On a Harry–Dym-Type Hierarchy: Trigonal Curve and Quasi-Periodic Solutions

Abstract

Resorting to the characteristic polynomial of Lax matrix for a Harry–Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker–Akhiezer function and meromorphic function are introduced. With the help of 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 obtain the quasi-periodic solutions for the entire Harry–Dym-type hierarchy.

1. Introduction

The Harry–Dym equation

$$\begin{array}{c}{u}_t={(u^{-\frac{1}{2}})}_{xxx},\hfill \end{array}$$

(1)

was first discovered in an unpublished work by Harry–Dym [1] and rediscovered in a more general form within the classical string problem by Sabatier [2] and Li [3]. It was shown that the Harry–Dym equation admits many properties typical for soliton equations, such as inverse scattering transform, bi-Hamiltonian structure, and an infinite number of conservation laws and symmetries (see [4,5,6,7] and references therein). The quasi-periodic and involutive solutions of Harry–Dym equation were also discussed in [8,9,10,11]. Moreover, the integrable extensions of the Harry–Dym equation have attracted the attention of many researchers [12,13,14,15,16,17,18,19,20,21].

Konopelchenko and Dubrovsky [12,14] found the following Harry–Dym-type equation

$$\begin{array}{c}{v}_t={(v^{-\frac{2}{3}})}_{xxxxx},\hfill \end{array}$$

(2)

from the reduction of a 2 + 1 dimensional system. The authors in Refs. [22,23] derived a hierarchy of Harry–Dym-type equations and discussed their parametric solutions through the method of nonlinearization. Furthermore, the Harry–Dym-type equation can be linked with the Kaup–Kupershmidt or Sawada–Kotera equation by hodograph and Miura transformations [24]. The principal aim of the present paper is to study the algebro-geometric constructions and quasi-periodic solutions [25,26,27,28,29] of the Harry–Dym-type hierarchy, with the aid of the theory of trigonal curve [30,31].

The outline of the present paper is as follows. In Section 2, in view of the Lenard recursion equations and the zero-curvature equation, we derive the Harry–Dym-type hierarchy. In Section 3, we introduce 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 defined with the help of the characteristic polynomial of Lax matrix for the Harry–Dym-type hierarchy. After this, the Harry–Dym-type hierarchy is decomposed into a system of Dubrovin-type equations. In Section 4, by introducing three kinds of Abelian differentials, we present the Riemann theta function representations of the meromorphic function, and in particular, that of the potential for the entire Harry–Dym-type hierarchy.

2. Harry–Dym-Type Hierarchy

In this section, we shall recall the derivation of the Harry–Dym-type hierarchy, in which the first nontrivial member includes Equation (2). To this end, we introduce two sets of Lenard recursion equations

$$\begin{array}{c}\hfill K{g}_{j-1}=J{g}_j,j\ge 0,\end{array}$$

(3)

$$\begin{array}{c}\hfill K{\widehat{g}}_{j-1}=J{\widehat{g}}_j,j\ge 0,\end{array}$$

(4)

with two starting points

$$g_{-1}=\left(\begin{array}{c}{v}^{-\frac{2}{3}}\\ 0\end{array}\right),{\widehat{g}}_{-1}=\left(\begin{array}{c}-\frac{1}{3}{v}^{-1}{(v^{-\frac{1}{3}})}_{xx}+\frac{1}{6}{v}^{-\frac{2}{3}}[{(v^{-\frac{1}{3}}{)}_x]}^2\\ v^{-\frac{1}{3}}\end{array}\right),$$

and two operators are defined as

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

It is easy to see that

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

In order to generate a hierarchy of Harry–Dym-type equations associated with the $3\times 3$ matrix spectral problem [22,23]

$$\begin{array}{c}\hfill {\psi }_x=U\psi ,\psi =\left(\begin{array}{c}{\psi }_1\\ {\psi }_2\\ {\psi }_3\end{array}\right),U=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ \lambda v& 0& 0\end{array}\right),\end{array}$$

(5)

where $v(\ne 0)$ is a potential and $\lambda$ is a constant spectral parameter, we solve the stationary zero-curvature equation

$$\begin{array}{c}\hfill V_x-[U,V]=0,V=\lambda \left(\begin{array}{ccc}{V}_{11}& V_{12}& V_{13}\\ V_{21}& V_{22}& V_{23}\\ V_{31}& V_{32}& V_{33}\end{array}\right),\end{array}$$

(6)

which is equivalent to

$$\begin{array}{ccc}\hfill & & V_{11,x}+\lambda v{V}_{13}-V_{21}=0,\hfill \\ \hfill & & V_{12,x}+V_{11}-V_{22}=0,\hfill \\ \hfill & & V_{13,x}+V_{12}-V_{23}=0,\hfill \\ \hfill & & V_{21,x}+\lambda v{V}_{23}-V_{31}=0,\hfill \\ \hfill & & V_{22,x}+V_{21}-V_{32}=0,\hfill \\ \hfill & & V_{23,x}+V_{22}-V_{33}=0,\hfill \\ \hfill & & V_{31,x}-\lambda v(V_{11}-V_{33})=0,\hfill \\ \hfill & & V_{32,x}-\lambda v{V}_{12}+V_{31}=0,\hfill \\ \hfill & & V_{33,x}-\lambda v{V}_{13}+V_{32}=0,\hfill \end{array}$$

(7)

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

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

(8)

Substituting (8) into (7) and expanding the functions a and b into the Laurent series in $\lambda$

$$\begin{array}{c}\hfill a={\displaystyle \sum _{j\ge 0}}{a}_{j-1}{\lambda }^{-2j},b={\displaystyle \sum _{j\ge 0}}{b}_{j-1}{\lambda }^{-2j},\end{array}$$

(9)

we obtain the recursion equations

$$\begin{array}{c}\hfill K{G}_{j-1}=J{G}_j,J{G}_{-1}=0,j\ge 0,\end{array}$$

(10)

with $G_j={(a_j,b_j)}^T$. Since equation $J{G}_{-1}=0$ has the general solution

$$\begin{array}{c}\hfill G_{-1}={\alpha }_0{g}_{-1}+{\beta }_0{\widehat{g}}_{-1},\end{array}$$

(11)

