$\displaystyle\tilde{D}_{j}\hat{A}^{ij}-\frac{2}{3}\Psi^{6}\tilde{D}^{i}K=$
$\displaystyle\ 8\pi\Psi^{10}p^{i}.$ (32)

We parameterize the energy and momentum densities as

$\displaystyle E=$ $\displaystyle\
\Psi^{-12}\tilde{\eta}\bar{\tilde{\eta}}+\Psi^{-4}\tilde{D}^{i}\Phi\tilde{D}_{i}\Phi+V(|\Phi|),$
(33) $\displaystyle p^{i}=$ $\displaystyle\
-\Psi^{-10}(\tilde{\eta}\tilde{D}^{i}\bar{\Phi}+\bar{\tilde{\eta}}\tilde{D}^{i}\Phi),$
(34)

where $\eta=\mathcal{L}_{n}\Phi$ and $\eta=\Psi^{6}\tilde{\eta}$, following
what was done in Ref. Corman and East (2022). In addition to the usual
conformal metric quantities that are free data in the CTS formalism, we
specify $\Phi$ and $\tilde{\eta}$ (as opposed to a conformally rescaled energy
and momentum density) as free data when solving the constraints.

Following Ref. East  _et al._ (2012a), the free data for the CTS constraint
equations is constructed from the plain superposition of the metric variables
and scalar field degrees of freedom of two isolated star solutions:

$\displaystyle\Phi=$ $\displaystyle\ \Phi_{(1)}+\Phi_{(2)},$
$\displaystyle\tilde{\eta}=\tilde{\eta}_{(1)}+\tilde{\eta}_{(2)}.$ (35)

We displace and boost the isolated solutions along vectors $\beta_{i}^{(1)}$
and $\beta_{i}^{(2)}$ with coordinate velocity $v_{(1)}$ and $v_{(2)}$,
respectively, prior to superposing the stars’ solutions. The numerical
construction of the isolated BS solutions used in this work is outlined in
Ref. Siemonsen and East (2021). The elliptic constraint equations, subject to
asymptotically flat boundary conditions, are then solved using Newton-Raphson
relaxation combined with the multigrid method East  _et al._ (2012a).

###  D.2 Numerical evolution

Given initial data for a scalar binary BS, we evolve the Einstein-Klein-Gordon
equations (following from the scalar action (3)) forward in time employing the
generalized harmonic formulation of the Einstein evolution equations Pretorius
(2005). To that end, we utilize fourth-order accurate finite difference
stencils over a compactified Cartesian grid containing spatial infinity. There
we impose asymptotically flat boundary conditions on the metric variables and
set the scalar field to zero. This is paired with a fourth-order accurate
Runge-Kutta time-integration. Of particular importance is the use of adaptive
mesh refinement, with refinement ratio 2:1, to track the stars as they move
across the Cartesian grid (see Ref. East  _et al._ (2012b) for details). The
compactness of the stars sets the number of levels required to resolve the
stars sufficiently; for low-compactness solutions [typically stars in the
repulsive scalar model, (5)], we require five to six refinement levels, while
for the high-compactness solutions [usually those in the solitonic scalar
model, (4)], we require six to seven levels. In the cases with black hole
formation, we add refinement levels dynamically to resolve the gravitational
collapse and apparent horizon (this requires seven to nine levels). The
resolution on the finest mesh refinement level for the binary evolutions
presented in Sec. II.4 is $\Delta x/M_{0}=0.15$. The resolution for the
solitonic cases shown in Sec. III.2 is $\Delta x/M_{0}=0.075$ on the finest
refinement level, while for the binaries in the repulsive model it is $\Delta
x/M_{0}=0.2$. Throughout, we use the standard damped harmonic gauge condition
to set the generalized harmonic source functions $H_{\mu}$ Choptuik and
Pretorius (2010); Lindblom and Szilagyi (2009).

###  D.3 Convergence tests

Figure 17: Here we consider the convergence behavior of the binary BS in the
$\sigma=0.05$ solitonic scalar model, with properties summarized in Table 2,
with decreasing grid spacing. The quantities $\mathcal{C}$ and
$\mathcal{I}_{\mathcal{C}}$ (defined in the text) are a positive definite
measure of the constraint violation, which we track throughout the simulation.
The rapid variation of the constraints is driven by gauge dynamics at early
times. The maximum of the constraint violation $\mathcal{C}$ occurs during the
merger of the binary at around $t/M_{0}\approx 75$. The binary merges earlier
with increasing resolution, and only the medium and high resolutions capture
small-scale features present in the remnant after merger. The quantity
$\max\mathcal{C}$ converges to zero roughly at third order, as expected, since
it is primarily set by the third-order accurate time interpolations on the
mesh refinement boundaries. On the other hand, the integrated quantity
$\mathcal{I}_{\mathcal{C}}$ converges at the expected forth order, as it is
largely insensitive to the lower-order time interpolations.

We present resolution studies of two exemplary binary mergers. First, we focus
on the $\sigma=0.05$ solitonic scalar model and the binary with parameters
given in Table 2. We consider three resolutions, corresponding to $\Delta x$,
$3\Delta x/4$, and $\Delta x/2$, where the lowest resolution corresponds to a
grid spacing of $\Delta x/M_{0}\approx 0.1$ on the finest level, and the
medium resolution is the default resolution for all simulations discussed in
Sec. III. In order to explicitly demonstrate that we are solving the
Hamiltonian and momentum constraints, we track the violations of the
constraints, given by $C_{\mu}=H_{\mu}-\square x_{\mu}$, in time. In Figure
17, we plot the evolution of the constraints at these different resolutions of
the binary with parameters given in Table 2. To track the constraint
violations, we define $\mathcal{C}=\sum_{\mu}|(C_{\mu})^{2}|/4$, and consider
the global maximum $\max\mathcal{C}$ in each time-slice, as well as the
integrated norm $\mathcal{I}_{\mathcal{C}}=\int
d^{3}x\sqrt{\gamma}\mathcal{C}$. In Figure 18, we show the convergence
behavior of the total $U(1)$-charge of the system. Overall, the constraint
violations converge to zero at the expected forth order of our numerical
methods. The violation of the conservation of the $U(1)$ charge $Q$, shown in
Figure 18, also converges towards zero. Likely, due to the compactness
($C=0.13$) of the BSs, rapid exponential decay of the scalar field outside the
stars, i.e., $\Phi\sim\exp(-\sqrt{\mu^{2}-\omega^{2}}r)$, with
$\omega/\mu=0.25$, and the large initial separation (of $D=40M_{0}$), the low
and medium resolutions exhibit relatively large drifts in the total conserved
charge. Hence, the scalar field gradients on the surface of the stars, as well
as the spatial scales of perturbations, require relatively high resolution.

Figure 18: We consider the convergence behavior of the global maximum of
$|\Phi|$, the total $U(1)$-charge $Q$, and the azimuthal mode $C_{5}$ of the
scalar field for the binary BS shown in Figure 17. The total charge $Q$ is
calculated in a coordinate sphere of radius $100M_{0}$ around the center of
mass of the system. We normalize $Q$ by $Q_{\infty}$, the sum of the BSs’
isolated charges $Q_{\infty}=Q_{1}+Q_{2}$. As the initial separation between
the two stars increases, the total charge approaches the superposed charge
Siemonsen and East (prep): $Q\rightarrow Q_{\infty}$. Lastly, we also show the
convergence behavior of the $C_{5}$ mode [defined in (19)] during the binary
evolution. The $m=5$ perturbations remaining after the merger (and the
formation of an $m=1$ rotating remnant) at around $t/M_{0}\approx 75$ are
converging towards zero with increasing resolution at roughly the expected
fourth order. Figure 19: We consider the convergence behavior of the
$\alpha=\pi/2$ case of Sec. II.4 with decreasing grid spacing $\Delta x$. The
quantities $\mathcal{C}$ and $\mathcal{I}_{\mathcal{C}}$ are defined in the
text. The low resolution evolution is based on a different mesh-refinement
layout (as discussed in the text,) and, hence, exhibits slightly different
convergence behavior. At early times, the convergence orders of these
quantities are the same as those discussed in the caption of Figure 17.

Secondly, we discuss the numerical convergence of one of the binaries
considered in Sec. II.4. In particular, we focus on the $\alpha=\pi/2$ case,
and compare its convergence behavior to that of the $\alpha=\pi$ binary
evolution. In Figure 19, we present the convergence of the constraint
violations with increasing resolution of the $\alpha=\pi/2$ evolution. Again,
this demonstrates explicitly that we are solving the Hamiltonian and momentum
constraints consistently within the $t=0$ slice. In the subsequent evolution
up to $t/M_{0}=100$, the constraints converge at the expected orders. For
numerical stability purposes, we have to increase the size of the second
coarsest mesh-refinement level in the lowest resolution run, moving the outer
boundary of this level from $|x_{i}|/M_{0}=100$ to $|x_{i}|/M_{0}\approx 241$.
This explains the disagreement between the $\Delta x$ and the $3\Delta x/4$ as
well as $\Delta x/2$ resolutions in Figure 19, after $t/M_{0}\approx 100$ (as
at this time constraint violations propagating outward reach the mesh-
refinement boundary in the medium and high resolution runs, but not yet in the
low-resolution case). Furthermore, this different mesh-refinement layout in
the low resolution case alters the convergence behavior, such that this case
mergers much earlier compared with the medium and high resolution runs.
However, we have checked explicitly that the merger delay between the
$\alpha=\pi/2$ and $\alpha=\pi$ cases increases from low (of $\Delta
t/M_{0}\approx 43$) to medium resolution evolutions (of $\Delta t/M_{0}\approx
262$). Hence, the dephasing, delayed merger and black hole collapse discussed
in Sec. II.4 are physical, and we are likely underestimating their impact on
the GWs. Notice also, identical numerical setups were used for all cases
presented Sec. II.4, both for the initial data construction and evolution.
Therefore, while absolute differences are not resolved, this is suggestive
that the relative difference in amplitude in the GW waveform between the
$\alpha$-cases are driven by the scalar interactions, rather than numerical
truncation error.

##  Appendix E Vortex ejection as an artifact of numerical resolution

Figure 20: The evolution of the scalar field modes $C_{m}$ (dotted and solid
lines corresponding to $m=1$ and 2, respectively) defined in (19) for the
binary BS merger specified in Table 2 with phase variation $\alpha/\pi=63/64$.
The merger occurs roughly at $t/M_{0}\approx 75$, after which the even-$m$
modes promptly begin to grow exponentially in the evolution with the lowest
resolution (the $m=0$ mode is representative of all even-$m$ modes). This
apparent instability is an artifact of low numerical resolution, and
disappears with increasing resolution.

We find that in our simulations of the rotating BS formed from the merger of
two non-rotating BS with a phase variation of $63/64\geq\alpha/\pi\geq 7/8$
exhibit a growing perturbation leading to vortex ejection at low resolutions,
but that this behavior disappears at sufficiently high resolution. In order to
understand this behavior, it is instructive to consider an azimuthal mode
decomposition of the real part of the scalar field,
$\Phi_{R}=\text{Re}(\Phi)$, defined in (19). In Figure 20, we show the scalar
field modes $C_{m}$ during the merger of the binary BS specified in Table 2
with initial phase variation $\alpha/\pi=63/64$. During, and shortly after,
the merger around $t/M_{0}=75$, the $m=1$ mode is the most dominant mode
representing the formation of a $m=1$ rotating BS, and indicating the
formation of a $q=1$ central vortex. Additionally, the amplitude of the
even-$m$ modes right after merger is consistent across resolutions. On the
other hand, the even-$m$ modes begin to grow exponentially right after
formation of the rotating remnant (the representative $m=0$ mode is shown in
Figure 20) in the evolution with lowest resolution. Furthermore, we find that
with increasing $\alpha$, the amplitude of the even-$m$ modes after merger
decreases, but in all cases the artificial instability appears at lowest
resolution; in fact, even in the $\alpha=\pi$ case, where the even-$m$ modes
are seeded at amplitudes consistent with floating point roundoff, we find this
behavior. In all cases considered, this growing perturbation at low resolution
saturates in the vortex ejection of the solution. However, we performed higher
resolution evolutions in the binaries with
$\alpha/\pi\in\\{63/64,31/32,7/8\\}$ and explicitly checked that the unstable
behavior disappears. This is illustrated for $\alpha/\pi=63/64$ in Figure 20.

