# Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

Yu Hou  Engui Fan111Corresponding author and e-mail address:
<EMAIL_ADDRESS>
(  School of Mathematical Sciences, Shanghai Center for Mathematical Sciences
and Key Laboratory of Mathematics for Nonlinear Science,
Fudan University, Shanghai 200433, P.R. China)

###### Abstract

This paper is dedicated to provide theta function representations of algebro-
geometric solutions and related crucial quantities for the two-component
Hunter-Saxton (HS2) hierarchy through studying an algebro-geometric initial
value problem. Our main tools include the polynomial recursive formalism, the
hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions,
the meromorphic function, the Dubrovin-type equations for auxiliary divisors,
and the associated trace formulas. With the help of these tools, the explicit
representations of the algebro-geometric solutions are obtained for the entire
HS2 hierarchy.

##  1 Introduction

In this paper, we consider the following integrable two-component Hunter-
Saxton (HS2) system:

$\left\\{\begin{array}[]{ll}m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0,\\\
\rho_{t}+(u\rho)_{x}=0,\end{array}\right.$ (1.1)

where $m=-u_{xx}$, $\sigma=\pm 1$, which was recently introduced by Constantin
and Ivanov in [9]. The variable $u(x,t)$ can be interpreted as the horizontal
fluid velocity and the variable $\rho(x,t)$ describes the horizontal deviation
of the surface from equilibrium, all measured in dimensionless units [9].

The HS2 system arises in the short-wave (or high-frequency) limits, obtained
via the space-time scaling $(x,t)\mapsto(\varepsilon x,\varepsilon t)$ and
letting $\varepsilon$ tend to zero in the resulting equation, of the two-
component integrable Camassa-Holm system [9, 17, 39, 43]. This system, reading
as (1.1) with $m$ replaced by $(1-\partial_{xx}^{2})u$, was derived from the
Green-Naghdi equations, which are approximations to the governing equations
for water waves. It has recently been the object of intensive study, see,
e.g., [8, 9, 25, 26, 45]. The HS2 system (1.1) is integrable, it has Lax pair
[9] and a bi-Hamiltonian structure [9, 39]. It is also a particular case of
the Gurevich-Zybin system describing the dynamics in a model of nondissipative
dark matter [24, 40, 44]. The mathematical properties of Eq.(1.1) have been
studied further in many works, see, e.g., [9, 23, 24, 36, 42, 43, 44].

For $\rho=0$, the system (1.1) becomes the Hunter-Saxton (HS) equation [30],
which models the propagation of weakly nonlinear orientation waves in a
massive nematic liquid crystal director field. Here, $u(x,t)$ stands for the
director field of a nematic liquid crystal, $x$ is a space variable in a
reference frame moving with the linearized wave velocity, and $t$ is a slow
time variable. The field of unit vectors $(\cos{u(x,t)},\sin{u(x,t)})$
describes the orientation of the molecules [23, 30, 44].

The HS equation also describes the high-frequency limit [13, 31] of the
Camassa-Holm (CH) equation –a model equation for shallow water waves [7, 11,
32] and a re-expression of the geodesic flow on the diffeomorphism group of
the circle [10] with a bi-Hamiltonian structure [18] which is completely
integrable [12]. The HS equation is also a completely integrable system with a
bi-Hamiltonian structure, and hence, it possesses a Lax pair, an infinite
family of commuting Hamiltonian flows, as well as an associated sequence of
conservation laws [5, 30, 31, 39, 41]. It also describes the geodesic flow on
the homogeneous space related to the Virasoro group [33, 34, 35]. Recently,
algebro-geometric solutions for the HS hierarchy was investigated in [28].
Moreover, peakon solutions, global weak solutions and the Cauchy problem of
the system (1.1) were discussed in [9, 23, 36, 44]. However, within the
knowledge of the authors, the algebro-geometric solutions of the entire HS2
hierarchy are not studied yet.

The principal subject of this paper concerns algebro-geometric quasi-periodic
solutions of the whole HS2 hierarchy, of which (1.1) ($\sigma=1$) is just the
first of infinitely many members. Algebro-geometric solution, as an important
feature of integrable system, is a kind of explicit solution closely related
to the inverse spectral theory [1, 6], [14]-[16]. In a degenerated case of the
algebro-geometric solution, the multi-soliton solution and periodic solution
in elliptic function type may be obtained [6, 37, 38]. A systematic approach,
proposed by Gesztesy and Holden to construct algebro-geometric solutions for
integrable equations, has been extended to the whole (1+1) dimensional
integrable hierarchy, such as the AKNS hierarchy, the CH hierarchy, etc.
[19]-[22]. Recently, we investigated algebro-geometric solutions for the
modified CH hierarchy and the Degasperis-Procesi hierarchy [27, 29].

The outline of the present paper is as follows.

In section 2, based on the polynomial recursion formalism, we derive the HS2
hierarchy, associated with the $2\times 2$ spectral problem. A hyperelliptic
curve $\mathcal{K}_{n}$ of arithmetic genus $n$ is introduced with the help of
the characteristic polynomial of Lax matrix $V_{n}$ for the stationary HS2
hierarchy.

In Section 3, we decompose the stationary HS2 equations into a system of
Dubrovin-type equations. Moreover, we obtain the stationary trace formulas for
the HS2 hierarchy.

In Section 4, we present the first set of our results, the explicit theta
function representations of the potentials $u,\rho$ for the entire stationary
HS2 hierarchy. Furthermore, we study the initial value problem on an algebro-
geometric curve for the stationary HS2 hierarchy.

In Sections 5 and 6, we extend the analyses of Sections 3 and 4, respectively,
to the time-dependent case. Each equation in the HS2 hierarchy is permitted to
evolve in terms of an independent time parameter $t_{r}$. As initial data, we
use a stationary solution of the $n$th equation and then construct a time-
dependent solution of the $r$th equation of the HS2 hierarchy. The Baker-
Akhiezer function, the analogs of the Dubrovin-type equations, the trace
formulas, and the theta function representations in Section 4 are all extended
to the time-dependent case.

Finally, we remark that although our focus in this paper is on Eq.(1.1) with
$\sigma=1$, all of the arguments presented here can be adapted, without
obvious modifications, to study the corresponding equation with $\sigma=-1$.

##  2 The HS2 hierarchy

In this section, we provide the construction of HS2 hierarchy and derive the
corresponding sequence of zero-curvature pairs using a polynomial recursion
formalism. Moreover, we introduce the underlying hyperelliptic curve in
connection with the stationary HS2 hierarchy.

Throughout this section, we make the following hypothesis.

######  Hypothesis 2.1.

In the stationary case, we assume that

$\begin{split}&u,\rho\in
C^{\infty}(\mathbb{R}),~{}\partial_{x}^{k}u,\partial_{x}^{k}\rho\in
L^{\infty}(\mathbb{R}),~{}k\in\mathbb{N}_{0}.\end{split}$ (2.1)

In the time-dependent case, we suppose

$\begin{split}&u(\cdot,t),\rho(\cdot,t)\in C^{\infty}(\mathbb{R}),\ \
\partial_{x}^{k}u(\cdot,t),\partial_{x}^{k}\rho(\cdot,t)\in
L^{\infty}(\mathbb{R}),~{}~{}k\in\mathbb{N}_{0},~{}t\in\mathbb{R},\\\
&u(x,\cdot),u_{xx}(x,\cdot),\rho(x,\cdot),\rho_{x}(x,\cdot)\in
C^{1}(\mathbb{R}),\quad x\in\mathbb{R}.\\\ \end{split}$ (2.2)

We first introduce the basic polynomial recursion formalism. Define
$\\{f_{l}\\}_{l\in\mathbb{N}_{0}}$, $\\{g_{l}\\}_{l\in\mathbb{N}_{0}}$, and
$\\{h_{l}\\}_{l\in\mathbb{N}_{0}}$ recursively by

$\begin{split}&f_{0}=\frac{1}{2},\\\
&f_{l,x}=-2\mathcal{G}\left(2\rho^{2}f_{l-2,x}+2u_{xx}f_{l-1,x}+2\rho\rho_{x}f_{l-2}+u_{xxx}f_{l-1}\right),\quad
l\in\mathbb{N},\\\ &g_{l}=\frac{1}{2}f_{l+1,x},\quad l\in\mathbb{N}_{0},\\\
&h_{l}=-g_{l+1,x}-\rho^{2}f_{l}-u_{xx}f_{l+1},\quad l\in\mathbb{N}_{0},\\\
\end{split}$ (2.3)

where $\mathcal{G}$ is given by

$\begin{split}&\mathcal{G}:L^{\infty}(\mathbb{R})\rightarrow
L^{\infty}(\mathbb{R}),\\\
&(\mathcal{G}v)(x)=\int_{-\infty}^{x}\int_{-\infty}^{x_{1}}v(y)~{}dydx_{1},\quad
x\in\mathbb{R},~{}v\in L^{\infty}(\mathbb{R}).\end{split}$ (2.4)

One observes that $\mathcal{G}$ is the resolvent of the one-dimensional
Laplacian operator, that is,

$\mathcal{G}=\Big{(}\frac{d^{2}}{dx^{2}}\Big{)}^{-1}.$ (2.5)

Explicitly, one computes

$\begin{split}&f_{0}=\frac{1}{2},\\\ &f_{1}=-u+c_{1},\\\
&f_{2}=\mathcal{G}(u_{x}^{2}-\rho^{2}+2uu_{xx})+2c_{1}(-u)+c_{2},\\\
&g_{0}=-\frac{1}{2}u_{x},\\\
&g_{1}=\frac{1}{2}\mathcal{G}(4u_{x}u_{xx}+2uu_{xxx}-2\rho\rho_{x})+2c_{1}(-\frac{1}{2}u_{x}),\\\
&h_{0}=-\frac{1}{2}f_{2,xx}-\rho^{2}f_{0}-u_{xx}f_{1},~{}\mathrm{etc}.,\end{split}$
(2.6)

where $\\{c_{l}\\}_{l\in\mathbb{N}}\subset\mathbb{C}$ are integration
constants.

Next, it is convenient to introduce the corresponding homogeneous coefficients
$\hat{f}_{l},\hat{g}_{l},$ and $\hat{h}_{l},$ defined by the vanishing of the
integration constants $c_{k},k=1,\ldots,l,$

$\begin{split}&\hat{f}_{0}=f_{0}=\frac{1}{2},\quad\hat{f}_{l}=f_{l}|_{c_{k}=0,~{}k=1,\ldots,l},\quad
l\in\mathbb{N},\quad\\\
&\hat{g}_{0}=g_{0}=-\frac{1}{2}u_{x},\quad\hat{g}_{l}=g_{l}|_{c_{k}=0,~{}k=1,\ldots,l},\quad
l\in\mathbb{N},\\\
&\hat{h}_{0}=h_{0},\quad\quad\hat{h}_{l}=h_{l}|_{c_{k}=0,~{}k=1,\ldots,l},\quad
l\in\mathbb{N}.\end{split}$ (2.7)

Hence,

$f_{l}=\sum_{k=0}^{l}2c_{l-k}\hat{f}_{k},\quad
g_{l}=\sum_{k=0}^{l}2c_{l-k}\hat{g}_{k},\quad
h_{l}=\sum_{k=0}^{l}2c_{l-k}\hat{h}_{k},\quad l\in\mathbb{N}_{0},$ (2.8)

defining

$c_{0}=\frac{1}{2}.$ (2.9)

Now, given Hypothesis 2.1, one introduces the following $2\times 2$ matrix $U$
by

$\psi_{x}=U(z,x)\psi=\left(\begin{array}[]{cc}0&1\\\
-z^{-2}\rho^{2}-z^{-1}u_{xx}&0\\\ \end{array}\right)\psi,$ (2.10)

and for each $n\in\mathbb{N}_{0},$ the following $2\times 2$ matrix $V_{n}$ by

$\psi_{t_{n}}=V_{n}(z)\psi,$ (2.11)

with

$V_{n}(z)=\left(\begin{array}[]{cc}-G_{n}(z)&F_{n}(z)\\\
z^{-2}H_{n}(z)&G_{n}(z)\\\ \end{array}\right),\quad
z\in\mathbb{C}\setminus\\{0\\},\quad n\in\mathbb{N}_{0},$ (2.12)

assuming $F_{n}$, $G_{n}$, and $H_{n}$ to be polynomials222$F_{n},G_{n},H_{n}$
are polynomials of degree $n+1,n,n+1$, respectively. with respect to $z$ and
$C^{\infty}$ in $x$. The compatibility condition of linear system (2.10) and
(2.11) yields the stationary zero-curvature equation

$-V_{n,x}+[U,V_{n}]=0,$ (2.13)

which is equivalent to

$\displaystyle F_{n,x}=2G_{n},$ (2.14) $\displaystyle
H_{n,x}=2(\rho^{2}+zu_{xx})G_{n},$ (2.15) $\displaystyle
z^{2}G_{n,x}=-H_{n}-(\rho^{2}+zu_{xx})F_{n}.$ (2.16)

From (2.14)-(2.16), one infers that

$\frac{d}{dx}\mathrm{det}(V_{n}(z,x))=-\frac{1}{z^{2}}\frac{d}{dx}\Big{(}z^{2}G_{n}(z,x)^{2}+F_{n}(z,x)H_{n}(z,x)\Big{)}=0,$
(2.17)

and hence

$z^{2}G_{n}(z,x)^{2}+F_{n}(z,x)H_{n}(z,x)=R_{2n+2}(z),$ (2.18)

where the polynomial $R_{2n+2}$ of degree $2n+2$ is $x$-independent. In
another way, one can write $R_{2n+2}$ as

$R_{2n+2}(z)=\left(\frac{1}{4}u_{x}^{2}+\frac{1}{2}h_{0}\right)\prod_{m=0}^{2n+1}(z-E_{m}),\quad\\{E_{m}\\}_{m=0,\ldots,2n+1}\in\mathbb{C}.$
(2.19)

Here, we emphasize that the coefficient
$(\frac{1}{4}u_{x}^{2}+\frac{1}{2}h_{0})$ is a constant. In fact, equation
(2.17) is equivalent to

$2z^{2}G_{n}G_{n,x}+F_{n}H_{n,x}+H_{n}F_{n,x}=0.$ (2.20)

Then comparing the coefficient of powers $z^{2n+2}$ yields

$2g_{0}g_{0,x}+f_{0}h_{0,x}+h_{0}f_{0,x}=0,$ (2.21)

which indicates

$\frac{1}{2}u_{x}u_{xx}+\frac{1}{2}h_{0,x}=0.$ (2.22)

Therefore,

$\frac{1}{4}u_{x}^{2}+\frac{1}{2}h_{0}=\partial^{-1}\left(\frac{1}{2}u_{x}u_{xx}+\frac{1}{2}h_{0,x}\right)=\mathrm{constant}.$
(2.23)

For simplicity, we denote it by $a^{2}$, $a\in\mathbb{C}.$ Then, $R_{2n+2}(z)$
can be rewritten as

$R_{2n+2}(z)=a^{2}\prod_{m=0}^{2n+1}(z-E_{m}),\quad\\{E_{m}\\}_{m=0,\ldots,2n+1}\in\mathbb{C}.$
(2.24)

Next, we compute the characteristic polynomial $\mathrm{det}(yI-zV_{n})$ of
Lax matrix $zV_{n}$,

$\displaystyle\mathrm{det}(yI-zV_{n})$
$\displaystyle=y^{2}-z^{2}G_{n}(z)^{2}-F_{n}(z)H_{n}(z)$
$\displaystyle=y^{2}-R_{2n+2}(z)=0,$ (2.25)

and then introduce the (possibly singular) hyperelliptic curve
$\mathcal{K}_{n}$ of arithmetic genus $n$ defined by

$\mathcal{K}_{n}:\mathcal{F}_{n}(z,y)=y^{2}-R_{2n+2}(z)=0.$ (2.26)

In the following, we will occasionally impose further constraints on the zeros
$E_{m}$ of $R_{2n+2}$ introduced in (2.24) and assume that

$E_{m}\in\mathbb{C},\quad E_{m}\neq E_{m^{\prime}},\quad\forall m\neq
m^{\prime},\quad m,m^{\prime}=0,\ldots,2n+1.$ (2.27)

The stationary zero-curvature equation (2.13) implies polynomial recursion
relations (2.3). Introducing the following polynomials $F_{n}(z),G_{n}(z)$,
and $H_{n}(z)$ with respect to the spectral parameter $z$,

$\displaystyle F_{n}(z)=\sum_{l=0}^{n+1}f_{l}z^{n+1-l},$ (2.28) $\displaystyle
G_{n}(z)=\sum_{l=0}^{n}g_{l}z^{n-l},$ (2.29) $\displaystyle
H_{n}(z)=\sum_{l=0}^{n+1}h_{l}z^{n+1-l}.$ (2.30)

Inserting (2.28)-(2.30) into (2.14)-(2.16) then yields the recursion relations
(2.3) for $f_{l},$ $l=0,\ldots,n+1,$ and $g_{l}$, $l=0,\ldots,n.$ For fixed
$n\in\mathbb{N}_{0}$, we obtain the recursion relations for $h_{l},$
$l=0,\ldots,n-1$ in (2.3) and

$h_{n}=-\rho^{2}f_{n}-u_{xx}f_{n+1},\quad h_{n+1}=-\rho^{2}f_{n+1}.$ (2.31)

Moreover, from (2.15), one infers that

$\begin{split}&-h_{n,x}+\rho^{2}f_{n,x}+u_{xx}f_{n+1,x}=0,\quad
n\in\mathbb{N}_{0},\\\ &-h_{n+1,x}+\rho^{2}f_{n+1,x}=0,\quad
n\in\mathbb{N}_{0}.\\\ \end{split}$ (2.32)

Then using (2.31) and (2.32) permits one to write the stationary HS2 hierarchy
as

$\textrm{s-HS2}_{n}(u,\rho)=\left(\begin{array}[]{c}2u_{xx}f_{n+1,x}+u_{xxx}f_{n+1}+2\rho\rho_{x}f_{n}+2\rho^{2}f_{n,x}\\\
-2\rho\rho_{x}f_{n+1}-2\rho^{2}f_{n+1,x}\\\ \end{array}\right)=0,\quad
n\in\mathbb{N}_{0}.$ (2.33)

We record the first equation explicitly,

$\begin{split}&\textrm{s-HS2}_{0}(u,\rho)=\left(\begin{array}[]{c}-2u_{x}u_{xx}-uu_{xxx}+\rho\rho_{x}+c_{1}u_{xxx}\\\
\rho_{x}u+\rho u_{x}-c_{1}\rho_{x}\\\ \end{array}\right)=0.\end{split}$ (2.34)

By definition, the set of solutions of (2.33) represents the class of algebro-
geometric HS2 solutions, with $n$ ranging in $\mathbb{N}_{0}$ and $c_{l}$ in
$\mathbb{C},~{}l\in\mathbb{N}$. We call the stationary algebro-geometric HS2
solutions $u,$ $\rho$ as HS2 potentials at times.

