\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 91, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/91\hfil Reality conditions]
{Reality conditions of loop solitons genus $g$:
hyperelliptic am functions}

\author[S. Matsutani\hfil EJDE-2007/91\hfilneg]
{Shigeki Matsutani} 

\address{Shigeki Matsutani \newline
8-21-1 Higashi-Linkan, Sagamihara, 228-0811, Japan}
\email{rxb01142@nifty.com}

\thanks{Submitted March 16, 2006. Published June 16, 2007.}
\subjclass[2000]{37K20, 35Q53, 14H45, 14H70}
\keywords{Loop soliton; elastica; reality condition; hyperelliptic functions}

\begin{abstract}
 This article is devoted to an investigation of a reality condition
 of a hyperelliptic loop soliton of higher genus. In the
 investigation, we have a natural extension of Jacobi am-function
 for an elliptic curves to that for a hyperelliptic curve. We also
 compute winding numbers of loop solitons.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction}

In this article, we  investigate a reality
condition of loop solitons with genus $g$.
Here the loop soliton is defined as follows.

\begin{definition} \label{def1} \rm
For a real parameter $t_2\in \mathbb{R}$,
let us consider a smooth immersion of a curve in $\mathbb{C}$
parameterized by $t_1 \in \mathbb{R}$
and its smooth deformation by $t_2$,
$Z_{t_2} : \mathbb{R} \hookrightarrow \mathbb{C}$
$(t_1 \mapsto Z(t_1, t_2):=Z_{t_2}(t_1) = X^1 + {\sqrt{-1}} X^2)$
with $\partial_{t_1} Z = \mathrm{e}^{{\sqrt{-1}} \phi(t_1, t_2)}$.
We call the deformation of the curve {\it loop soliton}
if its real tangential angle $\phi(t_1, t_2)$
 is characterized by a solution of MKdV equation
\begin{gather}
   \partial_{t_2} \phi
 + \frac{1}{4}
 (\partial_{t_1} \phi)^3
 +\partial_{t_1}^3 \phi=0.
\label{eq:MKdV}
\end{gather}
\end{definition}

The loop soliton or geometry of MKdV equation
 has been studied by several researchers
from viewpoints of a connection between integrable system
and classical differential geometry, and a relation
between
algebraic geometry and differential geometry (\cite{P} and
references therein). From a historical point of view,
simple loop solitons appeared in Euler's book \cite{E}
as solutions of an elastica problem which was
proposed by James Bernoulli as a problem
in mathematical science \cite{T}.
In \cite{Ma0},
 we have proposed a problem of statistical mechanics of elasticas
 as a generalization of the elastica problem,
which we sometimes call quantized elastica using
similarity between quantum mechanics and statistical mechanics.
The new problem is related to large polymers in a heat bath.
In \cite{Ma0} we show that the equi-energy state of
quantized elastica is given by the loop soliton.
It means that the loop soliton is directly related to
(low energy) physics.
Thus we have studied the loop soliton and quantized
elastica in a series of works \cite{Ma0, Ma1, Ma2}.


In \cite{Ma1},
we gave  explicit solutions of
loop solitons in terms of hyperelliptic functions
based upon theories of Baker's \cite{Ba, Ma1}
and Weierstrass's \cite{W} as follows.
For a hyperelliptic curve $C_g$ given by
an affine equation,
\begin{gather}
\split
C_g:\quad y^2 &=  x^{2g+1}+ \lambda_{2g} x^{2g}
         +\lambda_{2g-1} x^{2g-1}+\dots
       +\lambda_2 x^2 +\lambda_1 x  +\lambda_0 \\
     &=(x-e_1)(x-e_2) (x-e_3)\dots(x-e_{2g})(x-e_{2g+1}),
       \label{eq:curve-g}
 \endsplit
\end{gather}
where each $e_a$ is a complex number $\mathbb{C}$,
we have a coordinate system in a complex
vector space $J^\infty_g :=\mathbb{C}^g$
as maps from Abelian universal
covering of symmetric product of $C_g$, ${\mathcal{U}\mathrm{Sym}}^g(C_g)$ to $J^\infty_g$:
\begin{gather}
         u_{g-1} =\sum_{i=1}^g u_{g-1}^{(i)},
         \quad
          u_{g} = \sum_{i=1}^g u_{g}^{(i)}, \label{eq:J_g}
\\
             u_{g-1}^{(i)} = \int^{(x^{(i)}, y^{(i)})}_\infty
               \frac{ x^{g-2}d x }{2 y},
         \quad
         u_g^{(i)} =  \int^{(x^{(i)}, y^{(i)})}_\infty
               \frac{x^{g-1} d x }{2 y}.
\label{eq:ugi}
\end{gather}

\begin{proposition}\label{prop:loopg}
A hyperelliptic solution of the loop soliton of genus $g$
 is give by
\begin{equation}
         \partial_{t_1} Z^{(a)} = \prod_{i=1}^g (x^{(i)} - e_a),
\label{eq:loop-g}
\end{equation}
where $t_1= K u_g$ and $t_2 =K ( u_{g-1} - (\lambda_{2g} + e_a)^{-1} u_g)$
for a constant positive number $K$, if
the curve  (\ref{eq:curve-g}) and integrals contours
which satisfy the reality condition,
\begin{enumerate}
\item $|\partial_{u_g} Z^{(a)}| = R$
for a constant positive number $R$,

\item $u_g \in \mathbb{R}$.
\end{enumerate}
\end{proposition}

The proof or this proposition can be found in \cite[Proposition 3.4]{Ma1}.

However, we did not deal with explicit expression of its reality
conditions in \cite{Ma1}
Thus we will concentrate on the reality condition
of loop soliton in this article.
The reality condition of soliton equations were investigated
well [\cite{Na, GN} and references therein]
but these investigations can not be directly
applied to our problem.
On the other hand in \cite{Mu},
Mumford gave natural results on the reality condition
of the elastica and a loop soliton of genus one.
In other words,
he showed the moduli of loop solitons of genus one as elasticas
in terms of $\theta$
functions, or the geometry of
the Abelian varieties of genus one.
However when one considers its straightforward extension to
general genus case, he encounters a difficulty.
In the higher genus case,
there appears a problem that
 the moduli of the Abelian varieties differs
from the moduli of Jacobian varieties, {\it i.e.},
a problem that there are excess parameters in the Abelian
varieties.
On the other hand,
on the investigation of loop soliton even with
higher genus,
we have chosen the strategy that we use only the
data of curves themselves to avoid the
problem of excess parameters, and give some explicit
results in \cite{Ma1, Ma2}. Thus we will go on to follow the strategy
to investigate the reality condition.

