# Explicit quasi-periodic wave solutions and asymptotic analysis to the
supersymmetric Ito’s equation

Engui Fana111 Electronic mail<EMAIL_ADDRESS>Y. C. Honb,
a. School of Mathematics Sciences, Fudan University, Shanghai 200433, PR China
b. Department of Mathematics, City University of Hong Kong, Hong Kong, PR
China

Abstract: Based on a Riemann theta function and the super-Hirota bilinear
form, we propose a key formula for explicitly constructing quasi-periodic wave
solutions of the supersymmetric Ito’s equation in superspace
$\mathbb{C}_{\Lambda}^{2,1}$. Once a nonlinear equation is written in bilinear
forms, then the quasi-periodic wave solutions can be directly obtained from
our formula. The relations between the periodic wave solutions and the well-
known soliton solutions are rigorously established. It is shown that the
quasi-periodic wave solutions tends to the soliton solutions under small
amplitude limits.

##  1\. Introduction

The Ito’s equation takes the form

$u_{tt}+6(u_{x}u_{t})_{x}+u_{xxxt}=0,$ $None$

which was first proposed by Ito, and its bilinear Bäcklund transformation, Lax
representation and multi-soliton solutions were obtained [1] . The other
integrable properties of this equation such as the nonlinear superposition
formula, Kac-Moody algebra, bi-Hamiltonian structure have been further found
[2]-[5] . Recently, Liu, Hu and Liu proposed the following supersymmetric
Its’s equation [6]

$\displaystyle\mathfrak{D}_{t}F_{t}+6(F_{x}(\mathfrak{D}_{t}F))_{x}+\mathfrak{D}_{t}F_{xxx}=0,$
$None$

and obtained its one-, two- and three-soliton solutions, where
$F=F(x,t,\theta)$ is fermionic superfield depending on usual even independent
variable $x$, $t$ and odd Grassman variable $\theta$. The differential
operator $\mathfrak{D}_{t}=\partial_{\theta}+\theta\partial_{t}$ is the super
derivative.

The bilinear derivative method developed by Hirota is a powerful approach for
constructing exact solution of nonlinear equations[7]–[13] . Based on the
Hirota bilinear form and the Riemann theta functions, Nakamura presented an
approach to directly construct a kind of quasi-periodic solutions of nonlinear
equation [14, 15] , where the periodic wave solutions of the KdV equation and
the Boussinesq equation were obtained. This method not only conveniently
obtains periodic solutions of a nonlinear equation, but also directly gives
the explicit relations among frequencies, wave-numbers, phase shifts and
amplitudes of the wave. Recently, this method is further developed to
investigate the discrete Toda lattice, (2+1)-dimensional Kadomtsev-
Petviashvili equation and Bogoyavlenskii’s breaking soliton equation[16]-[20]
.

Our present paper will considerably improve the key steps of the above method
so as to make the method much more lucid and straightforward for applying a
class of nonlinear supersymmetric equations. First, the above method will be
generalized into the supersymmetric context. The quasi-periodic solutions of
supersymmetric equations still seem not investigated to our acknowledge.
Second, we a formula that the Riemannn theta functions satisfy a super-Hirota
bilinear equation. This formula actually provides us an uniform method which
can be used to construct quasi-periodic wave solutions of nonlinear
differential, difference and supersymmetric equations. Once a nonlinear
equation is written in bilinear forms, then the quasi-periodic wave solutions
of the nonlinear equation can be obtained directly by using the formula. As
illustrative example, we shall construct quasi-periodic wave solutions to the
supersymmetric Ito’s equation (1.2). Moreover, we also establish the relations
between our quasi-periodic wave solutions and the soliton solutions that were
obtained by Liu and Hu [5] .

The organization of this paper is as follows. In section 2, we briefly
introduce a super-Hirota bilinear that will be suitable for constructing
quasi-periodic solutions of the equation (1.2). And then introduce a general
Riemann theta function and provide a key formula for constructing periodic
wave solutions. In section 3, as application of our formula, we construct one-
periodic wave solutions to the equation (1.2). We further present a simple and
effective limiting procedure to analyze asymptotic behavior of the one-
periodic wave solutions. It is rigorously shown that the quasi-periodic wave
solutions tends to the known soliton solutions obtained by Liu and Hu under
“small amplitude” limits. At last, we briefly discuss the conditions on the
construction of multi-periodic wave solutions of the equation (1.2) in section
4.

##  2\.  The superspace, Hirota bilinear form and the Riemann theta functions

To fix the notations and make our presentation self-contained, we briefly
recall some properties about superanalysis and super-Hirota bilinear
operators. The details about superanalysis refer, for instance, to
Vladimirov’s work [22, 23].

A linear space $\Lambda$ is called $Z_{2}$-graded if it represented as a
direct sum of two subspaces

$\Lambda=\Lambda_{0}\oplus\Lambda,$

where elements of the spaces $\Lambda_{0}$ and $\Lambda_{1}$ are homogeneous.
We assume that $\Lambda_{0}$ is a subspace consisting of even elements and
$\Lambda_{1}$ is a subspace consisting of odd elements. For the element
$f\in\Lambda$ we denote by $f_{0}$ and $f_{1}$ its even and odd components. A
parity function is introduced on the $\Lambda$, namely,

$|f|=\left\\{\begin{matrix}0,\ \ {\rm if}\ \ f\in\Lambda_{0},\\\ \ 1,\ \ {\rm
if}\ \ f\in\Lambda_{1}.\end{matrix}\right.$

We introduce an annihilator of the set of odd elements by setting

${}^{\perp}\Lambda_{1}=\\{\lambda\in\Lambda:\lambda\Lambda_{1}=0\\}.$

A superalgebra is a $Z_{2}$-graded space $\Lambda=\Lambda_{0}\oplus\Lambda$ in
which, besides usual operations of addition and multiplication by numbers, a
product of elements is defined with the usual distribution law:

$a(\alpha b+\beta c)=\alpha ab+\beta ac,\ \ (\alpha b+\beta c)a=\alpha
ba+\beta ca,$

where $a,b,c\in\Lambda$ and $\alpha,\beta\in\mathbb{C}.$ Moreover, a structure
on $\Lambda$ is introduced of an associative algebra with a unite $e$ and even
multiplication i.e., the product of two even and two odd elements is an even
element and the product of an even element by an odd one is an odd element:
$|ab|=|a|+|b|$ mod (2).