######  Remark 2.2.

Here, we emphasize that if $u$, $\rho$ satisfy one of the stationary HS$2$
equations in $(\ref{2.34})$ for a particular value of $n$, then they satisfy
infinitely many such equations of order higher than $n$ for certain choices of
integration constants $c_{l}$. This is a common characteristic of the general
integrable soliton equations such as the KdV, AKNS, and CH hierarchies [21].

Next, we introduce the corresponding homogeneous polynomials
$\widehat{F}_{l},\widehat{G}_{l},\widehat{H}_{l}$ by

$\displaystyle\widehat{F}_{l}(z)=F_{l}(z)|_{c_{k}=0,~{}k=1,\dots,l}=\sum_{k=0}^{l}\hat{f}_{k}z^{l-k},~{}~{}l=0,\ldots,n+1,$
(2.35)
$\displaystyle\widehat{G}_{l}(z)=G_{l}(z)|_{c_{k}=0,~{}k=1,\dots,l}=\sum_{k=0}^{l}\hat{g}_{k}z^{l-k},~{}~{}l=0,\ldots,n,$
(2.36)
$\displaystyle\widehat{H}_{l}(z)=H_{l}(z)|_{c_{k}=0,~{}k=1,\dots,l}=\sum_{k=0}^{l}\hat{h}_{k}z^{l-k},~{}~{}l=0,\ldots,n-1,$
(2.37)
$\displaystyle\widehat{H}_{n}(z)=-\rho^{2}\hat{f}_{n}-u_{xx}\hat{f}_{n+1}+\sum_{k=0}^{n-1}\hat{h}_{k}z^{n-k},$
(2.38)
$\displaystyle\widehat{H}_{n+1}(z)=-\rho^{2}\hat{f}_{n+1}+z\widehat{H}_{n}(z).$
(2.39)

In accordance with our notation introduced in (2.7) and (2.35)-(2.39), the
corresponding homogeneous stationary HS2 equations are then defined by

$\textrm{s-}\widehat{\mathrm{HS2}}_{n}(u,\rho)=\textrm{s-HS2}_{n}(u,\rho)|_{c_{l}=0,~{}l=1,\dots,n}=0,\quad
n\in\mathbb{N}_{0}.$ (2.40)

At the end of this section, we turn to the time-dependent HS2 hierarchy. In
this case, $u$, $\rho$ are considered as functions of both space and time. We
introduce a deformation parameter $t_{n}\in\mathbb{R}$ in $u$ and $\rho$,
replacing $u(x),\rho(x)$ by $u(x,t_{n}),\rho(x,t_{n})$, for each equation in
the hierarchy. In addition, the definitions (2.10), (2.12), and (2.28)-(2.30)
of $U,$ $V_{n}$ and $F_{n},G_{n}$, and $H_{n},$ respectively, still apply. The
corresponding zero-curvature equation reads

$U_{t_{n}}-V_{n,x}+[U,V_{n}]=0,\quad n\in\mathbb{N}_{0},$ (2.41)

which results in the following set of equations

$\displaystyle F_{n,x}=2G_{n},$ (2.42) $\displaystyle
z^{2}G_{n,x}=-H_{n}-(\rho^{2}+zu_{xx})F_{n},$ (2.43)
$\displaystyle-2\rho\rho_{t_{n}}-zu_{xxt_{n}}-H_{n,x}+2(\rho^{2}+zu_{xx})G_{n}=0.$
(2.44)

For fixed $n\in\mathbb{N}_{0}$, inserting the polynomial expressions for
$F_{n}$, $G_{n}$, and $H_{n}$ into (2.42)-(2.44), respectively, first yields
recursion relations (2.3) for $f_{l}|_{l=0,\ldots,n+1}$,
$g_{l}|_{l=0,\ldots,n}$, $h_{l}|_{l=0,\ldots,n-1}$ and

$h_{n}=-\rho^{2}f_{n}-u_{xx}f_{n+1},\quad h_{n+1}=-\rho^{2}f_{n+1}.$ (2.45)

Moreover, using (2.44), one finds

$\begin{split}&-u_{xxt_{n}}-h_{n,x}+\rho^{2}f_{n,x}+u_{xx}f_{n+1,x}=0,\quad
n\in\mathbb{N}_{0},\\\ &2\rho\rho_{t_{n}}+h_{n+1,x}-\rho^{2}f_{n+1,x}=0,\quad
n\in\mathbb{N}_{0}.\end{split}$ (2.46)

Hence, using (2.45) and (2.46) permits one to write the time-dependent HS2
hierarchy as

$\textrm{HS2}_{n}(u,\rho)=\left(\begin{array}[]{c}-u_{xxt_{n}}+2u_{xx}f_{n+1,x}+u_{xxx}f_{n+1}+2\rho\rho_{x}f_{n}+2\rho^{2}f_{n,x}\\\
2\rho\rho_{t_{n}}-2\rho\rho_{x}f_{n+1}-2\rho^{2}f_{n+1,x}\\\
\end{array}\right)=0,\quad n\in\mathbb{N}_{0}.$ (2.47)

For convenience, we record the first equation in this hierarchy explicitly,

$\begin{split}&\mathrm{HS2}_{0}(u,\rho)=\left(\begin{array}[]{c}-u_{xxt_{0}}-2u_{x}u_{xx}-uu_{xxx}+\rho\rho_{x}+c_{1}u_{xxx}\\\
\rho_{t_{0}}+\rho_{x}u+\rho u_{x}-c_{1}\rho_{x}\\\
\end{array}\right)=0.\end{split}$ (2.48)

The first equation $\mathrm{HS2}_{0}(u,\rho)=0$ (with $c_{1}=0$) in the
hierarchy represents the HS2 system as discussed in section 1\. Similarly, one
can introduce the corresponding homogeneous HS2 hierarchy by

$\widehat{\mathrm{HS2}}_{n}(u,\rho)=\mathrm{HS2}_{n}(u,\rho)|_{c_{l}=0,~{}l=1,\ldots,n}=0,\quad
n\in\mathbb{N}_{0}.$ (2.49)

In fact, since the Lenard recursion formalism is almost universally adopted in
the contemporary literature, we thought it might be worthwhile to use the
Gesztesy’s method, the polynomial recursion formalism, to construct the HS2
hierarchy.

##  3 The stationary HS2 formalism

This section is devoted to a detailed study of the stationary HS2 hierarchy.
We first define a fundamental meromorphic function $\phi(P,x)$ on the
hyperelliptic curve $\mathcal{K}_{n}$, using the polynomial recursion
formalism described in section 2, and then study the properties of the Baker-
Akhiezer function $\psi(P,x,x_{0})$, Dubrovin-type equations, and trace
formulas.

For major parts of this section, we assume (2.1), (2.3), (2.6), (2.10)-(2.16),
(2.26)-(2.30), and (2.33), keeping $n\in\mathbb{N}_{0}$ fixed.

Recall the hyperelliptic curve $\mathcal{K}_{n}$

$\begin{split}&\mathcal{K}_{n}:\mathcal{F}_{n}(z,y)=y^{2}-R_{2n+2}(z)=0,\\\
&R_{2n+2}(z)=a^{2}\prod_{m=0}^{2n+1}(z-E_{m}),\quad\\{E_{m}\\}_{m=0,\ldots,2n+1}\in\mathbb{C},\end{split}$
(3.1)

which is compactified by joining two points at infinity $P_{\infty_{\pm}}$,
with $P_{\infty_{+}}\neq P_{\infty_{-}}$. But for notational simplicity, the
compactification is also denoted by $\mathcal{K}_{n}$. Hence,
$\mathcal{K}_{n}$ becomes a two-sheeted Riemann surface of arithmetic genus
$n$. Points $P$ on $\mathcal{K}_{n}\backslash\\{P_{\infty\pm}\\}$ are denoted
by $P=(z,y(P))$, where $y(\cdot)$ is the meromorphic function on
$\mathcal{K}_{n}$ satisfying $\mathcal{F}_{n}(z,y(P))=0.$

The complex structure on $\mathcal{K}_{n}$ is defined in the usual way by
introducing local coordinates

$\zeta_{Q_{0}}:P\rightarrow(z-z_{0})$

near points $Q_{0}=(z_{0},y(Q_{0}))\in\mathcal{K}_{n},$ which are neither
branch nor singular points of $\mathcal{K}_{n}$; near the branch and singular
points $Q_{1}=(z_{1},y(Q_{1}))\in\mathcal{K}_{n}$, the local coordinates are

$\zeta_{Q_{1}}:P\rightarrow(z-z_{1})^{1/2};$

near the points $P_{\infty_{\pm}}\in\mathcal{K}_{n}$, the local coordinates
are

$\zeta_{P_{\infty_{\pm}}}:P\rightarrow z^{-1}.$

The holomorphic map $\ast,$ changing sheets, is defined by

$\displaystyle\ast:\begin{cases}\mathcal{K}_{n}\rightarrow\mathcal{K}_{n},\\\
P=(z,y_{j}(z))\rightarrow P^{\ast}=(z,y_{j+1(\mathrm{mod}~{}2)}(z)),\quad
j=0,1,\end{cases}$ $\displaystyle
P^{\ast\ast}:=(P^{\ast})^{\ast},\quad\mathrm{etc}.,$ (3.2)

where $y_{j}(z),\,j=0,1$ denote the two branches of $y(P)$ satisfying
$\mathcal{F}_{n}(z,y)=0$, namely,

$(y-y_{0}(z))(y-y_{1}(z))=y^{2}-R_{2n+2}(z)=0.$ (3.3)

Taking into account (3.3), one easily finds

$\begin{split}&y_{0}+y_{1}=0,\\\ &y_{0}y_{1}=-R_{2n+2}(z),\\\
&y_{0}^{2}+y_{1}^{2}=2R_{2n+2}(z).\\\ \end{split}$ (3.4)

Moreover, positive divisors on $\mathcal{K}_{n}$ of degree $n$ are denoted by

$\mathcal{D}_{P_{1},\ldots,P_{n}}:\begin{cases}\mathcal{K}_{n}\rightarrow\mathbb{N}_{0},\\\
P\rightarrow\mathcal{D}_{P_{1},\ldots,P_{n}}=\begin{cases}\textrm{ $k$ \quad
if $P$ occurs $k$ times in $\\{P_{1},\ldots,P_{n}\\},$}\\\ \textrm{ $0$ \quad
if $P\notin$$\\{P_{1},\ldots,P_{n}\\}.$}\end{cases}\end{cases}$ (3.5)

Next, we define the stationary Baker-Akhiezer function $\psi(P,x,x_{0})$ on
$\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}=(0,i\rho
f_{n+1})\\}$ by

$\begin{split}&\psi(P,x,x_{0})=\left(\begin{array}[]{c}\psi_{1}(P,x,x_{0})\\\
\psi_{2}(P,x,x_{0})\\\ \end{array}\right),\\\
&\psi_{x}(P,x,x_{0})=U(u(x),\rho(x),z(P))\psi(P,x,x_{0}),\\\
&zV_{n}(u(x),\rho(x),z(P))\psi(P,x,x_{0})=y(P)\psi(P,x,x_{0}),\\\
&\psi_{1}(P,x_{0},x_{0})=1;\\\
&P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}=(0,i\rho
f_{n+1})\\},~{}(x,x_{0})\in\mathbb{R}^{2}.\end{split}$ (3.6)

Closely related to $\psi(P,x,x_{0})$ is the following meromorphic function
$\phi(P,x)$ on $\mathcal{K}_{n}$ defined by

$\phi(P,x)=z\frac{\psi_{1,x}(P,x,x_{0})}{\psi_{1}(P,x,x_{0})},\quad
P\in\mathcal{K}_{n},~{}x\in\mathbb{R}$ (3.7)

such that

$\psi_{1}(P,x,x_{0})=\mathrm{exp}\left(z^{-1}\int_{x_{0}}^{x}\phi(P,x^{\prime})~{}dx^{\prime}\right),\quad
P\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\}.$ (3.8)

Then, based on (3.6) and (3.7), a direct calculation shows that

$\displaystyle\phi(P,x)$ $\displaystyle=\frac{y+zG_{n}(z,x)}{F_{n}(z,x)}$
$\displaystyle=\frac{H_{n}(z,x)}{y-zG_{n}(z,x)},$ (3.9)

and

$\psi_{2}(P,x,x_{0})=\psi_{1}(P,x,x_{0})\phi(P,x)/z.$ (3.10)

In the following, the roots of polynomials $F_{n}$ and $H_{n}$ will play a
special role, and hence, we introduce on $\mathbb{C}\times\mathbb{R}$

$F_{n}(z,x)=\frac{1}{2}\prod_{j=0}^{n}(z-\mu_{j}(x)),\quad
H_{n}(z,x)=h_{0}\prod_{l=0}^{n}(z-\nu_{l}(x)).$ (3.11)

Moreover, we introduce

$\hat{\mu}_{j}(x)=(\mu_{j}(x),-\mu_{j}(x)G_{n}(\mu_{j}(x),x))\in\mathcal{K}_{n},~{}j=0,\ldots,n,~{}x\in\mathbb{R},$
(3.12)

and

$\hat{\nu}_{l}(x)=(\nu_{l}(x),\nu_{l}(x)G_{n}(\nu_{l}(x),x))\in\mathcal{K}_{n},~{}l=0,\ldots,n,~{}x\in\mathbb{R}.$
(3.13)

Due to assumption (2.1), $u$ and $\rho$ are smooth and bounded, and hence,
$F_{n}(z,x)$ and $H_{n}(z,x)$ share the same property. Thus, one concludes

$\mu_{j},\nu_{l}\in C(\mathbb{R}),\quad j,l=0,\dots,n,$ (3.14)

taking multiplicities (and appropriate reordering) of the zeros of $F_{n}$ and
$H_{n}$ into account. From (3), the divisor $(\phi(P,x))$ of $\phi(P,x)$ is
given by

$(\phi(P,x))=\mathcal{D}_{\hat{\nu}_{0}(x)\underline{\hat{\nu}}(x)}(P)-\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}(P).$
(3.15)

Here, we abbreviated

$\underline{\hat{\mu}}=\\{\hat{\mu}_{1},\ldots,\hat{\mu}_{n}\\},\quad\underline{\hat{\nu}}=\\{\hat{\nu}_{1},\ldots,\hat{\nu}_{n}\\}\in\mathrm{Sym}^{n}(\mathcal{K}_{n}).$
(3.16)

Further properties of $\phi(P,x)$ are summarized as follows.

######  Lemma 3.1.

Suppose $(\ref{2.1})$, assume the $n$th stationary $HS2$ equation
$(\ref{2.34})$ holds, and let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\},$
$(x,x_{0})\in\mathbb{R}^{2}$. Then $\phi$ satisfies the Riccati-type equation

$\phi_{x}(P)+z^{-1}\phi(P)^{2}=-z^{-1}\rho^{2}-u_{xx},$ (3.17)

as well as

$\displaystyle\phi(P)\phi(P^{\ast})=-\frac{H_{n}(z)}{F_{n}(z)},$ (3.18)
$\displaystyle\phi(P)+\phi(P^{\ast})=\frac{2zG_{n}(z)}{F_{n}(z)},$ (3.19)
$\displaystyle\phi(P)-\phi(P^{\ast})=\frac{2y}{F_{n}(z)}.$ (3.20)

Proof. Equation (3.17) follows using the definition (3) of $\phi$ as well as
relations (2.14)-(2.16). Relations (3.18)-(3.20) are clear from (2.18), (3.4),
and (3). $\square$

The properties of $\psi(P,x,x_{0})$ are summarized in the following lemma.