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# Differential Equations
of Genus Four Hyperelliptic $\wp$ Functions

Masahito Hayashi
Osaka Institute of Technology, Osaka 535-8585, Japan
Kazuyasu Shigemoto
Tezukayama University, Nara 631-8501, Japan
Takuya Tsukioka
Bukkyo University, Kyoto 603-8301, Japan
masahito.hayashi@oit.ac.jpshigemot<EMAIL_ADDRESS>
In order to find higher dimensional integrable models, we study differential
equations of hyperelliptic $\wp$ functions up to genus four. For genus two,
differential equations of hyperelliptic $\wp$ functions can be written in the
Hirota form. If genus is more than two, we have KdV and another KdV equations,
and if genus becomes more than three, there appear differential equations
which cannot be written in the Hirota form, which means that the Hirota form
is not enough to characterize the integrable differential equations. We have
shown that some of differential equations are satisfied for general genus. We
can obtain differential equations for general genus step by step.

##  1 Introduction

Through studies of soliton system, we have solved non-linear problems of very
interesting phenomena. Starting from the inverse scattering method [1, 2, 3],
many interesting developments have been done including the AKNS formulation
[4], the Bäcklund transformation [5, 6, 7], the Hirota equation [8, 9], the
Sato theory [10], the vertex construction of the soliton solution [11, 12,
13], and the Schwarzian type mKdV/KdV equation [14]. Soliton theory is, in
some sense, the prototype of the superstring theory, because the Möbius
transformation, vertex construction and AdS structure are used to understand
the structure of soliton system. Our understanding of the soliton has been
still in progress.

In our previous papers, we have revealed that the two dimensional integrable
models such as KdV/mKdV/sinh-Gordon are the consequence of the SO(2,1)$\cong$
SL(2,$\mathbb{R}$) Lie group structure [15, 16, 17, 18, 19].

Here we would like to to study higher-dimensional integrable models.
KdV/mKdV/sinh-Gordon equations and KP equations are typically understood as
two- and three-dimensional integrable models, respectively. First, we would
like to know whether there exists a universality of the integrable models,
that is, whether any two- and three-dimensional integrable models always
contain KdV/mKdV/sinh-Gordon equations and KP equations, respectively.

For higher-dimensional integrable models, there is a soliton type approach of
Kyoto school [10, 11, 12, 13] where they use the special fermion, which
generates $N$-soliton solutions. Starting with the fermionic bilinear identity
of $\mathfrak{gl}(\infty,\mathbb{R})$, they have obtained KP hierarchy and
finite higher-dimensional Hirota forms by the reduction of KP hierarchy.
Another systematic approach to high-dimensional integrable models is to find
differential equations for higher genus hyperelliptic functions by using the
analogy of differential equation of Weierstrass $\wp$ function. By solving the
Jacobi’s inversion problem, the integrability of hyperelliptic functions are
automatically guaranteed, since the integrability condition and the single-
valuedness are equivalent for hyperelliptic functions. So far, only for genus
one, two [22] and three [23, 24, 25] cases are studied because it becomes
difficult to solve the Jacobi’s inversion problem and obtain differential
equations for higher genus cases. In this paper, we study to obtain
differential equations of genus four case. In the approach, we would like to
examine the connections between i) higher-dimensional integrable differential
equations, ii) higher-rank Lie group structure and iii) higher genus
hyperelliptic functions.

##  2 Formulation of Differential Equations in General Genus and the Review of
Genus Two and Three Cases

###  2.1 Formulation of differential equations in general genus

We summarize the formulation of hyperelliptic $\wp$ function according to
Baker’s work [20, 21, 22, 23]. We consider the genus $g$ hyperelliptic curve

$C:\quad y_{i}^{2}=\sum_{k=0}^{2g+2}\lambda_{k}x_{i}^{k},\qquad
i=1,2,\cdots,g.$ (2.1)

The Jacobi’s inversion problem consists of solving the following system

${\rm d}u_{1}=\sum_{i=1}^{g}\frac{{\rm d}x_{i}}{y_{i}},\quad{\rm
d}u_{2}=\sum_{i=1}^{g}\frac{x_{i}{\rm d}x_{i}}{y_{i}},\quad\cdots,\quad{\rm
d}u_{g-1}=\sum_{i=1}^{g}\frac{x_{i}^{g-2}{\rm d}x_{i}}{y_{i}},\quad{\rm
d}u_{g}=\sum_{i=1}^{g}\frac{x_{i}^{g-1}{\rm d}x_{i}}{y_{i}}.$ (2.2)

From these equations, we have

$\frac{\partial x_{i}}{\partial
u_{j}}=\frac{y_{i}\chi_{g-j}\left(x_{i};x_{1},x_{2},\cdots,x_{g}\right)}{F^{\prime}(x_{i})},$
(2.3)

by using the relation

$\sum_{i=1}^{g}\frac{x_{i}^{k-1}\chi_{g-j}(x_{i};x_{1},x_{2},\cdots,x_{g})}{F^{\prime}(x_{i})}=\delta_{kj},\quad(1\leq
j\leq g).$ (2.4)

We define $\displaystyle{F(x)=\prod_{i=1}^{g}(x-x_{i})}$ and denote
$F^{\prime}(x_{i})$ as $\displaystyle{F^{\prime}(x_{i})=\frac{{\rm
d}F(x)}{{\rm d}x}\Big{|}_{x=x_{i}}}$. For example,
$F^{\prime}(x_{1})=(x_{1}-x_{2})(x_{1}-x_{3})\cdots(x_{1}-x_{g})$ . For
$\chi_{g-j}(x_{i};x_{1},x_{2},\cdots,x_{g})$, we first define the following
generalized function

$\displaystyle\chi_{g-j}(x;x_{1},\cdots,x_{p})=$ $\displaystyle\
x^{g-j}-h_{1}(x_{1},\cdots,x_{p})x^{g-j-1}$
$\displaystyle+h_{2}(x_{1},x_{2},\cdots,x_{p})x^{g-j-2}+\cdots+(-1)^{g-j}h_{g-j}(x_{1},\cdots,x_{p}),$
(2.5)

where $h_{j}(x_{1},\cdots,x_{p})$ is the $j$-th fundamental symmetric
polynomial basis of $\\{x_{1},\cdots,x_{p}\\}$, i.e.

$\prod_{i=1}^{p}(x-x_{i})=x^{p}+\sum_{j=1}^{p}(-1)^{j}h_{j}(x_{1},x_{2},\cdots,x_{p})x^{p-j}.$
(2.6)

Putting $p=g$ and $x=x_{k}$ in $\chi_{g-j}(x;x_{1},x_{2},\cdots,x_{p})$, we
have $\chi_{g-j}(x_{i};x_{1},x_{2},\cdots,x_{g})$ in the following form

$\displaystyle\chi_{g-j}(x_{i};x_{1},x_{2},\cdots,x_{g})=$ $\displaystyle\
x_{i}^{g-j}-h_{1}(x_{1},x_{2},\cdots,x_{g})x_{i}^{g-j-1}$
$\displaystyle+h_{2}(x_{1},x_{2},\cdots,x_{g})x_{i}^{g-j-2}+\cdots+(-1)^{g-j}h_{g-j}(x_{1},x_{2},\cdots,x_{g}).$
(2.7)

For example

$\displaystyle\chi_{0}(x_{1};x_{1},x_{2},\cdots,x_{g})$ $\displaystyle=1,$
$\displaystyle\chi_{1}(x_{1};x_{1},x_{2},\cdots,x_{g})$
$\displaystyle=x_{1}-(x_{1}+x_{2}+\cdots+x_{g})=-h_{1}(x_{2},x_{3},\cdots,x_{g}),$
$\displaystyle\chi_{2}(x_{1};x_{1},x_{2},\cdots,x_{g})$
$\displaystyle=x_{1}^{2}-(x_{1}+x_{2}+\cdots+x_{g})x_{1}+(x_{1}x_{2}+x_{1}x_{3}+\cdots)$
$\displaystyle=x_{2}x_{3}+x_{2}x_{4}+\cdots=h_{2}(x_{2},x_{3},\cdots,x_{g}),$
$\displaystyle\ \vdots$

From Eq.(2.6), we have

$x_{i}^{g}-h_{1}(x_{1},x_{2},\cdots,x_{g})x_{i}^{g-1}+h_{2}(x_{1},x_{2},\cdots,x_{g})x_{i}^{g-2}+\cdots+(-1)^{g}h_{g}(x_{1},x_{2},\cdots,x_{g})=0.$
(2.8)

The $\zeta_{j}$ functions are given from the hyperelliptic curve in the
following way [20]

${\rm d}(-\zeta_{j})=\sum_{i=1}^{g}\frac{{\rm
d}x_{i}}{y_{i}}\sum_{k=j}^{2g+1-j}(k+1-j)\lambda_{k+1+j}x_{i}^{k}-2{\rm
d}\left(\sum_{i=1}^{g}\frac{y_{i}\chi_{g-j-1}(x_{i};x_{1},\cdots,\widecheck{x}_{i},\cdots,x_{g})}{F^{\prime}(x_{i})}\right),$
(2.9)

where $\widecheck{x}_{j}$ denotes that the $x_{j}$ variable is missing. In
this expression, we can show ${\rm d}(-\zeta_{0})=0$ in the following way

$\displaystyle{\rm d}(-\zeta_{0})$ $\displaystyle=\sum_{i=1}^{g}\frac{{\rm
d}x_{i}}{y_{i}}\sum_{k=0}^{2g+1}(k+1)\lambda_{k+1}x_{i}^{k}-2{\rm
d}\left(\sum_{i=1}^{g}\frac{y_{i}\chi_{g-1}(x_{i};x_{1},\cdots,\widecheck{x}_{i},\cdots,x_{g})}{F^{\prime}(x_{i})}\right)$
$\displaystyle=\sum_{i=1}^{g}\frac{1}{y_{i}}{\rm
d}\left(\sum_{l=0}^{2g+2}\lambda_{l}x_{i}^{l}\right)-2{\rm
d}\left(\sum_{i=1}^{g}y_{i}\right)=\sum_{i=1}^{g}\frac{1}{y_{i}}{\rm
d}\left(y_{i}^{2}\right)-2{\rm d}\left(\sum_{i=1}^{g}y_{i}\right)$
$\displaystyle=0,$ (2.10)

where we use
$\chi_{g-1}(x_{i};x_{1},x_{2},\cdots,\widecheck{x}_{i},\cdots,x_{g})=F^{\prime}(x_{i})$.
These $\zeta_{j}(u_{1},u_{2},\cdots,u_{g})$ satisfy the integrability
condition

$\frac{\partial\left(-\zeta_{j}(u_{1},u_{2},\cdots,u_{g})\right)}{\partial
u_{k}}=\frac{\partial\left(-\zeta_{k}(u_{1},u_{2},\cdots,u_{g})\right)}{\partial
u_{j}}.$ (2.11)

In the Baker’s textbook [20], the expression of the second term of the r.h.s
of Eq.(2.9) is misleading. $\wp_{jk}(u_{1},u_{2},\cdots,u_{g})$ functions are
given from the above $\zeta_{j}(u_{1},u_{2},\cdots,u_{g})$ functions in the
form

$\wp_{jk}(u_{1},u_{2},\cdots,u_{g})=\wp_{kj}(u_{1},u_{2},\cdots,u_{g})=\displaystyle{\frac{\partial\left(-\zeta_{j}(u_{1},u_{2},\cdots,u_{g})\right)}{\partial
u_{k}}}.$ (2.12)

These $\zeta_{j}$, $\wp_{jk}$ and $\wp_{jklm}$ are given by the hyperelliptic
$\sigma$ function in the form

