# Algebro-geometric solutions for the two-component Camassa-Holm Dym 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
Camassa-Holm Dym (CHD2) hierarchy. 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 CHD2 hierarchy.
Key words: two-component Camassa-Holm Dym, hyperelliptic curve, algebro-
geometric solutions

##  1 Introduction

In this paper we consider the following integrable two-component Camassa-Holm
Dym (CHD2) system:

$\left\\{\begin{array}[]{ll}\rho_{t}+\left(\displaystyle\frac{m}{\rho^{2}}\right)_{x}=0,\\\
m_{t}-\left(\displaystyle(1-\partial_{x}^{2})\frac{1}{\rho}\right)_{x}=0,\end{array}\right.$
(1.1)

where $m=u-u_{xx}$, which was recently introduced by Holm and Ivanov in [18].
This coupled nonlinear system is at the position in the two component Camassa-
Holm hierarchy [5, 6, 10, 15, 16, 17, 21, 22, 27] that corresponds to the
modified Dym equation, first introduced as a tri-Hamiltonian system in [25].
The CHD2 equation combines to produce the nonlinear wave equation

$m_{tt}=(1-\partial_{x}^{2})\left(\partial_{x}\frac{1}{\rho^{2}}\partial_{x}\frac{1}{\rho^{2}}\right)m.$
(1.2)

Linearizing this equation around $m=0$ and $\rho=1$ yields the dispersion
relation for a plane wave $\mathrm{exp}(i(\kappa x-wt))$ with wave number
$\kappa$ and frequency $w$ as

$w^{2}(\kappa)=(1+\kappa^{2})\kappa^{2}.$

Accordingly, the phase speed of the linearized plane waves is
$w/\kappa=\sqrt{1+\kappa^{2}}$, so the higher wave numbers travel faster [18].
This type of dispersion relation is the same as for time-dependent Euler-
Bernoulli theory for an elastic beam with both bending and vibration response
[18, 26].

The CHD2 equation is also a completely integrable system with a bi-Hamiltonian
structure, and hence, it possesses a Lax pair and conservation laws [18].
Recently, soliton solutions and travelling wave solutions of the system (1.1)
were investigated in [18]. However, within the knowledge of the authors, the
algebro-geometric solutions of the entire CHD2 hierarchy are not studied yet.

The principal subject of this paper concerns algebro-geometric quasi-periodic
solutions of the whole CHD2 hierarchy, of which (1.1) 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]-[4], [7]-[9]. In a degenerated case of the
algebro-geometric solution, the multi-soliton solution and periodic solution
in elliptic function type may be obtained [4, 23, 24]. 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 Camassa-Holm (CH)
hierarchy etc. [11]-[14]. Recently, we investigated algebro-geometric
solutions for the modified CH hierarchy and the Degasperis-Procesi hierarchy
[19, 20].

The outline of the present paper is as follows.

In section 2, based on the polynomial recursion formalism, we derive the CHD2
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 CHD2
hierarchy.

In Section 3, we decompose the stationary CHD2 equations into a system of
Dubrovin-type equations. Moreover, we obtain the stationary trace formulas for
the CHD2 hierarchy.

In Section 4, we present the first set of our results, the explicit theta
function representations of the Baker-Akhiezer function, the meromorphic
function, and in particular, that of the potentials $u,\rho$ for the entire
stationary CHD2 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 CHD2 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 CHD2 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, it should perhaps be noted that the system (1.1) was obtained by
using the condition $u\rho=-2$ in [18]. At this point, we will occasionally
use this relation in the following sections.

##  2 The CHD2 hierarchy

In this section, we provide the construction of CHD2 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 CHD2 hierarchy.

Throughout this section, we make the following hypothesis.

######  Hypothesis 2.1.

In the stationary case, we assume that

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

In the time-dependent case, we suppose

$\begin{split}&u(\cdot,t),\rho(\cdot,t)\in C^{\infty}(\mathbb{R}),\ \
\partial_{x}^{k}u(\cdot,t),\partial_{x}^{k}\rho(\cdot,t)\in
L^{\infty}(\mathbb{R}),~{}~{}k\in\mathbb{N}_{0},~{}t\in\mathbb{R},\\\
&u(x,\cdot),u_{xx}(x,\cdot),\rho(x,\cdot),\rho_{x}(x,\cdot)\in
C^{1}(\mathbb{R}),\quad x\in\mathbb{R},\\\ &u(x,t)\neq 0,\rho(x,t)\neq
0,\quad(x,t)\in\mathbb{R}^{2}.\end{split}$ (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

$\begin{split}&f_{0}=-u,\\\
&f_{l}=\mathcal{G}\left(-f_{l-2,xxx}+4(u-u_{xx})f_{l-1,x}+f_{l-2,x}+2(u_{x}-u_{xxx})f_{l-1}\right),\quad
l\in\mathbb{N},\\\ &g_{l}=\frac{1}{2}f_{l,x},\quad l\in\mathbb{N}_{0},\\\
&h_{l}=-g_{l-2,x}-\rho^{2}f_{l}+(u-u_{xx})f_{l-1}+\frac{1}{4}f_{l-2},\quad
l\in\mathbb{N}_{0},\\\ \end{split}$ (2.3)

where $\mathcal{G}$ is given by

$\begin{split}&\mathcal{G}:L^{\infty}(\mathbb{R})\rightarrow
L^{\infty}(\mathbb{R}),\\\
&(\mathcal{G}v)(x)=\frac{1}{4}\rho(x)^{-1}\int_{-\infty}^{x}\rho(y)^{-1}v(y)~{}dy,\quad
x\in\mathbb{R},~{}v\in L^{\infty}(\mathbb{R}).\end{split}$ (2.4)

One observes that

$\mathcal{G}=\left(2\partial_{x}\rho^{2}+2\rho^{2}\partial_{x}\right)^{-1}.$
(2.5)

Explicitly, one computes

$\begin{split}&f_{0}=-u,\\\
&f_{1}=\mathcal{G}(-4u_{x}(u-u_{xx})-2u(u_{x}-u_{xxx}))+c_{1}(-u),\\\
&g_{0}=-\frac{1}{2}u_{x},\\\
&g_{1}=\frac{1}{2}\mathcal{G}(-4u_{xx}(u-u_{xx})-6u_{x}(u_{x}-u_{xxx})-2u(u_{xx}-u_{xxxx}))+c_{1}(-\frac{1}{2}u_{x}),\\\
&h_{0}=u\rho^{2},\\\ &h_{1}=-u(u-u_{xx})-\rho^{2}f_{1},~{}\mathrm{etc}.,\\\
\end{split}$ (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 $c_{k},k=1,\ldots,l,$

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

Hence,

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

defining

$c_{0}=1.$ (2.9)

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

$\psi_{x}=U(z,x)\psi=\left(\begin{array}[]{cc}0&1\\\
-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}&0\\\ \end{array}\right)\psi,$ (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

$V_{n}(z)=\left(\begin{array}[]{cc}-zG_{n}(z)&zF_{n}(z)\\\
zH_{n}(z)&zG_{n}(z)\\\ \end{array}\right),\quad
z\in\mathbb{C}\setminus\\{0\\},\quad n\in\mathbb{N}_{0},$ (2.12)

assuming $F_{n}$, $G_{n}$, and $H_{n}$ to be polynomials222$F_{n},G_{n},H_{n}$
are polynomials of degree $n,n,n+2$, 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) $\displaystyle
H_{n,x}=-2\left(-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}\right)G_{n},$ (2.15)
$\displaystyle
G_{n,x}=-H_{n}+\left(-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}\right)F_{n}.$
(2.16)

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

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

and hence

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

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

Next, we compute the characteristic polynomial $\mathrm{det}(yI-z^{-1}V_{n})$
of Lax matrix $z^{-1}V_{n}$,

$\displaystyle\mathrm{det}(yI-z^{-1}V_{n})$
$\displaystyle=y^{2}-G_{n}(z)^{2}-F_{n}(z)H_{n}(z)$
$\displaystyle=y^{2}-R_{2n+2}(z)=0,$ (2.20)

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

$\mathcal{K}_{n}:\mathcal{F}_{n}(z,y)=y^{2}-R_{2n+2}(z)=0.$ (2.21)

In the following, we will occasionally impose further constraints on the zeros
$E_{m}$ of $R_{2n+2}$ introduced in (2.19) and assume that

$E_{m}\in\mathbb{C},\quad E_{m}\neq E_{m^{\prime}},\quad\forall m\neq
m^{\prime},\quad m,m^{\prime}=0,\ldots,2n+1.$ (2.22)

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 $z$,

$\displaystyle F_{n}(z)=\sum_{l=0}^{n}f_{l}z^{n-l},$ (2.23) $\displaystyle
G_{n}(z)=\sum_{l=0}^{n}g_{l}z^{n-l},$ (2.24) $\displaystyle
H_{n}(z)=\sum_{l=0}^{n+2}h_{l}z^{n+2-l}.$ (2.25)

Inserting (2.23)-(2.25) into (2.14)-(2.16) then yields the recursion relations
(2.3) for $f_{l},$ $l=0,\ldots,n,$ and $g_{l}$, $l=0,\ldots,n.$ For fixed
$n\in\mathbb{N}_{0}$, we obtain the recursion relations for $h_{l},$
$l=0,\ldots,n$ in (2.3) and

$\begin{split}&h_{n+1}=-\frac{1}{2}f_{n-1,xx}+(u-u_{xx})f_{n}+\frac{1}{4}f_{n-1},\\\
&h_{n+2}=-\frac{1}{2}f_{n,xx}+\frac{1}{4}f_{n}.\end{split}$ (2.26)