A commutative superalgebra with unit $e=1$ is called a finite-dimensional
Grassmann algebra if it contains a system of anticommuting generators
$\sigma_{j},j=1,\cdots,n$ with the property:
$\sigma_{j}\sigma_{k}+\sigma_{k}\sigma_{j}=0,\ j,k=1,2,\cdots,n$, in
particular, $\sigma_{j}^{2}=0$. The Grassmann algebra will be denote by
$G_{n}=G_{n}(\sigma_{1},\cdots,\sigma_{n})$.

The monomials $\\{e_{0},e_{i}=\sigma_{j_{1}}\cdots\sigma_{j_{n}}\\}$,
$j=(j_{1}<\cdots<j_{n})$ form a basis in the Grassmann algebra $G_{n}$, $\dim
G_{n}=2^{n}$. Then it follows that any element of $G_{n}$ is a linear
combination of monomials $\sigma_{j_{1}}\cdots\sigma_{j_{k}},\
j_{1}<\cdots<j_{k}$, that is,

$f=f_{0}+\sum_{k\geq 0}\sum_{j_{1}<\cdots<j_{k}}f_{j_{1}\cdots
j_{k}}\sigma_{j_{1}}\cdots\sigma_{j_{k}},$

where the coefficients $f_{j_{1}\cdots j_{k}}\in\mathbb{C}$.

Definition 1. Let $\Lambda=\Lambda_{0}\oplus\Lambda$ be a commutative Banach
superalgebra, then the Banach space

$\mathbb{C}_{\Lambda}^{m,n}=\Lambda_{0}^{m}\times\Lambda_{1}^{n}$

is called a superspace of dimension $(m,n)$ over $\Lambda$. In particular, if
$\Lambda_{0}=\mathbb{C}$ and $\Lambda_{1}=0$, then
$\mathbb{C}_{\Lambda}^{m,n}=\mathbb{C}^{m}.$

A function $f(\boldsymbol{x}):\mathbb{C}_{\Lambda}^{m,n}\rightarrow\Lambda$ is
said to be superdifferentiable at the point $x\in\mathbb{C}_{\Lambda}^{m,n}$,
if there exist elements $F_{j}(\boldsymbol{x})$ in $\Lambda,\ j=1,\cdots,m+n$,
such that

$f(\boldsymbol{x}+\boldsymbol{h})=f(\boldsymbol{x})+\sum_{j=1}^{m+n}\langle
F_{j}(\boldsymbol{x}),h_{j}\rangle+o(\boldsymbol{x},\boldsymbol{h}),$

where $\boldsymbol{x}=(x_{1},\cdots,x_{m},x_{m+1},\cdots,x_{n})$ with
components $x_{j},j=1,\cdots,m$ being even variable and
$x_{m+j}=\theta_{j},j=1,\cdots,n$ being Grassmann odd ones. The vector
$\boldsymbol{h}=(h_{1},\cdots,h_{m}$, $h_{m+1},\cdots,h_{m+n})$ with
$(h_{1},\cdots,h_{m})\in\Lambda_{0}^{m}$ and
$(h_{m+1},\cdots,h_{m+n})\in\Lambda_{1}^{n}$. Moreover,

$\lim_{\parallel\boldsymbol{h}\parallel\rightarrow 0}\frac{\parallel
o(\boldsymbol{x},\boldsymbol{h})\parallel}{\parallel\boldsymbol{h}\parallel}\longrightarrow
0.$

The $F_{j}(\boldsymbol{x})$ are called the super partial derivative of $f$
with respect to $x_{j}$ at the point $\boldsymbol{x}$ and are denoted,
respectively, by

$\frac{\partial f(\boldsymbol{x})}{\partial x_{j}}=F_{j}(\boldsymbol{x}),\
j=1,\cdots,m+n.$

The derivatives $\frac{\partial f(\boldsymbol{x})}{\partial x_{j}}$ with
respect to even variables $x_{j},\ j=1,2,\cdots n$ are uniquely defined. While
the derivatives $\frac{\partial f(\boldsymbol{x})}{\partial\theta_{j}}$ to odd
variables $\theta_{j}=x_{j+n},\ j=1,2,\cdots m$ are not uniquely defined, but
with an accuracy to within an addition constant
$c\sigma_{1}\cdots\sigma_{n},c\in\mathbb{C}$ from an annihilator
${}^{\perp}G_{n}$ of finite-dimensional Grassmann algebra $G_{n}$.

The super derivative also satisfies Leibniz formula

$\frac{\partial(f(\boldsymbol{x})g(\boldsymbol{x}))}{\partial
x_{j}}=\frac{\partial f(\boldsymbol{x})}{\partial
x_{j}}g(\boldsymbol{x})+(-1)^{|x_{j}||f|}f(\boldsymbol{x})\frac{\partial
g(\boldsymbol{x})}{\partial x_{j}},\ j=1,\cdots,m+n.$ $None$

Denote by $\mathcal{P}(\Lambda_{1}^{n},\Lambda)$ the set of polynomials
defined on $\Lambda_{1}^{n}$ with value in $\Lambda$. We say that a super
integral is a map $I:\mathcal{P}(\Lambda_{1}^{n},\Lambda)\rightarrow\Lambda$
satisfying the following condition is an super integral about Grassmann
variable

(1) A linearity: $I(\mu f+\nu g)=\mu I(f)+\nu I(g),\ \mu,\nu\in\Lambda,\
f,g\in\mathcal{P}(\Lambda_{1}^{n},\Lambda);$

(2) translation invariance: $I(f_{\xi})=I(f)$, where
$f_{\xi}=f(\boldsymbol{\theta}+\boldsymbol{\xi})$ for all
$\boldsymbol{\xi}\in\Lambda_{1}^{n}$,
$f\in\mathcal{P}(\Lambda_{1}^{n},\Lambda).$

We denote $I(\theta^{\varepsilon})=I_{\varepsilon}$, where $\varepsilon$
belongs to the set of multiindices
$N_{n}=\\{\boldsymbol{\epsilon}=(\varepsilon_{1},\cdots,\varepsilon_{n}),\varepsilon_{j}=0,1,\boldsymbol{\theta}^{\varepsilon}=\theta_{1}^{\varepsilon_{1}}\cdots\theta_{n}^{\varepsilon_{n}}\not\equiv
0\\}$. In the case when $I_{\varepsilon}=0,\varepsilon\in
N_{n},|\varepsilon|\leq n=n-1$, such kind of integral has the form