######  Lemma 3.2.

Suppose $(\ref{2.1})$, assume the $n$th stationary $HS2$ equation
$(\ref{2.34})$ holds, and let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\},$
$(x,x_{0})\in\mathbb{R}^{2}$. Then $\psi_{1}(P,x,x_{0}),\psi_{2}(P,x,x_{0})$
satisfy

$\displaystyle\psi_{1}(P,x,x_{0})=\Big{(}\frac{F_{n}(z,x)}{F_{n}(z,x_{0})}\Big{)}^{1/2}\mathrm{exp}\left(\frac{y}{z}\int_{x_{0}}^{x}F_{n}(z,x^{\prime})^{-1}dx^{\prime}\right),$
(3.21)
$\displaystyle\psi_{1}(P,x,x_{0})\psi_{1}(P^{\ast},x,x_{0})=\frac{F_{n}(z,x)}{F_{n}(z,x_{0})},$
(3.22)
$\displaystyle\psi_{2}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})=-\frac{H_{n}(z,x)}{z^{2}F_{n}(z,x_{0})},$
(3.23)
$\displaystyle\psi_{1}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})+\psi_{1}(P^{\ast},x,x_{0})\psi_{2}(P,x,x_{0})=2\frac{G_{n}(z,x)}{F_{n}(z,x_{0})},$
(3.24)
$\displaystyle\psi_{1}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})-\psi_{1}(P^{\ast},x,x_{0})\psi_{2}(P,x,x_{0})=\frac{-2y}{zF_{n}(z,x_{0})}.$
(3.25)

Proof. Equation (3.21) is a consequence of (2.14), (3.8), and (3). Equation
(3.22) is clear from (3.21) and (3.23) is a consequence of (3.10), (3.18), and
(3.22). Equation (3.24) follows using (3.10), (3.19), and (3.22). Finally,
(3.25) follows from (3.10), (3.20), and (3.22). $\square$

In Lemma 3.2, we denote by

$\psi_{1}(P)=\psi_{1,+},~{}\psi_{1}(P^{\ast})=\psi_{1,-},~{}\psi_{2}(P)=\psi_{2,+},~{}\psi_{2}(P^{\ast})=\psi_{2,-},$

and then (3.22)-(3.25) imply

$(\psi_{1,+}\psi_{2,-}-\psi_{1,-}\psi_{2,+})^{2}=(\psi_{1,+}\psi_{2,-}+\psi_{1,-}\psi_{2,+})^{2}-4\psi_{1,+}\psi_{2,-}\psi_{1,-}\psi_{2,+},$
(3.26)

which is equivalent to the basic identity (2.18),
$z^{2}G_{n}^{2}+F_{n}H_{n}=R_{2n+2}$. This fact reveals the relations between
our approach and the algebro-geometric solutions of the HS2 hierarchy.

######  Remark 3.3.

The Baker-Akhiezer function $\psi$ of the stationary $HS2$ hierarchy is
formally analogous to that defined in the context of KdV or AKNS hierarchies.
However, its actual properties in a neighborhood of its essential singularity
will feature characteristic differences to standard Baker-Akhiezer functions
(cf. Remark 4.2).

Next, we derive Dubrovin-type equations, that is, first-order coupled systems
of differential equations that govern the dynamics of $\mu_{j}(x)$ and
$\nu_{l}(x)$ with respect to variations of $x$.

######  Lemma 3.4.

Assume $(\ref{2.1})$ and the $n$th stationary $HS2$ equation $(\ref{2.34})$
holds subject to the constraint $(\ref{2.26a})$.

  * _(i)_

Suppose that the zeros $\\{\mu_{j}(x)\\}_{j=0,\ldots,n}$ of $F_{n}(z,x)$
remain distinct for $x\in\Omega_{\mu},$ where
$\Omega_{\mu}\subseteq\mathbb{R}$ is an open interval, then
$\\{\mu_{j}(x)\\}_{j=0,\ldots,n}$ satisfy the system of differential
equations,

$\mu_{j,x}=4\frac{y(\hat{\mu}_{j})}{\mu_{j}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq j}^{n}(\mu_{j}(x)-\mu_{k}(x))^{-1},\quad
j=0,\ldots,n,$ (3.27)

with initial conditions

$\\{\hat{\mu}_{j}(x_{0})\\}_{j=0,\ldots,n}\in\mathcal{K}_{n},$ (3.28)

for some fixed $x_{0}\in\Omega_{\mu}$. The initial value problem
$(\ref{3.38})$, $(\ref{3.39})$ has a unique solution satisfying

$\hat{\mu}_{j}\in C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),\quad
j=0,\ldots,n.$ (3.29)

  * _(ii)_

Suppose that the zeros $\\{\nu_{l}(x)\\}_{l=0,\ldots,n}$ of $H_{n}(z,x)$
remain distinct for $x\in\Omega_{\nu},$ where
$\Omega_{\nu}\subseteq\mathbb{R}$ is an open interval, then
$\\{\nu_{l}(x)\\}_{l=0,\ldots,n}$ satisfy the system of differential
equations,

$\nu_{l,x}=-2\frac{(\rho^{2}+u_{xx}\nu_{l})y(\hat{\nu}_{l})}{h_{0}~{}\nu_{l}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq l}^{n}(\nu_{l}(x)-\nu_{k}(x))^{-1},\quad
l=0,\ldots,n,$ (3.30)

with initial conditions

$\\{\hat{\nu}_{l}(x_{0})\\}_{l=0,\ldots,n}\in\mathcal{K}_{n},$ (3.31)

for some fixed $x_{0}\in\Omega_{\nu}$. The initial value problem
$(\ref{3.41})$, $(\ref{3.42})$ has a unique solution satisfying

$\hat{\nu}_{l}\in C^{\infty}(\Omega_{\nu},\mathcal{K}_{n}),\quad
l=0,\ldots,n.$ (3.32)

Proof. It suffices to prove (3.27) and (3.29) since the proof of (3.30) and
(3.32) follow in an identical manner. Differentiating (3.11) with respect to
$x$ then yields

$F_{n,x}(\mu_{j})=-\frac{1}{2}\mu_{j,x}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq j}^{n}(\mu_{j}(x)-\mu_{k}(x)).$ (3.33)

On the other hand, taking into account equation (2.14), one finds

$F_{n,x}(\mu_{j})=2G_{n}(\mu_{j})=2\frac{y(\hat{\mu}_{j})}{-\mu_{j}}.$ (3.34)

Then combining equation (3.33) with (3.34) leads to (3.27). The proof of
smoothness assertion (3.29) is analogous to the KdV case in [21]. $\square$

Next, we turn to the trace formulas of the HS2 invariants, that is,
expressions of $f_{l}$ and $h_{l}$ in terms of symmetric functions of the
zeros $\mu_{j}$ and $\nu_{l}$ of $F_{n}$ and $H_{n}$, respectively. For
simplicity, we just record the simplest case.

######  Lemma 3.5.

Suppose $(\ref{2.1})$, assume the $n$th stationary $HS2$ equation
$(\ref{2.34})$ holds, and let $x\in\mathbb{R}.$ Then

$u(x)=\frac{1}{2}\sum_{j=0}^{n}\mu_{j}(x)-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}.$
(3.35)

Proof. Equation (3.35) follows by considering the coefficient of $z^{n}$ in
$F_{n}$ in (2.28) and (3.11), which yields

$-u+c_{1}=-\frac{1}{2}\sum_{j=0}^{n}\mu_{j}.$ (3.36)

The constant $c_{1}$ can be determined by a long straightforward calculation
considering the coefficient of $z^{2n+1}$ in (2.18), which results in

$c_{1}=-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}.$ (3.37)

##  4 Stationary algebro-geometric solutions of HS2 hierarchy

In this section, we obtain explicit Riemann theta function representations for
the meromorphic function $\phi$, and especially, for the solutions $u,\rho$ of
the stationary HS2 hierarchy.

We begin with the asymptotic properties of $\phi$ and $\psi_{j},j=1,2$.

######  Lemma 4.1.

Suppose $(\ref{2.1})$, assume the $n$th stationary $HS2$ equation
$(\ref{2.34})$ holds, and let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\},$
$(x,x_{0})\in\mathbb{R}^{2}$. Then

$\displaystyle\phi(P)\underset{\zeta\rightarrow 0}{=}-u_{x}(x)+O(\zeta),\quad
P\rightarrow P_{\infty_{\pm}},\quad\zeta=z^{-1},$ (4.1)
$\displaystyle\phi(P)\underset{\zeta\rightarrow
0}{=}i\rho(x)+\frac{iu_{xx}(x)-\rho_{x}(x)}{2\rho(x)}\zeta+O(\zeta^{2}),\quad
P\rightarrow P_{0},\quad\zeta=z,$ (4.2)

and

$\displaystyle\psi_{1}(P,x,x_{0})\underset{\zeta\rightarrow
0}{=}\mathrm{exp}\left((u(x_{0})-u(x))\zeta+O(\zeta^{2})\right),\quad
P\rightarrow P_{\infty_{\pm}},\quad\zeta=z^{-1},$ (4.3)
$\displaystyle\psi_{2}(P,x,x_{0})\underset{\zeta\rightarrow
0}{=}O(\zeta)~{}\mathrm{exp}\left((u(x_{0})-u(x))\zeta+O(\zeta^{2})\right),\quad
P\rightarrow P_{\infty_{\pm}},\quad\zeta=z^{-1},$ (4.4)
$\displaystyle\psi_{1}(P,x,x_{0})\underset{\zeta\rightarrow
0}{=}\mathrm{exp}\left(\frac{i}{\zeta}\int_{x_{0}}^{x}dx^{\prime}~{}\rho(x^{\prime})+O(1)\right),\quad
P\rightarrow P_{0},\quad\zeta=z,$ (4.5)
$\displaystyle\psi_{2}(P,x,x_{0})\underset{\zeta\rightarrow
0}{=}O(\zeta^{-1})~{}\mathrm{exp}\left(\frac{i}{\zeta}\int_{x_{0}}^{x}dx^{\prime}~{}\rho(x^{\prime})+O(1)\right),\quad
P\rightarrow P_{0},\quad\zeta=z.$ (4.6)

Proof. The existence of the asymptotic expansions of $\phi$ in terms of the
appropriate local coordinates $\zeta=z^{-1}$ near $P_{\infty_{\pm}}$ and
$\zeta=z$ near $P_{0}$ is clear from its explicit expression in (3). Next, we
compute the coefficients of these expansions utilizing the Riccati-type
equation (3.17). Indeed, inserting the ansatz

$\phi\underset{z\rightarrow\infty}{=}\phi_{0}+\phi_{1}z^{-1}+O(z^{-2})$ (4.7)

into (3.17) and comparing the same powers of $z$ then yields (4.1). Similarly,
inserting the ansatz

$\phi\underset{z\rightarrow 0}{=}\phi_{0}+\phi_{1}z+O(z^{2})$ (4.8)

into (3.17) and comparing the same powers of $z$ then yields (4.2). Finally,
expansions (4.3)-(4.6) follow from (3.8), (3.10), (4.1), and (4.2). $\square$

######  Remark 4.2.

We note the unusual fact that $P_{0}$, as opposed to $P_{\infty\pm}$, is the
essential singularity of $\psi_{j}$, $j=1,2$. In addition, one easily finds
the leading-order exponential term in $\psi_{j}$, $j=1,2,$ near $P_{0}$ is
$x$-dependent, which makes matters worse. This is in sharp contrast to
standard Baker-Akhiezer functions that typically feature a linear behavior
with respect to $x$ in connection with their essential singularities of the
type $\mathrm{exp}(c(x-x_{0})\zeta^{-1})$ near $\zeta=0$.

Next, we introduce the holomorphic differentials $\eta_{l}(P)$ on
$\mathcal{K}_{n}$

$\eta_{l}(P)=\frac{a~{}z^{l-1}}{y(P)}dz,\quad l=1,\ldots,n,$ (4.9)

and choose a homology basis $\\{a_{j},b_{j}\\}_{j=1}^{n}$ on $\mathcal{K}_{n}$
in such a way that the intersection matrix of the cycles satisfies

$a_{j}\circ b_{k}=\delta_{j,k},\quad a_{j}\circ a_{k}=0,\quad b_{j}\circ
b_{k}=0,\quad j,k=1,\ldots,n.$

Associated with $\mathcal{K}_{n}$, one introduces an invertible matrix $E\in
GL(n,\mathbb{C})$

$\begin{split}&E=(E_{j,k})_{n\times n},\quad E_{j,k}=\int_{a_{k}}\eta_{j},\\\
&\underline{c}(k)=(c_{1}(k),\ldots,c_{n}(k)),\quad
c_{j}(k)=(E^{-1})_{j,k},\end{split}$ (4.10)

and the normalized holomorphic differentials

$\omega_{j}=\sum_{l=1}^{n}c_{j}(l)\eta_{l},\quad\int_{a_{k}}\omega_{j}=\delta_{j,k},\quad\int_{b_{k}}\omega_{j}=\tau_{j,k},\quad
j,k=1,\ldots,n.$ (4.11)

Apparently, the matrix $\tau$ is symmetric and has a positive-definite
imaginary part.

We choose a fixed base point
$Q_{0}\in\mathcal{K}_{n}\setminus\\{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)\\}$. The
Abel maps $\underline{A}_{Q_{0}}(\cdot)$ and
$\underline{\alpha}_{Q_{0}}(\cdot)$ are defined by

$\begin{split}&\underline{A}_{Q_{0}}:\mathcal{K}_{n}\rightarrow
J(\mathcal{K}_{n})=\mathbb{C}^{n}/L_{n},\\\
&P\mapsto\underline{A}_{Q_{0}}(P)=(A_{Q_{0},1}(P),\ldots,A_{Q_{0},n}(P))=\left(\int_{Q_{0}}^{P}\omega_{1},\ldots,\int_{Q_{0}}^{P}\omega_{n}\right)(\mathrm{mod}~{}L_{n})\end{split}$
(4.12)

and

$\begin{split}&\underline{\alpha}_{Q_{0}}:\mathrm{Div}(\mathcal{K}_{n})\rightarrow
J(\mathcal{K}_{n}),\\\
&\mathcal{D}\mapsto\underline{\alpha}_{Q_{0}}(\mathcal{D})=\sum_{P\in\mathcal{K}_{n}}\mathcal{D}(P)\underline{A}_{Q_{0}}(P),\end{split}$
(4.13)

where
$L_{n}=\\{\underline{z}\in\mathbb{C}^{n}|~{}\underline{z}=\underline{N}+\tau\underline{M},~{}\underline{N},~{}\underline{M}\in\mathbb{Z}^{n}\\}.$

Let $\Phi_{n}^{(j)}(\bar{\mu}),$ $\Psi_{n+1}(\bar{\mu})$ be the symmetric
functions of $\\{\mu_{j}\\}_{j=0,\ldots,n},$

$\Phi_{n}^{(j)}(\bar{\mu})=(-1)^{n}\prod_{\scriptstyle p=0\atop\scriptstyle
p\neq
j}^{n}\mu_{p},\quad\Psi_{n+1}(\bar{\mu})=(-1)^{n+1}\prod_{p=0}^{n}\mu_{p}.$
(4.14)

Here,
$\bar{\mu}(x)=(\mu_{0}(x),\mu_{1}(x),\ldots,\mu_{n}(x))=\mu_{0}(x)\underline{\mu}(x).$

The following result shows the nonlinearity of the Abel map with respect to
the variable $x$, which indicates a characteristic difference between the HS2
hierarchy and other completely integrable systems such as the KdV and AKNS
hierarchies.

######  Theorem 4.3.

Assume $(\ref{2.26a})$ and suppose that
$\\{\hat{\mu}_{j}(x)\\}_{j=0,\ldots,n}$ satisfies the stationary Dubrovin
equations $(\ref{3.38})$ on an open interval $\Omega_{\mu}\subseteq\mathbb{R}$
such that $\mu_{j}(x),j=0,\ldots,n,$ remain distinct and nonzero for
$x\in\Omega_{\mu}.$ Introducing the associated divisor
$\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$, one computes

$\partial_{x}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)})=-\frac{4a}{\Psi_{n+1}(\bar{\mu}(x))}\underline{c}(1),\quad
x\in\Omega_{\mu}.$ (4.15)

In particular, the Abel map does not linearize the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$ on $\Omega_{\mu}$.

