1. Introduction

The Toda lattice [1], [2]

$$ \frac{\partial^2y}{\partial{t^2}}=\exp(y^{-}-y)-\exp(y-y^{+}),\qquad y=y(n,t),\quad (n,t)\in\mathbb{Z}\times\mathbb{R},$$
(1.1)

is an absolutely integrable equation with exponential interaction that was discovered in the course of seeking a system with rigorous periodic solutions; its exponential interaction was used to explain the nonergodic character of the famous Fermi–Pasta–Ulam problem [3]. It has abundant mathematical structures and is regarded as a model of physical phenomena, with the well-known equations such as the nonlinear Schrödinger (NLS) and Korteweg–de Vries equations being closely related to it or deduced from it by appropriate limit procedures [4], [5]. In addition, it can describe the motion of a chain of particles with nearest-neighbor interaction in constructing different mathematical models; the Toda lattice model of DNA is also a typical representative in biology [6].

It is worth mentioning that by the variable transformation \(\varpi=-\exp(y-y^{+})\) and \(x=y_t\), the Toda lattice can be rewritten in the form

$$ \varpi_t=\varpi(x-x^{+}),\qquad x_t=\varpi-\varpi^{-}.$$
(1.2)

With the increase in scholars’ attention to the Toda lattice, a variety of important methods were applied to it and numerous results have been achieved since it was proposed [7]–[12]. As one of the most effective research tools, the algebro-geometric methods were extensively applied to the Toda lattices. With the development of the finite-gap integration method in the works of Novikov, Matveev, Its, and others, the mathematical theory of the algebro-geometric method has been systematically developed since the early 1970s [13], [14]. The solutions can not only describe the integrability properties of the equations but also reveal the internal structure of the solutions for soliton equations [15]–[18].

The algebro-geometric solutions for numerous soliton equations related to a \(2\times 2\) matrix spectral problem have been obtained using the theta functions of hyperelliptic curves in a series of studies [19]–[21]. However, the studies of algebro-geometric solutions of 3rd-order soliton equations are very few. In the course of studying the algebro-geometric solutions of the 3rd-order soliton equations, the most classical findings originates in the Boussinesq equation, whose 3rd-order differential operators were studied in terms of the reduction theory of Riemann theta functions [22]; finite-gap solutions of the NLS equation were also confirmed smoothly by means of a special algorithm [23]. In 1999, based on the algebro-geometric method, Dickson, Gesztesy, and Unterkofler proposed a unified framework that yields all algebro-geometric solutions of the entire Boussinesq hierarchy related to a 3rd-order differential operator [24]. Based on the framework proposed previously, a systematic method for introducing a trigonal curve was developed with the help of the characteristic polynomial of the Lax matrix associated with the higher-order matrix spectral problem, from which the algebro-geometric method was successfully generalized to yield algebro-geometric solutions of the continuous hierarchies related to \(3\times 3\) matrices [25], [26]. Then the algebro-geometric method was further extended to 3rd-order discrete hierarchies [27], [28].

In this paper, we introduce the trigonal curve to define the Baker–Akhiezer function \(\Xi\) and the corresponding meromorphic function \(\Theta\). The soliton equations can then be separated into solvable Dubrovin-type ordinary differential equations. Based on the above step, the characteristics of the functions can be further analyzed. With the systematic algebro-geometric theory as support, we discuss the application of the algebro-geometric methods to the discrete hierarchy of a 3rd-order generalized Toda lattice

$$ \begin{alignedat}{3} &q_t=2rq^{--}q^{-}-\frac{s^{+}q}{s},&\qquad &r_t=\frac{s}{s^{-}}-\frac{s^{+}}{s}+2(qv^{+}-q^{-}v), \\ &s_t=-qsv-rs,&\qquad &v_t=\frac{qs}{s^{-}}-2rv+2v^{-}, \end{alignedat}$$
(1.3)

which becomes is the Toda lattice mentioned above (1.2) if \(q=0 \), \(r=\varpi^{+}\), \(x=s/s^{-}\), and \(v=0\). The Hamiltonian system for (1.3) was constructed in [29].

The paper is organized as follows. In Sec. 2, the difference operators \(K_n\) and \(J_n\) are deduced in accordance with the Lenard recurrence relations and hierarchy (1.3) is then derived from the zero-curvature equation. In Sec. 3, the trigonal curve \(\mathcal K_{l-1}\) is defined for the characteristic polynomial of the Lax pair for hierarchy (1.3), whence the functions \(\Xi\) and \(\Theta\) can be defined. In Sec. 4, in the stationary case, we analyze the characteristics of the functions and introduce the Abel differentials; the potentials of the Lax pair are then expressed in terms of the Riemann theta function. In Sec. 5, we apply the analysis in last two sections to the time-dependent case and separate hierarchy (1.3) into solvable Dubrovin-type ordinary differential equations. Then we straighten out the flows and obtain the Riemann theta representation. On the whole, the algebro-geometric solutions of hierarchy (1.3) are obtained and we rewrite the Riemann theta representation of the potentials for low genera. We summarize and conclude in Sec. 6.

2. The hierarchy of a generalized Toda lattice

We suppose that \(q\), \(r\), \(s\), and \(v\) satisfy the following conditions: in the stationary case,

$$q(n,\,{\cdot}\,), r(n,\,{\cdot}\,), s(n,\,{\cdot}\,), v(n,\,{\cdot}\,)\in C^1(\mathbb{R}),$$

and in the time-dependent case,

$$q(\,{\cdot}\,,t), r(\,{\cdot}\,,t), s(\,{\cdot}\,,t), v(\,{\cdot}\,,t)\in\mathbb{C}^{\,\mathbb{Z}},\qquad t\in\mathbb{R},$$

where \(\mathbb{C}^{\,\mathbb{Z}}\) is the set of all complex-valued functions of a variable in \(\mathbb{Z}\).

On the complex-valued sequence \(\hbar=\{\hbar(n)\}_{n\in\mathbb{Z}}\), we define the shift operators \(E^{\pm}\) as

$$(E^{\pm}\hbar)(n)=\hbar(n\pm 1),\qquad n\in\mathbb{Z},$$

and write \(\hbar^{\pm}=E^{\pm}\hbar\) with \( \hbar\in\mathbb{C}^{\,\mathbb{Z}}\).

We consider the discrete \(3\times 3\) matrix spectral problem [29]

$$ E\Xi=U\Xi,\qquad \Xi=\begin{pmatrix} \Xi_1 \\ \Xi_2 \\ \,\Xi_3\, \end{pmatrix},\qquad U=\begin{pmatrix} 1 & q & 0 \\ v & \lambda+r & s \\ 0 & -1/s & 0 \end{pmatrix},$$
(2.1)

where \(q\), \(r\), \(s\), and \(v\) are potentials and \(\lambda\) is a constant. The Lenard recurrence relations are

$$ \begin{alignedat}{3} &K_n\tilde g_j=J_n\tilde g_{j+1}, & \qquad &\tilde g_j=(\tilde a_j,\tilde b_j,\tilde c_j,\tilde d_j)^{\mathrm T}, \\ & K_n\bar g_j=J_n\bar g_{j+1}, & \qquad &\bar g_j=(\bar a_j,\bar b_j,\bar c_j,\bar d_j)^{\mathrm T} \end{alignedat}$$
(2.2)

We then introduce the starting points

$$ \tilde g_0=(1,0,-1,0)^{\mathrm T},\qquad \bar g_0=(-1,0,2,0)^{\mathrm T},$$
(2.3)

and define two difference operators \(J_n\) and \(K_n\) as

$$\begin{aligned} \, &J_n=\begin{pmatrix} 0 & E & 0 & 0 \\ 0 & 0 & E-1 & 0 \\ 0 & 0 & 0 & s^2E \\ -\dfrac{1}{q}(E-1) & -\dfrac{v}{q}E & 0 & 0 \end{pmatrix}, \\ &K_n=\begin{pmatrix} -qE & K_{12} & q & \dfrac{1}{s}Eq{s}^2E \\ -qE\dfrac{1}{q}(E-1) & v-qE\dfrac{v}{q}E & -r(E-1) & s-\dfrac{1}{s}E{s}^2E \\ -s & vsE & -sE-s & K_{34} \\ K_{41} & K_{42} & -vE & -sE^{-1}vE \end{pmatrix}, \end{aligned}$$

where

$$ \begin{aligned} \, &K_{12}=1+\frac{1}{s}EsE-rE,\qquad K_{34}=vq{s}^2E-r{s}^2E, \\ &K_{41}=(r-E)\frac{1}{q}(E-1)+\frac{s(E^{-1}-1)}{s^{-}q^{-}}+v, \\ &K_{42}=-E\frac{v}{q}+\frac{rv}{q}E-\frac{sE^{-1}vE}{s^{-}q^{-}}. \end{aligned}$$
(2.4)

Hence, \(\tilde g_j\) and \(\bar g_j\) can be found using the operators \(K_n\) and \(J_n\); the first two members are given by

$$ \tilde g_1=\biggl(1,-2q^{-},0,\frac{1}{s^{-}}\biggr)^{\!\mathrm T},\qquad \bar g_1=\biggr(1,3q^{-},1,-\frac{3}{s^{-}}\biggr)^{\!\mathrm T}.$$
(2.5)

To deduce the hierarchy related to spectral problem (2.1), we introduce the stationary zero-curvature equation

$$ (E\Gamma)U-U\Gamma=0,\qquad \Gamma=(\Gamma_{ij})_{3\times 3}=\begin{pmatrix} \Gamma_{11} & \Gamma_{12} & \Gamma_{13} \\ \Gamma_{21} & \Gamma_{22} & \Gamma_{23} \\ \Gamma_{31} & \Gamma_{32} & -\Gamma_{11}-\Gamma_{22} \end{pmatrix},$$
(2.6)

which is equivalent to

$$ \begin{aligned} \, &E\Gamma_{11}+vE\Gamma_{12}-\Gamma_{11}-q\Gamma_{21}=0, \\ &qE\Gamma_{11}+{(r+\lambda)}E\Gamma_{12}-\frac{1}{s}E\Gamma_{13}-\Gamma_{12}-q\Gamma_{22}=0, \\ &sE\Gamma_{12}-\Gamma_{13}-q\Gamma_{23}=0, \\ &E\Gamma_{21}+vE\Gamma_{22}-v\Gamma_{11}-(r+\lambda)\Gamma_{21}-s\Gamma_{31}=0, \\ &qE\Gamma_{21}+(r+\lambda)E\Gamma_{22}-\frac{1}{s}E\Gamma_{23}-v\Gamma_{12}-(r+\lambda)\Gamma_{22}-s\Gamma_{32}=0, \\ &sE\Gamma_{22}-v\Gamma_{13}-(r+\lambda)\Gamma_{23}+s(\Gamma_{11}+\Gamma_{22})=0, \\ &E\Gamma_{31}+vE\Gamma_{32}+\frac{1}{s}\Gamma_{21}=0, \\ &qE\Gamma_{31}+(r+\lambda)E\Gamma_{32}+\frac{1}{s}E(\Gamma_{11}+\Gamma_{22})+\frac{1}{s}\Gamma_{22}=0, \\ &sE\Gamma_{32}+\frac{1}{s}\Gamma_{23}=0. \end{aligned}$$
(2.7)

where each element \(\Gamma_{ij}=\Gamma_{ij}(a,b,c,d)\) is a Laurent expansion in \(\lambda\),

$$\begin{alignedat}{5} &\Gamma_{11}=a, & \quad &\Gamma_{12}=b, & \quad &\phantom{-}\Gamma_{13}=sEb+q{s}^2Ed, \nonumber\\ &\Gamma_{21}=\frac{1}{q}(E-1)a+\frac{v}{q}Eb, & \quad &\Gamma_{22}=c, & \quad &\phantom{-}\Gamma_{23}=-{s}^2Ed, \\ &\Gamma_{31}=\frac{(E^{-1}-1)}{s^{-}q^{-}}a-\frac{E^{-1}vE}{s^{-}q^{-}}b-E^{-1}vEd, & \quad &\Gamma_{32}=d, & \quad &-\Gamma_{11}-\Gamma_{11}=-a-c, \nonumber \end{alignedat}$$
(2.8)

with

$$ a=\sum_{j\ge 0}{a_j\lambda ^{-j}},\qquad b=\sum_{j\ge 0}b_j\lambda ^{-j},\qquad c=\sum_{j\ge 0}{c_j\lambda ^{-j}},\qquad d=\sum_{j\ge 0}{d_j\lambda ^{-j}}.$$
(2.9)

We can show by direct calculation that Eqs. (2.6) and (2.7) imply the Lenard equation

$$ K_nG=\lambda J_nG,\qquad G=(a,b,c,d)^{\mathrm T}.$$
(2.10)

We substitute (2.8) in (2.9) and compare the powers of \(\lambda\) to deduce the recurrence relations

$$ K_nG_j=\lambda J_nG_{j+1},\quad J_nG_0=0,\qquad j\ge 0,$$
(2.11)

where \(G_j=(a_j,b_j,c_j,d_j)^{\mathrm T}\). It is evident that \(\operatorname{ker}{J_n}=\{\alpha_0 \tilde g_0+\beta_0\bar g_0\mid\alpha_0,\beta_0\in\mathbb{R}\}\) and \(G_j\) has the expansion

$$ G_j=\alpha_0\tilde g_j+\beta_0\bar g_j+\cdots+\alpha_j\tilde g_0+\beta_j\bar g_0,\qquad j\ge 0,$$
(2.12)

where \(\alpha_j\) and \(\beta_j\) are constants.

Assuming that \(\Xi\) satisfies the discrete matrix spectral problem (2.1), we have

$$ \Xi_{t_m}=\widehat\Gamma^{(m)}\Xi ,\qquad \widehat\Gamma^{(m)}=(\widehat\Gamma_{ij}^{(m)})_{3\times3}=\begin{pmatrix} \widehat\Gamma_{11}^{(m)} & \widehat\Gamma_{12}^{(m)} & \widehat\Gamma_{13}^{(m)} \\ \widehat\Gamma_{21}^{(m)} & \widehat\Gamma_{22}^{(m)} & \widehat\Gamma_{23}^{(m)} \\ \widehat\Gamma_{31}^{(m)} & \widehat\Gamma_{32}^{(m)} & -\widehat\Gamma_{11}^{(m)}-\widehat\Gamma_{22}^{(m)} \end{pmatrix},$$
(2.13)

where \(\widehat\Gamma_{ij}^{(m)}=\widehat\Gamma_{ij}(\hat a^{(m)},\hat b^{(m)},\hat c^{(m)},\hat d^{(m)})\) and

$$ \begin{alignedat}{3} & \hat a^{(m)}=\sum_{j=0}^{m}\hat a_j\lambda ^{m-j}, & \qquad & \hat b^{(m)}=\sum_{j=0}^{m}\hat b_j\lambda ^{m-j}, \\ & \hat c^{(m)}=\sum_{j=0}^{m}\hat c_j\lambda^{m-j}, & \qquad & \hat d^{(m)}=\sum_{j=0}^{m}\hat d_j\lambda^{m-j}. \end{alignedat}$$
(2.14)

Similarly, the elements \(\hat a_j\), \(\hat b_j\), \(\hat c_j\), and \(\hat d_j\) can be determined as

$$ \widehat G_j=\hat\alpha_0\tilde g_j+\hat\beta_0\bar g_j+\cdots+\hat\alpha_j\tilde g_0+\hat\beta_j\bar g_0,\qquad j\ge 0,$$
(2.15)

where \(\widehat G_j\) are also solutions of (2.10). We note, importantly, that \(\hat\alpha_j\), \(\hat\beta_j\) and \(\alpha_j\), \(\beta_j\) in (2.12) are absolute of each other. The zero-curvature equation \(U_{t_m}=(E\widehat\Gamma^m)U-U\widehat\Gamma^m\) is generated by the compatibility condition of Eqs. (2.1) and (2.12), which is equivalent to discrete hierarchy (1.3),

$$ (q_{t_m},r_{t_m},s_{t_m},v_{t_m})^{\mathrm T}=\widehat{\mathcal I}_m,\qquad m\ge 0,$$
(2.16)

and the vector can be represented as

$$\widehat{\mathcal I}_j=K_n\widehat G_j=J_n\widehat G_{j+1},\qquad \widehat{\mathcal I}_j=\mathcal I(q,r,s,v, \underline {\hat\alpha}^{(j)}, \underline {\hat\beta}^{(j)}),$$

where \( \underline {\hat\alpha}^{(j)}=(\hat\alpha_0,\ldots,\hat\alpha_j)\), and \( \underline {\hat\beta}^{(j)}=(\hat\beta_0,\ldots,\hat\beta_j)\) for \(j\ge 0\).