$I(f)=J(f)I(1,\cdots,1),$

where

$J(f)=\frac{\partial^{n}f(0)}{\partial\theta_{1}\cdots\partial\theta_{n}}.$

Since the derivative is defined with an accurcy to with an additive constant
form the annihilator ${}^{\perp}L_{n}$,
$L_{n}=\\{\theta_{1}\cdots\theta_{n},\boldsymbol{\theta}\in\Lambda_{1}^{n}\\}$,
it follows that $J:\mathcal{P}\rightarrow\Lambda/^{\perp}L_{n}$ is single-
valued mapping. This mapping also satisfies the conditions 1 and 2, and
therefore we shall call it an integral and denote

$J(f)=\int
f(\boldsymbol{\theta})d\boldsymbol{\theta}=\int\theta_{1}\cdots\theta_{n}d\theta_{1}\cdots
d\theta_{n},$

which has properties:

$\displaystyle\int\theta_{1}\cdots\theta_{n}d\theta_{1}\cdots d\theta_{n}=1,$
$None$ $\displaystyle\int\frac{\partial
f}{\partial\theta_{j}}d\theta_{1}\cdots d\theta_{n}=0,\ j=1,\cdots,n.$
$\displaystyle\int f(\boldsymbol{\theta})\frac{\partial
g(\boldsymbol{\theta})}{\partial\theta_{j}}d\boldsymbol{\theta}=(-1)^{1+|g|}\int\frac{\partial
f(\boldsymbol{\theta})}{\partial\theta_{j}}g(\boldsymbol{\theta})d\boldsymbol{\theta}.$

In this paper, we consider functions with two ordinary even variables $x,t$
and a Grassmann odd variable $\theta$. The associated space
$\mathbb{C}_{\Lambda}^{2,1}=\Lambda_{0}^{2}\times\Lambda_{1}$ (we may take
$\Lambda_{0}=\mathbb{R}$ or $\mathbb{C}$) is a superspace over Grassmann
algebra $G_{1}(\sigma)=G_{1,0}\oplus G_{1,1}$, whose elements have the form

$f=f_{0}+f_{1}\sigma.$

where $e=1$ is a unit, $\sigma$ is anticommuting generator. The monomials
$\\{1,\sigma\\}$ form a basis of the $G_{1}(\sigma)$, dim$G_{1}(\sigma)=2$.
Under traveling wave frame in space $\mathbb{C}_{\Lambda}^{2,1}$, the phase
variable should have the form

$\xi=\alpha x+\omega t+\theta\sigma.$

Now we consider the bilinear form of the equation (1.2). By the dependent
variable transformation

$F=\partial_{x}\ln f(x,t,\theta),$ $None$

where
$f(x,t,\theta):\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{1,0}$
is a superdifferential function, the equation (1.2) is then transformed into a
bilinear form

$(S_{t}D_{t}+S_{t}D_{x}^{3})f(x,t,\theta)\cdot f(x,t,\theta)=0.$ $None$

where the Hirota bilinear differential operators $D_{x}$ and $D_{t}$ are
defined by

$\displaystyle D_{x}^{m}D_{t}^{n}f(x,t,\theta)\cdot
g(x,t,\theta)=(\partial_{x}-\partial_{x^{\prime}})^{m}(\partial_{t}-\partial_{t^{\prime}})^{n}f(x,t,\theta)g(x^{\prime},t^{\prime},\theta^{\prime})|_{x^{\prime}=x,t^{\prime}=t,\theta^{\prime}=\theta}.$

The super-Hirota bilinear operator is defined as [21]

$S_{t}^{N}f(x,t,\theta)\cdot
g(x,t,\theta)=\sum_{j=0}^{N}(-1)^{j|f|+\frac{1}{2}j(j+1)}\left[\begin{matrix}N\\\
j\end{matrix}\right]\mathfrak{D}_{t}^{N-j}f(x,t,\theta)\mathfrak{D}_{t}^{j}g(x,t,\theta),$

where the super binomial coefficients are defined by

$\left[\begin{matrix}N\\\
j\end{matrix}\right]=\left\\{\begin{matrix}\left(\begin{matrix}[N/2]\cr[j/2]\end{matrix}\right),{\rm
if}\ \ (N,j)\not=(0,1)\ \ {\rm mod}\ \ 2,\\\ 0,\ \ {\rm
otherwise}.\end{matrix}\right.$

$[k]$ is the integer part of the real number $k$ ($[k]\leq k\leq[k]+1$).

Following the Hirota bilinear theory, It is easy to find that the equation
(1.2) admits one-soliton solution (also called one-supersoliton solution)

$F_{1}=\partial_{x}\ln(1+e^{\eta}),$ $None$

with phase variable $\eta=kx-k^{3}t+\theta\zeta+\gamma$ and
$k,\gamma\in\Lambda_{0}$, $\zeta\in\Lambda_{1}$.

To apply the Hirota bilinear method for constructing periodic wave solutions
of the equation (1.2), we hope to add two odd variables $F_{0}$, $c$ and
consider a more general form than the bilinear equation (1.4)

$F=\partial_{\theta}^{-1}F_{0}+\partial_{x}\ln f(x,t,\theta),$ $None$

where
$F_{0}=F_{0}(\theta):\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{0,1}$
is an odd special solution of the equation (1.2). Substituting (2.1) into
(1.2) and integrating with respect to $x$, we then get the following bilinear
form

$\displaystyle G(S_{t},D_{x},D_{t})f\cdot
f=(S_{t}D_{t}+S_{t}D_{x}^{3}+3F_{0}D_{x}^{2}+c)f\cdot f=0,$ $None$

where
$c=c(\theta,t):\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{0,1}$
is an odd integration constant. For the bilinear equation (2.2), we are
interested in its multi-periodic solutions in terms of the Riemann theta
functions.