$-\zeta_{j}=\displaystyle\frac{\partial(-\log\sigma)}{\partial
u_{j}},\quad\wp_{jk}=\displaystyle\frac{\partial^{2}(-\log\sigma)}{\partial
u_{j}\partial
u_{k}},\quad\textrm{and}\quad\wp_{jklm}=\displaystyle\frac{\partial^{4}(-\log\sigma)}{\partial
u_{j}\partial u_{k}\partial u_{l}\partial u_{m}},\qquad\textrm{etc.}.$

For the Weierstrass type, i.e. $\lambda_{2g+2}=0$, we have $\displaystyle{{\rm
d}(-\zeta_{g})=\lambda_{2g+1}\sum_{i=1}^{g}\frac{x_{i}^{g}{\rm
d}x_{i}}{y_{i}}}$, which gives

$\displaystyle\widehat{\wp}_{gg}(u_{1},u_{2},\cdots,u_{g})$
$\displaystyle=\frac{1}{\lambda_{2g+1}}\wp_{gg}(u_{1},u_{2},\cdots,u_{g})=h_{1}(x_{1},x_{2},\cdots,x_{g}),$
(2.13) $\displaystyle\widehat{\wp}_{g,g-1}(u_{1},u_{2},\cdots,u_{g})$
$\displaystyle=\frac{1}{\lambda_{2g+1}}\wp_{g,g-1}(u_{1},u_{2},\cdots,u_{g})=-h_{2}(x_{1},x_{2},\cdots,x_{g}),$
(2.14) $\displaystyle\ \vdots$
$\displaystyle\widehat{\wp}_{g1}(u_{1},u_{2},\cdots,u_{g})$
$\displaystyle=\frac{1}{\lambda_{2g+1}}\wp_{g1}(u_{1},u_{2},\cdots,u_{g})=(-1)^{g-1}h_{g}(x_{1},x_{2},\cdots,x_{g}),$
(2.15)

by using

$\sum_{i=1}^{g}\frac{x_{i}^{g}\chi_{g-j}(x_{i};x_{1},x_{2},\cdots,x_{g})}{F^{\prime}(x_{i})}=(-1)^{g-j}h_{g-j+1}(x_{1},x_{2},\cdots,x_{g}).$
(2.16)

Then we have

$x_{i}^{g}=\sum_{j=1}^{g}\widehat{\wp}_{gj}x_{i}^{j-1}=\widehat{\wp}_{gg}x_{i}^{g-1}+\widehat{\wp}_{g,g-1}x_{i}^{g-2}+\cdots+\widehat{\wp}_{g2}x_{i}+\widehat{\wp}_{g1}.$
(2.17)

We can easily show Eq.(2.4) and Eq.(2.16) by using Eq.(2.7) , Eq.(2.8) and the
following relation [26]

$\sum_{i=1}^{g}\frac{x_{i}^{j-1}}{F^{\prime}(x_{i})}=\delta_{jg},\qquad(1\leq
j\leq g).$ (2.18)

In this way, we have $\displaystyle{{\rm
d}(-\zeta_{g})=\sum_{j=1}^{g}\wp_{gj}{\rm d}u_{j}}$. For other $\wp_{ij}$, we
must use $\zeta_{j}$, which satisfies the integrability condition Eq.(2.11).

###  2.2 Differential equations of genus two hyperelliptic $\wp$ functions

We here review the genus two hyperelliptic $\wp$ function. The hyperelliptic
curve in this case is given by

$C:\quad
y_{i}^{2}=\lambda_{6}x_{i}^{6}+\lambda_{5}x_{i}^{5}+\lambda_{4}x_{i}^{4}+\lambda_{3}x_{i}^{3}+\lambda_{2}x_{i}^{2}+\lambda_{1}x_{i}+\lambda_{0}.$
(2.19)

The Jacobi’s inversion problem consists of solving the following system

${\rm d}u_{1}=\frac{{\rm d}x_{1}}{y_{1}}+\frac{{\rm d}x_{2}}{y_{2}},\qquad{\rm
d}u_{2}=\frac{x_{1}{\rm d}x_{1}}{y_{1}}+\frac{x_{2}{\rm d}x_{2}}{y_{2}}.$
(2.20)

Then we have

$\frac{\partial x_{1}}{\partial
u_{2}}=\frac{y_{1}}{x_{1}-x_{2}},\qquad\frac{\partial x_{2}}{\partial
u_{2}}=-\frac{y_{2}}{x_{1}-x_{2}},\qquad\frac{\partial x_{1}}{\partial
u_{1}}=-\frac{x_{2}y_{1}}{x_{1}-x_{2}},\qquad\frac{\partial x_{2}}{\partial
u_{1}}=\frac{x_{1}y_{2}}{x_{1}-x_{2}}.$ (2.21)

In this case,

$\displaystyle{\rm d}(-\zeta_{2})$
$\displaystyle=\sum_{i=1}^{2}\frac{\left(2\lambda_{6}x_{i}^{3}+\lambda_{5}x_{i}^{2}\right){\rm
d}x_{i}}{y_{i}},$ (2.22) $\displaystyle{\rm d}(-\zeta_{1})$
$\displaystyle=\sum_{i=1}^{2}\frac{\left(4\lambda_{6}x_{i}^{4}+3\lambda_{5}x_{i}^{3}+2\lambda_{4}x_{i}^{2}+\lambda_{3}x_{i}\right){\rm
d}x_{i}}{y_{i}}-2{\rm d}\left(\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\right).$ (2.23)

For these $\zeta_{1},\zeta_{2}$, we have checked the integrability condition
$\partial\zeta_{1}/\partial u_{2}=\partial\zeta_{2}/\partial u_{1}$. We use
the useful functions
$\widehat{\wp}_{22},\widehat{\wp}_{21},\widehat{\wp}_{11}$ of the form

$\displaystyle\widehat{\wp}_{22}$
$\displaystyle=\frac{1}{\lambda_{5}}\wp_{22}=\frac{1}{\lambda_{5}}\frac{\partial(-\zeta_{2})}{\partial
u_{2}}=x_{1}+x_{2}+\frac{2\lambda_{6}}{\lambda_{5}}\left(x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}\right),$
(2.24) $\displaystyle\widehat{\wp}_{21}$
$\displaystyle=\frac{1}{\lambda_{5}}\wp_{21}=\frac{1}{\lambda_{5}}\frac{\partial(-\zeta_{2})}{\partial
u_{1}}=-x_{1}x_{2}-\frac{2\lambda_{6}}{\lambda_{5}}x_{1}x_{2}\left(x_{1}+x_{2}\right),$
(2.25) $\displaystyle\widehat{\wp}_{11}$
$\displaystyle=\frac{1}{\lambda_{5}}\wp_{11}=\frac{1}{\lambda_{5}}\frac{\partial(-\zeta_{1})}{\partial
u_{1}}=\frac{1}{\lambda_{5}}\frac{F(x_{1},x_{2})-2y_{1}y_{2}}{(x_{1}-x_{2})^{2}}+\frac{2\lambda_{6}}{\lambda_{5}}x_{1}^{2}x_{2}^{2},$
(2.26)

where

$\displaystyle F(x_{1},x_{2})=$ $\displaystyle
2\lambda_{6}x_{1}^{3}x_{2}^{3}+\lambda_{5}x_{1}^{2}x_{2}^{2}(x_{1}+x_{2})+2\lambda_{4}x_{1}^{2}x_{2}^{2}$
$\displaystyle+\lambda_{3}x_{1}x_{2}(x_{1}+x_{2})+2\lambda_{2}x_{1}x_{2}+\lambda_{1}(x_{1}+x_{2})+2\lambda_{0}.$

Defining
$\mathrm{\mathring{\wp}}_{22}=x_{1}+x_{2},\mathrm{\mathring{\wp}}_{21}=-x_{1}x_{2}$,
we have

$\displaystyle\widehat{\wp}_{22}$
$\displaystyle=\mathrm{\mathring{\wp}}_{22}+\frac{2\lambda_{6}}{\lambda_{5}}(\mathrm{\mathring{\wp}}_{22}^{2}+\mathrm{\mathring{\wp}}_{21}),$
(2.27) $\displaystyle\widehat{\wp}_{21}$
$\displaystyle=\mathrm{\mathring{\wp}}_{21}+\frac{2\lambda_{6}}{\lambda_{5}}\mathrm{\mathring{\wp}}_{21}\mathrm{\mathring{\wp}}_{22}.$
(2.28)

Then we can express $\mathrm{\mathring{\wp}}_{22},\
\mathrm{\mathring{\wp}}_{21}$ as infinite power series of
$\widehat{\wp}_{22},\ \widehat{\wp}_{21}$. We have the differential equation
for $\widehat{\wp}_{22}$ in the form

$\displaystyle\frac{\partial^{2}\widehat{\wp}_{22}}{\partial u_{2}^{2}}=$
$\displaystyle\frac{3}{2}\lambda_{5}\widehat{\wp}_{22}^{2}+\lambda_{4}\widehat{\wp}_{22}+\lambda_{5}\widehat{\wp}_{21}+3\lambda_{6}\widehat{\wp}_{11}+\frac{1}{2}\lambda_{3}$
$\displaystyle+\frac{2\lambda_{6}}{\lambda_{5}}\left(\lambda_{6}\left(3\mathrm{\mathring{\wp}}_{22}^{4}+6\mathrm{\mathring{\wp}}_{22}^{2}\mathrm{\mathring{\wp}}_{21}-3\mathrm{\mathring{\wp}}_{21}^{2}\right)+\lambda_{5}\left(3\mathrm{\mathring{\wp}}_{22}^{3}+3\mathrm{\mathring{\wp}}_{22}\mathrm{\mathring{\wp}}_{21}\right)+3\lambda_{4}\mathrm{\mathring{\wp}}_{22}^{2}+3\lambda_{3}\mathrm{\mathring{\wp}}_{22}+2\lambda_{2}\right).$
(2.29)

In order that the differential equation becomes the polynomial type of
$\widehat{\wp}_{22},\ \widehat{\wp}_{21}$ but not infinite series of these, we
must put $\lambda_{6}=0$. Even if $\lambda_{6}\neq 0$, $\zeta_{2},\zeta_{1}$
satisfies the integrability condition, we must put $\lambda_{6}=0$ in order
that the differential equation is of polynomial type. Then we have

$\displaystyle\frac{1}{\lambda_{5}}\wp_{22}$
$\displaystyle=\widehat{\wp}_{22}=\mathrm{\mathring{\wp}}_{22}=x_{1}+x_{2},$
(2.30) $\displaystyle\frac{1}{\lambda_{5}}\wp_{21}$
$\displaystyle=\widehat{\wp}_{21}=\mathrm{\mathring{\wp}}_{21}=-x_{1}x_{2},$
(2.31) $\displaystyle\frac{1}{\lambda_{5}}\wp_{11}$
$\displaystyle=\widehat{\wp}_{11}=\frac{1}{\lambda_{5}}\frac{F(x_{1},x_{2})|_{\lambda_{6}=0}-2y_{1}y_{2}}{(x_{1}-x_{2})^{2}}.$
(2.32)

By using the analogy of the differential equation of Weierstrass $\wp$
function in the form $d^{2}\wp(x)/dx^{2}=6\wp(x)^{2}-g_{2}/2$, we have the
following differential equations [22]

$\displaystyle 1)\quad$
$\displaystyle\wp_{2222}-\frac{3}{2}\wp_{22}^{2}=\lambda_{5}\wp_{21}+\lambda_{4}\wp_{22}+\frac{1}{2}\lambda_{5}\lambda_{3},$
(2.33) $\displaystyle 2)\quad$
$\displaystyle\wp_{2221}-\frac{3}{2}\wp_{22}\wp_{21}=-\frac{1}{2}\lambda_{5}\wp_{11}+\lambda_{4}\wp_{21},$
(2.34) $\displaystyle 3)\quad$
$\displaystyle\wp_{2211}-\wp_{21}^{2}-\frac{1}{2}\wp_{22}\wp_{11}=\frac{1}{2}\lambda_{3}\wp_{21},$
(2.35) $\displaystyle 4)\quad$
$\displaystyle\wp_{2111}-\frac{3}{2}\wp_{21}\wp_{11}=\lambda_{2}\wp_{21}-\frac{1}{2}\lambda_{1}\wp_{22}-\lambda_{5}\lambda_{0},$
(2.36) $\displaystyle 5)\quad$
$\displaystyle\wp_{1111}-\frac{3}{2}\wp_{11}^{2}=\lambda_{2}\wp_{11}+\lambda_{1}\wp_{21}-3\lambda_{0}\wp_{22}+\frac{1}{2}\lambda_{3}\lambda_{1}-2\lambda_{4}\lambda_{0}.$
(2.37)

In addition to $\lambda_{6}=0$, which is necessary to obtain differential
equations of polynomial type, we can always put $\lambda_{0}=0$ by the
constant shift of $x_{i}$ in Eq.(2.19), i.e. $x_{i}\rightarrow x_{i}+a$ with
$\sum_{j=0}^{5}\lambda_{j}a^{j}=0$. Then, in the standard form of
$\lambda_{0}=0$, we have some dual symmetry Eq.(2.33) $\leftrightarrow$
Eq.(2.37) , Eq.(2.34) $\leftrightarrow$ Eq.(2.36) , Eq.(2.35)
$\leftrightarrow$ Eq.(2.35) under ${\rm d}u_{2}\leftrightarrow\pm{\rm
d}u_{1}$, $\lambda_{1}\leftrightarrow\lambda_{5}$,
$\lambda_{2}\leftrightarrow\lambda_{4}$,
$\lambda_{3}\leftrightarrow\lambda_{3}$.