To use the strategy, we will, first, interpret the results of Mumford
in terms of the language of the curve in the
case of genus one.
After then, we will apply the scheme to
the reality condition of higher genus
case.
Section two is devoted to the reinterpretation
of Mumford results.
Section three gives the moduli of the
loop solitons of genus two, which can be easily
generalized to higher genus cases as in \S 4.
As we will show in Theorem \ref{theorem:g}, the reality condition
is reduced to the following conditions.

\begin{theorem}\label{theorem:gIn}
Let a set of the  zero points $e_b$ of $y$ in (\ref{eq:curve-g}) be denoted
by ${\mathcal{B}}$. $Z^{(a)}$ satisfies the reality condition
if and only if the following conditions satisfy,

\begin{enumerate}
\item each $e_c \in {\mathcal{B}}$ is real,

\item there exists $g$ pairs $(e_{c_j}, e_{d_j})_{j=1, \dots, g}$
satisfies $(e_{c_j}-e_a)(e_{d_j}-e_a)=e_a^2$ for
negative $e_a$,

\item the contour in the integral $u_g$
in (\ref{eq:J_g}) satisfies a certain condition.
\end{enumerate}
\end{theorem}

Using our result of this article, we give in principle
explicit solutions of the loop solitons,
even though the numerical problems might remain
to illustrate its shape graphically.
Though \cite{TW} illustrated
shapes of large polymers in terms of elliptic functions
as approximations, our results of this article
promises to steps to exact solutions of such shapes.

In the investigation, we have a natural
extension of Jacobi am-function for an
elliptic curves to that for a hyperelliptic
curve. We also compute winding numbers
of loop soliton.

As there are so many open problems related to
this as in \cite{Ma2, P}, this result could be
applied to them.




\section{Genus One}

First we  consider the genus one case using
data from the  curve given by
\begin{gather}
\split
 y^2 &=x^3
       +\lambda_2 x^2 +\lambda_1 x  +\lambda_0 \\
     &=(x-e_1)(x-e_2) (x-e_3).
 \endsplit
\label{eq:curve-g1}
\end{gather}
The coordinate $u$ of the complex plane $J_1^\infty:=\mathbb{C}$
is given by,
\begin{gather}
        \int^{(x,y)} d u, \quad d u = \frac{ d x }{2 y}.
         \label{eq:u-g1}
\end{gather}

It is known that a shape of
the (classical) elastica, {\it i.e.},
 a loop soliton with genus one, $Z : \mathbb{R} \hookrightarrow \mathbb{C}$
$(u \mapsto Z(u) = X^1(u) + {\sqrt{-1}} X^2(u))$
with $\partial_u Z = \mathrm{e}^{{\sqrt{-1}} \phi}$
satisfies the differential equation,
\begin{gather}
   a\partial_{u} (\phi)
 + \frac{1}{3}(\partial_{u}\phi)^3 +\partial_{u}^3 \phi=0,
 \label{eq:SMKdV}
\end{gather}
where $\partial_u := d /du$.

\begin{proposition}[Euler \cite{E}] \label{prop2.1}
A solution of (\ref{eq:SMKdV}) is given by
$$
        \partial_u Z^{(a)} = (x - e_a),
$$
for an elliptic curve given by the form
(\ref{eq:curve-g1}). If it is a loop soliton
if an only if it satisfies
the  reality condition:
\begin{enumerate}
\item $|\partial_u Z^{(a)}| = 1$.

\item $u \in \mathbb{R}$.
\end{enumerate}
\end{proposition}

For a proof of the above propositions, see  \cite[Proposition 3.4]{Ma1}.


\begin{proposition}[Mumford \cite{Mu}] \label{Mum1}
The moduli $\Lambda$
of elastica or loop soliton of genus one is given by
the following subspace in the upper half plane
$\mathbb{H}:=\{z \in \mathbb{C} \ | \ \Im z >0\}$ modulo $\mathrm{PSL}(2,\mathbb{Z})$,
$$
\Lambda := {\sqrt{-1}} \mathbb{R}_{>0} \cup \left(\frac{1}{2} + {\sqrt{-1}} \mathbb{R}_{>0}\right)
\cup \infty
\quad\mathrm{modulo}\quad \mathrm{PSL}(2,\mathbb{Z}).
$$
Here $\mathbb{R}_{>0}$ is $\{ x \in \mathbb{R} \ |\ x >0\}$.
\end{proposition}

Though Mumford led this result using the geometry of Abelian
variety of genus one \cite{Mu}, we will give another proof only
using the language of curve itself as mentioned in Introduction.
The purpose of this section is to  give its proof using
only the data of the curve itself.

\begin{lemma} \label{lem2.1}
For different numbers $a$, $b$ and $c$ in $\{1, 2, 3\}$,
let $\mathrm{e}^{2{\sqrt{-1}}\varphi_a} :=(x-e_a)/c_{cba}$,
$e_{ab} := e_a - e_b$ and $c_{cba}:=\sqrt{e_{ca}e_{ba}}$.
The  elliptic differential of the first kind (\ref{eq:u-g1})
up to sign is
$$
        du = \frac{ d \varphi_a}
         {\sqrt{(\sqrt{e_{ba}}-\sqrt{e_{ca}})^2
            +4 \sqrt{e_{ba}e_{ca}} \sin^2 \varphi_a}}.
$$
\end{lemma}

\begin{proof}
Direct computations give
\begin{align*}
d x &= 2 c_{c b a} {\sqrt{-1}} \mathrm{e}^{2 {\sqrt{-1}} \varphi_a} d \varphi_a,\\
   y &=c_{c b a} {\sqrt{-1}} \mathrm{e}^{2 {\sqrt{-1}} \varphi_a}
         \sqrt{ e_{ba}(\mathrm{e}^{-2{\sqrt{-1}}\varphi_a}- c_{c b a} e_{ba}^{-1})
                    (\mathrm{e}^{2{\sqrt{-1}}\varphi_a}- c_{c b a}^{-1} e_{c a})}\\
    &= c_{c b a}{\sqrt{-1}} \mathrm{e}^{2 {\sqrt{-1}} \varphi_a}
         \sqrt{ e_{ba}+e_{c a}
 - 2 \sqrt{e_{ba}e_{c a}} \cos 2\varphi_a},
\end{align*}
up to sign.
The addition formula $\cos(2\varphi) = 1 - 2 \sin^2\varphi$
leads the result.
\end{proof}