$G_j$ can be expressed as

$$\begin{array}{c}{G}_j={\alpha }_0{g}_j+{\beta }_0{\widehat{g}}_j+\cdots +{\alpha }_j{g}_0+{\beta }_j{\widehat{g}}_0+{\alpha }_{j+1}{g}_{-1}+{\beta }_{j+1}{\widehat{g}}_{-1},j\ge 0,\hfill \end{array}$$

(12)

where ${\alpha }_j$ and ${\beta }_j$ are arbitrary constants. Let $\psi$ satisfy the spectral problem (5) and an auxiliary problem

$$\begin{array}{c}\hfill {\psi }_{t_r}={\tilde{V}}^{(r)}\psi ,{\tilde{V}}^{(r)}=\lambda \left(\begin{array}{ccc}{\tilde{V}}_{11}^{(r)}& {\tilde{V}}_{12}^{(r)}& {\tilde{V}}_{13}^{(r)}\\ {\tilde{V}}_{21}^{(r)}& {\tilde{V}}_{22}^{(r)}& {\tilde{V}}_{23}^{(r)}\\ {\tilde{V}}_{31}^{(r)}& {\tilde{V}}_{32}^{(r)}& {\tilde{V}}_{33}^{(r)}\end{array}\right),\end{array}$$

(13)

where

$$\begin{array}{ccc}\hfill & & {\tilde{V}}_{ij}^{(r)}=V_{ij}({\tilde{a}}^{(r)},{\tilde{b}}^{(r)}),{\tilde{a}}^{(r)}={\displaystyle \sum _{j=0}^r}{\tilde{a}}_{j-1}{\lambda }^{2(r-j)},{\tilde{b}}^{(r)}={\displaystyle \sum _{j=0}^r}{\tilde{b}}_{j-1}{\lambda }^{2(r-j)},\hfill \\ \hfill & & {({\tilde{a}}_j,{\tilde{b}}_j)}^T={\tilde{\alpha }}_0{g}_j+{\tilde{\beta }}_0{\widehat{g}}_j+\cdots +{\tilde{\alpha }}_j{g}_0+{\tilde{\beta }}_j{\widehat{g}}_0+{\tilde{\alpha }}_{j+1}{g}_{-1}+{\tilde{\beta }}_{j+1}{\widehat{g}}_{-1},j\ge -1,\hfill \end{array}$$

(14)

and the constants ${\tilde{\alpha }}_j,{\tilde{\beta }}_j$ are independent of ${\alpha }_j,{\beta }_j.$ Then the compatibility condition of (5) and (13) yields the zero-curvature equation, $U_{t_r}-{\tilde{V}}_x^{(r)}+[U,{\tilde{V}}^{(r)}]=0$, which is equivalent to

$$\begin{array}{c}\hfill \left(\begin{array}{c}{v}_{t_r}\\ 0\end{array}\right)=K\left(\begin{array}{c}{\tilde{a}}_{r-1}\\ {\tilde{b}}_{r-1}\end{array}\right)=J\left(\begin{array}{c}{\tilde{a}}_r\\ {\tilde{b}}_r\end{array}\right).\end{array}$$

(15)

The first component in (15) gives rise to a hierarchy of nonlinear evolution equations

$$\begin{array}{c}\hfill v_{t_r}=-\frac{1}{3}{\partial }^5{\tilde{a}}_{r-1}=-(\partial v+2v\partial ){\tilde{b}}_r,\end{array}$$

(16)

in which the first nontrivial member is

$$\begin{array}{ccc}\hfill & & v_{t_0}=-\frac{1}{3}{\partial }^5\{{\tilde{\alpha }}_0{v}^{-\frac{2}{3}}+{\tilde{\beta }}_0[-\frac{1}{3}{v}^{-1}{(v^{-\frac{1}{3}})}_{xx}+\frac{1}{6}{v}^{-\frac{2}{3}}({v^{-\frac{1}{3}}{)}_x}^2]\}.\hfill \end{array}$$

(17)

If choosing ${\tilde{\alpha }}_0=-3,{\tilde{\beta }}_0=0,t_0=t$ in the Equation (17), it reduces to the Harry–Dym-type Equation (2).

 Remark 1.

1.

The second component of (15) infers that ${\tilde{b}}_r=(-2{\partial }^{-2}v-{\partial }^{-3}v\partial ){\tilde{a}}_r.$ Substituting it into (16), two Hamiltonian operators for the Harry–Dym-type hierarchy read as $H_1={\partial }^5$ and $H_2=\partial v{\partial }^{-3}v\partial +4v{\partial }^{-1}v+2v{\partial }^{-2}v\partial +2\partial v{\partial }^{-2}v,$ which immediately gives recursion operator $R=H_1{H}_2^{-1}.$

2.

Here, we only show the first nontrivial member (17) of the Harry–Dym-type hierarchy (16) for $r=0$, since the higher flows for $r\ge 1$ are very complicated. If you are interested, you can use some mathematical softwares to compute.

3. The Trigonal Curve and Dubrovin-Type Equations

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

We introduce the vector-valued Baker–Akhiezer function

$$\begin{array}{ccc}\hfill & & {\psi }_x(P,x,x_0,t_r,t_{0,r})=U(v,\lambda )\psi (P,x,x_0,t_r,t_{0,r}),\hfill \\ \hfill & & {\psi }_{t_r}(P,x,x_0,t_r,t_{0,r})={\tilde{V}}^{(r)}(v,\lambda )\psi (P,x,x_0,t_r,t_{0,r}),\hfill \\ \hfill & & {\lambda }^{-1}{V}^{(n)}(v,\lambda )\psi (P,x,x_0,t_r,t_{0,r})=y(P)\psi (P,x,x_0,t_r,t_{0,r}),\hfill \\ \hfill & & {\psi }_1(P,x_0,x_0,t_{0,r},t_{0,r})=1,P=(\lambda ,y),x,t_r\in \mathbb{C}.\hfill \end{array}$$

(18)

Here $V^{(n)}=\lambda {\left(\begin{array}{c}{V}_{ij}^{(n)}\end{array}\right)}_{3\times 3},V_{ij}^{(n)}=V_{ij}(a^{(n)},b^{(n)}),$ and

$$\begin{array}{c}\hfill a^{(n)}={\displaystyle \sum _{j=0}^n}{a}_{j-1}{\lambda }^{2(n-j)},b^{(n)}={\displaystyle \sum _{j=0}^n}{b}_{j-1}{\lambda }^{2(n-j)}.\end{array}$$