In the following, we introduce a super one-dimensional Riemann theta function
on super space $\mathbb{C}_{\Lambda}^{2,1}$ and discuss its quasi-periodicity,
which plays a central role in this paper. The Riemann theta function reads

$\vartheta(\mathbf{\xi},\varepsilon,s|\tau)=\sum_{n\in\mathbb{Z}}\exp[2\pi
i(\xi+\varepsilon)(n+s)-\pi\tau({n}+{s})^{2}].$ $None$

Here the integer value $n\in\mathbb{Z}$, $s,\varepsilon\in\mathcal{C}$, and
complex phase variables $\xi=\alpha x+\omega t+\theta\sigma+\delta$ is
dependent of even variable $x,t$ and odd $\theta$; The $\tau>0$ is called the
period matrix of the Riemann theta function. It is obvious that the Riemann
theta function (2.3) converges absolutely and superdifferentiable on
superspace $\mathbb{C}_{\Lambda}^{2,1}$. For the simplicity, in the case when
$s=\omega=0$, we denote

$\vartheta(\mathbf{\xi},\tau)=\vartheta(\mathbf{\xi},0,0|\tau).$

Definition 2. A function
$f(\xi):\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{1,0}$ is
said to be quasi-periodic in $\xi=\alpha x+\omega t+\theta\sigma+\delta$ with
fundamental periods $T$, if there exist certain constants $a,b\in\Lambda_{0}$,
such that

$f(\xi+T)=f(\xi)+a\xi+b.$

An example of this is the ordinary Weierstrass zeta function, where

$\zeta(\xi+\omega)=\zeta(\xi)+\eta,$

for a fixed constant $\eta$ when $\omega$ is a period of the corresponding
Weierstrass elliptic $\wp$ function.

Proposition 2. [24] The Riemann theta function $\vartheta(\xi,\tau)$ defined
above has the periodic properties

$\displaystyle\vartheta(\xi+1+i\tau,\tau)=\exp(-2\pi
i\xi+\pi\tau)\vartheta(\xi,\tau).$ $None$

Now we turn to see the periodicity of the solution (2.4), we take
$f(x,t,\theta)$ in the bilinear equation (2.2) as

$f(x,t,\theta)=\vartheta(\xi,\tau),$

where phase variable $\xi=\alpha x+\omega t+\theta\sigma+\delta$. By using
(2.4), it is easy to see that

$\displaystyle\frac{\vartheta^{\prime}_{\xi}(\xi+i\tau,\tau)}{\vartheta(\xi+i\tau,\tau)}=-2\pi
i+\frac{\vartheta^{\prime}_{\xi}(\xi,\tau)}{\vartheta(\xi,\tau)},$

that is,

$\displaystyle\partial_{\xi}\ln\vartheta(\xi+i\tau,\tau)=-2\pi
i+\partial_{\xi}\ln\vartheta(\xi,\tau).$ $None$

According to the differential relation, we have

$F(x,t,\theta)=F(\xi)=\partial_{\theta}^{-1}F_{0}+\alpha\partial_{\xi}\ln\vartheta(\xi,\tau).$
$None$

The equations (2.5) and (2.6) demonstrate that

$F(\xi+1+i\tau)=\partial_{\theta}^{-1}F_{0}+\alpha\partial_{\xi}\ln\vartheta(\xi+1+i\tau,\tau)=-2\pi
i\alpha+F(\xi).$

Therefore the solution $F(\xi)$ is a quasi-periodic function with two
fundamental periods $1$ and $i\tau$.

In following, we establish uniform formula on the Riemannn theta function,
which will play a key role in the construction of the periodic wave solutions.

Proposition 2. [21] Suppose that $f(x,t,\theta),g(x,t,\theta)$ are super
differentiable on space $\mathbb{C}_{\Lambda}^{2,1}$. Then the Hirota bilinear
operators $D_{x},D_{t}$ and super-Hirota bilinear operator $S_{x}$ have
properties

$\displaystyle S_{x}^{2N}f\cdot g=D_{x}^{N}f\cdot g,$ $None$ $\displaystyle
D_{x}^{m}D_{t}^{n}e^{\xi_{1}}\cdot
e^{\xi_{2}}=(\alpha_{1}-\alpha_{2})^{m}(\omega_{1}-\omega_{2})^{n}e^{\xi_{1}+\xi_{2}},$
$\displaystyle S_{x}e^{\xi_{1}}\cdot
e^{\xi_{2}}=[\sigma_{1}-\sigma_{2}+\theta(\alpha_{1}-\alpha_{2})]e^{\xi_{1}+\xi_{2}},$

where $\xi_{j}=\alpha_{j}x+\omega_{j}t+\theta\sigma_{j}+\delta_{j},j=1,2$.
More generally, we have

$\displaystyle F(S_{x},D_{x},D_{t})e^{\xi_{1}}\cdot
e^{\xi_{2}}=F(\sigma_{1}-\sigma_{2}+\theta(\alpha_{1}-\alpha_{2}),\alpha_{1}-\alpha_{2},\omega_{1}-\omega_{2})e^{\xi_{1}+\xi_{2}},$
$None$

where $G(S_{t},D_{x},D_{t})$ is a polynomial about $S_{t},D_{x}$ and $D_{t}$.
This properties will be utilized later to explore the quasi-periodic wave
solutions of the equation (1.2).

Proposition 3. The Hirota bilinear operators $D_{x},D_{t}$ and super-Hirota
bilinear operator $S_{x}$ have properties when they act on the Riemann theta
functions

$D_{x}\vartheta(\xi,\varepsilon^{\prime},s^{\prime}|\tau)\cdot\vartheta(\xi,\varepsilon,s|\tau)=\sum_{\mu=0,1}\partial_{x}\vartheta(2\xi,\varepsilon^{\prime}-\varepsilon,(s^{\prime}-s-\mu)/2|2\tau)|_{\xi=0}\vartheta(2\xi,\varepsilon^{\prime}+\varepsilon,(s^{\prime}+s+\mu)/2|2\tau),$
$None$
$S_{t}\vartheta(\xi,\varepsilon^{\prime},s^{\prime}|\tau)\cdot\vartheta(\xi,\varepsilon,s|\tau)=\sum_{\mu=0,1}\mathfrak{D}_{t}\vartheta(2\xi,\varepsilon^{\prime}-\varepsilon,(s^{\prime}-s-\mu)/2|2\tau)|_{\xi=0}\vartheta(2\xi,\varepsilon^{\prime}+\varepsilon,(s^{\prime}+s+\mu)/2|2\tau),$
$None$

where $\sum_{\mu=0,1}$ indicates sum with respective to $\mu=0,1$.