The first nontrivial member of hierarchy (2.16) is given by as

$$\widehat{\mathcal I}_0=K_n\widehat G_0=K_n(\alpha_0\tilde g_0+\beta_0\bar g_0),$$

whence, with \(\alpha_0=1\) and \(\beta_0=0\), we have

$$ q_{t_0}=-2q,\qquad r_{t_0}=0,\qquad s_{t_0}=s,\qquad v_{t_0}=2v.$$
(2.17)

Similarly, for \(j=2\) and \(\widehat{\mathcal I}_1=K_n(\alpha_0\tilde g_1+\alpha_1\tilde g_0+\beta_0\bar g_1+\beta_1\bar g_0)\) with \(\alpha_0=1\), \(\beta_0=1\), and \(t_0=t\), we obtain the hierarchy that we study in what follows:

$$ \begin{alignedat}{3} &q_t=2rq^{--}q^{-}-\frac{s^{+}q}{s}, & \qquad &r_t=\frac{s}{s^{-}}-\frac{s^{+}}{s}+2(qv^{+}-q^{-}v), \\ &s_t=-qsv-rs, & \qquad &v_t=\frac{qs}{s^{-}}-2rv+2v^{-}. \end{alignedat}$$
(2.18)

If \(q=0 \), \(r=\varpi^{+}\), \(x=s/s^{-}\), and \(v=0\), Eqs. (2.18) become the Toda lattice (1.2).

3. The stationary meromorphic function

We consider hierarchy (1.3) in the stationary case \(\mathcal I_p=\mathcal I(q,r,s,v; \underline {\alpha}^{(p)}\), \( \underline {\beta}^{(p)})=0\), \( \underline {\alpha}^{(p)}=(\alpha_0\ldots\alpha_p)\), and \( \underline {\beta}^{(p)}=(\beta_0\ldots\beta_p)\). It is then equivalent to the stationary zero-curvature equation

$$ (E\Gamma^{(p)})U-U\Gamma^{(p)}=0,\qquad\Gamma^{(p)}=(\lambda^{p}\Gamma)_{+}=(\Gamma_{ij}^{(p)})_{3\times3},$$
(3.1)

with \(\Gamma_{ij}^{(p)}=\Gamma_{ij}(a^{(p)},b^{(p)},c^{(p)},d^{(p)})\),

$$ a^{(p)}=\sum_{j=0}^{p}a_j\lambda^{p-j},\quad b^{(p)}=\sum_{j=0}^{p}b_j\lambda^{p-j},\quad c^{(p)}=\sum_{j=0}^{p}c_j\lambda^{p-j},\quad d^{(p)}=\sum_{j=0}^{p}d_j\lambda^{p-j}.$$
(3.2)

Direct calculation indicates that the characteristic polynomial \(\digamma_l(\lambda,f)=\det( fI-\Gamma^{(p)})\) of \(\Gamma^{(p)}\) also satisfies zero-curvature equation (3.1) and is a constant independent of \(n\). It has the expansion

$$ \det(fI-\Gamma^{(p)})=f^3-f^2X_l(\lambda)+fY_l(\lambda)-Z_l(\lambda),$$
(3.3)

where \(X_l(\lambda),Y_l(\lambda)\) and \(Z_l(\lambda)\) are constant-coefficient polynomials in \(\lambda\),

$$ \begin{aligned} \, X_l(\lambda)&=t_r\Gamma^{(p)}=\Gamma_{11}^{(p)}+\Gamma_{22}^{(p)}+(-\Gamma_{11}^{(p)}-\Gamma_{22}^{(p)})=0, \\ Y_l(\lambda)&= \begin{vmatrix} \Gamma_{11}^{(p)} & \Gamma_{12}^{(p)} \\ \Gamma_{21}^{(p)} & \Gamma_{22}^{(p)} \end{vmatrix}+ \begin{vmatrix} \Gamma_{11}^{(p)} & \Gamma_{13}^{(p)} \\ \Gamma_{31}^{(p)} & -\Gamma_{11}^{(p)}-\Gamma_{22}^{(p)} \end{vmatrix}+ \begin{vmatrix} \Gamma_{22}^{(p)} & \Gamma_{23}^{(p)} \\ \Gamma_{32}^{(p)} & -\Gamma_{11}^{(p)}-\Gamma_{22}^{(p)} \end{vmatrix}= \\ &=(-\alpha_0^2+\alpha_0\beta_0-3\beta_0^2)\lambda^{2p}+\cdots, \\ Z_l(\lambda)&=\det\Gamma_n^{(p)}= \begin{vmatrix} \Gamma_{11}^{(p)} & \Gamma_{12}^{(p)} & \Gamma_{13}^{(p)} \\ \Gamma_{21}^{(p)} & \Gamma_{22}^{(p)} & \Gamma_{23}^{(p)} \\ \Gamma_{31}^{(p)} & \Gamma_{32}^{(p)} & -\Gamma_{11}^{(p)}-\Gamma_{22}^{(p)} \end{vmatrix}= \\ &=({\alpha_0}^2\beta_0-3\alpha_0{\beta_0}^2+2{\beta_0}^3)\lambda^{3p}+\cdots{}. \end{aligned}$$
(3.4)

Then the trigonal curve \(\digamma_l(\lambda,f)=0\) whose degree is \(l=3p\) for \(\alpha_0\beta_0\neq 0\) can be introduced as

$$ \mathcal K_{l-1}\colon\, \digamma_l(\lambda ,f)=f^3-f^2X_l(\lambda)+fY_l(\lambda)-Z_l(\lambda)=0.$$
(3.5)

Under the condition \(l=3p\), it is obvious that the trigonal curve \(\mathcal K_{l-1}\) can be compactified by adding different infinite points \(u_{\infty'}\) and \(u_{\infty''}\) based on (3.2) and (3.4), where we choose \(u_{\infty''}\) as a double branch point. We still use \(\mathcal K_{l-1}\) to denote the compactified curve. The discriminant of (3.5) is

$$ \Delta(\lambda)=27{Z_l}^2-18X_lS_lZ_l+4{Y_l}^3-{X_l}^2{Y_l}^2+4{X_l}^3Z_l.$$
(3.6)

By the Riemann–Hurwitz formula, we can obtain that the arithmetic genus of \(\mathcal K_{l-1}\) is \(l-1\). Therefore, \(\mathcal K_{l-1}\) turns into a three-sheet Riemann surface of genus \(l-1\) if the curve is irreducible and

$$\biggl(\frac{\partial\digamma_l(\lambda,f)}{\partial \lambda},\frac{\partial\digamma_l(\lambda,f)}{\partial f}\biggr)\bigg|_{(\lambda,f)=(\lambda_0,f_0)}\neq 0$$

for any \(u_0=(\lambda_0,f_0)\in\mathcal K_{l-1}\).

We introduce the stationary Baker–Akhiezer function \(\Xi\) as

$$ \begin{aligned} \, &E\Xi(u,n,n_0)=U(q(n),r(n),s(n),v(n);\lambda(u))\Xi(u,n,n_0), \\ &\Gamma^{(p)}(q(n),r(n),s(n),v(n);\lambda(u))\Xi(u,n,n_0)=f(u)\Xi(u,n,n_0), \\ &\Xi_1(u,n_0,n_0)=1,\qquad u=(\lambda,f)\in\mathcal K_{l-1}\backslash\{u_{\infty'},u_{\infty''}\},\quad n,n_0\in\mathbb{Z}. \end{aligned}$$
(3.7)

Based on the function \(\Xi\), the meromorphic function \(\Theta\) on \(\mathcal K_{l-1}\) is defined as

$$ \Theta(u,n)=\frac{\Xi_2(u,n,n_0)}{\Xi_1(u,n,n_0)},\qquad u\in\mathcal K_{l-1},\quad n\in\mathbb{Z},$$
(3.8)

whence we have

$$ \Xi_1(u,n,n_0)= \begin{cases} \displaystyle\prod_{n'=n_0}^{n-1}(1+q(n)\Theta(u,n')), & n\ge n_0+1,\\ \quad 1, & n=n_0, \\ \displaystyle\prod_{n'=n_0}^{n-1}(1+q(n)\Theta(u,n'))^{-1}, & n\le n_0-1. \end{cases}$$
(3.9)

The meromorphic function \(\Theta(u,n)\) obtained in accordance with (3.7) and (3.8) is

$$\begin{aligned} \, \Theta(u,n)&=\frac{y\Gamma_{23}^{(p)}+A_l(\lambda,n)}{f\Gamma_{13}^{(p)}+B_l(\lambda,n)}= \frac{E_{l-1}(\lambda,n)}{f^2\Gamma_{23}^{(p)}-fA_l(\lambda,n)+C_l(\lambda,n)}= \nonumber\\ &=\frac{f^2\Gamma_{13}^{(p)}-fB_l(\lambda,n)+D_l(\lambda,n)}{F_{l-1}(\lambda,n)}, \end{aligned}$$
(3.10)

where

$$ \begin{aligned} \, &A_l=\Gamma_{13}^{(p)}\Gamma_{21}^{(p)}-\Gamma_{11}^{(p)}\Gamma_{23}^{(p)},\qquad B_l=\Gamma_{12}^{(p)}\Gamma_{23}^{(p)}-\Gamma_{13}^{(p)}\Gamma_{22}^{(p)}, \\ &C_l=\Gamma_{21}^{(p)}(\Gamma_{13}^{(p)}\Gamma_{22}^{(p)}-\Gamma_{12}^{(p)}\Gamma_{23}^{(p)})- \Gamma_{23}^{(p)}(\Gamma_{11}^{(p)}\Gamma_{22}^{(p)}+(\Gamma_{22}^{(p)})^2+\Gamma_{23}^{(p)}\Gamma_{32}^{(p)}), \\ &D_l=\Gamma_{12}^{(p)}(\Gamma_{11}^{(p)}\Gamma_{23}^{(p)}-\Gamma_{13}^{(p)}\Gamma_{21}^{(p)})-\Gamma_{13}^{(p)}((\Gamma_{11}^{(p)})^2+ \Gamma_{11}^{(p)}\Gamma_{22}^{(p)}+\Gamma_{13}^{(p)}\Gamma_{31}^{(p)}), \\ &E_{l-1}=(\Gamma_{23}^{(p)})^2\Gamma_{31}^{(p)}+\Gamma_{21}^{(p)}\Gamma_{23}^{(p)}(2\Gamma_{11}^{(p)}+ \Gamma_{22}^{(p)})-(\Gamma_{21}^{(p)})^2\Gamma_{13}^{(p)}, \\ &F_{l-1}=(\Gamma_{13}^{(p)})^2\Gamma_{32}^{(p)}+\Gamma_{12}^{(p)}\Gamma_{13}^{(p)}(2\Gamma_{22}^{(p)}+ \Gamma_{11}^{(p)})-(\Gamma_{12}^{(p)})^2\Gamma_{23}^{(p)}. \end{aligned}$$
(3.11)

Moreover, we introduce two other elements

$$ \begin{aligned} \, &G_l=\Gamma_{13}^{(p)}\Gamma_{32}^{(p)}+\Gamma_{12}^{(p)}(\Gamma_{11}^{(p)}+\Gamma_{22}^{(p)}), \\ &H_l=\Gamma_{12}^{(p)}(\Gamma_{11}^{(p)}\Gamma_{22}^{(p)}-\Gamma_{12}^{(p)}\Gamma_{21}^{(p)})+ \Gamma_{13}^{(p)}(\Gamma_{11}^{(p)}\Gamma_{32}^{(p)}-\Gamma_{12}^{(p)}\Gamma_{31}^{(p)}). \end{aligned}$$
(3.12)

Obviously, we can find various relations among polynomials (3.11), (3.12) and \(X_l\), \(Y_l\), \(Z_l\). We list some of them:

$$ \begin{aligned} \, &F_{l-1}=\Gamma_{13}^{(p)}G_l-\Gamma_{12}^{(p)}B_l, \\ &B_lE_{l-1}=(\Gamma_{23}^{(p)})^2Z_l+A_lC_l,\qquad A_lF_{l-1}=(\Gamma_{13}^{(p)})^2Z_l+B_lD_l, \\ &\Gamma_{13}^{(p)}E_{l-1}=\Gamma_{23}^{(p)}C_l-(\Gamma_{23}^{(p)})^2Y_l-A_l^2,\qquad \Gamma_{23}^{(p)}F_{l-1}=\Gamma_{13}^{(p)}B_l-(\Gamma_{13}^{(p)})^2Y_l-B_l^2, \\ &E_{l-1}^{-}=-F_{l-1},\qquad G_l=A_l^{-},\qquad H_l=C_l^{-}. \end{aligned}$$
(3.13)

Using (3.1), (3.2), (3.11), and (3.13), we find that \(F_{l-1}\) and \(E_{l-1}\) are polynomials of degree \(l-1\) and can therefore be represented as

$$ \begin{aligned} \, &F_{l-1}(\lambda ,n)=F_{l-1,0}\prod_{j=1}^{l-1}(\lambda-\mu_j(n)), \\ &E_{l-1}(\lambda ,n)=-F_{l-1,0}\prod_{j=1}^{l-1}(\lambda-\mu_j^{+}(n)). \end{aligned}$$
(3.14)

On the trigonal curve \(\mathcal K_{l-1}\), we define \(\{\tilde\mu_j(n)\}_{j=\overline{1,l-1}}\) and \(\{\tilde\mu_j^{+}(n)\}_{j=\overline{1,l-1}}\) as

$$ \begin{aligned} \, &\tilde\mu_j(n)=\bigl(\mu_j(n),y(\hat{\mu}_j(n))\bigr)=\biggl(\mu_j(n)-\frac{B_l(\lambda,n)}{\Gamma_{32}^{(p)}(\mu_j(n))}\biggr), \\ &\tilde\mu_j^{+}(n)=\bigl(\mu_j^{+}(n),y(\hat{\mu}_j^{+}(n))\bigr)=\biggl(\mu_j^{+}(n)-\frac{A_l(\lambda,n)}{\Gamma_{32}^{(p)}(\mu_j^{+}(n))}\biggr). \end{aligned}$$
(3.15)