Moreover, from (2.15), one infers that

$\begin{split}&-h_{n+1,x}-(u-u_{xx})f_{n,x}-\frac{1}{4}f_{n-1,x}=0,\quad
n\in\mathbb{N}_{0},\\\ &-h_{n+2,x}-\frac{1}{4}f_{n,x}=0,\quad
n\in\mathbb{N}_{0}.\\\ \end{split}$ (2.27)

Then using (2.26) and (2.27) permits one to write the stationary CHD2
hierarchy as

$\textrm{s-CHD2}_{n}(u,\rho)=\left(\begin{array}[]{c}-2(u-u_{xx})f_{n,x}-(u_{x}-u_{xxx})f_{n}-\frac{1}{2}f_{n-1,x}+\frac{1}{2}f_{n-1,xxx}\\\
-\frac{1}{2}f_{n,x}+\frac{1}{2}f_{n,xxx}\\\ \end{array}\right)=0,\quad
n\in\mathbb{N}_{0}.$ (2.28)

We record the first equation explicitly,

$\begin{split}&\textrm{s-CHD2}_{0}(u,\rho)=\left(\begin{array}[]{c}2u_{x}(u-u_{xx})+u(u_{x}-u_{xxx})\\\
\frac{1}{2}u_{x}-\frac{1}{2}u_{xxx}\\\ \end{array}\right)=0.\end{split}$
(2.29)

By definition, the set of solutions of (2.28) represents the class of algebro-
geometric CHD2 solutions, with $n$ ranging in $\mathbb{N}_{0}$ and $c_{l}$ in
$\mathbb{C},~{}l\in\mathbb{N}$. We call the stationary algebro-geometric CHD2
solutions $u,$ $\rho$ as CHD2 potentials at times.

######  Remark 2.2.

Here, we emphasize that if $u$, $\rho$ satisfy one of the stationary CHD$2$
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 [13].

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

$\displaystyle\widehat{F}_{l}(z)=F_{l}(z)|_{c_{k}=0,~{}k=1,\dots,l}=\sum_{k=0}^{l}\hat{f}_{k}z^{l-k},~{}~{}l=0,\ldots,n,$
(2.30)
$\displaystyle\widehat{G}_{l}(z)=G_{l}(z)|_{c_{k}=0,~{}k=1,\dots,l}=\sum_{k=0}^{l}\hat{g}_{k}z^{l-k},~{}~{}l=0,\ldots,n,$
(2.31)
$\displaystyle\widehat{H}_{l}(z)=H_{l}(z)|_{c_{k}=0,~{}k=1,\dots,l}=\sum_{k=0}^{l}\hat{h}_{k}z^{l-k},~{}~{}l=0,\ldots,n,$
(2.32)
$\displaystyle\widehat{H}_{n+1}(z)=-\hat{g}_{n-1,x}+(u-u_{xx})\hat{f}_{n}+\frac{1}{4}\hat{f}_{n-1}+\sum_{k=0}^{n}\hat{h}_{k}z^{n+1-k},$
(2.33)
$\displaystyle\widehat{H}_{n+2}(z)=-\hat{g}_{n,x}+\frac{1}{4}\hat{f}_{n}+z\widehat{H}_{n+1}(z).$
(2.34)

In accordance with our notation introduced in (2.7) and (2.30)-(2.34), the
corresponding homogeneous stationary CHD2 equations are then defined by

$\textrm{s-}\widehat{\mathrm{CHD2}}_{n}(u,\rho)=\textrm{s-CHD2}_{n}(u,\rho)|_{c_{l}=0,~{}l=1,\dots,n}=0,\quad
n\in\mathbb{N}_{0}.$ (2.35)

At the end of this section, we turn to the time-dependent CHD2 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 (2.23)-(2.25)
of $U,$ $V_{n}$ and $F_{n},G_{n}$, and $H_{n},$ respectively, still apply. The
corresponding zero-curvature equation reads

$U_{t_{n}}-V_{n,x}+[U,V_{n}]=0,\quad n\in\mathbb{N}_{0},$ (2.36)

which results in the following set of equations

$\displaystyle F_{n,x}=2G_{n},$ (2.37) $\displaystyle
G_{n,x}=-H_{n}+\left(-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}\right)F_{n},$
(2.38)
$\displaystyle-2z\rho\rho_{t_{n}}+(u_{t_{n}}-u_{xxt_{n}})-H_{n,x}-2\left(-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}\right)G_{n}=0.$
(2.39)

For fixed $n\in\mathbb{N}_{0}$, inserting the polynomial expressions for
$F_{n}$, $G_{n}$, and $H_{n}$ into (2.37)-(2.39), respectively, first yields
recursion relations (2.3) for $f_{l}|_{l=0,\ldots,n}$,
$g_{l}|_{l=0,\ldots,n}$, $h_{l}|_{l=0,\ldots,n}$ and

$\begin{split}&h_{n+1}=-\frac{1}{2}f_{n-1,xx}+(u-u_{xx})f_{n}+\frac{1}{4}f_{n-1},\\\
&h_{n+2}=-\frac{1}{2}f_{n,xx}+\frac{1}{4}f_{n}.\end{split}$ (2.40)

Moreover, using (2.39), one finds

$\begin{split}&-2\rho\rho_{t_{n}}-h_{n+1,x}-(u-u_{xx})f_{n,x}-\frac{1}{4}f_{n-1,x}=0,\quad
n\in\mathbb{N}_{0},\\\
&u_{t_{n}}-u_{xxt_{n}}-h_{n+2,x}-\frac{1}{4}f_{n,x}=0,\quad
n\in\mathbb{N}_{0}.\\\ \end{split}$ (2.41)

Hence, using (2.40) and (2.41) permits one to write the time-dependent CHD2
hierarchy as

$\displaystyle\textrm{CHD2}_{n}(u,\rho)$
$\displaystyle=\left(\begin{array}[]{c}-2\rho\rho_{t_{n}}-2(u-u_{xx})f_{n,x}-(u_{x}-u_{xxx})f_{n}-\frac{1}{2}f_{n-1,x}+\frac{1}{2}f_{n-1,xxx}\\\
u_{t_{n}}-u_{xxt_{n}}-\frac{1}{2}f_{n,x}+\frac{1}{2}f_{n,xxx}\\\
\end{array}\right)=0,$ (2.44)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n\in\mathbb{N}_{0}.$
(2.45)

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

$\begin{split}&\mathrm{CHD2}_{0}(u,\rho)=\left(\begin{array}[]{c}-2\rho\rho_{t_{0}}+2u_{x}(u-u_{xx})+u(u_{x}-u_{xxx})\\\
u_{t_{0}}-u_{xxt_{0}}+\frac{1}{2}u_{x}-\frac{1}{2}u_{xxx}\\\
\end{array}\right)=0.\end{split}$ (2.46)

The first equation $\mathrm{CHD2}_{0}(u,\rho)=0$ in the hierarchy is
equivalent to the CHD2 system as discussed in section 1, taking into account
$u\rho=-2$ and the differential relation arising in the $z^{n+2}$ term in
(2.39), that is, $-2u_{x}\rho^{2}=2u\rho\rho_{x}$. Similarly, one can
introduce the corresponding homogeneous CHD2 hierarchy by

$\widehat{\mathrm{CHD2}}_{n}(u,\rho)=\mathrm{CHD2}_{n}(u,\rho)|_{c_{l}=0,~{}l=1,\ldots,n}=0,\quad
n\in\mathbb{N}_{0}.$ (2.47)

######  Remark 2.3.

The Lenard recursion formalism for the CHD2 hierarchy can be set up as
follows. Define the two Lenard operators

$K=\left(\begin{array}[]{cc}\partial^{3}-\partial&0\\\
0&-2\partial\rho^{2}-2\rho^{2}\partial\\\ \end{array}\right),\quad
J=\left(\begin{array}[]{cc}2\partial
m+2m\partial&-2\partial\rho^{2}-2\rho^{2}\partial\\\
-2\partial\rho^{2}-2\rho^{2}\partial&0\\\ \end{array}\right),$

then the Lenard recursion sequence is given by

$K\vartheta_{j}=J\vartheta_{j+1},\quad J\vartheta_{0}=0,$

where $m=u-u_{xx}$, $\vartheta_{j}=(f_{j},f_{j+1})^{T}$, $j=0,\ldots,n-1$.
Hence, using the zero-curvature equation $(\ref{2.41})$, one can obtain the
CHD2 hierarchy.

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 CHD2
hierarchy.

##  3 The stationary CHD2 formalism

This section is devoted to a detailed study of the stationary CHD2 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),
(2.21)-(2.25), and (2.28), keeping $n\in\mathbb{N}_{0}$ fixed.