If we differentiate Eq.(2.33) with $u_{2}$, and identify
$\wp_{22}(u_{1},u_{2})\rightarrow u(x,t)$, ${\rm d}u_{2}\rightarrow{\rm d}x$
and ${\rm d}u_{1}\rightarrow{\rm d}t$, we have

$u_{xxx}-3uu_{x}=\lambda_{5}u_{t}+\lambda_{4}u_{x}.$ (2.38)

We can eliminate $\lambda_{4}u_{x}$ by the constant shift of $u\rightarrow
u-\lambda_{4}/3$, which gives the KdV equation
$\lambda_{5}u_{t}-u_{xxx}+3uu_{x}=0$. In the standard form of $\lambda_{0}=0$,
as the result of some dual symmetry, by identifying
$\wp_{11}(u_{1},u_{2})\rightarrow u(x,t)$, ${\rm d}u_{1}\rightarrow{\rm d}x$,
${\rm d}u_{2}\rightarrow{\rm d}t$, we have another KdV equation

$u_{xxx}-3uu_{x}=\lambda_{2}u_{x}+\lambda_{1}u_{t}$ (2.39)

from Eq.(2.37).

We must notice that $u(x,t)=\wp_{xx}(x,t)=\partial_{x}^{2}(-\log\sigma(x,t))$,
expressed with the genus two hyperelliptic $\sigma$ function, is the solution
but not the wave type solution, because $x$ and $t$ comes in the combination
$X=x-vt\ (v:\text{const.})$ in the wave type solution.

In this way, we have the KdV equation and another KdV equation. As the Lie
group structure of genus two hyperelliptic differential equations, we have sub
structure of SO(2,1) and another SO(2,1) because each KdV equations have the
SO(2,1) Lie group structure [15, 16, 17, 18, 19].

###  2.3 Differential equations of genus three hyperelliptic $\wp$ functions

We now move to the genus three case. The hyperelliptic curve in this case is
given by

$C:\quad y_{i}^{2}=\sum_{k=0}^{8}\lambda_{k}x_{i}^{k}.$ (2.40)

The Jacobi’s inversion problem consists of solving the following system

${\rm d}u_{1}=\sum_{i=1}^{3}\frac{{\rm d}x_{i}}{y_{i}},\qquad{\rm
d}u_{2}=\sum_{i=1}^{3}\frac{x_{i}{\rm d}x_{i}}{y_{i}},\qquad{\rm
d}u_{3}=\sum_{i=1}^{3}\frac{x_{i}^{2}{\rm d}x_{i}}{y_{i}}.$ (2.41)

Then we have

$\frac{\partial x_{1}}{\partial
u_{3}}=\frac{y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})},\quad\frac{\partial
x_{1}}{\partial
u_{2}}=-\frac{(x_{2}+x_{3})y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})},\quad\frac{\partial
x_{1}}{\partial u_{1}}=\frac{x_{2}x_{3}y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})},$
(2.42)

and $\\{x_{1},x_{2},x_{3}\\},\\{y_{1},y_{2},y_{3}\\}$ cyclic permutation. In
this case,

$\displaystyle{\rm d}(-\zeta_{3})=$
$\displaystyle\sum_{i=1}^{3}\frac{\left(2\lambda_{8}x_{i}^{4}+\lambda_{7}x_{i}^{3}\right){\rm
d}x_{i}}{y_{i}},$ (2.43) $\displaystyle{\rm d}(-\zeta_{2})=$
$\displaystyle\sum_{i=1}^{3}\frac{\left(4\lambda_{8}x_{i}^{5}+3\lambda_{7}x_{i}^{4}+2\lambda_{6}x_{i}^{3}+\lambda_{5}x_{i}^{2}\right){\rm
d}x_{i}}{y_{i}}$ $\displaystyle-2{\rm
d}\left(\frac{y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})}+\frac{y_{2}}{(x_{2}-x_{1})(x_{2}-x_{3})}+\frac{y_{3}}{(x_{3}-x_{1})(x_{3}-x_{2})}\right),$
(2.44) $\displaystyle{\rm d}(-\zeta_{1})=$
$\displaystyle\sum_{i=1}^{3}\frac{\left(6\lambda_{8}x_{i}^{6}+5\lambda_{7}x_{i}^{5}+4\lambda_{6}x_{i}^{4}+3\lambda_{5}x_{i}^{3}+2\lambda_{4}x_{i}^{2}+\lambda_{3}x_{i}\right){\rm
d}x_{i}}{y_{i}}$ $\displaystyle-2{\rm
d}\left(\frac{(x_{1}-x_{2}-x_{3})y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})}+\frac{(x_{2}-x_{3}-x_{1})y_{2}}{(x_{2}-x_{1})(x_{2}-x_{3})}+\frac{(x_{3}-x_{1}-x_{2})y_{3}}{(x_{3}-x_{1})(x_{3}-x_{2})}\right).$
(2.45)

For these $\zeta_{3},\zeta_{2},\zeta_{1}$, we have checked the integrability
condition $\partial\zeta_{i}/\partial u_{j}=\partial\zeta_{j}/\partial u_{i}$,
$(1\leq i<j\leq 3)$. Just as the same as the genus two case, in order that
differential equations become of the polynomial type, we must put
$\lambda_{8}=0$. In this case, we have

${\rm d}(-\zeta_{3})=\lambda_{7}\sum_{i=1}^{3}\frac{x_{i}^{3}{\rm
d}x_{i}}{y_{i}}=\sum_{j=1}^{3}\wp_{3j}{\rm d}u_{j}.$ (2.46)

which gives

$\displaystyle\widehat{\wp}_{33}$
$\displaystyle=\frac{1}{\lambda_{7}}\wp_{33}=\frac{1}{\lambda_{7}}\frac{\partial(-\zeta_{3})}{\partial
u_{3}}=x_{1}+x_{2}+x_{3},$ (2.47) $\displaystyle\widehat{\wp}_{32}$
$\displaystyle=\frac{1}{\lambda_{7}}\wp_{32}=\frac{1}{\lambda_{7}}\frac{\partial(-\zeta_{3})}{\partial
u_{2}}=-(x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{1}),$ (2.48)
$\displaystyle\widehat{\wp}_{31}$
$\displaystyle=\frac{1}{\lambda_{7}}\wp_{31}=\frac{1}{\lambda_{7}}\frac{\partial(-\zeta_{3})}{\partial
u_{1}}=x_{1}x_{2}x_{3}.$ (2.49)

Then we have the following differential equations[23, 24, 25]

$\displaystyle 1)\quad$
$\displaystyle\wp_{3333}-\frac{3}{2}\wp_{33}^{2}=\lambda_{7}\wp_{32}+\lambda_{6}\wp_{33}+\frac{1}{2}\lambda_{7}\lambda_{5},$
(2.50) $\displaystyle 2)\quad$
$\displaystyle\wp_{3332}-\frac{3}{2}\wp_{33}\wp_{32}=\frac{3}{2}\lambda_{7}\wp_{31}-\frac{1}{2}\lambda_{7}\wp_{22}+\lambda_{6}\wp_{32},$
(2.51) $\displaystyle 3)\quad$
$\displaystyle\wp_{3331}-\frac{3}{2}\wp_{33}\wp_{31}=-\frac{1}{2}\lambda_{7}\wp_{21}+\lambda_{6}\wp_{31},$
(2.52) $\displaystyle 4)\quad$
$\displaystyle\wp_{3322}-\frac{1}{2}\wp_{33}\wp_{22}-\wp_{32}^{2}=-\frac{1}{2}\lambda_{7}\wp_{21}+\lambda_{6}\wp_{31}+\frac{1}{2}\lambda_{5}\wp_{32},$
(2.53) $\displaystyle 5)\quad$
$\displaystyle\wp_{3321}-\frac{1}{2}\wp_{33}\wp_{21}-\wp_{32}\wp_{31}=\frac{1}{2}\lambda_{5}\wp_{31},$
(2.54) $\displaystyle 6)\quad$
$\displaystyle\wp_{3311}-\frac{1}{2}\wp_{33}\wp_{11}-\wp_{31}^{2}=\frac{1}{2}\Delta,$
(2.55) $\displaystyle 7)\quad$
$\displaystyle\wp_{3222}-\frac{3}{2}\wp_{32}\wp_{22}=-\frac{3}{2}\lambda_{7}\wp_{11}+\lambda_{5}\wp_{31}+\lambda_{4}\wp_{32}-\frac{1}{2}\lambda_{3}\wp_{33}-\lambda_{7}\lambda_{2},$
(2.56) $\displaystyle 8)\quad$
$\displaystyle\wp_{3221}-\frac{1}{2}\wp_{31}\wp_{22}-\wp_{32}\wp_{21}=-\frac{1}{2}\Delta+\lambda_{4}\wp_{31}-\frac{1}{2}\lambda_{7}\lambda_{1},$
(2.57) $\displaystyle 9)\quad$
$\displaystyle\wp_{3211}-\frac{1}{2}\wp_{32}\wp_{11}-\wp_{31}\wp_{21}=\frac{1}{2}\lambda_{3}\wp_{31}-\lambda_{7}\lambda_{0},$
(2.58) $\displaystyle 10)\quad$
$\displaystyle\wp_{3111}-\frac{3}{2}\wp_{31}\wp_{11}=\lambda_{2}\wp_{31}-\frac{1}{2}\lambda_{1}\wp_{32}+\lambda_{0}\wp_{33},$
(2.59) $\displaystyle 11)\quad$
$\displaystyle\wp_{2222}-\frac{3}{2}\wp_{22}^{2}=3\Delta-3\lambda_{6}\wp_{11}+\lambda_{5}\wp_{21}+\lambda_{4}\wp_{22}+\lambda_{3}\wp_{32}-3\lambda_{2}\wp_{33}$
$\displaystyle-2\lambda_{6}\lambda_{2}+\frac{1}{2}\lambda_{5}\lambda_{3}-\frac{3}{2}\lambda_{7}\lambda_{1},$
(2.60) $\displaystyle 12)\quad$
$\displaystyle\wp_{2221}-\frac{3}{2}\wp_{22}\wp_{21}=-\frac{1}{2}\lambda_{5}\wp_{11}+\lambda_{4}\wp_{21}+\lambda_{3}\wp_{31}-\frac{3}{2}\lambda_{1}\wp_{33}-2\lambda_{7}\lambda_{0}-\lambda_{6}\lambda_{1},$
(2.61) $\displaystyle 13)\quad$
$\displaystyle\wp_{2211}-\frac{1}{2}\wp_{22}\wp_{11}-\wp_{21}^{2}=\frac{1}{2}\lambda_{3}\wp_{21}+\lambda_{2}\wp_{31}-\frac{1}{2}\lambda_{1}\wp_{32}-2\lambda_{0}\wp_{33}-2\lambda_{6}\lambda_{0},$
(2.62) $\displaystyle 14)\quad$
$\displaystyle\wp_{2111}-\frac{3}{2}\wp_{21}\wp_{11}=\lambda_{2}\wp_{21}+\frac{3}{2}\lambda_{1}\wp_{31}-\frac{1}{2}\lambda_{1}\wp_{22}-2\lambda_{0}\wp_{32}-\lambda_{5}\lambda_{0},$
(2.63) $\displaystyle 15)\quad$
$\displaystyle\wp_{1111}-\frac{3}{2}\wp_{11}^{2}=\lambda_{2}\wp_{11}+\lambda_{1}\wp_{21}+4\lambda_{0}\wp_{31}-3\lambda_{0}\wp_{22}-2\lambda_{4}\lambda_{0}+\frac{1}{2}\lambda_{3}\lambda_{1},$
(2.64)

where
$\Delta=\wp_{32}\wp_{21}-\wp_{31}\wp_{22}-\wp_{33}\wp_{11}+\wp_{31}^{2}$.

