1}^{\mathrm{\infty }}{\kappa }_{j+1}\left(x,{t}_{r}\right){\zeta }^{j}\right)}_{{t}_{r}}.$
(4.28)

Using (3.2), (4.2), and comparing coefficients of ζ in (4.28), we should obtain

$\begin{array}{r}{\delta }_{j,x}\left(x,{t}_{r}\right)={\kappa }_{j+1,{t}_{r}}\left(x,{t}_{r}\right),\phantom{\rule{1em}{0ex}}j=1,2,\dots \\ {\delta }_{1,x}\left(x,{t}_{r}\right)={\kappa }_{2,{t}_{r}}\left(x,{t}_{r}\right)=\frac{2}{3}{u}_{{t}_{r}}\left(x,{t}_{r}\right)=-{\overline{\stackrel{˜}{b}}}_{r,x}^{\left(r,s\right)}\left(x,{t}_{r}\right),\\ {\delta }_{2,x}\left(x,{t}_{r}\right)={\kappa }_{3,{t}_{r}}\left(x,{t}_{r}\right)=\frac{1}{3}{\left(-u\left(x,{t}_{r}\right)+v\left(x,{t}_{r}\right)\right)}_{{t}_{r}}=\frac{1}{2}{\overline{\stackrel{˜}{b}}}_{r,xx}^{\left(r,s\right)}\left(x,{t}_{r}\right)-{\overline{\stackrel{˜}{a}}}_{r,x}^{\left(r,s\right)}\left(x,{t}_{r}\right),\\ {\delta }_{3,x}\left(x,{t}_{r}\right)={\kappa }_{4,{t}_{r}}\left(x,{t}_{r}\right)={\left(\frac{1}{9}{u}_{xx}\left(x,{t}_{r}\right)-\frac{1}{3}{v}_{x}\left(x,{t}_{r}\right)\right)}_{{t}_{r}}=-\frac{1}{6}{\overline{\stackrel{˜}{b}}}_{r,xxx}^{\left(r,s\right)}\left(x,{t}_{r}\right)+{\overline{\stackrel{˜}{a}}}_{r,xx}^{\left(r,s\right)}\left(x,{t}_{r}\right).\end{array}$
(4.29)

That is,

$\begin{array}{c}{\delta }_{1}\left(x,{t}_{r}\right)={\gamma }_{1}\left({t}_{r}\right)-{\overline{\stackrel{˜}{b}}}_{r}^{\left(r,s\right)}\left(x,{t}_{r}\right),\hfill \\ {\delta }_{2}\left(x,{t}_{r}\right)={\gamma }_{2}\left({t}_{r}\right)+\frac{1}{2}{\overline{\stackrel{˜}{b}}}_{r,x}^{\left(r,s\right)}\left(x,{t}_{r}\right)-{\overline{\stackrel{˜}{a}}}_{r}^{\left(r,s\right)}\left(x,{t}_{r}\right),\hfill \\ {\delta }_{3}\left(x,{t}_{r}\right)={\gamma }_{3}\left({t}_{r}\right)-\frac{1}{6}{\overline{\stackrel{˜}{b}}}_{r,xx}^{\left(r,s\right)}\left(x,{t}_{r}\right)+{\overline{\stackrel{˜}{a}}}_{r,x}^{\left(r,s\right)}\left(x,{t}_{r}\right),\hfill \end{array}$
(4.30)

with ${\gamma }_{1}\left({t}_{r}\right)$, ${\gamma }_{2}\left({t}_{r}\right)$, ${\gamma }_{3}\left({t}_{r}\right)$ being integration constants. From the definition of ${\overline{I}}_{s}$, the power series for ${\varphi }_{1}\left(P,x,{t}_{r}\right)$ and the coefficients of $\overline{\stackrel{˜}{a}}\left(\zeta ,x,{t}_{r}\right)$, $\overline{\stackrel{˜}{b}}\left(\zeta ,x,{t}_{r}\right)$, we deduce that ${\gamma }_{1}\left({t}_{r}\right)={\gamma }_{2}\left({t}_{r}\right)={\gamma }_{3}\left({t}_{r}\right)=0$. Hence one concludes