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

$\begin{split}&\mathcal{K}_{n}:\mathcal{F}_{n}(z,y)=y^{2}-R_{2n+2}(z)=0,\\\
&R_{2n+2}(z)=-4\prod_{m=0}^{2n+1}(z-E_{m}),\quad\\{E_{m}\\}_{m=0,\ldots,2n+1}\in\mathbb{C},\end{split}$
(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
$\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}=(0,y(0))\\}$
by

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

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

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

such that

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

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

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

and

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

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

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

Moreover, we introduce

$\hat{\mu}_{j}(x)=(\mu_{j}(x),G_{n}(\mu_{j}(x),x))\in\mathcal{K}_{n},\quad
j=1,\ldots,n,~{}x\in\mathbb{R},$ (3.12)

and

$\hat{\nu}_{l}(x)=(\nu_{l}(x),-G_{n}(\nu_{l}(x),x))\in\mathcal{K}_{n},\quad
l=1,\ldots,n+2,~{}x\in\mathbb{R}.$ (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

$\mu_{j},\nu_{l}\in C(\mathbb{R}),\quad j=1,\ldots,n,~{}l=1,\dots,n+2,$ (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

$(\phi(P,x))=\mathcal{D}_{\hat{\nu}_{1}(x)\hat{\nu}_{2}(x)\underline{\hat{\nu}}(x)}(P)-\mathcal{D}_{P_{\infty_{+}}P_{\infty_{-}}\underline{\hat{\mu}}(x)}(P).$
(3.15)

Here, we abbreviated

$\underline{\hat{\mu}}=\\{\hat{\mu}_{1},\ldots,\hat{\mu}_{n}\\},\quad\underline{\hat{\nu}}=\\{\hat{\nu}_{3},\ldots,\hat{\nu}_{n+2}\\}\in\mathrm{Sym}^{n}(\mathcal{K}_{n}).$
(3.16)

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

######  Lemma 3.1.

Suppose $(\ref{2.1})$, assume the $n$th stationary $CHD2$ equation
$(\ref{2.34})$ holds, and let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\},$
$(x,x_{0})\in\mathbb{R}^{2}$. Then $\phi$ satisfies the Riccati-type equation

$\phi_{x}(P)+\phi(P)^{2}=-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4},$ (3.17)

as well as

$\displaystyle\phi(P)\phi(P^{\ast})=-\frac{H_{n}(z)}{F_{n}(z)},$ (3.18)
$\displaystyle\phi(P)+\phi(P^{\ast})=\frac{2G_{n}(z)}{F_{n}(z)},$ (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 $CHD2$ 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

$\displaystyle\psi_{1}(P,x,x_{0})=\Big{(}\frac{F_{n}(z,x)}{F_{n}(z,x_{0})}\Big{)}^{1/2}\mathrm{exp}\left(y\int_{x_{0}}^{x}F_{n}(z,x^{\prime})^{-1}dx^{\prime}\right),$
(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)
$\displaystyle\psi_{2}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})=-\frac{H_{n}(z,x)}{F_{n}(z,x_{0})},$
(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)
$\displaystyle\psi_{1}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})-\psi_{1}(P^{\ast},x,x_{0})\psi_{2}(P,x,x_{0})=\frac{-2y}{F_{n}(z,x_{0})}.$
(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),
$G_{n}^{2}+F_{n}H_{n}=R_{2n+2}$. This fact reveals the relations between our
approach and the algebro-geometric solutions of the CHD2 hierarchy.

######  Remark 3.3.

The Baker-Akhiezer function $\psi$ of the stationary $CHD2$ 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 $CHD2$ equation $(\ref{2.34})$
holds subject to the constraint $(\ref{2.26a})$.

  * _(i)_

Suppose that the zeros $\\{\mu_{j}(x)\\}_{j=1,\ldots,n}$ of $F_{n}(z,x)$
remain distinct for $x\in\Omega_{\mu},$ where
$\Omega_{\mu}\subseteq\mathbb{R}$ is an open interval, then
$\\{\mu_{j}(x)\\}_{j=1,\ldots,n}$ satisfy the system of differential
equations,

$\mu_{j,x}=-2\frac{y(\hat{\mu}_{j})}{f_{0}}\prod_{\scriptstyle
k=1\atop\scriptstyle k\neq j}^{n}(\mu_{j}(x)-\mu_{k}(x))^{-1},\quad
j=1,\ldots,n,$ (3.27)

with initial conditions

$\\{\hat{\mu}_{j}(x_{0})\\}_{j=1,\ldots,n}\in\mathcal{K}_{n},$ (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

$\hat{\mu}_{j}\in C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),\quad
j=1,\ldots,n.$ (3.29)

  * _(ii)_

Suppose that the zeros $\\{\nu_{l}(x)\\}_{l=1,\ldots,n+2}$ of $H_{n}(z,x)$
remain distinct for $x\in\Omega_{\nu},$ where
$\Omega_{\nu}\subseteq\mathbb{R}$ is an open interval, then
$\\{\nu_{l}(x)\\}_{l=1,\ldots,n+2}$ satisfy the system of differential
equations,

$\nu_{l,x}=2\frac{(\nu_{l}^{2}\rho^{2}-(u-u_{xx})\nu_{l}-\frac{1}{4})y(\hat{\nu}_{l})}{h_{0}}\prod_{\scriptstyle
k=1\atop\scriptstyle k\neq l}^{n+2}(\nu_{l}(x)-\nu_{k}(x))^{-1},\quad
l=1,\ldots,n+2,$ (3.30)

with initial conditions

$\\{\hat{\nu}_{l}(x_{0})\\}_{l=1,\ldots,n+2}\in\mathcal{K}_{n},$ (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

$\hat{\nu}_{l}\in C^{\infty}(\Omega_{\nu},\mathcal{K}_{n}),\quad
l=1,\ldots,n+2.$ (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

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

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

$F_{n,x}(\mu_{j})=2G_{n}(\mu_{j})=2y(\hat{\mu}_{j}).$ (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 [13]. $\square$

Next, we turn to the trace formulas of the CHD2 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 $CHD2$ equation
$(\ref{2.34})$ holds, and let $x\in\mathbb{R}.$ Then

$u^{-1}\mathcal{G}(-4u_{x}(u-u_{xx})-2u(u_{x}-u_{xxx}))=\sum_{j=1}^{n}\mu_{j}(x)-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}.$
(3.35)

Proof. Equation (3.35) follows by considering the coefficient of $z^{n-1}$ in
$F_{n}$ in (2.23) and (3.11), which yields

$\mathcal{G}(-4u_{x}(u-u_{xx})-2u(u_{x}-u_{xxx}))-c_{1}u=-f_{0}\sum_{j=1}^{n}\mu_{j}.$
(3.36)

The constant $c_{1}$ can be determined by 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 CHD2 hierarchy

In this section, we obtain explicit Riemann theta function representations for
the meromorphic function $\phi$, the Baker-Akhiezer function $\psi$, and
especially, for the solutions $u,\rho$ of the stationary CHD2 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 $CHD2$ 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

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

and

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

$\phi\underset{z\rightarrow\infty}{=}\phi_{-1}z+\phi_{0}+O(z^{-1})$ (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 fact that $P_{\infty\pm}$, are the essential singularities of
$\psi_{j}$, $j=1,2$. In addition, one easily finds the leading-order
exponential term in $\psi_{j}$, $j=1,2,$ near $P_{\infty_{\pm}}$ 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
$\mathcal{K}_{n}$

$\eta_{l}(P)=\frac{z^{l-1}}{y(P)}dz,\quad l=1,\ldots,n,$ (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
$Q_{0}\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\}$. 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
$L_{n}=\\{\underline{z}\in\mathbb{C}^{n}|~{}\underline{z}=\underline{N}+\tau\underline{M},~{}\underline{N},~{}\underline{M}\in\mathbb{Z}^{n}\\}.$

The following result shows the nonlinearity of the Abel map with respect to
the variable $x$, which indicates a characteristic difference between the CHD2
hierarchy and other completely integrable systems such as the KdV and AKNS
hierarchies.

######  Theorem 4.3.

Assume $(\ref{2.26a})$ and suppose that
$\\{\hat{\mu}_{j}(x)\\}_{j=1,\ldots,n}$ satisfies the stationary Dubrovin
equations $(\ref{3.38})$ on an open interval $\Omega_{\mu}\subseteq\mathbb{R}$
such that $\mu_{j}(x),j=1,\ldots,n,$ remain distinct and nonzero for
$x\in\Omega_{\mu}.$ Introducing the associated divisor
$\mathcal{D}_{\underline{\hat{\mu}}(x)}\in\mathrm{Sym}^{n}(\mathcal{K}_{n})$,
one computes

$\partial_{x}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)})=\frac{2}{u(x)}\underline{c}(n),\qquad
x\in\Omega_{\mu}.$ (4.14)

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

Proof. Let $x\in\Omega_{\mu}.$ One finds

$\displaystyle\partial_{x}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)})$
$\displaystyle=\partial_{x}\left(\sum_{j=1}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}}\underline{\omega}\right)=\sum_{j=1}^{n}\mu_{j,x}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{y(\hat{\mu}_{j})}$
$\displaystyle=\sum_{j=1}^{n}\sum_{k=1}^{n}-\frac{2}{f_{0}}\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=1\atop\scriptstyle l\neq j}^{n}(\mu_{j}-\mu_{l})}\underline{c}(k)$
$\displaystyle=-\frac{2}{f_{0}}\sum_{k=1}^{n}\underline{c}(k)\delta_{k,n}=-\frac{2}{f_{0}}\underline{c}(n),$
(4.15)

where we used the notation
$\underline{\omega}=(\omega_{1},\ldots,\omega_{n})$, and a special case of
Lagrange’s interpolation formula (cf. Theorem E.1 [13]),

$\sum_{j=1}^{n}\mu_{j}^{k-1}\prod_{\scriptstyle l=1\atop\scriptstyle l\neq
j}^{n}(\mu_{j}-\mu_{l})^{-1}=\delta_{k,n},\quad j,k=1,\ldots,n.$ (4.16)

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

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

$\omega_{P_{\infty_{+}},\hat{\nu}_{1}(x)}^{(3)}(P)=\frac{y-G_{n}(\nu_{1})}{z-\nu_{1}}\frac{dz}{2y}-\frac{1}{2y}\prod_{j=1}^{n}(z-\lambda_{j})dz$
(4.17)

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

$\displaystyle\omega_{P_{\infty_{+}},\hat{\nu}_{1}(x)}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}(\zeta^{-1}+O(1))d\zeta,\quad\textrm{as $P\rightarrow P_{\infty_{+}},$}$
(4.18)
$\displaystyle\omega_{P_{\infty_{+}},\hat{\nu}_{1}(x)}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}(-\zeta^{-1}+O(1))d\zeta,\quad\textrm{as
$P\rightarrow\hat{\nu}_{1}(x),$}$ (4.19)

where $\zeta$ in (4.18) denotes the local coordinate

$\zeta=z^{-1}~{}~{}\textrm{for $P$ near $P_{\infty_{+}}$},$

near $P_{\infty_{+}}$, and analogously, $\zeta$ in (4.19) that near
$\hat{\nu}_{1}(x).$ The constants $\\{\lambda_{j}\\}_{j=1,\ldots,n}$ in (4.17)
are determined by the normalization condition