The compatibility conditions of the first three equations in (18) yield that

$$\begin{array}{c}{U}_{t_r}-{\tilde{V}}_x^{(r)}+[U,{\tilde{V}}^{(r)}]=0,\hfill \end{array}$$

(19)

$$\begin{array}{c}-V_x^{(n)}+[U,V^{(n)}]=0,\hfill \end{array}$$

(20)

$$\begin{array}{c}-V_{t_r}^{(n)}+[{\tilde{V}}^{(r)},V^{(n)}]=0.\hfill \end{array}$$

(21)

A direct calculation shows that $yI-{\lambda }^{-1}{V}^{(n)}$ satisfies (20) and (21), which ensures that the characteristic polynomial of Lax matrix ${\lambda }^{-1}{V}^{(n)}$ for the Harry-Dym-type, i.e., ${\mathcal{F}}_m(\lambda ,y)=\det (yI-{\lambda }^{-1}{V}^{(n)})$, is a constant independent of variables x and $t_r$. Hence, ${\mathcal{F}}_m(\lambda ,y)=0$ naturally leads to a trigonal curve

$$\begin{array}{c}\hfill {\mathcal{K}}_{m-1}:{\mathcal{F}}_m(\lambda ,y)=y^3+y{S}_m(\lambda )-T_m(\lambda )=0,\end{array}$$

(22)

where

$$\begin{array}{ccc}\hfill & & S_m(\lambda )=\left|\begin{array}{cc}{V}_{11}^{(n)}& V_{12}^{(n)}\\ V_{21}^{(n)}& V_{22}^{(n)}\end{array}\right|+\left|\begin{array}{cc}{V}_{11}^{(n)}& V_{13}^{(n)}\\ V_{31}^{(n)}& V_{33}^{(n)}\end{array}\right|+\left|\begin{array}{cc}{V}_{22}^{(n)}& V_{23}^{(n)}\\ V_{32}^{(n)}& V_{33}^{(n)}\end{array}\right|,\hfill \\ \\ \hfill & & T_m(\lambda )=\left|\begin{array}{ccc}{V}_{11}^{(n)}& V_{12}^{(n)}& V_{13}^{(n)}\\ V_{21}^{(n)}& V_{22}^{(n)}& V_{23}^{(n)}\\ V_{31}^{(n)}& V_{32}^{(n)}& V_{33}^{(n)}\end{array}\right|=\left\{\begin{array}{c}{\beta }_0^3{\lambda }^{6n+4}+\cdots ,{\beta }_0\ne 0,{\alpha }_0\in \mathbb{R},\hfill \\ -8{\alpha }_0^3{\lambda }^{6n+2}+\cdots ,{\beta }_0=0,{\alpha }_0\ne 0.\hfill \end{array}\right.\hfill \end{array}$$

(23)

It is evident that $T_m(\lambda )$ is a polynomial of degree $m=6n+4=3(2n+1)+1$ and $m=6n+2=3(2n)+2$ as ${\beta }_0\ne 0$, ${\alpha }_0\in \mathbb{R}$ and ${\beta }_0=0,{\alpha }_0\ne 0$, respectively. We compactify the trigonal curve ${\mathcal{K}}_{m-1}$ to be a three-sheeted Riemann surface of arithmetic genus $m-1$, and still denote it by the same symbol ${\mathcal{K}}_{m-1}$.

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

$$\begin{array}{c}\hfill {\displaystyle \varphi (P)=\varphi (P,x,t_r)=v^{-\frac{1}{3}}\frac{{\psi }_{1,x}(P,x,x_0,t_r,t_{0,r})}{{\psi }_1(P,x,x_0,t_r,t_{0,r})}=v^{-\frac{1}{3}}\frac{{\psi }_2(P,x,x_0,t_r,t_{0,r})}{{\psi }_1(P,x,x_0,t_r,t_{0,r})},P\in {\mathcal{K}}_{m-1},}\end{array}$$

(24)

which, together with (18), infers that

$$\begin{array}{c}\varphi (P)=v^{-\frac{1}{3}}\frac{{\displaystyle y{V}_{23}^{(n)}+C_m}}{{\displaystyle y{V}_{13}^{(n)}+A_m}}=\frac{{\displaystyle v^{-\frac{1}{3}}{F}_m}}{{\displaystyle y^2{V}_{23}^{(n)}-y{C}_m+D_m}}=\frac{{\displaystyle y^2{V}_{13}^{(n)}-y{A}_m+B_m}}{{\displaystyle v^{\frac{1}{3}}{E}_{m-1}}},\hfill \end{array}$$

(25)

where

$$\begin{array}{c}{A}_m=V_{12}^{(n)}{V}_{23}^{(n)}-V_{13}^{(n)}{V}_{22}^{(n)},C_m=V_{13}^{(n)}{V}_{21}^{(n)}-V_{11}^{(n)}{V}_{23}^{(n)},\hfill \\ B_m=V_{13}^{(n)}(V_{11}^{(n)}{V}_{33}^{(n)}-V_{13}^{(n)}{V}_{31}^{(n)})+V_{12}^{(n)}(V_{11}^{(n)}{V}_{23}^{(n)}-V_{13}^{(n)}{V}_{21}^{(n)}),\hfill \\ D_m=V_{23}^{(n)}(V_{22}^{(n)}{V}_{33}^{(n)}-V_{23}^{(n)}{V}_{32}^{(n)})+V_{21}^{(n)}(V_{13}^{(n)}{V}_{22}^{(n)}-V_{12}^{(n)}{V}_{23}^{(n)}),\hfill \\ E_{m-1}={(V_{13}^{(n)})}^2{V}_{32}^{(n)}+V_{12}^{(n)}{V}_{13}^{(n)}(V_{22}^{(n)}-V_{33}^{(n)})-{(V_{12}^{(n)})}^2{V}_{23}^{(n)},\hfill \\ F_m={(V_{23}^{(n)})}^2{V}_{31}^{(n)}+V_{21}^{(n)}{V}_{23}^{(n)}(V_{11}^{(n)}-V_{33}^{(n)})-V_{13}^{(n)}{(V_{21}^{(n)})}^2.\hfill \end{array}$$

(26)