Let us use the standard representations,
$$
k :=\frac{2{\sqrt{-1}} \root4\of{e_{ba}e_{ca}}}
{\sqrt{e_{ba}}-\sqrt{e_{ca}}}
$$
and then
\begin{gather}
        d u = \frac{ d \varphi_a}
         {(\sqrt{e_{ba}}-\sqrt{e_{c a}})
           \sqrt{1-k^2 \sin^2 \varphi_a}}.
      \label{eq:am-g1}
\end{gather}
By letting $w:=\sin(\varphi_a)$, (\ref{eq:am-g1}) becomes
\begin{equation}
        d u =\frac{d w}
         {(\sqrt{e_{ba}}-\sqrt{e_{c a}})\sqrt{(1-w^2)(1 - k^2 w^2)}}
\label{eq:w-g1}
\end{equation}

\begin{remark}\label{remark:am1} {\rm

(1) Due to the (\ref{eq:am-g1}), we have the following
elliptic integral $u(\varphi_a)$
$$
         u(\varphi_a) = \int^{\varphi_a}_{0} \frac{d \varphi}{H_a^{[1]}(\varphi)},
$$
and its inverse function $\varphi_a(u)$  gives
$$
         \exp({\sqrt{-1}}\varphi_a(u) ) = \sqrt{x-e_a}.
$$
As $\sqrt{(e_3-e_1)/(x-e_3)}$ is sn-function, $\varphi_a(u)$
is essentially the same as Jacobi-am
function $\mathrm{am}(u)$ \cite{PS}, though we need Landen-transformation.

\noindent(2)  Behind (\ref{eq:w-g1}), there is a kinematic system
with an energy
$$
       E = \dot w^2 + (1-w^2)(1 - k^2 w^2).
$$
}
\end{remark}


Due to the reality condition, Proposition \ref{prop2.1} (1), 
$\varphi_a$ belongs to a subregion of
a real number.
For any $\varphi_a$ in a certain region $[\varphi_l, \varphi_u]$,
the reality condition, Proposition \ref{prop2.1} (2), requires that
the denominator in (\ref{eq:w-g1}) should be real and thus
that $k^2$, or $\sqrt{e_{ba}e_{ca}}$ and $(\sqrt{e_{ba}}-\sqrt{e_{ca}})^2$,
should be real;
$$
\Im \sqrt{e_{ba}}= \Im\sqrt{e_{ca}},\quad
\arg(e_{ba})= -\arg(e_{ca}),
$$
where $\arg(a):=\Im\log(a)$ for $a\in\mathbb{C}$.
Accordingly introducing an
 expression $e_{ba}=:\beta_{ba}\mathrm{e}^{\sqrt{-1}\alpha_{ba}}$,
using $\alpha_{ba}\in [0,\pi)$ and
$\beta_{ba} \in \mathbb{R}$,\footnote{Here we defined $\beta_{ba} \in \mathbb{R}$
rather than $\beta_{ba} \in \mathbb{R}_{\ge 0}$
due to the domain of $\alpha_{ba}$.}  the reality condition
of the loop soliton $Z^{(a)}$
require alternative cases:
\begin{enumerate}
\item $\alpha_{ba}$ and $\alpha_{ca}$vanish,
{\it i.e.}, $e_{ba}$ and $e_{ca}$ belong to $\mathbb{R}$, or
\item
$\alpha_{ba}=-\alpha_{ca}$ and $\beta_{ba}=\beta_{ca}$.
\end{enumerate}
However, the second case means that
$(\sqrt{e_{ba}}-\sqrt{e_{ca}})^2$ vanishes and
corresponds to $k=\infty$.\footnote{Though it is not
important, it is interesting that the second case
can be reduced to the first case,
{\it i.e.}, $\alpha_{ca}=0$,
by transforming $\varphi_{a}$ to $\varphi_{a}-\alpha_{ca}$
due to the formula in the proof in Lemma \ref{lem2.1}.}
Thus we find the following lemma.

\begin{lemma}\label{lemma:2-2}
The reality condition  of
the loop soliton $Z^{(a)}$
is reduced to two alternative cases:
\begin{enumerate}
\item[I-1] $e_{ba}>0$ and $e_{ca}>0$,
{\it i.e.}, $k\in {\sqrt{-1}} \mathbb{R}_{\ge0}$,
$w\equiv\sin \varphi_a \in [-1, 1]$.


\item[I-2] $e_{ba}\le0$ and $e_{ca}\le0$, {\it i.e.},
$k>1$ and $w\equiv\sin \varphi_a \in [1/k, 1]$ or
$w\equiv\sin \varphi_a \in [-1, -1/k]$.

\end{enumerate}
\end{lemma}

\begin{proof}
For general $\varphi_a\in\mathbb{R}$, $u$ must be real. Hence the
candidates of $e_{ba}$'s are followings:
(I-0) $e_{ba}<0$ and $e_{ca}>0$, or $e_{ba}>0$ and $e_{ca}<0$,
(I-1) $e_{ba}>0$ and $e_{ca}>0$,
and (I-2) $e_{ba}\le0$ and $e_{ca}\le0$.

