Algebro-geometric Constructions to the Dym-type Hierarchy

TL;DR

This paper develops algebro-geometric methods using trigonal curves and Riemann theta functions to construct solutions for the Dym-type hierarchy, advancing the understanding of integrable systems.

Contribution

It introduces a novel algebro-geometric framework for the Dym-type hierarchy based on trigonal curves and Baker-Akhiezer functions, providing explicit solution representations.

Findings

Explicit Riemann theta function representations of meromorphic functions.

Algebro-geometric constructions for the entire Dym-type hierarchy.

Connection between trigonal curves and integrable systems.

Abstract

Resorting to the characteristic polynomial of Lax matrix for the Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker-Akhiezer function and meromorphic function are introduced. Based on the theory of trigonal curve and three kinds of Abelian differentials, we obtain the explicit Riemann theta function representations of the meromorphic function, from which we get the algebro-geometric constructions for the entire Dym-type hierarchy

Equations

$$\begin{array}[]{lcl}u_{t}=(u^{-\frac{1}{2}})_{xxx},\end{array}$$

$$\begin{array}[]{lcl}v_{t}=-\frac{1}{3}(v^{-\frac{2}{3}})_{xxxxx}.\end{array}$$

$$\begin{array}[]{rcl}\psi_{x}=U\psi,\quad\psi=\left(\begin{matrix}\psi_{1}\cr\psi_{2}\cr\psi_{3}\end{matrix}\right),\quad U=\left(\begin{matrix}0&1&0\cr 0&0&1\cr\lambda v&0&0\end{matrix}\right),\end{array}$$

$$\begin{array}[]{rcl}Kg_{j-1}=Jg_{j},\quad j\geq 0,\end{array}$$

$$\begin{array}[]{rcl}K\hat{g}_{j-1}=J\hat{g}_{j},\quad j\geq 0,\end{array}$$

$$\begin{array}[]{rcl}g_{-1}=\left(\begin{matrix}v^{-\frac{2}{3}}\cr 0\end{matrix}\right),\quad\hat{g}_{-1}=\left(\begin{matrix}-\frac{1}{3}v^{-1}(v^{-\frac{1}{3}})_{xx}+\frac{1}{6}v^{-\frac{2}{3}}[({v^{-\frac{1}{3}})_{x}}]^{2}\cr v^{-\frac{1}{3}}\end{matrix}\right),\end{array}$$

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

$$\ker J=\{{\alpha }_0{g}_{-1}+{\beta }_0{\hat{g}}_{-1}|\forall {\alpha }_0,{\beta }_0\in \mathbb{R}\}.$$

$$\begin{array}[]{rcl}V_{x}-[U,V]=0,\ \ V=\lambda\left(\begin{matrix}V_{11}&V_{12}&V_{13}\cr V_{21}&V_{22}&V_{23}\cr V_{31}&V_{32}&V_{33}\end{matrix}\right),\end{array}$$

$$\begin{array}[]{rcl}&&V_{11,x}+\lambda vV_{13}-V_{21}=0,\\ &&V_{12,x}+V_{11}-V_{22}=0,\\ &&V_{13,x}+V_{12}-V_{23}=0,\\ &&V_{21,x}+\lambda vV_{23}-V_{31}=0,\\ &&V_{22,x}+V_{21}-V_{32}=0,\\ &&V_{23,x}+V_{22}-V_{33}=0,\\ &&V_{31,x}-\lambda v(V_{11}-V_{33})=0,\\ &&V_{32,x}-\lambda vV_{12}+V_{31}=0,\\ &&V_{33,x}-\lambda vV_{13}+V_{32}=0,\\ \end{array}$$

$$\begin{array}[]{lcl}V_{11}=-\frac{1}{3}\partial^{2}a-\lambda\partial b,\quad V_{12}=\partial a+\lambda b,\quad V_{13}=-2a,\\ V_{21}=-\frac{1}{3}\partial^{3}a-\lambda\partial^{2}b-2\lambda va,\ \ V_{22}=\frac{2}{3}\partial^{2}a,\ \ V_{23}=-\partial a+\lambda b,\\ V_{31}=-\frac{1}{3}\partial^{4}a+\lambda^{2}vb,\ \ V_{32}=\frac{1}{3}\partial^{3}a-\lambda\partial^{2}b-2\lambda va,\quad V_{33}=-\frac{1}{3}\partial^{2}a+\lambda\partial b.\end{array}$$

$$\begin{array}[]{rcl}a=\sum\limits_{j\geq 0}a_{j-1}\lambda^{-2j},\quad b=\sum\limits_{j\geq 0}b_{j-1}\lambda^{-2j},\end{array}$$