$\int_{a_{k}}\omega_{P_{\infty_{+}},\hat{\nu}_{1}(x)}^{(3)}=0,\qquad
k=1,\ldots,n.$

Similarly, let $\omega_{P_{\infty_{-}},\hat{\nu}_{2}(x)}^{(3)}(P)$ be another
normalized differential of the third kind holomorphic on
$\mathcal{K}_{n}\setminus\\{P_{\infty_{-}},\hat{\nu}_{2}(x)\\}$ with simple
poles at $P_{\infty_{-}}$ and $\hat{\nu}_{2}(x)$ and residues $1$ and $-1$,
respectively,

$\displaystyle\omega_{P_{\infty_{-}},\hat{\nu}_{2}(x)}^{(3)}(P)=\frac{y-G_{n}(\nu_{2})}{z-\nu_{2}}\frac{dz}{2y}+\frac{1}{2y}\prod_{j=1}^{n}(z-\gamma_{j})dz,$
(4.20)
$\displaystyle\omega_{P_{\infty_{-}},\hat{\nu}_{2}(x)}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}(\zeta^{-1}+O(1))d\zeta,\quad\textrm{as $P\rightarrow P_{\infty_{-}},$}$
(4.21)
$\displaystyle\omega_{P_{\infty_{-}},\hat{\nu}_{2}(x)}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}(-\zeta^{-1}+O(1))d\zeta,\quad\textrm{as
$P\rightarrow\hat{\nu}_{2}(x),$}$ (4.22)

where $\zeta$ in (4.21) denotes the local coordinate

$\zeta=z^{-1}~{}~{}\textrm{for $P$ near $P_{\infty_{-}}$},$

near $P_{\infty_{-}}$, and analogously, $\zeta$ in (4.22) that near
$\hat{\nu}_{2}(x).$ The constants $\\{\gamma_{j}\\}_{j=1,\ldots,n}$ in (4.20)
are determined by the normalization condition

$\int_{a_{k}}\omega_{P_{\infty_{-}},\hat{\nu}_{2}(x)}^{(3)}=0,\qquad
k=1,\ldots,n.$

We define

$\Omega^{(3)}=\omega_{P_{\infty_{+}},\hat{\nu}_{1}(x)}^{(3)}+\omega_{P_{\infty_{-}},\hat{\nu}_{2}(x)}^{(3)}.$
(4.23)

Then

$\int_{Q_{0}}^{P}\Omega^{(3)}\underset{\zeta\rightarrow
0}{=}\mathrm{ln}\zeta+d_{0}+O(\zeta),\quad\textrm{as $P\rightarrow
P_{\infty_{\pm}},$ }$ (4.24)

for some constant $d_{0}\in\mathbb{C}$.

Next, let $\omega_{P_{\infty_{\pm}},0}^{(2)}$ be normalized differentials of
the second kind, satisfying

$\displaystyle\int_{a_{k}}\omega_{P_{\infty_{\pm}},0}^{(2)}=0,\quad
k=1,\ldots,n,$ (4.25)
$\displaystyle\omega_{P_{\infty_{\pm}},0}^{(2)}\underset{\zeta\rightarrow
0}{=}(\zeta^{-2}+O(1))d\zeta,\quad\textrm{as $P\rightarrow P_{\infty_{\pm}}.$
}$ (4.26)

We introduce

$\Omega_{0}^{(2)}=\omega_{P_{\infty_{-}},0}^{(2)}-\omega_{P_{\infty_{+}},0}^{(2)}.$
(4.27)

Then

$\int_{Q_{0}}^{P}\Omega_{0}^{(2)}\underset{\zeta\rightarrow
0}{=}\pm(\zeta^{-1}+e_{0,0}+O(\zeta)),\quad\textrm{as $P\rightarrow
P_{\infty_{\pm}},$ }$ (4.28)

for some constant $e_{0,0}\in\mathbb{C}$. In the following, it will be
convenient to introduce the abbreviations

$\displaystyle\underline{z}(P,\underline{Q})=\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{Q}}),$
$\displaystyle
P\in\mathcal{K}_{n},\,\underline{Q}=(Q_{1},\ldots,Q_{n})\in\mathrm{Sym}^{n}(\mathcal{K}_{n}),$
(4.29)

where $\underline{\Xi}_{Q_{0}}$ is the vector of Riemann constants (cf.(A.45)
[13]). It turns out that $\underline{z}(\cdot,\underline{Q})$ is independent
of the choice of base point $Q_{0}$ (cf.(A.52), (A.53) [13]).

Based on above preparations, we will give explicit representations for the
meromorphic function $\phi$, the Baker-Akhiezer function $\psi$, and the
stationary CHD2 solutions $u,\rho$ in terms of the Riemann theta function
associated with $\mathcal{K}_{n}$.

######  Theorem 4.4.

Suppose $(\ref{2.1})$, and assume the $n$th stationary $CHD2$ equation
$(\ref{2.34})$ holds on $\Omega$ subject to the constraint $(\ref{2.26a})$.
Moreover, let $P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}}\\}$ and
$x,x_{0}\in\Omega$, 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, $\phi$, $\psi$, $u,$ and $\rho$ admit the following
representations

$\displaystyle\phi(P,x)=i\rho(x)\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(x)))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x)))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x)))\theta(\underline{z}(P,\underline{\hat{\mu}}(x)))}\mathrm{exp}\left(d_{0}-\int_{Q_{0}}^{P}\Omega^{(3)}\right),$
(4.30)
$\displaystyle\psi_{1}(P,x,x_{0})=\frac{\theta(\underline{z}(P,\underline{\hat{\mu}}(x)))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x_{0})))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x)))\theta(\underline{z}(P,\underline{\hat{\mu}}(x_{0})))}\mathrm{exp}\left(\int_{x_{0}}^{x}dx^{\prime}~{}i\rho(x^{\prime})\int_{Q_{0}}^{P}\Omega_{0}^{(2)}\right),$
(4.31)
$\displaystyle\psi_{2}(P,x,x_{0})=i\rho(x)\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(x)))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x_{0})))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x)))\theta(\underline{z}(P,\underline{\hat{\mu}}(x_{0})))}\mathrm{exp}\left(d_{0}-\int_{Q_{0}}^{P}\Omega^{(3)}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times~{}\mathrm{exp}\left(\int_{x_{0}}^{x}dx^{\prime}~{}i\rho(x^{\prime})\int_{Q_{0}}^{P}\Omega_{0}^{(2)}\right),$
(4.32) $\displaystyle
u(x)=-4i\frac{\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x)))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x)))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x)))\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x)))},$
(4.33)
$\displaystyle\rho(x)=-\frac{i}{2}\frac{\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x)))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x)))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x)))\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x)))}.$
(4.34)

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 $x\in\widetilde{\Omega}$},$ (4.35)

for appropriate $\widetilde{\Omega}\subseteq\Omega$. Since by (3.15),
$\mathcal{D}_{\hat{\nu}_{1}\hat{\nu}_{2}\underline{\hat{\nu}}}\sim\mathcal{D}_{P_{\infty_{+}}P_{\infty_{-}}\underline{\hat{\mu}}}$,
and
$(P_{\infty_{+}})^{\ast}=P_{\infty_{-}}\notin\\{\hat{\mu}_{1},\ldots,\hat{\mu}_{n}\\}$
by hypothesis, one can use Theorem A.31 [13] 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 $\phi$, $\psi$, $u$, and $\rho$ in
terms of the Riemann theta function. A special case of Riemann’s vanishing
theorem (cf. Theorem A.26 [13]) 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 $P\in\\{Q_{1},\ldots,Q_{n}\\}$}.$ (4.36)

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

$C(x)\frac{\theta(\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x)}))}{\theta(\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)}))}\mathrm{exp}\Big{(}d_{0}-\int_{Q_{0}}^{P}\Omega^{(3)}\Big{)},$
(4.37)

where $C(x)$ is independent of $P\in\mathcal{K}_{n}$. Then taking into account
the asymptotic expansion of $\phi(P,x)$ near $P_{\infty_{+}}$ in (4.1), we
obtain (4.30).

