Algebro-Geometric Solutions of the Sine-Gordon Hierarchy

Abstract

On the basis of two sets of Lenard recursion sequences and zero-curvature equation associated with a matrix spectral problem, we derive the entire sine-Gordon hierarchy, which is composed of all the positive and negative flows. Using the theory of hyperelliptic curves, the Abel-Jacobi coordinates are introduced, from which the corresponding positive and negative flows are linearized. The algebro-geometric solutions of the entire sine-Gordon hierarchy are constructed by using the asymptotic properties of the meromorphic function.

1 Introduction

Soliton equations are infinite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the integrability of soliton equations has been found by the inverse scattering transformation method [1,2,3], which has greatly promoted the universal understanding of soliton equations, thus making the soliton theory get rapid development. Subsequently, several systematic approaches have been developed to obtain algebro-geometric solutions of soliton equations associated with $2\times 2$ matrix spectral problems. For example, an analog of the inverse scattering theory for Hill’s equation was developed, from which ones gave an explicit formula of the periodic potentials with the finite number of gaps in the spectrum and derived the periodic N-soliton solution of the KdV equation was constructed explicitly [4,5,6,7]. Another effective method is the algebraic curve method that combine the spectral theory of differential equations, the algebraic curve of compact Riemann surfaces and their Jacobians, the Riemann theta functions and inverse problems (see, e.g., [8,9,10,11] and references therein), by which algebro-geometric solutions of many soliton equations associated with $2\times 2$ matrix spectral problems are obtained such as the KdV equation, nonlinear Schrödinger equation, mKdV equation, sine-Gordon equation, relativistic Toda lattice, and so on [4,5,6,7,8,9,10,11]. Matveev and Yavor gave complex finite-band multiphase solutions of the Kaup-Boussinesq equation and used the degeneracy procedure to find multisoliton solutions [12]. The nonlinearization approach of Lax pairs in Ref. [13] has been developed and applied to construct the analytical Riemann theta function solutions of soliton equations [14,15,16,17,18,19,20,21], which is generalized to the spatial part and the temporal parts of the Lax pairs and the adjoint Lax pairs are constrained as finite-dimensional Liouville integrable Hamiltonian systems and integrable symplectic maps [22, 23]. Gesztesy and Ratnaseelan have proposed an alternative systematic approach based on elementary algebraic methods to get analytical Riemann theta function solutions of the AKNS hierarchy [24]. In addition, this method was further developed to deal with soliton equations such as the modified Boussinesq, the Kaup-Kupershmidt, the coupled modified KdV hierarchies, and so on [25,26,27,28,29,30].

The main aim of the present paper is to construct analytical Riemann theta function solutions of the entire local sine-Gordon hierarchy, which is composed of all the positive and negative flows, on the basis of the algebraic curve method. It is well known that the higher-order sine-Gordon equations in the sine-Gordon hierarchy are nonlocal in the light-cone coordinates, which makes the problem extremely complicated. To overcome this difficulty, Boiti et al. use the fact that a nonlinear evolution equation in one field can be written as a system of coupled nonlinear evolution equations in two fields. They found that the suitable spectral problem to realize their idea is the Boiti-Tu spectral problem [31] with some compatiable reductions [32]. They derive a hierarchy of local nonlinear evolution equations generated by a recursion operator and its explicit inverse. This hierarchy satisfies a canonical geometrical scheme and contains the sine-Gordon equation and Liouville equation for special cases in laboratory coordinates. A generalization of the Bäcklund transformation and a nonlinear superposition formula for the sine-Gordon equation are also obtained. The positive and negative order hierarchy generated from the Boiti-Tu spectral problem was first proposed by Qiao in 1994 [33]. Regarding the negative flows of the sine-Gordon hierarchy is also given by Qiao in [34].

This paper is organized as follows. In Sect. 2, we introduce the Lenard gradient sequences and derive the entire local sine-Gordon hierarchy, which is composed of all the positive and negative flows, with the aid of the zero-curvature equation associated with a $2\times 2$ matrix spectral problem. In Sect. 3, we define a Lax matrix from which the elliptic variables and the hyperelliptic Riemann surface of arithmetic genus $2N+1$ are derived, the meromorphic function $\phi$ is introduced and its properties on some points are discussed. In Sect. 4, under the Able-Jacobi coordinates, the spatial and temporal flows of the entire local sine-Gordon hierarchy are linearized. Finally, analytical Riemann theta function solutions of the entire local sine-Gordon hierarchy are constructed by using the asymptotic properties of the meromorphic function $\phi$.

2 The Entire Local Sine-Gordon Hierarchy

In this section, we shall derive the entire local sine-Gordon hierarchy associated with the Boiti-Tu spectral problem [31]

$$\begin{aligned} \varphi _{x}=U\varphi ,\quad \varphi =\left( \begin{array}{c} \varphi _{1}\\ \varphi _{2} \end{array}\right) ,\quad U=\left( \begin{array}{cc} -\mathrm{i}\lambda +\mathrm{i}s{\lambda }^{-1} &{} u+\mathrm{i}v{\lambda }^{-1} \\ u-\mathrm{i}v{\lambda }^{-1} &{} \mathrm{i}\lambda -\mathrm{i}s{\lambda }^{-1} \\ \end{array}\right) , \end{aligned}$$

(1)

where u, v, s are potentials with the condition $s^{2}-v^{2}=s_{0}^{2}$, $s_{0}$ is a nonzero constant and $\lambda$ is a constant spectral parameter. To this end, we introduce the Lenard gradient sequences

$$\begin{aligned}&KS_{j,+}=JS_{j+1,+},~~~ JS_{0,+}=0,\quad j\ge 0, \end{aligned}$$

(2)

$$\begin{aligned}&\quad JS_{j,-}=KS_{j+1,-},~~~ KS_{0,-}=0,\quad j\ge 0, \end{aligned}$$

(3)

with starting points

$$\begin{aligned} S_{0,+}=\left( \begin{matrix}2u\\ 0\\ -\mathrm{i}\end{matrix}\right) ,\quad S_{0,-}=\left( \begin{matrix}0\\ -2\mathrm{i}v\\ -\mathrm{i}s\end{matrix}\right) , \end{aligned}$$

where $S_{j,\pm }=(c_{j,\pm },b_{j,\pm },a_{j,\pm })^{T}$, and two operators K and J are defined by

$$\begin{aligned} K=\begin{pmatrix} \frac{1}{2}\partial &{} -\mathrm{i}s &{} 2\mathrm{i}v \\ -s &{} 0 &{} 0 \\ -\mathrm{i}v &{} su-\frac{1}{2}v\partial &{} s\partial -2uv \\ \end{pmatrix},\quad J=\begin{pmatrix} 0 &{} -\mathrm{i} &{} 0 \\ -1 &{} \frac{1}{2}\mathrm{i}\partial &{} 2\mathrm{i}u \\ -\mathrm{i}v &{} su-\frac{1}{2}v\partial &{} s\partial -2uv \\ \end{pmatrix}. \end{aligned}$$

It is easy to see that

$$\begin{aligned} \hbox {Ker} J=\{\beta _{+} S_{0,+} \mid \beta _{+}\in {\mathbb {C}}\} ,\\ \hbox {Ker} K=\{ \beta _{-}S_{0,-} \mid \beta _{-}\in {\mathbb {C}}\} , \end{aligned}$$

where $\beta _{\pm }$ are arbitrary constants. Then $S_{j,+}(j\ge 0)$ and $S_{j,-}(j\le 0)$ are uniquely determined by (2) and (3), respectively, if choosing all constants of integration to be zero. For example, the first two members are

$$\begin{aligned} S_{1,+}= & {} \left( \begin{matrix}-\frac{1}{2}u_{xx}-v_{x}+2us+u^{3}\\ \mathrm{i}(u_{x}+2v)\\ -\frac{1}{2}\mathrm{i}u^{2}\end{matrix}\right) .\\ S_{1,-}= & {} \left( \begin{matrix}-\frac{v_{x}}{s}-2u\\ \frac{\mathrm{i}}{s^{2}_{0}}(\frac{1}{2}v_{xx}-\frac{3s_{x}v_{x}}{4s}-2vs-u^{2}v+u_{x}s-us_{x})\\ -\frac{\mathrm{i}}{s^{2}_{0}}(v^{2}+\frac{1}{2}u^{2}s-\frac{1}{2}(u_{x}v-uv_{x})+\frac{1}{4}s_{xx}-\frac{s_{x}v_{x}}{8v})\end{matrix}\right) . \end{aligned}$$

Assume that the time evolution of the eigenfunction $\varphi$ obeys the differential equation

$$\begin{aligned} \varphi _{t_{\underline{m}}}=V^{(\underline{m})}\varphi ,\quad V^{(\underline{m})}=(V^{(\underline{m})}_{ij})_{2\times 2},\quad \end{aligned}$$