For convenience, we let \(u\), \(u^*\), and \(u^{**}\) denote points on each of the three different sheets of the Riemann surface \(\mathcal K_{l-1}\) and suppose that \(f_i(\lambda)\) (\(i=1,2,3\)) are three roots of \(\digamma_l(\lambda,f)=0\):

$$ \bigl(f-f_1(\lambda)\bigr)\bigl(f-f_2(\lambda)\bigr)\bigl(f-f_3(\lambda)\bigr)=f^3-f^2X_l+fY_l-Z_l=f^3+fY_l-Z_l=0.$$
(3.16)

Then the three points \((\lambda,f_1(\lambda))\), \((\lambda, f_2(\lambda))\), and \((\lambda,f_3(\lambda))\) are also on the Riemann surface \(\mathcal K_{l-1}\). Let \(\{u,u^*,u^{**}\}=\{(\lambda,f_i(\lambda)),\,i=1,2,3\}\) be any one of the three points. From (3.16), the following system can easily be obtained:

$$\begin{aligned} \, &f_1+f_2+f_3=X_l=0,\qquad f_1f_2+f_2f_3+f_3f_1=Y_l,\qquad f_1f_2f_3=Z_l, \\ &f_1^2+f_2^2+f_3^2=-2Y_l,\qquad f_1^3+f_2^3+f_3^3=3Z_l,\qquad f_1^2f_2^2+f_1^2f_3^2+f_2^2f_3^2=X_l^2, \\ &(f_1+f_2)f_3^2+(f_2+f_3)f_1^2+(f_1+f_3)f_2^2=-3Z_l. \end{aligned}$$

The function \(\Theta(u,n)\) then satisfies the relations

$$ \begin{aligned} \, &\Theta(u,n)\Theta(u^*,n)\Theta(u^{**},n)=-\frac{E_{l-1}(\lambda,n)}{F_{l-1}(\lambda,n)}, \\ &\Theta(u,n)+\Theta(u^*,n)+\Theta(u^{**},n )=\frac{3D_l(\lambda,n)-2\Gamma_{32}^{(p)}Y_l(\lambda)}{F_{l-1}(\lambda,n)}, \\ &\frac{1}{\Theta(u,n)}+\frac{1}{\Theta(u^*,n)}+\frac{1}{\Theta(u^{**},n)}= \frac{3C_l(\lambda,n)-2\Gamma_{12}^{(p)}(\lambda,n)Y_l(\lambda)}{E_{l-1}(\lambda,n)}. \end{aligned}$$
(3.17)

4. Algebro-geometric solutions of the stationary hierarchy

We analyze the asymptotic behavior of the functions \(\Theta(u,n)\) and \(\Xi(u,n)\), and then introduce the Abel differential and the Riemann theta function. As a result, we obtain algebro-geometric solutions in the stationary case, whereby the potentials \(q\), \(r\), \(s\), and \(v\) can be expressed as in terms of the Riemann theta function.

First, it follows by direct calculation that \(\Theta(u,n)\) satisfies the Riccati-type equation

$$\begin{aligned} \, q^{-}(n)q(n)&\Theta^{+}(u,n)\Theta(u,n)\Theta^{-}(u,n)= \biggl(v(n)q^{-}(n)-\frac{s(n)}{s^{-}(n)}\biggr)\Theta^{-}(u,n)+{} \nonumber\\ &+(\lambda+r(n))\Theta(u,n)-\Theta^{+}(u,n)+v(n)+(\lambda+r(n))q^{-}(n)\Theta^{-}(u,n)\Theta(u,n)-{} \nonumber\\ &-q(n)\Theta(u,n)\Theta^{+}(u,n)-q^{-}(n)\Theta^{-}(u,n)\Theta^{+}(u,n). \end{aligned}$$
(4.1)

Introducing the local coordinate \(\varsigma=\lambda^{-1}\) near \(u_{\infty'}\) and comparing the powers of \(\varsigma\), we have the formula

$$\Theta=\sum_{j=1}^{\infty}\delta_j\varsigma^j,\qquad u\to u_{\infty'},$$

with

$$ \begin{aligned} \, &\delta_1=-v,\qquad\delta_2=v^{+}-\frac{s}{s^{-}}v^{-}+rv, \\ &\delta_3=v(r+r^{+}+1-qv+{r}^2)+v^{+}(2+r-qv-2q^{-}v^{-}-r^{+})-v^{++}. \end{aligned}$$
(4.2)

As at the preceding step, we introduce the local coordinate \(\lambda=\eta^{-2}\) near \(u_{\infty''}\) and compare the powers of \(\eta\), which yields

$$\Theta=\sum_{j=0}^{\infty}\kappa_j\eta^j,\qquad u\to u_{\infty''},$$

with

$$ \kappa_{0}=1,\qquad\kappa_1=-q^{-}-v,\qquad \kappa_2=1-r+{q^{-}}^2+q^{-}v+q^{-}q^{--}+\frac{s}{s^{-}}.$$
(4.3)

The divisors [16] of the meromorphic function are

$$ (\Theta(u,n))=\mathcal D_{u_{\infty'},\tilde\mu_1^{+}(n),\ldots,\tilde\mu_{l-1}^{+}(n)}(u)- \mathcal D_{u_{\infty''},\tilde\mu_1^{+}(n),\ldots,\tilde\mu_{l-1}^{+}(n)}(u),$$
(4.4)

whence it follows that \(\Theta(u,n)\) has \(l\) zeros, \(u_{\infty'}\), \(\tilde\mu_1^{+}(n),\ldots,\tilde\mu_{l-1}^{+}(n)\), and \(l\) poles \(u_{\infty''}\), \(\tilde\mu_1(n),\ldots,\tilde\mu_{l-1}(n)\). Besides, according to (3.8), (4.2) and (4.3), we have

$$ \Xi_1(u,n,n_0)\underset{\varsigma\to 0}{=}\Upsilon(n,n_0)\varsigma^{n-n_0}(1+O(\varsigma)),\qquad u\to u_{\infty'},\quad\varsigma=\lambda^{-1},$$
(4.5)

where

$$\Upsilon(n,n_0)=\begin{cases} \displaystyle\prod_{n'=n_0}^{n-1}-v(n'),& n\ge n_0+1,\\ \quad 1, & n=n_0,\\ \displaystyle\prod_{n'=n_0}^{n_0-1}(-v(n'))^{-1},&n\le n_0-1, \end{cases}$$

and

$$ \Xi_1(u,n,n_0)\underset{\eta\to 0}{=}\eta^{n-n_0}(1+O(\eta)),\qquad u\to u_{\infty''},\quad \eta=\lambda^{1/2}.$$
(4.6)

The divisors of the Baker–Akhiezer function \(\Xi_1(u,n,n_0)\) are

$$ (\Xi_1(u,n,n_0))=\mathcal D_{\tilde\mu_1(n),\ldots,\tilde\mu_{l-1}(n)}- \mathcal D_{\tilde\mu_1(n_0),\ldots,\tilde\mu_{l-1}(n_0)}+ (n-n_0)(\mathcal D_{u_{\infty'}}-\mathcal D_{u_{\infty''}}).$$
(4.7)

The Riemann surface \(\mathcal K_{l-1}\) has a canonical basis of cycles \(\mathbf w_1,\ldots,\mathbf w_{l-1}\) and \(\mathbf o_1,\ldots,\mathbf o_{l-1}\) whose intersection numbers are

$$ \mathbf w_j\circ\mathbf o_\sigma=0,\quad \mathbf w_j\circ\mathbf w_\sigma=0,\quad \mathbf o_j\circ\mathbf o_\sigma=0,\qquad j,\sigma=1,\ldots,l-1.$$
(4.8)

On \(\mathcal K_{l-1}\), we define

$$ \tilde\omega_h(u)=\frac{1}{3f^2+Y_l}= \begin{cases} \lambda^{h-1}d\lambda, & 1\le h\le 2p-1,\\ f\lambda^{h-2p-2}, & 2p\le h\le l-1, \end{cases}$$
(4.9)

and set

$$\mathbb{O}_{ij}=\int_{\mathbf w_j}\tilde\omega_i,\qquad \mathbb{P}_{ij}=\int_{\mathbf o_j}\tilde\omega_i,$$

where the matrices \(\mathbb{O}\) and \(\mathbb{P}\) are invertible. Now, we introduce new matrices \(\mathbb{Q}\) and \(\tau\) such that \(\mathbb{Q}=\mathbb{O}^{-1}\) and \(\tau=\mathbb{O}^{-1}\mathbb{P}\). It is easy to see that \(\tau\) is symmetric \((\tau_{ij}=\tau_{ji})\) and its imaginary part is positive definite (\(\operatorname{Im}\tau>0\)).

Transforming \(\tilde\omega_h\) into the new basis \(\omega_j\),

$$ \omega_j=\sum_{h=1}^{l-1}\mathbb{Q}_{jh}\tilde\omega_h,\qquad j=1,\ldots,l-1,$$
(4.10)

we have

$$\begin{aligned} \, &\int_{\mathbf w_\sigma}\omega_j=\sum_{h=1}^{l-1}\mathbb{Q}_{jh}\int_{\mathbf w_i}\tilde\omega_h= \sum_{h=1}^{l-1}\mathbb{Q}_{jh}\mathbb{P}_{h\sigma}=\gamma_{j\sigma}, \\ &\int_{\mathbf o_\sigma}\omega_j=\sum_{h=1}^{l-1}\mathbb{Q}_{jh}\int_{\mathbf o_i}\tilde\omega_h= \sum_{h=1}^{l-1}\mathbb{Q}_{jh}\mathbb{P}_{h\sigma}=\tau_{j\sigma}. \end{aligned}$$

We define the third kind holomorphic differential on \(\mathcal K_{l-1}\backslash\{Q',Q''\}\) as \(\omega_{Q',Q''}^{(3)}\). It has poles at \(Q_k\) with the residues \((-1)^{k+1}\), \(k=1,2\). In particular,

$$\int_{{\mathbf w}_j}\omega_{Q',Q''}^{(3)}=0,\qquad \int_{\mathbf o_j}\omega_{Q',Q''}^{(3)}=2\pi i\int_{Q''}^{Q'}\omega_j,\qquad j=1,\ldots,l-1.$$

For \(\omega_{u_{\infty'},u_{\infty''}}^{(3)}\), we have

$$ \begin{alignedat}{5} &\omega_{u_{\infty'},u_{\infty''}}^{(3)}\underset{\varsigma\to 0}{=} (\varsigma^{-1}+\omega_{0}^{\infty'}\varsigma^0+O(\varsigma))\,d\varsigma,&\qquad & u\to u_{\infty'},&\quad &\varsigma=\lambda^{-1}, \\ & \omega_{u_{\infty'},u_{\infty''}}^{(3)}\underset{\eta\to 0}{=} (-\eta^{-1}+\omega_{0}^{\infty''}\eta^0+O(\eta))\,d\eta,&\qquad & u\to u_{\infty''},&\quad &\eta=\lambda^{-1/2}, \end{alignedat}$$
(4.11)

whence

$$ \begin{alignedat}{3} &\int_{Q_0}^{u}\omega_{u_{\infty'},u_{\infty''}}^{(3)}\underset{\varsigma\to 0}{=} \ln\varsigma+\ell_1(Q_0)+\omega_{0}^{\infty'}\varsigma+O(\varsigma^2), &\qquad & u\to u_{\infty'}, \\ &\int_{Q_0}^{u}\omega_{u_{\infty'},u_{\infty''}}^{(3)}\underset{\eta\to 0}{=} -\ln\eta+\ell_2(Q_0)+\omega_{0}^{\infty''}\eta+O(\eta^2), &\qquad & u\to u_{\infty''}, \end{alignedat}$$
(4.12)

where \(Q_0\) is a variable base point on \(\mathcal K_{l-1}\{u_{\infty'},u_{\infty''}\}\), and \(\ell_1(Q_0)\), \(\ell_2(Q_0)\), \(\omega_{0}^{\infty'}\), and \(\omega_{0}^{\infty''}\) are constants.

Let \(\mathcal T_{l-1}\) be the period lattice \(\{ \underline {z}\in\mathbb{C}^{l-1}| \underline {z}= \underline {\mathcal F}+ \underline {\mathcal H}\tau, \underline {\mathcal F}, \underline {\mathcal H}\in\mathbb{Z}^{l-1}\}\). On \(\mathcal K_{l-1}\), we regard \(\mathcal J_{l-1}=\mathbb{C}^{l-1}/\mathcal T_{l-1}\) as the Jacobian variety. We can then introduce the Abel map \( \underline {\mathcal A}\colon\mathcal K_{l-1}\to\mathcal J_{l-1}\),

$$\underline {\mathcal A}(u)=\bigl(\mathcal A_1(u),\ldots,\mathcal A_{l-1}(u)\bigr)= \biggl(\,\int_{Q_0}^{u}{\omega_1},\ldots,\int_{Q_0}^{u}{\omega_{l-1}}\biggr)\quad (\mathrm{mod}\;\;\mathcal T_{l-1}).$$

We define the divisors group \(\operatorname{Div}(\mathcal K_{l-1})\) and continue the above equation to it by linearity:

$$\underline {\mathcal A}\biggl(\sum h_\sigma u_\sigma\biggr)=\sum h_\sigma \underline {\mathcal A}(u_\sigma).$$

We consider the nonspecial divisor \(\mathcal D_{ \underline {\tilde\mu}(n)}=\sum_{\sigma=1}^{l-1}\tilde\mu_\sigma(n)\) and define

$$ \underline {\rho}(n)= \underline {\mathcal A} \biggl(\,\sum_{\sigma=1}^{l-1}\tilde\mu_\sigma(n)\biggr)= \sum_{\sigma=1}^{l-1} \underline {\mathcal A}(\tilde\mu_\sigma(n))= \sum_{\sigma=1}^{l-1}\int_{Q_0}^{\tilde\mu_\sigma(n)} \underline {\omega}\,,$$
(4.13)

where \( \underline {\rho}=(\rho_1(n),\ldots,\rho_{l-1}(n))\) and \( \underline {\omega}=(\omega_1,\ldots,\omega_{l-1})\). We define the Riemann theta function \(\theta( \underline {z})\) on \(\mathcal K_{l-1}\) as

$$ \begin{aligned} \, &\theta( \underline {z})=\sum_{ \underline {\mathcal F}\in\mathbb{Z}^{l-1}} \exp\{2\pi i\langle\kern1pt \underline {\mathcal F}, \underline {z}\kern1pt\rangle+ \pi i\langle\kern1pt \underline {\mathcal F}, \underline {\mathcal F\tau}\rangle\},\qquad \underline {z}=(z_1,\ldots,z_{l-1})\in\mathbb{C}^{l-1}, \\ &\langle\kern1pt \underline {\mathcal F}, \underline {z}\kern1pt\rangle=\sum_{j=1}^{l-1}\mathcal F_jz_j,\qquad \langle\kern1pt \underline {\mathcal F}, \underline {\mathcal F\tau}\kern1pt\rangle= \sum_{j,\sigma=1}^{l-1}\tau_{j\sigma}\mathcal F_j\mathcal F_\sigma. \end{aligned}$$
(4.14)

We then introduce the function

$$ \begin{aligned} \, &\theta( \underline {z}(u,\tilde{ \underline {\mu}}(n)))=\theta( \underline {\Lambda}-\mathcal A(u)+ \underline {\rho}(n)), \\ &u\in\mathcal K_{l-1},\qquad \underline {\tilde\mu}(n)=\{\tilde\mu_1(n),\ldots,\tilde\mu_{l-1}(n)\}\in\sigma^{l-1}\mathcal K_{l-1}, \end{aligned}$$
(4.15)

where \(\sigma^{l-1}\mathcal K_{l-1}\) is the \((l-1)\)th symmetric power of \(\mathcal K_{l-1}\), and the expression of the vector \(\Lambda\) depending on the base point \(Q_0\) is

$$\Lambda_j=\frac{1}{2}(1+\tau_{jj})- \sum_{\substack{\sigma=1,\\ \!\!\sigma\neq j}}^{l-1}\int_{\mathbf w_\sigma}{\omega_\sigma}\int_{Q_0}^{u}\omega_j,\qquad j=1,\ldots,l-1.$$

Theorem 1.