Proof. Let $x\in\Omega_{\mu}.$ Then, using

$\frac{1}{\mu_{j}}=\frac{\prod_{\scriptstyle p=0\atop\scriptstyle p\neq
j}^{n}\mu_{p}}{\prod_{p=0}^{n}\mu_{p}}=-\frac{\Phi_{n}^{(j)}(\bar{\mu})}{\Psi_{n+1}(\bar{\mu})},\quad
j=0,\ldots,n,$ (4.16)

one obtains

$\displaystyle\partial_{x}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)})$
$\displaystyle=\partial_{x}\left(\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}}\underline{\omega}\right)=\sum_{j=0}^{n}\mu_{j,x}\sum_{k=1}^{n}\underline{c}(k)\frac{a~{}\mu_{j}^{k-1}}{y(\hat{\mu}_{j})}$
$\displaystyle=\sum_{j=0}^{n}\sum_{k=1}^{n}\frac{4a~{}\mu_{j}^{k-1}}{\mu_{j}}\frac{1}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq j}^{n}(\mu_{j}-\mu_{l})}\underline{c}(k)$
$\displaystyle=-\frac{4a}{\Psi_{n+1}(\bar{\mu})}\sum_{j=0}^{n}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq j}^{n}(\mu_{j}-\mu_{l})}\Phi_{n}^{(j)}(\bar{\mu})$
$\displaystyle=-\frac{4a}{\Psi_{n+1}(\bar{\mu})}\sum_{j=0}^{n}\sum_{k=1}^{n}\underline{c}(k)(U_{n+1}(\bar{\mu}))_{k,j}(U_{n+1}(\bar{\mu}))_{j,1}^{-1}$
$\displaystyle=-\frac{4a}{\Psi_{n+1}(\bar{\mu})}\sum_{k=1}^{n}\underline{c}(k)\delta_{k,1}$
$\displaystyle=-\frac{4a}{\Psi_{n+1}(\bar{\mu})}\underline{c}(1),$ (4.17)

where we used the notation
$\underline{\omega}=(\omega_{1},\ldots,\omega_{n})$, and the relations
(cf.(E.13), (E.14) [21]),

$U_{n+1}(\bar{\mu})=\left(\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq j}^{n}(\mu_{j}-\mu_{l})}\right)_{\scriptstyle
j=0\atop\scriptstyle k=1}^{n},\quad
U_{n+1}(\bar{\mu})^{-1}=\left(\Phi_{n}^{(j)}(\bar{\mu})\right)_{\scriptstyle
j=0}^{n}.$ (4.18)

The analogous results hold for the corresponding divisor
$\mathcal{D}_{\hat{\nu}_{0}(x)\underline{\hat{\nu}}(x)}$ associated with
$\phi(P,x).$

Next, we introduce

$\begin{split}&~{}~{}\underline{\widehat{B}}_{Q_{0}}:\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\}\rightarrow\mathbb{C}^{n},\\\
&~{}~{}~{}~{}~{}P\mapsto\underline{\widehat{B}}_{Q_{0}}(P)=(\widehat{B}_{Q_{0},1},\ldots,\widehat{B}_{Q_{0},n})\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\left\\{\begin{array}[]{ll}\int_{Q_{0}}^{P}\tilde{\omega}_{P_{\infty_{+}},P_{\infty_{-}}}^{(3)},&\hbox{$n=1$,}\\\
\Big{(}\int_{Q_{0}}^{P}\eta_{2},\ldots,\int_{Q_{0}}^{P}\eta_{n},\int_{Q_{0}}^{P}\tilde{\omega}_{P_{\infty_{+}},P_{\infty_{-}}}^{(3)}\Big{)},&\hbox{$n\geq
2$,}\end{array}\right.\end{split}$ (4.19)

where $\tilde{\omega}_{P_{\infty_{+}},P_{\infty_{-}}}^{(3)}=a~{}z^{n}dz/y(P)$
(cf.(F.53) [21]) and

$\begin{split}&\underline{\hat{\beta}}_{Q_{0}}:\mathrm{Sym}^{n}(\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\})\rightarrow\mathbb{C}^{n},\\\
&\mathcal{D}_{\underline{Q}}\mapsto\underline{\hat{\beta}}_{Q_{0}}(\mathcal{D}_{\underline{Q}})=\sum_{j=1}^{n}\underline{\widehat{B}}_{Q_{0}}(Q_{j}),\quad\underline{Q}=\\{Q_{1},\ldots,Q_{n}\\}\in\mathrm{Sym}^{n}(\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\}),\end{split}$
(4.20)

choosing identical paths of integration from $Q_{0}$ to $P$ in all integrals
in (4.19) and (4.20). Then, one obtains the following result.

######  Corollary 4.4.

Assume $(\ref{2.26a})$ and suppose that
$\\{\hat{\mu}_{j}(x)\\}_{j=0,\ldots,n}$ satisfies the stationary Dubrovin
equations $(\ref{3.38})$ on an open interval $\Omega_{\mu}\subseteq\mathbb{R}$
such that $\mu_{j}(x),j=0,\ldots,n,$ remain distinct and nonzero for
$x\in\Omega_{\mu}.$ Then one computes

$\displaystyle\partial_{x}\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}(x)}\eta_{1}=-\frac{4a}{\Psi_{n+1}(\bar{\mu}(x))},\quad
x\in\Omega_{\mu},$ (4.21)
$\displaystyle\partial_{x}\underline{\hat{\beta}}(\mathcal{D}_{\underline{\hat{\mu}}(x)})=\left\\{\begin{array}[]{ll}4a,&\hbox{$n=1$,}\\\
4a~{}(0,\ldots,0,1),&\hbox{$n\geq 2$,}\end{array}\right.\quad
x\in\Omega_{\mu}.$ (4.24)

Proof. Equation (4.21) is a special case (4.15). Equation (4.24) follows from
(4). $\square$

The fact that the Abel map does not provide the proper change of variables to
linearize the divisor $\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$
in the HS2 context is in sharp contrast to standard integrable soliton
equations such as the KdV and AKNS hierarchies. However, the change of
variables

$x\mapsto\tilde{x}=\int^{x}dx^{\prime}\Big{(}\frac{4a}{\Psi_{n+1}(\bar{\mu}(x^{\prime}))}\Big{)}$
(4.25)

linearizes the Abel map
$\underline{A}_{Q_{0}}(\mathcal{D}_{\hat{\tilde{\mu}}_{0}(\tilde{x})\underline{\hat{\tilde{\mu}}}(\tilde{x})}),$
$\tilde{\mu}_{j}(\tilde{x})=\mu_{j}(x),j=0,\ldots,n.$ The intricate relation
between the variable $x$ and $\tilde{x}$ is detailed in (4.5).

Next, given the Riemann surface $\mathcal{K}_{n}$ and the homology basis
$\\{a_{j},b_{j}\\}_{j=1,\ldots,n}$, one introduces the Riemann theta function
by

$\theta(\underline{z})=\sum_{\underline{n}\in\mathbb{Z}^{n}}\exp\Big{(}2\pi
i(\underline{n},\underline{z})+\pi
i(\underline{n},\tau\underline{n})\Big{)},~{}~{}\underline{z}\in\mathbb{C}^{n},$

where $(\underline{A},\underline{B})=\sum_{j=1}^{n}\overline{A}_{j}B_{j}$
denotes the scalar product in $\mathbb{C}^{n}.$

Let

$\omega_{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)}^{(3)}(P)=\frac{a}{y}\prod_{j=1}^{n}(z-\lambda_{j})dz$
(4.26)

be the normalized differential of the third kind holomorphic on
$\mathcal{K}_{n}\setminus\\{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)\\}$ with simple
poles at $\hat{\mu}_{0}(x)$ and $\hat{\nu}_{0}(x)$ and residues $1$ and $-1$,
respectively,

$\displaystyle\omega_{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}(\zeta^{-1}+O(1))d\zeta,\quad\textrm{as $P\rightarrow\hat{\mu}_{0}(x),$}$
(4.27)
$\displaystyle\omega_{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}(-\zeta^{-1}+O(1))d\zeta,\quad\textrm{as
$P\rightarrow\hat{\nu}_{0}(x),$}$ (4.28)

where $\zeta$ in (4.27) and (4.28) denotes the corresponding local coordinate
near $\hat{\mu}_{0}(x)$ and $\hat{\nu}_{0}(x)$ (cf. Sect.3, also Appendix C
[21]). The constants $\\{\lambda_{j}\\}_{j=1,\ldots,n}$ in (4.26) are
determined by the normalization condition

$\int_{a_{k}}\omega_{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)}^{(3)}=0,\quad
k=1,\ldots,n.$

Then

$\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}\mathrm{ln}\zeta+e_{0}+O(\zeta),\quad\textrm{as
$P\rightarrow\hat{\mu}_{0}(x),$ }$ (4.29)
$\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}-\mathrm{ln}\zeta+d_{0}+O(\zeta),\quad\textrm{as
$P\rightarrow\hat{\nu}_{0}(x),$}$ (4.30)

for some constants $e_{0},d_{0}\in\mathbb{C}$. We also record

$\underline{A}_{Q_{0}}(P)-\underline{A}_{Q_{0}}(P_{\infty_{\pm}})\underset{\zeta\rightarrow
0}{=}\pm\underline{U}\zeta+O(\zeta^{2}),\quad\textrm{as $P\rightarrow
P_{\infty_{\pm}},$}\quad\underline{U}=\underline{c}(n).$ (4.31)

In the following, it will be convenient to introduce the abbreviations

$\displaystyle\underline{z}(P,\underline{Q})=\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{Q}}),$
$\displaystyle
P\in\mathcal{K}_{n},\,\underline{Q}=(Q_{1},\ldots,Q_{n})\in\mathrm{Sym}^{n}(\mathcal{K}_{n}),$
(4.32)

where $\underline{\Xi}_{Q_{0}}$ is the vector of Riemann constants (cf.(A.45)
[21]). It turns out that $\underline{z}(\cdot,\underline{Q})$ is independent
of the choice of base point $Q_{0}$ (cf.(A.52), (A.53) [21]).

Based on above preparations, we will give explicit representations for the
meromorphic function $\phi$ and the stationary HS2 solutions $u,\rho$ in terms
of the Riemann theta function associated with $\mathcal{K}_{n}$.

######  Theorem 4.5.

Suppose $(\ref{2.1})$, and assume the $n$th stationary $HS2$ equation
$(\ref{2.34})$ holds on $\Omega$ subject to the constraint $(\ref{2.26a})$.
Moreover, let $P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{0}\\}$ and
$x\in\Omega$, where $\Omega\subseteq\mathbb{R}$ is an open interval. In
addition, suppose that $\mathcal{D}_{\underline{\hat{\mu}}(x)}$, or
equivalently, $\mathcal{D}_{\underline{\hat{\nu}}(x)}$ is nonspecial for
$x\in\Omega$. Then, $\phi$, $u,$ and $\rho$ admit the following
representations

$\displaystyle\phi(P,x)=i\rho(x)\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(x)))\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x)))}{\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x)))\theta(\underline{z}(P,\underline{\hat{\mu}}(x)))}\mathrm{exp}\left(e_{0}-\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)}^{(3)}\right),$
(4.33) $\displaystyle
u(x)=-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}+\frac{1}{2}\sum_{j=1}^{n}\lambda_{j}-\frac{1}{2}\sum_{j=1}^{n}U_{j}\partial_{\omega_{j}}\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x))+\underline{\omega})}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x))+\underline{\omega})}\right)\Big{|}_{\underline{\omega}=0},$
(4.34)
$\displaystyle\rho(x)=iu_{x}(x)\frac{\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x)))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x)))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x)))\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x)))}.$
(4.35)

Moreover, let $\widetilde{\Omega}\subseteq\Omega$ be such that $\mu_{j},$
$j=0,\ldots,n,$ are nonvanishing on $\widetilde{\Omega}.$ Then, the constraint

$\displaystyle 4a(x-x_{0})$
$\displaystyle=-4a\int_{x_{0}}^{x}\frac{dx^{\prime}}{\prod_{k=0}^{n}\mu_{k}(x^{\prime})}\sum_{j=1}^{n}\Big{(}\int_{a_{j}}\tilde{\omega}_{P_{\infty_{+}}P_{\infty_{-}}}^{(3)}\Big{)}c_{j}(1)$
$\displaystyle+\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x_{0})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x)))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x_{0})))\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x)))}\right),\
\ x,x_{0}\in\widetilde{\Omega}$ (4.36)

holds, with

$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)})$
$\displaystyle=\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x_{0})\underline{\hat{\mu}}(x_{0})})-4a\int_{x_{0}}^{x}\frac{dx^{\prime}}{\Psi_{n+1}(\bar{\mu}(x^{\prime}))}\underline{c}(1)$
$\displaystyle=\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x_{0})})-\underline{c}(1)(\tilde{x}-\tilde{x}_{0}),\
\ x\in\widetilde{\Omega}.$ (4.37)

Proof. First, we temporarily assume that

$\mu_{j}(x)\neq\mu_{j^{\prime}}(x),\quad\nu_{k}(x)\neq\nu_{k^{\prime}}(x)\quad\textrm{for
$j\neq j^{\prime},k\neq k^{\prime}$ and $x\in\widetilde{\Omega}$},$ (4.38)

for appropriate $\widetilde{\Omega}\subseteq\Omega$. Since by (3.15),
$\mathcal{D}_{\hat{\nu}_{0}\underline{\hat{\nu}}}\sim\mathcal{D}_{\hat{\mu}_{0}\underline{\hat{\mu}}}$,
and $(\hat{\mu}_{0})^{\ast}\notin\\{\hat{\mu}_{1},\ldots,\hat{\mu}_{n}\\}$ by
hypothesis, one can use Theorem A.31 [21] to conclude that
$\mathcal{D}_{\underline{\hat{\nu}}}\in\textrm{Sym}^{n}(\mathcal{K}_{n})$ is
nonspecial. This argument is of course symmetric with respect to
$\underline{\hat{\mu}}$ and $\underline{\hat{\nu}}$. Thus,
$\mathcal{D}_{\underline{\hat{\mu}}}$ is nonspecial if and only if
$\mathcal{D}_{\underline{\hat{\nu}}}$ is.

Next, we derive the representations of $\phi$, $u$, and $\rho$ in terms of the
Riemann theta function. A special case of Riemann’s vanishing theorem (cf.
Theorem A.26 [21]) yields

$\theta(\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{Q}}))=0\quad\textrm{
if and only if $P\in\\{Q_{1},\ldots,Q_{n}\\}$}.$ (4.39)

Therefore, the divisor (3.15) shows that $\phi(P,x)$ has expression of the
type

$C(x)\frac{\theta(\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x)}))}{\theta(\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)}))}\mathrm{exp}\Big{(}e_{0}-\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x),\hat{\nu}_{0}(x)}^{(3)}\Big{)},$
(4.40)

where $C(x)$ is independent of $P\in\mathcal{K}_{n}$. Then taking into account
the asymptotic expansion of $\phi(P,x)$ near $P_{0}$ in (4.2), we obtain
(4.33). The representation (4.34) for $u$ on $\widetilde{\Omega}$ follows from
trace formula (3.35) and the expression (F.59)[21] for
$\sum_{j=0}^{n}\mu_{j}$. The representation (4.35) for $\rho$ on
$\widetilde{\Omega}$ is clear from (4.1) and (4.33). By continuity, (4.33),
(4.34), and (4.35) extend from $\widetilde{\Omega}$ to $\Omega$. Assuming
$\mu_{j}\neq 0$, $j=0,\ldots,n$, the constraint (4.5) follows by combining
(4.21), (4.24), and (F.58) [21]. Equation (4.5) is clear from (4.15). Finally,
(4.5) and (4.5) extend to $\widetilde{\Omega}$ by continuity. $\square$

######  Remark 4.6.

We note that the stationary $HS2$ solutions $u,\rho$ in $(\ref{4.35})$ and
$(\ref{4.35a1})$ are meromorphic quasi-periodic functions with respect to the
new variable $\tilde{x}$ in $(\ref{4.33})$. In addition, the Abel map in
$(\ref{4.37})$ linearizes the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$ with respect to
$\tilde{x}$ under the constraint $(\ref{4.36})$.

######  Remark 4.7.

Since by $(\ref{3.14})$ $\mathcal{D}_{\hat{\nu}_{0}\underline{\hat{\nu}}}$ and
$\mathcal{D}_{\hat{\mu}_{0}\underline{\hat{\mu}}}$ are linearly equivalent,
that is

$\underline{A}_{Q_{0}}(\hat{\mu}_{0}(x))+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)})=\underline{A}_{Q_{0}}(\hat{\nu}_{0}(x))+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x)}).$
(4.41)

Then one infers

$\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x)})=\underline{\Delta}+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)}),\quad\underline{\Delta}=\underline{A}_{\hat{\nu}_{0}(x)}(\hat{\mu}_{0}(x)).$
(4.42)

Hence, one can eliminate $\mathcal{D}_{\underline{\hat{\nu}}(x)}$ in
$(\ref{4.34})$, in terms of $\mathcal{D}_{\underline{\hat{\mu}}(x)}$ using

$\underline{z}(P,\underline{\hat{\nu}})=\underline{z}(P,\underline{\hat{\mu}})+\underline{\Delta},\quad
P\in\mathcal{K}_{n}.$ (4.43)

######  Remark 4.8.

We emphasized in Remark 4.2 that $\psi$ in $(\ref{3.6})$ and $(\ref{3.8})$
differs from standard Baker-Akhiezer functions. Hence, one can not expect the
usual theta function representation of $\psi_{j}$, $j=1,2,$ in terms of ratios
of theta functions times an exponential term containing a meromorphic
differential with a pole at the essential singularity of $\psi_{j}$ multiplied
by $(x-x_{0})$. However, using $(E.3)$ and
$(F.59)~{}\cite[cite]{[\@@bibref{}{15}{}{}]}$, one computes

$\displaystyle F_{n}(z)$
$\displaystyle=\frac{1}{2}z^{n+1}+\frac{1}{2}\sum_{k=0}^{n}\Psi_{n+1-k}(\bar{\mu}(x))z^{k}$
$\displaystyle=\frac{1}{2}z^{n+1}+\frac{1}{2}\sum_{k=1}^{n}\Big{(}\Psi_{n+1-k}(\underline{\lambda})-\sum_{j=1}^{n}c_{j}(k)\partial_{\omega_{j}}\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x))+\underline{\omega})}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x))+\underline{\omega})}\right)\Big{|}_{\underline{\omega}=0}\Big{)}z^{k}$
$\displaystyle=\frac{1}{2}\prod_{j=0}^{n}(z-\lambda_{j})-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}c_{j}(k)\partial_{\omega_{j}}\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x))+\underline{\omega})}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x))+\underline{\omega})}\right)\Big{|}_{\underline{\omega}=0}z^{k},$
(4.44)

and hence obtains the theta function representation of $\psi_{1}$ upon
inserting $(\ref{4.45})$ into $(\ref{3.26})$. Then, the corresponding theta
function representation of $\psi_{2}$ follows by $(\ref{3.8})$ and
$(\ref{4.34})$.