(4)

where $\underline{m}=(m_{1},m_{2}),$ and

$$\begin{aligned} V^{(\underline{m})}_{11}= & {} -V^{(\underline{m})}_{22}=a^{(\underline{m})},\\ V^{(\underline{m})}_{12}= & {} \frac{1}{2}(\lambda b^{(\underline{m})}+c^{(\underline{m})}),\\ V^{(\underline{m})}_{21}= & {} \frac{1}{2}(-\lambda b^{(\underline{m})}+c^{(\underline{m})}), \end{aligned}$$

with

$$\begin{aligned} \begin{array}{l} a^{(\underline{m})}=\sum _{j=0}^{m_{1}}{\tilde{a}}_{j,+}\lambda ^{2m_{1}-2j}+\sum ^{m_{2}-1}_{j=0}{\tilde{a}}_{j,-}\lambda ^{-2m_{2}+2j},\quad {\tilde{a}}_{j,\pm }=\sum _{k=0}^{j}\tilde{\alpha }_{j-k,\pm }a_{k,\pm },\\ b^{(\underline{m})}=\sum _{j=0}^{m_{1}}{\tilde{b}}_{j,+}\lambda ^{2m_{1}-2j}+\sum ^{m_{2}-1}_{j=0}{\tilde{b}}_{j,-}\lambda ^{-2m_{2}+2j},\quad {\tilde{b}}_{j,\pm }=\sum _{k=0}^{j}\tilde{\alpha }_{j-k,\pm }b_{k,\pm },\\ c^{(\underline{m})}=\sum _{j=0}^{m_{1}}{\tilde{c}}_{j,+}\lambda ^{2m_{1}-2j}+\sum ^{m_{2}-1}_{j=0}{\tilde{c}}_{j,-}\lambda ^{-2m_{2}+2j},\quad {\tilde{c}}_{j,\pm }=\sum _{k=0}^{j}\tilde{\alpha }_{j-k,\pm }c_{k,\pm },\\ \end{array} \end{aligned}$$

(5)

and $\tilde{\alpha }_{k,\pm }$ are constants. Then the compatibility condition of (1) and (4) yields the zero curvature equation, $U_{t_{\underline{m}}}-V^{(\underline{m})}_{x}+[U,V^{(\underline{m})}]=0,$ which is equivalent to the hierarchy of nonlinear evolution equations

$$\begin{aligned} u_{t_{\underline{m}}}=\mathrm{i}({\tilde{b}}_{m_{2}-1,-}-{\tilde{b}}_{m_{1}+1,+}),\quad v_{t_{\underline{m}}}=s({\tilde{c}}_{m_{2},-}-{\tilde{c}}_{m_{1},+}),\quad m_{1}\ge 0, m_{2}\ge 1. \end{aligned}$$

(6)

In [33], Qiao first proposed this hierarchy and obtained the commutator representation of it. Two special examples of nonlinear evolution equations in (6) are given as follows.

• (i) Let $(m_{1},m_{2})=(0,1), t_{\underline{m}}=t, \tilde{\alpha }_{0,-}=\tilde{\alpha }_{0,+}=1, \tilde{\alpha }_{1,+}=0$. Then the first nontrivial member in the hierarchy is as follow

  $$\begin{aligned} \left\{ \begin{aligned} u_{t}&=u_{x}+4v,\\ v_{t}&=-v_{x}-4us. \end{aligned}\right. \end{aligned}$$

  (7)

  Introducing the new function w(x, t) via

  $$\begin{aligned} s=s_{0}\cos w,\quad v=\mathrm{i}s_{0}\sin w, \end{aligned}$$

  then (7) is reduced to

  $$\begin{aligned} w_{tt}-w_{xx}+16s_{0}\sin w=0, \end{aligned}$$

  which is the sine-Gordon equation in laboratory coordinates.

• (ii) Let $(m_{1},m_{2})=(1,2), t_{\underline{m}}=t, \tilde{\alpha }_{0,-}=-s_{0}^{2}, \tilde{\alpha }_{0,+}=1, \tilde{\alpha }_{j,\pm }=0,\quad j\ge 1$. Then the second nontrivial member in the hierarchy reads

  $$\begin{aligned} \left\{ \begin{aligned} u_{t}&=-\frac{1}{4}u_{xxx}+\frac{3}{2}u^{2}u_{x}+3u_{x}s-\frac{3}{4}v_{x}\bigg (\frac{s_{x}}{s}\bigg ),\\ v_{t}&=-\frac{1}{4}s(\frac{v_{x}}{s})_{xx}+\frac{3}{4}uv_{x}(\frac{v_{x}}{s}+2u)+3sv_{x}+\frac{1}{8}v_{x}\bigg (\frac{v_{x}}{s}\bigg )^{2}. \end{aligned}\right. \end{aligned}$$

3 Elliptic Variables

Let $\chi =(\chi _{1},\chi _{2})^{T}$ and $\psi =(\psi _{1},\psi _{2})^{T}$ be two basic solutions of (1) and (4). We introduce a Lax matrix

$$\begin{aligned} W=\frac{1}{2}(\chi \psi ^{T}+\psi \chi ^{T})\left( \begin{array}{cc} 0 &{} -1 \\ 1 &{} 0 \\ \end{array}\right) =\left( \begin{array}{cc} G &{} F \\ H &{} -G \\ \end{array}\right) \end{aligned}$$

which satisfies the Lax equations

$$\begin{aligned} W_{x}=[U,W],\quad W_{t_{\underline{m}}}=[V^{(\underline{m})},W]. \end{aligned}$$

(8)

Therefore, detW is a constant independent of x and $t_{\underline{m}}$. Equation (8) can be written as

$$\begin{aligned} G_{x}= & {} u(H-F)+\mathrm{i}v{\lambda }^{-1}(F+H),\nonumber \\ F_{x}= & {} -2\mathrm{i}(\lambda -s{\lambda }^{-1})F-2(u+\mathrm{i}v{\lambda }^{-1})G,\nonumber \\ H_{x}= & {} 2(u-\mathrm{i}v{\lambda }^{-1})G+2\mathrm{i}(\lambda -s{\lambda }^{-1})H, \end{aligned}$$

(9)

and

$$\begin{aligned} \begin{aligned} G_{t_{\underline{m}}}&=V_{12}^{(\underline{m})}H-V_{21}^{(\underline{m})}F,\\ F_{t_{\underline{m}}}&=2(V_{11}^{(\underline{m})}F-V_{12}^{(\underline{m})}G),\\ H_{t_{\underline{m}}}&=2(V_{21}^{(\underline{m})}G-V_{11}^{(\underline{m})}H). \end{aligned} \end{aligned}$$

(10)

Let

$$\begin{aligned} G=g,\quad F=\frac{1}{2}(f+\lambda ^{-1}h),\quad H=-\frac{1}{2}(f-\lambda ^{-1}h), \end{aligned}$$

(11)

where g, f and h are finite-order polynomials in $\lambda$ and $\lambda ^{-1}$:

$$\begin{aligned} \begin{aligned} g&=\sum _{j=0}^{N_{1}+1}g_{j,+}\lambda ^{2N_{1}+2-2j}+\sum ^{N_{2}-1}_{j=0}g_{j,-}\lambda ^{-2N_{2}+2j},\\ f&=\sum _{j=0}^{N_{1}+1}f_{j,+}\lambda ^{2N_{1}+2-2j}+\sum ^{N_{2}-1}_{j=0}f_{j,-}\lambda ^{-2N_{2}+2j},\\ h&=\sum _{j=0}^{N_{1}+1}h_{j,+}\lambda ^{2N_{1}+2-2j}+\sum ^{N_{2}-1}_{j=0}h_{j,-}\lambda ^{-2N_{2}+2j},\\ \end{aligned} \end{aligned}$$

(12)

Substituting (12) into (9) yields

$$\begin{aligned}&\begin{array}{l} KE_{j,+}=JE_{j+1,+},\quad JE_{0,+}=0,\quad 0\le j\le N_{1},\\ JE_{j,-}=KE_{j+1,-},\quad KE_{0,-}=0.\quad 0\le j\le N_{2}-2,\\ \end{array}\nonumber \\&\quad KE_{N_{1}+1,+}-JE_{N_{2}-1,-}=0. \end{aligned}$$

(13)

where $E_{j,\pm }=(h_{j,\pm },f_{j,\pm },g_{j,\pm })^{T}$. It is easy to see that the equations $JE_{0,+}=0$ and have the general solution

$$\begin{aligned} E_{0,+}= & {} \alpha _{0,+} S_{0,+}, \end{aligned}$$

(14)

$$\begin{aligned} E_{0,-}= & {} \alpha _{0,-} S_{0,-}, \end{aligned}$$

(15)

where $\alpha _{0,+}$ and $\alpha _{0,-}$ are constants. Acting with the operators $(J^{-1}K)^{k}$ and $(K^{-1}J)^{k}$ respectively, on (14) and (15), we obtain from (13) and (2) that

$$\begin{aligned} E_{k,+}= & {} \sum \limits _{j=0}^{k}\alpha _{j,+}S_{k-j,+},\quad 0\le k\le N_{1}+1,\nonumber \\ E_{k,-}= & {} \sum \limits _{j=0}^{k}\alpha _{j,-}S_{k-j,-},\quad 0\le k\le N_{2}-1, \end{aligned}$$

(16)

where $\alpha _{1,\pm },\alpha _{2,\pm },\ldots ,\alpha _{k,\pm }$ are constants of integration. The first member in (16) is

$$\begin{aligned} E_{1,+}=\left( \begin{matrix}\alpha _{0,+}(-\frac{1}{2}u_{xx}-v_{x}+2us+u^{3})+2\alpha _{1,+}u \\ \mathrm{i}\alpha _{0,+}(u_{x}+2v) \\ -\frac{1}{2}\mathrm{i}\alpha _{0,+}u^{2}-\mathrm{i}\alpha _{1,+} \end{matrix}\right) . \end{aligned}$$

By inspection of (11) and (12) , we write G, F and H as follows:

$$\begin{aligned} \begin{aligned} G&=\lambda ^{-2N_{2}}\sum _{j=0}^{2(N_{1}+N_{2})+2}G_{j}\lambda ^{2(N_{1}+N_{2})+2-j},\\ F&=\lambda ^{-2N_{2}}\sum _{j=0}^{2(N_{1}+N_{2})+1}F_{j}\lambda ^{2(N_{1}+N_{2})+1-j},\\ H&=\lambda ^{-2N_{2}}\sum _{j=0}^{2(N_{1}+N_{2})+1}H_{j}\lambda ^{2(N_{1}+N_{2})+1-j},\\ \end{aligned} \end{aligned}$$