$$\begin{array}[]{rcl}KG_{j-1}=JG_{j},\quad JG_{-1}=0,\ \ j\geq 0,\end{array}$$

$$\begin{array}[]{rcl}G_{-1}=\alpha_{0}g_{-1}+\beta_{0}\hat{g}_{-1},\end{array}$$

$$\begin{array}[]{lcl}&&G_{j}=\alpha_{0}g_{j}+\beta_{0}\hat{g}_{j}+\cdots+\alpha_{j}g_{0}+\beta_{j}\hat{g}_{0}\\ &&\quad\quad\quad+\alpha_{j+1}g_{-1}+\beta_{j+1}\hat{g}_{-1},\quad j\geq 0,\end{array}$$

$$\begin{array}[]{rcl}\psi_{t_{r}}=\widetilde{V}^{(r)}\psi,\quad\widetilde{V}^{(r)}=\lambda\left(\begin{matrix}\widetilde{V}_{11}^{(r)}&\widetilde{V}_{12}^{(r)}&\widetilde{V}_{13}^{(r)}\cr\widetilde{V}_{21}^{(r)}&\widetilde{V}_{22}^{(r)}&\widetilde{V}_{23}^{(r)}\cr\widetilde{V}_{31}^{(r)}&\widetilde{V}_{32}^{(r)}&\widetilde{V}_{33}^{(r)}\end{matrix}\right),\end{array}$$

$$\begin{array}[]{rcl}&&\widetilde{V}_{11}^{(r)}=-\frac{1}{3}\partial^{2}\tilde{a}^{(r)}-\lambda\partial\tilde{b}^{(r)},\quad\widetilde{V}_{12}^{(r)}=\partial\tilde{a}^{(r)}+\lambda\tilde{b}^{(r)},\quad\widetilde{V}_{13}^{(r)}=-2\tilde{a}^{(r)},\\ &&\widetilde{V}_{21}^{(r)}=-\frac{1}{3}\partial^{3}\tilde{a}^{(r)}-\lambda\partial^{2}\tilde{b}^{(r)}-2\lambda v\tilde{a}^{(r)},\quad\widetilde{V}_{22}^{(r)}=\frac{2}{3}\partial^{2}\tilde{a}^{(r)},\quad\widetilde{V}_{23}^{(r)}=-\partial\tilde{a}^{(r)}+\lambda\tilde{b}^{(r)},\\ &&\widetilde{V}_{31}^{(r)}=-\frac{1}{3}\partial^{4}\tilde{a}^{(r)}+\lambda^{2}v\tilde{b}^{(r)},\ \ \widetilde{V}_{32}^{(r)}=\frac{1}{3}\partial^{3}\tilde{a}^{(r)}-\lambda\partial^{2}\tilde{b}^{(r)}-2\lambda v\tilde{a}^{(r)},\\ &&\widetilde{V}_{33}^{(r)}=-\frac{1}{3}\partial^{2}\tilde{a}^{(r)}+\lambda\partial\tilde{b}^{(r)},\ \ \tilde{a}^{(r)}=\sum\limits^{r}_{j=0}\tilde{a}_{j-1}\lambda^{2(r-j)},\quad\tilde{b}^{(r)}=\sum\limits^{r}_{j=0}\tilde{b}_{j-1}\lambda^{2(r-j)},\\ &&\widetilde{G}_{j}=(\tilde{a}_{j},\tilde{b}_{j})^{T}=\tilde{\alpha}_{0}g_{j}+\tilde{\beta}_{0}\hat{g}_{j}+\cdots+\tilde{\alpha}_{j}g_{0}+\tilde{\beta}_{j}\hat{g}_{0}+\tilde{\alpha}_{j+1}g_{-1}+\tilde{\beta}_{j+1}\hat{g}_{-1},\quad j\geq-1,\end{array}$$

$$\begin{array}[]{rcl}v_{t_{r}}=-\frac{1}{3}\partial^{5}\tilde{a}_{r-1}=-(\partial v+2v\partial)\tilde{b}_{r}.\end{array}$$

$$\begin{array}[]{rcl}&&v_{t_{0}}=-\frac{1}{3}\partial^{5}\{\tilde{\alpha}_{0}v^{-\frac{2}{3}}+\tilde{\beta}_{0}[-\frac{1}{3}v^{-1}(v^{-\frac{1}{3}})_{xx}+\frac{1}{6}v^{-\frac{2}{3}}({v^{-\frac{1}{3}})_{x}}^{2}]\},\\ \end{array}$$