Taking (22) and (25) 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$:

$$\begin{array}{c}{V}_{13}^{(n)}{F}_m=V_{23}^{(n)}{D}_m-{(V_{23}^{(n)})}^2{S}_m-C_m^2,\hfill \\ A_m{F}_m={(V_{23}^{(n)})}^2{T}_m+C_m{D}_m,\hfill \end{array}$$

(27)

$$\begin{array}{c}{V}_{23}^{(n)}{E}_{m-1}=V_{13}^{(n)}{B}_m-{(V_{13}^{(n)})}^2{S}_m-A_m^2,\hfill \\ C_m{E}_{m-1}={(V_{13}^{(n)})}^2{T}_m+A_m{B}_m,\hfill \end{array}$$

(28)

$$\begin{array}{c}{V}_{23}^{(n)}{B}_m+V_{13}^{(n)}{D}_m-V_{13}^{(n)}{V}_{23}^{(n)}{S}_m+A_m{C}_m=0,\hfill \\ V_{13}^{(n)}{V}_{23}^{(n)}{T}_m+V_{23}^{(n)}{A}_m{S}_m+V_{13}^{(n)}{C}_m{S}_m-B_m{C}_m-A_m{D}_m=0,\hfill \\ V_{23}^{(n)}{A}_m{T}_m+V_{13}^{(n)}{C}_m{T}_m-B_m{D}_m+E_{m-1}{F}_m=0,\hfill \end{array}$$

(29)

$$\begin{array}{c}{E}_{m-1,x}=-2V_{13}^{(n)}{S}_m+3B_m,\hfill \\ V_{23}^{(n)}{F}_{m,x}=-3V_{22}^{(n)}{F}_m+V_{21}^{(n)}(2V_{23}^{(n)}{S}_m-3D_m).\hfill \end{array}$$

(30)

In what follows, we shall present some properties of the meromorphic function $\varphi (P,x,t_r)$ without proofs.

$$\begin{array}{c}{[v^{\frac{1}{3}}\varphi (P)]}_{xx}+3v^{\frac{1}{3}}\varphi (P){[v^{\frac{1}{3}}\varphi (P)]}_x+v{\varphi }^3(P)=\lambda v,\hfill \end{array}$$

(31)

$$\begin{array}{c}{[v^{\frac{1}{3}}\varphi (P)]}_{t_r}=\lambda {\left({\tilde{V}}_{11}^{(r)}+v^{\frac{1}{3}}{\tilde{V}}_{12}^{(r)}\varphi (P)+{\tilde{V}}_{13}^{(r)}[{(v^{\frac{1}{3}}\varphi (P))}_x+v^{\frac{2}{3}}{\varphi }^2(P)]\right)}_x,\hfill \end{array}$$

(32)

$$v\varphi (P)\varphi (P^{*})\varphi (P^{**})=-\frac{F_m}{E_{m-1}},$$

(33)

$$v^{\frac{1}{3}}[\varphi (P)+\varphi (P^{*})+\varphi (P^{**})]=\frac{E_{m-1,x}}{E_{m-1}},$$

(34)

$$\begin{array}{c}\frac{1}{\varphi (P)}+\frac{1}{\varphi (P^{*})}+\frac{1}{\varphi (P^{**})}=v^{\frac{1}{3}}\left[-3\frac{V_{22}^{(n)}}{V_{21}^{(n)}}-\frac{V_{23}^{(n)}{F}_{m,x}}{V_{21}^{(n)}{F}_m}\right].\hfill \end{array}$$

(35)

Here $P=(\lambda ,y),P^{*}=(\lambda ,y^{*}),P^{**}=(\lambda ,y^{**})$, and $y,y^{*},y^{**}$ denote three branches of y satisfying ${\mathcal{F}}_m(\lambda ,y)=0.$

Lemma 1.

Assume that (18) and (19) hold, and let $(\lambda ,x,t_r)\in {\mathbb{C}}^3$. Then

$$\begin{array}{c}\hfill E_{m-1,t_r}=\lambda E_{m-1,x}[{\tilde{V}}_{12}^{(r)}-\frac{{\tilde{V}}_{13}^{(r)}}{V_{13}^{(n)}}{V}_{12}^{(n)}]+3\lambda E_{m-1}[{\tilde{V}}_{11}^{(r)}-\frac{{\tilde{V}}_{13}^{(r)}}{V_{13}^{(n)}}{V}_{11}^{(n)}],\end{array}$$

(36)

$$\begin{array}{c}{F}_{m,t_r}=\lambda F_{m,x}[{\tilde{V}}_{23}^{(r)}-\frac{{\tilde{V}}_{21}^{(r)}}{V_{21}^{(n)}}{V}_{23}^{(n)}]+3\lambda F_m[{\tilde{V}}_{22}^{(r)}-\frac{{\tilde{V}}_{21}^{(r)}}{V_{21}^{(n)}}{V}_{22}^{(n)}].\hfill \end{array}$$

(37)

Proof. 

Differentiating (34) with respect to $t_r$ and using (32) and (34), we have

$$\begin{array}{c}{\partial }_x{\partial }_{t_r}(ln{E}_{m-1})=\lambda {\left[({\tilde{V}}_{12}^{(r)}-\frac{{\tilde{V}}_{13}^{(r)}}{V_{13}^{(n)}}{V}_{12}^{(n)})\frac{E_{m-1,x}}{E_{m-1}}+3({\tilde{V}}_{11}^{(r)}-\frac{{\tilde{V}}_{13}^{(r)}}{V_{13}^{(n)}}{V}_{11}^{(n)})\right]}_x.\hfill \end{array}$$

Integrating the above equation with respect to x, and choosing the integration constant as zero to imply (36). Differentiating (33) with respect to $t_r$, an analogous process shows (37). □

By observing (12) and (26), we find that $E_{m-1}$ and $F_m$ are polynomials with respect to $\lambda$ of degree $m-1$ and m, respectively. Therefore,

$$E_{m-1}(\lambda ,x,t_r)=-ϵ(m)v^{-1}\prod _{j=1}^{m-1}(\lambda -{\mu }_j(x,t_r)),$$

(38)

$$F_m(\lambda ,x,t_r)=ϵ(m)\prod _{l=0}^{m-1}(\lambda -{\nu }_l(x,t_r)),$$

(39)

with

$$\begin{array}{c}\hfill ϵ(m)=\left\{\begin{array}{c}{\beta }_0^3,m=6n+4,\hfill \\ 8{\alpha }_0^3,m=6n+2.\hfill \end{array}\right.\end{array}$$