Comparing the coefficients of $\lambda$ with the same power, we arrive at

$$\begin{aligned} G_{2j+1}= & {} 0,\quad 0\le j\le N_{1}+N_{2},\\ G_{2j}= & {} g_{j,+},\quad 0\le j\le N_{1}+1,\\ G_{2j}= & {} g_{N_{1}+N_{2}+1-j,-}, N_{1}+2\le j\le N_{1}+N_{2}+1,\\ F_{2j+1}= & {} -H_{2j+1}=\frac{1}{2}f_{j+1,+},\quad 0\le j\le N_{1},\\ F_{2j+1}= & {} -H_{2j+1}=\frac{1}{2}f_{N_{1}+N_{2}-j,-}, \quad N_{1}+1\le j\le N_{1}+N_{2},\\ F_{2j}= & {} H_{2j}=\frac{1}{2}h_{j,+},\quad 0\le j\le N_{1}+1,\\ F_{2j}= & {} H_{2j}=\frac{1}{2}h_{N_{1}+N_{2}+1-j,-}, \quad N_{1}+2\le j\le N_{1}+N_{2},\\ G_{0}= & {} g_{0,+}=-i\alpha _{0,+},F_{0}=\frac{1}{2}h_{0,+}=\alpha _{0,+}u,H_{0}=\frac{1}{2}h_{0,+}=\alpha _{0,+}u. \end{aligned}$$

From the relation above, one can easily find that $H(\lambda )=-F(-\lambda )$. So we can write F and H as finite products which take the form:

$$\begin{aligned} F=\alpha _{0,+}u\lambda ^{-2N_{2}}\prod \limits _{j=1}^{2N+1}(\lambda -\mu _{j}),\quad H=\alpha _{0,+}u\lambda ^{-2N_{2}}\prod \limits _{j=1}^{2N+1}(\lambda +\mu _{j}),\quad N=N_{1}+N_{2}, \end{aligned}$$

(17)

where $\{\mu _{j}\}_{j=1}^{2N+1}$ are called elliptic variables.

Since detW depends only on $\lambda$, whose coefficients are constants independent of x and $t_{m}$, we have

$$\begin{aligned} \mathrm {det}W=-G^{2}-FH=\left( \frac{\alpha _{0,+}}{\lambda ^{2N_{2}}}\right) ^{2}\prod \limits _{j=1}^{2N+2}(\lambda ^{2}-\lambda _{j})=R(\lambda ), \end{aligned}$$

(18)

from which one is naturally led to introduce the hyperelliptic curve ${\mathcal {K}}_{2N+1}$ of arithmetic genus $2N+1$ defined by

$$\begin{aligned} {\mathcal {K}}_{2N+1}:y^{2}-\left( \frac{\lambda ^{2N_{2}}}{\alpha _{0,+}}\right) ^{2}R(\lambda )=y^{2}-\prod \limits _{j=1}^{2N+2}(\lambda ^{2}-\lambda _{j})=0. \end{aligned}$$

(19)

The curve ${\mathcal {K}}_{2N+1}$ can be compactified by joining two points at infinity, $P_{\infty \pm }$, where $P_{\infty +}\ne P_{\infty -}$. For notational simplicity the compactification of the curve ${\mathcal {K}}_{2N+1}$ is also denoted by ${\mathcal {K}}_{2N+1}$. Here we assume that $\lambda _{j}$ of $R(\lambda )$ in (18) are mutually distinct, which means $\lambda _{j}\ne \lambda _{k}$, for $j\ne k$, $1\le j,k\le 2N+2$. Then the hyperelliptic curve ${\mathcal {K}}_{2N+1}$ becomes nonsingular. According to the definition of ${\mathcal {K}}_{2N+1}$, we can lift the roots $\mu _{j}$ to ${\mathcal {K}}_{2N+1}$ by introducing

$$\begin{aligned} \hat{\mu }_{j}(x,t_{\underline{m}})=\left( \mu _{j}(x,t_{\underline{m}}),\mathrm{i}G(\mu _{j}(x,t_{\underline{m}}),x,t_{\underline{m}})\mu ^{2N_{2}}_{j}(x,t_{\underline{m}})/\alpha _{0,+}\right) , \end{aligned}$$

(20)

where $j=1,\ldots ,2N+1,$ $(x,t_{\underline{m}})\in {\mathbb {R}}^{2}$. We also introduce the points $P_{0\pm }$ by

$$\begin{aligned} P_{0\pm }=(0,y(P_{0\pm }))=\left( 0,\pm \frac{\alpha _{0,-}}{\alpha _{0,+}}s_{0}\right) \in {\mathcal {K}}_{2N+1},\quad \frac{\alpha _{0,-}^{2}}{\alpha _{0,+}^{2}}=\prod \limits _{j=1}^{2N+2}\lambda _{j},\quad \alpha _{0,\pm }\in {\mathbb {C}}\backslash 0, \end{aligned}$$

where we emphasize that $P_{0\pm }$ and $P_{\infty \pm }$ are not necessarily on the same sheet of ${\mathcal {K}}_{2N+1}$.

From (18) and (19) we can define the meromorphic function $\phi (P,x,t_{\underline{m}})$ on ${\mathcal {K}}_{2N+1}$

$$\begin{aligned} \phi (P,x,t_{\underline{m}})=\frac{\mathrm{i}\alpha _{0,+}\lambda ^{-2N_{2}}y-G}{F}=\frac{H}{\mathrm{i}\alpha _{0,+}\lambda ^{-2N_{2}}y+G}, \end{aligned}$$

(21)

where $P=(\lambda ,y)\in {\mathcal {K}}_{2N+1}\backslash \{P_{\infty +},P_{\infty -}\}$.

Lemma 1

Suppose that $u(x,t_{\underline{m}}), v(x,t_{\underline{m}})\in C^{\infty }({\mathbb {R}}^{2})$ satisfy the hierarchy (6). Let $\lambda _{j}\in {\mathbb {C}}\backslash \{0\}$, $j=1,2,\cdots , 4N+4$, and $P=(\lambda ,y)\in {\mathcal {K}}_{2N+1}\backslash \{P_{\infty +},P_{\infty -}\}$. Then

$$\begin{aligned} \phi \mathop {=}\limits _{\zeta \rightarrow 0}\left\{ \begin{array}{ll} \frac{\mathrm{i}}{2}u\zeta +\frac{u_{x}-2v}{2}\zeta ^{2}+O(\zeta ^{3}),~~as~~P\rightarrow P_{\infty +},\\ \frac{2\mathrm{i}}{u}\zeta ^{-1}+\frac{u_{x}+2v}{u^{2}}+O(\zeta ),~~as~~P\rightarrow P_{\infty -},\\ \end{array} \right. \qquad \zeta =\lambda ^{-1}, \end{aligned}$$

(22)

$$\begin{aligned} \phi \mathop {=}\limits _{\zeta \rightarrow 0}\left\{ \begin{array}{ll} \frac{-s-s_{0}}{v}+O(\zeta ),~~as~~P\rightarrow P_{0+},\\ \frac{-s+s_{0}}{v}+O(\zeta ),~~as~~P\rightarrow P_{0-},\\ \end{array} \right. \qquad \zeta =\lambda . \end{aligned}$$

(23)

Proof

According to (9) and (21), we can get that $\phi$ satisfies the Riccati-type equation

$$\begin{aligned} \phi _{x}(P)+(u+\mathrm{i}v\lambda ^{-1})\phi (P)^{2}-2\mathrm{i}(\lambda -s\lambda ^{-1})\phi (P)=u-\mathrm{i}v\lambda ^{-1}. \end{aligned}$$

(24)

By inserting the ansatz

$$\begin{aligned} \phi \mathop {=}\limits _{\zeta \rightarrow 0}\phi _{1}\zeta +\phi _{2}\zeta ^{2}+O(\zeta ^{3}) \end{aligned}$$

into (24), a comparison of powers of $\zeta$ then proves the first line in (22). Similarly, after the ansatz

$$\begin{aligned} \phi \mathop {=}\limits _{\zeta \rightarrow 0}\phi _{-1}\zeta ^{-1}+\phi _{0}+O(\zeta ) \end{aligned}$$

is inserted into (24), comparing powers of $\zeta$ proves the second line in (22). In exactly the same manner, inserting the ansatz

$$\begin{aligned} \phi \mathop {=}\limits _{\zeta \rightarrow 0}\phi _{0}+O(\zeta ) \end{aligned}$$

into (24) immediately yields (23). $\square$

Hence the divisor of $\phi (P,x,t_{\underline{m}})$ is

$$\begin{aligned} (\phi (P,x,t_{\underline{m}}))=D_{P_{\infty +},-\hat{\mu }_{1}(x,t_{\underline{m}}),\ldots ,-\hat{\mu }_{2N+1}(x,t_{\underline{m}})}-D_{P_{\infty -},\hat{\mu }_{1}(x,t_{\underline{m}}),\ldots ,\hat{\mu }_{2N+1}(x,t_{\underline{m}})}. \end{aligned}$$