Let \(u=(\lambda,f)\in\mathcal K_{l-1}\backslash\{u_{\infty'},u_{\infty''}\}\) , \((n,n_0)\in\mathbb{Z}^2\) , and \(\mathcal D_{ \underline {\tilde\mu}(n)}\) be a nonspecial divisor. Then

$$ \begin{aligned} \, &\Theta(u,n)=\frac{\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}(n)))\theta( \underline {z}(u, \underline {\tilde\mu}^{+}(n)))} {\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}^{+}(n)))\theta( \underline {z}(u, \underline {\tilde\mu}(n)))} \exp\biggl(\,\int_{Q_0}^{u}{\omega_{u_{\infty'},u_{\infty''}}^{(3)}-\ell_2(Q_0)}\biggr), \\ &\Xi_1(u,n,n_0)=\frac{\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}(n_0)))\theta( \underline {z}(u, \underline {\tilde\mu}(n)))} {\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}(n)))\theta( \underline {z}(u, \underline {\tilde\mu}(n_0)))}\times{} \\ &\kern90pt \times\exp\biggl((n-n_0)\biggl(\,\int_{Q_0}^{u}{\omega_{u_{\infty',\infty''}}^{(3)}-\ell_2(Q_0)}\biggr)\!\biggr). \end{aligned}$$
(4.16)

The divisor \(\mathcal D_{ \underline {\tilde\mu}}\) can be linearized as follows under the Abel map:

$$ \underline {\rho}(n)= \underline {\rho}(n_0)+(n-n_0)\bigl( \underline {\mathcal A}(u_{\infty''})- \underline {\mathcal A}(u_{\infty'})\bigr).$$
(4.17)

Proof.

Using the Abel theorem, we can obtain (4.17) from (4.7) and deduce the equations by (4.10),

$$ \begin{alignedat}{3} &\exp\biggl(\,\int_{Q_0}^{u}\omega_{u_{\infty'},u_{\infty''}}^{(3)}-\ell_2(Q_0)\biggr)\underset{\varsigma\to 0}{=} \varsigma\exp(\ell_1(Q_0)-\ell_2(Q_0)+O(\varsigma^2)), &\quad & u\to u_{\infty'}, \\ &\exp\biggl(\,\int_{Q_0}^{u}\omega_{u_{\infty'},u_{\infty''}}^{(3)}-\ell_2(Q_0)\biggr)\underset{\eta\to 0}{=} \eta^{-1}+O(1),&\quad & u\to u_{\infty''}. \end{alignedat}$$
(4.18)

Letting \(\phi\) denote the right-hand side of (4.16), we find that \(\phi\) and \(\Theta\) have the same zeros and poles. According to the Riemann–Roch theorem and Eqs. (4.3) and (4.18), we have

$$\frac{\phi}{\Theta}\underset{\eta\to 0}{=}\frac{(1+O(\eta))(\eta^{-1}+O(1))}{\eta^{-1}+O(1)}=1+O(\eta),\qquad u\to u_{\infty''}.$$

Hence, the Riemann theta representation of \(\Theta(u,n)\) can be proved and the representation of \(\Xi_1(u,n,n_0)\) can also be proved by (3.8) and the representation of \(\Theta\).

Theorem 2.

Let \(n\in\mathbb{Z}\) , \(\mathcal D_{ \underline {\tilde\mu}(n)}\) be a nonspecial divisor. Then the potentials \(q\) , \(r\) , \(s\) , and \(v\) can be expressed in terms of the Riemann theta function as

$$ \begin{aligned} \, &q(n)=-\,\omega_{0}^{\infty''}- \sum_{j=1}^{l-1}\biggl(\frac{1}{3\beta_0}\mathbb{Q}_{j,l-1}+\frac{1}{\alpha_0\beta_0}\mathbb{Q}_{j,2p-1}\biggr) \frac{\partial}{\partial_{zj}} \ln\frac{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{++}(n)}))}{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{+}(n)}))}-{} \\ &\hphantom{q(n)={}} -\frac{\theta( \underline {z}(u_{\infty'},\tilde{ \underline {\mu}}(n)))\theta( \underline {z}(u_{\infty'},{ \underline {\mu}}^{++}(n)))} {\theta( \underline {z}(u_{\infty''},\tilde{ \underline {\mu}}^{++}(n)))\theta( \underline {z}(u_{\infty'},\tilde{ \underline {\mu}}^{+}(n)))} \exp(\ell_1(Q_0)-\ell_2(Q_0)), \\ &r(n)=-\,\omega_{0}^{\infty'}+ \sum_{j=1}^{l-1}\biggl(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{\alpha^2}\mathbb{Q}_{j,2p-1}\biggr) \frac{\partial}{\partial_{zj}} \ln\frac{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{+}(n)}))} {\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}(n)}))}, \\ &\frac{s(n)}{s^{-}(n)}=\omega_{0}^{\infty'}- \sum_{j=1}^{l-1}\biggl(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{\alpha^2}\mathbb{Q}_{j,2p-1}\biggr) \frac{\partial}{\partial_{zj}} \ln\frac{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{+}(n)}))} {\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}(n)}))}, \\ &v(n)=-\frac{\theta( \underline {z}(u_{\infty''},{ \underline {\mu}}(n)))\theta( \underline {z}(u_{\infty'},{ \underline {\mu}}^{+}(n)))} {\theta( \underline {z}(u_{\infty'},\tilde{ \underline {\mu}}(n)))^2} \exp(\ell_1(Q_0)-\ell_2(Q_0)). \end{aligned}$$
(4.19)

Proof.

According to the Abel theorem and (4.6), we have

$$\underline {\rho}^{+}(n)+ \underline {\mathcal A}(u_{\infty'})= \underline {\rho}(n)+ \underline {\mathcal A}(u_{\infty''}),$$

so

$$\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}^{+}(n)))=\theta( \underline {z}(u_{\infty'}, \underline {\tilde\mu}(n))),$$

whence using \(f=\Gamma_{11}^{(p)}+\Gamma_{12}^{(p)}+\Gamma_{13}^{(p)}/\Theta^{-1}\) we have

$$f\underset{\stackrel{\scriptstyle\varsigma\to 0,}{\eta\to 0}}{=} \begin{cases} \varsigma^{-p-1}[\alpha_0-\alpha_1\varsigma^2+O(\varsigma^3)], & u\to u_{\infty'},\;\;\varsigma=\lambda^{-1}, \\ \eta^{-2p-1}[\beta_0-\beta_1\eta^2+O(\eta^3)], & u\to u_{\infty''},\;\;\eta=\lambda^{-1/2}. \end{cases}$$

Using (4.9) and (4.10), we deduce the equality

$$\omega_j=\sum_{h=1}^{l-1}\mathbb{Q}_{ij}\hat{\omega}_h= \sum_{h=1}^{2p-1}\mathbb{Q}_{ij}\frac{\lambda^{h-1}d\lambda}{3f^2+Y_l}+ \sum_{h=2p}^{l-1}\mathbb{Q}_{ij}\frac{f\lambda^{h-2p-2}d\lambda}{3f^2+Y_l},\qquad j=1,\ldots,l-1.$$

The expression of \(\omega_j\) can then be obtained by direct calculation:

$$\omega_j\underset{\stackrel{\scriptstyle\varsigma\to 0,}{\eta\to 0}}{=} \begin{cases} \phantom{-}\dfrac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\dfrac{1}{{\alpha_0}^2}\mathbb{Q}_{j,2p-1}+O(\varsigma)d\varsigma, & u\to u_{\infty'}, \\ -\dfrac{1}{3\beta_0}\mathbb{Q}_{j,l-1}-\dfrac{1}{{\alpha_0\beta_0}}\mathbb{Q}_{j,2p-1}+O(\eta)d\eta, & u\to u_{\infty''}. \end{cases}$$

With the Riemann theta representation of \(\Theta(u,n)\) in (4.16), we have

$$\begin{aligned} \, &\frac{\theta( \underline {z}(u, \underline {\tilde\mu}^{+}(n)))}{\theta( \underline {z}(u, \underline {\tilde\mu}(n)))}= \frac{\theta( \underline {\Lambda}- \underline {\mathcal A}(u)+ \underline {\rho}^{+}(n))} {\theta( \underline {\Lambda}- \underline {\mathcal A}(u)+ \underline {\rho}(n))}= \frac{\theta( \underline {\Lambda}- \underline {\mathcal A}(u_{\infty'})+ \underline {\rho}^{+}(n)+\int_{u}^{u_{\infty'}}{ \underline {\omega}})} {\theta( \underline {\Lambda}- \underline {\mathcal A}(u_{\infty'})+ \underline {\rho}(n)+\int_{u}^{u_{\infty'}}{ \underline {\omega}})} \underset{\varsigma\to 0}{=} \\ &\;\;\underset{\varsigma\to 0}{=} \frac{\theta(\ldots,\Lambda_j-\mathcal A_j(u_{\infty'})+\rho_{j}^{+}(n)- \bigl(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{{\alpha_0}^2}\mathbb{Q}_{j,2p-1}\bigr)\varsigma+O(\varsigma^2),\ldots)} {\theta(\cdots,\Lambda_j-\mathcal A_j(u_{\infty'})+\rho_{j}(n)- \bigl(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{{\alpha_0}^2}\mathbb{Q}_{j,2p-1}\bigr)\varsigma+O(\varsigma^2),\ldots)} \underset{\varsigma\to 0}{=} \\ &\;\;\underset{\varsigma\to 0}{=} \frac{\theta( \underline {z}(u_{\infty'}, \underline {\tilde\mu}^{+}(n)))- \sum_{j=1}^{l-1}\bigl(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{{\alpha_0}^2}\mathbb{Q}_{j,2p-1}\bigr) \frac{\partial}{\partial z_j}\ln\theta( \underline {z}(u_{\infty'}, \underline {\tilde\mu}(n)))\varsigma+ O(\varsigma^2)} {\theta( \underline {z}(u_{\infty'}, \underline {\tilde\mu}(n)))- \sum_{j=1}^{l-1}\bigl(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{{\alpha_0}^2}\mathbb{Q}_{j,2p-1}\bigr) \frac{\partial}{\partial z_j}\ln\theta( \underline {z}(u_{\infty'}, \underline {\tilde\mu}^{+}(n)))\varsigma+ O(\varsigma^2)} \underset{\varsigma\to 0}{=} \\ &\;\;\underset{\varsigma\to 0}{=} \frac{\theta_{+}^{'}}{\theta'} \biggl(1-\sum_{j=1}^{l-1}(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}- \frac{1}{{\alpha_0}^2}\mathbb{Q}_{j,2p-1}) \frac{\partial}{\partial z_j}\ln\frac{\theta_{+}^{'}}{\theta'}\varsigma+O(\varsigma^2)\biggr), \end{aligned}$$

where \(u\to u_{\infty'}\) and \(\theta'=\theta( \underline {z}(u_{\infty'}, \underline {\tilde\mu}(n)))\), \(\theta_{+}^{'}=\theta( \underline {z}(u_{\infty'}, \underline {\tilde\mu}^{+}(n)))\).

Similarly to the previous steps, we obtain

$$\frac{\theta( \underline {z}(u, \underline {\tilde\mu}^{+}(n)))}{\theta( \underline {z}(u, \underline {\tilde\mu}(n)))}= \frac{\theta_{+}^{''}}{\theta''} \biggl(1-\sum_{j=1}^{l-1}(-\frac{1}{3\beta_0}\mathbb{Q}_{j,l-1}- \frac{1}{{\alpha_0\beta_0}}\mathbb{Q}_{j,2p-1})\frac{\partial}{\partial z_j}\ln\frac{\theta_{+}^{''}}{\theta''}\eta+O(\eta^2)\biggr),$$

where \(\eta\to 0\), \(\theta''=\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}(n)))\), and \(\theta_{+}^{''}=\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}^{+}(n)))\). Hence, we have the following formulas as \(\eta\to 0\) and \(\varsigma\to 0\):

$$\Theta(u,n)\underset{\stackrel{\scriptstyle\varsigma\to 0,}{\eta\to 0}}{=} \begin{cases} \Bigl(\varsigma+\Bigl(\omega_{0}^{\infty'}- \sum_{j=1}^{l-1}\Bigl(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{{\alpha_0}^2}\mathbb{Q}_{j,2p-1}\bigr) \frac{\partial}{\partial z_j}\ln\frac{\theta_{+}^{'}}{\theta'}\Bigr)\varsigma^2+O(\varsigma^3)\Bigl) \\ \quad\times\frac{\theta''\theta_{+}^{'}}{\theta_{+}^{''}\theta'}\exp(\ell_1(Q_0)-\ell_2(Q_0)), & \kern-40pt u\to u_{\infty'}, \\ \eta^{-1}+\omega_{0}^{\infty''}+ \sum_{j=1}^{l-1}(\frac{1}{3\beta_0}\mathbb{Q}_{j,l-1}+ \frac{1}{{\alpha_0\beta_0}}\mathbb{Q}_{j,2p-1})\frac{\partial}{\partial z_j} \ln\frac{\theta_{+}^{''}}{\theta''}+O(\eta), & \\ &\kern-40pt u\to u_{\infty''}. \end{cases}$$

In addition, in accordance with (4.2) and (4.3), we have

$$\Theta(u,n)\underset{\stackrel{\scriptstyle\varsigma\to 0,}{\eta\to 0}}{=}\ \begin{cases} -v\varsigma+\biggl(v^{+}-\dfrac{s}{s^{-}}v^{-}+rv\biggr)\varsigma^2+O(\varsigma^3), & u\to u_{\infty'}, \\ \eta^{-1}-q^{-}+v+O(\eta), & u\to u_{\infty''}. \end{cases}$$

Formulas (4.19) are thus proved.