In general, for a polynomial operator $G(S_{t},D_{x},D_{t})$ about
$S_{t},D_{x}$ and $D_{t}$, we have

$\displaystyle
G(S_{t},D_{x},D_{t})\vartheta(\xi,\tau)\cdot\vartheta(\xi,\tau)=\sum_{\mu=0,1}C(\alpha,\omega,\sigma,\mu)\vartheta(2\xi,\mu/2|2\tau),$
$None$

where

$\displaystyle\xi=\alpha x+\omega t+\theta\sigma+\gamma.$ $None$
$\displaystyle C(\alpha,\omega,\sigma,\mu|\tau)=\sum_{n\in Z}G\left\\{4\pi
i(n-\mu/2)\alpha,4\pi i(n-\mu/2)\omega,\right.$ $\displaystyle\left.4\pi
i(n-\mu/2)(\sigma+\theta\omega)\right\\}\times\exp\left[-2\pi\tau(n-\mu/2)^{2}\right].$

Proof. By using (2.7), we have

$\displaystyle\Gamma=S_{t}\vartheta(\xi,\varepsilon^{\prime},s^{\prime}|\tau)\cdot\vartheta(\xi,\varepsilon,s|\tau)$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}S_{t}\exp\\{2\pi
i(m^{\prime}+s^{\prime})(\xi+\varepsilon^{\prime})-\pi(m^{\prime}+s^{\prime})^{2}\tau\\}\cdot\exp\\{2\pi
i(m+s)(\xi+\varepsilon)-\pi(m+s)^{2}\tau\\},$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}2\pi
i(\sigma+\theta\omega)(m^{\prime}-m+s^{\prime}-s)\exp\left\\{2\pi
i(m^{\prime}+m+s^{\prime}+s)\xi-2\pi
i[(m^{\prime}+s^{\prime})\varepsilon^{\prime}+(m+s)\varepsilon]\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.-\pi\tau[(m^{\prime}+s^{\prime})^{2}+(m+s)^{2}]\right\\}$
$\displaystyle\stackrel{{\scriptstyle
m=l^{\prime}-m^{\prime}}}{{=}}\sum_{l^{\prime},m^{\prime}\in\mathbb{Z}}2\pi
i(\sigma+\theta\omega)(2m^{\prime}-l^{\prime}+s^{\prime}-s)\exp\left\\{2\pi
i(l^{\prime}+s^{\prime}+s)\xi-2\pi
i[(m^{\prime}+s^{\prime})\varepsilon^{\prime}\right.$ $\displaystyle\ \ \ \ \
\ \ \ \ \ \ \ \ \
\left.+(l^{\prime}-m^{\prime}+s)\varepsilon]-\pi[(m^{\prime}+s^{\prime})^{2}+(l^{\prime}-m^{\prime}+s)^{2}]\tau\right\\}$
$\displaystyle\stackrel{{\scriptstyle
l^{\prime}=2l+\mu}}{{=}}\sum_{\mu=0,1}\sum_{l,m^{\prime}\in\mathbb{Z}}2\pi
i(\sigma+\theta\omega)(2m^{\prime}-2l+s^{\prime}-s-\mu)\exp\\{4\pi
i\xi[l+(s^{\prime}+s+\mu)/2]$ $\displaystyle\ \ \ \ \ -2\pi
i[(m^{\prime}+s^{\prime})\varepsilon^{\prime}-(m^{\prime}-2l-s-\mu)\varepsilon]-\pi[(m^{\prime}+s^{\prime})^{2}+(m^{\prime}-2l-s-\mu)^{2}]\tau\\}$

Let $m^{\prime}=n+l$, and using the relations

$\displaystyle
n+l+s^{\prime}=[n+(s^{\prime}-s-\mu)/2]+[l+(s^{\prime}+s+\mu)/2],$
$\displaystyle n-l-s-\mu=[n+(s^{\prime}-s-\mu)/2]-[l+(s^{\prime}+s+\mu)/2],$

we finally obtain that

$\displaystyle\Gamma=\sum_{\mu=0,1}\left[\sum_{n\in\mathbb{Z}}4\pi
i(\sigma+\theta\omega)[n+(s^{\prime}-s-\mu)/2]\exp\\{-2\pi
i[n+(s^{\prime}-s-\mu)/2](\varepsilon^{\prime}-\varepsilon)-2\pi\tau[n+(s^{\prime}-s-\mu)/2]^{2}\\}\right]$
$\displaystyle\ \ \ \ \ \ \ \ \times\left[\sum_{l\in\mathbb{Z}}\exp\\{2\pi
i[l+(s^{\prime}+s+\mu)/2](2\xi+\varepsilon^{\prime}+\varepsilon)-2\pi\tau[l+(s^{\prime}+s+\mu)/2]^{2}\right]$
$\displaystyle=\sum_{\mu=0,1}\mathfrak{D}_{t}\vartheta(2\xi,\varepsilon^{\prime}-\varepsilon,(s^{\prime}-s-\mu)/2|2\tau)|_{\xi=0}\vartheta(2\xi,\varepsilon^{\prime}+\varepsilon,(s^{\prime}+s+\mu)/2|2\tau).$

In a similar way, we can prove the formulae (2.9). As a special case when
$\varepsilon=s=0$ of the Riemann theta function (2.3), by using (2.9) an
(2.10), we can prove the formula (2.11). $\Box$

From the formulae (2.11) and (2.12), it is seen that if the following
equations are satisfied

$C(\alpha,\omega,\sigma,\mu|\tau)=0,$

for $\mu=0,1$, then $\vartheta(\xi,\tau)$ is a solution of the bilinear
equation

$G(S_{t},D_{x},D_{t})\vartheta(\xi,\tau)\cdot\vartheta(\xi,\tau)=0.$

##  3\.  Quasi-periodic waves and asymptotic properties

In this section, we consider periodic wave solutions of the equation (1.2). As
a simple case of the theta function (2.3) when $N=1,s=0$, we take
$f(x,t,\theta)$ as

$f(x,t,\theta)=\vartheta(\xi,\tau)=\sum_{n\in\mathbb{Z}}\exp({2\pi in\xi-\pi
n^{2}\tau}),$ $None$

where the phase variable $\xi=\alpha x+\omega t+\theta\sigma+\delta$, and the
parameter $\tau>0$.