$$\begin{array}[]{rcl}&&\psi_{x}(P,x,x_{0},t_{r},t_{0,r})=U(v,\lambda)\psi(P,x,x_{0},t_{r},t_{0,r}),\\ &&\psi_{t_{r}}(P,x,x_{0},t,t_{0,r})=\widetilde{V}^{(r)}(v,\lambda)\psi(P,x,x_{0},t_{r},t_{0,r}),\\ &&\lambda^{-1}V^{(n)}(v,\lambda)\psi(P,x,x_{0},t,t_{0,r})=y(P)\psi(P,x,x_{0},t_{r},t_{0,r}),\\ &&\psi_{1}(P,x_{0},x_{0},t_{0,r},t_{0,r})=1,\quad P=(\lambda,y),\ x,t_{r}\in\mathbb{C}.\end{array}$$

$$\begin{array}[]{rcl}&&V^{(n)}_{11}=-\frac{1}{3}\partial^{2}a^{(n)}-\lambda\partial b^{(n)},\quad V^{(n)}_{12}=\partial a^{(n)}+\lambda b^{(n)},\quad V^{(n)}_{13}=-2a^{(n)},\\ &&V^{(n)}_{21}=-\frac{1}{3}\partial^{3}a^{(n)}-\lambda\partial^{2}b^{(n)}-2\lambda va^{(n)},\quad V^{(n)}_{22}=\frac{2}{3}\partial^{2}a^{(n)},\quad V^{(n)}_{23}=-\partial a^{(n)}+\lambda b^{(n)},\\ &&V^{(n)}_{31}=-\frac{1}{3}\partial^{4}a^{(n)}+\lambda^{2}vb^{(n)},\ \ V^{(n)}_{32}=\frac{1}{3}\partial^{3}a^{(n)}-\lambda\partial^{2}b^{(n)}-2\lambda va^{(n)},\\ &&V^{(n)}_{33}=-\frac{1}{3}\partial^{2}a^{(n)}+\lambda\partial b^{(n)},\ \ a^{(n)}=\sum\limits_{j=0}^{n}a_{j-1}\lambda^{2(n-j)},\quad b^{(n)}=\sum\limits_{j=0}^{n}b_{j-1}\lambda^{2(n-j)},\end{array}$$

$$\begin{array}[]{lcl}U_{t_{r}}-\widetilde{V}^{(r)}_{x}+[U,\widetilde{V}^{(r)}]=0,\end{array}$$

$$\begin{array}[]{lcl}-V^{(n)}_{x}+[U,V^{(n)}]=0,\end{array}$$

$$\begin{array}[]{lcl}-V^{(n)}_{t_{r}}+[\widetilde{V}^{(r)},V^{(n)}]=0.\end{array}$$

$$\begin{array}[]{rcl}{\mathcal{F}}_{m}(\lambda,y)={\rm det}(yI-\lambda^{-1}V^{(n)})=y^{3}+yS_{m}(\lambda)-T_{m}(\lambda),\end{array}$$

$$\begin{array}[]{rcl}&&S_{m}(\lambda)=\left|\begin{matrix}V^{(n)}_{11}&V^{(n)}_{12}\cr V^{(n)}_{21}&V^{(n)}_{22}\cr\end{matrix}\right|+\left|\begin{matrix}V^{(n)}_{11}&V^{(n)}_{13}\cr V^{(n)}_{31}&V^{(n)}_{33}\cr\end{matrix}\right|+\left|\begin{matrix}V^{(n)}_{22}&V^{(n)}_{23}\cr V^{(n)}_{32}&V^{(n)}_{33}\cr\end{matrix}\right|,\\ \\ &&T_{m}(\lambda)=\left|\begin{matrix}V^{(n)}_{11}&V^{(n)}_{12}&V^{(n)}_{13}\cr V^{(n)}_{21}&V^{(n)}_{22}&V^{(n)}_{23}\cr V^{(n)}_{31}&V^{(n)}_{32}&V^{(n)}_{33}\cr\end{matrix}\right|=\left\{\begin{array}[]{lcl}\beta_{0}^{3}\lambda^{6n+4}+\cdots,\ \ \beta_{0}\neq 0,\alpha_{0}\in\mathbb{R},\\ -8\alpha_{0}^{3}\lambda^{6n+2}+\cdots,\ \ \beta_{0}=0,\alpha_{0}\neq 0.\end{array}\right.\\ \end{array}$$

$$\begin{array}[]{rcl}{\mathcal{K}}_{m-1}:\quad{\mathcal{F}}_{m}(\lambda,y)=y^{3}+yS_{m}(\lambda)-T_{m}(\lambda)=0,\end{array}$$