At the end of this section, we turn to the initial value problem for the
stationary HS2 hierarchy. We will show that the solvability of the Dubrovin
equations (3.27) on $\Omega_{\mu}\subseteq\mathbb{R}$ in fact implies the
stationary HS2 equation (2.33) on $\Omega_{\mu}$.

######  Theorem 4.9.

Fix $n\in\mathbb{N}_{0}$, assume $(\ref{2.26a})$, and suppose that
$\\{\hat{\mu}_{j}\\}_{j=0,\ldots,n}$ satisfies the stationary Dubrovin
equations $(\ref{3.38})$ on $\Omega_{\mu}$ such that $\mu_{j}$,
$j=0,\ldots,n$, remain distinct and nonzero on $\Omega_{\mu},$ where
$\Omega_{\mu}\subseteq\mathbb{R}$ is an open interval. Then, $u,\rho\in
C^{\infty}(\Omega_{\mu})$, defined by

$u=-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}+\frac{1}{2}\sum_{j=0}^{n}\mu_{j},$
(4.45)

and

$\rho^{2}=u_{x}^{2}+2uu_{xx}-\frac{d^{2}}{dx^{2}}\left(\Psi_{2}(\bar{\mu})-u\sum_{m=0}^{2n+1}E_{m}\right),$
(4.46)

satisfy the $n$th stationary $HS2$ equation $(\ref{2.34}),$ that is,

$\mathrm{s}\textrm{-}\mathrm{HS2}_{n}(u,\rho)=0~{}\textrm{on $\Omega_{\mu}.$}$
(4.47)

Proof. Given the solutions $\hat{\mu}_{j}=(\mu_{j},y(\hat{\mu}_{j}))\in
C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),j=0,\ldots,n$ of (3.27), we introduce

$\displaystyle\begin{split}&F_{n}(z)=\frac{1}{2}\prod_{j=0}^{n}(z-\mu_{j}),\\\
&G_{n}(z)=\frac{1}{2}F_{n,x}(z),\end{split}$ (4.48)

on $\mathbb{C}\times\Omega_{\mu}$. Taking into account (4.48), the Dubrovin
equations (3.27) imply

$y(\hat{\mu}_{j})=\frac{1}{4}\mu_{j}\mu_{j,x}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq
j}^{n}(\mu_{j}-\mu_{k})=-\frac{1}{2}\mu_{j}F_{n,x}(\mu_{j})=-\mu_{j}G_{n}(\mu_{j}).$
(4.49)

Hence

$R_{2n+2}(\mu_{j})^{2}-\mu_{j}^{2}G_{n}(\mu_{j})^{2}=y(\hat{\mu}_{j})^{2}-\mu_{j}^{2}G_{n}(\mu_{j})^{2}=0,\quad
j=0,\ldots,n.$ (4.50)

Next, we define a polynomial $H_{n}$ on $\mathbb{C}\times\Omega_{\mu}$ such
that

$R_{2n+2}(z)-z^{2}G_{n}(z)^{2}=F_{n}(z)H_{n}(z).$ (4.51)

Such a polynomial $H_{n}$ exists since the left-hand side of (4.51) vanishes
at $z=\mu_{j},~{}j=0,\cdots,n$, by (4.50). To determine the degree of $H_{n}$,
using (4.48), one computes

$R_{2n+2}(z)-z^{2}G_{n}(z)^{2}\underset{|z|\rightarrow\infty}{=}\frac{1}{2}h_{0}z^{2n+2}+O(z^{2n+1}).$
(4.52)

Then combining (4.48), (4.51), and (4.52), one infers that $H_{n}$ has degree
$n+1$ with respect to $z$. Hence, we may write

$H_{n}(z)=h_{0}\prod_{l=0}^{n}(z-\nu_{l}),\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$}.$ (4.53)

Next, one defines the polynomial $P_{n}$ by

$P_{n}(z)=H_{n}(z)+(\rho^{2}+u_{xx}z)F_{n}(z)+z^{2}G_{n,x}(z).$ (4.54)

Using (4.48) and (4.53), one infers that indeed $P_{n}$ has degree at most
$n$. Differentiating (4.51) with respect to $x$ yields

$2z^{2}G_{n}(z)G_{n,x}(z)+F_{n,x}(z)H_{n}(z)+F_{n}(z)H_{n,x}(z)=0.$ (4.55)

Then multiplying (4.54) by $G_{n}$ and replacing the term $G_{n}G_{n,x}$ with
(4.55) leads to

$\displaystyle
G_{n}(z)P_{n}(z)=F_{n}(z)((\rho^{2}+u_{xx}z)G_{n}(z)-\frac{1}{2}H_{n,x}(z))+(G_{n}(z)-\frac{1}{2}F_{n,x}(z))H_{n}(z),$
(4.56)

and hence

$G_{n}(\mu_{j})P_{n}(\mu_{j})=0,\quad j=0,\ldots,n$ (4.57)

on $\Omega_{\mu}$. Restricting $x\in\Omega_{\mu}$ temporarily to
$x\in\widetilde{\Omega}_{\mu}$, where

$\displaystyle\widetilde{\Omega}_{\mu}$ $\displaystyle=\\{x\in\Omega_{\mu}\mid
G(\mu_{j}(x),x)=-\frac{y(\hat{\mu}_{j}(x))}{\mu_{j}(x)}\neq
0,\,j=0,\ldots,n\\}$
$\displaystyle=\\{x\in\Omega_{\mu}\mid\mu_{j}(x)\notin\\{E_{m}\\}_{m=0,\ldots,2n+1},\,j=0,\ldots,n\\},$
(4.58)

one infers that

$P_{n}(\mu_{j}(x),x)=0,\quad j=0,\ldots,n,\,~{}x\in\widetilde{\Omega}_{\mu}.$
(4.59)

Since $P_{n}(z)$ has degree at most $n$, (4.59) implies

$P_{n}=0\quad\textrm{on $\mathbb{C}\times\widetilde{\Omega}_{\mu}$},$ (4.60)

and hence (2.16) holds, that is,

$z^{2}G_{n,x}(z)=-H_{n}(z)-(\rho^{2}+u_{xx}z)F_{n}(z)\quad$ (4.61)

$\textrm{on $\mathbb{C}\times\widetilde{\Omega}_{\mu}$}.$ Inserting (4.61) and
(4.48) into (4.55) yields

$F_{n}(z)(H_{n,x}(z)-2(\rho^{2}+u_{xx}z)G_{n}(z))=0,$ (4.62)

namely

$H_{n,x}(z)=2(\rho^{2}+u_{xx}z)G_{n}(z)$ (4.63)

on $\mathbb{C}\times\widetilde{\Omega}_{\mu}$. Thus, we obtain the fundamental
equations (2.14)-(2.16) and (2.18) on
$\mathbb{C}\times\widetilde{\Omega}_{\mu}$. In order to extend these results
to $\Omega_{\mu}$, we next investigate the case where $\hat{\mu}_{j}$ hits a
branch point $(E_{m_{0}},0)$. Hence, we suppose

$\mu_{j_{1}}(x)\rightarrow E_{m_{0}}\quad\textrm{as $x\rightarrow
x_{0}\in\Omega_{\mu}$}$ (4.64)

for some $j_{1}\in\\{0,\ldots,n\\},\,m_{0}\in\\{0,\ldots,2n+1\\}$. Introducing

$\begin{split}&\zeta_{j_{1}}(x)=\sigma(\mu_{j_{1}}(x)-E_{m_{0}})^{1/2},\quad\sigma=\pm
1,\quad\\\ &\mu_{j_{1}}(x)=E_{m_{0}}+\zeta_{j_{1}}(x)^{2},\end{split}$ (4.65)

for some $x$ in an open interval centered near $x_{0}$, the Dubrovin equation
(3.27) for $\mu_{j_{1}}$ becomes

$\displaystyle\zeta_{j_{1},x}(x)$
$\displaystyle=c(\sigma)\frac{2a}{E_{m_{0}}}\Big{(}\prod_{\scriptstyle
m=0\atop\scriptstyle m\neq m_{0}}^{2n+1}(E_{m_{0}}-E_{m})\Big{)}^{1/2}$
$\displaystyle~{}~{}~{}~{}~{}\times\prod_{\scriptstyle k=0\atop\scriptstyle
k\neq j_{1}}^{n}(E_{m_{0}}-\mu_{k}(x))^{-1}(1+O(\zeta_{j_{1}}(x)^{2}))$ (4.66)

for some $|c(\sigma)|=1$. Hence, (4.60)-(4.63) extend to $\Omega_{\mu}$ by
continuity. We have now established relations (2.14)-(2.16) on
$\mathbb{C}\times\Omega_{\mu}$, and one can proceed as in Section 2 to obtain
(4.47). $\square$

######  Remark 4.10.

Although we formulated Theorem 4.9 in terms of $\\{\mu_{j}\\}_{j=0,\ldots,n}$
only, the analogous result (and strategy of proof) obviously works in terms of
$\\{\nu_{j}\\}_{j=0,\ldots,n}$.

######  Remark 4.11.

A closer look at Theorem 4.9 reveals that $u,\rho$ are uniquely determined in
an open neighborhood $\Omega$ of $x_{0}$ by $\mathcal{K}_{n}$ and the initial
condition
$(\hat{\mu}_{0}(x_{0}),\hat{\mu}_{1}(x_{0}),\ldots,\hat{\mu}_{n}(x_{0}))$, or
equivalently, by the auxiliary divisor
$\mathcal{D}_{\hat{\mu}_{0}(x_{0})\underline{\hat{\mu}}(x_{0})}$ at $x=x_{0}$.
Conversely, given $\mathcal{K}_{n}$ and $u,\rho$ in an open neighborhood
$\Omega$ of $x_{0}$, one can construct the corresponding polynomial
$F_{n}(z,x)$, $G_{n}(z,x)$ and $H_{n}(z,x)$ for $x\in\Omega$, and then recover
the auxiliary divisor $\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$
for $x\in\Omega$ from the zeros of $F_{n}(z,x)$ and from $(\ref{3.10})$. In
this sense, once the curve $\mathcal{K}_{n}$ is fixed, elements of the
isospectral class of the $HS2$ potentials $u,\rho$ can be characterized by
nonspecial auxiliary divisors
$\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$.

##  5 The time-dependent HS2 formalism

In this section, we extend the algebro-geometric analysis of Section 3 to the
time-dependent HS2 hierarchy.

Throughout this section, we assume (LABEL:2.2) holds.

The time-dependent algebro-geometric initial value problem of the HS2
hierarchy is to solve the time-dependent $r$th HS2 flow with a stationary
solution of the $n$th equation as initial data in the hierarchy. More
precisely, given $n\in\mathbb{N}_{0}$, based on the solution
$u^{(0)},\rho^{(0)}$ of the $n$th stationary HS2 equation
$\textrm{s-HS2}_{n}(u^{(0)},\rho^{(0)})=0$ associated with $\mathcal{K}_{n}$
and a set of integration constants
$\\{c_{l}\\}_{l=1,\ldots,n}\subset\mathbb{C}$, we want to construct a solution
$u,\rho$ of the $r$th HS2 flow $\mathrm{HS2}_{r}(u,\rho)=0$ such that
$u(t_{0,r})=u^{(0)},$ $\rho(t_{0,r})=\rho^{(0)},$ for some
$t_{0,r}\in\mathbb{R},~{}r\in\mathbb{N}_{0}$.

To emphasize that the integration constants in the definitions of the
stationary and the time-dependent HS2 equations are independent of each other,
we indicate this by adding a tilde on all the time-dependent quantities.
Hence, we employ the notation $\widetilde{V}_{r},$ $\widetilde{F}_{r},$
$\widetilde{G}_{r},$ $\widetilde{H}_{r},$ $\tilde{f}_{s}$, $\tilde{g}_{s},$
$\tilde{h}_{s}$, $\tilde{c}_{s}$ in order to distinguish them from $V_{n},$
$F_{n},$ $G_{n},$ $H_{n},$ $f_{l},$ $g_{l},$ $h_{l}$, $c_{l}$ in the
following. In addition, we mark the individual $r$th HS2 flow by a separate
time variable $t_{r}\in\mathbb{R}$.

Summing up, we are seeking a solution $u,\rho$ of the time-dependent algebro-
geometric initial value problem

$\displaystyle\mathrm{HS2}_{r}(u,\rho)=\left(\begin{array}[]{c}-u_{xxt_{r}}+2u_{xx}\tilde{f}_{r+1,x}+u_{xxx}\tilde{f}_{r+1}+2\rho\rho_{x}\tilde{f}_{r}+2\rho^{2}\tilde{f}_{r,x}\\\
2\rho\rho_{t_{r}}-2\rho\rho_{x}\tilde{f}_{r+1}-2\rho^{2}\tilde{f}_{r+1,x}\\\
\end{array}\right)=0,$ (5.3)
$\displaystyle(u,\rho)|_{t_{r}=t_{0,r}}=(u^{(0)},\rho^{(0)}),$
$\displaystyle\textrm{s-HS2}_{n}(u^{(0)},\rho^{(0)})=\left(\begin{array}[]{c}2u_{xx}f_{n+1,x}+u_{xxx}f_{n+1}+2\rho\rho_{x}f_{n}+2\rho^{2}f_{n,x}\\\
-2\rho\rho_{x}f_{n+1}-2\rho^{2}f_{n+1,x}\\\ \end{array}\right)=0,$ (5.6)

for some $t_{0,r}\in\mathbb{R},$ $n,r\in\mathbb{N}_{0}$, where $u=u(x,t_{r}),$
$\rho=\rho(x,t_{r})$ satisfy (LABEL:2.2), and the curve $\mathcal{K}_{n}$ is
associated with the initial data $(u^{(0)},\rho^{(0)})$ in (5.6). Noticing
that the HS2 flows are isospectral, we further assume that (5.6) holds not
only for $t_{r}=t_{0,r}$, but also for all $t_{r}\in\mathbb{R}$. Hence, we
start with the zero-curvature equations

$U_{t_{r}}-\widetilde{V}_{r,x}+[U,\widetilde{V}_{r}]=0,$ (5.7)
$-V_{n,x}+[U,V_{n}]=0,$ (5.8)

where

$\begin{split}&U(z)=\left(\begin{array}[]{cc}0&1\\\
-z^{-2}\rho^{2}-z^{-1}u_{xx}&0\\\ \end{array}\right),\\\
&V_{n}(z)=\left(\begin{array}[]{cc}-G_{n}(z)&F_{n}(z)\\\
z^{-2}H_{n}(z)&G_{n}(z)\\\ \end{array}\right),\\\
&\widetilde{V}_{r}(z)=\left(\begin{array}[]{cc}-\widetilde{G}_{r}(z)&\widetilde{F}_{r}(z)\\\
z^{-2}\widetilde{H}_{r}(z)&\widetilde{G}_{r}(z)\\\
\end{array}\right),\end{split}$ (5.9)

and

$\displaystyle
F_{n}(z)=\sum_{l=0}^{n+1}f_{l}z^{n+1-l}=f_{0}\prod_{j=0}^{n}(z-\mu_{j}),$
(5.10) $\displaystyle G_{n}(z)=\sum_{l=0}^{n}g_{l}z^{n-l},$ (5.11)
$\displaystyle
H_{n}(z)=\sum_{l=0}^{n+1}h_{l}z^{n+1-l}=h_{0}\prod_{l=0}^{n}(z-\nu_{l}),$
(5.12)
$\displaystyle\widetilde{F}_{r}(z)=\sum_{s=0}^{r+1}\tilde{f}_{s}z^{r+1-s},$
(5.13) $\displaystyle\widetilde{G}_{r}(z)=\sum_{s=0}^{r}\tilde{g}_{s}z^{r-s},$
(5.14)
$\displaystyle\widetilde{H}_{r}(z)=\sum_{s=0}^{r+1}\tilde{h}_{s}z^{r+1-s},$
(5.15)

for fixed $n,r\in\mathbb{N}_{0}$. Here, $\\{f_{l}\\}_{l=0,\ldots,n+1},$
$\\{g_{l}\\}_{l=0,\ldots,n}$, $\\{h_{l}\\}_{l=0,\ldots,n+1}$,
$\\{\tilde{f}_{s}\\}_{s=0,\ldots,r+1},$ $\\{\tilde{g}_{s}\\}_{s=0,\ldots,r}$,
and $\\{\tilde{h}_{s}\\}_{s=0,\ldots,r+1}$ are defined as in (2.3), with
$u(x),\rho(x)$ replaced by $u(x,t_{r}),\rho(x,t_{r})$ etc., and with
appropriate integration constants. Explicitly, (5.7) and (5.8) are equivalent
to

$\displaystyle-2\rho\rho_{t_{r}}-zu_{xxt_{r}}-\widetilde{H}_{r,x}+2(\rho^{2}+zu_{xx})\widetilde{G}_{r}=0,$
(5.16) $\displaystyle\widetilde{F}_{r,x}=2\widetilde{G}_{r},$ (5.17)
$\displaystyle
z^{2}\widetilde{G}_{r,x}=-\widetilde{H}_{r}-(\rho^{2}+zu_{xx})\widetilde{F}_{r}$
(5.18)

and

$\displaystyle F_{n,x}=2G_{n},$ (5.19) $\displaystyle
H_{n,x}=2(\rho^{2}+zu_{xx})G_{n},$ (5.20) $\displaystyle
z^{2}G_{n,x}=-H_{n}-(\rho^{2}+zu_{xx})F_{n}.$ (5.21)

From (5.19)-(5.21), one finds

$\frac{d}{dx}\mathrm{det}(V_{n}(z))=-\frac{1}{z^{2}}\frac{d}{dx}\Big{(}z^{2}G_{n}(z)^{2}+F_{n}(z)H_{n}(z)\Big{)}=0,$
(5.22)

and meanwhile (see Lemma 5.2)

$\frac{d}{dt_{r}}\mathrm{det}(V_{n}(z))=-\frac{1}{z^{2}}\frac{d}{dt_{r}}\Big{(}z^{2}G_{n}(z)^{2}+F_{n}(z)H_{n}(z)\Big{)}=0.$
(5.23)

Hence, $z^{2}G_{n}(z)^{2}+F_{n}(z)H_{n}(z)$ is independent of variables both
$x$ and $t_{r}$, which implies the fundamental identity (2.18) holds,

$z^{2}G_{n}(z)^{2}+F_{n}(z)H_{n}(z)=R_{2n+2}(z),$ (5.24)

and the hyperelliptic curve $\mathcal{K}_{n}$ is still given by (2.26).