(4.31)

On the other hand, we will get

$\begin{array}{rcl}{\overline{I}}_{s+3}\left(P,x,{t}_{r}\right)& =& {\zeta }^{-3}{\overline{I}}_{s}+\left({\overline{\stackrel{˜}{a}}}_{r}^{\left(r+1,s+3\right)}-\frac{1}{2}{\overline{\stackrel{˜}{b}}}_{r,x}^{\left(r+1,s+3\right)}\right){\varphi }_{1}+{\overline{\stackrel{˜}{b}}}_{r}^{\left(r+1,s+3\right)}\left({\varphi }_{1,x}+{\varphi }_{1}^{2}-u\right)\\ +\frac{1}{6}{\overline{\stackrel{˜}{b}}}_{r,xx}^{\left(r+1,s+3\right)}-\frac{1}{3}u{\overline{\stackrel{˜}{b}}}_{r}^{\left(r+1,s+3\right)}-{\overline{\stackrel{˜}{a}}}_{r,x}^{\left(r+1,s+3\right)}\\ =& {\zeta }^{-s-3}+O\left(\zeta \right).\end{array}$
(4.32)

□

By (3.1) one knows that

(4.33)

Thus

(4.34)

Furthermore, integrating (4.23) yields

(4.35)

where ${e}_{s+1}^{\left(2\right)}\left({P}_{0}\right)$ is a constant. Combing (4.34) and (4.35) indicates

(4.36)

Given these preparations, the theta function representation of ${\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)$ reads as follows.

Theorem 4.5 Let $P=\left(\lambda ,y\right)\in {\mathcal{K}}_{m-1}\setminus \left\{{P}_{\mathrm{\infty }}\right\}$ and let $\left(x,{t}_{r}\right),\left({x}_{0},{t}_{0,r}\right)\in {\mathrm{\Omega }}_{\mu }$, where ${\mathrm{\Omega }}_{\mu }\subseteq {\mathbb{C}}^{2}$ is open and connected. Suppose that ${\mathcal{D}}_{\stackrel{ˆ}{\underline{\mu }}\left(x,{t}_{r}\right)}$, or equivalently, ${\mathcal{D}}_{\stackrel{ˆ}{\underline{\nu }}\left(x,{t}_{r}\right)}$ is nonspecial for $\left(x,{t}_{r}\right)\in {\mathrm{\Omega }}_{\mu }$. Then

$\begin{array}{rcl}{\psi }_{1}\left(P,x,{x}_{0},{t}_{r},{t}_{0,r}\right)& =& \frac{\theta \left(\underline{\lambda }\left(P,\stackrel{ˆ}{\underline{\mu }}\left(x,{t}_{r}\right)\right)\right)\theta \left(\underline{\lambda }\left({P}_{\mathrm{\infty }},\stackrel{ˆ}{\underline{\mu }}\left({x}_{0},{t}_{0,r}\right)\right)\right)}{\theta \left(\underline{\lambda }\left({P}_{\mathrm{\infty }},\stackrel{ˆ}{\underline{\mu }}\left(x,{t}_{r}\right)\right)\right)\theta \left(\underline{\lambda }\left(P,\stackrel{ˆ}{\underline{\mu }}\left({x}_{0},{t}_{0,r}\right)\right)\right)}\\ ×exp\left(\left(x-{x}_{0}\right)\left({e}_{2}^{\left(2\right)}\left({P}_{0}\right)-{\int }_{{P}_{0}}^{P}{\omega }_{{P}_{\mathrm{\infty }},2}^{\left(2\right)}\right)\\ +\left({t}_{r}-{t}_{0,r}\right)\left({e}_{s+1}^{\left(}\end{array}$