In (I-0) case
$(\sqrt{e_{ba}}-\sqrt{e_{ca}})$ has a non-trivial
angle in the complex plane, which
cannot be cancelled by the other factors. We remove (I-0) case.
(I-1) is obvious. The region of $\sin \phi_a$ must be a subset
of $[-1, 1]$.  On the case (I-2),
noting that prefactor
$1/(\sqrt{e_{ba}}-\sqrt{e_{ca}})$ generates
the factor ${\sqrt{-1}}$,
we conclude that $k>1$ and
$\sin \phi_a \in [1/k, 1]$ or
$\sin \phi_a \in [-1, -1/k]$.
\end{proof}


\begin{figure}
\begin{center}
\includegraphics[width=0.9\textwidth]{fig1}
\caption{Geometry of Contours: $\alpha$ and $\beta$ are
Homology basis of the elliptic curves.}
\label{fig:1}
\end{center}
\end{figure}




\begin{proof}[Proof of Proposition \ref{Mum1}]
Let us consider the geometry of the integration.
Fig.1 gives an illustration of our situations,
where Fig.1 (a) corresponds to case I-1 and
(b) to case I-2 in Lemma \ref{lemma:2-2}

\noindent\textbf{I-1:}
The periodicity $(4\omega, 2\omega')$ of
$\sqrt{(x-e_a)}$ is given by
\begin{gather*}
        \omega =  \int^1_0
\frac{d w}
 {\sqrt{(1-w^2)((\sqrt{e_{ba}}-\sqrt{e_{c a}})^2
            +4 \sqrt{e_{ba}e_{c a}} w^2)}},\\
        \omega' = ( \int^0_1+\int_0^{{\sqrt{-1}}/|k|})
\frac{d w}
 {\sqrt{(1-w^2)((\sqrt{e_{ba}}-\sqrt{e_{c a}})^2
            +4 \sqrt{e_{ba}e_{c a}} w^2)}}.
\end{gather*}
Thus $\omega' = \omega + {\sqrt{-1}} L[k]$ for
general $k$ with a certain real valued function $L$.
On the other hand, for $k \to 0$, $L\to \infty$
and for $k \to \infty$, $L$ vanishes. Further
$L[k]$ is a continuous function of $k$ and its range is $\mathbb{R}_{>0}$.
Hence $\tau = 2\omega'/4\omega \in (1/2+ {\sqrt{-1}} \mathbb{R}_{>0})$.

\noindent\textbf{I-2:}
The periodicity $(4\omega, 2\omega')$ of
$\sqrt{(x-e_a)}$ is given by
\begin{gather*}
        \omega = 2 \int^{1/k}_0
\frac{d w}
 {\sqrt{(1-w^2)((\sqrt{e_{ba}}-\sqrt{e_{c a}})^2
            +4 \sqrt{e_{ba}e_{c a}} w^2)}},\\
        \omega' = \int^1_{1/k}
\frac{d w}
 {\sqrt{(1-w^2)((\sqrt{e_{ba}}-\sqrt{e_{c a}})^2
            +4 \sqrt{e_{ba}e_{c a}} w^2)}}.
\end{gather*}
On the other hand, for $k \to 0$, $\omega\to \infty$
and for $k \to \infty$, $\omega$ vanishes
while $\omega'$ is a finite number. Further
$\omega[k]$ and $\omega'[k]$ are  continuous in $k$.
Hence $\tau = 2\omega'/4\omega \in {\sqrt{-1}} \mathbb{R}_{>0}$.

Since theory of the Jacobi elliptic functions
gives the fact that $k' :=\sqrt{1-k^2}$ gives
the inversion of moduli $\tau \to -1/\tau$,
the constraint $k>1$ in Lemma \ref{lemma:2-2}
is less important.

We note that the periodicity of $\sqrt{(x-e_a)}$
differs from $\partial_u Z^{(a)}$ by twice but
the difference is not so significant.
Hence we have a complete proof of Proposition
\ref{Mum1} based upon geometry of elliptic curve itself
instead of geometry of Abelian variety as
a domain of elliptic theta function.
\end{proof}

\begin{remark}{\rm
(1)
We list its special cases for $a=1$:
\begin{enumerate}
\item[(a)] $k=0$ in I-1: its shape is a circle
and its related curve is $y^2 = (x-e_1)^2(x-e_2)$

\item[(b)] $k=\infty$ in I-2: its shape is a
loop soliton solution,
and its related curve is $y^2 = (x-e_1)(x-e_2)^2$
\end{enumerate}

\noindent(2)
Since $\partial_s Z\equiv \mathrm{e}^{{\sqrt{-1}} \phi}$
can be regarded as a harmonic
map: $\partial_s Z: S^1 \to S^1$ with energy
$$
        E = \oint d s |\partial_s \phi|^2.
$$

\noindent(3) Above Lemma \ref{lemma:2-2}, we argued the angle of $e_{ba}$'s.
However the geometry of the integrals depends only on
$\sqrt{e_{ba}e_{ca}}$ and $\sqrt{e_{ba}}-\sqrt{e_{ca}}$
rather than $e_{ba}$'s themselves.
}
\end{remark}

For the map $\partial_u Z: S^1 \to S^1$,
  we can find index as a winding number
as shown in Fig.2.
  We call it index($\partial_u Z$).


\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}
\caption{The behavior of $\varphi$}
\label{fig:2}
\end{center}
\end{figure}


\begin{corollary} \label{coro2.1}
The $\mathop{\rm index}(\partial_u Z)$ is given as follows.
\begin{enumerate}
\item[I-1] $\mathop{\rm index}(\partial_u Z)=\pm1$.

\item[I-2] $\mathop{\rm index}(\partial_u Z)=0$.
\end{enumerate}
\end{corollary}

\begin{proof}
In the case I-1,
since the contours $w\equiv \sin \varphi_a$ is $[-1,1]$
which is identified with the range of sine function,
$\varphi_a$ becomes a monotonic
 increasing function of $u$.
 In fact passing by $w=\pm1$ changes the
 sign of $\sqrt{1-w^2}$ or $\cos\varphi_a$.
By paying attentions on the orientation of the contour,
we have the sign of the index.
On the other hand, in the case I-2,
$\varphi$ does not wind around $S^1$ like Fig. 2(b).
The branch point $(1/k, 0)$ does not have an effect
of the sign of $\sqrt{1-w^2}$.
\end{proof}