The following elementary results are benefit for our main work. Let $\{\lambda _{j}\}_{j=1,\ldots ,2N+2}\subset {\mathbb {C}}$ for some $N\in {\mathbb {N}}$ and $\xi \in {\mathbb {C}}$, such that $\vert \xi \vert < \mathrm{min} \{\vert \lambda _{1}\vert ^{-1},\ldots ,\vert \lambda _{2N+2}\vert ^{-1}\}$ , and abbreviate $\underline{\lambda }=(\lambda _{1},\ldots ,\lambda _{2N+2})$, $\underline{\lambda }^{-1}=(\lambda _{1}^{-1},\ldots ,\lambda _{2N+2}^{-1})$. Then

$$\begin{aligned} \left( \prod \limits _{j=1}^{2N+2}(1-\lambda _{j}^{\pm 1}\xi )\right) ^{-\frac{1}{2}}=\sum \limits _{k=0}^{\infty }\hat{\alpha }_{k}(\underline{\lambda }^{\pm 1})\xi ^{k}, \end{aligned}$$

where $\hat{\alpha }_{0}(\underline{\lambda }^{\pm 1})=1,\hat{\alpha }_{1}(\underline{\lambda }^{\pm 1})=\frac{1}{2}\sum \nolimits _{j=1}^{2N+2}\lambda _{j}^{\pm 1},\ldots ,$

$$\begin{aligned} \hat{\alpha }_{k}(\underline{\lambda }^{\pm 1})=\sum \limits ^{k}_{{\mathop {j_1+\ldots +j_{2N+2}=k}\limits ^{j_1,\ldots ,j_{2N+2}=0}}}\frac{(2j_1)!\ldots (2j_{2N+2})!\lambda _1^{\pm j_1}\ldots \lambda _{2N+2}^{\pm j_{2N+2}}}{2^{2k}(j_1!)^2\ldots (j_{2N+2}!)^2},\qquad k\in {\mathbb {N}}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \left( \prod \limits _{j=1}^{2N+2}(1-\lambda _{j}^{\pm 1}\xi )\right) ^{\frac{1}{2}}=\sum \limits _{k=0}^{\infty }\alpha _{k}(\underline{\lambda }^{\pm 1})\xi ^{k}, \end{aligned}$$

where $\alpha _{0}(\underline{\lambda }^{\pm 1})=1,\alpha _{1}(\underline{\lambda }^{\pm 1})=-\frac{1}{2}\sum \limits _{j=1}^{2N+2}\lambda _{j}^{\pm 1},\ldots ,$

$$\begin{aligned} \alpha _{k}(\underline{\lambda }^{\pm 1})=\sum \limits ^{k}_{{\mathop {j_1+\ldots +j_{2N+2}=k}\limits ^{j_1,\ldots ,j_{2N+2}=0}}}\frac{(2j_1)!\ldots (2j_{2N+2})!\lambda _1^{\pm j_1}\ldots \lambda _{2N+2}^{\pm j_{2N+2}}}{2^{2k}(j_1!)^2\ldots (j_{2N+2}!)^2(2j_1-1)\ldots (2j_{2N+2}-1)},\qquad k\in {\mathbb {N}}. \end{aligned}$$

We infer for the $\lambda _{j}$-dependent summation constants $\alpha _{0,+},\alpha _{1,+},\ldots ,\alpha _{N_{1},+}$ and $\alpha _{0,-},\alpha _{1,-}$, $\ldots ,\alpha _{N_{2}-1,-}$ in $G(\lambda )$, $F(\lambda )$, $H(\lambda )$ that

$$\begin{aligned} \begin{aligned} \alpha _{k,+}=\alpha _{0,+}\alpha _{k}(\underline{\lambda }),\qquad k=0,\ldots ,N_{1}+1,\\ \alpha _{k,-}=\alpha _{0,-}\alpha _{k}(\underline{\lambda }^{-1}),\qquad k=0,\ldots ,N_{2}-1. \end{aligned} \end{aligned}$$

In addition, we have

$$\begin{aligned} \begin{aligned} f_{l,+}&=\alpha _{0,+}\sum \limits _{k=0}^{l}\alpha _{l-k}(\underline{\lambda })b_{k,+},\\ f_{l,-}&=\alpha _{0,-}\sum \limits _{k=0}^{l}\alpha _{l-k}(\underline{\lambda }^{-1})b_{k,-},\\ h_{l,+}&=\alpha _{0,+}\sum \limits _{k=0}^{l}\alpha _{l-k}(\underline{\lambda })c_{k,+},\\ h_{l,-}&=\alpha _{0,-}\sum \limits _{k=0}^{l}\alpha _{l-k}(\underline{\lambda }^{-1})c_{k,-},\\ \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \alpha _{0,+}b_{l,+}&=\sum \limits _{k=0}^{l\wedge (N_{1}+1)}\hat{\alpha }_{l-k}(\underline{\lambda })f_{k,+},\\ \alpha _{0,-}b_{l,-}&=\sum \limits _{k=0}^{l\wedge (N_{2}-1)}\hat{\alpha }_{l-k}(\underline{\lambda }^{-1})f_{k,-},\\ \alpha _{0,+}c_{l,+}&=\sum \limits _{k=0}^{l\wedge (N_{1}+1)}\hat{\alpha }_{l-k}(\underline{\lambda })h_{k,+}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \alpha _{0,-}c_{l,-}=\sum \limits _{k=0}^{l\wedge (N_{2}-1)}\hat{\alpha }_{l-k}(\underline{\lambda }^{-1})h_{k,-}, \end{aligned}$$

where

$$\begin{aligned} l\in {\mathbb {N}}_{0},l\wedge N_{1}=\min \{l,N_{1}+1\},\quad l\wedge (N_{2}-1)=\min \{l,N_{2}-1\}. \end{aligned}$$

4 Straightening Out the Flows

In order to straightening out the spatial and temporal flows of the full local sine-Gordon hierarchy, we consider the Riemann surface ${\mathcal {K}}_{2N+1}$ and equip ${\mathcal {K}}_{2N+1}$ with canonical basis cycles: ${\mathbf {a}}_{1},\ldots ,{\mathbf {a}}_{2N+1}$; ${\mathbf {b}}_{1},\ldots ,{\mathbf {b}}_{2N+1}$, which are independent and having intersection numbers as follows

$$\begin{aligned} {\mathbf {a}}_{j}\circ {\mathbf {a}}_{k}=0,\quad {\mathbf {b}}_{j}\circ {\mathbf {b}}_{k}=0,\quad {\mathbf {a}}_{j}\circ {\mathbf {b}}_{k}=\delta _{jk}. \end{aligned}$$

For the present, we will choose our basis as the following set [22, 23]

$$\begin{aligned} \tilde{\omega }_{l}=\frac{\lambda ^{l-1}d\lambda }{y(P)},\quad 1\le l\le 2N+1, \end{aligned}$$

which are $2N+1$ linearly independent homomorphic differentials on ${\mathcal {K}}_{2N+1}$. Then the period matrices A and B can be constructed from

$$\begin{aligned} A_{kj}=\int _{{\mathbf {a}}_{j}}\tilde{\omega }_{k},\quad B_{kj}=\int _{{\mathbf {b}}_{j}}\tilde{\omega }_{k}. \end{aligned}$$

It is possible to show that matrices A and B are invertible [35, 36]. Now we define the matrices C and $\tau$ by $C=A^{-1}$, $\tau =A^{-1}B$. The matrix $\tau$ can be shown to be symmetric ($\tau _{kj}=\tau _{jk}$), and it has positive definite imaginary part (Im$\tau >0$). If we normalize $\tilde{\omega }_{l}$ into the new basis $\omega _{j}$,

$$\begin{aligned} \omega _{j}=\sum \limits _{l=1}^{2N+1}C_{jl}\tilde{\omega }_{l},\quad 1\le j\le 2N+1, \end{aligned}$$

then we obtain

$$\begin{aligned} \int _{{\mathbf {a}}_{k}}\omega _{j}=\sum \limits _{l=1}^{2N+1}C_{jl}\int _{{\mathbf {a}}_{k}}\tilde{\omega }_{l}=\delta _{jk},\int _{{\mathbf {b}}_{k}}\omega _{j}=\tau _{jk}. \end{aligned}$$

Let ${\mathcal {T}}$ be the lattice generated by $4N+2$ vectors $\sigma _{j}$, $\tau _{j}$ where $\sigma _{j}=(\underbrace{0,\ldots ,0}\limits _{j-1},1,\underbrace{0,\ldots ,0}\limits _{2N+1-j})^{T},$ and $\tau _{j}=\tau \sigma _{j}$. The complex torus ${\mathscr {T}}={\mathbb {C}}^{2N+1}/{\mathcal {T}}$ is called the Jacobian variety of ${\mathcal {K}}_{2N+1}$. Now we introduce the Abel map $\underline{{\mathcal {A}}}(P):\mathrm {Div}({\mathcal {K}}_{2N+1})\rightarrow {\mathscr {T}}$

$$\begin{aligned} \underline{{\mathcal {A}}}(P)=\int _{Q_{0}}^{P}\underline{\omega },\quad \underline{{\mathcal {A}}}\left( \sum n_{k}P_{k}\right) =\sum n_{k}\underline{{\mathcal {A}}}(P_{k}), \end{aligned}$$

where P, $P_{k}\in {\mathcal {K}}_{2N+1}$, $\underline{\omega }=(\omega _{1},\ldots ,\omega _{2N+1})$. Considering two special divisors $\sum \limits _{k=1}^{2N+1}P_{k}^{(r)},~r=1,2$, we define the Abel-Jacobi coordinates as follows

$$\begin{aligned} \underline{{\mathcal {A}}}\left( \sum _{k=1}^{2N+1}P_{k}^{(r)}\right) =\sum _{k=1}^{2N+1}\underline{{\mathcal {A}}}(P_{k}^{(r)})=\sum _{k=1}^{2N+1}\int _{Q_{0}}^{P_{k}^{(r)}}\underline{\omega }=\underline{\rho }^{(r)}(x,t_{\underline{m}}), \end{aligned}$$

with $P_{k}^{(1)}=\hat{\mu }_{k}(x,t_{\underline{m}}),$ and $P_{k}^{(2)}=-\hat{\mu }_{k}(x,t_{\underline{m}}),$ whose components are

$$\begin{aligned} \sum _{k=1}^{2N+1}\int _{Q_{0}}^{P_{k}^{(r)}}\omega _{j}=\rho ^{(r)}_{j}(x,t_{\underline{m}}),\quad 1\le j\le 2N+1,\quad r=1,2. \end{aligned}$$

(25)

Without loss of generality, we choose the branch point $Q_{0}=(\sigma \lambda _{j_{0}}^{\frac{1}{2}},0)$, $\sigma =1$ or $\sigma =-1$, $j_{0}\in \{1,\ldots ,2N+2\}$, as a convenient base point, and $\lambda (Q_{0})$ is its local coordinate.