Let us denote

$$\begin{array}{ccc}\hfill {\widehat{\mu }}_j(x,t_r)& =& ({\mu }_j(x,t_r),y({\widehat{\mu }}_j(x,t_r)))\hfill \\ & =& \left({\mu }_j(x,t_r),-\frac{A_m({\mu }_j(x,t_r),x,t_r)}{V_{13}^{(n)}({\mu }_j(x,t_r),x,t_r)}\right)\in {\mathcal{K}}_{m-1},1\le j\le m-1,\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill {\widehat{\nu }}_l(x,t_r)& =& ({\nu }_l(x,t_r),y({\widehat{\nu }}_l(x,t_r)))\hfill \\ & =& \left({\nu }_l(x,t_r),-\frac{C_m({\nu }_l(x,t_r),x,t_r)}{V_{23}^{(n)}({\nu }_l(x,t_r),x,t_r)}\right)\in {\mathcal{K}}_{m-1},0\le l\le m-1,\hfill \end{array}$$

(41)

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

Lemma 2.

Suppose the zeros ${\left\{{\mu }_j(x,t_r)\right\}}_{j=1}^{m-1}$ and ${\left\{{\nu }_l(x,t_r)\right\}}_{l=0}^{m-1}$ of $E_{m-1}(\lambda ,x,t_r)$ and $F_m(\lambda ,x,t_r)$ remain distinct for $(x,t_r)\in {\mathsf{\Omega }}_{\mu }$ and $(x,t_r)\in {\mathsf{\Omega }}_{\nu }$, respectively, where ${\mathsf{\Omega }}_{\mu },{\mathsf{\Omega }}_{\nu }\subseteq {\mathbb{C}}^2$ are open and connected. Then, ${\left\{{\mu }_j(x,t_r)\right\}}_{j=1}^{m-1}$ and ${\left\{{\nu }_l(x,t_r)\right\}}_{l=0}^{m-1}$ satisfy the Dubrovin-type equations

$$\begin{array}{c}{\mu }_{j,x}=\frac{v{V}_{13}^{(n)}({\mu }_j)[3y^2({\widehat{\mu }}_j)+S_m({\mu }_j)]}{ϵ(m){\prod }_{\begin{array}{c}k=1\\ k\ne j\end{array}}^{m-1}({\mu }_j-{\mu }_k)},1\le j\le m-1,\hfill \end{array}$$

(42)

$$\begin{array}{c}{\mu }_{j,t_r}=\frac{v{\mu }_j[V_{13}^{(n)}({\mu }_j){\tilde{V}}_{12}^{(r)}({\mu }_j)-{\tilde{V}}_{13}^{(r)}({\mu }_j)V_{12}^{(n)}({\mu }_j)][3y^2({\widehat{\mu }}_j)+S_m({\mu }_j)]}{ϵ(m){\prod }_{\begin{array}{c}k=1\\ k\ne j\end{array}}^{m-1}({\mu }_j-{\mu }_k)},1\le j\le m-1,\hfill \end{array}$$

(43)

$$\begin{array}{c}{\nu }_{l,x}=\frac{V_{21}^{(n)}({\nu }_l)[3y^2({\widehat{\nu }}_l)+S_m({\nu }_l)]}{ϵ(m){\prod }_{\begin{array}{c}k=0\\ k\ne l\end{array}}^{m-1}({\nu }_l-{\nu }_k)},0\le l\le m-1,\hfill \end{array}$$

(44)

$$\begin{array}{c}{\nu }_{l,t_r}=\frac{{\nu }_l[V_{21}^{(n)}({\nu }_l){\tilde{V}}_{23}^{(r)}({\nu }_l)-{\tilde{V}}_{21}^{(r)}({\nu }_l)V_{23}^{(n)}({\nu }_l)][3y^2({\widehat{\nu }}_l)+S_m({\nu }_l)]}{ϵ(m){\prod }_{\begin{array}{c}k=0\\ k\ne l\end{array}}^{m-1}({\nu }_l-{\nu }_k)},0\le l\le m-1.\hfill \end{array}$$

(45)

Proof. 

We just need to prove that (42) for the proofs of (43)–(45) are similar to (42). Substituting $\lambda ={\mu }_j$ into the first expression in (30), and using (28) and (41), we obtain

$$\begin{array}{c}{E}_{m-1,x}({\mu }_j,x,t_r)=V_{13}^{(n)}({\mu }_j,x,t_r)[3y^2({\widehat{\mu }}_j)+S_m({\mu }_j)].\hfill \end{array}$$

(46)

On the other hand, differentiating (38) with respect to x and inserting $\lambda ={\mu }_j$ into it give rise to

$$\begin{array}{c}{E}_{m-1,x}({\mu }_j,x,t_r)=ϵ(m)v^{-1}{\mu }_{j,x}\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{m-1}({\mu }_j-{\mu }_k).\hfill \end{array}$$

(47)

A comparison of (46) and (47) yields (42). □

4. Quasi-Periodic Solutions

In this section, we shall derive Riemann theta function representations of the meromorphic function and potentials for the entire Harry–Dym-type hierarchy.

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

$${\mathfrak{a}}_j\circ {\mathfrak{b}}_k={\delta }_{j,k},{\mathfrak{a}}_j\circ {\mathfrak{a}}_k=0,{\mathfrak{b}}_j\circ {\mathfrak{b}}_k=0,j,k=1,\cdots ,m-1,$$

and the basis of holomorphic differentials

$$\begin{array}{ccc}\hfill & & {\tilde{\omega }}_l(P)=\frac{1}{3y^2(P)+S_m(\lambda )}\left\{\begin{array}{c}{\lambda }^{l-1}d\lambda ,1\le l\le m-2n-2,\hfill \\ y(P){\lambda }^{l+2n-m+1}d\lambda ,m-2n-1\le l\le m-1,\hfill \end{array}\right.m=6n+4,\hfill \\ \hfill & & {\tilde{\omega }}_l(P)=\frac{1}{3y^2(P)+S_m(\lambda )}\left\{\begin{array}{c}{\lambda }^{l-1}d\lambda ,1\le l\le m-2n-1,\hfill \\ y(P){\lambda }^{l+2n-m}d\lambda ,m-2n\le l\le m-1,\hfill \end{array}\right.m=6n+2.\hfill \end{array}$$

(48)