Now, let $\Psi$ denote the right-hand side of (4.31). We intend to prove
$\psi_{1}=\Psi$ with $\psi_{1}$ given by (3.8). For that purpose we first
investigate the local zeros and poles of $\psi_{1}$. Since they can only come
from zeros of $F_{n}(z,x^{\prime})$ in (3.8), one computes using (3.12), the
definition (3) of $\phi$, and the Dubrovin equations (3.27),

$\displaystyle\phi(P,x)$
$\displaystyle=\frac{y+G_{n}}{F_{n}}=\frac{y}{F_{n}}+\frac{F_{n,x}}{2F_{n}}$
$\displaystyle=-\frac{1}{2}\frac{\mu_{j,x}}{z-\mu_{j}}-\frac{1}{2}\frac{\mu_{j,x}}{z-\mu_{j}}+O(1)$
$\displaystyle=-\frac{\mu_{j,x}}{z-\mu_{j}}+O(1),\quad\textrm{as
$z\rightarrow\mu_{j}(x)$,}$ (4.38)

where

$y\rightarrow y(\hat{\mu}_{j}(x))=G_{n}(\mu_{j}(x)),\quad\textrm{as
$z\rightarrow\mu_{j}(x)$.}$

More concisely,

$\phi(P,x)=\frac{\partial}{\partial
x}\mathrm{ln}(z-\mu_{j}(x))+O(1)\quad\textrm{for $P$ near
$\hat{\mu}_{j}(x)$},$ (4.39)

which together with (3.8) yields

$\displaystyle\psi_{1}(P,x,x_{0})$
$\displaystyle=\mathrm{exp}\left(\int_{x_{0}}^{x}dx^{\prime}\left(\frac{\partial}{\partial
x^{\prime}}\mathrm{ln}(z-\mu_{j}(x^{\prime}))+O(1)\right)\right)$
$\displaystyle=\frac{z-\mu_{j}(x)}{z-\mu_{j}(x_{0})}O(1)$
$\displaystyle=\left\\{\begin{array}[]{ll}(z-\mu_{j}(x))O(1)&\hbox{for $P$
near $\hat{\mu}_{j}(x)\neq\hat{\mu}_{j}(x_{0})$,}\\\ O(1)&\hbox{for $P$ near
$\hat{\mu}_{j}(x)=\hat{\mu}_{j}(x_{0})$,}\\\
(z-\mu_{j}(x_{0}))^{-1}O(1)&\hbox{for $P$ near
$\hat{\mu}_{j}(x_{0})\neq\hat{\mu}_{j}(x)$,}\end{array}\right.$ (4.43)

where $O(1)\neq 0$ in (4.43). Consequently, $\psi_{1}$ and $\Psi$ have
identical zeros and poles on
$\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\}$, which are all
simple by hypothesis (4.35). It remains to identify the behavior of $\psi_{1}$
and $\Psi$ near $P_{\infty_{\pm}}$. Taking into account (3.8), (4.3), (4.28),
and the expression (4.31) for $\Psi$, one observes that $\psi_{1}$ and $\Psi$
have identical exponential behavior up to order $O(1)$ near
$P_{\infty_{\pm}}$. Thus, $\psi_{1}$ and $\Psi$ share the same singularities
and zeros, and the Riemann-Roch-type uniqueness result (cf. Lemma C.2 [13])
then completes the proof $\psi_{1}=\Psi.$ The representation (4.32) for
$\psi_{2}$ on $\widetilde{\Omega}$ follows using (3.10), (4.30), and (4.31).
The representation (4.34) for $\rho$ on $\widetilde{\Omega}$ is clear from
(4.2) and (4.30). The representation (4.33) for $u$ on $\widetilde{\Omega}$
follows from (4.34) and the relation $u\rho=-2$. Finally, the extension of all
these results from $x\in\widetilde{\Omega}$ to $x\in\Omega$ then simply
follows from the continuity of $\underline{\alpha}_{O_{0}}$ and the hypothesis
of $\mathcal{D}_{\underline{\hat{\mu}}(x)}$ being nonspecial for $x\in\Omega$.
$\square$

######  Remark 4.5.

Since by $(\ref{3.14})$
$\mathcal{D}_{\hat{\nu}_{1}\hat{\nu}_{2}\underline{\hat{\nu}}}$ and
$\mathcal{D}_{P_{\infty_{+}}P_{\infty_{-}}\underline{\hat{\mu}}}$ are linearly
equivalent, that is

$\underline{A}_{Q_{0}}(P_{\infty_{+}})+\underline{A}_{Q_{0}}(P_{\infty_{-}})+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)})=\underline{A}_{Q_{0}}(\hat{\nu}_{1}(x))+\underline{A}_{Q_{0}}(\hat{\nu}_{2}(x))+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x)}).$
(4.44)

Then one infers

$\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)})=\underline{\Delta}+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x)}),\quad\underline{\Delta}=\underline{A}_{P_{\infty_{+}}}(\hat{\nu}_{1}(x))+\underline{A}_{P_{\infty_{-}}}(\hat{\nu}_{2}(x)).$
(4.45)

Hence, one can eliminate $\mathcal{D}_{\underline{\hat{\mu}}(x)}$ in
$(\ref{4.34})$, $(\ref{4.35})$, and $(\ref{4.35a1})$, in terms of
$\mathcal{D}_{\underline{\hat{\nu}}(x)}$ using

$\underline{z}(P,\underline{\hat{\mu}})=\underline{z}(P,\underline{\hat{\nu}})+\underline{\Delta},\qquad
P\in\mathcal{K}_{n}.$ (4.46)

##  5 The time-dependent CHD2 formalism

In this section, we extend the algebro-geometric analysis of Section 3 to the
time-dependent CHD2 hierarchy.

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

The time-dependent algebro-geometric initial value problem of the CHD2
hierarchy is to solve the time-dependent $r$th CHD2 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 CHD2 equation
$\textrm{s-CHD2}_{n}(u^{(0)},\rho^{(0)})=0$ 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 CHD2 flow $\mathrm{CHD2}_{r}(u,\rho)=0$ 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 CHD2 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
CHD2 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

$\displaystyle\mathrm{CHD2}_{r}(u,\rho)=\left(\begin{array}[]{c}-2\rho\rho_{t_{r}}-2(u-u_{xx})\tilde{f}_{r,x}-(u_{x}-u_{xxx})\tilde{f}_{r}-\frac{1}{2}\tilde{f}_{r-1,x}+\frac{1}{2}\tilde{f}_{r-1,xxx}\\\
u_{t_{r}}-u_{xxt_{r}}-\frac{1}{2}\tilde{f}_{r,x}+\frac{1}{2}\tilde{f}_{r,xxx}\\\
\end{array}\right)=0,$ (5.3)
$\displaystyle(u,\rho)|_{t_{r}=t_{0,r}}=(u^{(0)},\rho^{(0)}),$
$\displaystyle\textrm{s-CHD2}_{n}(u^{(0)},\rho^{(0)})=\left(\begin{array}[]{c}-2(u-u_{xx})f_{n,x}-(u_{x}-u_{xxx})f_{n}-\frac{1}{2}f_{n-1,x}+\frac{1}{2}f_{n-1,xxx}\\\
-\frac{1}{2}f_{n,x}+\frac{1}{2}f_{n,xxx}\\\ \end{array}\right)=0,$ (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 CHD2 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

$\begin{split}&U(z)=\left(\begin{array}[]{cc}0&1\\\
-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}&0\\\ \end{array}\right),\\\
&V_{n}(z)=\left(\begin{array}[]{cc}-zG_{n}(z)&zF_{n}(z)\\\
zH_{n}(z)&zG_{n}(z)\\\ \end{array}\right),\\\
&\widetilde{V}_{r}(z)=\left(\begin{array}[]{cc}-z\widetilde{G}_{r}(z)&z\widetilde{F}_{r}(z)\\\
z\widetilde{H}_{r}(z)&z\widetilde{G}_{r}(z)\\\ \end{array}\right),\end{split}$
(5.9)

and

$\displaystyle
F_{n}(z)=\sum_{l=0}^{n}f_{l}z^{n-l}=f_{0}\prod_{j=1}^{n}(z-\mu_{j}),$ (5.10)
$\displaystyle G_{n}(z)=\sum_{l=0}^{n}g_{l}z^{n-l},$ (5.11) $\displaystyle
H_{n}(z)=\sum_{l=0}^{n+2}h_{l}z^{n+2-l}=h_{0}\prod_{l=1}^{n+2}(z-\nu_{l}),$
(5.12) $\displaystyle\widetilde{F}_{r}(z)=\sum_{s=0}^{r}\tilde{f}_{s}z^{r-s},$
(5.13) $\displaystyle\widetilde{G}_{r}(z)=\sum_{s=0}^{r}\tilde{g}_{s}z^{r-s},$
(5.14)
$\displaystyle\widetilde{H}_{r}(z)=\sum_{s=0}^{r+2}\tilde{h}_{s}z^{r+2-s},$
(5.15)

for fixed $n,r\in\mathbb{N}_{0}$. Here, $\\{f_{l}\\}_{l=0,\ldots,n},$
$\\{g_{l}\\}_{l=0,\ldots,n}$, $\\{h_{l}\\}_{l=0,\ldots,n+2}$,
$\\{\tilde{f}_{s}\\}_{s=0,\ldots,r},$ $\\{\tilde{g}_{s}\\}_{s=0,\ldots,r}$,
and $\\{\tilde{h}_{s}\\}_{s=0,\ldots,r+2}$ 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

$\displaystyle-2z\rho\rho_{t_{r}}+(u_{t_{r}}-u_{xxt_{r}})-\widetilde{H}_{r,x}-2\Big{(}-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}\Big{)}\widetilde{G}_{r}=0,$
(5.16) $\displaystyle\widetilde{F}_{r,x}=2\widetilde{G}_{r},$ (5.17)
$\displaystyle\widetilde{G}_{r,x}=-\widetilde{H}_{r}+\Big{(}-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}\Big{)}\widetilde{F}_{r}$
(5.18)

and

$\displaystyle F_{n,x}=2G_{n},$ (5.19) $\displaystyle
H_{n,x}=-2\Big{(}-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}\Big{)}G_{n},$ (5.20)
$\displaystyle
G_{n,x}=-H_{n}+\Big{(}-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4}\Big{)}F_{n}.$
(5.21)

From (5.19)-(5.21), one finds

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

and meanwhile (see Lemma 5.2)

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

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

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

and the hyperelliptic curve $\mathcal{K}_{n}$ is still given by (2.21).