\section{Genus Two}

In this section, we will investigate the reality condition
associated with a hyperelliptic curve $C_2$ of genus two
 expressed by
\begin{gather}
\split
 y^2 &=  x^5 + \lambda_4 x^4 +\lambda_3 x^3
       +\lambda_2 x^2 +\lambda_1 x  +\lambda_0 \\
     &=(x-e_1)(x-e_2) (x-e_3)(x-e_4)(x-e_5),
       \label{eq:curve-g2}
 \endsplit
\end{gather}
where each $e_a$ is a complex number $\mathbb{C}$.
We have the coordinate system of the  complex vector
space $J^\infty_2 :=\mathbb{C}^2$;
\begin{gather}
         u_1 =u_1^{(1)}+u_1^{(2)},
         \quad
          u_2 = u_2^{(1)}+u_2^{(2)}, \label{eq:J_2}
\\
             u_1^{(i)} = \int^{(x^{(i)}, y^{(i)})}_\infty
               \frac{ d x }{2 y},
         \quad
         u_2^{(i)} =  \int^{(x^{(i)}, y^{(i)})}_\infty
               \frac{x d x }{2 y}.
\end{gather}
Let the Abelian map  $\mathrm{Sym}^2(C_2)\to J_2:=J^\infty_2/\Lambda$ be
denoted by $\omega'_A$ where $\Lambda$ is a lattice in $J^\infty_2$
associated with $C_2$. Considering winding numbers,
we will denote the Abelian universal covering of $\mathrm{Sym}^2(C_2)$
by ${\mathcal{U}\mathrm{Sym}}^2(C_2)$ and its map from ${\mathcal{U}\mathrm{Sym}}^2(C_2)$
to $J^\infty_2$ by $\omega_A$.


The loop soliton solution
of (\ref{eq:curve-g2}) is given by
$\partial_{t_1}Z^{(a)}=(x^{(1)}-e_a)(x^{(2)}-e_a)$
if it satisfies the reality condition.

\begin{lemma} \label{lemma:g2gene}
For different numbers $a$, $b$ and $c$ of $\{1, 2, 3, 4, 5\}$,
let $\mathrm{e}^{2{\sqrt{-1}}\varphi_a^{(i)}} :=(x^{(i)}-e_a)/c_{cba}$,
$e_{ab} := e_a - e_b$ and $c_{cba}:=\sqrt{e_{ba}e_{ca}}$.
In general, the following relation up to sign holds:
\begin{align*}
        du_2^{(i)}
&= \frac{ {\sqrt{-1}}(c_{cba}\mathrm{e}^{{\sqrt{-1}}\varphi_a^{(i)}}
              +e_a\mathrm{e}^{-{\sqrt{-1}}\varphi_a^{(i)}})
                 d \varphi_a^{(i)}}
         {\sqrt{((\sqrt{e_{ba}}-\sqrt{e_{ca}})^2
            +4 \sqrt{e_{ba}e_{ca}} \sin^2 \varphi_a^{(i)})
        c_{cba}e_{da}(\mathrm{e}^{-2{\sqrt{-1}}\varphi_a^{(i)}} -c_{cba}e_{da}^{-1})
}}\\
&\quad\times\frac1{\sqrt{
                   (\mathrm{e}^{2{\sqrt{-1}}\varphi_a^{(i)}} -c_{cba}^{-1}e_{ea})}}.
\end{align*}
\end{lemma}

\begin{proof}
Direct computations lead the formula.
\end{proof}

We will find a subspace $(\Gamma,\omega_A(\Gamma)) \subset
{\mathcal{U}\mathrm{Sym}}^2(C_2)\times J_2^\infty$ which satisfies
the reality condition.
We note that since the reality condition is local,
we need not pay attentions upon
 the difference between $\mathrm{Sym}^2(C_2)$
and ${\mathcal{U}\mathrm{Sym}}^2(C_2)$.

\begin{lemma} \label{lemma:RC02}
The reality condition  of
the loop soliton $Z^{(a)}$
satisfies if and only if
$(x^{(1)}, x^{(2)}) \in {\mathcal{U}\mathrm{Sym}}^2(C_2)$ and $\lambda$'s
satisfy the following relations:
\begin{enumerate}
\item $|(x^{(i)}-e_a)| = K_i$ of a real constant $K_i$, $(i=1,2)$,

\item $u_2^{(i)} \in \mathbb{R}$ for $i=1,2$.
\end{enumerate}
\end{lemma}


\begin{proof}
Proposition \ref{prop:loopg} leads to
$(x^{(1)}, x^{(2)}) \in \Gamma \subset {\mathcal{U}\mathrm{Sym}}^2(C_2)$ satisfying
the reality conditions is given by
\begin{equation}
         |x^{(2)}-e_a| = \frac{K}{|x^{(1)}-e_a|},    \label{eq:RC-04}
\end{equation}
for a real constant $K$ and
\begin{gather}
        \Im u_2^{(2)}(x^{(2)}) = - \Im u_2^{(1)}(x^{(1)}).
       \label{eq:RC-05}
\end{gather}
When (\ref{eq:RC-04}) is trivial,
{\it i.e.}, $|(x^{(i)}-e_a)| = K_i$ of a real constant $K_i$, $(i=1,2)$,
 and both sides in
(\ref{eq:RC-05}) vanish, we obtain above conditions
as sufficient conditions.

Thus we will consider its necessity condition.
Assume that both conditions
 (\ref{eq:RC-04}) and (\ref{eq:RC-05}) are not
trivial, {\it i.e.},
$x^{(2)}$ and $x^{(1)}$ are not independent.
Since these conditions
 (\ref{eq:RC-04}) and (\ref{eq:RC-05})
are real analytic ones, we must also deal with their
complex conjugate
$\overline{x}^{(1)}$,
$\overline{x}^{(2)}$, and so on. Due to the conditions, for example,
$x^{(2)}$ is a function of $x^{(1)}$,
$\overline{x}^{(1)}$, and $\overline{x}^{(2)}$.
Of course, there is no guarantee whether there exists
such a function $x^{(2)}(x^{(1)}, \overline{x}^{(1)},
\overline{x}^{(2)})$ and even continuity but we can assume
that they exist, at least, locally.
The reality condition locally determines an open subspace
$\omega_A(\Gamma)$ in $J_2^\infty$.
 Due to the dependence
between $x^{(1)}$ and $x^{(2)}$ or $u_2^{(1)}$ and  $u_2^{(2)}$,
$u_1$, $u_2$
$\overline{u_1}$ and $\overline{u_2}$ are neither independent
 over $\omega_A(\Gamma)$.
Hence
${\partial}/{\partial x^{(1)}}|_{x^{(2)}}$ nor
${\partial}/{\partial u_1}|_{u_2}$ do not behave well
as differential operators among sections
over $\Gamma$ and $\omega_A(\Gamma)$,
and  should be replaced with covariant derivatives.
For example, ${\partial}/{\partial u_1}|_{u_2}$
is replaced with
${\partial}/{\partial u_1} - A_{u_1}(u_1, u_2, \overline{u_1},
\overline{u_2})$
using an appropriate connection $A_{u_1}$.