We let the \(\mathbf o\)-periods of \(\omega_{u_{\infty'},u_{\infty''}}^{(3)}\) be denoted as

$$ \underline {\mathcal A}^{(3)}=(\mathcal A_{1}^{(3)},\ldots,\mathcal A_{l-1}^{(3)}),\qquad \mathcal A_\sigma^{(3)}=\frac{1}{2\pi i} \int_{\mathbf o_\sigma}\omega_{u_{\infty'},u_{\infty''}}^{(3)},\quad \sigma=1,\ldots,l-1.$$
(4.20)

Combining (4.11), (4.17), (4.19), and (4.20) shows that the Riemann theta representation for \(q(n)\), \(r(n)\), \(s(n)\), and \(v(n)\) has a remarkable linearity in \(n\in\mathbb{Z}\times\mathbb{R}\). As a matter of fact, Eqs. (4.19) can be rewritten as

$$\begin{aligned} \, &q(n)=-\,\omega_{0}^{\infty''}+ \sum_{j=1}^{l-1}\mathbb{K}_{j,0}^{(\infty'')} \frac{\partial}{\partial_{z_j}}\ln\frac{\theta( \underline {\mathcal B}_2+ \underline {\mathcal A}^{(3)}n)}{\theta( \underline {\mathcal B}_1+ \underline {A}^{(3)}n)}-\frac{\theta( \underline {\mathcal B}_0+ \underline {\mathcal A}^{(3)}n)\theta( \underline {\mathcal B}_2+ \underline {\mathcal A}^{(3)}n)} {\theta( \underline {\mathcal B}'_2+ \underline {\mathcal A}^{(3)}n)\theta( \underline {\mathcal B}_1+ \underline {\mathcal A}^{(3)}n)} \exp(\ell_1(Q_0)-\ell_2(Q_0)), \\ &r(n)=-\,\omega_{0}^{\infty'}+ \sum_{j=1}^{l-1}\mathbb{K}_{j,0}^{(\infty')} \frac{\partial}{\partial_{z_j}}\ln\frac{\theta( \underline {\mathcal B}_1+ \underline {\mathcal A}^{(3)}n)}{\theta( \underline {\mathcal B}_0+ \underline {\mathcal A}^{(3)}n)}, \\ &\frac{s(n)}{s^{-}(n)}=\omega_{0}^{\infty'}- \sum_{j=1}^{l-1}\mathbb{K}_{j,0}^{(\infty')} \frac{\partial}{\partial_{z_j}}\ln\frac{\theta( \underline {\mathcal B}_1+ \underline {\mathcal A}^{(3)}n)}{\theta( \underline {\mathcal B}_0+ \underline {\mathcal A}^{(3)}n)}, \\ &v(n)=-\,\frac{\theta( \underline {\mathcal B}'_{0}+ \underline {\mathcal A}^{(3)}n)\theta( \underline {\mathcal B}_1+ \underline {\mathcal A}^{(3)}n)} {\theta( \underline {\mathcal B}_0+ \underline {\mathcal A}^{(3)}n)^2}\exp(\ell_1(Q_0)-\ell_2(Q_0)). \end{aligned}$$

where

$$\begin{aligned} \, & \underline {\mathcal B}_0= \underline {\mathcal{S}}- \underline {\mathcal A}^{(3)},\qquad \underline {\mathcal B}'_0= \underline {\mathcal{S}}'- \underline {\mathcal A}^{(3)},\qquad \underline {\mathcal B}_1= \underline {\mathcal{S}}+ \underline {\mathcal A}^{(3)}, \\ & \underline {\mathcal B}'_1= \underline {\mathcal{S}}'+ \underline {\mathcal A}^{(3)},\qquad \underline {\mathcal B}_2= \underline {\mathcal{S}}+2 \underline {\mathcal A}^{(3)},\qquad \underline {\mathcal B}'_2= \underline {\mathcal{S}}'+2 \underline {\mathcal A}^{(3)}, \\ &\mathbb{K}_{j,0}^{(\infty')}=\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{\alpha^2}\mathbb{Q}_{j,2p-1},\qquad \mathbb{K}_{j,0}^{(\infty'')}=-\frac{1}{3\beta_0}\mathbb{Q}_{j,l-1}-\frac{1}{\alpha_0\beta_0}\mathbb{Q}_{j,2p-1}, \\ & \underline {\mathcal{S}}= \underline {\Lambda}- \underline {\mathbb{O}}(u_{\infty'})+ \underline {\rho}(n_0)- \underline {\mathcal A}^{(3)}n_0,\qquad \underline {\mathcal{S}}'= \underline {\Lambda}- \underline {\mathbb{O}}(u_{\infty''})+ \underline {\rho}(n_0)- \underline {\mathcal A}^{(3)}n_0. \end{aligned}$$

5. Algebro-geometric solutions of the hierarchy in the time-dependent case

In this section, we discuss the algebro-geometric solutions of (1.3) in the time-dependent case. We first define the time-dependent Baker–Akhiezer function

$$ \begin{aligned} \, &E\Xi(u,n,n_0,t_m,t_{0m})= U(q(n,t_m),r(n,t_m),s(n,t_m),v(n,t_m);\lambda(u))\Xi(u,n,n_0,t_m,t_{0m}), \\ &\Xi_{tm}(u,n,n_0,t_m,t_{0m})= \widehat\Gamma^mU(q(n,t_m),r(n,t_m),s(n,t_m),v(n,t_m);\lambda(u))\Xi(u,n,n_0,t_m,t_{0m}), \\ &\Gamma^{(p)}(q(n,t_m),r(n,t_m),s(n,t_m),v(n,t_m);\lambda(u))\Xi(u,n,n_0,t_m,t_{0m}) =f(u)\Xi(u,n,n_0,t_m,t_{0m}), \\ &\Xi_1(u,n_0,n_0,t_{0m},t_{0m})=1, \\ &u\in\mathcal K_{l-1}\backslash\{u_{\infty'},u_{\infty''}\},\quad (n,t_m),(n_0,t_{0m})\in{\mathbb{Z}\times\mathbb{R}}. \end{aligned}$$
(5.1)

From the compatibility condition for Eqs. (5.1), we have

$$ U_{t_m}-(E\widehat\Gamma^m)U+U\widehat\Gamma^m=0,\quad\;\; (E\Gamma^{(p)})U-U\Gamma^{(p)}=0,\quad\;\; \Gamma_{t_m}^{(p)}-[\widehat\Gamma^m,\Gamma^{(p)}]=0.$$
(5.2)

Direct calculation shows that \(\digamma_l(\lambda,f)=\det(f I-\Gamma^{(p)})\) satisfies the stationary zero-curvature equation. The Lax pair \(\Gamma^{(p)}\) characteristic polynomial is a constant independent of \(n\) and \(t_m\), and we have

$$\det(fI-\Gamma^{(p)})=f^3-f^2X_l(\lambda)+fY_l(\lambda)-Z_l(\lambda).$$

Then the trigonal curve \(\mathcal K_{l-1}\) is naturally defined in the time-dependent case as

$$\mathcal K_{l-1}\colon\,f_l(\lambda,f)=f^3-f^2X_l(\lambda)+fY_l(\lambda)-Z_l(\lambda).$$

The meromorphic function \(\Theta (u,n,t_m)\) on \(\mathcal K_{l-1}\) is defined as

$$ \Theta(u,n,t_m)=\frac{\Xi_2(u,n,n_0,t_m,t_{0m})}{\Xi_1(u,n,n_0,t_m,t_{0m})},\qquad u\in\mathcal K_{l-1},\quad (n,t_m)\in{\mathbb{Z}\times\mathbb{R}},$$
(5.3)

whence we have

$$\Xi_1(u,n,n_0,t_0,t_{0m})=\begin{cases} \displaystyle\prod_{n'=n_0}^{n-1}(1+q(n,t_m)\Theta(u,n',t_m)),& n\ge n_0+1,\\ \quad 1, & n=n_0, \\ \displaystyle\prod_{n'=n_0}^{n-1}(1+q(n,t_m)\Theta(u,n',t_m))^{-1}, & n\le n_0-1. \end{cases}$$

From (5.3), we have

$$\begin{aligned} \, \Theta(u,n,t_m)&=\frac{f\Gamma_{23}^m(\lambda,n,t_m)+A_l(\lambda,n,t_m)}{f\Gamma_{13}^m(\lambda,n,t_m)+B_l(\lambda,n,t_m)}= \nonumber\\ &=\frac{E_{l-1}(\lambda,n,t_m)}{f^2\Gamma_{23}^m(\lambda,n,t_m)-fA_l(\lambda,n,t_m)+C_l(\lambda,n,t_m)}= \nonumber\\ &=\frac{f^2\Gamma_{13}^m(\lambda,n,t_m)-fB_l(\lambda,n,t_m)+D_l(\lambda,n,t_m)}{F_{l-1}(\lambda,n,t_m)}, \end{aligned}$$
(5.4)

where \(u=(\lambda,f)\) and the elements such as \(A_l(\lambda,n,t_m)\) are defined the same as in the stationary case. Similarly,

$$\begin{aligned} \, &F_{l-1}(\lambda,n,t_m)=F_{l-1,0}\prod_{j=1}^{l-1}(\lambda-\mu_j(n,t_m)), \\ &E_{l-1}(\lambda ,n,t_m)=-F_{l-1,0}\prod_{j=1}^{l-1}(\lambda-\mu_j^{+}(n,t_m)). \end{aligned}$$

We give the expressions for \(\{\tilde\mu_j(n,t_m)\}_{j=1,\ldots,l-1}\subset\mathcal K_{l-1}\) and \(\{\tilde\mu_j^{+}(n,t_m)\}_{j=1,\ldots,l-1}\subset\mathcal K_{l-1}\) in the form

$$ \begin{aligned} \, \tilde\mu_j(n,t_m)&=\bigl(\mu_j(n,t_m),f(\hat{\mu}_j(n,t_m))\bigr)= \\ &=\biggl(\mu_j(n,t_m)-\frac{B_l(\mu_j(n,t_m),n,t_m)}{\Gamma_{32}^m(\mu_j(n,t_m),n,t_m)}\biggr), \\ \tilde\mu_j^{+}(n,t_m)&=\bigl(\mu_j^{+}(n,t_m),f(\hat{\mu}_j^{+}(n,t_m))\bigr)= \\ &=\biggl(\mu_j^{+}(n,t_m)-\frac{A_l(\mu_j(n,t_m),n,t_m)}{\Gamma_{32}^m(\mu_j^{+}(n,t_m),n,t_m)}\biggr),\qquad(n,t_m)\in\mathbb{Z}\times\mathbb{R}. \end{aligned}$$
(5.5)

From (5.4), the divisor of \(\Theta(u,n,t_m)\) can be expressed as

$$ (\Theta(u,n,t_m))=\mathcal D_{u_{\infty'},\tilde\mu_1^{+}(n,t_m),\ldots,\tilde\mu_{l-1}^{+}(n,t_m)}(u)- \mathcal D_{u_{\infty''},\tilde\mu_1^{+}(n,t_m),\ldots,\tilde\mu_{l-1}^{+}(n,t_m)}(u),$$
(5.6)

and hence \(\Theta(u,n,t_m)\) still has \(l\) zeros, \(u_{\infty'},\tilde\mu_1^{+}(n,t_m),\ldots,\tilde\mu_{l-1}^{+}(n,t_m)\), and \(l\) poles \(u_{\infty''},\tilde\mu_1(n,t_m),\ldots, [0]\tilde\mu_{l-1}(n,t_m)\).

By the same calculation, it is clear that \(\Theta(u,n,t_m)\) satisfies the Riccati-type equation

$$\begin{aligned} \, q^{-}&(n,t_m)q(n,t_m)\Theta^{+}(u,n,t_m)\Theta(u,n,t_m)\Theta^{-}(u,n,t_m)= \nonumber\\ &=\biggl(v(n,t_m)q^{-}(n,t_m)-\frac{s(n,t_m)}{s^{-}(n,t_m)}\biggr)\Theta^{-}(u,n,t_m)+{} \nonumber\\ &\quad +(\lambda+r(n,t_m))\Theta(u,n,t_m)-\Theta^{+}(u,n,t_m)+v(n,t_m)+{} \nonumber\\ &\quad +(\lambda+r(n,t_m))q^{-}(n,t_m)\Theta^{-}(u,n,t_m)\Theta(u,n,t_m)-{} \nonumber\\ &\quad -q(n,t_m)\Theta(u,n,t_m)\Theta^{+}(u,n,t_m)-{}q^{-}(n,t_m)\Theta^{-}(u,n,t_m)\Theta^{+}(u,n,t_m). \end{aligned}$$
(5.7)

Also similarly to the preceding subsection, it can be shown that the function \(\Theta(u,n,t_m)\) satisfies the system of equations

$$ \begin{aligned} \, &\Theta(u,n,t_m)\Theta(u^*,n,t_m)\Theta(u^{**},n,t_m)=-\frac{E_{l-1}(\lambda,n,t_m)}{F_{l-1}(\lambda,n,t_m)}, \\ &\Theta(u,n,t_m)+\Theta(u^*,n,t_m)+\Theta(u^{**},n,t_m)= \frac{3D_l(\lambda,n,t_m)-2\Gamma_{32}^m(\lambda,n,t_m)Y_l(\lambda,n,t_m)}{F_{l-1}(\lambda,n,t_m)}, \\ &\frac{1}{\Theta(u,n,t_m)}+\frac{1}{\Theta(u^*,n,t_m)}+\frac{1}{\Theta(u^{**},n,t_m)}= \frac{3C_l(\lambda,n,t_m)-2\Gamma_{12}^m(\lambda,n,t_m)(\lambda,n,t_m)Y_l(\lambda,n,t_m)}{E_{l-1}(\lambda,n,t_m)}. \end{aligned}$$
(5.8)

Differentiating the meromorphic function with respect to \(t_m\), we have

$$\begin{aligned} \, \Theta_{t_m}=\biggl(\frac{\Xi_1^{+}}{\Xi_1}\biggr)_{t_m}&= \frac{\Xi_{1,t_m}^{+}\Xi_1-\Xi_1^{+}\Xi_{1,t_m}}{\Xi_1^2}= \frac{\Xi_1^{+}}{\Xi_1}\biggl(\frac{\Xi_{1,t_m}^{+}}{\Xi_1^{+}}-\frac{\Xi_{1,t_m}}{\Xi_1}\biggr)= \\ &=\Theta\Delta\frac{\Xi_{1,t_m}}{\Xi_1}= \Theta\Delta\biggl(\widehat\Gamma_{11}^m+\widehat\Gamma_{12}^m\Theta+\widehat\Gamma_{13}^m\frac{1}{\Theta^{-}}\biggr), \end{aligned}$$

whence

$$ \frac{\Theta(u,n,t_m)_{t_m}}{\Theta(u,n,t_m)}= \Delta\biggl(\widehat\Gamma_{11}^m(\lambda,n,t_m)+\widehat\Gamma_{12}^m(\lambda,n,t_m)\Theta+ \widehat\Gamma_{13}^m(\lambda,n,t_m)\frac{1}{\Theta^{-}(u,n,t_m)}\biggr),$$
(5.9)

where \(\Delta\) is the difference operator and \(\Delta=E-1\).

The dynamics of \(\mu_j(n,t_m)\) of \(F_{l-1}(\lambda,n,t_m)\) can be described by Dubrovin-type equations in accordance with the following lemma.

Lemma 1.

Let \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\) . The zeros \(\{\mu_j(n,t_m)\}_{j=\overline{1,l-1}}\) of \(F_{l-1}(\lambda,n,t_m)\) satisfy the equations

$$\begin{aligned} \, \mu_{j,t_m}(n,t_m)=[&\,\widehat\Gamma_{12}^m(\mu_j(n,t_m),n,t_m)\Gamma_{13}^{(p)}(\mu_j(n,t_m),n,t_m)-{} \nonumber\\ &-\widehat\Gamma_{13}^m(\mu_j(n,t_m),n,t_m)\Gamma_{12}^{(p)}(\mu_j(n,t_m),n,t_m)] \frac{3f^2(\hat{\mu}_j(n,t_m))+Y_l(\mu_j(n,t_m))}{F_{{l-1},0}\prod_{\substack{\sigma=1,\\ \sigma\neq j}}^{l-1}(\mu_j(n,t_m)-\mu_\sigma(n,t_m))}, \end{aligned}$$
(5.10)

where \(1\le j\le l-1\) .