Next, we define the time-dependent Baker-Akhiezer function
$\psi(P,x,x_{0},t_{r},t_{0,r})$ on
$\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\}$ by

$\begin{split}&\psi(P,x,x_{0},t_{r},t_{0,r})=\left(\begin{array}[]{c}\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\\\
\psi_{2}(P,x,x_{0},t_{r},t_{0,r})\\\ \end{array}\right),\\\
&\psi_{x}(P,x,x_{0},t_{r},t_{0,r})=U(u(x,t_{r}),\rho(x,t_{r}),z(P))\psi(P,x,x_{0},t_{r},t_{0,r}),\\\
&\psi_{t_{r}}(P,x,x_{0},t_{r},t_{0,r})=\widetilde{V}_{r}(u(x,t_{r}),\rho(x,t_{r}),z(P))\psi(P,x,x_{0},t_{r},t_{0,r}),\\\
&zV_{n}(u(x,t_{r}),\rho(x,t_{r}),z(P))\psi(P,x,x_{0},t_{r},t_{0,r})=y(P)\psi(P,x,x_{0},t_{r},t_{0,r}),\\\
&\psi_{1}(P,x_{0},x_{0},t_{0,r},t_{0,r})=1;\\\
&P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\},~{}(x,t_{r})\in\mathbb{R}^{2}.\end{split}$
(5.25)

Closely related to $\psi(P,x,x_{0},t_{r},t_{0,r})$ is the following
meromorphic function $\phi(P,x,t_{r})$ on $\mathcal{K}_{n}$ defined by

$\phi(P,x,t_{r})=z\frac{\psi_{1,x}(P,x,x_{0},t_{r},t_{0,r})}{\psi_{1}(P,x,x_{0},t_{r},t_{0,r})},\quad
P\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\},~{}(x,t_{r})\in\mathbb{R}^{2}$
(5.26)

such that

$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})$
$\displaystyle=\mathrm{exp}\Big{(}\int_{t_{0,r}}^{t_{r}}ds(z^{-1}\widetilde{F}_{r}(z,x_{0},s)\phi(P,x_{0},s)$
$\displaystyle~{}~{}~{}-\widetilde{G}_{r}(z,x_{0},s))+z^{-1}\int_{x_{0}}^{x}dx^{\prime}\phi(P,x^{\prime},t_{r})\Big{)},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\}.$
(5.27)

Then, using (5.25) and (5.26), one infers that

$\displaystyle\phi(P,x,t_{r})$
$\displaystyle=\frac{y+zG_{n}(z,x,t_{r})}{F_{n}(z,x,t_{r})}$
$\displaystyle=\frac{H_{n}(z,x,t_{r})}{y-zG_{n}(z,x,t_{r})},$ (5.28)

and

$\psi_{2}(P,x,x_{0},t_{r},t_{0,r})=\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\phi(P,x,t_{r})/z.$
(5.29)

In analogy to (3.12) and (3.13), we introduce

$\displaystyle\hat{\mu}_{j}(x,t_{r})=(\mu_{j}(x,t_{r}),-\mu_{j}(x,t_{r})G_{n}(\mu_{j}(x,t_{r}),x,t_{r}))\in\mathcal{K}_{n},$
(5.30)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}j=0,\ldots,n,~{}(x,t_{r})\in\mathbb{R}^{2},$
$\displaystyle\hat{\nu}_{l}(x,t_{r})=(\nu_{l}(x,t_{r}),\nu_{l}(x,t_{r})G_{n}(\nu_{l}(x,t_{r}),x,t_{r}))\in\mathcal{K}_{n},$
(5.31)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}l=0,\ldots,n,~{}(x,t_{r})\in\mathbb{R}^{2}.$

The regularity properties of $F_{n}$, $H_{n}$, $\mu_{j}$, and $\nu_{l}$ are
analogous to those in Section 3 due to assumptions (LABEL:2.2). Similar to
(3.15), the divisor $(\phi(P,x,t_{r}))$ of $\phi(P,x,t_{r})$ reads

$(\phi(P,x,t_{r}))=\mathcal{D}_{\hat{\nu}_{0}(x,t_{r})\underline{\hat{\nu}}(x,t_{r})}(P)-\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}(P)$
(5.32)

with

$\underline{\hat{\mu}}=\\{\hat{\mu}_{1},\ldots,\hat{\mu}_{n}\\},\quad\underline{\hat{\nu}}=\\{\hat{\nu}_{1},\ldots,\hat{\nu}_{n}\\}\in\mathrm{Sym}^{n}(\mathcal{K}_{n}).$
(5.33)

The properties of $\phi(P,x,t_{r})$ are summarized as follows.

######  Lemma 5.1.

Assume $(\ref{2.2})$ and suppose that $(\ref{5.3})$, $(\ref{5.4})$ hold.
Moreover, let $P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\}$
and $(x,t_{r})\in\mathbb{R}^{2}.$ Then $\phi$ satisfies

$\displaystyle\phi_{x}(P)+z^{-1}\phi(P)^{2}=-z^{-1}\rho^{2}-u_{xx},$ (5.34)
$\displaystyle\phi_{t_{r}}(P)=(-z\widetilde{G}_{r}(z)+\widetilde{F}_{r}(z)\phi(P))_{x}$
(5.35)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=z^{-1}\widetilde{H}_{r}(z)+(z^{-1}\rho^{2}+u_{xx})\widetilde{F}_{r}(z)+(\widetilde{F}_{r}(z)\phi(P))_{x},$
$\displaystyle\phi_{t_{r}}(P)=z^{-1}\widetilde{H}_{r}(z)+2\widetilde{G}_{r}(z)\phi(P)-z^{-1}\widetilde{F}_{r}(z)\phi(P)^{2},$
(5.36) $\displaystyle\phi(P)\phi(P^{\ast})=-\frac{H_{n}(z)}{F_{n}(z)},$ (5.37)
$\displaystyle\phi(P)+\phi(P^{\ast})=2\frac{zG_{n}(z)}{F_{n}(z)},$ (5.38)
$\displaystyle\phi(P)-\phi(P^{\ast})=\frac{2y}{F_{n}(z)}.$ (5.39)

Proof. Equations (5.34) and (5.37)-(5.39) can be proved as in Lemma 3.1. Using
(5.25) and (5.26), one infers that

$\displaystyle\phi_{t_{r}}=z(\mathrm{ln}\psi_{1})_{xt_{r}}=z(\mathrm{ln}\psi_{1})_{t_{r}x}=z\Big{(}\frac{\psi_{1,t_{r}}}{\psi_{1}}\Big{)}_{x}$
$\displaystyle~{}~{}~{}~{}=z\Big{(}\frac{-\widetilde{G}_{r}\psi_{1}+\widetilde{F}_{r}\psi_{2}}{\psi_{1}}\Big{)}_{x}=(-z\widetilde{G}_{r}+\widetilde{F}_{r}\phi)_{x}.$
(5.40)

Insertion of (5.18) into (5) then yields (5.35). To prove (5.36), one observes
that

$\displaystyle\phi_{t_{r}}=$ $\displaystyle
z\Big{(}\frac{\psi_{2}}{\psi_{1}}\Big{)}_{t_{r}}=z\Big{(}\frac{\psi_{2,t_{r}}}{\psi_{1}}-\frac{\psi_{2}\psi_{1,t_{r}}}{\psi_{1}^{2}}\Big{)}$
$\displaystyle=$ $\displaystyle
z\Big{(}\frac{z^{-2}\widetilde{H}_{r}\psi_{1}+\widetilde{G}_{r}\psi_{2}}{\psi_{1}}-z^{-1}\phi\frac{-\widetilde{G}_{r}\psi_{1}+\widetilde{F}_{r}\psi_{2}}{\psi_{1}}\Big{)}$
$\displaystyle=$ $\displaystyle
z^{-1}\widetilde{H}_{r}+2\widetilde{G}_{r}\phi-z^{-1}\widetilde{F}_{r}\phi^{2},$
(5.41)

which leads to (5.36). Alternatively, one can also insert (5.16)-(5.18) into
(5.35) to obtain (5.36). $\square$

Next, we determine the time evolution of $F_{n}$, $G_{n}$, and $H_{n}$, using
relations (5.16)-(5.18) and (5.19)-(5.21).

######  Lemma 5.2.

Assume $(\ref{2.2})$ and suppose that $(\ref{5.3})$, $(\ref{5.4})$ hold. Then

$\displaystyle F_{n,t_{r}}=2(G_{n}\widetilde{F}_{r}-\widetilde{G}_{r}F_{n}),$
(5.42) $\displaystyle
z^{2}G_{n,t_{r}}=\widetilde{H}_{r}F_{n}-H_{n}\widetilde{F}_{r},$ (5.43)
$\displaystyle H_{n,t_{r}}=2(H_{n}\widetilde{G}_{r}-G_{n}\widetilde{H}_{r}).$
(5.44)

Equations $(\ref{5.38})$–$(\ref{5.40})$ are equivalent to

$-V_{n,t_{r}}+[\widetilde{V}_{r},V_{n}]=0.$ (5.45)

Proof. Differentiating (5.39) with respect to $t_{r}$ naturally yields

$(\phi(P)-\phi(P^{\ast}))_{t_{r}}=-2yF_{n,t_{r}}F_{n}^{-2}.$ (5.46)

On the other hand, using (5.36), (5.38), and (5.39), the left-hand side of
(5.46) can be expressed as

$\displaystyle\phi(P)_{t_{r}}-\phi(P^{\ast})_{t_{r}}=$ $\displaystyle
2\widetilde{G}_{r}(\phi(P)-\phi(P^{\ast}))-z^{-1}\widetilde{F}_{r}(\phi(P)^{2}-\phi(P^{\ast})^{2})$
$\displaystyle=$ $\displaystyle
4y(\widetilde{G}_{r}F_{n}-\widetilde{F}_{r}G_{n})F_{n}^{-2}.$ (5.47)

Combining (5.46) and (5) then proves (5.42). Similarly, differentiating (5.38)
with respect to $t_{r}$, one finds

$(\phi(P)+\phi(P^{\ast}))_{t_{r}}=2z(G_{n,t_{r}}F_{n}-G_{n}F_{n,t_{r}})F_{n}^{-2}.$
(5.48)

Meanwhile, the left-hand side of (5.48) also equals

$\displaystyle\phi(P)_{t_{r}}+\phi(P^{\ast})_{t_{r}}=$ $\displaystyle
2\widetilde{G}_{r}(\phi(P)+\phi(P^{\ast}))-z^{-1}\widetilde{F}_{r}(\phi(P)^{2}+\phi(P^{\ast})^{2})+2z^{-1}\widetilde{H}_{r}$
$\displaystyle=$
$\displaystyle-2zG_{n}F_{n}^{-2}F_{n,t_{r}}+2z^{-1}F_{n}^{-1}(\widetilde{H}_{r}F_{n}-\widetilde{F}_{r}H_{n}),$
(5.49)

using (5.36), (5.37), and (5.38). Equation (5.43) is clear from (5.48) and
(5). Then, (5.44) follows by differentiating (2.18), that is,
$z^{2}G_{n}^{2}+F_{n}H_{n}=R_{2n+2}(z)$, with respect to $t_{r}$, and using
(5.42) and (5.43). Finally, a direct calculation shows (5.45) holds. $\square$

Basic properties of $\psi(P,x,x_{0},t_{r},t_{0,r})$ are summarized as follows.

######  Lemma 5.3.

Assume $(\ref{2.2})$ and suppose that $(\ref{5.3})$, $(\ref{5.4})$ hold.
Moreover, let $P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\}$
and $(x,x_{0},t_{r},t_{0,r})\in\mathbb{R}^{4}.$ Then, the Baker-Akhiezer
function $\psi$ satisfies

$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})=\Big{(}\frac{F_{n}(z,x,t_{r})}{F_{n}(z,x_{0},t_{0,r})}\Big{)}^{1/2}\mathrm{exp}\Bigg{(}\frac{y}{z}\int_{t_{0,r}}^{t_{r}}ds\widetilde{F}_{r}(z,x_{0},s)F_{n}(z,x_{0},s)^{-1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{y}{z}\int_{x_{0}}^{x}dx^{\prime}F_{n}(z,x^{\prime},t_{r})^{-1}\Bigg{)},$
(5.50)
$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\psi_{1}(P^{\ast},x,x_{0},t_{r},t_{0,r})=\frac{F_{n}(z,x,t_{r})}{F_{n}(z,x_{0},t_{0,r})},$
(5.51)
$\displaystyle\psi_{2}(P,x,x_{0},t_{r},t_{0,r})\psi_{2}(P^{\ast},x,x_{0},t_{r},t_{0,r})=-\frac{H_{n}(z,x,t_{r})}{z^{2}F_{n}(z,x_{0},t_{0,r})},$
(5.52)
$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\psi_{2}(P^{\ast},x,x_{0},t_{r},t_{0,r})+\psi_{1}(P^{\ast},x,x_{0},t_{r},t_{0,r})\psi_{2}(P,x,x_{0},t_{r},t_{0,r})=2\frac{G_{n}(z,x,t_{r})}{F_{n}(z,x_{0},t_{0,r})},$
(5.53)
$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\psi_{2}(P^{\ast},x,x_{0},t_{r},t_{0,r})-\psi_{1}(P^{\ast},x,x_{0},t_{r},t_{0,r})\psi_{2}(P,x,x_{0},t_{r},t_{0,r})=-\frac{2y}{zF_{n}(z,x_{0},t_{0,r})}.$
(5.54)

Proof. To prove (5.50), we first consider the part of time variable in the
definition (5), that is,

$\mathrm{exp}\left(\int_{t_{0,r}}^{t_{r}}ds~{}(z^{-1}\widetilde{F}_{r}(z,x_{0},s)\phi(P,x_{0},s)-\widetilde{G}_{r}(z,x_{0},s))\right).$
(5.55)

The integrand in the above integral equals

$\displaystyle
z^{-1}\widetilde{F}_{r}(z,x_{0},s)\phi(P,x_{0},s)-\widetilde{G}_{r}(z,x_{0},s)$
$\displaystyle~{}~{}~{}~{}~{}=z^{-1}\widetilde{F}_{r}(z,x_{0},s)\frac{y+zG_{n}(z,x_{0},s)}{F_{n}(z,x_{0},s)}-\widetilde{G}_{r}(z,x_{0},s)$
$\displaystyle~{}~{}~{}~{}~{}=\frac{y}{z}\widetilde{F}_{r}(z,x_{0},s)F_{n}(z,x_{0},s)^{-1}+(\widetilde{F}_{r}(z,x_{0},s)G_{n}(z,x_{0},s)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\widetilde{G}_{r}(z,x_{0},s)F_{n}(z,x_{0},s))F_{n}(z,x_{0},s)^{-1}$
$\displaystyle~{}~{}~{}~{}~{}=\frac{y}{z}\widetilde{F}_{r}(z,x_{0},s)F_{n}(z,x_{0},s)^{-1}+\frac{1}{2}\frac{F_{n,s}(z,x_{0},s)}{F_{n}(z,x_{0},s)},$
(5.56)

using (5) and (5.42). Hence, (5.55) can be expressed as

$\Big{(}\frac{F_{n}(z,x_{0},t_{r})}{F_{n}(z,x_{0},t_{0,r})}\Big{)}^{1/2}\mathrm{exp}\left(\frac{y}{z}\int_{t_{0,r}}^{t_{r}}ds\widetilde{F}_{r}(z,x_{0},s)F_{n}(z,x_{0},s)^{-1}\right).$
(5.57)

On the other hand, the part of space variable in (5) can be written as

$\Big{(}\frac{F_{n}(z,x,t_{r})}{F_{n}(z,x_{0},t_{r})}\Big{)}^{1/2}\mathrm{exp}\left(\frac{y}{z}\int_{x_{0}}^{x}dx^{\prime}F_{n}(z,x^{\prime},t_{r})^{-1}\right),$
(5.58)

using the similar procedure in Lemma 3.2. Then combining (5.57) and (5.58)
readily leads to (5.50). Evaluating (5.50) at the points $P$ and $P^{\ast}$
and multiplying the resulting expressions yields (5.51). The remaining
statements are direct consequences of (5.29), (5.37)-(5.39), and (5.51).
$\square$

In analogy to Lemma 3.4, the dynamics of the zeros
$\\{\mu_{j}(x,t_{r})\\}_{j=0,\ldots,n}$ and
$\\{\nu_{l}(x,t_{r})\\}_{l=0,\ldots,n}$ of $F_{n}(z,x,t_{r})$ and
$H_{n}(z,x,t_{r})$ with respect to $x$ and $t_{r}$ are described in terms of
the following Dubrovin-type equations.