Just as in genus two case, if we take $\lambda_{0}=0$ as the standard form of
the hyperelliptic curve, the set of differential equations have some dual
symmetry Eq.(2.50) $\leftrightarrow$ Eq.(2.64) , Eq.(2.51) $\leftrightarrow$
Eq.(2.63), Eq.(2.52) $\leftrightarrow$ Eq.(2.59), etc. under
$u_{3}\leftrightarrow\pm u_{1}$, $u_{2}\leftrightarrow\pm u_{2}$,
$\lambda_{1}\leftrightarrow\lambda_{7}$,
$\lambda_{2}\leftrightarrow\lambda_{6}$,
$\lambda_{3}\leftrightarrow\lambda_{5}$,
$\lambda_{4}\leftrightarrow\lambda_{4}$. In this standard form of
$\lambda_{0}=0$, Eq.(2.50) and Eq.(2.64) become KdV equation Eq.(2.38) with
$\lambda_{j}\rightarrow\lambda_{j+2}$ and another KdV equation Eq.(2.39).

While if we take $\lambda_{1}=0$ as the standard form, by identifying
$\wp_{11}\rightarrow u$, ${\rm d}u_{1}\rightarrow{\rm d}x$, ${\rm
d}u_{2}\rightarrow{\rm d}y$, ${\rm d}u_{3}\rightarrow{\rm d}t$, we have KP
equation

$\big{(}u_{xxx}-3uu_{x}-\lambda_{2}u_{x}-4\lambda_{0}u_{t}\big{)}_{x}=-3\lambda_{0}u_{yy},$
(2.65)

from Eq.(2.64). In this way, Eq.(2.64) becomes the KdV equation in the
$\lambda_{0}=0$ standard form, and the same Eq.(2.64) becomes the KP equation
in the $\lambda_{1}=0$ standard form. Then the difference of the KdV equation
and the KP equation comes from the choice of standard form of the
hyperelliptic curve. Therefore, the KdV equation and the KP equation belongs
to the same family in this approach.

By differentiating Eq.(2.60) with $u_{2}$ twice, we have the following three
variables differential equation

$\big{(}u_{xxx}-3uu_{x}-\lambda_{4}u_{x}-\lambda_{5}u_{t}\big{)}_{x}=3\Delta_{xx}-3\lambda_{6}u_{tt}+\lambda_{3}u_{xy}-3\lambda_{2}u_{yy},$
(2.66)

by identifying $\wp_{22}\rightarrow u$, ${\rm d}u_{1}\rightarrow{\rm d}t$,
${\rm d}u_{2}\rightarrow{\rm d}x$, ${\rm d}u_{3}\rightarrow{\rm d}y$. If we
consider the special hyperelliptic curve with $\lambda_{6}=0$,
$\lambda_{3}=0$, Eq.(2.65) becomes the KP equation except $\Delta_{xx}$ term
in the form

$\big{(}u_{xxx}-3uu_{x}-\lambda_{4}u_{x}-\lambda_{5}u_{t}\big{)}_{x}+3\lambda_{2}u_{yy}=3\Delta_{xx},$
(2.67)

and we have checked that $\Delta_{xx}\neq 0$ even for this special
hyperelliptic curve. Then we have three variables new type integrable
differential equation, which is KP type but is different from KP equation
itself.

##  3 Differential Equations of Genus Four Hyperelliptic $\wp$ Functions

Now let us consider the genus four case. The hyperelliptic curve in this case
is given by

$C:\quad y_{i}^{2}=\sum_{k=0}^{10}\lambda_{k}x_{i}^{k}.$ (3.1)

The Jacobi’s inversion problem consists of solving the following system

$\displaystyle{\rm d}u_{1}=\sum_{i=1}^{4}\frac{{\rm d}x_{i}}{y_{i}},\qquad{\rm
d}u_{2}=\sum_{i=1}^{4}\frac{x_{i}{\rm d}x_{i}}{y_{i}},\qquad{\rm
d}u_{3}=\sum_{i=1}^{4}\frac{x_{i}^{2}{\rm d}x_{i}}{y_{i}},\qquad{\rm
d}u_{4}=\sum_{i=1}^{4}\frac{x_{i}^{3}{\rm d}x_{i}}{y_{i}}.$ (3.2)

Then we have

$\displaystyle\frac{\partial x_{1}}{\partial u_{4}}$
$\displaystyle=\frac{y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})},$
$\displaystyle\frac{\partial x_{1}}{\partial u_{3}}$
$\displaystyle=-\frac{(x_{2}+x_{3}+x_{4})y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})},$
$\displaystyle\frac{\partial x_{1}}{\partial u_{2}}$
$\displaystyle=\frac{(x_{2}x_{3}+x_{3}x_{4}+x_{4}x_{2})y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})},$
$\displaystyle\frac{\partial x_{1}}{\partial u_{1}}$
$\displaystyle=-\frac{x_{2}x_{3}x_{4}y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})},$
(3.3)

and $\\{x_{1},x_{2},x_{3},x_{4}\\},\\{y_{1},y_{2},y_{3},y_{4}\\}$ cyclic
permutation. In this case,

$\displaystyle{\rm d}(-\zeta_{4})=$
$\displaystyle\sum_{i=1}^{4}\frac{\left(2\lambda_{10}x_{i}^{5}+\lambda_{9}x_{i}^{4}\right){\rm
d}x_{i}}{y_{i}},$ (3.4) $\displaystyle{\rm d}(-\zeta_{3})=$
$\displaystyle\sum_{i=1}^{4}\frac{\left(4\lambda_{10}x_{i}^{6}+3\lambda_{9}x_{i}^{5}+2\lambda_{8}x_{i}^{4}+\lambda_{7}x_{i}^{3}\right){\rm
d}x_{i}}{y_{i}}$ $\displaystyle-2{\rm
d}\bigg{(}\frac{y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})}+\frac{y_{2}}{(x_{2}-x_{1})(x_{2}-x_{3})(x_{2}-x_{4})}$
$\displaystyle\hskip
28.45274pt+\frac{y_{3}}{(x_{3}-x_{1})(x_{3}-x_{2})(x_{3}-x_{4})}+\frac{y_{4}}{(x_{4}-x_{1})(x_{4}-x_{2})(x_{4}-x_{3})}\bigg{)},$
(3.5) $\displaystyle{\rm d}(-\zeta_{2})=$
$\displaystyle\sum_{i=1}^{4}\frac{\left(6\lambda_{10}x_{i}^{7}+5\lambda_{9}x_{i}^{6}+4\lambda_{8}x_{i}^{5}+3\lambda_{7}x_{i}^{4}+2\lambda_{6}x_{i}^{3}+\lambda_{5}x_{i}^{2}\right){\rm
d}x_{i}}{y_{i}}$ $\displaystyle-2{\rm
d}\bigg{(}\frac{(x_{1}-x_{2}-x_{3}-x_{4})y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})}+\frac{(x_{2}-x_{3}-x_{4}-x_{1})y_{2}}{(x_{2}-x_{1})(x_{2}-x_{3})(x_{2}-x_{4})}$
$\displaystyle\hskip
28.45274pt+\frac{(x_{3}-x_{4}-x_{1}-x_{2})y_{3}}{(x_{3}-x_{1})(x_{3}-x_{2})(x_{3}-x_{4})}+\frac{(x_{4}-x_{1}-x_{2}-x_{3})y_{4}}{(x_{4}-x_{1})(x_{4}-x_{2})(x_{4}-x_{3})}\bigg{)},$
(3.6) $\displaystyle{\rm d}(-\zeta_{1})=$
$\displaystyle\sum_{i=1}^{4}\frac{\left(8\lambda_{10}x_{i}^{8}+7\lambda_{9}x_{i}^{7}+6\lambda_{8}x_{i}^{6}+5\lambda_{7}x_{i}^{5}+4\lambda_{6}x_{i}^{4}+3\lambda_{5}x_{i}^{3}+2\lambda_{4}x_{i}^{2}+\lambda_{3}x_{i}\right){\rm
d}x_{i}}{y_{i}}$ $\displaystyle-2{\rm
d}\left(\frac{\left(x_{1}^{2}-x_{1}(x_{2}+x_{3}+x_{4})+(x_{2}x_{3}+x_{3}x_{4}+x_{4}x_{2})\right)y_{1}}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})}\right.$
$\displaystyle\hskip
28.45274pt+\frac{\left(x_{2}^{2}-x_{2}(x_{3}+x_{4}+x_{1})+(x_{3}x_{4}+x_{4}x_{1}+x_{1}x_{3})\right)y_{2}}{(x_{2}-x_{1})(x_{2}-x_{3})(x_{2}-x_{4})}$
$\displaystyle\hskip
28.45274pt+\frac{\left(x_{3}^{2}-x_{3}(x_{4}+x_{1}+x_{2})+(x_{4}x_{1}+x_{1}x_{2}+x_{2}x_{4})\right)y_{3}}{(x_{3}-x_{1})(x_{3}-x_{2})(x_{3}-x_{4})}$
$\displaystyle\hskip
28.45274pt\left.+\frac{\left(x_{4}^{2}-x_{4}(x_{1}+x_{2}+x_{3})+(x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{1})\right)y_{4}}{(x_{4}-x_{1})(x_{4}-x_{2})(x_{4}-x_{3})}\right).$
(3.7)

For these $\zeta_{4},\zeta_{3},\zeta_{2},\zeta_{1}$, we have checked the
integrability condition $\partial\zeta_{i}/\partial
u_{j}=\partial\zeta_{j}/\partial u_{i}$, $(1\leq i<j\leq 4)$. Just as the same
as the genus two and genus three cases, in order that differential equations
become of the polynomial type, we must take $\lambda_{8}=0$. In this case, we
have