Proof.

From (3.10), (3.11), and (5.2), we have

$$\begin{aligned} \, F_{{l-1},t_m}(\lambda,n,t_m)&= \bigl((\Gamma_{13}^{(p)})^2\Gamma_{32}^{(p)}+ \Gamma_{12}^{(p)}\Gamma_{13}^{(p)}(\Gamma_{22}^{(p)}- \Gamma_{33}^{(p)})-(\Gamma_{12}^{(p)})^2\Gamma_{23}^{(p)}\bigr)_{t_m}= \\ &=3\widehat\Gamma_{11}^mF_{l-1}+ 3(\widehat\Gamma_{12}^mA_l-\widehat\Gamma_{13}^mG_l)- 2(\widehat\Gamma_{12}^m\Gamma_{13}^{(p)}-\widehat\Gamma_{13}^m\Gamma_{12}^{(p)})Y_l= \\ &=3\widehat\Gamma_{11}^mF_{l-1}+3\widehat\Gamma_{12}^m(\Gamma_{23}^{(p)}G_l- \Gamma_{22}^{(p)}B_l)-3\widehat\Gamma_{13}^m(\Gamma_{32}^{(p)}B_l-\Gamma_{33}^{(p)}G_l)+{} \\ &\quad +(\widehat\Gamma_{12}^m\Gamma_{13}^{(p)}-\widehat\Gamma_{13}^m\Gamma_{12}^{(p)})Y_l. \end{aligned}$$

With (3.12) and (5.5), we then have

$$\frac{B_l}{\Gamma_{13}^{(p)}}\biggl|_{\lambda=\mu_j(n,t_m)}= \frac{G_l}{\Gamma_{12}^{(p)}}\biggl|_{\lambda=\mu_j(n,t_m)}=-f(\tilde\mu_j(n,t_m)),$$

whence

$$\begin{aligned} \, &\widehat\Gamma_{12}^m(\Gamma_{23}^{(p)}G_l-\Gamma_{22}^{(p)}B_l)|_{\lambda=\mu_j(n,t_m)}= f^2(\hat{\mu}_j(n,t_m))\widehat\Gamma_{12}^m\Gamma_{13}^{(p)}|_{\lambda=\mu_j(n,t_m)}, \\ &\widehat\Gamma_{13}^m(\Gamma_{32}^{(p)}B_l-\Gamma_{33}^{(p)}G_l)|_{\lambda=\mu_j(n,t_m)}= f^2(\hat{\mu}_j(n,t_m))\widehat\Gamma_{13}^m\Gamma_{12}^{(p)}|_{\lambda=\mu_j(n,t_m)}, \\ &(\widehat\Gamma_{12}^mB_l-\widehat\Gamma_{13}^mG_l)|_{\lambda=\mu_j(n,t_m)}= -f(\hat{\mu}_j(n,t_m))(\widehat\Gamma_{12}^m\Gamma_{13}^{(p)}-\widehat\Gamma_{13}^m\Gamma_{12}^{(p)})\big|_{\lambda=\mu_j(n,t_m)}. \end{aligned}$$

Therefore,

$$\begin{aligned} \, F_{{l-1},t_m}(\lambda,n,t_m)|_{\lambda=\mu_j(n,t_m)}&=-\mu_{j,t_m}(n,t_m)F_{{l-1},0} \prod_{\substack{\sigma=1,\\ \sigma\neq j\;}}^{l-1}(\mu_j(n,t_m)-\mu_\sigma(n,t_m))= \\ &=\bigl(3f^2(\hat{\mu}_j(n,t_m))+Y_l(\mu_j(n,t_m))\bigr) (\widehat\Gamma_{12}^m\Gamma_{13}^{(p)}-\widehat\Gamma_{13}^m\Gamma_{12}^{(p)})\big|_{\lambda=\mu_j(n,t_m)}, \end{aligned}$$

and Eq. (5.10) is thus proved.

Moreover, in accordance with (5.1), we have

$$\begin{aligned} \, \Xi_1(u,n,n_0,t_m,t_{0m})={}& \exp\biggl(\,\int_{t_{0m}}^{t_m} \biggl(\widehat\Gamma_{11}^m(\lambda,n,t_m)+\widehat\Gamma_{12}^m(\lambda,n,t_m)\Theta+ \widehat\Gamma_{13}^m(\lambda,n,t_m)\frac{1}{\Theta^{-}(u,n,t_m)}\biggr)dt'\biggr)\times{} \nonumber\\ &\times\begin{cases} \displaystyle\prod_{n'=n_0}^{n-1}(1+q(n,t_m)\Theta(u,n',t_m)),& n\ge n_0+1,\\ \quad 1,& n=n_0,\\ \displaystyle\prod_{n'=n}^{n_0-1}(1+q(n,t_m)\Theta(u,n',t_m))^{-1},& n\le n_0-1, \end{cases} \end{aligned}$$
(5.11)

and

$$ \Xi_1(u,n,n_0,t_m,t_{0m})=\Xi_1(u,n,n_0,t_m,t_m)\Xi_1(u,n_0,n_0,t_m,t_{0m}),$$
(5.12)

where \(u=(\lambda,f)\in\mathcal K_{l-1}\backslash\{u_{\infty'},u_{\infty''}\}\), and \((n,t_m),(n_0,t_{0m})\in\mathbb{Z}\times\mathbb{R}\). Using the function in (5.11), we define \(\Pi_m(u,n,t_m)\) as

$$\Pi_m(u,n,t_m)=\widehat\Gamma_{11}^m(\lambda,n,t_m)+ \widehat\Gamma_{12}^m(\lambda,n,t_m)\Theta(u,n,t_m)+\widehat\Gamma_{13}^m\frac{1}{\Theta^{-}(u,n,t_m)},$$

whence

$$\begin{aligned} \, \widetilde\Pi_m^{(k)}(u,n,t_m)&=\widetilde{\widehat\Gamma}_{11}^{{}_{{}_{\scriptstyle(m,k)}}}(\lambda,n,t_m)+ \widetilde{\widehat\Gamma}_{12}^{{}_{{}_{\scriptstyle(m,k)}}}(\lambda,n,t_m)\Theta(u,n,t_m)+ \widetilde{\widehat\Gamma}_{13}^{{}_{{}_{\scriptstyle(m,k)}}}\frac{1}{\Theta^{-}(u,n,t_m)}, \end{aligned}$$
(5.13)

where

$$\widetilde{\widehat\Gamma}_{1j}^{{}_{{}_{\scriptstyle(m,1)}}}=\tilde{\Gamma}_{1j}^m|_{\hat\alpha_0=1,\hat\beta_1=0},\qquad \widetilde{\widehat\Gamma}_{1j}^{{}_{{}_{\scriptstyle(m,2)}}}=\tilde{\Gamma}_{1j}^m|_{\hat\alpha_0=1,\hat\beta_1=0},$$

and \(\hat\alpha_1=\cdots=\hat\alpha_m=\hat\beta_1=\cdots=\hat\beta_m=0\). Hence,

$$ \Pi_m(u,n,t_m)=\sum_{h=0}^{m}\hat\alpha_{m-h}\widetilde\Pi_{h}^{(1)}(u,n,t_m)+ \sum_{h=0}^{m}\hat\beta_{m-h}\widetilde\Pi_{h}^{(2)}(u,n,t_m).$$
(5.14)

Lemma 2.

Let \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\) , and let \(\varsigma=\lambda^{-1}\) and \(\eta=\lambda^{1/2}\) be local coordinates near \(u_{\infty'}\) and \(u_{\infty''}\) . Then

$$ \begin{aligned} \, &\widetilde\Pi_m^{(1)}(u,n,t_m)\underset{\stackrel{\scriptstyle\varsigma\to 0,}{\eta\to 0}}{=} \begin{cases} \varsigma^{-(m+1)}+O(\varsigma),& u\to u_{\infty'},\\ -\eta^{-(m+1)}-O(\eta), & u\to u_{\infty''}, \end{cases} \\ &\widetilde\Pi_m^{(2)}(u,n,t_m)\underset{\stackrel{\scriptstyle\varsigma\to 0,}{\eta\to 0}}{=} \begin{cases} -\bar b_m(n,t_m)-\bar b_{m+1}^{+}(n,t_m)\bar d_{m+1}^{+}(n,t_m)+O(\varsigma), & u\to u_{\infty'}, \\ 2\dfrac{\bar b_{m+1}(n,t_m)}{\bar b_{m+1}^{++}(n,t_m)}+\dfrac{1}{3}\bar d_{m+1}^{+}(n,t_m)+O(\eta), & u\to u_{\infty''}. \end{cases} \end{aligned}$$
(5.15)

Proof.

We set \(\tilde\alpha^m=\hat\alpha^m|_{\hat\alpha_0=1,\hat\beta_0=0}\). From (5.1) and (5.13), we then have

$$\begin{aligned} \, \widetilde\Pi_m^{(1)}(u,n,t_m)={}& \widetilde{\widehat\Gamma}_{11}^{{}_{{}_{\scriptstyle(m,1)}}}(\lambda,n,t_m)+ \widetilde{\widehat\Gamma}_{12}^{{}_{{}_{\scriptstyle(m,1)}}}(\lambda,n,t_m)\Theta(u,n,t_m)+ \frac{\widetilde{\widehat\Gamma}_{13}^{{}_{{}_{\scriptstyle(m,1)}}}(\lambda,n,t_m)}{\Theta^{-}(u,n,t_m)}= \\ ={}&\tilde a^m+\biggl(\Theta(u,n,t_m)+s(n,t_m)E\frac{1}{\Theta^{-}(u,n,t_m)}\biggr)\tilde b^m+ \\ &+q(n,t_m){s(n,t_m)}^2E\tilde d^m\frac{1}{\Theta^{-}(u,n,t_m)}. \end{aligned}$$

Using (4.2) and (4.3), we have the following result as \(m=0\):

$$\widetilde\Pi_{0}^{(1)}(u,n,t_m)\underset{\stackrel{\scriptstyle\varsigma\to 0,}{\eta\to 0}}{=} \begin{cases} \varsigma^{-1}+O(\varsigma), & u\to u_{\infty'},\\ -\eta^{-1}-O(\eta), &u\to u_{\infty''}, \end{cases}$$

We now suppose that

$$\widetilde\Pi_m^{(1)}(u,n,t_m)\underset{\stackrel{\scriptstyle\varsigma\to 0,}{\eta\to 0}}{=} \begin{cases} \varsigma^{-(m+1)}+\sum_{j=0}^{\infty}{\delta_j}(n,t_m)\varsigma^j, & u\to u_{\infty'},\\ -\eta^{-(m+1)}-\sum_{j=0}^{\infty}{\kappa_j}(n,t_m)\eta^j, & u\to u_{\infty''}, \end{cases}$$

where the coefficients of \(\{\delta_j(n,t_m)\}_{j\in\mathbb{Q}_0}\) and \(\{\kappa_j(n,t_m)\}_{j\in\mathbb{Q}_0}\) can be determined. In accordance with (5.9) and (5.13), we obtain

$$\Theta(u,n,t_m)_{t_m}=\Theta(u,n,t_m)\Delta\widetilde\Pi_m^{(1)}(u,n,t_m).$$

Comparing the coefficients of \(\varsigma\) and \(\eta\), we have

$$\begin{aligned} \, &\delta_{j,t_m}=\delta_1\Delta\varrho_{j-1}+\delta_2\Delta\varrho_{j-2}+\cdots+\delta_j\Delta\varrho_{0}, \\ &\kappa_{j,t_m}=\kappa_0\Delta\chi_{j}+\kappa_2\Delta\chi_{j-1}+\cdots+\kappa_{j-1}\Delta\chi_{1},\qquad j\ge 0, \end{aligned}$$

whence

$$\begin{aligned} \, &\Delta\chi_0=0,\qquad \Delta\chi_1=\kappa_{1,t_m}=\Delta\bar b_{m+1}, \\ &\Delta\varrho_0=\frac{\Theta_{1,t_m}}{\Theta_1}=\frac{v_{t_m}}{v}=2\tilde a_{m+1}, \\ &\Delta\varrho_1=\frac{1}{\Theta_1}\Theta_{2,t_m}-\frac{\Theta_2}{\Theta_1}\Delta\varrho_0=\Delta(\tilde c_{m+1}+{(E-1)}^{-1}\tilde a_{m+1}). \end{aligned}$$

It then follows that

$$\begin{aligned} \, &\chi_0(n,t_m)=0,\qquad \chi_1(n,t_m)=\bar b_{m+1}(n,t_m), \\ &\varrho_0(n,t_m)=2(E-1)^{-1}\tilde a_{m+1}(n,t_m), \\ &\varrho_1(n,t_m)=-\tilde c_{m+1}(n,t_m)+{(E-1)}^{-1}\tilde a_{m+1}(n,t_m). \end{aligned}$$

Therefore, in view of \(\Delta\Delta^{-1}=\Delta^{-1}\Delta=1\), the following results can be deduced:

$$\begin{aligned} \, \widetilde\Pi_{m+1}^{(1)}(u,n,t_m)&\underset{\varsigma\to 0}{=} \widetilde\Pi_m^{(1)}(u,n,t_m)\varsigma^{-2}+\biggl(\Theta(u,n,t_m)+s_nE\frac{1}{\Theta^{-}(u,n,t_m)}\biggr)\tilde b_{m+1}+{} \\ &\qquad +\tilde a_{m+1}(n,t_m)+q(n,t_m){s(n,t_m)}^2E\tilde d_{m+1}\frac{1}{\Theta^{-}(u,n,t_m)}= \\ &\;\,=\varsigma^{-(m+1)}+\varsigma^{-1}\bigl(\varrho_0-2\tilde a_{m+1}(n,t_m)\bigr)+\varrho_1+\tilde c_{m+1}(n,t_m)-{} \\ &\qquad-(E-1)^{-1}\tilde a_{m+1}(n,t_m)= \\ &\;\,=\varsigma^{-(m+1)}+O(\varsigma),\qquad u\to u_{\infty'}, \\ \widetilde\Pi_{m+1}^{(1)}(u,n,t_m)&\underset{\eta\to 0}{=} \widetilde\Pi_m^{(1)}(u,n,t_m)\eta^{-2}+\biggl(\Theta(u,n,t_m)+s(n,t_m)E\frac{1}{\Theta^{-}(u,n,t_m)}\biggr)\tilde b_{m+1}+{} \\ &\qquad+\tilde a_{m+1}(n,t_m)+q(n,t_m){s(n,t_m)}^2E\tilde d_{m+1}\frac{1}{\Theta^{-}(u,n,t_m)}= \\ &\;\,=-\eta^{-(m+1)}-\eta^{-1}\chi_0-\chi_1+\bar b_{m+1}(n,t_m)= \\ &\;\,=-\eta^{-(m+1)}-O(\eta),\qquad u\to u_{\infty''}. \end{aligned}$$

We have proved (5.15) for \(\widetilde\Pi_{m+1}^{(k)}\) for \(k=1\); the proof for \(k=2\) is similar.