From (20), we obtain

$$\begin{aligned}&G(\mu _{j})=-\mathrm{i}\alpha _{0,+}\mu _{j}^{-2N_{2}}y(\hat{\mu _{j}}),\quad 1\le j\le 2N+1. \end{aligned}$$

Noticing (9), (10) and (17), we get

$$\begin{aligned} F_{x}(\mu _{j})= & {} -\alpha _{0,+}u\mu _{j,x}\mu _{j}^{-2N_{2}}\prod \limits _{ \begin{array}{l} r=1\\ r\ne j\\ \end{array} }^{2N+1}(\mu _{j}-\mu _{r})=-2(u+\mathrm{i}v\mu _{j}^{-1})G(\mu _{j}).\\ F_{t_{\underline{m}}}(\mu _{j})= & {} -\alpha _{0,+}u\mu _{j,t_{\underline{m}}}\mu _{j}^{-2N_{2}}\prod \limits _{ \begin{array}{l} r=1\\ r\ne j\\ \end{array} }^{2N+1}(\mu _{j}-\mu _{r})=-2V_{12}^{(\underline{m})}G(\mu _{j}). \end{aligned}$$

hence we arrive at the evolution of $\mu _{j}$ along the x and $t_{\underline{m}}$ flow:

$$\begin{aligned} \begin{aligned} \mu _{j,x}=-\frac{2\mathrm{i}(u+\mathrm{i}v/\mu _{j})y(\hat{\mu _{j}})}{u\prod \nolimits _{ \begin{array}{l} r=1\\ r\ne j\\ \end{array} }^{2N+1}(\mu _{j}-\mu _{r})},\quad \mu _{j,t_{\underline{m}}}=\frac{-2\mathrm{i}V_{12}^{(\underline{m})}(\mu _{j})y(\hat{\mu _{j}})}{u\prod \nolimits _{ \begin{array}{l} r=1\\ r\ne j\\ \end{array} }^{2N+1}(\mu _{j}-\mu _{r})},\quad 1\le j\le 2N+1. \end{aligned} \end{aligned}$$

(26)

Assuming $N\in {\mathbb {N}}$ to be fixed and introducing

$$\begin{aligned} {\mathbf {S}}_{k}= & {} \{\underline{l}=(l_{1},\ldots ,l_{k})\in {\mathbb {N}}^{k}\mid 1\le l_{1}<\cdots <l_{1}\le 2N+1\}, \quad k=1,\ldots ,2N+1,\\ {\mathbf {I}}_{k}^{(j)}= & {} \{\underline{l}=(l_{1},\ldots ,l_{k})\in {\mathbf {S}}_{k}\mid l_{r}\ne j, r=1,\ldots , k\}, \quad k=1,\ldots ,2N, \quad j=1,\ldots ,2N+1, \end{aligned}$$

one defines the symmetric functions

$$\begin{aligned} \begin{aligned} \Psi _{0}(\underline{\mu })=1,\qquad \Psi _{k}(\underline{\mu })=(-1)^{k}\sum \limits _{\underline{l}\in {\mathbf {S}}_{k}}\mu _{l_{1}\cdots \mu _{l_{k}}},\qquad k=1,\ldots ,2N+1,\\ \Phi _{k}^{(j)}(\underline{\mu })=(-1)^{k}\sum \limits _{\underline{l}\in {\mathbf {I}}_{k}^{(j)}}\mu _{l_{1}\cdots \mu _{l_{k}}},\qquad j=1,\ldots ,2N,\qquad k=1,\ldots ,2N+1,\\ \Phi _{0}^{(j)}(\underline{\mu })=1,\qquad \Phi _{2N+1}^{(j)}(\underline{\mu })=0,\qquad j=1,\ldots ,2N+1, \end{aligned} \end{aligned}$$

(27)

where $\underline{\mu }=(\mu _{1},\ldots ,\mu _{2N+1})$. Then $\prod \nolimits _{j=1}^{2N+1}(\lambda -\mu _{j})=\sum \nolimits _{l=0}^{2N+1}\Psi _{l}(\underline{\mu })\lambda ^{2N+1-l}$.

According to the Lagrange interpolation theorem, we have the following important facts (these lemmas were proven in detail in [26], Lemma E.2 and Lemma E.3).

Lemma 2

Assume that $\mu _{1},\ldots ,\mu _{2N+1}$ are $2N+1$ distinct complex numbers. Then,

$$\begin{aligned} \sum \limits _{j=0}^{k}\Psi _{k-l}(\underline{\mu })\mu _{j}^{l}=\Phi _{k}^{(j)}(\underline{\mu }),\qquad j=1,\ldots ,2N+1,\qquad j=1,\ldots ,2N+1. \end{aligned}$$

(28)

Lemma 3

Assume that $\mu _{j}\ne \mu _{j'}$ for $j\ne j'$, we introduce the $N\times N$ matrix ${\mathbf {U}}_{2N+1}(\underline{\mu })$ by

$$\begin{aligned} \begin{aligned} {\mathbf {U}}_{1}(\underline{\mu })=1,\qquad {\mathbf {U}}_{2N+1}(\underline{\mu })=\left( \frac{\mu _{k}^{j-1}}{\prod \nolimits _{ \begin{array}{l} r=1\\ r\ne k\\ \end{array} }^{2N+1}(\mu _{k}-\mu _{r})}\right) ^{2N+1}_{j,k=1}, \end{aligned} \end{aligned}$$

then

$$\begin{aligned} {\mathbf {U}}_{2N+1}(\underline{\mu })^{-1}=\left( \Phi _{2N+1-k}^{(j)}(\underline{\mu })\right) ^{2N+1}_{j,k=1}. \end{aligned}$$

Theorem 4

(Straightening out of the flows) Assume that $\mu _{j}\ne \mu _{j'}$ for $j\ne j', j,j'=1,\ldots ,2N+1$. Then

$$\begin{aligned} \underline{\rho }^{(1)}(x,t_{\underline{m}})= & {} \underline{U}x+\underline{{\mathbf {Y}}}^{(\underline{m})}t_{\underline{m}}+\underline{\rho }^{(1)}(x_{0},t_{0,\underline{m}}), \end{aligned}$$

(29)

$$\begin{aligned} \underline{\rho }^{(2)}(x,t_{\underline{m}})= & {} -\underline{U}x-\underline{{\mathbf {Y}}}^{(\underline{m})}t_{\underline{m}}+\underline{\rho }^{(2)}(x_{0},t_{0,\underline{m}}), \end{aligned}$$

(30)

where $\underline{U}=-2\mathrm{i}(\underline{C}_{2N+1}+\frac{\alpha _{0,+}}{\alpha _{0,-}}\underline{C}_{1})$, $\underline{{\mathbf {Y}}}^{(\underline{m})}=-2\mathrm{i}(\underline{{\mathbf {Y}}}^{(m_{1})}-\frac{\alpha _{0,+}}{\alpha _{0,-}}\underline{{\mathbf {Y}}}^{(m_{2})})$, constants $\underline{\rho }^{(r)}(x_{0},t_{0,\underline{m}})\in {\mathbb {C}}^{2N+1},r=1,2,$ $\underline{C}_k=(C_{1k},\ldots ,C_{2N+1,k})$, $1\le k\le 2N+1$, and

$$\begin{aligned}&\underline{{\mathbf {Y}}}^{(m_{1})}\mathop {=}\left\{ \begin{array}{ll} 0,\qquad m_{1}=0,\\ \sum \nolimits _{k=0}^{m_{1}}\widetilde{\alpha }_{m_{1}-k,+}\sum \nolimits _{s=0\vee (l-N_{1})}^{l}\hat{\alpha }_{s}(\underline{\lambda })C_{j,2N+1-2l-2s},\qquad m_{1}\ge 1,\\ \end{array} \right. \\&\underline{{\mathbf {Y}}}^{(m_{2})}=\sum \nolimits _{s=0\vee (l+1-N_{2})}^{l-1}\hat{\alpha }_{s}(\underline{\lambda }^{-1})C_{j,2l-2s+1},\qquad m_{2}\ge 1. \end{aligned}$$

Proof

Here we only give the proof of (29). Equation (30) can be proved in a similar way. From (25) and (26), we get

$$\begin{aligned} \begin{aligned} \partial _{x}\rho _{j}^{(1)}&=\sum \limits _{l=1}^{2N+1}\sum \limits _{k=1}^{2N+1}C_{jl}\frac{\mu _{k}^{l-1}\mu _{k,x}}{y(\hat{\mu _{k}})} =\sum \limits _{l=1}^{2N+1}\sum \limits _{k=1}^{2N+1}\frac{-2\mathrm{i}C_{jl}(u+\mathrm{i}v/\mu _{k})\mu _{k}^{l-1}}{u\prod \nolimits _{ \begin{array}{l} r=1\\ r\ne k\\ \end{array} }^{2N+1}(\mu _{k}-\mu _{r})}\\&=-2\mathrm{i}C_{j,2N+1}-2\mathrm{i}C_{j,1}\frac{\alpha _{0,+}}{\alpha _{0,-}}=\underline{U},\quad j=1,2,\ldots ,2N+1, \end{aligned} \end{aligned}$$