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

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

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

such that

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

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

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

and

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

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

$\displaystyle\hat{\mu}_{j}(x,t_{r})=(\mu_{j}(x,t_{r}),G_{n}(\mu_{j}(x,t_{r}),x,t_{r}))\in\mathcal{K}_{n},\quad
j=1,\ldots,n,~{}(x,t_{r})\in\mathbb{R}^{2},$ (5.30)
$\displaystyle\hat{\nu}_{l}(x,t_{r})=(\nu_{l}(x,t_{r}),-G_{n}(\nu_{l}(x,t_{r}),x,t_{r}))\in\mathcal{K}_{n},\quad
l=1,\ldots,n+2,~{}(x,t_{r})\in\mathbb{R}^{2}.$ (5.31)

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

$(\phi(P,x,t_{r}))=\mathcal{D}_{\hat{\nu}_{1}(x,t_{r})\hat{\nu}_{2}(x,t_{r})\underline{\hat{\nu}}(x,t_{r})}(P)-\mathcal{D}_{P_{\infty_{+}}P_{\infty_{-}}\underline{\hat{\mu}}(x,t_{r})}(P)$
(5.32)

with

$\underline{\hat{\mu}}=\\{\hat{\mu}_{1},\ldots,\hat{\mu}_{n}\\},\quad\underline{\hat{\nu}}=\\{\hat{\nu}_{3},\ldots,\hat{\nu}_{n+2}\\}\in\mathrm{Sym}^{n}(\mathcal{K}_{n}).$
(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

$\displaystyle\phi_{x}(P)+\phi(P)^{2}=-z^{2}\rho^{2}+z(u-u_{xx})+\frac{1}{4},$
(5.34)
$\displaystyle\phi_{t_{r}}(P)=(-\widetilde{G}_{r}(z)+\widetilde{F}_{r}(z)\phi(P))_{x}$
(5.35)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=\widetilde{H}_{r}(z)+\Big{(}z^{2}\rho^{2}-z(u-u_{xx})-\frac{1}{4}\Big{)}\widetilde{F}_{r}(z)+(\widetilde{F}_{r}(z)\phi(P))_{x},$
$\displaystyle\phi_{t_{r}}(P)=\widetilde{H}_{r}(z)+2\widetilde{G}_{r}(z)\phi(P)-\widetilde{F}_{r}(z)\phi(P)^{2},$
(5.36) $\displaystyle\phi(P)\phi(P^{\ast})=-\frac{H_{n}(z)}{F_{n}(z)},$ (5.37)
$\displaystyle\phi(P)+\phi(P^{\ast})=\frac{2G_{n}(z)}{F_{n}(z)},$ (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

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

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

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

$-V_{n,t_{r}}+[z^{-1}\widetilde{V}_{r},V_{n}]=0.$ (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

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

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

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

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

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

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

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

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

The integrand in the above integral equals

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

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

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

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

$\Big{(}\frac{F_{n}(z,x,t_{r})}{F_{n}(z,x_{0},t_{r})}\Big{)}^{1/2}\mathrm{exp}\left(y\int_{x_{0}}^{x}dx^{\prime}F_{n}(z,x^{\prime},t_{r})^{-1}\right),$
(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
$\\{\mu_{j}(x,t_{r})\\}_{j=1,\ldots,n}$ and
$\\{\nu_{l}(x,t_{r})\\}_{l=1,\ldots,n+2}$ 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 $\\{\mu_{j}(x,t_{r})\\}_{j=1,\ldots,n}$ 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
$\\{\mu_{j}(x,t_{r})\\}_{j=1,\ldots,n}$ satisfy the system of differential
equations,

$\displaystyle\mu_{j,x}=-2\frac{y(\hat{\mu}_{j})}{f_{0}}\prod_{\scriptstyle
k=1\atop\scriptstyle k\neq j}^{n}(\mu_{j}-\mu_{k})^{-1},\quad j=1,\ldots,n,$
(5.59)
$\displaystyle\mu_{j,t_{r}}=-2\frac{\widetilde{F}_{r}(\mu_{j})y(\hat{\mu}_{j})}{f_{0}}\prod_{\scriptstyle
k=1\atop\scriptstyle k\neq j}^{n}(\mu_{j}-\mu_{k})^{-1},\quad j=1,\ldots,n,$
(5.60)

with initial conditions

$\\{\hat{\mu}_{j}(x_{0},t_{0,r})\\}_{j=1,\ldots,n}\in\mathcal{K}_{n},$ (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

$\hat{\mu}_{j}\in C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),\quad
j=1,\ldots,n.$ (5.62)

  * _(ii)_

Suppose that the zeros $\\{\nu_{l}(x,t_{r})\\}_{l=1,\ldots,n+2}$ 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
$\\{\nu_{l}(x,t_{r})\\}_{l=1,\ldots,n+2}$ satisfy the system of differential
equations,

$\displaystyle\nu_{l,x}=2\frac{(\nu_{l}^{2}\rho^{2}-(u-u_{xx})\nu_{l}-\frac{1}{4})y(\hat{\nu}_{l})}{h_{0}}\prod_{\scriptstyle
k=1\atop\scriptstyle k\neq l}^{n+2}(\nu_{l}-\nu_{k})^{-1},\quad
l=1,\ldots,n+2,$ (5.63)
$\displaystyle\nu_{l,t_{r}}=-2\frac{\widetilde{H}_{r}(\nu_{l})y(\hat{\nu}_{l})}{h_{0}}\prod_{\scriptstyle
k=1\atop\scriptstyle k\neq l}^{n+2}(\nu_{l}-\nu_{k})^{-1},\quad
l=1,\ldots,n+2,$ (5.64)

with initial conditions

$\\{\hat{\nu}_{l}(x_{0},t_{0,r})\\}_{l=1,\ldots,n+2}\in\mathcal{K}_{n},$
(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

$\hat{\nu}_{l}\in C^{\infty}(\Omega_{\nu},\mathcal{K}_{n}),\quad
l=1,\ldots,n+2.$ (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

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

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

$F_{n,t_{r}}(\mu_{j})=2G_{n}(\mu_{j})\widetilde{F}_{r}(\mu_{j})=2y(\hat{\mu}_{j})\widetilde{F}_{r}(\mu_{j}).$
(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 CHD2 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,

$u^{-1}\mathcal{G}(-4u_{x}(u-u_{xx})-2u(u_{x}-u_{xxx}))=\sum_{j=1}^{n}\mu_{j}(x,t_{r})-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}.$
(5.69)

##  6 Time-dependent algebro-geometric solutions of CHD2 hierarchy

In our final section, we extend the results of section 4 from the stationary
CHD2 hierarchy, to the time-dependent case. We obtain Riemann theta function
representations for the meromorphic function $\phi$, the Baker-Akhiezer
function $\psi$, and especially, for the algebro-geometric solutions $u,\rho$
of the whole CHD2 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,

$\displaystyle\phi(P)\underset{\zeta\rightarrow 0}{=}\pm
i\rho\zeta^{-1}+\frac{\mp i(u-u_{xx})-\rho_{x}}{2\rho}+O(\zeta),\quad
P\rightarrow P_{\infty_{\pm}},\quad\zeta=z^{-1},$ (6.1)
$\displaystyle\phi(P)\underset{\zeta\rightarrow
0}{=}\frac{1}{2}+(u-u_{x})\zeta+O(\zeta^{2}),\quad P\rightarrow
P_{0},\quad\zeta=z.$ (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
$\underline{\mu}=(\mu_{1},\ldots,\mu_{n})\in\mathbb{C}^{n}$, we define the
following symmetric functions by

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

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

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

where
$\mathcal{T}_{k}^{(j)}=\\{\underline{l}=(l_{1},\ldots,l_{k})\in\mathcal{S}_{k}~{}|~{}l_{m}\neq
j\\},$ $j=1,\ldots,n.$ For the properties of $\Psi_{k}(\underline{\mu})$ and
$\Phi_{k}^{(j)}(\underline{\mu})$, we refer to Appendix E [13].

Introducing

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

for a given set of constants
$\\{\tilde{c}_{l}\\}_{l=1,\ldots,r}\subset\mathbb{C}$, the corresponding
homogeneous and nonhomogeneous quantities $\widehat{F}_{r}(\mu_{j})$ and
$\widetilde{F}_{r}(\mu_{j})$ in the CHD2 case are then given by 333$m\wedge
n=\mathrm{min}\\{m,n\\}$, $m\vee n=\mathrm{max}\\{m,n\\}$

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

using (D.59) and (D.60) [13]. Here, $\hat{c}_{s}(\underline{E})$,
$s\in\mathbb{N}_{0}$, is defined by (D.2) [13].

We now state the analog of Theorem 4.3, which indicates marked differences
between the CHD2 hierarchy and other completely integrable systems such as the
KdV and AKNS hierarchies.

######  Theorem 6.2.

Assume $(\ref{2.26a})$ and suppose that $\\{\hat{\mu}_{j}\\}_{j=1,\ldots,n}$
satisfies the Dubrovin equations $(\ref{5.56})$, $(\ref{5.57})$ on an open set
$\Omega_{\mu}\subseteq\mathbb{R}^{2}$ such that $\mu_{j}$, $j=1,\ldots,n,$
remain distinct and nonzero on $\Omega_{\mu}$ and that
$\widetilde{F}_{r}(\mu_{j})\neq 0$ on $\Omega_{\mu}$, $j=1,\ldots,n$.
Introducing the associated divisor
$\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})}\in\mathrm{Sym}^{n}(\mathcal{K}_{n})$,
one computes,

$\displaystyle\partial_{x}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})})=\frac{2}{u(x,t_{r}))}\underline{c}(n),\quad(x,t_{r})\in\Omega_{\mu},$
(6.7)
$\displaystyle\partial_{t_{r}}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})})=-2\Big{(}\sum_{\ell=0\vee(n-r)}^{n}\tilde{d}_{r,n-\ell}(\underline{E})\underline{c}(\ell)\Big{)},\quad(x,t_{r})\in\Omega_{\mu}.$
(6.8)

In particular, the Abel map dose not linearize the divisor
$\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})}$ 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. Using (5.60), (6.6),
and (E.4) [13], one infers that