Thus, the period matrices A and B constructed by

$$\begin{array}{c}\hfill {\displaystyle A_{jk}={\int }_{{\mathfrak{a}}_k}{\tilde{\omega }}_j,B_{jk}={\int }_{{\mathfrak{b}}_k}{\tilde{\omega }}_j,}\end{array}$$

(49)

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 a positive definite imaginary part (Im $\tau >0$) [32,33]. If we normalize $\tilde{\omega }=({\tilde{\omega }}_1,\cdots ,{\tilde{\omega }}_{m-1})$ into new basis $\omega =({\omega }_1,\cdots ,{\omega }_{m-1})$

$$\begin{array}{c}\hfill {\omega }_j={\displaystyle \sum _{l=1}^{m-1}}{C}_{jl}{\tilde{\omega }}_l,\end{array}$$

(50)

then, we have

$${\displaystyle {\int }_{{\mathfrak{a}}_k}{\omega }_j={\delta }_{jk},{\int }_{{\mathfrak{b}}_k}{\omega }_j={\tau }_{jk},j,k=1,\cdots ,m-1.}$$

The Riemann theta function $\theta (\underline{z})$ [32,33] on ${\mathcal{K}}_{m-1}$ is defined as

$$\theta (\underline{z})=\sum _{\underline{N}\in {\mathbb{Z}}^{m-1}}exp(2\pi i<\underline{z},\underline{N}>+\pi i<\tau \underline{N},\underline{N}>),\underline{z}\in {\mathbb{C}}^{m-1},$$

where $<·,·>$ stands for the Euclidean scalar product.

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

$$\begin{array}{c}\hfill {\displaystyle {\int }_{{\mathfrak{a}}_j}{\omega }_{P_{\infty },2}^{(2)}(P)=0,j=1,\cdots ,m-1,}\end{array}$$

(51)

$$\begin{array}{c}\hfill {\omega }_{P_{\infty },2}^{(2)}(P)=({\zeta }^{-2}+O(1))d\zeta ,P\to P_{\infty },\end{array}$$

(52)

where $\zeta ={\lambda }^{-\frac{1}{3}}$ is a local coordinate near $P_{\infty }$. The b-periods of the differential ${\omega }_{P_{\infty },2}^{(2)}$ are denoted by

$$\begin{array}{c}\hfill {\displaystyle U_2^{(2)}=(U_{2,1}^{(2)},\cdots ,U_{2,m-1}^{(2)}),U_{2,j}^{(2)}=\frac{1}{2\pi i}{\int }_{{\mathfrak{b}}_j}{\omega }_{P_{\infty },2}^{(2)}(P).}\end{array}$$

(53)

Furthermore, let ${\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}(P)$ denote the normalized Abelian differential of the third kind defined by

$$\begin{array}{c}\hfill {\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}(P)=-\frac{y^2(P)+2y^2({\widehat{\nu }}_0(x,t_r))+S_m({\nu }_0(x,t_r))}{\lambda -{\nu }_0(x,t_r)}\frac{d\lambda }{3y^2(P)+S_m(\lambda )}+{\displaystyle \sum _{j=1}^{m-1}}{\gamma }_j{\omega }_j,\end{array}$$

(54)

which is holomorphic on ${\mathcal{K}}_{m-1}\setminus \{P_{\infty },{\widehat{\nu }}_0(x,t_r)\}$ and has simple poles at $P_{\infty }$ and ${\widehat{\nu }}_0(x,t_r)$, with corresponding residues $+1$ and $-1$. The constants ${\left\{{\gamma }_j\right\}}_{j=1}^{m-1}$ are determined by the normalization condition

$$\begin{array}{c}\hfill {\displaystyle {\int }_{{\mathfrak{a}}_j}{\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}(P)=0,j=1,\cdots ,m-1.}\end{array}$$

(55)

In terms of the local coordinate $\zeta ={\lambda }^{-\frac{1}{3}}$ near $P_{\infty }$ and $\zeta =\lambda -{\nu }_0(x,t_r)$ near ${\widehat{\nu }}_0(x,t_r)$, respectively, a direct calculation shows that

$$\begin{array}{c}\hfill {\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}(P)\underset{\zeta \to 0}{=}\left\{\begin{array}{c}({\zeta }^{-1}-\delta (m)+O(\zeta ))d\zeta ,P\to P_{\infty },\hfill \\ (-{\zeta }^{-1}+O(1))d\zeta ,P\to {\widehat{\nu }}_0(x,t_r),\hfill \end{array}\right.\end{array}$$

(56)

with

$$\begin{array}{c}\hfill \delta (m)=\left\{\begin{array}{c}\hfill \frac{1}{{\beta }_0}{\displaystyle \sum _{j=1}^{m-1}}{\gamma }_j{C}_{j,m-1},m=6n+4,\\ \frac{1}{4{\alpha }_0^2}{\displaystyle \sum _{j=1}^{m-1}}{\gamma }_j{C}_{j,4n+1},m=6n+2.\end{array}\right.\end{array}$$

(57)

Then,

$$\begin{array}{c}\hfill {\displaystyle {\int }_{Q_0}^P{\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}(P)\underset{\zeta \to 0}{=}\left\{\begin{array}{c}ln\zeta +e_{\infty }^{(3)}(Q_0)-\delta (m)\zeta +O({\zeta }^2),P\to P_{\infty },\hfill \\ -ln\zeta +e_0^{(3)}(Q_0)+O(\zeta ),P\to {\widehat{\nu }}_0(x,t_r),\hfill \end{array}\right.}\end{array}$$

(58)

with $Q_0$ as a chosen base point on ${\mathcal{K}}_{m-1}\setminus \{P_{\infty },{\widehat{\nu }}_0(x,t_r)\}$ 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 $\mathcal{A}:{\mathcal{K}}_{m-1}\to {\mathcal{J}}_{m-1}$ is defined as

$${\displaystyle \mathcal{A}(P)=\left({\mathcal{A}}_1(P),\cdots ,{\mathcal{A}}_{m-1}(P)\right)=({\int }_{Q_0}^P{\omega }_1,\cdots ,{\int }_{Q_0}^P{\omega }_{m-1)}(\mathrm{mod}{\mathcal{T}}_{m-1}),}$$