Let \(\omega_{u_{\infty k},j}^{(2)}\), \(j\ge 2\), be the normalized differential of the second kind that is holomorphic on \(\mathcal K_{l-1}\backslash\{u_{\infty k}\}\) and has a \(j\)th-order pole at \(u_{\infty k}\) (\(k=1,2\)),

$$\begin{alignedat}{5} &\omega_{u_\infty',j}^{(2)}\underset{\varsigma\to 0}{=}(\varsigma^{-j}+O(1))\,d\varsigma, &\qquad &u\to u_{\infty'},&\quad &\varsigma=\lambda^{-1}, \\ &\omega_{u_\infty'',j}^{(2)}\underset{\eta\to 0}{=}(\eta^{-j}+O(1))\,d\eta, &\qquad &u\to u_{\infty''},&\quad &\eta=\lambda^{-1/2}, \end{alignedat}$$

and has the \(\mathbf w\)-periods \(\int_{\mathbf w_\sigma}\omega_{u_{\infty k},j}^{(2)}=0\), \(\sigma=1,\ldots,l-1\). Let

$$ \widehat{\mho}_m^{(2)}=-\sum_{h=0}^{m}\hat\alpha_{m-h}(h+1){\omega}_{u_{\infty'},h+2}^{(2)}+ \sum_{h=0}^{m}\hat\beta_{m-h}(2h+1)\hat{\omega}_{u_{\infty''},2h+2}^{(2)}.$$
(5.16)

Integrating (5.16), we obtain

$$\begin{alignedat}{3} &\int_{Q_0}^{u}\widehat{\mho}_m^{(2)}\underset{\varsigma\to 0}{=} \sum_{h=0}^{m}\hat\alpha_{m-h}\varsigma^{-h-1}+\hat{\ell}_{1}^{(2)}(Q_0)+O(\varsigma),&\qquad &u\to u_{\infty'}, \\ &\int_{Q_0}^{u}\widehat{\mho}_m^{(2)}\underset{\eta\to 0}{=} -\sum_{h=0}^{m}\hat\beta_{m-h}\eta^{-2h-1}+\hat{\ell}_2^{(2)}(Q_0)+O(\eta),&\qquad &u\to u_{\infty''}. \end{alignedat}$$

We next find the explicit Riemann theta function representations for the functions \(\Theta(u,n,t_m)\) and \(\Xi_1(u,n,n_0,t_m,t_{0m})\).

Theorem 3.

Let \(u=(\lambda,f)\in\mathcal K_{l-1}\backslash\{u_{\infty'},u_{\infty''}\}\) , \((n,n_0,t_m,t_{0m})\in\mathbb{Z}^2\times\mathbb{R}^2\) . If \(\mathcal D_{ \underline {\tilde\mu}(n,t_m)}\) is nonspecial and \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\) , then \(\Theta(u,n,t_m)\) and \(\Xi_1(u,n,n_0,t_m,t_{0m})\) can be represented as

$$ \Theta(u,n,t_m)= \frac{\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}(n)))\theta( \underline {z}(u, \underline {\tilde\mu}^{+}(n)))} {\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}^{+}(n)))\theta( \underline {z}(u, \underline {\tilde\mu}(n)))} \exp\biggl(\,\int_{Q_0}^{u}\omega_{u_{\infty'},u_{\infty''}}^{(3)}-\ell_2(Q_0)\biggr),$$
(5.17)

and

$$\begin{aligned} \, \Xi_1(u,n,n_0,t_m,t_{0m})={}& \frac{\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}(n_0,t_{0m})))\theta( \underline {z}(u, \underline {\tilde\mu}^{+}(n,t_m)))} {\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}^{+}(n,t_m)))\theta( \underline {z}(u, \underline {\tilde\mu}(n_0,t_{0m})))} \exp\biggl((n-n_0)\biggl(\,\int_{Q_0}^{u}{\omega_{u_{\infty',\infty''}}^{(3)}-\ell_2(Q_0)}\biggr)\!\biggr)+{} \nonumber\\ &+(t_m-t_{0m})(\hat{\ell}_2^{(2)}(Q_0))-\int_{Q_0}^{u}\widehat{\mho}_m^{(2)}. \end{aligned}$$
(5.18)

Proof.

For \(t_{0m}=t_m\), \(\Xi_1(u,n,n_0,t_m,t_m)\) has the form

$$\begin{aligned} \, \Xi_1(u,n,n_0,t_m,t_m)&= \frac{\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}(n_0,t_{0m})))\theta( \underline {z}(u, \underline {\tilde\mu}^{+}(n,t_m)))} {\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}^{+}(n,t_m)))\theta( \underline {z}(u, \underline {\tilde\mu}(n_0,t_{0m})))} \exp\biggl((n-n_0)\biggl(\,\int_{Q_0}^{u}{\omega_{u_{\infty',\infty''}}^{(3)}-\ell_2(Q_0)}\biggr)\!\biggr). \end{aligned}$$

We also need to verify that

$$\Xi_1(u,n_0,n_0,t_m,t_{0m})=\exp\biggl(\,\int_{t_{0m}}^{t_m}\Pi_m(u,n_0,t')dt'\biggr).$$

We let \(\mathcal W_1(u,n_0,n_0,t_m,t_m)\) denote the right-hand side of (5.18). Then

$$\begin{aligned} \, \mathcal W_1(u,n_0,n_0,t_m,t_m)={}& \frac{\theta( \underline {z}(u_\infty'', \underline {\tilde\mu}(n_0,t_{0m})))\theta( \underline {z}(u, \underline {\tilde\mu}^{+}(n_0,t_m)))} {\theta( \underline {z}(u_\infty'', \underline {\tilde\mu}^{+}(n_0,t_m)))\theta( \underline {z}(u, \underline {\tilde\mu}(n_0,t_{0m})))} \exp\biggl((t_m-t_{0m})\biggl(\hat{\ell}_2^{(2)}(Q_0)-\int_{Q_0}^{u}{\hat{\omega}_m^{(2)}}\biggr)\!\biggr). \end{aligned}$$

Next, we prove that

$$\Xi_1(u,n_0,n_0,t_m,t_{0m})=\mathcal W_1(u,n_0,n_0,t_m,t_{0m}).$$

First, we use (3.12), (5.4), and (5.13) to obtain the formula

$$\begin{aligned} \, \Pi_m(u,n,t_m)&= \widehat\Gamma_{11}^m(\lambda,n,t_m)+ \widehat\Gamma_{12}^m(\lambda,n,t_m)\Theta(u,n,t_m)+ \widehat\Gamma_{13}^m\frac{1}{\Theta^{-}(u,n,t_m)}= \\ &=\widehat\Gamma_{11}^m+\widehat\Gamma_{12}^m\frac{f^2\Gamma_{12}^{(p)}-fB_l+A_l}{F_{l-1}}- \widehat\Gamma_{13}^m\frac{f^2\Gamma_{12}^{(p)}-fG_l+H_l}{F_{l-1}}= \\ &=\frac{1}{F_{l-1}} \biggl(\frac{1}{3}F_{l-1,t_m}+ (\widehat\Gamma_{12}^m\Gamma_{13}^{(p)}-\widehat\Gamma_{13}^m\Gamma_{12}^{(p)})\biggl(f^2+\frac{2}{3}Y_l\biggr)- (\widehat\Gamma_{12}^mB_l-\widehat\Gamma_{12}^mG_l)f\biggr)= \\ &=-\frac{\mu_{j,t_m}(n,t_m)}{\lambda-\mu_j(n,t_m)}+O(1)= \partial_{t_m}\ln(\lambda-\mu_j(n,t_m))+O(1),\quad \lambda\to\mu_j(n,t_m), \end{aligned}$$

where \(O(1)\neq 0\). Consequently,

$$\begin{aligned} \, \Xi_1(u,n_0,n_0,t_m,t_{0m})&= \exp\biggl(\,\int_{t_{0m}}^{t_m}\partial_{t'}\ln(\lambda-\mu_j(n_0,t')\,dt')\biggr)= \frac{\lambda-\mu_j(n_0,t_m)}{\lambda-\mu_j(n_0,t_{0m})}\,O(1)= \\ &=\begin{cases} (\lambda-\mu_j(n_0,t_m))O(1), & u\to \tilde\mu_j(n_0,t_m)\neq\tilde\mu_j(n_0,t_{0m}),\\ O(1),& u\to \tilde\mu_j(n_0,t_m)=\tilde\mu_j(n_0,t_{0m}),\\ (\lambda-\mu_j(n_0,t_{0m}))^{-1}O(1), & u\to \tilde\mu_j(n_0,t_{0m})\neq\tilde\mu_j(n_0,t_m). \end{cases} \end{aligned}$$

Hence, \(\Xi_1(u,n_0,n_0,t_m,t_{0m})\) and \(\mathcal W_1(u,n_0,n_0,t_m,t_{0m})\) have the same poles and zeros on \(\mathcal K_{l-1}\). In addition, we can find that \(\mathcal K_{l-1}\), \(\Xi_1(u,n_0,n_0,t_m,t_{0m})\) and \(\mathcal W_1(u,n_0,n_0,t_m,t_{0m})\) have the identical essential singularities. Because of \(\mathcal D_{\tilde{ \underline {\mu}}(n,t_m)}\) is nonspecial, Eqs. (5.17) and (5.18) have been proved.

We let the \(\mathbf o\)-periods of \(\widehat{\mho}_m^{(2)}\) be denoted as

$$ \underline {\widehat{\mathcal A}}_m^{(2)}=(\widehat{\mathcal A}_{m,1}^{(2)},\ldots,\widehat{\mathcal A}_{m,l-1}^{(2)}),\qquad \widehat{\mathcal A}_{m,\sigma}^{(2)}=\frac{1}{2\pi i}\int_{\mathbf o_j}\widehat{\mho}_m^{(2)},\quad \sigma=1,\ldots,l-1.$$
(5.19)

Theorem 4 (straightening out of the flows).

The following equality holds :

$$ \underline {\rho}(n,t_m)= \underline {\rho}(n_0,t_{0m})+ \underline {\mathcal A}^{(3)}(n-n_0)+ \underline {\widehat{\mathcal A}}_m^{(2)}(t_m-t_{0m}).$$
(5.20)

Proof.

Introducing the meromorphic differential

$$\mho(n,n_0,t_m,t_{0m})=\frac{\partial}{\partial\lambda}\ln(\Xi_1(u,n,n_0,t_m,t_{0m}))\,d\lambda,$$

we use (5.18) to obtain

$$\mho(n,n_0,t_m,t_{0m})=(n-n_0)\omega_{u_{\infty'},u_{\infty''}}^{(3)}-(t_m-t_{0m})\widehat{\mho}_m^{(2)}+ \sum_{j=1}^{l-1}\omega_{\tilde\mu_j(n,t_m),\mu_j(n_0,t_{0m})}^{(3)}+\sum_{j=1}^{l-1}\check{\ell}_j\omega_j,$$

where \(\check{\ell}\in\mathbb{C}\), \(j=1,\ldots,l-1\). On \(\mathcal K_{l-1}\), any of the \(\mathbf w\)-periods and \(\mathbf o\)-periods is an integer multiple of \(2\pi i\) because \(\Xi_1(u,n,n_0,t_m,t_{0m})\) is single-valued, and hence

$$2\pi i\mathcal B_\sigma=\int_{\mathbf w_\sigma}\mho(n,n_0,t_m,t_{0m})= \int_{\mathbf w_\sigma}{\sum_{j=1}^{l-1}\check{\ell}_j\omega_j=\check{\ell}_\sigma},\qquad\sigma=1,\ldots,l-1,$$

where \(\mathcal B_\sigma\in\mathbb{Z}\). Similarly, for \(\mathcal{C}_\sigma\in\mathbb{Z}\) (\(\sigma=1,\ldots,l-1\)), we have

$$\begin{aligned} \, 2\pi i\mathcal{C}_\sigma&=\int_{\mathbf o_\sigma}{\mho(n,n_0,t_m,t_{0m})}= \\[1mm] &=(n-n_0)\int_{\mathbf o_\sigma}{\omega_{u_{\infty'},u_{\infty''}}^{(3)}-(t_m-t_{0m})}\int_{\mathbf o_\sigma}{\widehat{\mho}_m^{(2)}}+{} \\ &\quad +\sum_{j=1}^{l-1}\int_{\mathbf o_\sigma}{\omega_{\tilde\mu_j(n,t_m),\tilde\mu_j(n_0,t_{0m})}}+ \int_{\mathbf o_\sigma}{\sum_{j=1}^{l-1}\check{\ell}_j\omega_j}= \\[1mm] &=2\pi i(n-n_0)\mathcal A_{\sigma}^{(3)}-2\pi i(t_m-t_{0m})\int_{\mathbf o_\sigma}\widehat{\mho}_m^{(2)}+{} \\ &\quad +2\pi i\sum_{j=1}^{l-1}\int_{\tilde\mu_j(n_0,t_{0m})}^{\tilde\mu_j(n,t_m)}{\omega_\sigma}+ 2\pi i\sum_{j=1}^{l-1}\mathcal B_j\int_{\mathbf o_\sigma}\omega_j= \\[1mm] &=2\pi i(n-n_0)\mathcal A_{\sigma}^{(3)}-2\pi i(t_m-t_{0m})\widehat{\mathcal A}_{m,\sigma}^{(2)}+{} \\ &\quad +2\pi i\biggl(\,\sum_{j=1}^{l-1}\int_{Q_0}^{\tilde\mu_j(n,t_m)}\omega_\sigma- \sum_{j=1}^{l-1}\int_{Q_0}^{\tilde\mu_j(n_0,t_{0m})}\omega_\sigma\biggr)+ 2\pi i\sum_{j=1}^{l-1}\mathcal B_j\tau_{j,\sigma}, \end{aligned}$$

whence

$$ \underline {\mathcal{C}}=(n-n_0) \underline {\mathcal A}^{(3)}-(t_m-t_{0m}) \underline {\widehat{\mathcal A}}_m^{(2)}+ \sum_{j=1}^{l-1}\int_{Q_0}^{\tilde\mu_j(n,t_m)}\!\! \underline {\omega}- \sum_{j=1}^{l-1}\int_{Q_0}^{\tilde\mu_j(n_0,t_{0m})}\!\! \underline {\omega}+ \underline {\mathcal B}\tau,$$
(5.21)

where \( \underline {\mathcal{C}}=(\mathcal{C}_1,\ldots,\mathcal{C}_{l-1})\in\mathbb{Z}^{l-1}\) and \( \underline {\mathcal B}=(\mathcal B_1,\ldots,\mathcal B_{l-1})\in\mathbb{Z}^{l-1}\). Therefore, we have proved (5.20) because (5.21) is equivalent to (5.20).