$$\begin{array}[]{rcl}\phi(P)=\phi(P,x,t_{r})=v^{-\frac{1}{3}}\displaystyle\frac{\psi_{1,x}(P,x,x_{0},t_{r},t_{0,r})}{\psi_{1}(P,x,x_{0},t_{r},t_{0,r})}=v^{-\frac{1}{3}}\displaystyle\frac{\psi_{2}(P,x,x_{0},t_{r},t_{0,r})}{\psi_{1}(P,x,x_{0},t_{r},t_{0,r})},\quad P\in{\mathcal{K}}_{m-1}.\end{array}$$

$$\begin{array}[]{lcl}\phi(P)=v^{-\frac{1}{3}}\frac{\displaystyle yV^{(n)}_{23}+C_{m}}{\displaystyle yV^{(n)}_{13}+A_{m}}=\frac{\displaystyle v^{-\frac{1}{3}}F_{m}}{\displaystyle y^{2}V^{(n)}_{23}-yC_{m}+D_{m}}=\frac{\displaystyle y^{2}V^{(n)}_{13}-yA_{m}+B_{m}}{\displaystyle v^{\frac{1}{3}}E_{m-1}},\end{array}$$

$$\begin{array}[]{lcl}A_{m}=V^{(n)}_{12}V^{(n)}_{23}-V^{(n)}_{13}V^{(n)}_{22},\ \ C_{m}=V^{(n)}_{13}V^{(n)}_{21}-V^{(n)}_{11}V^{(n)}_{23},\\ B_{m}=V^{(n)}_{13}(V^{(n)}_{11}V^{(n)}_{33}-V^{(n)}_{13}V^{(n)}_{31})+V^{(n)}_{12}(V^{(n)}_{11}V^{(n)}_{23}-V^{(n)}_{13}V^{(n)}_{21}),\\ D_{m}=V^{(n)}_{23}(V^{(n)}_{22}V^{(n)}_{33}-V^{(n)}_{23}V^{(n)}_{32})+V^{(n)}_{21}(V^{(n)}_{13}V^{(n)}_{22}-V^{(n)}_{12}V^{(n)}_{23}),\\ E_{m-1}=(V^{(n)}_{13})^{2}V^{(n)}_{32}+V^{(n)}_{12}V^{(n)}_{13}(V^{(n)}_{22}-V^{(n)}_{33})-(V^{(n)}_{12})^{2}V^{(n)}_{23},\\ F_{m}=(V^{(n)}_{23})^{2}V^{(n)}_{31}+V^{(n)}_{21}V^{(n)}_{23}(V^{(n)}_{11}-V^{(n)}_{33})-V^{(n)}_{13}(V^{(n)}_{21})^{2}.\\ \end{array}$$

Full text

Algebro-geometric Constructions to the Dym-type Hierarchy

Lihua Wu1, Guoliang He2, and Xianguo Geng3

       1 Department of Mathematics, Huaqiao University, Quanzhou 362021, P. R. China

       2 Department of Mathematics and Information Science, Zhengzhou University of Light

         Industry, Zhengzhou 450002, China

       3 School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, P. R. China

       Correspondence to be sent to: wulihua@hqu.edu.cn

Resorting to the characteristic polynomial of Lax matrix for the Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker-Akhiezer function and meromorphic function are introduced. Based on the theory of trigonal curve and three kinds of Abelian differentials, we obtain the explicit Riemann theta function representations of the meromorphic function, from which we get the algebro-geometric constructions for the entire Dym-type hierarchy.

1 Introduction

As is well-known, there exist several methods to study the algebro-geometric solutions of soliton equations and from which many soliton equations associated with $2\times 2$ matrix spectral problems were discussed ([1, 4, 6, 7, 10, 11, 14, 16, 17, 20, 22, 26, 28, 29, 30, 31, 33, 34, 36, 39, 40] and references therein). However, the research of soliton equations associated with $3\times 3$ matrix spectral problems is very few, which is also much more difficult and complicated for the underlying algebraic curve is trigonal curve. It should be pointed out that this trigonal curve considerably complicates the analysis and hence makes it a rather challenging problem. More recently, according to a unified framework [5], algebro-geometric solutions for a lot of soliton hierarchies associated with $3\times 3$ matrix spectral problems have been discussed, such as modified Boussinesq hierarchy [12], Kaup-Kuperschmidt hierarchy [13], three-wave resonant interaction hierarchy [19], and others [15, 38].