On the other hand, the loop soliton $\partial_{t_1} Z^{(a)}$
is a meromorphic function over $\Gamma$ and $\omega_A(\Gamma)$.
However it is a restricted section of the $J^\infty_2$ at $\omega_A(\Gamma)$
in (\ref{eq:J_2}) and satisfies the MKdV equation (\ref{eq:MKdV})
with respect {\it only} to the differentials of
$u_1$ and $u_2$ over there as
mentioned in Proposition \ref{prop:loopg}.
However the connection $A_{u_1}$ prevents
that the angle part of $\partial_{t_1} Z^{(a)}$
does satisfy the MKdV equation (\ref{eq:MKdV}).
Hence $A_{u_1}$ and $A_{u_2}$ must vanish.

However, the condition that $A_{u_1}$ vanishes means
that $\omega_A(\Gamma)$ is a flat real plane in $J_2^\infty=\mathbb{C}^2$
and  $x^{(2)}$ is independent of $x^{(1)}$.
Hence we prove this Lemma.
\end{proof}

\begin{remark} \label{rmk3.3} {\rm
 By letting an appropriate immersion
$\iota: S^1\hookrightarrow C_2$,
$\partial_{u_2} Z\circ \omega_A\circ \iota$ is a analytic map from
$S^1$ to $S^1$.
}\end{remark}



\begin{lemma} \label{lemma3.4}
For the situation of Lemma \ref{lemma:g2gene},
the reality condition  of
the loop soliton $Z^{(a)}$
needs
 $e_a = - c_{cba}$ and $c_{eda}=c_{cba}$, and then
we have the relation up to sign,
\begin{align*}
&  d u_2^{(i)} \\
&= \frac{ 2\sqrt{c_{c b a}}\sin\varphi_a^{(i)}
                 d \varphi_a^{(i)}}
         {\sqrt{((\sqrt{e_{ba}}-\sqrt{e_{c a}})^2
            +4 \sqrt{e_{ba}e_{c a}} \sin^2 \varphi_a^{(i)})
           ((\sqrt{e_{d a}}-\sqrt{e_{e a}})^2
            +4 \sqrt{e_{d a}e_{e a}} \sin^2 \varphi_a^{(i)})}}.
\end{align*}
\end{lemma}

\begin{proof} Due to the Lemma \ref{lemma:RC02},
$\varphi_a^{(i)}$ is real and each factor must be
real. Hence the imaginary parts should be
canceled locally. It means the conditions.
\end{proof}


Let us introduce a representation as an extension
of  the standard representation (\ref{eq:am-g1}),
$$
k_1 := \frac{2{\sqrt{-1}}\root4\of{e_{ba}e_{ca}}}
{\sqrt{e_{ba}}-\sqrt{e_{ca}}}, \quad
k_2 := \frac{2{\sqrt{-1}}\root4\of{e_{da}e_{ea}}}
{\sqrt{e_{da}}-\sqrt{e_{ea}}},
$$
and then
\begin{equation}
        d u_2^{(i)}= \frac{2\root4\of{e_{ba}e_{c a}}
           \sin\varphi_a^{(i)} d \varphi_a^{(i)}}
         {(\sqrt{e_{ba}}-\sqrt{e_{c a}})(\sqrt{e_{d a}}-\sqrt{e_{e a}})
           \sqrt{1-k_1^2 \sin^2 \varphi_a^{(i)}}
           \sqrt{1-k_2^2 \sin^2 \varphi_a^{(i)}}}.
\label{eq:am-g2}
\end{equation}
By letting $w:=\sin(\varphi_a^{(i)})$, we have
\begin{gather}
\split
 &       d u_2^{(i)}\\
 &= \frac{ \root4\of{e_{ba}e_{c a}}w d w}
 {\sqrt{(1-w^2)((\sqrt{e_{ba}}-\sqrt{e_{c a}})^2
            +4 \sqrt{e_{ba}e_{c a}} w^2)
((\sqrt{e_{d a}}-\sqrt{e_{e a}})^2
            +4 \sqrt{e_{d a}e_{e a}} w^2)}}\\
     &=\frac{2\root4\of{e_{ba}e_{c a}}w d w}
         {(\sqrt{e_{ba}}-\sqrt{e_{c a}})
           (\sqrt{e_{d a}}-\sqrt{e_{e a}})
          \sqrt{(1-w^2)(1 - k_1^2 w^2)(1 - k_2^2 w^2)}
}.
\endsplit \label{eq:am-w2}
\end{gather}

\begin{remark}\label{remark:genus-two} {\rm
(1)
(\ref{eq:am-w2}) is an elliptic integral
by $u=w^2$ due to a specialty of genus two.
It cannot be generalized to higher genus case.

\noindent (2)
Due to the remark \ref{remark:am1},
we should be regard that (\ref{eq:am-g2}) gives the
integral as a function $u_2^{(i)}$ of $\varphi_a^{(i)}$,
$$
     u_2^{(i)} = \int^{\varphi_a^{(i)}}_0
        \frac{d\varphi_a^{(i)\prime}}
           {H_a^{[2]}(\varphi_a^{(i)\prime})}
$$
for an appropriate function $H_2^{[2]}$.
Hence the inverse function $\varphi_a^{(i)}(u_2^{(i)})$
gives the relation,
$$
        \exp({\sqrt{-1}} \varphi_a^{(i)}(u_2^{(i)}))
         = \sqrt{(x^{(i)}-e_a)/c_{cba}}.
$$
Further $\varphi_a:=\varphi_a^{(1)}(u_2^{(1)})
          + \varphi_a^{(2)}(u_2^{(2)})$
gives the al-function of $u_2 :=u_2^{(1)}+u_2^{(2)}$ \cite{Ba, W},
$$
        \exp({\sqrt{-1}} \varphi_a(u_2)) = \mathrm{al}_a(u_2).
$$
Accordingly, we should regard this $\varphi_a$ as a
hyperelliptic am-function of genus two.