(31)

with the aid of the following equalities:

$$\begin{aligned} \begin{aligned}&\sum \limits _{k=1}^{2N+1}\frac{\mu _{k}^{l-1}}{\prod \nolimits _{r\ne k}(\mu _{k}-\mu _{r})} =\left\{ \begin{array}{ll} \delta _{l,2N+1},\quad l=1,2,\ldots ,2N+1,\\ \sum \limits _{r_{1}+\cdots +r_{2N+1}=l-1-2N,r_{j}\ge 0}\mu _{1}^{r_{1}}\cdots \mu _{2N+1}^{r_{2N+1}},\quad l>2N+1, \end{array}\right. \\&\sum \limits _{k=1}^{2N+1}\frac{\mu _{k}^{-1}}{\prod \nolimits _{r\ne k}(\mu _{k}-\mu _{r})}=\frac{1}{\prod \nolimits _{r=1}^{2N+1}\mu _{r}}. \end{aligned} \end{aligned}$$

In order to evaluate the $t_{\underline{m}}$-derivative of $\rho _{j}^{(1)}$, we introduce polynomials ${\hat{G}}_{l,\pm }(\lambda )$, ${\hat{F}}_{l,\pm }(\lambda )$, ${\hat{H}}_{l,\pm }(\lambda )$ by ($l\in {\mathbb {N}}$):

$$\begin{aligned} \begin{aligned} {\hat{G}}_{0,+}(\lambda )&=-\mathrm{i}, \quad {\hat{G}}_{l,+}(\lambda )=\sum \limits _{k=0}^{l}a_{l-k,+}\lambda ^{2k},\quad {\hat{G}}_{0,-}(\lambda )=0,\quad {\hat{G}}_{l,-}(\lambda )=\sum \limits _{k=1}^{l}a_{l-k,-}\lambda ^{-2k},\\ {\hat{F}}_{0,+}(\lambda )&=0, \quad {\hat{F}}_{l,+}(\lambda )=\sum \limits _{k=0}^{l-1}b_{l-k,+}\lambda ^{2k},\quad {\hat{F}}_{0,-}(\lambda )=0,\quad {\hat{F}}_{l,-}(\lambda )=\sum \limits _{k=1}^{l}b_{l-k,-}\lambda ^{-2k},\\ {\hat{H}}_{0,+}(\lambda )&=2u, \quad {\hat{H}}_{l,+}(\lambda )=\sum \limits _{k=0}^{l}c_{l-k,+}\lambda ^{2k},\quad {\hat{H}}_{0,-}(\lambda )=0,\quad {\hat{H}}_{l,-}(\lambda )=\sum \limits _{k=1}^{l-1}c_{l-k,-}\lambda ^{-2k},\\ \end{aligned} \end{aligned}$$

(32)

One immediately finds that

$$\begin{aligned} \begin{aligned} a^{(\underline{m})}(\lambda )&=\sum \limits _{k=0}^{m_{1}}\widetilde{\alpha }_{m_{1}-k,+}{\hat{G}}_{k,+}(\lambda )+\sum \limits _{k=0}^{m_{2}}\widetilde{\alpha }_{m_{2}-k,-}{\hat{G}}_{k,-}(\lambda ),\\ b^{(\underline{m})}(\lambda )&=\sum \limits _{k=0}^{m_{1}}\widetilde{\alpha }_{m_{1}-k,+}{\hat{F}}_{k,+}(\lambda )+\sum \limits _{k=0}^{m_{2}}\widetilde{\alpha }_{m_{2}-k,-}{\hat{F}}_{k,-}(\lambda ),\\ c^{(\underline{m})}(\lambda )&=\sum \limits _{k=0}^{m_{1}}\widetilde{\alpha }_{m_{1}-k,+}{\hat{H}}_{k,+}(\lambda )+\sum \limits _{k=0}^{m_{2}}\widetilde{\alpha }_{m_{2}-k,-}{\hat{H}}_{k,-}(\lambda ). \end{aligned} \end{aligned}$$

(33)

From (11), (17) and (27) we have

$$\begin{aligned} \lambda ^{2N_{2}}F(\lambda )=\alpha _{0,+}u\prod \limits _{j=1}^{2N+1}(\lambda -\mu _{j})=\alpha _{0,+}u\sum \limits _{l=0}^{2N+1}\Psi _{l}(\underline{\mu })\lambda ^{2N+1-l}. \end{aligned}$$

Noticing that $f_{0,+}=b_{0,+}=0,h_{0,-}=c_{0,-}=0$, using (11), we arrive at

$$\begin{aligned} \begin{aligned} f_{l,+}&=2\alpha _{0,+}u\Psi _{2l-1}(\underline{\mu }),\quad b_{l,+}=2u\sum \limits _{k=1}^{l\wedge (N_{1}+1)}\hat{\alpha }_{l-k}(\underline{\lambda })\Psi _{2k-1}(\underline{\mu }),\quad l\ge 1,\\ f_{l,-}&=2\alpha _{0,+}u\Psi _{2N+1-2l}(\underline{\mu }),\quad b_{l,-}=2u\frac{\alpha _{0,+}}{\alpha _{0,-}}\sum \limits _{k=0}^{l\wedge (N_{2}-1)}\hat{\alpha }_{l-k}(\underline{\lambda }^{-1})\Psi _{2N+1-2k}(\underline{\mu }),\quad l\ge 0,\\ h_{l,+}&=2\alpha _{0,+}u\Psi _{2l}(\underline{\mu }),\quad c_{l,+}=2u\sum \limits _{k=0}^{l\wedge (N_{1}+1)}\hat{\alpha }_{l-k}(\underline{\lambda })\Psi _{2k}(\underline{\mu }),\quad l\ge 0,\\ h_{l,-}&=2\alpha _{0,+}u\Psi _{2N+2-2l}(\underline{\mu }),\quad c_{l,-}=2u\frac{\alpha _{0,+}}{\alpha _{0,-}}\sum \limits _{k=0}^{l\wedge (N_{2}-1)}\hat{\alpha }_{l-k}(\underline{\lambda }^{-1})\Psi _{2N+2-2k}(\underline{\mu }),\quad l\ge 1. \end{aligned} \end{aligned}$$

Using (32) and (28), we obtain

$$\begin{aligned} \begin{aligned} \mu _{j}{\hat{F}}_{l,+}(\mu _{j})+{\hat{H}}_{l,+}(\mu _{j})&=2u\sum \limits _{s=0\vee (l-N_{1})}^{l}\hat{\alpha }_{s}(\underline{\lambda })\Phi _{2l-2s}^{(j)}(\underline{\mu }),\quad 1\le l\le m_{1},\\ \mu _{j}{\hat{F}}_{l,-}(\mu _{j})+{\hat{H}}_{l,-}(\mu _{j})&=-2u\frac{\alpha _{0,+}}{\alpha _{0,-}}\sum \limits _{s=0\vee (l+1-N_{2})}^{l-1}\hat{\alpha }_{s}(\underline{\lambda }^{-1})\Phi _{2N-2l+2s}^{(j)}(\underline{\mu }),\quad 1\le l\le m_{2}-1. \end{aligned} \end{aligned}$$

Thus, according to (5) and (33), we have

$$\begin{aligned} \begin{aligned}&V_{12}^{(\underline{m})}(\mu _{j})\\&=\frac{1}{2}\sum \limits _{k=0}^{m_{1}}\widetilde{\alpha }_{m_{1}-k,+}(\mu _{j}{\hat{F}}_{k,+}(\mu _{j})+{\hat{H}}_{k,+}(\mu _{j})) +\frac{1}{2}\sum \limits _{k=0}^{m_{2}}\widetilde{\alpha }_{m_{2}-k,-}({\hat{F}}_{k,-}(\mu _{j})+{\hat{H}}_{k,-}(\mu _{j}))\\&=u\sum \limits _{k=0}^{m_{1}}\widetilde{\alpha }_{m_{1}-k,+}\sum \limits _{s=0\vee (l-N_{1})}^{l}\hat{\alpha }_{s}(\underline{\lambda })\Phi _{2l-2s}^{(j)}(\underline{\mu }) -u\frac{\alpha _{0,+}}{\alpha _{0,-}}\sum \limits _{s=0\vee (l+1-N_{2})}^{l-1}\hat{\alpha }_{s}(\underline{\lambda }^{-1})\Phi _{2N-2l+2s}^{(j)}(\underline{\mu })\\&=\quad u\sum \limits _{k=0}^{m_{1}}\widetilde{\alpha }_{m_{1}-k,+}\sum \limits _{s=0\vee (l-N_{1})}^{l}\hat{\alpha }_{s}(\underline{\lambda }){\mathbf {U}}_{2N+1}(\underline{\mu })^{-1}_{j,2N+1-2l-2s}\\&\quad -u\frac{\alpha _{0,+}}{\alpha _{0,-}}\sum \limits _{s=0\vee (l+1-N_{2})}^{l-1}\hat{\alpha }_{s}(\underline{\lambda }^{-1}){\mathbf {U}}_{2N+1}(\underline{\mu })^{-1}_{j,2l-2s+1}. \end{aligned} \end{aligned}$$