To let the Riemann theta function (3.1) be a solution of the bilinear equation
(2.2), according to the formula (2.11), the following equations only need to
be satisfied

$\displaystyle\sum_{n\in\mathbb{Z}}\left[-16\pi^{2}(n-\mu/2)^{2}(\sigma+\theta\omega)\omega+256\pi^{4}(n-\mu/2)^{4}(\sigma+\theta\omega)\alpha^{3}\right.$
$None$ $\displaystyle\left.\ \ \ \ \ \ \ \ \
-48\pi^{2}(n-\mu/2)^{2}\alpha^{2}F_{0}+c\right]\exp[-2\pi(n-\mu/2)^{2}\tau]=0,\
\mu=0,1.$

We introduce the notations by

$\displaystyle\lambda=e^{-\pi\tau/2},\quad\vartheta_{1}(\xi,\lambda)=\vartheta(2\mathbf{\xi},2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{4n^{2}}\exp(4i\pi
n\xi),$ $None$
$\displaystyle\vartheta_{2}(\xi,\lambda)=\vartheta(2\xi,0,-1/2,2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{(2n-1)^{2}}\exp[2i\pi(2n-1)\xi].$

By using formula (3.3), the equation (3.2) can be written as a linear system

$\displaystyle\theta\vartheta_{1}^{\prime\prime}\omega^{2}+(\sigma\vartheta_{1}^{\prime\prime}+\alpha^{3}\theta\vartheta_{1}^{(4)})\omega+\vartheta_{1}c+\sigma\alpha^{3}\vartheta_{1}^{(4)}+3\alpha^{2}F_{0}\vartheta_{1}^{\prime\prime}=0,$
$None$
$\displaystyle\theta\vartheta_{2}^{\prime\prime}\omega^{2}+(\sigma\vartheta_{2}^{\prime\prime}+\alpha^{3}\theta\vartheta_{2}^{(4)})\omega+\vartheta_{2}c+\sigma\alpha^{3}\vartheta_{2}^{(4)}+3\alpha^{2}F_{0}\vartheta_{2}^{\prime\prime}=0,$

where $\omega\in\Lambda_{0}$ is even and
$c,F_{0}:\mathbb{C}_{\Lambda}^{2,1}\rightarrow\mathbb{C}_{\Lambda}^{0,1}$ are
odd. In addition, we have denoted derivatives of $\vartheta_{j}(\xi,\lambda)$
at $\xi=0$ by simple notations

$\vartheta_{j}^{(k)}=\vartheta_{j}^{(k)}(0,\lambda)=\frac{d^{k}\vartheta_{j}(\xi,\lambda)}{d\xi^{k}}|_{\xi=0},\
\ j=1,2,k=0,1,2,\cdots$

Moreover, these functions are independent of Grassmann variable $\theta$ and
$\sigma$.

We show there existence real solutions to the system (3.4). Since
$c=c(\theta,t)$ and $F=F_{0}(\theta)$ are function of Grassmann variable
$\theta$, we can expand them in the form

$c=c_{1}+c_{2}\theta,\ \ F_{0}=f_{1}+f_{2}\theta,$ $None$

where $c_{1},f_{1}\in\Lambda_{1}$ are odd and $c_{2},f_{2}\in\Lambda_{0}$ are
even. Substituting (3.5) into (3.4) leads to

$\displaystyle(\sigma\vartheta_{1}^{\prime\prime}\omega+\vartheta_{1}c_{1}+\sigma\alpha^{3}\vartheta_{1}^{(4)}+3\alpha^{2}\vartheta_{1}^{\prime\prime}f_{1})+\theta(3\alpha^{2}\vartheta_{1}^{\prime\prime}f_{2}+\vartheta_{1}c_{2}+\vartheta_{1}^{\prime\prime}\omega^{2}+\alpha^{3}\vartheta_{1}^{(4)}\omega)=0,$
$None$
$\displaystyle(\sigma\vartheta_{2}^{\prime\prime}\omega+\vartheta_{2}c_{1}+\sigma\alpha^{3}\vartheta_{2}^{(4)}+3\alpha^{2}\vartheta_{2}^{\prime\prime}f_{1})+\theta(3\alpha^{2}\vartheta_{2}^{\prime\prime}f_{2}+\vartheta_{2}c_{2}+\vartheta_{2}^{\prime\prime}\omega^{2}+\alpha^{3}\vartheta_{2}^{(4)}\omega)=0,$

where $\omega,\ c_{1},\ c_{2},f_{1}$ and $f_{2}$ are parameters to be
determined.

Since $\theta$ is a Grassmann variable, the system (3.6) will be satisfied
provided that

$\displaystyle\sigma\vartheta_{1}^{\prime\prime}\omega+\vartheta_{1}c_{1}+\sigma\alpha^{3}\vartheta_{1}^{(4)}+3\alpha^{2}\vartheta_{1}^{\prime\prime}f_{1}=0,$
$None$
$\displaystyle\sigma\vartheta_{2}^{\prime\prime}\omega+\vartheta_{2}c_{1}+\sigma\alpha^{3}\vartheta_{2}^{(4)}+3\alpha^{2}\vartheta_{2}^{\prime\prime}f_{1}=0$

and

$\displaystyle
3\alpha^{2}\vartheta_{1}^{\prime\prime}f_{2}+\vartheta_{1}c_{2}+\vartheta_{1}^{\prime\prime}\omega^{2}+\alpha^{3}\vartheta_{1}^{(4)}\omega=0,$
$None$ $\displaystyle
3\alpha^{2}\vartheta_{2}^{\prime\prime}f_{2}+\vartheta_{2}c_{2}+\vartheta_{2}^{\prime\prime}\omega^{2}+\alpha^{3}\vartheta_{2}^{(4)}\omega=0.$

In the systems (3.7) and (3.8), it is obvious that vectors
$(\vartheta_{1},\vartheta_{2})^{T}$ and
$(\vartheta_{1}^{\prime\prime},\vartheta_{2}^{\prime\prime})^{T}$ are linear
independent, and $(\vartheta_{1}^{(4)},\vartheta_{2}^{(4)})^{T}\not=0$.
Therefore the system (3.7) admits a solution

$\displaystyle\omega=-3\beta\alpha^{2}+\frac{(\vartheta_{2}^{(4)}\vartheta_{1}-\vartheta_{1}^{(4)}\vartheta_{2})\alpha^{3}}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}}\in\Lambda_{0},\
\ \
c_{1}=\frac{(\vartheta_{2}^{(4)}\vartheta_{1}^{\prime\prime}-\vartheta_{1}^{(4)}\vartheta_{2}^{\prime\prime})\alpha^{3}\sigma}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}}\in\Lambda_{1},$
$None$

here we have taken $f_{1}=\beta\sigma,\ \beta\in R$ for simplicity, and other
parameters $\alpha,\tau,\sigma$, $\beta$ are free.