${\rm d}(-\zeta_{4})=\lambda_{9}\sum_{i=1}^{4}\frac{x_{i}^{4}{\rm
d}x_{i}}{y_{i}}=\sum_{j=1}^{4}\wp_{4j}{\rm d}u_{j},$ (3.8)

which gives

$\displaystyle\widehat{\wp}_{44}$
$\displaystyle=\frac{1}{\lambda_{9}}\wp_{44}=\frac{1}{\lambda_{9}}\frac{\partial(-\zeta_{4})}{\partial
u_{4}}=x_{1}+x_{2}+x_{3}+x_{4},$ (3.9) $\displaystyle\widehat{\wp}_{43}$
$\displaystyle=\frac{1}{\lambda_{9}}\wp_{43}=\frac{1}{\lambda_{9}}\frac{\partial(-\zeta_{4})}{\partial
u_{3}}=-(x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4}),$
(3.10) $\displaystyle\widehat{\wp}_{42}$
$\displaystyle=\frac{1}{\lambda_{9}}\wp_{42}=\frac{1}{\lambda_{9}}\frac{\partial(-\zeta_{4})}{\partial
u_{2}}=x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4},$
(3.11) $\displaystyle\widehat{\wp}_{41}$
$\displaystyle=\frac{1}{\lambda_{9}}\wp_{41}=\frac{1}{\lambda_{9}}\frac{\partial(-\zeta_{4})}{\partial
u_{1}}=-x_{1}x_{2}x_{3}x_{4}.$ (3.12)

Then we have the following differential equations

$\displaystyle 1)\quad$
$\displaystyle\wp_{4444}-\frac{3}{2}\wp_{44}^{2}=\lambda_{9}\wp_{43}+\lambda_{8}\wp_{44}+\frac{1}{2}\lambda_{9}\lambda_{7},$
(3.13) $\displaystyle 2)\quad$
$\displaystyle\wp_{4443}-\frac{3}{2}\wp_{44}\wp_{43}=\frac{3}{2}\lambda_{9}\wp_{42}-\frac{1}{2}\lambda_{9}\wp_{33}+\lambda_{8}\wp_{43},$
(3.14) $\displaystyle 3)\quad$
$\displaystyle\wp_{4442}-\frac{3}{2}\wp_{44}\wp_{42}=\frac{3}{2}\lambda_{9}\wp_{41}-\frac{1}{2}\lambda_{9}\wp_{32}+\lambda_{8}\wp_{42},$
(3.15) $\displaystyle 4)\quad$
$\displaystyle\wp_{4441}-\frac{3}{2}\wp_{44}\wp_{41}=-\frac{1}{2}\lambda_{9}\wp_{31}+\lambda_{8}\wp_{41},$
(3.16) $\displaystyle 5)\quad$
$\displaystyle\wp_{4433}-\frac{1}{2}\wp_{44}\wp_{33}-\wp_{43}^{2}=\frac{3}{2}\lambda_{9}\wp_{41}-\frac{1}{2}\lambda_{9}\wp_{32}+\lambda_{8}\wp_{42}+\frac{1}{2}\lambda_{7}\wp_{43},$
(3.17) $\displaystyle 6)\quad$
$\displaystyle\wp_{4432}-\frac{1}{2}\wp_{44}\wp_{32}-\wp_{43}\wp_{42}=-\frac{1}{2}\lambda_{9}\wp_{31}+\lambda_{8}\wp_{41}+\frac{1}{2}\lambda_{7}\wp_{42},$
(3.18) $\displaystyle 7)\quad$
$\displaystyle\wp_{4431}-\frac{1}{2}\wp_{44}\wp_{31}-\wp_{43}\wp_{41}=\frac{1}{2}\lambda_{7}\wp_{41},$
(3.19) $\displaystyle 8)\quad$
$\displaystyle\wp_{4422}-\frac{3}{2}\wp_{42}^{2}=\frac{1}{2}\Delta_{1}+\frac{1}{2}\lambda_{7}\wp_{41},$
(3.20) $\displaystyle 9)\quad$
$\displaystyle\wp_{4421}-\frac{3}{2}\wp_{42}\wp_{41}=\frac{1}{2}\Delta_{8},$
(3.21) $\displaystyle 10)\quad$
$\displaystyle\wp_{4411}-\frac{3}{2}\wp_{41}^{2}=\frac{1}{2}\Delta_{9},$
(3.22) $\displaystyle 11)\quad$
$\displaystyle\wp_{4333}-\frac{3}{2}\wp_{43}\wp_{33}=\frac{3}{2}\lambda_{9}\wp_{31}-\frac{3}{2}\lambda_{9}\wp_{22}+\lambda_{8}\wp_{41}+\lambda_{7}\wp_{42}+\lambda_{6}\wp_{43}$
$\displaystyle-\frac{1}{2}\lambda_{5}\wp_{44}-\lambda_{9}\lambda_{4},$ (3.23)
$\displaystyle 12)\quad$
$\displaystyle\wp_{4332}-\frac{1}{2}\wp_{42}\wp_{33}-\wp_{43}\wp_{32}=-\frac{1}{2}\Delta_{2}-\lambda_{9}\wp_{21}+\lambda_{7}\wp_{41}+\lambda_{6}\wp_{42}-\frac{1}{2}\lambda_{9}\lambda_{3},$
(3.24) $\displaystyle 13)\quad$
$\displaystyle\wp_{4331}-\frac{1}{2}\wp_{41}\wp_{33}-\wp_{43}\wp_{31}=\frac{1}{2}\Delta_{3}+\frac{1}{2}\lambda_{9}\wp_{11}+\lambda_{6}\wp_{41},$
(3.25) $\displaystyle 14)\quad$
$\displaystyle\wp_{4322}-\frac{1}{2}\wp_{43}\wp_{22}-\wp_{42}\wp_{32}=\frac{1}{2}\Delta_{3}-\lambda_{9}\wp_{11}+\lambda_{6}\wp_{41}+\frac{1}{2}\lambda_{5}\wp_{42}-\lambda_{9}\lambda_{2},$
(3.26) $\displaystyle 15)\quad$
$\displaystyle\wp_{4321}-\frac{1}{2}\wp_{43}\wp_{21}-\frac{1}{2}\wp_{42}\wp_{31}-\frac{1}{2}\wp_{41}\wp_{32}=\frac{1}{2}\lambda_{5}\wp_{41}-\frac{1}{2}\lambda_{9}\lambda_{1},$
(3.27) $\displaystyle 16)\quad$
$\displaystyle\wp_{4311}-\frac{3}{2}\wp_{41}\wp_{31}=\frac{1}{2}\Delta_{10}-\lambda_{9}\lambda_{0},$
(3.28) $\displaystyle 17)\quad$
$\displaystyle\wp_{4222}-\frac{3}{2}\wp_{42}\wp_{22}=\frac{3}{2}\Delta_{4}+\lambda_{5}\wp_{41}+\lambda_{4}\wp_{42}-\frac{1}{2}\lambda_{3}\wp_{43}+\lambda_{2}\wp_{44}-\lambda_{9}\lambda_{1},$
(3.29) $\displaystyle 18)\quad$
$\displaystyle\wp_{4221}-\frac{1}{2}\wp_{41}\wp_{22}-\wp_{42}\wp_{21}=\frac{1}{2}\Delta_{5}+\lambda_{4}\wp_{41}+\frac{1}{2}\lambda_{1}\wp_{44}-\lambda_{9}\lambda_{0},$
(3.30) $\displaystyle 19)\quad$
$\displaystyle\wp_{4211}-\frac{1}{2}\wp_{42}\wp_{11}-\wp_{41}\wp_{21}=\frac{1}{2}\lambda_{3}\wp_{41}+\lambda_{0}\wp_{44},$
(3.31) $\displaystyle 20)\quad$
$\displaystyle\wp_{4111}-\frac{3}{2}\wp_{41}\wp_{11}=\lambda_{2}\wp_{41}-\frac{1}{2}\lambda_{1}\wp_{42}+\lambda_{0}\wp_{43},$
(3.32) $\displaystyle 21)\quad$
$\displaystyle\wp_{3333}-\frac{3}{2}\wp_{33}^{2}=3\Delta_{2}-3\lambda_{9}\wp_{21}+4\lambda_{8}\wp_{31}-3\lambda_{8}\wp_{22}+\lambda_{7}\wp_{32}+\lambda_{6}\wp_{33}+\lambda_{5}\wp_{43}$
$\displaystyle-3\lambda_{4}\wp_{44}+\frac{1}{2}\lambda_{7}\lambda_{5}-2\lambda_{8}\lambda_{4}-\frac{3}{2}\lambda_{9}\lambda_{3},$
(3.33) $\displaystyle 22)\quad$
$\displaystyle\wp_{3332}-\frac{3}{2}\wp_{33}\wp_{32}=-\frac{3}{2}\Delta_{3}-\frac{3}{2}\lambda_{9}\wp_{11}-2\lambda_{8}\wp_{21}+\frac{3}{2}\lambda_{7}\wp_{31}-\frac{1}{2}\lambda_{7}\wp_{22}+\lambda_{6}\wp_{32}$
$\displaystyle+\lambda_{5}\wp_{42}-\frac{3}{2}\lambda_{3}\wp_{44}-\lambda_{8}\lambda_{3}-2\lambda_{9}\lambda_{2},$
(3.34) $\displaystyle 23)\quad$
$\displaystyle\wp_{3331}-\frac{3}{2}\wp_{33}\wp_{31}=\frac{3}{2}\Delta_{4}+\lambda_{8}\wp_{11}-\frac{1}{2}\lambda_{7}\wp_{21}+\lambda_{6}\wp_{31}+\lambda_{5}\wp_{41}-\lambda_{9}\lambda_{1},$
(3.35) $\displaystyle 24)\quad$
$\displaystyle\wp_{3322}-\frac{1}{2}\wp_{33}\wp_{22}-\wp_{32}^{2}=-\frac{3}{2}\Delta_{4}-2\lambda_{8}\wp_{11}-\frac{1}{2}\lambda_{7}\wp_{21}+\lambda_{6}\wp_{31}$
$\displaystyle+\frac{1}{2}\lambda_{5}\wp_{41}+\frac{1}{2}\lambda_{5}\wp_{32}+\lambda_{4}\wp_{42}-\frac{1}{2}\lambda_{3}\wp_{43}-2\lambda_{2}\wp_{44}-2\lambda_{8}\lambda_{2}-2\lambda_{9}\lambda_{1},$
(3.36) $\displaystyle 25)\quad$
$\displaystyle\wp_{3321}-\frac{1}{2}\wp_{33}\wp_{21}-\wp_{32}\wp_{31}=\frac{1}{2}\Delta_{5}+\frac{1}{2}\lambda_{5}\wp_{31}+\lambda_{4}\wp_{41}-\lambda_{1}\wp_{44}$
$\displaystyle-\lambda_{8}\lambda_{1}-2\lambda_{9}\lambda_{0},$ (3.37)
$\displaystyle 26)\quad$
$\displaystyle\wp_{3311}-\frac{3}{2}\wp_{31}^{2}=\frac{1}{2}\Delta_{6}+\frac{1}{2}\lambda_{3}\wp_{41}-2\lambda_{0}\wp_{44}-2\lambda_{8}\lambda_{0},$
(3.38) $\displaystyle 27)\quad$
$\displaystyle\wp_{3222}-\frac{3}{2}\wp_{32}\wp_{22}=-\frac{3}{2}\Delta_{5}-\frac{3}{2}\lambda_{7}\wp_{11}+\lambda_{5}\wp_{31}+\lambda_{4}\wp_{32}+\frac{3}{2}\lambda_{3}\wp_{42}$
$\displaystyle-\frac{1}{2}\lambda_{3}\wp_{33}-2\lambda_{2}\wp_{43}-\frac{3}{2}\lambda_{1}\wp_{44}-2\lambda_{8}\lambda_{1}-\lambda_{7}\lambda_{2}-3\lambda_{9}\lambda_{0},$
(3.39) $\displaystyle 28)\quad$
$\displaystyle\wp_{3221}-\frac{1}{2}\wp_{31}\wp_{22}-\wp_{32}\wp_{21}=-\frac{1}{2}\Delta_{7}+\lambda_{4}\wp_{31}+\lambda_{3}\wp_{41}-\lambda_{1}\wp_{43}-\lambda_{0}\wp_{44}$
$\displaystyle-2\lambda_{8}\lambda_{0}-\frac{1}{2}\lambda_{7}\lambda_{1},$
(3.40) $\displaystyle 29)\quad$
$\displaystyle\wp_{3211}-\frac{1}{2}\wp_{32}\wp_{11}-\wp_{31}\wp_{21}=\frac{1}{2}\lambda_{3}\wp_{31}+\lambda_{2}\wp_{41}-\frac{1}{2}\lambda_{1}\wp_{42}-\lambda_{0}\wp_{43}-\lambda_{7}\lambda_{0},$
(3.41) $\displaystyle 30)\quad$
$\displaystyle\wp_{3111}-\frac{3}{2}\wp_{31}\wp_{11}=\lambda_{2}\wp_{31}+\frac{3}{2}\lambda_{1}\wp_{41}-\frac{1}{2}\lambda_{1}\wp_{32}-3\lambda_{0}\wp_{42}+\lambda_{0}\wp_{33},$
(3.42) $\displaystyle 31)\quad$
$\displaystyle\wp_{2222}-\frac{3}{2}\wp_{22}^{2}=3\Delta_{7}-3\lambda_{6}\wp_{11}+\lambda_{5}\wp_{21}+\lambda_{4}\wp_{22}+\lambda_{3}\wp_{32}+4\lambda_{2}\wp_{42}$
$\displaystyle-3\lambda_{2}\wp_{33}-3\lambda_{1}\wp_{43}-3\lambda_{0}\wp_{44}-4\lambda_{8}\lambda_{0}-\frac{3}{2}\lambda_{7}\lambda_{1}-2\lambda_{6}\lambda_{2}+\frac{1}{2}\lambda_{5}\lambda_{3},$
(3.43) $\displaystyle 32)\quad$
$\displaystyle\wp_{2221}-\frac{3}{2}\wp_{22}\wp_{21}=-\frac{1}{2}\lambda_{5}\wp_{11}+\lambda_{4}\wp_{21}+\lambda_{3}\wp_{31}+\lambda_{2}\wp_{41}+\frac{3}{2}\lambda_{1}\wp_{42}-\frac{3}{2}\lambda_{1}\wp_{33}$
$\displaystyle-3\lambda_{0}\wp_{43}-2\lambda_{7}\lambda_{0}-\lambda_{6}\lambda_{1},$
(3.44) $\displaystyle 33)\quad$
$\displaystyle\wp_{2211}-\frac{1}{2}\wp_{22}\wp_{11}-\wp_{21}^{2}=\frac{1}{2}\lambda_{3}\wp_{21}+\lambda_{2}\wp_{31}+\frac{3}{2}\lambda_{1}\wp_{41}-\frac{1}{2}\lambda_{1}\wp_{32}+\lambda_{0}\wp_{42}$
$\displaystyle-2\lambda_{0}\wp_{33}-2\lambda_{6}\lambda_{0},$ (3.45)
$\displaystyle 34)\quad$
$\displaystyle\wp_{2111}-\frac{3}{2}\wp_{21}\wp_{11}=\lambda_{2}\wp_{21}+\frac{3}{2}\lambda_{1}\wp_{31}-\frac{1}{2}\lambda_{1}\wp_{22}+3\lambda_{0}\wp_{41}-2\lambda_{0}\wp_{32}-\lambda_{5}\lambda_{0},$
(3.46) $\displaystyle 35)\quad$
$\displaystyle\wp_{1111}-\frac{3}{2}\wp_{11}^{2}=\lambda_{2}\wp_{11}+\lambda_{1}\wp_{21}+4\lambda_{0}\wp_{31}-3\lambda_{0}\wp_{22}-2\lambda_{4}\lambda_{0}+\frac{1}{2}\lambda_{3}\lambda_{1},$
(3.47)