From Theorem 4, we have

$$ \begin{aligned} \, &\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}(n,t_m)}))=\theta( \underline {\widehat{\mathcal B}}_0+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m), \\[1mm] & \theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{+}(n,t_m)}))=\theta( \underline {\widehat{\mathcal B}}_1+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m), \\[1mm] &\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{++}(n,t_m)}))=\theta( \underline {\widehat{\mathcal B}}_2+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m), \\[1mm] & \theta( \underline {z}(u_{\infty''},{ \underline {\mu}}(n,t_m)))=\theta( \underline {\widehat{\mathcal B}}'_{0}+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m), \\[1mm] &\theta( \underline {z}(u_{\infty''},{ \underline {\mu}}^{+}(n,t_m)))=\theta( \underline {\widehat{\mathcal B}}'_1+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m), \\[1mm] &\theta( \underline {z}(u_{\infty''},{ \underline {\mu}}^{++}(n,t_m)))=\theta( \underline {\widehat{\mathcal B}}'_2+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m), \end{aligned}$$
(5.22)

where

$$\begin{aligned} \, &\begin{alignedat}{5} & \underline {\widehat{\mathcal B}}_0= \underline {\widehat{\mathcal S}}- \underline {\mathcal A}^{(3)}, &\qquad & \underline {\widehat{\mathcal B}}'_0= \underline {\widehat{\mathcal S}}'- \underline {\mathcal A}^{(3)}, &\qquad & \underline {\widehat{\mathcal B}}_1= \underline {\widehat{\mathcal S}}+ \underline {\mathcal A}^{(3)}, \\[1mm] & \underline {\widehat{\mathcal B}}'_1= \underline {\widehat{\mathcal S}}'+ \underline {\mathcal A}^{(3)}, &\qquad & \underline {\widehat{\mathcal B}}_2= \underline {\widehat{\mathcal S}}+2 \underline {\mathcal A}^{(3)}, &\qquad & \underline {\widehat{\mathcal B}}'_2= \underline {\widehat{\mathcal S}}'+2 \underline {\mathcal A}^{(3)}, \end{alignedat} \\[1mm] &\begin{aligned} \, & \underline {\widehat{\mathcal S}}= \underline {\Lambda}- \underline {\mathbb{O}}(u_{\infty'})+ \underline {\rho}(n_0,t_{0m})- \underline {\mathcal A}^{(3)}n_0- \underline {\mathcal A}_m^{(2)}t_{0m}, \\[1mm] & \underline {\widehat{\mathcal S}}'= \underline {\Lambda}- \underline {\mathbb{O}}(u_{\infty''})+ \underline {\rho}(n_0,t_{0m})- \underline {\mathcal A}^{(3)}n_0- \underline {\mathcal A}_m^{(2)}t_{0m}. \end{aligned} \end{aligned}$$

Theorem 5.

Let \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\) and let the divisor \(\mathcal D_{\tilde{ \underline {\mu}}(n,t_m)}\) be nonspecial. Then

$$\begin{aligned} \, &q(n,t_m)=-\,\omega_{0}^{\infty''}- \sum_{j=1}^{l-1}\biggl(\frac{1}{3\beta_0}\mathbb{Q}_{j,l-1}+\frac{1}{\alpha_0\beta_0}\mathbb{Q}_{j,2p-1}\biggr) \frac{\partial}{\partial_{zj}} \ln\frac{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{++}(n,t_m)}))}{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{+}(n,t_m)}))}-{} \nonumber\\ &\hphantom{q(n,t_m)={}} -\frac{\theta( \underline {z}(u_{\infty'},\tilde{ \underline {\mu}}(n,t_m)))\theta( \underline {z}(u_{\infty'},{ \underline {\mu}}^{++}(n,t_m)))} {\theta( \underline {z}(u_{\infty''},\tilde{ \underline {\mu}}^{++}(n,t_m)))\theta( \underline {z}(u_{\infty'},\tilde{ \underline {\mu}}^{+}(n,t_m)))}\exp(\ell_1(Q_0)-\ell_2(Q_0)), \nonumber\\ &r(n,t_m)=-\,\omega_{0}^{\infty'}{+}\, \sum_{j=1}^{l-1}\biggl(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{\alpha^2}\mathbb{Q}_{j,2p-1}\!\biggr) \frac{\partial}{\partial_{z_j}} \ln\frac{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{+}(n,t_m)}))}{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}(n,t_m)}))}, \\ &\frac{s(n,t_m)}{s^{-}(n,t_m)}=\omega_{0}^{\infty'}- \sum_{j=1}^{l-1}\biggl(\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{\alpha^2}\mathbb{Q}_{j,2p-1}\biggr) \frac{\partial}{\partial_{z_j}} \ln\frac{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}^{+}(n,t_m)}))}{\theta( \underline {z}(u_{\infty'},{ \underline {\tilde\mu}(n,t_m)}))}, \nonumber\\ &v(n,t_m)= -\frac{\theta( \underline {z}(u_{\infty''},{ \underline {\mu}}(n,t_m)))\theta( \underline {z}(u_{\infty'},{ \underline {\mu}}^{+}(n,t_m)))} {\theta( \underline {z}(u_{\infty'},\tilde{ \underline {\mu}}(n,t_m)))^2}\exp(\ell_1(Q_0)-\ell_2(Q_0)). \nonumber \end{aligned}$$
(5.23)

Combining (5.20) and (5.23) shows that the Riemann theta representation for \(q(n,t_m)\), \(r(n,t_m)\), \(s(n,t_m)\) and \(v(n,t_m)\) has a remarkable linearity in \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\). Expressions (5.23) can then be rewritten as

$$\begin{aligned} \, &q(n,t_m)=-\,\omega_{0}^{\infty''}+ \sum_{j=1}^{l-1}\mathbb{K}_{j,0}^{(\infty'')} \frac{\partial}{\partial_{z_j}} \ln\frac{\theta( \underline {\widehat{\mathcal B}}_2+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)} {\theta( \underline {\widehat{\mathcal B}}_1+ \underline {A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)}-{} \nonumber\\ &\hphantom{q(n,t_m)={}} -\frac{\theta( \underline {\widehat{\mathcal B}}_0+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)\theta( \underline {\widehat{\mathcal B}}_2+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)} {\theta( \underline {\widehat{\mathcal B}}'_2+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)\theta( \underline {\widehat{\mathcal B}}_1+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)} \exp(\ell_1(Q_0)-\ell_2(Q_0)), \nonumber\\ &r(n,t_m)=-\,\omega_{0}^{\infty'}+ \sum_{j=1}^{l-1}\mathbb{K}_{j,0}^{(\infty')} \frac{\partial}{\partial_{z_j}} \ln\frac{\theta( \underline {\widehat{\mathcal B}}_1+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)} {\theta( \underline {\widehat{\mathcal B}}_0+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)}, \\ &\frac{s(n,t_m)}{s^{-}(n,t_m)}=\omega_{0}^{\infty'}- \sum_{j=1}^{l-1}\mathbb{K}_{j,0}^{(\infty')} \frac{\partial}{\partial_{z_j}} \ln\frac{\theta( \underline {\widehat{\mathcal B}}_1+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)} {\theta( \underline {\widehat{\mathcal B}}_0+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)}, \nonumber\\ &v(n,t_m)=-\frac{\theta( \underline {\widehat{\mathcal B}}'_{0}+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)\theta( \underline {\widehat{\mathcal B}}_1+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)} {\theta( \underline {\widehat{\mathcal B}}_0+ \underline {\mathcal A}^{(3)}n+ \underline {\widehat{\mathcal A}}_m^{(2)}t_m)^2} \exp(\ell_1(Q_0)-\ell_2(Q_0)). \nonumber \end{aligned}$$
(5.24)

Hence, formulas (5.23) and (5.24). give algebro-geometric solutions of the discrete hierarchy of the generalized Toda lattice (1.3).

To clarify the algebro-geometric solutions, we consider a simpler example of the Riemann theta representation under the condition \(p=1\). The genus of \(\mathcal K_2\) is therefore equal to 2, and we can obtain the following results by direct calculation:

$$\begin{alignedat}{5} & \Gamma_{11}^{(1)}=\alpha_0+\beta_0+\alpha_1-\beta_1,&\qquad & \Gamma_{12}^{(1)}=(3\beta_0-2\alpha_0)q^{-},&\qquad & \Gamma_{13}^{(1)}=-\alpha_0sq, \\ & \Gamma_{21}^{(1)}=(3\beta_0-2\alpha_0)v,&\qquad & \Gamma_{22}^{(1)}=\beta_0-\alpha_1+2\beta_1,&\qquad & \Gamma_{23}^{(1)}=(3\beta_0-\alpha_0)s, \\ & \Gamma_{31}^{(1)}=(2\alpha_0-3\beta_0)\frac{v^{-}}{s^{-}q^{-}},&\qquad & \Gamma_{32}^{(1)}=(\alpha_0-3\beta_0)\frac{1}{s^{-}},&\qquad & -\Gamma_{11}^{(1)}-\Gamma_{22}^{(1)}=-\alpha_0-\beta_0-\beta_1. \end{alignedat}$$

The trigonal curve \(\digamma_3(\lambda,f)=0\), whose degree is \(l=3\) (\(\alpha_0\beta_0\neq 0\)), can then be defined as

$$\mathcal K_2\colon\, \digamma_3(\lambda ,f)=f^3-f^2X_3(\lambda)+fY_3(\lambda)-Z_3(\lambda)=0,$$

where

$$\begin{aligned} \, & X_3(\lambda)=0, \\ & Y_3(\lambda)=(-{\alpha_0}^2+\alpha_0\beta_0-3{\beta_0}^2){\lambda}^2+\imath_1\lambda-({a_2}^2+a_2c_2+{c_2}^2), \\ & Z_3(\lambda)=({\alpha_0}^2\beta_0-3\alpha_0{\beta_0}^2+2{\beta_0}^3)\lambda^3+\imath_2\lambda^2+\imath_3\lambda-({a_2}^2c_2+a_2{c_2}^2), \end{aligned}$$

and \(\imath_1\), \(\imath_2\), and \(\imath_3\) are arbitrary constants. Therefore, the polynomials of \(F_2\) and \(E_2\) can be reexpressed as

$$\begin{aligned} \, & F_2(\lambda,n,t_m)=(3\beta_0-\alpha_0)(\lambda-\mu_1(n))(\lambda-\mu_2(n)), \\ & E_2(\lambda,n,t_m)=(\alpha_0-3\beta_0)(\lambda-\mu_1^{+}(n))(\lambda-\mu_2^{+}(n)). \end{aligned}$$

The Riemann theta representations of the potentials in the case of genus is 2 can therefore be rewritten as

$$\begin{aligned} \, & q(n,t_m)=-\,\omega_{0}^{\infty''}+ \sum_{j=1}^2\mathbb{K}_{j,0}^{(\infty'')} \frac{\partial}{\partial_{z_j}} \ln\frac{\theta(\widehat{\mathcal B}_2+\mathcal A_2^{(3)}n+\widehat{\mathcal A}_{m,2}^{(2)}t_m)}{\theta(\widehat{\mathcal B}_1+\mathcal A_2^{(3)}n+ \widehat{\mathcal A}_{m,2}^{(2)}t_m)}-{} \\ &\hphantom{q(n,t_m)={}} -\frac{\theta(\widehat{\mathcal B}_0+\mathcal A_2^{(3)}n+\widehat{\mathcal A}_{m,2}^{(2)}t_m)\theta(\widehat{\mathcal B}_2+\mathcal A_2^{(3)}n+ \widehat{\mathcal A}_{m,2}^{(2)}t_m)} {\theta(\widehat{\mathcal B}'_2+\mathcal A_2^{(3)}n+ \widehat{\mathcal A}_{m,2}^{(2)}t_m)\theta(\widehat{\mathcal B}_1+ \mathcal A_2^{(3)}n+\widehat{\mathcal A}_m^{(2)}t_m)} \exp(\ell_1(Q_0)-\ell_2(Q_0)), \\ & r(n,t_m)=-\,\omega_{0}^{\infty'}+ \sum_{j=1}^2\mathbb{K}_{j,0}^{(\infty')} \frac{\partial}{\partial_{z_j}} \ln\frac{\theta(\widehat{\mathcal B}_1+\mathcal A_2^{(3)}n+\widehat{\mathcal A}_{m,2}^{(2)}t_m)}{\theta(\widehat{\mathcal B}_0+\mathcal A_2^{(3)}n+ \widehat{\mathcal A}_{m,2}^{(2)}t_m)}, \\ & \frac{s(n,t_m)}{s^{-}(n,t_m)}=\omega_{0}^{\infty'}- \sum_{j=1}^2\mathbb{K}_{j,0}^{(\infty')} \frac{\partial}{\partial_{z_j}} \ln\frac{\theta(\widehat{\mathcal B}_1+\mathcal A_2^{(3)}n+\widehat{\mathcal A}_{m,2}^{(2)}t_m)} {\theta(\widehat{\mathcal B}_0+\mathcal A^{(3)}n+\widehat{\mathcal A}_{m,2}^{(2)}t_m)},\ \\ & v(n,t_m)=-\frac{\theta(\widehat{\mathcal B}'_{0}+\mathcal A_2^{(3)}n+ \widehat{\mathcal A}_{m,2}^{(2)}t_m)\theta(\widehat{\mathcal B}_1+\mathcal A_2^{(3)}n+ \widehat{\mathcal A}_{m,2}^{(2)}t_m)} {\theta(\widehat{\mathcal B}_0+\mathcal A_2^{(3)}n+ \widehat{\mathcal A}_{m,2}^{(2)}t_m)^2} \exp(\ell_1(Q_0)-\ell_2(Q_0)), \end{aligned}$$

where

$$\begin{aligned} \, & \theta(z)=\sum_{\mathcal F_1\in\mathbb{Z}}\exp\{2\pi i\mathcal F_1 z+\pi i\tau_{11}\mathcal F_{1}^2\}, \\ &\begin{alignedat}{5} &\widehat{\mathcal B}_0=\widehat{\mathcal S}-\mathcal A_2^{(3)},&\qquad &\widehat{\mathcal B}'_0=\widehat{\mathcal S}'-\mathcal A_2^{(3)},&\qquad &\widehat{\mathcal B}_1=\widehat{\mathcal S}+\mathcal A_2^{(3)}, \\ &\widehat{\mathcal B}'_1=\widehat{\mathcal S}'+\mathcal A_2^{(3)},&\qquad &\widehat{\mathcal B}_2=\widehat{\mathcal S}+2\mathcal A_2^{(3)},&\quad &\widehat{\mathcal B}'_2=\widehat{\mathcal S}'+2\mathcal A_2^{(3)}, \end{alignedat} \\ &\begin{aligned} \, &\widehat{\mathcal S}=\Lambda_2-\mathbb{O}_2(u_{\infty'})+\rho_2(n_0,t_{0m})-\mathcal A_2^{(3)}n_0-\widehat{\mathcal A}_{m,2}^{(2)}t_{0m}, \\ &\widehat{\mathcal S}'=\Lambda_2-\mathbb{O}_2(u_{\infty''})+\rho_2(n_0,t_{0m})-\mathcal A_2^{(3)}n_0-\widehat{\mathcal A}_{r,2}^{(2)}t_{0m}, \end{aligned} \\ &\mathbb{K}_{j,0}^{(\infty')}=\frac{2}{3\alpha_0}\mathbb{Q}_{j,l-1}-\frac{1}{\alpha^2}\mathbb{Q}_{j,2p-1},\qquad \mathbb{K}_{j,0}^{(\infty'')}=-\frac{1}{3\beta_0}\mathbb{Q}_{j,l-1}-\frac{1}{\alpha_0\beta_0}\mathbb{Q}_{j,2p-1}. \end{aligned}$$

These formulas define algebro-geometric solutions of the discrete hierarchy of the generalized Toda lattice (1.3) in the case of genus 2.

6. Conclusions and Remarks

We have found algebro-geometric solutions of the hierarchy of generalized Toda lattices. The hierarchy was generated using the zero-curvature equation, and the functions \(\Xi\) and \(\Theta\) were introduced on the trigonal curve. Based on the Abel differential, the Riemann theta representations of the potentials were constructed in the stationary and time-dependent cases, and solutions of the hierarchy were obtained. Currently, increasingly many researchers focus on trigonal curves and the application of these methods is gaining in popularity. Discussing the algebro-geometric solutions of the 4th-order soliton equations is also interesting, and we plan to address this problem in the future. Equally important is the study of soliton solutions beyond the algebro-geometric solutions, such as the lump–soliton and breather solutions.