By using (3.9) and solving system (3.8), we obtain that

$\displaystyle f_{2}=-\beta\omega\in\Lambda_{0},\ \ \
c_{2}=\frac{(\vartheta_{1}^{(4)}\vartheta_{2}^{\prime\prime}-\vartheta_{2}^{(4)}\vartheta_{1}^{\prime\prime})\alpha^{3}\omega}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}}\in\Lambda_{0}.$
$None$

Noting that $\int\theta\ d\theta=1$ and $\ \int d\theta=0$, we have

$\partial^{-1}F_{0}=\int(\beta\sigma-\beta\omega\theta)\
d\theta=-\beta\omega.$

In this way, we indeed can get an explicit periodic wave solution of the
equation (1.12)

$F=-\beta\omega+\partial_{x}\ln\vartheta(\xi,\tau),$ $None$

with the theta function $\vartheta(\xi,\tau)$ given by (3.1) and parameters
$\omega$, $c_{1},\ c_{2}$ by (3.9) and (3.10), while other parameters
$\alpha,\sigma,\tau,\delta,\ \beta$ are free. Among them, the three parameters
$\alpha,\sigma$ and $\tau$ completely dominate a periodic wave. In summary,
the periodic wave (3.11) is real-valued and bounded for all complex variables
$(x,t,\theta)$. It is one-dimensional, i.e. there is a single phase variable
$\xi$, and has two fundamental periods $1$ and $i\tau$ in phase variable
$\xi$.

In the following, we further consider asymptotic properties of the periodic
wave solution. Interestingly, the relation between the one-periodic wave
solution (3.11) and the one-super soliton solution (1.5) can be established as
follows.

Theorem 1. Suppose that the $\omega\in\Lambda_{0}$ and $c\in\Lambda_{1}$ are
given given by (3.5), (3.9) and (3.10). For the one-periodic wave solution
(3.11), we let

$\alpha=\frac{k}{2\pi i},\ \ \sigma=\frac{\zeta}{2\pi i},\ \
\delta=\frac{\gamma+\pi\tau}{2\pi i},$ $None$

where the $k,\zeta$ and $\gamma$ are the same as those in (1.5). Then we have
the following asymptotic properties

$c\longrightarrow 0,\ \ \xi\longrightarrow\frac{\eta+\pi\tau}{2\pi i},\ \
\vartheta(\xi,\tau)\longrightarrow 1+e^{\eta},\ \ {\rm as}\ \
\lambda\rightarrow 0.$ $None$

In other words, the periodic solution (3.11) tends to the one-soliton solution
(1.5) under a small amplitude limit , that is,

$F\longrightarrow F_{1},\ \ {\rm as}\ \ \lambda\rightarrow 0.$ $None$

Proof. Here we will directly use the system (3.4) to analyze asymptotic
properties of one-periodic solution, which is more simple and effective than
our original method by solving the system [16]-[20] . Since the coefficients
of system (3.4) are power series about $\lambda$, its solution
$(\omega,c)^{T}$ also should be a series about $\lambda$.

We explicitly expand the coefficients of system (3.4) as follows

$\displaystyle\vartheta_{1}(0,\lambda)=1+2\lambda^{4}+\cdots,\quad\vartheta_{1}^{\prime\prime}(0,\lambda)=-32\pi^{2}\lambda^{4}+\cdots,$
$None$
$\displaystyle\vartheta_{1}^{(4)}(0,\lambda)=512\pi^{4}\lambda^{4}+\cdots,\ \
\vartheta_{2}(0,\lambda)=2+2\lambda^{8}+\cdots$
$\displaystyle\vartheta_{2}^{\prime\prime}(0,\lambda)=-8\pi^{2}-72\pi^{2}\lambda^{8}+\cdots,\
\vartheta_{2}^{(4)}(0,\lambda)=32\pi^{4}+2592\pi^{4}\lambda^{8}+\cdots.$

Let the solution of the system (3.4) be in the form

$\displaystyle\omega=\omega_{0}+\omega_{1}\lambda+\omega_{2}\lambda^{2}+\cdots=\omega_{0}+o(\lambda),$
$None$ $\displaystyle
c=b_{0}+b_{1}\lambda+b_{2}\lambda^{2}+\cdots=b_{0}+o(\lambda),$

where $\omega_{j}\in\Lambda_{0},\ b_{j}\in\Lambda_{1},\ j=0,1,2\cdots$

Substituting the expansions (3.11) and (3.12) into the system (3.5) and
letting $\lambda\longrightarrow 0$, we immediately obtain the following
relations

$\displaystyle b_{0}=0,\ \
-8\pi^{2}\sigma\omega_{0}+2b_{0}+32\pi^{4}\sigma\alpha^{3}=0,$

which has a solution

$b_{0}=0,\ \ w_{0}=4\pi^{2}\alpha^{3}.$

Then from the relations (3.12) and (3.16), we have

$c\longrightarrow 0,\ \ 2\pi i\omega\longrightarrow
8\pi^{3}i\alpha^{3}=-k^{3},\ \ {\rm as}\ \ \lambda\rightarrow 0,$

and thus

$\displaystyle\hat{\xi}=2\pi i\xi-\pi\tau=kx+2\pi i\omega
t+\theta\zeta+\gamma$ $None$ $\displaystyle\quad\longrightarrow
kx-k^{3}t+\theta\zeta+\gamma=\eta,\ \ {\rm as}\ \ \lambda\rightarrow 0,$

It remains to show that the one-periodic wave (3.11) possesses the same form
with the one-soliton solution (1.5) under the limit $\lambda\rightarrow 0$.
For this purpose, we first expand the Riemann theta function
$\vartheta(\xi,\tau)$ in the form

$\vartheta(\xi,\tau)=1+\lambda^{2}(e^{2\pi i\xi}+e^{-2\pi
i\xi})+\lambda^{8}(e^{4\pi i\xi}+e^{-4\pi i\xi})+\cdots.$