The Dym equation

[TABLE]

was first discovered by Harry Dym [23] and rediscovered by Li [24] and Sabatier [35]. It was shown that the Dym equation possesses many properties typical for integrable systems (see [3, 27, 37] and references therein). Moreover, the algebro-geometric solution of Dym equation was also discussed in [6, 31]. Meanwhile, the integrable extensions of Dym equation attract much attention of many researchers [2, 21, 25, 32].

By considering a $3\times 3$ matrix spectral problem, Prof. Geng [9] derived a hierarchy of Dym-type equations and discussed its nonlinearization. The first nontrivial member in the hierarchy is Dym-type equation

[TABLE]

The principal aim of the present paper is to study algebro-geometric constructions of the Dym-type flows. With the aid of the three kinds of Abelian differentials and asymptotic expansions, we arrive at the Riemann theta function representations of the meromorphic function, and solutions for the entire Dym-type hierarchy. In this process, an explicit expression of the third kind of Abelian differential proposed by us is of great importance.

The outline of the present paper is as follows. In section 2, based on the Lenard recursion equations and the zero-curvature equation, we deduce the Dym-type hierarchy. In section 3, we define the vector-valued Baker-Akhiezer function and the associated meromorphic function, from which a trigonal curve ${\mathcal{K}}_{m-1}$ of arithmetic genus $m-1$ is introduced with the help of the characteristic polynomial of Lax matrix for the Dym-type hierarchy. It is shown that the Dym-type hierarchy is decomposed into a system of Dubrovin-type equations. In section 4, by introducing three kinds of Abelian differentials, especially the explicit third kind, we present the Riemann theta function representations of the meromorphic function, and in particular, that of the potential $v$ for the entire Dym-type hierarchy.

2 Dym-type Hierarchy

In this section, we follow the Geng [9] and derive the Dym-type hierarchy associated with the $3\times 3$ matrix spectral problem

[TABLE]

where $v(\neq 0)$ is a potential and $\lambda$ is a constant spectral parameter. To this end, we introduce two sets of Lenard recursion equations

[TABLE]

[TABLE]

with two starting points

[TABLE]

and two operators are defined as

[TABLE]

It is easy to see that

[TABLE]

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

[TABLE]

which is equivalent to

[TABLE]

where each entry $V_{ij}=V_{ij}(a,b)$ is a Laurent expansion in $\lambda$:

[TABLE]

Substituting (2.7) into (2.6) and expanding the functions $a$ and $b$ into the Laurent series in $\lambda$

[TABLE]

we obtain the recursion equations

[TABLE]

with $G_{j}=(a_{j},b_{j})^{T}$. Since equation $JG_{-1}=0$ has the general solution

[TABLE]

$G_{j}$ can be expressed as

[TABLE]

where $\alpha_{j}$ and $\beta_{j}$ are arbitrary constants. Let $\psi$ satisfy the spectral problem (2.1) and an auxiliary problem

[TABLE]

where

[TABLE]

and the constants $\tilde{\alpha}_{j},\tilde{\beta}_{j}$ are independent of $\alpha_{j},\beta_{j}.$ Then the compatibility condition of (2.1) and (2.12) yields the zero-curvature equation, $U_{t_{r}}-\widetilde{V}_{x}^{(r)}+[U,\widetilde{V}^{(r)}]=0$, which is equivalent to the hierarchy of nonlinear evolution equations

[TABLE]

The first nontrivial member in the hierarchy (2.14) is

[TABLE]

which is just the Dym-type equation (1.2) for $\tilde{\alpha}_{0}=1,\tilde{\beta}_{0}=0,t_{0}=t.$

3 The Baker-Akhiezer Function

In this section, we shall introduce the vector-valued Baker-Akhiezer function, meromorphic function and trigonal curve associated with the Dym-type hierarchy. Then we derive a system of Dubrovin-type differential equations.

We introduce the vector-valued Baker-Akhiezer function

[TABLE]

Here $V^{(n)}=\lambda\left(\begin{matrix}V_{ij}^{(n)}\end{matrix}\right)_{3\times 3}$ and

[TABLE]

in which $a_{j},b_{j}$ are determined by (2.11). The compatibility conditions of the first three equations in (3.1) yield that

[TABLE]

[TABLE]

[TABLE]

A direct calculation shows that $yI-\lambda^{-1}V^{(n)}$ satisfies (3.3) and (3.4). Hence the characteristic polynomial of Lax matrix $\lambda^{-1}V^{(n)}$ for the Dym-type hierarchy is a constant independent of variables $x$ and $t_{r}$, and possesses following expansion

[TABLE]

where $S_{m}(\lambda)$ and $T_{m}(\lambda)$ are polynomials of $\lambda$ with constant coefficients

[TABLE]

It is evident that $T_{m}(\lambda)$ is a polynomial of degree $6n+4=3(2n+1)+1$ and $6n+2=3(2n)+2$ as $\beta_{0}\neq 0$, $\alpha_{0}\in\mathbb{R}$ and $\beta_{0}=0,\alpha_{0}\neq 0$, respectively. Then ${\mathcal{F}}_{m}(\lambda,y)=0$ naturally leads to a trigonal curve

[TABLE]

with $m=6n+4$ or $m=6n+2$.