\noindent (3) Behind the hyperelliptic
am-functions, there is also kinematic system
with a hamiltonian:
$$
E= \dot w^2+
(1-w^2)((\sqrt{e_{ba}}-\sqrt{e_{c a}})^2
            +4 \sqrt{e_{ba}e_{c a}} w^2)
((\sqrt{e_{d a}}-\sqrt{e_{e a}})^2
            +4 \sqrt{e_{d a}e_{e a}} w^2).
$$
}
\end{remark}

For each $\varphi_a^{(i)}$ in a region $[\varphi_l, \varphi_u]$,
the reality condition of
the loop soliton $Z^{(a)}$, Lemma \ref{lemma:RC02} (2), requires that
the denominator should be real and thus
that $k_d^2$, or $\sqrt{e_{ba}e_{ca}}$  and
$\sqrt{e_{ba}}-\sqrt{e_{ca}}$ should be also real.


\begin{theorem}
The reality condition of
the loop soliton $Z^{(a)}$ of genus two
is reduced to the conditions:
$e_a = - c_{cba}$ and $c_{eda}=c_{cba}$ with
 three alternative cases:
\begin{enumerate}
\item[II-1.] $e_{ba}>0$, $e_{ca}>0$
$e_{ea}>0$, $e_{da}>0$, {\it i.e.},
$k_1, k_2\in {\sqrt{-1}} \mathbb{R}$
and
$\sin \varphi_a \in [-1, 1]$.

\item[II-2.]
 $e_{ba}>0$, $e_{ca}>0$,
$e_{ea}\le0$ and $e_{da}\le0$, {\it i.e.},
$k_1 \in {\sqrt{-1}} \mathbb{R} $ and $k_2\in \mathbb{R}$
$\sin \varphi_a \in [1/k_2, 1]$ or
$\sin \varphi_a \in [-1, -1/k_2]$.

\item[II-3.] $e_{ba}\le0$, $e_{ca}\le0$
$e_{ea}\le0$, $e_{da}\le0$, {\it i.e.},
 $k_1, k_2\in  \mathbb{R}$, $(k_1<k_2)$,
\begin{enumerate}
\item
if $k_2<1$, $\sin \varphi_a \in [-1, 1]$.

\item
if $k_2>1$, $\sin \varphi_a \in [-1/k_2, 1/k_2]$.

\item
if $k_1>1$, $\sin \varphi_a \in [1/k_1, 1]$ or
$\sin \varphi_a \in [-1, -1/k_1]$.
\end{enumerate}
\end{enumerate}
\end{theorem}

\begin{proof}
As in the case of the elliptic curves, we have the results.
\end{proof}


\begin{figure}
\begin{center}
\includegraphics[width=0.9\textwidth]{fig3}
\caption{Geometry of Contours: $\alpha_1$, $\beta_1$,
$\alpha_2$ and $\beta_2$ are Homology basis of the
hyperelliptic curves.}
\label{fig:3}
\end{center}
\end{figure}


Fig.3 gives an illustration of our situation,
where Fig.3 (a) corresponds to II-1 and
(b) does to II-2 and (c) to II-3.

In this case,
 we show the index($\partial_{t_1} Z$).

\begin{corollary} \label{coro3.1}
The $\mathop{\rm index}(\partial_{t_1}Z)$ as a winding
number of the map
$\iota(S^1)$ to $S^1$ is
\begin{enumerate}
\item[II-1.] $\mathop{\rm index}(\partial_{t_1}Z)=0$ or $\pm 2$,

\item[II-2.] $\mathop{\rm index}(\partial_{t_1}Z)=0$,

\item[II-3.] (a) $\mathop{\rm index}(\partial_{t_1}Z)=0$ or $\pm 2$,
 and (b) (c) $\mathop{\rm index}(\partial_{t_1}Z)=0$.
\end{enumerate}
\end{corollary}

\begin{proof}
These indexes consist of those of each $2\varphi_{a}^{(i)}$.
If the index of $2\varphi_{a}^{(i)}$ is one,
that of $2\varphi_{a}$ is sum over $i=1,2$,
$\varphi_a = \pm \varphi_{a}^{(1)}\pm\varphi_{a}^{(2)}$.
Here $\pm$ depends upon the orientation of contours.
The computations of $\varphi_{a}$
are essentially the same as the genus one
illustrated in Fig. 2.
\end{proof}

\section{Genus $g$}

The computations of genus two are easily extended to
higher genus loop solitons.
Let us introduce the sets,
${A}:=\{1, 2, 3, \dots, 2g+1\}$,
${A}_a:={A} -\{a\}$ for $a\in {A}$,
${O}_1:=\{3, 5, \dots, 2g-1\}$,
and a bijection $\sigma_a: \{1, 2, \dots, 2g\} \to
{A}_a$ for $a\in {A}$ which determines
the order. We will fix the order $\sigma_a$ for
an $a \in {A}$.

Recalling the facts in genus two case,
the direct computations give the following lemmas.

\begin{lemma} \label{lemma:g-gene}
For $a\in {A}$,
let $\mathrm{e}^{2{\sqrt{-1}}\varphi_a^{(i)}} :=(x^{(i)}-e_a)/c_{cba}$,
$e_{ba} := e_{\sigma_a(b)} - e_a$ and
$c_{cba}:=\sqrt{e_{ba}e_{ca}}$,
\begin{align*}
D^{(i)}_{a,\sigma_a}(\varphi_a)
&:=
\Bigr((\sqrt{e_{1a}}-\sqrt{e_{2a}})^2
            +4 \sqrt{e_{1a}e_{2a}} \sin^2 \varphi_a^{(i)})\\
&\times
        \prod_{d\in {O}_1, e=d+1}
        c_{12a}e_{da}(\mathrm{e}^{-2{\sqrt{-1}}\varphi_a^{(i)}} -c_{12a}e_{da}^{-1})
                   (\mathrm{e}^{2{\sqrt{-1}}\varphi_a^{(i)}} -c_{12a}^{-1}e_{ea})
                    \Bigr)^{1/2},
\end{align*}
$$
N^{(i)}_{a,\sigma_a}(\varphi_a):=\left(
 {\sqrt{-1}}(c_{12a}\mathrm{e}^{{\sqrt{-1}}\varphi_a^{(i)}}
              +e_a\mathrm{e}^{-{\sqrt{-1}}\varphi_a^{(i)}})\right)^{g-1}.
$$
In general, (\ref{eq:ugi}) up to sign becomes
$$
        du_g^{(i)} =
 \frac{ N^{(i)}_{a,\sigma_a}d\varphi_a}{D^{(i)}_{a,\sigma_a}}.
$$
\end{lemma}