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

$$\mathcal{A}(\sum _k{n}_k{P}_k)=\sum _k{n}_k\mathcal{A}(P_k),$$

which enables us to give

$$\begin{array}{c}\hfill {\displaystyle \mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)})=\sum _{k=1}^{m-1}{\int }_{Q_0}^{{\widehat{\mu }}_k(x,t_r)}\omega ,\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)})=\sum _{k=1}^{m-1}{\int }_{Q_0}^{{\widehat{\nu }}_k(x,t_r)}\omega ,}\end{array}$$

(59)

where $\underline{\widehat{\mu }}(x,t_r)=({\widehat{\mu }}_1(x,t_r),\cdots ,{\widehat{\mu }}_{m-1}(x,t_r)),\underline{\widehat{\nu }}(x,t_r)=({\widehat{\nu }}_1(x,t_r),\cdots ,{\widehat{\nu }}_{m-1}(x,t_r))$, ${\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}={\displaystyle \sum _{k=1}^{m-1}}{\widehat{\mu }}_k(x,t_r)$ and ${\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}={\displaystyle \sum _{k=1}^{m-1}}{\widehat{\nu }}_k(x,t_r)$.

Taking the local coordinate $\zeta ={\lambda }^{-\frac{1}{3}}$ near $P_{\infty }\in {\mathcal{K}}_{m-1}$ in (31), the Laurent series of $\varphi (P,x,t_r)$ can be explicitly expressed as

$$\begin{array}{c}\hfill \varphi (P,x,t_r)\underset{\zeta \to 0}{=}\frac{1}{\zeta }{\displaystyle \sum _{j=0}^{\infty }}{\kappa }_j(x,t_r){\zeta }^j,P\to P_{\infty },\end{array}$$

(60)

where

$$\begin{array}{ccc}& & {\kappa }_0=1,{\kappa }_1={(v^{-\frac{1}{3}})}_x,{\kappa }_2=\frac{2}{9}{v}^{-\frac{5}{3}}{v}_{xx}-\frac{7}{27}{v}^{-\frac{8}{3}}{v}_x^2,{\kappa }_3=\frac{1}{3}{v}^{-\frac{2}{3}}{(v^{-\frac{1}{3}})}_{xxx},\hfill \\ & & {\displaystyle {\kappa }_j=-\frac{1}{3}[v^{-1}{(v^{\frac{1}{3}}{\kappa }_{j-2})}_{xx}+3v^{-\frac{2}{3}}\sum _{i=0}^{j-1}{\kappa }_{j-1-i}{(v^{\frac{1}{3}}{\kappa }_i)}_x+\sum _{i=1}^{j-1}{\kappa }_i{\kappa }_{j-i}}\hfill \\ & & +{\displaystyle \sum _{i=1}^{j-1}}{\displaystyle \sum _{l=0}^{j-i}}{\kappa }_i{\kappa }_l{\kappa }_{j-i-l}],(j\ge 2).\hfill \end{array}$$

Taken together with (25), Equation (60) shows that the divisor $(\varphi (P,x,t_r))$ of $\varphi (P,x,t_r)$ is

$$(\varphi (P,x,t_r))={\mathcal{D}}_{{\widehat{\nu }}_0(x,t_r),{\widehat{\nu }}_1(x,t_r),\cdots ,{\widehat{\nu }}_{m-1}(x,t_r)}(P)-{\mathcal{D}}_{P_{\infty },{\widehat{\mu }}_1(x,t_r),\cdots ,{\widehat{\mu }}_{m-1}(x,t_r)}(P),$$

(61)

which implies that ${\widehat{\nu }}_0(x,t_r),{\widehat{\nu }}_1(x,t_r),\cdots ,{\widehat{\nu }}_{m-1}(x,t_r)$ are m zeros and $P_{\infty },{\widehat{\mu }}_1(x,t_r),$$\cdots ,$${\widehat{\mu }}_{m-1}(x,t_r)$ are m poles of $\varphi (P,x,t_r)$. In view of (58) and (60), the Riemann and Riemann–Roch theorems ensure that the following theorem holds.

Theorem 1.

Let the curve ${\mathcal{K}}_{m-1}$ be nonsingular, $P=(\lambda ,y)\in {\mathcal{K}}_{m-1}\setminus \left\{P_{\infty }\right\}$, and $(x,t_r)$, $(x_0,t_{0,r})\in {\mathsf{\Omega }}_{\mu }$, where ${\mathsf{\Omega }}_{\mu }\subseteq {\mathbb{C}}^2$ is open and connected. Suppose also that ${\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}$, or equivalently, ${\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}$ is nonspecial for $(x,t_r)\in {\mathsf{\Omega }}_{\mu }$. Then, $\varphi (P,x,t_r)$ may be explicitly constructed by the formula

$$\begin{array}{c}\hfill {\displaystyle \varphi (P,x,t_r)=\frac{\theta (M-\mathcal{A}(P)+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))\theta (M-\mathcal{A}(P)+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}exp(e_{\infty }^{(3)}(Q_0)-{\int }_{Q_0}^P{\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}),}\end{array}$$

(62)

where M is the Riemann constant vector.

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

Theorem 2.

Assume that the curve ${\mathcal{K}}_{m-1}$ is nonsingular and let $(x,t_r)\in {\mathsf{\Omega }}_{\mu }$, where ${\mathsf{\Omega }}_{\mu }\subseteq {\mathbb{C}}^2$ is open and connected. Suppose also that ${\mathcal{D}}_{\underline{\widehat{\mu }}(x,t_r)}$, or equivalently, ${\mathcal{D}}_{\underline{\widehat{\nu }}(x,t_r)}$ is nonspecial for $(x,t_r)\in {\mathsf{\Omega }}_{\mu }$. Then, the Harry–Dym-type hierarchy admits quasi-periodic solutions

$$\begin{array}{c}{(v^{-\frac{1}{3}})}_x={\partial }_{U_2^{(2)}}ln\frac{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))}+\delta (m),\hfill \end{array}$$

(63)

where ${\partial }_{U_2^{(2)}}={\displaystyle \sum _{j=1}^{m-1}}{U}_{2,j}^{(2)}\frac{\partial }{\partial z_j}$, and $\delta (m)$ are defined in (57).

Proof. 