For the convenience, we denote the compactification of the curve ${\mathcal{K}}_{m-1}$ by the same symbol ${\mathcal{K}}_{m-1}$. Thus ${\mathcal{K}}_{m-1}$ becomes a three-sheeted Riemann surface of arithmetic genus $m-1$ if it is nonsingular or smooth, which means that $\left.(\frac{\partial{\mathcal{F}}_{m}(\lambda,y)}{\partial\lambda},\frac{\partial{\mathcal{F}}_{m}(\lambda,y)}{\partial y})\right.|_{(\lambda,y)=(\lambda^{\prime},y^{\prime})}\neq(0,0)$ at each point $P^{\prime}=(\lambda^{\prime},y^{\prime})\in{\mathcal{K}}_{m-1}$.

A meromorphic function $\phi(P,x,t_{r})$ on ${\mathcal{K}}_{m-1}$ is defined as

[TABLE]

It infers from (3.1) and (3.8) that

[TABLE]

where

[TABLE]

Taking (3.7) and (3.9) into account, we arrive at some important identities among polynomials $A_{m},B_{m},C_{m},D_{m},$ $E_{m-1},F_{m},S_{m},T_{m}$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now, we define the holomorphic mapping $\ast$, changing sheets, by

[TABLE]

where $y_{i}(\lambda),i=0,1,2$, denote the three branches of $y(P)$ satisfying ${\mathcal{F}}_{m}(\lambda,y)=0,$ namely,

[TABLE]

Consequently, we have

[TABLE]

In what follows, we shall summarize some properties of the meromorphic function $\phi(P,x,t_{r})$ without proofs.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Lemma 3.1. Assume that (3.1) and (3.2) hold, and let $(\lambda,x,t_{r})\in{\mathbb{C}}^{3}$. Then

[TABLE]

[TABLE]

Proof. Differentiating (3.21) with respect to $t_{r}$ and using (3.19), (3.21) and (3.23), we have

[TABLE]

Integrating the above equation with respect to $x$ and choosing the integration constant as zero imply the first equation in (3.24). Differentiating (3.20) with respect to $t_{r}$, an analogous process shows (3.25).

By observing (2.11) and (3.10), we can easily find that $E_{m-1}$ and $F_{m}$ are polynomials with respect to $\lambda$ of degree $m-1$ and $m$, respectively. Therefore

[TABLE]

[TABLE]

with

[TABLE]

Let us denote

[TABLE]

[TABLE]

then it is easy to see that the following Lemma holds.

Lemma 3.2. Suppose the zeros $\{\mu_{j}(x,t_{r})\}_{j=1}^{m-1}$ and $\{\nu_{l}(x,t_{r})\}_{l=0}^{m-1}$ of $E_{m-1}(\lambda,x,t_{r})$ and $F_{m}(\lambda,x,t_{r})$ remain distinct for $(x,t_{r})\in\Omega_{\mu}$ and $(x,t_{r})\in\Omega_{\nu}$, respectively, where $\Omega_{\mu},\Omega_{\nu}\subseteq\mathbb{C}^{2}$ are open and connected. Then $\{\mu_{j}(x,t_{r})\}_{j=1}^{m-1}$ and $\{\nu_{l}(x,t_{r})\}_{l=0}^{m-1}$ satisfy the Dubrovin-type equations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof. We just need to prove (3.30) for the proofs of (3.31)-(3.33) are similar to (3.30). Substituting $\lambda=\mu_{j}$ into the first expression in (3.14), and using (3.12) and (3.29), we get

[TABLE]

On the other hand, differentiating (3.26) with respect to $x$ and inserting $\lambda=\mu_{j}$ into it give rise to

[TABLE]

A comparison of (3.34) and (3.35) yields (3.30).

4 Algebro-geometric Constructions to the Dym-type Hierarchy

In this section, we shall derive explicit Riemann theta function representations for the meromorphic function $\phi(P,x,t_{r})$, and in particular, that of potential $v(x,t_{r})$ for the entire Dym-type hierarchy.