######  Lemma 5.4.

Assume $(\ref{2.2})$ and suppose that $(\ref{5.3})$, $(\ref{5.4})$ hold
subject to the constraint $(\ref{2.26a})$.

  * _(i)_

Suppose that the zeros $\\{\mu_{j}(x,t_{r})\\}_{j=0,\ldots,n}$ of
$F_{n}(z,x,t_{r})$ remain distinct for $(x,t_{r})\in\Omega_{\mu},$ where
$\Omega_{\mu}\subseteq\mathbb{R}^{2}$ is open and connected, then
$\\{\mu_{j}(x,t_{r})\\}_{j=0,\ldots,n}$ satisfy the system of differential
equations,

$\displaystyle\mu_{j,x}=4\frac{y(\hat{\mu}_{j})}{\mu_{j}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq j}^{n}(\mu_{j}-\mu_{k})^{-1},\quad j=0,\ldots,n,$
(5.59)
$\displaystyle\mu_{j,t_{r}}=4\frac{\widetilde{F}_{r}(\mu_{j})y(\hat{\mu}_{j})}{\mu_{j}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq j}^{n}(\mu_{j}-\mu_{k})^{-1},\quad j=0,\ldots,n,$
(5.60)

with initial conditions

$\\{\hat{\mu}_{j}(x_{0},t_{0,r})\\}_{j=0,\ldots,n}\in\mathcal{K}_{n},$ (5.61)

for some fixed $(x_{0},t_{0,r})\in\Omega_{\mu}$. The initial value problem
$(\ref{5.57})$, $(\ref{5.58})$ has a unique solution satisfying

$\hat{\mu}_{j}\in C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),\quad
j=0,\ldots,n.$ (5.62)

  * _(ii)_

Suppose that the zeros $\\{\nu_{l}(x,t_{r})\\}_{l=0,\ldots,n}$ of
$H_{n}(z,x,t_{r})$ remain distinct for $(x,t_{r})\in\Omega_{\nu},$ where
$\Omega_{\nu}\subseteq\mathbb{R}^{2}$ is open and connected, then
$\\{\nu_{l}(x,t_{r})\\}_{l=0,\ldots,n}$ satisfy the system of differential
equations,

$\displaystyle\nu_{l,x}=-2\frac{(\rho^{2}+u_{xx}\nu_{l})y(\hat{\nu}_{l})}{h_{0}~{}\nu_{l}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq l}^{n}(\nu_{l}-\nu_{k})^{-1},\quad l=0,\ldots,n,$
(5.63)
$\displaystyle\nu_{l,t_{r}}=2\frac{\widetilde{H}_{r}(\nu_{l})y(\hat{\nu}_{l})}{h_{0}~{}\nu_{l}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq l}^{n}(\nu_{l}-\nu_{k})^{-1},\quad l=0,\ldots,n,$
(5.64)

with initial conditions

$\\{\hat{\nu}_{l}(x_{0},t_{0,r})\\}_{l=0,\ldots,n}\in\mathcal{K}_{n},$ (5.65)

for some fixed $(x_{0},t_{0,r})\in\Omega_{\nu}$. The initial value problem
$(\ref{5.61})$, $(\ref{5.62})$ has a unique solution satisfying

$\hat{\nu}_{l}\in C^{\infty}(\Omega_{\nu},\mathcal{K}_{n}),\quad
l=0,\ldots,n.$ (5.66)

Proof. It suffices to prove (5.60) since the argument for (5.64) is analogous
and that for (5.59) and (5.63) has been given in the proof of Lemma 3.4.
Differentiating (5.10) with respect to $t_{r}$ yields

$F_{n,t_{r}}(\mu_{j})=-\frac{1}{2}\mu_{j,t_{r}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq j}^{n}(\mu_{j}-\mu_{k}).$ (5.67)

On the other hand, inserting $z=\mu_{j}$ into (5.42) and using (5.30), one
finds

$F_{n,t_{r}}(\mu_{j})=2G_{n}(\mu_{j})\widetilde{F}_{r}(\mu_{j})=2\frac{y(\hat{\mu}_{j})}{-\mu_{j}}\widetilde{F}_{r}(\mu_{j}).$
(5.68)

Combining (5.67) and (5.68) then yields (5.60). The rest is analogous to the
proof of Lemma 3.4. $\square$

Since the stationary trace formulas for HS2 invariants in terms of symmetric
functions of $\mu_{j}$ in Lemma 3.5 extend line by line to the corresponding
time-dependent setting, we next record the $t_{r}$-dependent trace formulas
without proof. For simplicity, we confine ourselves to the simplest one only.

######  Lemma 5.5.

Assume $(\ref{2.2})$, suppose that $(\ref{5.3})$, $(\ref{5.4})$ hold, and let
$(x,t_{r})\in\mathbb{R}^{2}$. Then,

$u(x,t_{r})=\frac{1}{2}\sum_{j=0}^{n}\mu_{j}(x,t_{r})-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}.$
(5.69)

##  6 Time-dependent algebro-geometric solutions of HS2 hierarchy

In our final section, we extend the results of section 4 from the stationary
HS2 hierarchy, to the time-dependent case. We obtain Riemann theta function
representations for the meromorphic function $\phi$, and especially, for the
algebro-geometric solutions $u,\rho$ of the whole HS2 hierarchy.

We first record the asymptotic properties of $\phi$ in the time-dependent
case.

######  Lemma 6.1.

Assume $(\ref{2.2})$ and suppose that $(\ref{5.3})$, $(\ref{5.4})$ hold.
Moreover, let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\}$,
$(x,t_{r})\in\mathbb{R}^{2}$. Then,

$\displaystyle\phi(P)\underset{\zeta\rightarrow
0}{=}-u_{x}(x,t_{r})+O(\zeta),\quad P\rightarrow
P_{\infty_{\pm}},\quad\zeta=z^{-1},$ (6.1)
$\displaystyle\phi(P)\underset{\zeta\rightarrow
0}{=}i\rho(x,t_{r})+\frac{iu_{xx}(x,t_{r})-\rho_{x}(x,t_{r})}{2\rho(x,t_{r})}\zeta+O(\zeta^{2}),\quad
P\rightarrow P_{0},\quad\zeta=z.$ (6.2)

Since the proof of Lemma 6.1 is identical to the corresponding stationary
results in Lemma 4.1, we omit the corresponding details.

Next, we investigate the properties of the Abel map. To do this, let
$\bar{\mu}=(\mu_{0},\ldots,\mu_{n})\in\mathbb{C}^{n+1}$, we define the
following symmetric functions by

$\Psi_{k+1}(\bar{\mu})=(-1)^{k+1}\sum_{\underline{l}\in\mathcal{S}_{k+1}}\mu_{l_{1}}\ldots\mu_{l_{k+1}},\quad
k=0,\ldots,n,$ (6.3)

where
$\mathcal{S}_{k+1}=\\{\underline{l}=(l_{1},\ldots,l_{k+1})\in\mathbb{N}_{0}^{k+1}~{}|~{}l_{1}<\cdots<l_{k+1}\leq
n\\}$;

$\Phi_{k+1}^{(j)}(\bar{\mu})=(-1)^{k+1}\sum_{\underline{l}\in\mathcal{T}_{k+1}^{(j)}}\mu_{l_{1}}\ldots\mu_{l_{k+1}},\quad
k=0,\ldots,n-1,$ (6.4)

where
$\mathcal{T}_{k+1}^{(j)}=\\{\underline{l}=(l_{1},\ldots,l_{k+1})\in\mathcal{S}_{k+1}~{}|~{}l_{m}\neq
j\\},$ $j=0,\ldots,n.$ For the properties of $\Psi_{k+1}(\bar{\mu})$ and
$\Phi_{k+1}^{(j)}(\bar{\mu})$, we refer to Appendix E [21].

Introducing

$\tilde{d}_{r+1,k}(\underline{E})=\sum_{s=0}^{r+1-k}\tilde{c}_{r+1-k-s}\hat{c}_{s}(\underline{E}),\quad
k=0,\ldots,r+1\wedge n+1,$ (6.5)

for a given set of constants
$\\{\tilde{c}_{l}\\}_{l=1,\ldots,r+1}\subset\mathbb{C}$, the corresponding
homogeneous and nonhomogeneous quantities $\widehat{F}_{r}(\mu_{j})$ and
$\widetilde{F}_{r}(\mu_{j})$ in the HS2 case are then given by 333$m\wedge
n=\mathrm{min}\\{m,n\\}$, $m\vee n=\mathrm{max}\\{m,n\\}$

$\begin{split}&\widehat{F}_{r}(\mu_{j})=\sum_{s=(r-n)\vee
0}^{r+1}\hat{c}_{s}(\underline{E})\Phi_{r+1-s}^{(j)}(\bar{\mu}),\\\
&\widetilde{F}_{r}(\mu_{j})=\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\widehat{F}_{s}(\mu_{j})=\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\Phi_{k}^{(j)}(\bar{\mu}),\quad
r\in\mathbb{N}_{0},~{}\tilde{c}_{0}=1,\end{split}$ (6.6)

using (D.59) and (D.60) [21]. Here, $\hat{c}_{s}(\underline{E})$,
$s\in\mathbb{N}_{0}$, is defined by (D.2) [21].

We now state the analog of Theorem 4.3, which indicates marked differences
between the HS2 hierarchy and other completely integrable systems such as the
KdV and AKNS hierarchies.

######  Theorem 6.2.

Assume $(\ref{2.26a})$ and suppose that $\\{\hat{\mu}_{j}\\}_{j=0,\ldots,n}$
satisfies the Dubrovin equations $(\ref{5.56})$, $(\ref{5.57})$ on an open set
$\Omega_{\mu}\subseteq\mathbb{R}^{2}$ such that $\mu_{j}$, $j=0,\ldots,n,$
remain distinct and nonzero on $\Omega_{\mu}$ and that
$\widetilde{F}_{r}(\mu_{j})\neq 0$ on $\Omega_{\mu}$, $j=0,\ldots,n$.
Introducing the associated divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$, one
computes,

$\displaystyle\partial_{x}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})})$
$\displaystyle=-\frac{4a}{\Psi_{n+1}(\bar{\mu}(x,t_{r}))}\underline{c}(1),\quad(x,t_{r})\in\Omega_{\mu},$
(6.7)
$\displaystyle\partial_{t_{r}}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})})$
$\displaystyle=-\frac{4a}{\Psi_{n+1}(\bar{\mu}(x,t_{r}))}\left(\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\Psi_{k}(\bar{\mu}(x,t_{r}))\right)\underline{c}(1)$
$\displaystyle+4a\left(\sum_{\ell=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-\ell}(\underline{E})\underline{c}(\ell)\right),\quad(x,t_{r})\in\Omega_{\mu}.$
(6.8)

In particular, the Abel map dose not linearize the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$ on
$\Omega_{\mu}$.

Proof. Let $(x,t_{r})\in\Omega_{\mu}$. It suffices to prove (6.8), since (6.7)
is proved as in the stationary context of Theorem 4.3. We first recall a
fundamental identity (E.10) [21], that is,

$\Phi_{k+1}^{(j)}(\bar{\mu})=\mu_{j}\Phi_{k}^{(j)}(\bar{\mu})+\Psi_{k+1}(\bar{\mu}),\quad
k=0,\ldots,n,~{}j=0,\ldots,n.$ (6.9)

Then, applying (4.16), (6.6), and (6.9), one finds

$\displaystyle\frac{\widetilde{F}_{r}(\mu_{j})}{\mu_{j}}$
$\displaystyle=\mu_{j}^{-1}\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Phi_{m}^{(j)}(\bar{\mu})$
(6.10)
$\displaystyle=\mu_{j}^{-1}\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Big{(}\mu_{j}\Phi_{m-1}^{(j)}(\bar{\mu})+\Psi_{m}(\bar{\mu})\Big{)}$
$\displaystyle=\sum_{m=1}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Phi_{m-1}^{(j)}(\bar{\mu})-\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Psi_{m}(\bar{\mu})\frac{\Phi_{n}^{(j)}(\bar{\mu})}{\Psi_{n+1}(\bar{\mu})}.$

Hence, using (5.60), (6.10), (E.4), (E.13), and (E.14) [21], one infers that

$\displaystyle\partial_{t_{r}}\Big{(}\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}}\underline{\omega}\Big{)}=\sum_{j=0}^{n}\mu_{j,t_{r}}\sum_{k=1}^{n}\underline{c}(k)\frac{a~{}\mu_{j}^{k-1}}{y(\hat{\mu}_{j})}$
$\displaystyle=4a\sum_{j=0}^{n}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq
j}^{n}(\mu_{j}-\mu_{l})}\frac{\widetilde{F}_{r}(\mu_{j})}{\mu_{j}}$
$\displaystyle=4a\sum_{j=0}^{n}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq j}^{n}(\mu_{j}-\mu_{l})}$
$\displaystyle\times\Big{(}-\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Psi_{m}(\bar{\mu})\frac{\Phi_{n}^{(j)}(\bar{\mu})}{\Psi_{n+1}(\bar{\mu})}+\sum_{m=1}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Phi_{m-1}^{(j)}(\bar{\mu})\Big{)}$
$\displaystyle=-4a\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\frac{\Psi_{m}(\bar{\mu})}{\Psi_{n+1}(\bar{\mu})}\sum_{k=1}^{n}\sum_{j=0}^{n}\underline{c}(k)(U_{n+1}(\bar{\mu}))_{k,j}(U_{n+1}(\bar{\mu}))_{j,1}^{-1}$
$\displaystyle~{}~{}~{}~{}~{}+4a\sum_{m=1}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\sum_{k=1}^{n}\sum_{j=0}^{n}\underline{c}(k)(U_{n+1}(\bar{\mu}))_{k,j}(U_{n+1}(\bar{\mu}))_{j,n-m+2}^{-1}$
$\displaystyle=-\frac{4a}{\Psi_{n+1}(\bar{\mu})}\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Psi_{m}(\bar{\mu})\underline{c}(1)+4a\sum_{m=1}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\underline{c}(n-m+2)$
$\displaystyle=-\frac{4a}{\Psi_{n+1}(\bar{\mu})}\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Psi_{m}(\bar{\mu})\underline{c}(1)+4a\sum_{m=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-m}(\underline{E})\underline{c}(m),$
(6.11)

which is equivalent to (6.8). $\square$

The analogous results hold for the corresponding divisor
$\mathcal{D}_{\hat{\nu}_{0}(x,t_{r})\underline{\hat{\nu}}(x,t_{r})}$
associated with $\phi(P,x,t_{r})$.

Next, recalling the definition of $\underline{\widehat{B}}_{Q_{0}}$ and
$\underline{\hat{\beta}}_{Q_{0}}$ in (4.19) and (4.20), one obtains the
following result.

######  Corollary 6.3.

Assume $(\ref{2.26a})$ and suppose that $\\{\hat{\mu}_{j}\\}_{j=0,\ldots,n}$
satisfies the Dubrovin equations $(\ref{5.56})$, $(\ref{5.57})$ on an open set
$\Omega_{\mu}\subseteq\mathbb{R}^{2}$ such that $\mu_{j}$, $j=0,\ldots,n,$
remain distinct and nonzero on $\Omega_{\mu}$ and that
$\widetilde{F}_{r}(\mu_{j})\neq 0$ on $\Omega_{\mu},$ $j=0,\ldots,n$. Then,
one computes

$\displaystyle\partial_{x}\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}(x,t_{r})}\eta_{1}=-\frac{4a}{\Psi_{n+1}(\bar{\mu}(x,t_{r}))},\quad(x,t_{r})\in\Omega_{\mu},$
(6.12)
$\displaystyle\partial_{x}\underline{\hat{\beta}}(\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})})=\begin{cases}4a,~{}~{}~{}~{}~{}~{}~{}~{}n=1,\\\
4a(0,\ldots,0,1),~{}~{}~{}~{}~{}~{}~{}~{}n\geq
2,\end{cases}\quad(x,t_{r})\in\Omega_{\mu},$ (6.13)
$\displaystyle\partial_{t_{r}}\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}(x,t_{r})}\eta_{1}=-\frac{4a}{\Psi_{n+1}(\bar{\mu}(x,t_{r}))}\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\Psi_{k}(\bar{\mu}(x,t_{r}))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+4a\tilde{d}_{r+1,n+1}(\underline{E})\delta_{n+1,r+1\wedge
n+1},\quad(x,t_{r})\in\Omega_{\mu},$ (6.14)
$\displaystyle\partial_{t_{r}}\underline{\hat{\beta}}(\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})})=4a\Big{(}\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\hat{c}_{s+1-n}(\underline{E}),\ldots,\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\hat{c}_{s+1}(\underline{E}),\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\hat{c}_{s}(\underline{E})\Big{)},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad\hat{c}_{-l}(\underline{E})=0,~{}l\in\mathbb{N},\quad(x,t_{r})\in\Omega_{\mu}.$
(6.15)

Proof. Equations (6.12) and (6.13) are proved as in the stationary context of
Corollary 4.4. Equation (6.14) is a special case of (6.8), and (6.15) follows
by (6.11), taking into account (E.4) [21]. $\square$.