$\displaystyle\partial_{t_{r}}\Big{(}\sum_{j=1}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}}\underline{\omega}\Big{)}$
$\displaystyle=\sum_{j=1}^{n}\mu_{j,t_{r}}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{y(\hat{\mu}_{j})}$
$\displaystyle=-2\sum_{j=1}^{n}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=1\atop\scriptstyle l\neq
j}^{n}(\mu_{j}-\mu_{l})}\frac{\widetilde{F}_{r}(\mu_{j})}{f_{0}}$
$\displaystyle=-\frac{2}{f_{0}}\sum_{j=1}^{n}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=1\atop\scriptstyle l\neq
j}^{n}(\mu_{j}-\mu_{l})}\Big{(}f_{0}\sum_{m=0}^{r\wedge
n}\tilde{d}_{r,m}(\underline{E})\Phi_{m}^{(j)}(\underline{\mu})\Big{)}$
$\displaystyle=-2\sum_{m=0}^{r\wedge
n}\tilde{d}_{r,m}(\underline{E})\sum_{k=1}^{n}\sum_{j=1}^{n}\underline{c}(k)(U_{n}(\underline{\mu}))_{k,j}(U_{n}(\underline{\mu}))_{j,n-m}^{-1}$
$\displaystyle=-2\sum_{m=0}^{r\wedge
n}\tilde{d}_{r,m}(\underline{E})\underline{c}(n-m)$
$\displaystyle=-2\sum_{m=0\vee(n-r)}^{n}\tilde{d}_{r,n-m}(\underline{E})\underline{c}(m),$
(6.9)

where we used the relations (cf.(E.13), (E.14) [13]),

$U_{n}(\underline{\mu})=\left(\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=1\atop\scriptstyle l\neq j}^{n}(\mu_{j}-\mu_{l})}\right)_{\scriptstyle
j,k=1}^{n},\quad
U_{n}(\underline{\mu})^{-1}=\left(\Phi_{n-k}^{(j)}(\underline{\mu})\right)_{j,k=1}^{n}.$
(6.10)

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

For subsequent purpose we note the following asymptotic spectral parameter
expansion of $F_{n}/y$ as $P\rightarrow P_{\infty_{\pm}}$,

$\frac{F_{n}(z)}{y}\underset{\zeta\rightarrow
0}{=}\pm\frac{1}{2i}\sum_{l=0}^{\infty}\hat{f}_{l}\zeta^{l+1},\quad\textrm{as
$P\rightarrow P_{\infty_{\pm}},\quad\zeta=z^{-1}.$}$ (6.11)

Here, $\hat{f}_{l}$ denote the homogenous coefficients, satisfying (2.3) with
vanishing integration constants.

Next, we shall provide the explicit representations of $\psi$, $\phi$, $u,$
and $\rho$ in terms of the Riemann theta function associated with
$\mathcal{K}_{n}$, assuming the affine part of $\mathcal{K}_{n}$ to be
nonsingular.

Let $\omega_{P_{\infty_{\pm}},r}^{(2)}$ be the normalized differentials of the
second kind with a unique pole at $P_{\infty_{\pm}}$, and principal part
$\zeta^{-2-r}d\zeta$ near $P_{\infty_{\pm}}$, and define

$\widetilde{\Omega}_{r}^{(2)}=\sum_{q=0}^{r}(q+1)\tilde{c}_{r-q}(\omega_{P_{\infty_{-}},q}^{(2)}-\omega_{P_{\infty_{+}},q}^{(2)}),\quad\tilde{c}_{0}=1,$
(6.12)

where $\tilde{c}_{q},$ $q=0,\ldots,r,$ are the constants introduced in the
definition of $\widetilde{F}_{r}(z)$. Hence, one infers

$\displaystyle\int_{a_{k}}\widetilde{\Omega}_{r}^{(2)}=0,\quad k=1,\ldots,n,$
(6.13)
$\displaystyle\int_{Q_{0}}^{P}\widetilde{\Omega}_{r}^{(2)}\underset{\zeta\rightarrow
0}{=}\pm\Big{(}\sum_{q=0}^{r}\tilde{c}_{r-q}\zeta^{-1-q}+\tilde{e}_{r,0}+O(\zeta)\Big{)},\quad\textrm{as
$P\rightarrow P_{\infty_{\pm}},$}$ (6.14)

for some constants $\tilde{e}_{r,0}\in\mathbb{C}$.

Recalling (4.17)-(4), the analog of Theorem 4.4 in the stationary case then
reads as follows.

######  Theorem 6.3.

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
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}}\\}$ 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, $\phi$, $\psi$, $u$, and $\rho$ admit the
representations

$\displaystyle\phi(P,x,t_{r})=i\rho(x,t_{r})\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x,t_{r})))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P,\underline{\hat{\mu}}(x,t_{r})))}\mathrm{exp}\left(d_{0}-\int_{Q_{0}}^{P}\Omega^{(3)}\right),$
(6.15)
$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})=\frac{\theta(\underline{z}(P,\underline{\hat{\mu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x_{0},t_{0,r})))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x,t_{r})))\theta(\underline{z}(P,\underline{\hat{\mu}}(x_{0},t_{0,r})))}$
(6.16)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times~{}\mathrm{exp}\left(\int_{x_{0}}^{x}dx^{\prime}~{}i\rho(x^{\prime},t_{r})\int_{Q_{0}}^{P}\Omega_{0}^{(2)}+2i(t_{r}-t_{0,r})\int_{Q_{0}}^{P}\widetilde{\Omega}_{r}^{(2)}\right),$
$\displaystyle\psi_{2}(P,x,x_{0},t_{r},t_{0,r})=i\rho(x,t_{r})\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x_{0},t_{0,r})))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P,\underline{\hat{\mu}}(x_{0},t_{0,r})))}\mathrm{exp}\left(d_{0}-\int_{Q_{0}}^{P}\Omega^{(3)}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times~{}\mathrm{exp}\left(\int_{x_{0}}^{x}dx^{\prime}~{}i\rho(x^{\prime},t_{r})\int_{Q_{0}}^{P}\Omega_{0}^{(2)}+2i(t_{r}-t_{0,r})\int_{Q_{0}}^{P}\widetilde{\Omega}_{r}^{(2)}\right),$
(6.17) $\displaystyle
u(x,t_{r})=-4i\frac{\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x,t_{r})))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x,t_{r})))},$
(6.18)
$\displaystyle\rho(x,t_{r})=-\frac{i}{2}\frac{\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\nu}}(x,t_{r})))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x,t_{r})))\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x,t_{r})))}.$
(6.19)

Proof. We start with the proof of the theta function representation (6.16) for
$\psi_{1}$. Without loss of generality it suffices to treat the homogenous
case $\tilde{c}_{0}=1$, $\tilde{c}_{q}=0$, $q=1,\ldots,r$. As in the
corresponding stationary case we temporarily assume

$\mu_{j}(x,t_{r})\neq\mu_{j^{\prime}}(x,t_{r}),\quad\textrm{for $j\neq
j^{\prime}$ and $(x,t_{r})\in\widetilde{\Omega}$}$ (6.20)

for appropriate $\widetilde{\Omega}\subseteq\Omega$, and define the right-hand
side of (6.16) to be $\Psi$. We intend to prove $\psi_{1}=\Psi$, with
$\psi_{1}$ given by (5). For that purpose we first investigate the local zeros
and poles of $\psi_{1}$. Using the definition (5) of $\phi$, (5.42), and
Dubrovin equations (5.60), one computes

$\displaystyle\widetilde{F}_{r}(z,x,t_{r})\phi(P,x,t_{r})-\widetilde{G}_{r}(z,x,t_{r})=\widetilde{F}_{r}(z,x,t_{r})\frac{y+G_{n}(z,x,t_{r})}{F_{n}(z,x,t_{r})}-\widetilde{G}_{r}(z,x,t_{r})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{y~{}\widetilde{F}_{r}(z,x,t_{r})}{F_{n}(z,x,t_{r})}+\frac{1}{2}\frac{F_{n,t_{r}}(z,x,t_{r})}{F_{n}(z,x,t_{r})}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=-\frac{1}{2}\frac{\mu_{j,t_{r}}}{z-\mu_{j}}-\frac{1}{2}\frac{\mu_{j,t_{r}}}{z-\mu_{j}}+O(1)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=-\frac{\mu_{j,t_{r}}}{z-\mu_{j}}+O(1),\quad\textrm{as
$z\rightarrow\mu_{j}(x,t_{r})$. }$ (6.21)

More concisely,

$\displaystyle\widetilde{F}_{r}(z,x_{0},s)\phi(P,x_{0},s)-\widetilde{G}_{r}(z,x_{0},s)=\frac{\partial}{\partial
s}\mathrm{ln}(z-\mu_{j}(x_{0},s))+O(1)$ for $P$ near $\hat{\mu}_{j}(x_{0},s)$.
(6.22)

Meanwhile, (4.39) gives

$\phi(P,x^{\prime})=\frac{\partial}{\partial
x^{\prime}}\mathrm{ln}(z-\mu_{j}(x^{\prime},t_{r}))+O(1)\quad\textrm{for $P$
near $\hat{\mu}_{j}(x^{\prime},t_{r})$}.$ (6.23)

Hence, combining (5), (6), and (6.23) yields

$\psi_{1}(P,x,x_{0},t_{r},t_{0,r})=\left\\{\begin{array}[]{ll}(z-\mu_{j}(x,t_{r}))O(1),&\hbox{{for
$P$ near $\hat{\mu}_{j}(x,t_{r})\neq\hat{\mu}_{j}(x_{0},t_{0,r}),$}}\\\
O(1),&\hbox{{for $P$ near
$\hat{\mu}_{j}(x,t_{r})=\hat{\mu}_{j}(x_{0},t_{0,r}),$}}\\\
(z-\mu_{j}(x_{0},t_{0,r}))^{-1}O(1),&\hbox{{for $P$ near
$\hat{\mu}_{j}(x_{0},t_{0,r})\neq\hat{\mu}_{j}(x,t_{r}),$}}\end{array}\right.$
(6.24)

with $O(1)\neq 0$. Consequently, $\psi_{1}$ and $\Psi$ have identical zeros
and poles on $\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\}$,
which are all simple by hypothesis (6.20). It remains to study the behavior of
$\psi_{1}$ near $P_{\infty_{\pm}}$. By (5), (5.42), (6.1), and (6.11), one
infers that