Taking the local coordinate $\zeta=\lambda^{-\frac{1}{3}}$ near $P_{\infty}\in{\mathcal{K}}_{m-1}$ in (3.18), the Laurent series of $\phi(P,x,t_{r})$ can be explicitly expressed as

[TABLE]

where

[TABLE]

Defining the positive divisors on ${\mathcal{K}}_{m-1}$ of degree $m-1$

[TABLE]

with $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$, one obtains from (3.9) and (4.1) that the divisor $(\phi(P,x,t_{r}))$ of $\phi(P,x,t_{r})$ is given by

[TABLE]

which implies that $\hat{\nu}_{0}(x,t_{r}),\hat{\nu}_{1}(x,t_{r}),\dots,\hat{\nu}_{m-1}(x,t_{r})$ are $m$ zeros and $P_{\infty},\hat{\mu}_{1}(x,t_{r}),$ $\dots,$ $\hat{\mu}_{m-1}(x,t_{r})$ are $m$ poles of $\phi(P,x,t_{r})$.

Equip the Riemann surface ${\mathcal{K}}_{m-1}$ with canonical basis of cycles $\{\mathfrak{a}_{j},\mathfrak{b}_{j}\}_{j=1}^{m-1}$, which admits intersection numbers

[TABLE]

and the basis of holomorphic differentials

[TABLE]

Thus the period matrices $A$ and $B$ constructed by

[TABLE]

are invertible. Defining the matrix $C=A^{-1},\tau=CB$, the Riemannian bilinear relation makes it possible to verify that the matrix $\tau$ is symmetric $(\tau_{jk}=\tau_{kj})$ and has positive definite imaginary part (Im $\tau>0$) ([8, 18]). If we normalize $\widetilde{\omega}=(\widetilde{\omega}_{1},\cdots,\widetilde{\omega}_{m-1})$ into new basis $\omega=(\omega_{1},\cdots,\omega_{m-1})$

[TABLE]

then we have

[TABLE]

A straightforward calculation yields the following asymptotic expansions:

[TABLE]

[TABLE]

[TABLE]

Let $\omega_{P_{\infty},2}^{(2)}(P)$ denote the normalized Abelian differential of the second kind, which is holomorphic on ${\mathcal{K}}_{m-1}\setminus\{P_{\infty}\}$ with a pole of order 2 at $P_{\infty}$ and satisfies

[TABLE]

[TABLE]

The $b$-periods of the differential $\omega_{P_{\infty},2}^{(2)}$ are denoted by

[TABLE]

Furthermore, let $\omega_{P_{\infty},\hat{\nu}_{0}(x,t_{r})}^{(3)}(P)$ denote the normalized Abelian differential of the third kind defined by

[TABLE]

which is holomorphic on ${\mathcal{K}}_{m-1}\setminus\{P_{\infty},\hat{\nu}_{0}(x,t_{r})\}$ and has simple poles at $P_{\infty}$ and $\hat{\nu}_{0}(x,t_{r})$ with corresponding residues $+1$ and $-1$. The constants $\{\gamma_{j}\}_{j=1}^{m-1}$ are determined by the normalization condition

[TABLE]

A direct calculation shows

[TABLE]

with

[TABLE]

Then

[TABLE]

with $Q_{0}$ a chosen base point on ${\mathcal{K}}_{m-1}\setminus\{P_{\infty},\hat{\nu}_{0}(x,t_{r})\}$ and $e_{\infty}^{(3)}(Q_{0}),e_{0}^{(3)}(Q_{0})$ two integration constants.

Let ${\mathcal{T}}_{m-1}=\{\underline{N}+\tau\underline{L},\underline{N},\underline{L}\in\mathbb{Z}^{m-1}\}$ be a period lattice. The complex torus ${\mathcal{J}}_{m-1}=\mathbb{C}^{m-1}/{\mathcal{T}}_{m-1}$ is called a Jacobian variety of ${\mathcal{K}}_{m-1}$. The Abelian mapping $\mathcal{A}:{\mathcal{K}}_{m-1}\rightarrow{\mathcal{J}}_{m-1}$ is defined as

[TABLE]

and is extended linearly to the divisor group $\hbox{Div}({\mathcal{K}}_{m-1})$

[TABLE]

which enables us to give the Abel-Jacobi coordinates for the nonspecial divisor ${\mathcal{D}}_{\hat{\underline{\mu}}(x,t_{r})}=\sum^{m-1}\limits_{k=1}\hat{\mu}_{k}(x,t_{r})$ and ${\mathcal{D}}_{\hat{\underline{\nu}}(x,t_{r})}=\sum^{m-1}\limits_{k=1}\hat{\nu}_{k}(x,t_{r})$:

[TABLE]

where $\underline{\hat{\mu}}(x,t_{r})=(\hat{\mu}_{1}(x,t_{r}),\cdots,\hat{\mu}_{m-1}(x,t_{r})),\underline{\hat{\nu}}(x,t_{r})=(\hat{\nu}_{1}(x,t_{r}),\cdots,\hat{\nu}_{m-1}(x,t_{r}))$.