The fact that the Abel map does not effect a linearization of the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$ in the
time-dependent HS2 context, which is well known and discussed (using different
approaches) by Constantin and McKean [12], Alber, Camassa, Fedorov, Holm, and
Marsden [2], Alber and Fedorov [3, 4]. The change of variables

$x\mapsto\tilde{x}=\int^{x}dx^{\prime}\left(\frac{4a}{\Psi_{n+1}(\bar{\mu}(x^{\prime},t_{r}))}\right)$
(6.16)

and

$\displaystyle t_{r}\mapsto\tilde{t}_{r}$
$\displaystyle=\int^{t_{r}}ds\Bigg{(}\frac{4a}{\Psi_{n+1}(\bar{\mu}(x,s))}\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\Psi_{k}(\bar{\mu}(x,s))$
$\displaystyle~{}~{}~{}~{}~{}-4a\sum_{\ell=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-\ell}(\underline{E})\frac{\underline{c}(\ell)}{\underline{c}(1)}\Bigg{)}$
(6.17)

linearizes the Abel map
$\underline{A}_{Q_{0}}(\mathcal{D}_{\hat{\tilde{\mu}}_{0}(\tilde{x},\tilde{t}_{r})\underline{\hat{\tilde{\mu}}}(\tilde{x},\tilde{t}_{r})})$,
$\tilde{\mu}_{j}(\tilde{x},\tilde{t}_{r})=\mu_{j}(x,t_{r})$, $j=0,\ldots,n$.
The intricate relation between the variables $(x,t_{r})$ and
$(\tilde{x},\tilde{t}_{r})$ is detailed in (6.21). Our approach follows a
route similar to Gesztesy and Holden’s treatment of the CH hierarchy [21].

Next, we shall provide the explicit representations of $\phi$ and $u,\rho$ in
terms of the Riemann theta function associated with $\mathcal{K}_{n}$,
assuming the affine part of $\mathcal{K}_{n}$ to be nonsingular. Recalling
(4.26)-(4), the analog of Theorem 4.5 in the stationary case then reads as
follows.

######  Theorem 6.4.

Assume $(\ref{2.2})$ and suppose that $(\ref{5.3})$, $(\ref{5.4})$ hold on
$\Omega$ subject to the constraint $(\ref{2.26a})$. In addition, let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{0}\\}$ and
$(x,t_{r}),(x_{0},t_{0,r})\in\Omega$, where $\Omega\subseteq\mathbb{R}^{2}$ is
open and connected. Moreover, suppose that
$\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})}$, or equivalently,
$\mathcal{D}_{\underline{\hat{\nu}}(x,t_{r})}$ is nonspecial for
$(x,t_{r})\in\Omega$. Then, $\phi$, $u$, and $\rho$ admit the representations

$\displaystyle\phi(P,x,t_{r})=i\rho(x,t_{r})\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x,t_{r})))}{\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P,\underline{\hat{\mu}}(x,t_{r})))}\mathrm{exp}\left(e_{0}-\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x,t_{r}),\hat{\nu}_{0}(x,t_{r})}^{(3)}\right),$
(6.18) $\displaystyle
u(x,t_{r})=-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}+\frac{1}{2}\sum_{j=1}^{n}\lambda_{j}-\frac{1}{2}\sum_{j=1}^{n}U_{j}\partial_{\omega_{j}}\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x,t_{r}))+\underline{\omega})}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x,t_{r}))+\underline{\omega})}\right)\Big{|}_{\underline{\omega}=0},$
(6.19)
$\displaystyle\rho(x,t_{r})=iu_{x}(x,t_{r})\frac{\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x,t_{r})))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x,t_{r})))}.$
(6.20)

Moreover, let $\widetilde{\Omega}\subseteq\Omega$ be such that $\mu_{j}$,
$j=0,\ldots,n,$ are nonvanishing on $\widetilde{\Omega}$. Then, the constraint

$\displaystyle
4a(x-x_{0})+4a(t_{r}-t_{0,r})\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\hat{c}_{s}(\underline{E})=\sum_{j=1}^{n}\left(\int_{a_{j}}\tilde{\omega}_{P_{\infty_{+}}P_{\infty_{-}}}^{(3)}\right)c_{j}(1)$
$\displaystyle~{}~{}~{}~{}~{}~{}\times~{}\left(-4a\int_{x_{0}}^{x}\frac{dx^{\prime}}{\prod_{k=0}^{n}\mu_{k}(x^{\prime},t_{r})}-4a\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\int_{t_{0,r}}^{t_{r}}\frac{\Psi_{k}(\bar{\mu}(x_{0},s))}{\Psi_{n+1}(\bar{\mu}(x_{0},s))}ds\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}+~{}4a(t_{r}-t_{0,r})\sum_{\ell=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-\ell}(\underline{E})\sum_{j=1}^{n}\left(\int_{a_{j}}\tilde{\omega}_{P_{\infty_{+}}P_{\infty_{-}}}^{(3)}\right)c_{j}(\ell)$
$\displaystyle~{}~{}~{}~{}~{}~{}+~{}\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x_{0},t_{0,r})))}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x_{0},t_{0,r})))}\right),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(x,t_{r}),(x_{0},t_{0,r})\in\widetilde{\Omega}$
(6.21)

holds, with

$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})})=\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x_{0},t_{r})\underline{\hat{\mu}}(x_{0},t_{r})})-4a\left(\int_{x_{0}}^{x}\frac{dx^{\prime}}{\Psi_{n+1}(\bar{\mu}(x^{\prime},t_{r}))}\right)\underline{c}(1)$
(6.22)
$\displaystyle~{}~{}=\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{0,r})\underline{\hat{\mu}}(x,t_{0,r})})-4a\left(\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\int_{t_{0,r}}^{t_{r}}\frac{\Psi_{k}(\bar{\mu}(x,s))}{\Psi_{n+1}(\bar{\mu}(x,s))}ds\right)\underline{c}(1)$
$\displaystyle~{}~{}+4a(t_{r}-t_{0,r})\Bigg{(}\sum_{\ell=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-\ell}(\underline{E})\underline{c}(\ell)\Bigg{)},$
(6.23)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(x,t_{r}),(x_{0},t_{0,r})\in\widetilde{\Omega}.$

Proof. We first assume that $\mu_{j}$, $j=0,\ldots,n$, are distinct and
nonvanishing on $\widetilde{\widetilde{\Omega}}$ and
$\widetilde{F}_{r}(\mu_{j})\neq 0$ on $\widetilde{\widetilde{\Omega}}$,
$j=0,\ldots,n,$ where $\widetilde{\widetilde{\Omega}}\subseteq\Omega$. Then,
the representation (6.18) for $\phi$ on $\widetilde{\widetilde{\Omega}}$
follows by combining (5.32), (6.1), (6.2), and Theorem A.26 [21]. The
representation (6.19) for $u$ on $\widetilde{\widetilde{\Omega}}$ follows from
the trace formulas (5.69) and (F.59) [21]. The representation (6.20) for
$\rho$ on $\widetilde{\widetilde{\Omega}}$ is clear from (6.18) and (6.1). In
fact, since the proofs of (6.18), (6.19), and (6.20) are identical to the
corresponding stationary results in Theorem 4.5, which can be extended line by
line to the time-dependent setting, here we omit the corresponding details. By
continuity, (6.18), (6.19), and (6.20) extend from
$\widetilde{\widetilde{\Omega}}$ to $\Omega$. The constraint (6.21) then holds
on $\widetilde{\widetilde{\Omega}}$ by combining (6.12)-(6.15) and (F.58)
[21]. Equations (6.22) and (6.23) are clear from (6.7) and (6.8). Again by
continuity, (6.21)-(6.23) extend from $\widetilde{\widetilde{\Omega}}$ to
$\widetilde{\Omega}$. $\square$

######  Remark 6.5.

One observes that $(\ref{6.22})$ and $(\ref{6.22a18})$ are equivalent to

$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})})=$
$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x_{0},t_{r})\underline{\hat{\mu}}(x_{0},t_{r})})-\underline{c}(1)(\tilde{x}-\tilde{x}_{0})$
(6.24) $\displaystyle=$
$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{0,r})\underline{\hat{\mu}}(x,t_{0,r})})-\underline{c}(1)(\tilde{t}_{r}-\tilde{t}_{0,r}),$
(6.25)

under the change of variables $x\mapsto\tilde{x}$ and
$t_{r}\mapsto\tilde{t}_{r}$ in $(\ref{6.17})$ and $(\ref{6.18})$. Hence, the
Abel map linearizes the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$ on
$\Omega$ with respect to $\tilde{x},\tilde{t}_{r}$.

######  Remark 6.6.

Remark 4.7 applies in the present time-dependent context. Moreover, to obtain
the theta function representation of $\psi_{j}$, $j=1,2,$, one can write
$\widetilde{F}_{r}$ in terms of $\Psi_{k}(\bar{\mu})$ and use $(\ref{5.46})$,
in analogy to the stationary case discussed in Remark 4.8. Here we omit
further details.

At the end of this section, we turn to the time-dependent algebro-geometric
initial value problem of HS2 hierarchy. We will show that the solvability of
the Dubrovin equations (5.59) and (5.60) on
$\Omega_{\mu}\subseteq\mathbb{R}^{2}$ in fact implies equations (5.7) and
(5.8) on $\Omega_{\mu}$.

######  Theorem 6.7.

Fix $n\in\mathbb{N}_{0}$, assume $(\ref{2.26a})$, and suppose that
$\\{\hat{\mu}_{j}\\}_{j=0,\ldots,n}$ satisfies the Dubrovin equations
$(\ref{5.56})$, $(\ref{5.57})$ on an open and connected set
$\Omega_{\mu}\subseteq\mathbb{R}^{2}$, with $\widetilde{F}_{r}(\mu_{j})$ in
$(\ref{5.57})$ expressed in terms of $\mu_{k}$, $k=0,\ldots,n$, by
$(\ref{6.6})$. Moreover, assume that $\mu_{j}$, $j=0,\ldots,n,$ remain
distinct and nonzero on $\Omega_{\mu}$. Then, $u,\rho\in
C^{\infty}(\Omega_{\mu})$, defined by

$u=-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}+\frac{1}{2}\sum_{j=0}^{n}\mu_{j},$
(6.26)

and

$\rho^{2}=u_{x}^{2}+2uu_{xx}-\frac{d^{2}}{dx^{2}}\Bigg{(}\Psi_{2}(\bar{\mu})-u\sum_{m=0}^{2n+1}E_{m}\Bigg{)},$
(6.27)

satisfy the $r$th $HS2$ equation $(\ref{5.1})$, that is,

$\mathrm{HS2}_{r}(u,\rho)=0\quad\textrm{on \,$\Omega_{\mu}$},$ (6.28)

with initial values satisfying the $n$th stationary $HS2$ equation
$(\ref{5.1a8})$.

Proof. Given the solutions $\hat{\mu}_{j}=(\mu_{j},y(\hat{\mu}_{j}))\in
C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),$ $j=0,\ldots,n$ of (5.59) and
(5.60), we define polynomials $F_{n},$ $G_{n}$, and $H_{n}$ on $\Omega_{\mu}$
as in the stationary case (cf. Theorem 4.9) with properties

$\displaystyle F_{n}(z)=\frac{1}{2}\prod_{j=0}^{n}(z-\mu_{j}),$ (6.29)
$\displaystyle G_{n}(z)=\frac{1}{2}F_{n,x}(z),$ (6.30) $\displaystyle
z^{2}G_{n,x}(z)=-H_{n}(z)-(\rho^{2}+u_{xx}z)F_{n}(z),$ (6.31) $\displaystyle
H_{n,x}(z)=2(\rho^{2}+u_{xx}z)G_{n}(z),$ (6.32) $\displaystyle
R_{2n+2}(z)=z^{2}G_{n}^{2}(z)+F_{n}(z)H_{n}(z),$ (6.33)

treating $t_{r}$ as a parameter. Define the polynomials $\widetilde{G}_{r}$
and $\widetilde{H}_{r}$ by

$\displaystyle\widetilde{G}_{r}(z)=\frac{1}{2}\widetilde{F}_{r,x}(z)\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$},$ (6.34)
$\displaystyle\widetilde{H}_{r}(z)=-z^{2}\widetilde{G}_{r,x}(z)-(\rho^{2}+u_{xx}z)\widetilde{F}_{r}(z)\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$},$ (6.35)

respectively. Next, we claim that

$F_{n,t_{r}}(z)=2(G_{n}(z)\widetilde{F}_{r}(z)-F_{n}(z)\widetilde{G}_{r}(z))\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$}.$ (6.36)

To prove (6.36), one computes from (5.59) and (5.60) that

$\displaystyle F_{n,x}(z)=-F_{n}(z)\sum_{j=0}^{n}\mu_{j,x}(z-\mu_{j})^{-1},$
(6.37) $\displaystyle
F_{n,t_{r}}(z)=-F_{n}(z)\sum_{j=0}^{n}\widetilde{F}_{r}(\mu_{j})\mu_{j,x}(z-\mu_{j})^{-1}.$
(6.38)

Using (6.30) and (6.34), one concludes that (6.36) is equivalent to

$\widetilde{F}_{r,x}(z)=\sum_{j=0}^{n}(\widetilde{F}_{r}(\mu_{j})-\widetilde{F}_{r}(z))\mu_{j,x}(z-\mu_{j})^{-1}.$
(6.39)

Equation (6.39) is proved in Lemma F.9 [21]. This in turn proves (6.36).

Next, differentiating (6.30) with respect to $t_{r}$ yields

$F_{n,xt_{r}}=2G_{n,t_{r}}.$ (6.40)

On the other hand, taking the derivative of (6.36) with respect to $x$, and
using (6.30), (6.31), (6.34), one obtains

$\displaystyle F_{n,t_{r}x}=$
$\displaystyle-2z^{-2}H_{n}\widetilde{F}_{r}-2(z^{-2}\rho^{2}+u_{xx}z^{-1})F_{n}\widetilde{F}_{r}+2G_{n}\widetilde{F}_{r,x}$
$\displaystyle-2\widetilde{G}_{r,x}F_{n}-4\widetilde{G}_{r}G_{n}.$ (6.41)

Combining (6.30), (6.34), (6.40), and (6), one concludes

$z^{2}G_{n,t_{r}}(z)=\widetilde{H}_{r}(z)F_{n}(z)-H_{n}(z)\widetilde{F}_{r}(z)\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$. }$ (6.42)

Next, differentiating (6.33) with respect to $t_{r}$, and using expressions
(6.36) and (6.42) for $F_{n,t_{r}}$ and $G_{n,t_{r}}$, respectively, one
obtains

$H_{n,t_{r}}(z)=2(H_{n}(z)\widetilde{G}_{r}(z)-G_{n}(z)\widetilde{H}_{r}(z))\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$. }$ (6.43)

Finally, taking the derivative of (6.42) with respect to $x$, and using
expressions (6.30), (6.32), and (6.34) for $F_{n,x}$, $H_{n,x}$, and
$\widetilde{F}_{r,x}$, respectively, one infers that

$z^{2}G_{n,t_{r}x}=F_{n}\widetilde{H}_{r,x}+2G_{n}\widetilde{H}_{r}-2(\rho^{2}+u_{xx}z)G_{n}\widetilde{F}_{r}-2H_{n}\widetilde{G}_{r}.$
(6.44)

On the other hand, differentiating (6.31) with respect to $t_{r}$, using
(6.36) and (6.43) for $F_{n,t_{r}}$ and $H_{n,t_{r}}$, respectively, leads to

$\displaystyle z^{2}G_{n,xt_{r}}$
$\displaystyle=2G_{n}\widetilde{H}_{r}-2H_{n}\widetilde{G}_{r}-(2\rho\rho_{t_{r}}+u_{xxt_{r}}z)F_{n}-2(\rho^{2}+u_{xx}z)(G_{n}\widetilde{F}_{r}-\widetilde{G}_{r}F_{n}).$
(6.45)

Combining (6.44) and (6.45) then yields

$-2\rho\rho_{t_{r}}-u_{xxt_{r}}z-\widetilde{H}_{r,x}+2(\rho^{2}+u_{xx}z)\widetilde{G}_{r}=0.$
(6.46)

Thus, we have proved (5.16)-(5.21) and (5.42)-(5.44) on
$\mathbb{C}\times\Omega_{\mu}$ and hence conclude that (6.28) holds on
$\mathbb{C}\times\Omega_{\mu}$. $\square$

######  Remark 6.8.

Again we formulated Theorem 6.7 in terms of $\\{\mu_{j}\\}_{j=0,\ldots,n}$
only. Obviously, the analogous result (and strategy proof) works in terms of
$\\{\nu_{j}\\}_{j=0,\ldots,n}$.

The analog of Remark 4.11 directly extends to the current time-dependent
setting.

## Acknowledgments

The work described in this paper was supported by grants from the National
Science Foundation of China (Project No.10971031; 11271079; 11075055),
Doctoral Programs Foundation of the Ministry of Education of China, and the
Shanghai Shuguang Tracking Project (Project 08GG01).

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