$\displaystyle\int_{x_{0}}^{x}dx^{\prime}\phi(P,x^{\prime},t_{r})+\int_{t_{0,r}}^{t_{r}}ds~{}(\widetilde{F}_{r}(\zeta^{-1},x_{0},s)\phi(P,x_{0},s)-\widetilde{G}_{r}(\zeta^{-1},x_{0},s))$
$\displaystyle\underset{\zeta\rightarrow 0}{=}\pm
i\zeta^{-1}\int_{x_{0}}^{x}\rho(x^{\prime},t_{r})~{}dx^{\prime}+\left\\{\begin{array}[]{ll}O(1)&\hbox{{for
$P\rightarrow P_{\infty_{+}}$ }}\\\ O(1)&\hbox{{for $P\rightarrow
P_{\infty_{-}}$ }}\end{array}\right.$ (6.27)
$\displaystyle~{}~{}~{}~{}~{}~{}+\int_{t_{0,r}}^{t_{r}}ds\left(\frac{y~{}\widetilde{F}_{r}(\zeta^{-1},x_{0},s)}{F_{n}(\zeta^{-1},x_{0},s)}+\frac{1}{2}\frac{F_{n,t_{r}}(\zeta^{-1},x_{0},s)}{F_{n}(\zeta^{-1},x_{0},s)}\right)$
$\displaystyle\underset{\zeta\rightarrow 0}{=}\pm
i\zeta^{-1}\int_{x_{0}}^{x}\rho(x^{\prime},t_{r})~{}dx^{\prime}+\left\\{\begin{array}[]{ll}O(1)&\hbox{{for
$P\rightarrow P_{\infty_{+}}$ }}\\\ O(1)&\hbox{{for $P\rightarrow
P_{\infty_{-}}$ }}\end{array}\right.$ (6.30)
$\displaystyle~{}~{}~{}~{}~{}~{}+\int_{t_{0,r}}^{t_{r}}ds\left(\pm
2i\zeta^{-r-1}\frac{\sum_{m=0}^{r}\tilde{f}_{m}(x_{0},s)\zeta^{m}}{\sum_{l=0}^{\infty}\tilde{f}_{l}(x_{0},s)\zeta^{l}}+\frac{1}{2}\frac{u_{t_{r}}(x_{0},s)}{u(x_{0},s)}+O(\zeta)\right)$
$\displaystyle\underset{\zeta\rightarrow 0}{=}\pm
i\zeta^{-1}\int_{x_{0}}^{x}\rho(x^{\prime},t_{r})~{}dx^{\prime}+\left\\{\begin{array}[]{ll}O(1)&\hbox{{for
$P\rightarrow P_{\infty_{+}}$ }}\\\ O(1)&\hbox{{for $P\rightarrow
P_{\infty_{-}}$ }}\end{array}\right.$ (6.33)
$\displaystyle~{}~{}~{}~{}~{}~{}+\int_{t_{0,r}}^{t_{r}}ds\left(\pm
2i\zeta^{-r-1}\mp\frac{2i\tilde{f}_{r+1}(x_{0},s)}{\tilde{f}_{0}(x_{0},s)}+\frac{1}{2}\frac{u_{t_{r}}(x_{0},s)}{u(x_{0},s)}+O(\zeta)\right)$
$\displaystyle\underset{\zeta\rightarrow 0}{=}\pm
i\zeta^{-1}\int_{x_{0}}^{x}\rho(x^{\prime},t_{r})~{}dx^{\prime}\pm
2i\zeta^{-r-1}(t_{r}-t_{0,r})+\left\\{\begin{array}[]{ll}O(1)&\hbox{{for
$P\rightarrow P_{\infty_{+}}$, }}\\\ O(1)&\hbox{{for $P\rightarrow
P_{\infty_{-}}$, }}\end{array}\right.$ (6.36)

where we used $\tilde{f}_{0}=-u$ and $u_{t_{r}}=\frac{1}{2}\tilde{f}_{r,x}$
(cf.(5.3)) in the homogeneous case $\tilde{c}_{0}=1$, $\tilde{c}_{q}=0$,
$q=1,\ldots,r$. A comparison of $\psi_{1}$ and $\Psi$ near $P_{\infty_{\pm}}$,
taking into account (5), (6.14), (6.16), and (6.36), then shows that
$\psi_{1}$ and $\Psi$ have identical exponential behavior up to order $O(1)$
near $P_{\infty_{\pm}}$. Thus, $\psi_{1}$ and $\Psi$ share the same
singularities and zeros, and the Riemann-Roch-type uniqueness result (cf.
Lemma C.2 [13]) then proves $\psi_{1}=\Psi$. The representation (6.15) for
$\phi$ on $\widetilde{\Omega}$ follows by combining (5.32), (6.1), and Theorem
A.26 [13]. The representation (6.19) for $\rho$ on $\widetilde{\Omega}$ is
clear from (6.15) and (6.2). The representation (6.18) for $u$ on
$\widetilde{\Omega}$ follows from (6.19) and the relation $u\rho=-2$. In fact,
since the proofs of (6.15), (6.18), and (6.19) are identical to the
corresponding stationary results in Theorem 4.4, which can be extended line by
line to the time-dependent setting, here we omit the corresponding details.
Finally, the extension of all these results from
$(x,t_{r})\in\widetilde{\Omega}$ to $(x,t_{r})\in\Omega$ then simply follows
from the continuity of $\underline{\alpha}_{Q_{0}}$ and the hypothesis of
$\mathcal{D}_{\hat{\mu}(x,t_{r})}$ being nonspecial for $(x,t_{r})\in\Omega$.
$\square$

Remark 4.5 applies in the present time-dependent context as well.

## 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).

## References

  * [1] M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The inverse scattering transform–Fourier analysis for nonlinear problems. Stud. Appl. Math. 53 (1974), 249–315.
  * [2] M. S. Alber, R. Camassa, Y. N. Fedorov, D. D. Holm, and J. E. Marsden, The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE s of shallow water and Dym type. Commun. Math. Phys. 221 (2001), 197–227.
  * [3] M. S. Alber, and Y. N. Fedorov, Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians. Inverse Problems 17 (2001), 1017–1042.
  * [4] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nolinear Integrable Equations. Springer, Berlin, (1994).
  * [5] M. Chen, S. Q. Liu, and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys. 75 (2006) 1–15.
  * [6] A. Constantin, and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A 372 (2008) 7129–7132.
  * [7] B. A. Dubrovin, Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties. Funct. Anal. Appl. 11 (1977), 265–277.
  * [8] B. A. Dubrovin, Theta functions and nonlinear equations. Russian Math. Surv. 36 (1981), 11–92.
  * [9] B. A. Dubrovin, Matrix finite-zone operators. Revs. Sci. Technol. 23 (1983), 20–50.
  * [10] J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa- Holm equation, Discrete Contin. Dyn. Syst. 19 (2007) 493–513.
  * [11] F. Gesztesy, and H. Holden, Algebro-geometric solutions of the Camassa-Holm hierarchy. Rev. Mat. Iberoam. 19 (2003) 73–142.
  * [12] F. Gesztesy, and H. Holden, Real-valued algebro-geometric solutions of the Camassa-Holm hierarchy. Philos. Trans. R. Soc. Lond. Ser. A. 366 (2008), 1025–1054.
  * [13] F. Gesztesy, and H. Holden, Soliton Equations and Their Algebro- Geometric Solutions. vol. I: (1+1)-Dimensional Continuous Models. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, (2003).
  * [14] F. Gesztesy, and R. Ratneseelan, An alternative approach to algebro-geometric solutions of the AKNS hierarchy. Rev. Math. Phys. 10 (1998), 345–391.
  * [15] C. Guan, and Y. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations 248 (2010) 2003–2014.
  * [16] G. L. Gui, and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal. 258 (2010) 4251–4278.
  * [17] D. J. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. B 12 (2009) 597–606.
  * [18] D. D. Holm, and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys. A: Math. Theor. 43 (2010) 492001.
  * [19] Y. Hou, E. G. Fan, and Z. J. Qiao, The algebro-geometric solutions for the modified Camassa-Holm hierarchy, preprint, (2012) arXiv: 1205.6062.
  * [20] Y. Hou, P. Zhao, E. G. Fan, and Z. J. Qiao, Algebro-geometric solutions for Degasperis-Procesi hierarchy, SIAM J. Math. Anal. 45 (2013) 1216–1266.
  * [21] R. I. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities, Z. Naturf. A 61 (2006) 133–138.
  * [22] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech. 457 (2002) 63–82.
  * [23] V. B. Matveev, and M. I. Yavor, Solutions presque p$\mathrm{\acute{e}}$riodiques et $\mathrm{\grave{a}}$ $N$-solitons de l’$\mathrm{\acute{e}}$quation hydrodynamique non lin$\mathrm{\acute{e}}$aire de Kaup. Ann. Inst. H. Poincar$\mathrm{\acute{e}}$ Sect. A 31 (1979), 25–41.
  * [24] S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Methods. Concultants Bureau, New York, (1984).
  * [25] P. Olver, and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E 53 (1996) 1900–1906.
  * [26] J. N. Reddy, Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci. 45 (2007) 288–307.
  * [27] J. F. Song, and C. Z. Qu, Geometric integrability of two-component Camassa-Holm and Hunter-Saxton systems, Commun. Theor. Phys. 55 (2011) 955–959.