Let $\theta(\underline{z}(\cdot,\cdot))$ denote the Riemann theta function ([8, 18]) on ${\mathcal{K}}_{m-1}$. Here $\underline{z}(\cdot,\cdot)$ is defined as

[TABLE]

where $M$ is the Riemann constant vector. Then the Riemann theta function representation of $\phi(P,x,t_{r})$ reads as follows.

Theorem 4.1. Let the curve ${\mathcal{K}}_{m-1}$ be nonsingular, $P=(\lambda,y)\in{\mathcal{K}}_{m-1}\setminus\{P_{\infty}\}$, and $(x,t_{r}),(x_{0},t_{0,r})\in\Omega_{\mu}$, where $\Omega_{\mu}\subseteq\mathbb{C}^{2}$ is open and connected. Suppose also that ${\mathcal{D}}_{\hat{\underline{\mu}}(x,t_{r})}$, or equivalently, ${\mathcal{D}}_{\hat{\underline{\nu}}(x,t_{r})}$ is nonspecial for $(x,t_{r})\in\Omega_{\mu}$. Then $\phi(P,x,t_{r})$ may be explicitly constructed by the formula

[TABLE]

Proof. Let $\Phi$ denote the right hand side of (4.20), we now have to show $\phi=\Phi$. The Riemann theorem and (4.17) allow us to conclude that $\Phi$ exactly has zeros at the points $\hat{\nu}_{0}(x,t_{r}),\hat{\nu}_{1}(x,t_{r})\cdots,\hat{\nu}_{m-1}(x,t_{r})$ and poles at $P_{\infty},\hat{\mu}_{1}(x,t_{r}),$ $\cdots,\hat{\mu}_{m-1}(x,t_{r}).$ It infers from (4.17) that

[TABLE]

which together with (4.1) gives

[TABLE]

Applying the Riemann-Roch theorem, we get $\frac{\Phi}{\phi}=1$, which completes the proof.

Based on the above results, we will obtain the Riemann theta function representations of solutions for the entire Dym-type hierarchy immediately.

**Theorem 4.2. ** Assume that the curve $\mathcal{K}_{m-1}$ is nonsingular and let $(x,t_{r})\in\Omega_{\mu}$, where $\Omega_{\mu}\subseteq\mathbb{C}^{2}$ is open and connected. Suppose also that $\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})}$, or equivalently, $\mathcal{D}_{\underline{\hat{\nu}}(x,t_{r})}$ is nonspecial for $(x,t_{r})\in\Omega_{\mu}$. Then the Dym-type hierarchy admits algebro-geometric solutions

[TABLE]

with $\delta(m)$ defined in (4.16).

Proof. From (4.9), (4.12), (4.18) and (4.19) it follows that

[TABLE]

Hence

[TABLE]

where $\partial_{U_{2}^{(2)}}\theta(\underline{z}(P_{\infty},\underline{\hat{\mu}}(x,t_{r})))=\left.[\partial_{U_{2}^{(2)}}\theta(M-\mathcal{A}(P_{\infty})+\rho^{(1)}(x,t_{r})-U^{(2)}_{2}\zeta+O(\zeta^{2}))]\right|_{\zeta=0}$ and $\partial_{U_{2}^{(2)}}=\sum\limits_{j=1}^{m-1}U_{2,j}^{(2)}\frac{\partial}{\partial z_{j}}$. Quite similarly, we get

[TABLE]

with $\partial_{U_{2}^{(2)}}\theta(\underline{z}(P_{\infty},\underline{\hat{\nu}}(x,t_{r})))=\left.[\partial_{U_{2}^{(2)}}\theta(M-\mathcal{A}(P_{\infty})+\rho^{(2)}(x,t_{r})-U^{(2)}_{2}\zeta+O(\zeta^{2}))]\right|_{\zeta=0}$.

By virtue of (4.17), (4.20), (4.24) and (4.25), we have

[TABLE]

Comparing (4.1) with (4.26), we arrive at (4.23).

Acknowledgments

This work was supported by National Natural Science Foundation of China (project nos.11331008, 11401230 and 11501526 ), Cultivation Program for Outstanding Young Scientific talents of the Higher Education Institutions of Fujian Province in 2015, and Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (project no. ZQN-PY301).