where

$\displaystyle\Delta_{1}$
$\displaystyle=\wp_{44}\wp_{31}-\wp_{43}\wp_{41}+\wp_{43}\wp_{32}-\wp_{42}\wp_{33},$
$\displaystyle\Delta_{2}$
$\displaystyle=\Delta_{1}-\wp_{44}\wp_{22}+\wp_{42}^{2},$
$\displaystyle\Delta_{3}$
$\displaystyle=\wp_{44}\wp_{21}-\wp_{42}\wp_{41}-\wp_{43}\wp_{31}+\wp_{41}\wp_{33},$
$\displaystyle\Delta_{4}$
$\displaystyle=\wp_{44}\wp_{11}-\wp_{41}^{2}-\wp_{42}\wp_{31}+\wp_{41}\wp_{32},$
$\displaystyle\Delta_{5}$
$\displaystyle=\wp_{43}\wp_{11}-\wp_{41}\wp_{31}-\wp_{42}\wp_{21}+\wp_{41}\wp_{22},$
$\displaystyle\Delta_{6}$
$\displaystyle=\wp_{42}\wp_{11}-\wp_{41}\wp_{21}+\wp_{32}\wp_{21}-\wp_{31}\wp_{22},$
$\displaystyle\Delta_{7}$
$\displaystyle=\Delta_{6}-\wp_{33}\wp_{11}+\wp_{31}^{2},$
$\displaystyle\Delta_{8}$ $\displaystyle=\wp_{43}\wp_{31}-\wp_{41}\wp_{33},$
$\displaystyle\Delta_{9}$ $\displaystyle=\wp_{42}\wp_{31}-\wp_{41}\wp_{32},$
$\displaystyle\Delta_{10}$ $\displaystyle=\wp_{42}\wp_{21}-\wp_{41}\wp_{22}.$

These $\Delta_{i}$ have the symmetry $\Delta_{1}\leftrightarrow\Delta_{6}$,
$\Delta_{2}\leftrightarrow\Delta_{7}$, $\Delta_{3}\leftrightarrow\Delta_{5}$,
$\Delta_{4}\leftrightarrow\Delta_{4}$, $\Delta_{8}\leftrightarrow\Delta_{10}$,
$\Delta_{9}\leftrightarrow\Delta_{9}$, under ${\rm
d}u_{1}\leftrightarrow\pm{\rm d}u_{4}$, ${\rm d}u_{2}\leftrightarrow\pm{\rm
d}u_{3}$.

Just as in genus two and three cases, in the standard form of the
hyperelliptic curve of $\lambda_{0}=0$, the set of differential equations have
the dual symmetry Eq.(3.13) $\leftrightarrow$ Eq.(3.47) , Eq.(3.14)
$\leftrightarrow$ Eq.(3.46), Eq.(3.15) $\leftrightarrow$ Eq.(3.42), etc.,
under $u_{4}\leftrightarrow\pm u_{1}$, $u_{3}\leftrightarrow\pm u_{2}$,
$\lambda_{1}\leftrightarrow\lambda_{9}$,
$\lambda_{2}\leftrightarrow\lambda_{8}$,
$\lambda_{3}\leftrightarrow\lambda_{7}$,
$\lambda_{4}\leftrightarrow\lambda_{6}$,
$\lambda_{5}\leftrightarrow\lambda_{5}$.

In the standard form of $\lambda_{0}=0$, the differential equation of
Eq.(3.13) and Eq.(3.47) are KdV equation Eq.(2.38) with
$\lambda_{j}\rightarrow\lambda_{j+4}$ and another KdV equation Eq.(2.39).
While in the standard form of $\lambda_{1}=0$, the differential equation
Eq.(3.47) is KP equation Eq.(2.65).

By differentiating Eq.(3.33) with $u_{3}$ twice, we have four variables
differential equation, which is KP type equation except the term
$(\Delta_{2})_{xx}(\neq 0)$ in the form

$\big{(}u_{xxx}-3uu_{x}-\lambda_{7}u_{t}-\lambda_{6}u_{x}\big{)}_{x}=3(\Delta_{2})_{xx}-3\lambda_{9}u_{zt}+4\lambda_{8}u_{zx}-3\lambda_{8}u_{tt}+\lambda_{5}u_{xy}-3\lambda_{4}u_{yy},$
(3.48)

by identifying $\wp_{33}\rightarrow u$, ${\rm d}u_{1}\rightarrow{\rm d}z$,
${\rm d}u_{2}\rightarrow{\rm d}t$, ${\rm d}u_{3}\rightarrow{\rm d}x$, ${\rm
d}u_{4}\rightarrow{\rm d}y$. Then we have four variables KP type new
integrable differential equation. Eq.(3.43) gives four variables another KP
type differential equation.

##  4 Properties of Hyperelliptic Differential Equations

###  4.1 Some dual symmetry for the set of differential equations

In the previous sections, we have explained the symmetry of differential
equations, that is, in the standard form of $\lambda_{2g+2}=0$ and
$\lambda_{0}=0$ in the hyperelliptic curve, the set of differential equations
have some dual symmetry under

$\wp_{jk}\leftrightarrow\wp_{g+1-j,g+1-k},\quad\wp_{jklm}\leftrightarrow\wp_{g+1-j,g+1-k,g+1-l,g+1-m},\quad\lambda_{k}\leftrightarrow\tilde{\lambda}_{k}=\lambda_{2g+2-k}.$
(4.1)

The standard form of the hyperelliptic curve is given by

$C:\quad
y_{i}^{2}=\lambda_{2g+1}x_{i}^{2g+1}+\lambda_{2g}x_{i}^{2g}+\cdots+\lambda_{2}x_{i}^{2}+\lambda_{1}x_{i}.$
(4.2)

If we change variables in the form
$\displaystyle{\tilde{x}_{i}=\frac{1}{x_{i}}}$,
$\displaystyle{\tilde{y}_{i}=\frac{y_{i}}{x_{i}^{g+1}}}$,
$\tilde{\lambda}_{k}=\lambda_{2g+2-k}$, we can rewrite the curve in the form

$\tilde{C}:\quad\tilde{y}_{i}^{2}=\tilde{\lambda}_{2g+1}\tilde{x}_{i}^{2g+1}+\tilde{\lambda}_{2g}\tilde{x}_{i}^{2g}+\cdots+\tilde{\lambda}_{2}\tilde{x}_{i}^{2}+\tilde{\lambda}_{1}\tilde{x}_{i}.$
(4.3)

Then we have

${\rm d}\tilde{u}_{j}=\sum_{i=1}^{g}\frac{\tilde{x}_{i}^{j-1}{\rm
d}\tilde{x}_{i}}{\tilde{y}_{i}}=-\sum_{i=1}^{g}\frac{x_{i}^{g-j}{\rm
d}x_{i}}{y_{i}}=-{\rm d}u_{g+1-j},$ (4.4)

that is, ${\rm d}\tilde{u}_{g}=-{\rm d}u_{1}$, ${\rm d}\tilde{u}_{g-1}=-{\rm
d}u_{2}$, $\cdots$, ${\rm d}\tilde{u}_{2}=-{\rm d}u_{g-1}$, and ${\rm
d}\tilde{u}_{1}=-{\rm d}u_{g}$.

From the curve Eq.(4.3), we construct hyperelliptic sigma function
$\tilde{\sigma}$. While we construct $\sigma$ from the curve Eq.(4.2). But the
difference between Eq.(4.2) and Eq.(4.3) is only the choice of the local
variable, so that $\sigma$ function and $\tilde{\sigma}$ function is
essentially the same, then we have
$\displaystyle{\frac{\partial(-\log\tilde{\sigma})}{\partial\tilde{u}_{j}}=\frac{\partial(-\log\sigma)}{\partial\tilde{u}_{j}}=(-\zeta_{\tilde{j}})}$.
Then ${\rm d}u_{j}\leftrightarrow-{\rm d}\tilde{u}_{j}$ is equivalent to
$\wp_{jk}\leftrightarrow(-1)^{2}\wp_{\tilde{j}\tilde{k}}=\wp_{\tilde{j}\tilde{k}}$,
$\wp_{jklm}\leftrightarrow(-1)^{4}\wp_{\tilde{j}\tilde{k}\tilde{l}\tilde{m}}=\wp_{\tilde{j}\tilde{k}\tilde{l}\tilde{m}}$.
Therefore, we conclude that the set of differential equations have some dual
symmetry under (4.1).