Finally, using (25), (26), and Lemma 3, we obtain

$$\begin{aligned} \begin{aligned} \partial _{t_{\underline{m}}}\rho _{j}^{(1)}=&\sum \limits _{l=1}^{2N+1}\sum \limits _{k=1}^{2N+1}C_{jl}\frac{\mu _{k}^{l-1}\mu _{k,t_{\underline{m}}}}{y(\hat{\mu _{k}})} =\sum \limits _{l=1}^{2N+1}\sum \limits _{k=1}^{2N+1}\frac{-2\mathrm{i}C_{jl}V_{12}^{(\underline{m})}(\mu _{k})\mu _{k}^{l-1}}{u\prod \nolimits _{ \begin{array}{l} r=1\\ r\ne k\\ \end{array} }^{2N+1}(\mu _{k}-\mu _{r})}\\ =&-2\mathrm{i}\sum \limits _{l=1}^{2N+1}\sum \limits _{k=1}^{2N+1}C_{jl}{\mathbf {U}}_{2N+1}(\underline{\mu })_{l,k}\frac{V_{12}^{(\underline{m})}(\mu _{k})}{u}\\ =&-2\mathrm{i}\left( \sum \limits _{k=0}^{m_{1}}\widetilde{\alpha }_{m_{1}-k,+}\sum \limits _{s=0\vee (l-N_{1})}^{l}\hat{\alpha }_{s}(\underline{\lambda })C_{j,2N+1-2l-2s}\right. \\&\left. -\frac{\alpha _{0,+}}{\alpha _{0,-}}\sum \limits _{s=0\vee (l+1-N_{2})}^{l-1}\hat{\alpha }_{s}(\underline{\lambda }^{-1})C_{j,2l-2s+1}\right) \\ =&-2\mathrm{i}(\underline{{\mathbf {Y}}}^{(m_{1})}-\frac{\alpha _{0,+}}{\alpha _{0,-}}\underline{{\mathbf {Y}}}^{(m_{2})})=\underline{{\mathbf {Y}}}^{(\underline{m})},\quad 1\le j\le 2N+1. \end{aligned} \end{aligned}$$

(34)

Finally, (29) immediately follows from (31) and (34). $\square$

5 Construction of the Theta Representation for the Solution

In the section, we shall construct quasi-periodic solutions of the full local sine-Gordon hierarchy (6). Let $\omega ^{(3)}_{P_{\infty -},P_{\infty +}}(P)$ denote the normalized Abelian differentials of the third kind holomorphic on ${\mathcal {K}}_{2N+1}\backslash \{P_{\infty -},P_{\infty +}\}$ with simple poles at $P_{\infty +}$ and $P_{\infty -}$ with residues $\pm 1$, respectively, which can be expressed as

$$\begin{aligned} \omega ^{(3)}_{P_{\infty -},P_{\infty +}}(P)=-\frac{1}{y}\prod \limits _{j=1}^{2N+1}(\epsilon _{j}-\lambda )\mathrm{d}\lambda , \end{aligned}$$

where $\epsilon _{j}\in {\mathbb {C}}$, $j=1,\ldots ,2N+1$, are constants that are determined by

$$\begin{aligned} \int _{{\mathbf {a}}_{j}}\omega ^{(3)}_{P_{\infty -},P_{\infty +}}(P)=0,\quad j=1,\ldots ,2N+1. \end{aligned}$$

If the local coordinate near $P_{\infty \pm }$ is given by $\zeta =\lambda ^{-1}$, then we have

$$\begin{aligned} \begin{aligned} \omega ^{(3)}_{P_{\infty -},P_{\infty +}}(P)&\mathop {=}\limits _{\zeta \rightarrow 0}\pm \zeta ^{2N+2}\left( 1-\frac{\alpha _{1,+}}{\alpha _{0,+}}\zeta ^{2}+O(\zeta ^{3})\right) \cdot \left( \zeta ^{-2N-3}\prod \limits _{j=1}^{2N+1}(1-\epsilon _{j}\zeta )\right) \mathrm{d}\zeta \\&\mathop {=}\limits _{\zeta \rightarrow 0}\pm \zeta ^{-1}\left( 1-\frac{\alpha _{1,+}}{\alpha _{0,+}}\zeta ^{2}+O(\zeta ^{3})\right) \cdot (1+O(\zeta ))\mathrm{d}\zeta \\&\mathop {=}\limits _{\zeta \rightarrow 0}\pm (\zeta ^{-1}+O(1))\mathrm{d}\zeta ~~~as~~~P\rightarrow P_{\infty \pm }. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned}&\int _{Q_{0}}^{P}\omega ^{(3)}_{P_{\infty -},P_{\infty +}}(P) \mathop {=}\limits _{\zeta \rightarrow 0}\left\{ \begin{array}{ll} \mathrm {ln}\zeta +\omega _0^{\infty +}+O(\zeta ),~~as~~P\rightarrow P_{\infty +}, \\ -\mathrm {ln}\zeta +\omega _0^{\infty -}+O(\zeta ),~~as~~P\rightarrow P_{\infty -}, \\ \end{array}\right. \\&\int _{Q_{0}}^{P}\omega ^{(3)}_{P_{\infty -},P_{\infty +}}(P) \mathop {=}\limits _{\zeta \rightarrow 0}\left\{ \begin{array}{ll} \omega _0^{0+}+O(\zeta ),~~as~~P\rightarrow P_{0+}, \\ \omega _0^{0-}+O(\zeta ),~~as~~P\rightarrow P_{0-}, \\ \end{array}\right. \end{aligned}$$

where $\omega _{0}^{\infty \pm }$ and $\omega _{0}^{0\pm }$ are integration constants.

Let $\theta (\underline{z})$ denote the Riemann theta function [35, 36] associated with ${\mathcal {K}}_{2N+1}$ equipped with homology basis and holomorphic differentials as before

$$\begin{aligned} \theta (\underline{z})=\sum _{\underline{L}\in {\mathbb {Z}}^{2N+1}}\{\mathrm{exp}{2\pi \mathrm{i}\langle \underline{L},\underline{z}\rangle +\pi \mathrm{i}\langle \underline{L}\tau ,\underline{L}\rangle }\}. \end{aligned}$$

Here $\underline{z}=(z_{1},\ldots ,z_{2N+1})\in {\mathbb {C}}^{2N+1}$ is a complex vector. The diamond brackets denote the scalar product $\langle \underline{L},\underline{z}\rangle =\overline{\underline{L}} z^T=\sum _{j=1}^{2N+1}{\overline{L}}_{j}z_{j}$. For brevity, let $\underline{Q}=\{Q_1,\cdots , Q_{2N+1}\}\in \sigma ^{2N+1}{\mathcal {K}}_{2N+1}$, $\sigma ^{2N+1}{\mathcal {K}}_{2N+1}$ denotes the nth symmetric power of ${\mathcal {K}}_{2N+1}$, then define the function $\underline{z}: {\mathcal {K}}_{2N+1}\times \sigma ^{2N+1}{\mathcal {K}}_{2N+1}\rightarrow {\mathbb {C}}^{2N+1}$,

$$\begin{aligned} \underline{z}(P,\underline{Q})=\underline{K}-\underline{{\mathcal {A}}}(P)+\sum _{Q'\in \underline{Q}}{\mathcal {D}}(Q')\underline{{\mathcal {A}}}(Q'), \end{aligned}$$

where $P\in {\mathcal {K}}_{2N+1},~{\mathcal {D}}(Q')=m(m\in {\mathbb {N}})$ as $Q'$ occurs m times in $\underline{Q}$, and $\underline{K}=(K_{1},\ldots ,K_{2N+1})$ is the vector of Riemann constant defined by

$$\begin{aligned} K_{j}=\frac{1}{2}(1+\tau _{jj})-\sum _{\begin{array}{l} k=1\\ k\ne j\\ \end{array}}^{2N+1}\int _{{\mathbf {a}}_{k}}\omega _{k}\int _{Q_{0}}^{P}\omega _{j},\quad j=1,\ldots ,2N+1. \end{aligned}$$

Noting Theorem 4, then we have

$$\begin{aligned} \begin{aligned} \theta (\underline{z}(P,\hat{\underline{\mu }}(x,t_m)))&=\theta (\underline{K}-\underline{{\mathcal {A}}}(P)+\underline{\rho }^{(1)}(x,t_{\underline{m}})),\\ \theta (\underline{z}(P,-\hat{\underline{\mu }}(x,t_m)))&=\theta (\underline{K}-\underline{{\mathcal {A}}}(P)+\underline{\rho }^{(2)}(x,t_{\underline{m}})) \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \hat{\underline{\mu }}(x,t_{\underline{m}})=\{\hat{\mu }_{1}(x,t_{\underline{m}}),\ldots ,\hat{\mu }_{1}(x,t_{\underline{m}})\},~ -\hat{\underline{\mu }}(x,t_{\underline{m}})=\{-\hat{\mu }_{1}(x,t_{\underline{m}}),\ldots ,-\hat{\mu }_{1}(x,t_{\underline{m}})\}. \end{aligned}$$