\begin{lemma} \label{lem4.1}
For the situations of Lemma \ref{lemma:g-gene},
the reality condition of
the loop soliton $Z^{(a)}$ requires the conditions
that
 $e_a = - c_{cba}$ for any $c \in {O}_1$, $b=c+1$ and then
we have
\begin{gather}
\split
D^{(i)}_{a,\sigma_a}(\varphi_a)
&=\Bigr(((\sqrt{e_{1a}}-\sqrt{e_{2a}})^2
            +4 \sqrt{e_{1a}e_{2a}} \sin^2 \varphi_a^{(i)})\\
 &\times \prod_{d\in {O}_1, e=d+1}
        ((\sqrt{e_{d a}}-\sqrt{e_{e a}})^2
            +4 \sqrt{e_{d a}e_{e a}} \sin^2 \varphi_a^{(i)})
            \Bigr)^{1/2},
             \endsplit
\end{gather}
$$
N^{(i)}_{a,\sigma_a}(\varphi_a)=
       \left( 2\sqrt{c_{1 2 a}}\sin\varphi_a^{(i)}\right)^{g-1}.
$$
\end{lemma}


These lemma can be proved along the line of the arguments
for the case of genus two.

Corresponding to Remark \ref{remark:genus-two}, we have
the following remarks:

\begin{remark}\label{remark:genus-g}{\rm
(1)
Let $\varphi_a:=\varphi_a^{(1)}+\varphi_a^{(2)}+\dots +\varphi_a^{(g)}$
and then  (\ref{eq:loop-g}) is expressed by
$$
         \partial_{t_1} Z^{(a)}=\mathrm{e}^{2{\sqrt{-1}}\varphi_a},
$$
as a function of $u_g:=u_g^{(1)}+u_g^{(2)}+\dots+u_g^{(g)}$.
The hyperelliptic al-function is written by
$$
         \mathrm{al}_a(u)=\mathrm{e}^{{\sqrt{-1}}\varphi_a(u)},
$$

\noindent (2) $\varphi_a$ can be regarded as hyperelliptic
am-function of genus $g$.
}
\end{remark}

We will state our main theorem as follows, which is also
proved along the line of the same arguments
in the case of genus two.


\begin{theorem}\label{theorem:g}
The reality condition of the loop soliton
$Z^{(a)}$ in (\ref{eq:loop-g}) can be reduced to the conditions that  there
are $g$ pairs
$(e_{b,a},e_{b+1,a})_{b\in {O}_1} \in \mathbb{R}^2$ satisfying
$-e_a = \sqrt{e_{b,a}e_{b+1,a}}\ge 0$,
and the contour of integral
of each $u^{(i)}_g$ of $i=1, \dots, g$ should be
chosen so that $u^{(i)}_g$ is real.
\end{theorem}

\subsection*{Acknowledgments}
The author wants to thank Prof. E. Previato,  Prof. J. McKay and
 Prof. Y. \^Onishi  for their helpful suggestions and
encouragement. Especially I am grateful to Prof. J. McKay
for refereing me to the book of Prasolov and Solovyev
\cite{PS}.


\begin{thebibliography}{00}

\bibitem{Ba}
  {H. F. Baker};
  \emph{On the hyperelliptic sigma functions},
  {Amer. J. of Math.}, Vol.XX  (1898),  301--384.


\bibitem{E} {L. Euler};
\emph{Methodus inveniendi lineas curvas
maximi minimive proprietate gaudentes 1744},
Leonhardi Euleri Opera Omnia Ser. I Vol. 14.

\bibitem{GN} {P. G. Grinevich and S. P. Novikov};
\emph{Topological charge of the real periodic finite-gap
sine-Gordon solutions}, {Comm. Pure. Appl. Math.} , Vol.56 (2003), 956--978.

\bibitem{Ma0} {S. Matstutani};
      \emph{Statistical Mechanics of Elastica on plane:
      Origin of MKdV hierarchy}
        {J. Phys. A}, Vol.31 (1998), 2705--2725.

\bibitem{Ma1} {S. Matstutani};
\emph{Hyperelliptic Loop Solitons with Genus $g$:
          Investigations of a Quantized Elastica},
        J. Geom. Phys., Vol. 43(2002), 146--162.

\bibitem{Ma2} {S. Matstutani};
\emph{On the Moduli of a Quantized Elastica in $\mathbb{P}$
and KdV Flows: Study of Hyperelliptic Curves as an
Extension of Euler's Perspective of Elastica I},
          Rev. Math. Phys., Vol. 15, 559-628.

 \bibitem{Mu} {D.~Mumford};
    \emph{Elastica and Computer Vision} in
    Algebraic Geometry and its Applications,
      edited by C.~Bajaj, Springer-Verlag, Berlin (1993), 507--518.

\bibitem{Na} {S. M. Natanzon};
\emph{Real nonsingular finite zone solutions of soliton equations},
Topics in topology and mathematical physics,
Amer. Math. Soc. Transl. Ser. 2, 170, (1995), 153--183.

\bibitem{P}   {E.~Previato};
    \emph{Geometry of the Modified KdV Equation} in
   Geometric and Quantum Aspects of Integrable System,
      edited by G. F. Helminck, Springer-Verlag, Berlin (1993)
      43--65.

\bibitem{PS}   {V. Prasolov and Y. Solovyev};
\emph{Elliptic Functions and Elliptic Integrals}
 (1991) translated by D. Leites, AMS (1997).


\bibitem{T} {C. Truesdell}
\emph{The rational mechanics of flexible or elastic bodies 1638-1788},
Leonhardi Euleri Opera Omnia Ser. II, Vol. 11 part 2,
Orell F\"ussli, Z\"urich, 1960.

\bibitem{TW} {H. Tsuru and M. Wadati};
\emph{Elastic Model of Highly Supercoiled DNA}
{Biopolymers}, Vol. 25 (1986) , 2083-2096.

\bibitem{W}  K. Weierstrass;
\emph{Zur Theorie der Abel'schen Functionen},
Aus dem Crelle'schen Journal, Vol. 47 (1854).

\end{thebibliography}

\end{document}