###  4.2 Some differential equations for general genus

Using

${\rm
d}(-\zeta_{g-1})=\sum_{i=1}^{g}\frac{\left(\lambda_{2g-1}x_{i}^{g-1}+2\lambda_{2g}x_{i}^{g}+3\lambda_{2g+1}x_{i}^{g+1}\right){\rm
d}x_{i}}{y_{i}}-2{\rm d}\left(\widehat{\wp}_{ggg}\right),$ (4.5)

with $\widehat{\wp}_{ggg}=\wp_{ggg}/\lambda_{2g+1}$, Buchstarber et.al. [24,
25] have shown that the differential equation of the KdV family

$\displaystyle\wp_{gggj}=$
$\displaystyle\frac{3}{2}\wp_{gg}\wp_{gj}+\frac{3}{2}\lambda_{2g+1}\wp_{g,j-1}-\frac{1}{2}\lambda_{2g+1}\wp_{g-1,j}+\lambda_{2g}\wp_{gj}$
$\displaystyle+\frac{1}{2}\lambda_{2g+1}\lambda_{2g-1}\delta_{j,g},\qquad(1\leq
j\leq g),$ (4.6)

is satisfied for general genus. Then in the standard form of $\lambda_{0}=0$,
another KdV equation

$\displaystyle\wp_{111,g+1-j}=$
$\displaystyle\frac{3}{2}\wp_{11}\wp_{1,g+1-j}+\frac{3}{2}\lambda_{1}\wp_{1,g+2-j}-\frac{1}{2}\lambda_{1}\wp_{2,g+1-j}$
$\displaystyle+\lambda_{2}\wp_{1,g+1-j}+\frac{1}{2}\lambda_{1}\lambda_{3}\delta_{g+1-j,1},\qquad(1\leq
j\leq g),$ (4.7)

is satisfied for general genus.

We can obtain other differential equations for general genus recursively. For
example, by using

$\displaystyle{\rm d}(-\zeta_{g-2})=$
$\displaystyle\sum_{i=1}^{g}\frac{\left(\lambda_{2g-3}x_{i}^{g-2}+2\lambda_{2g-2}x_{i}^{g-1}+3\lambda_{2g-1}x_{i}^{g}+4\lambda_{2g}x_{i}^{g+1}+5\lambda_{2g+1}x_{i}^{g+2}\right){\rm
d}x_{i}}{y_{i}}$ $\displaystyle-{\rm
d}\big{(}2\widehat{\wp}_{gg}\widehat{\wp}_{ggg}+4\widehat{\wp}_{gg,g-1}\big{)},$
(4.8)

we have

$\displaystyle\lambda_{2g+1}\widehat{\wp}_{g-2,j}+2\widehat{\wp}_{ggg}\widehat{\wp}_{ggj}+2\widehat{\wp}_{gg}\widehat{\wp}_{gggj}+4\widehat{\wp}_{gg,g-1,j}$
$\displaystyle=5\lambda_{2g+1}(\widehat{\wp}_{gg}^{2}\widehat{\wp}_{gj}+\widehat{\wp}_{g,g-1}\widehat{\wp}_{gj}+\widehat{\wp}_{gg}\widehat{\wp}_{g,j-1}+\widehat{\wp}_{g,j-2})+4\lambda_{2g}(\widehat{\wp}_{gg}\widehat{\wp}_{gj}+\widehat{\wp}_{g,j-1})$
$\displaystyle\quad+3\lambda_{2g-1}\widehat{\wp}_{gj}+2\lambda_{2g-2}\delta_{j,g}+\lambda_{2g-3}\delta_{j,g-1},\qquad(1\leq
j\leq g).$ (4.9)

This is another type differential equation, which is the different type from
the type of Eqs.(2.33)-(2.37), Eqs.(2.50)-(2.64), and Eqs.(3.13)-(3.47).

For another example, by using

${\rm d}(-\zeta_{1})=\sum_{i=1}^{g}\left(\frac{\lambda_{1}{\rm
d}x_{i}}{x_{i}y_{i}}+\frac{2\lambda_{0}{\rm
d}x_{i}}{x_{i}^{2}y_{i}}\right)+2{\rm
d}\left(\widehat{\wp}_{gg2}-\frac{\widehat{\wp}_{g2}\widehat{\wp}_{gg1}}{\widehat{\wp}_{g1}}\right),$
(4.10)

we have the differential equation for general genus

$\displaystyle\lambda_{2g+1}\widehat{\wp}_{1j}-2\widehat{\wp}_{gg2j}+\frac{2\widehat{\wp}_{g2}\widehat{\wp}_{gg1j}}{\widehat{\wp}_{g1}}+\frac{2\widehat{\wp}_{gg1}\widehat{\wp}_{g2j}}{\widehat{\wp}_{g1}}-\frac{2\widehat{\wp}_{g2}\widehat{\wp}_{gg1}\widehat{\wp}_{g1j}}{\widehat{\wp}_{g1}^{2}}$
$\displaystyle=-\frac{\lambda_{1}\widehat{\wp}_{g,j+1}}{\widehat{\wp}_{g1}}-\frac{2\lambda_{0}\widehat{\wp}_{g,j+2}}{\widehat{\wp}_{g1}}+\frac{2\lambda_{0}\widehat{\wp}_{g2}\widehat{\wp}_{g,j+1}}{\widehat{\wp}_{g1}^{2}}+\frac{2\lambda_{0}}{\widehat{\wp}_{g1}}\delta_{j,g-1}+\frac{\lambda_{1}}{\widehat{\wp}_{g1}}\delta_{j,g}-\frac{2\lambda_{0}\widehat{\wp}_{g2}}{\widehat{\wp}_{g1}^{2}}\delta_{j,g},\quad(1\leq
j\leq g).$ (4.11)

We have explicitly checked Eq.(4.9) and Eq.(4.11) for $g=3$ and $j=1,2,3$.

###  4.3 Hirota form differential equations

For genus two case, all differential equations Eqs.(2.33)-(2.37) are written
in the Hirota form, that is, bilinear differential equation with Hirota
derivative. For genus three case, though the left hand side can be written in
the Hirota form, but differential equations which contain $\Delta$ are not
written in the Hirota form. For genus four case, though the left hand side can
be written in the Hirota form, but differential equations which contain
$\Delta_{i}$ are not written in the Hirota form. As it is quite natural, Baker
already has used the Hirota derivative for the derivative of $(-\log\sigma)$,
that is, $\wp_{jk},\wp_{jklm}$ [22]. We use following relations

$\displaystyle(\log\tau)_{xy}=$
$\displaystyle\frac{D_{x}D_{y}\tau\cdot\tau}{2\tau^{2}},$ (4.12)
$\displaystyle(\log\tau)_{xyzt}=$
$\displaystyle\frac{D_{x}D_{y}D_{z}D_{t}\tau\cdot\tau}{2\tau^{2}}-\frac{(D_{x}D_{y}\tau\cdot\tau)(D_{z}D_{t}\tau\cdot\tau)}{2\tau^{4}}-\frac{(D_{x}D_{z}\tau\cdot\tau)(D_{y}D_{t}\tau\cdot\tau)}{2\tau^{4}}$
$\displaystyle-\frac{(D_{x}D_{t}\tau\cdot\tau)(D_{y}D_{z}\tau\cdot\tau)}{2\tau^{4}},$
(4.13)

where $D_{x},D_{y},D_{z},D_{t}$ are Hirota derivatives. Just as the
Weierstrass $\wp$ function solution in the KdV equation, we identify the
$\tau$ function in such a way as $(-\log\tau)$ is proportional to
$(-\log\sigma)$ [18]. Then we put
$\displaystyle{\wp_{jk}=(-\log\sigma)_{jk}=\alpha(-\log\tau)_{jk}}$ with
constant $\alpha$. We show that
$\displaystyle{I=\tau^{2}\Big{(}\wp_{xyzt}-\frac{1}{2}(\wp_{xy}\wp_{zt}+\wp_{xz}\wp_{yt}+\wp_{xt}\wp_{yz})\Big{)}}$
can be written in the Hirota form in the following way

$\displaystyle I=$
$\displaystyle\tau^{2}\Big{(}\wp_{xyzt}-\frac{1}{2}(\wp_{xy}\wp_{zt}+\wp_{xz}\wp_{yt}+\wp_{xt}\wp_{yz})\Big{)}$
$\displaystyle=$
$\displaystyle(-\alpha)\tau^{2}\Big{[}(\log\tau)_{xyzt}+\frac{\alpha}{2}\Big{(}(\log\tau)_{xy}(\log\tau)_{zt}+(\log\tau)_{xz}(\log\tau)_{yt}+(\log\tau)_{xt}(\log\tau)_{yz}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{2}\bigg{[}D_{x}D_{y}D_{z}D_{t}\tau\cdot\tau-\left(1-\frac{\alpha}{4}\right)\left(\frac{(D_{x}D_{y}\tau\cdot\tau)(D_{z}D_{t}\tau\cdot\tau)}{\tau^{2}}+\frac{(D_{x}D_{z}\tau\cdot\tau)(D_{y}D_{t}\tau\cdot\tau)}{\tau^{2}}\right.$
$\displaystyle\hskip
28.45274pt\left.+\frac{(D_{x}D_{t}\tau\cdot\tau)(D_{y}D_{z}\tau\cdot\tau)}{\tau^{2}}\right)\bigg{]}$
$\displaystyle=$ $\displaystyle-2D_{x}D_{y}D_{z}D_{t}\tau\cdot\tau.$ (4.14)

where in the last step we choose $\alpha=4$. For more general form, we have

$\displaystyle J=$
$\displaystyle\tau^{2}\left(\wp_{xyzt}-\frac{1}{2}(\wp_{xy}\wp_{zt}+\wp_{xz}\wp_{yt}+\wp_{xt}\wp_{yz})+a\wp_{xy}+b\right)$
$\displaystyle=$
$\displaystyle-2\left(D_{x}D_{y}D_{z}D_{t}\tau\cdot\tau+aD_{x}D_{y}\tau\cdot\tau-\frac{1}{2}b\tau^{2}\right).$
(4.15)

The l.h.s. of Eqs.(2.33)-(2.37), Eqs.(2.50)-(2.64), and Eqs.(3.13)-(3.47) and
the linear term of $\wp_{jk}$ and constant term in the r.h.s can be written in
the generalized Hirota form, which contains $\text{(const.)}\times\tau^{2}$
term, such as the Hirota form for Weierstrass $\wp$ solution in the KdV
equation [18]. Equations which contain $\Delta$, $\Delta_{i}$ cannot be
written as the Hirota bilinear differential form.

##  5 Summary and Discussions

In order to find higher dimensional integrable models, we have explicitly
studied how to obtain differential equations of genus four hyperelliptic $\wp$
function.

In the standard form of $\lambda_{0}=0$, we have KdV and another KdV equations
for genus being more than two. In the standard form of $\lambda_{1}=0$, if
genus is three, we have KP equation. The universality of integrable model is
guaranteed up to three dimensional integrable models. As the two- and three-
dimensional integrable models, KdV equation and KP equation come out,
respectively.

If genus is two, all differential equations are written in the Hirota form.
However, we obtain differential equations which cannot be written in the
Hirota form, if genus is more than three. This means that the Hirota form or
the fermionic bilinear form is not sufficient to characterize higher
dimensional integrable models.

From the series of investigations of genus two, three, and four, differential
equations for general genus will not be so complicated, but only the quadratic
term $\Delta_{j}$ of $\wp_{jk}$ becomes complicated.

We have also shown, in the standard form of $\lambda_{0}=0$, some duality for
the set of differential equations, which gives that KdV and another KdV
equations always exist for genus being more than two. In the standard form of
$\lambda_{1}=0$, there also exist duality for the KP equation for genus three
and four. We expect that the same expression Eq.(2.64) and/or Eq.(3.47) will
be satisfied for the general genus.

Since we have KdV, another KdV equation, and $(g-2)$ pieces of KP type
differential equation in the standard form of $\lambda_{0}=0$, where KP type
equation is similar to the KdV equation, we expect that genus $g$
hyperelliptic $\wp$ functions have rank $g$ Lie group structure. In some
special cases, we have given some differential equations for general genus. By
using our method step by step, we can show other differential equations for
general genus.

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