By using the (3.12) and (3.17), it follows that

$\displaystyle\vartheta(\xi,\tau)=1+e^{\hat{\xi}}+\lambda^{4}(e^{-\hat{\xi}}+e^{2\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}+e^{3\hat{\xi}})+\cdots$
$\displaystyle\quad\longrightarrow 1+e^{\hat{\xi}}\longrightarrow 1+e^{\eta},\
\ {\rm as}\ \ \lambda\rightarrow 0,$

which implies (3.13) and (3.14). Therefore we conclude that the one-periodic
solution (3.11) just goes to the one-soliton solution (1.5) as the amplitude
$\lambda\rightarrow 0$. $\square$

##  4\.  Discussion on the conditions of $N$-periodic wave solutions

In this section, we consider condition for $N$-periodic wave solutions of the
equation (1.2). The theta function is taken the form

$\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=\vartheta(\xi_{1},\cdots,\xi_{N},\tau)=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}\exp\\{2\pi
i<\boldsymbol{\xi},\boldsymbol{n}>-\pi<\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}>\\},$
$None$

where $\boldsymbol{n}=(n_{1},\cdots,n_{N})^{T}\in\mathbb{Z}^{N},\ \
\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})^{T}\in\mathcal{C}^{N},\ \
\xi_{i}=\alpha_{j}x+\omega_{j}t+\theta\sigma_{j}+\delta_{j},\ \ j=1,\cdots,N$,
$\tau$ is a ${N\times N}$ symmetric positive definite matrix.

To make the theta function (4.1) satisfy the bilinear equation (2.2), we
obtain that according to the formula (2.11)

$\displaystyle\sum_{\mu=0,1}\ \
\sum_{n_{1},\cdots,n_{N}=-\infty}^{\infty}G\left\\{4\pi
i\sum_{j=1}^{N}(n_{j}-\mu_{j}/2)\alpha_{j},4\pi
i\sum_{j=1}^{N}(n_{j}-\mu_{j}/2)\omega_{j},\right.$ $None$
$\displaystyle\left.4\pi
i\sum_{j=1}^{N}(n_{j}-\mu_{j}/2)(\sigma_{j}+\theta\omega_{j})\right\\}\times\exp\left[-2\pi\sum_{j,k=1}^{N}(n_{j}-\mu_{j}/2)\tau_{jk}(n_{k}-\mu_{k}/2)\right]=0.$

Now we consider the number of equation and some unknown parameters. Obviously,
in the case of supersymmetric equations, the number of constraint equations of
the type (4.2) is $2^{N+1}$, which is two times of the constraint equations
needed in the case of ordinary equations [16]-[20] . On the other hand we have
parameters
$\tau_{ij}=\tau_{ji},c_{1},c_{2},f_{1},f_{2},\alpha_{i},\omega_{i}$, whose
total number is $\frac{1}{2}N(N+1)+2N+4$. Among them, $2N$ parameters
$\tau_{ii},\omega_{i}$ are taken to be the given parameters related to the
amplitudes and wave numbers (or frequencies) of $N$-periodic waves;
$\frac{1}{2}N(N+1)$ parameters $\tau_{ij}$ implicitly appear in series form,
which is general can not to be solved explicit. Hence, the number of the
explicit unknown parameters is only $N+4$. The number of equations is larger
than the unknown parameters in the case when $N\geq 2$. In this paper, we
consider one-periodic wave solution of the equation (1.2), which belongs to
the cases when $N=1$. There are still certain difficulties in the calculation
for the case $N\geq 2$, which will be considered in our future work.

## Acknowledgment

I would like to express my special thanks to the referee for constructive
suggestions which have been followed in the present improved version of the
paper. The work described in this paper was supported by grants from the
Research Grants Council of Hong Kong (No.9041473), the National Science
Foundation of China (No.10971031), Shanghai Shuguang Tracking Project
(No.08GG01) and Innovation Program of Shanghai Municipal Education Commission
(No.10ZZ131).

## References

  * [1] M. Ito, J. Phys. Soc. Jpn. 49 (1980) 771
  * [2] X. B. Hu and Y Li: J. Phys. A 24 (1991) 1979.
  * [3] M. Jimbo and T. Miwa: Publ. RIMS Kyoto Univ. 19 (1983) 943.
  * [4] V. G. Drinfeld and V. Sokolov: J. Sov. Math. 30 (1985) 1975.
  * [5] Q. P. Liu: Phys. Lett. A 277 (2000), 31.
  * [6] S. Q. Liu, X. B. Hu and Q. P. Liu, J. Phys. Soc. Jpn 75 (2006), 064004
  * [7] R. Hirota and J. Satsuma: Prog. Theor. Phys. 57 (1977) 797.
  * [8] R. Hirota: Direct methods in soliton theory (Springer-verlag, Berlin, 2004).
  * [9] X. B. Hu: P. A. Clarkson, J. Phys. A 28 (1995) 5009.
  * [10] X. B. Hu: C X Li, J. J. C. Nimmo, G. F. Yu, J. Phys. A, 38 (2005) 195.
  * [11] R Hirota and Y. Ohta: J. Phys. Soc. Jpn. 60 (1991) 798\.
  * [12] D. J. Zhang: J. Phys. Soc. Jpn. 71 (2002) 2649.
  * [13] K Sawada and T Kotera: Prog. Theor. Phys. 51 (1974) 1355.
  * [14] A. Nakamura, J. Phys. Soc. Jpn. 47, 1701-1705 (1979).
  * [15] A. Nakamura, J. Phys. Soc. Jpn. 48(1980), 1365.
  * [16] H. H. Dai, E. G. Fan and X. G. Geng, arxiv.org/pdf/nlin/0602015
  * [17] Y. C. Hon and E. G. Fan, Modern Phys Lett B, 22 (2008), 547.
  * [18] E. G. Fan and Y. C. Hon, Phys Rev E, 78 (2008), 036607.
  * [19] W. X. Ma, R. G. Zhou and L. Gao, Modern Phys. Lett. A, 21 (2009), 1677.
  * [20] E. G. Fan, J. Phys A, 42 (2009), 095206.
  * [21] A. S. Carstea, Nonlinearity 13 (2000), 1645.
  * [22] V. S. Vladimirov, Theor. Math. Phys. 59 (1984), 3.
  * [23] V. S. Vladimirov, Theor. Math. Phys. 60 (1984), 169\.
  * [24] D. Mumford, Tata Lectures on Theta II Progress in Mathmatics, Vol. 43 ( Boston: Birkhäuser, 1984).