A direct calculation gives the following asymptotic expressions near $P_{\infty }(P\to P_{\infty })$

$$\begin{array}{c}\hfill \frac{\theta (M-\mathcal{A}(P)+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}\underset{\zeta \to 0}{=}1-[{\partial }_{U_2^{(2)}}ln\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))]\zeta +O({\zeta }^2),\\ \frac{\theta (M-\mathcal{A}(P)+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))}{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))}\underset{\zeta \to 0}{=}1-[{\partial }_{U_2^{(2)}}ln\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))]\zeta +O({\zeta }^2),\end{array}$$

which, together with (58) and (62), imply that

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill \varphi (P,x,t_r)& \underset{\zeta \to 0}{=}& [{\zeta }^{-1}+{\partial }_{U_2^{(2)}}ln\frac{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))}+\delta (m)+O(\zeta )],\hfill \end{array}P\to P_{\infty }.\end{array}$$

(64)

Comparing (60) with (64), we arrive at (63). □

Next, we shall give 1-genus and 3-genus quasi-periodic solutions of the Equation (16) for $n=0$.

(i) ${\alpha }_0\ne 0,{\beta }_0=0,$ through direct calculation, we have

$$\begin{array}{ccc}& & m=2,{\mathcal{K}}_1:{\mathcal{F}}_1=y^3+{\eta }_{1}y+8{\alpha }_0^3{\lambda }^2+{\eta }_2=0,g=1,\hfill \\ & & E_1=-8{\alpha }_0^3{v}^{-1}(x,t_r)(\lambda -{\mu }_1(x,t_r)),F_2=8{\alpha }_0^3(\lambda -{\nu }_0(x,t_r))(\lambda -{\nu }_1(x,t_r)),\hfill \\ & & {\displaystyle {\tilde{\omega }}_1=\frac{d\lambda }{3y^2(P)+{\eta }_1},{\omega }_1=C_{11}{\tilde{\omega }}_1,\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)})={\int }_{Q_0}^{{\widehat{\nu }}_1(x,t_r)}{\omega }_1,\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)})={\int }_{Q_0}^{{\widehat{\mu }}_1(x,t_r)}{\omega }_1,}\hfill \\ & & {\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}(P)=-\frac{y^2(P)+2y^2({\widehat{\nu }}_0(x,t_r))+{\eta }_1}{\lambda -{\nu }_0(x,t_r)}\frac{d\lambda }{3y^2(P)+{\eta }_1}+{\gamma }_1{\omega }_1,\hfill \\ & & {\displaystyle \varphi (P,x,t_r)=\frac{\theta (M-\mathcal{A}(P)+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))\theta (M-\mathcal{A}(P)+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}exp(e_{\infty }^{(3)}(Q_0)-{\int }_{Q_0}^P{\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}),}\hfill \\ & & {[v^{-\frac{1}{3}}(x,t_r)]}_x={\partial }_{U_2^{(2)}}ln\frac{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))}+\frac{{\gamma }_1{C}_{11}}{4{\alpha }_0^2},\hfill \end{array}$$

where ${\eta }_1,{\eta }_2$ are arbitrary constants and ${\gamma }_1$ is determined by ${\displaystyle {\int }_{{\mathfrak{a}}_1}{\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}(P)=0.}$

(ii) ${\alpha }_0\in \mathbb{R},{\beta }_0\ne 0,$ tedious computing indicates that

$${\displaystyle \begin{array}{c}m=4,{\mathcal{K}}_4:{\mathcal{F}}_4=y^3+(6{\alpha }_0{\beta }_0{\lambda }^2+{\eta }_1)y-{\beta }_0{\lambda }^4+8{\alpha }_0^3{\lambda }^2+{\eta }_2=0,g=3,\hfill \\ E_3=-{\beta }_0^3{v}^{-1}(x,t_r)\prod _{j=1}^3(\lambda -{\mu }_j(x,t_r)),F_4={\beta }_0^3\prod _{j=1}^4(\lambda -{\nu }_j(x,t_r)),\hfill \\ {\tilde{\omega }}_1=\frac{d\lambda }{3y^2(P)+6{\alpha }_0{\beta }_0{\lambda }^2+{\eta }_1},{\tilde{\omega }}_2=\frac{\lambda d\lambda }{3y^2(P)+6{\alpha }_0{\beta }_0{\lambda }^2+{\eta }_1},{\tilde{\omega }}_3=\frac{y(P)d\lambda }{3y^2(P)+6{\alpha }_0{\beta }_0{\lambda }^2+{\eta }_1},\hfill \\ {\omega }_j={\displaystyle \sum _{l=1}^3}{C}_{jl}{\tilde{\omega }}_l,j=1,2,3,\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)})={\displaystyle \sum _{k=1}^3}{\int }_{Q_0}^{{\widehat{\nu }}_k(x,t_r)}\omega ,\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)})={\displaystyle \sum _{k=1}^3}{\int }_{Q_0}^{{\widehat{\mu }}_1(x,t_r)}\omega ,\hfill \\ {\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}(P)=-\frac{y^2(P)+2y^2({\widehat{\nu }}_0(x,t_r))+6{\alpha }_0{\beta }_0{\nu }_0^2(x,t_r)+{\eta }_1}{\lambda -{\nu }_0(x,t_r)}\frac{d\lambda }{3y^2(P)+6{\alpha }_0{\beta }_0{\lambda }^2+{\eta }_1}+{\displaystyle \sum _{j=1}^3}{\gamma }_j{\omega }_j,\hfill \\ \varphi (P,x,t_r)=\frac{\theta (M-\mathcal{A}(P)+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))\theta (M-\mathcal{A}(P)+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}exp(e_{\infty }^{(3)}(Q_0)-{\int }_{Q_0}^P{\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}),\hfill \\ {\left[v^{-\frac{1}{3}}(x,t_r)\right]}_x={\partial }_{U_2^{(2)}}ln\frac{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\mu }}(x,t_r)}))}{\theta (M-\mathcal{A}(P_{\infty })+\mathcal{A}({\mathcal{D}}_{\widehat{\underline{\nu }}(x,t_r)}))}+\frac{1}{{\beta }_0}{\displaystyle \sum _{j=1}^3}{\gamma }_j{C}_{j3},\hfill \end{array}}$$

where η₁, η₂ are arbitrary constants, ω = (ω₁, ω₂, ω₃) and γ_j, j = 1, 2, 3, are determined by ${\displaystyle {\int }_{{\mathfrak{a}}_j}{\omega }_{P_{\infty },{\widehat{\nu }}_0(x,t_r)}^{(3)}(P)=0.}$