Theorem 5

Let $P=(\lambda ,y)\in {\mathcal {K}}_{2N+1}\backslash \{P_{\infty +},P_{\infty -}\}$, $(x,t_{\underline{m}})\in M$, where $M\subseteq {\mathbb {R}}^{2}$ is open and connected. Suppose $u(x,t_{\underline{m}}),v(x,t_{\underline{m}})\in C^{\infty }(M)$ satisfy (6), and assume that $\lambda _{j}\in {\mathbb {C}}\backslash \{0\}$, $j=1,2,\cdots ,4N+4$, in (18) satisfy $\lambda _{j}\in {\mathbb {C}}\backslash \{0\}$, and $\lambda _{j}\ne \lambda _{k}$ as $j\ne k$. Moreover, suppose that $D_{\underline{{\hat{\mu }}}(x,t_{\underline{m}})}$ is nonspecial for $(x,t_{\underline{m}})\in M$. Then $\phi (P,x,t_{\underline{m}})$, $u(x,t_{\underline{m}})$ and $v(x,t_{\underline{m}})$ admit the following representation

$$\begin{aligned} \begin{aligned} \phi (P,x,t_{\underline{m}})&=\left( -\mathrm {exp}(-\omega _{0}^{\infty -}-\omega _{0}^{\infty +})\frac{\theta (\underline{z}(P_{\infty +}, \underline{\hat{\mu }}(x,t_{\underline{m}})))\theta (\underline{z}(P_{\infty -},\underline{\hat{\mu }}(x,t_{\underline{m}})))}{\theta (\underline{z}(P_{\infty +},-\underline{\hat{\mu }}(x,t_{\underline{m}})))\theta (\underline{z}(P_{\infty -}, -\underline{\hat{\mu }}(x,t_{\underline{m}})))}\right) ^{\frac{1}{2}} \\&\quad \times \frac{\theta (\underline{z}(P,-\underline{\hat{\mu }} (x,t_{\underline{m}})))}{\theta (\underline{z}(P,\underline{\hat{\mu }}(x,t_{\underline{m}})))} \mathrm {exp}\left( \int _{Q_{0}}^{P}\omega ^{(3)}_{P_{\infty -},P_{\infty +}}(P)\right) , \end{aligned}\nonumber \\ \begin{aligned} u(x,t_{\underline{m}})&=2\left( \mathrm {exp}(\omega _{0}^{\infty +}-\omega _{0}^{\infty -}) \frac{\theta (\underline{z}(P_{\infty +},-{\underline{\hat{\mu }}}(x,t_{\underline{m}})))\theta (\underline{z}(P_{\infty -},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P_{\infty +},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))\theta (\underline{z}(P_{\infty -},{-\underline{\hat{\mu }}(x,t_{\underline{m}}))})}\right) ^{\frac{1}{2}},\\ \end{aligned} \end{aligned}$$

(35)

$$\begin{aligned} \begin{aligned} v(x,t_{\underline{m}})&=\frac{1}{2s_{0}}\left( -\mathrm {exp}(-\omega _{0}^{\infty -}-\omega _{0}^{\infty +})\frac{\theta (\underline{z}(P_{\infty +},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))\theta (\underline{z}(P_{\infty -},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P_{\infty +},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))\theta (\underline{z}(P_{\infty -},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}\right) ^{-\frac{1}{2}} \\&\quad \times \left( \mathrm {exp}(\omega _{0}^{0-})\frac{\theta (\underline{z}(P_{0-},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P_{0-},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}-\mathrm {exp}(\omega _{0}^{0+})\frac{\theta (\underline{z}(P_{0+},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P_{0+},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}\right) ^{-1},\\ \end{aligned} \end{aligned}$$

(36)

where

$$\begin{aligned} \begin{aligned}&\theta (\underline{z}(P_{\infty \pm },\hat{\underline{\mu }}(x,t_m)))=\theta (\underline{\Delta }_{\infty \pm }+\underline{U}x+\underline{{\mathbf {Y}}}^{(\underline{m})}t_{\underline{m}}),\\&\theta (\underline{z}(P_{\infty \pm },-\hat{\underline{\mu }}(x,t_m)))=\theta (\underline{\Upsilon }_{\infty \pm }-\underline{U}x-\underline{{\mathbf {Y}}}^{(\underline{m})}t_{\underline{m}}),\\&\theta (\underline{z}(P_{0\pm },\hat{\underline{\mu }}(x,t_m)))=\theta (\underline{\Delta }_{0\pm }+\underline{U}x+\underline{{\mathbf {Y}}}^{(\underline{m})}t_{\underline{m}}),\\&\theta (\underline{z}(P_{0\pm },-\hat{\underline{\mu }}(x,t_m)))=\theta (\underline{\Upsilon }_{0\pm }-\underline{U}x-\underline{{\mathbf {Y}}}^{(\underline{m})}t_{\underline{m}}),\\&\underline{\Delta }_{\infty \pm }=\underline{K}+\underline{\rho }^{(1)}(x_{0},t_{0,\underline{m}})-\underline{{\mathcal {A}}}(P_{\infty \pm }),\underline{\Upsilon }_{\infty \pm }=\underline{K}+\underline{\rho }^{(2)}(x_{0},t_{0,\underline{m}})-\underline{{\mathcal {A}}}(P_{\infty \pm }),\\&\underline{\Delta }_{0\pm }=\underline{K}+\underline{\rho }^{(1)}(x_{0},t_{0,\underline{m}})-\underline{{\mathcal {A}}}(P_{0\pm }),\underline{\Upsilon }_{0\pm }=\underline{K}+\underline{\rho }^{(2)}(x_{0},t_{0,\underline{m}})-\underline{{\mathcal {A}}}(P_{0\pm }). \end{aligned} \end{aligned}$$

Proof

We assume that

$$\begin{aligned} \mu _{j}(x,t_{\underline{m}})\ne \mu _{k}(x,t_{\underline{m}})~\mathrm {for}~j\ne k~\mathrm {and}~(x,t_{\underline{m}})\in {\tilde{M}} \end{aligned}$$

for appropriate ${\tilde{M}}\subseteq M$, where ${\tilde{M}}$ is open and connected. According to the Riemann vanishing theorem [24], the definition and asymptotic properties of the meromorphic function $\phi (P,x,t_{\underline{m}})$ has the following form

$$\begin{aligned} \phi (P,x,t_{\underline{m}})=N(x,t_{\underline{m}})\frac{\theta (\underline{z}(P,{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P,{\underline{\hat{\mu }}(x,t_{\underline{m}})}))} \mathrm {exp}\left( \int _{Q_{0}}^{P}\omega ^{(3)}_{P_{\infty -},P_{\infty +}}(P)\right) , \end{aligned}$$

(37)

where $N(x,t_{\underline{m}})$ is independent of $P\in {\mathcal {K}}_{2N+1}$.

Considering the asymptotic expansions of $\phi (P,x,t_{\underline{m}})$ near $P_{\infty \pm }$, we have

$$\begin{aligned} \begin{aligned} \frac{\mathrm{i}u}{2}&=N(x,t_{\underline{m}})\mathrm {exp}(\omega _{0}^{\infty +}) \frac{\theta (\underline{z}(P_{\infty +},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P_{\infty +},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}, \end{aligned}\nonumber \\ \begin{aligned} \frac{2\mathrm{i}}{u}&=N(x,t)\mathrm {exp}(\omega _{0}^{\infty -})\frac{\theta (\underline{z} (P_{\infty -},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z} (P_{\infty -},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}, \end{aligned} \end{aligned}$$

(38)

Multiplying (37) and (38), we can derive

$$\begin{aligned} \begin{aligned} N(x,t_{\underline{m}})=\left( -\mathrm {exp}(-\omega _{0}^{\infty -}-\omega _{0}^{\infty +})\frac{\theta (\underline{z}(P_{\infty +},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))\theta (\underline{z}(P_{\infty -},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P_{\infty +},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))\theta (\underline{z}(P_{\infty -},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}\right) ^{\frac{1}{2}}, \end{aligned} \end{aligned}$$

which proves (35).

Similarly, divided (37) by (38), we can get

$$\begin{aligned} \begin{aligned} u(x,t_{\underline{m}})&=2\left( \mathrm {exp}(\omega _{0}^{\infty +}-\omega _{0}^{\infty -})\frac{\theta (\underline{z}(P_{\infty +},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))\theta (\underline{z}(P_{\infty -},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P_{\infty +},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))\theta (\underline{z}(P_{\infty -},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}\right) ^{\frac{1}{2}}.\\ \end{aligned} \end{aligned}$$

Considering the asymptotic expansions of $\phi (P,x,t_{\underline{m}})$ near $P_{0\pm }$, we have

$$\begin{aligned} \begin{aligned} \frac{-s-s_{0}}{v}=N(x,t_{\underline{m}})\mathrm {exp}(\omega _{0}^{0+})\frac{\theta (\underline{z}(P_{0+},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P_{0+},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \frac{-s+s_{0}}{v}=N(x,t_{\underline{m}})\mathrm {exp}(\omega _{0}^{0-})\frac{\theta (\underline{z}(P_{0-},{-\underline{\hat{\mu }}(x,t_{\underline{m}})}))}{\theta (\underline{z}(P_{0-},{\underline{\hat{\mu }}(x,t_{\underline{m}})}))}, \end{aligned} \end{aligned}$$

by direct calculation, we can derive (36). Hence, we prove this theorem on ${\tilde{M}}$. The extension of all these results from ${\tilde{M}}$ to M then follows by continuity of the Abel map and the nonspecial nature of $D_{\underline{\hat{\mu }}(x,t_{\underline{m}})}$ on M. $\square$

6 Conclusion

In this paper, algebro-geometric solutions are constructed for the entire sine-Gordon hierarchy, which is composed of all the positive and negative flows. The results are different from the sine-Gordon hierarchy in light-cone coordinates which contains complicated nonlocal higher order sine-Gordon equations. Because this work involves hyperelliptic curves, it is very difficult to study the algebro-geometric solutions of the entire sine-Gordon hierarchy. Using the theory of hyperelliptic curves, the Abel-Jacobi coordinates are introduced, from which the corresponding positive and negative flows are linearized. The algebro-geometric solutions of the entire sine-Gordon hierarchy are constructed by using the asymptotic properties of the meromorphic function.