Bilinear approach to the quasi-periodic wave solutions of supersymmetric
equations in superspace $\mathbb{R}_{\Lambda}^{2,1}$
Engui Fan 111 E-mail address<EMAIL_ADDRESS>
School of Mathematical Sciences and Key Laboratory of Mathematics for
Nonlinear Science,
Fudan University, Shanghai, 200433, P.R. China

Abstract

We devise a lucid and straightforward way for explicitly constructing quasi-
periodic wave solutions (also called multi-periodic wave solutions) of
supersymmetric equations in superspace $\mathbb{R}_{\Lambda}^{2,1}$ over two-
dimensional Grassmann algebra $G_{1}(\sigma)$. Once a nonlinear equation is
written in a bilinear form, its quasi-periodic wave solutions can be directly
obtained by using a formula. Moreover, properties of these solutions are
investigated in detail by analyzing their structures, plots and asymptotic
behaviors. The relations between the quasi-periodic wave solutions and soliton
solutions are rigorously established. It is shown that the soliton solutions
can be obtained only as limiting cases of the quasi-periodic wave solutions
under small amplitude limits in superspace $\mathbb{R}_{\Lambda}^{2,1}$. We
find that, in contrast to the purely bosonic case, there is an interesting
influencing band occurred among the quasi-periodic waves under the presence of
the Grassmann variable. The quasi-periodic waves are symmetric about the band
but collapse along with the band. Furthermore, the amplitudes of the quasi-
periodic waves increase as the waves move away from the band. The efficiency
of our proposed method can be demonstrated on a class variety of
supersymmetric equations such as those considered in this paper,
$\mathcal{N}=1$ supersymmetric KdV, Sawada-Kotera-Ramani and Ito’s equations,
as well as $\mathcal{N}=2$ supersymmetric KdV equation.
Keywords: supersymmetric equations; super-Hirota’s bilinear form; Riemann
theta function; quasi-periodic wave solutions; soliton solutions.
PACS numbers: 11.30.Pb; 05.45.Yv; 02.30.Gp; 45.10.-b.

1\. Introduction

The algebro-geometric solutions or finite gap solutions of nonlinear equations
were originally obtained on the KdV equation based on inverse spectral theory
and algebro-geometric method developed by pioneers such as Novikov, Dubrovin,
Mckean, Lax, Its and Matveev et al. [2]-[6] in the late 1970s. In fact, such a
solution is an expression written in terms of the Riemann theta functions.
Hence it is also called a quasi-periodic solution due to the quasi-periodicity
of the theta functions. By now this theory has been extended to a large class
of nonlinear integrable equations including sine-Gordon equation, Camassa-Holm
equation, Thirring model equation, Kadomtsev-Petviashvili equation, Ablowitz-
Ladik lattice and Toda lattice [7]-[17].

The quasi-periodic solutions have important applications in physics. For
instance, they can describe the nonlinear interaction of several modes. All
the main physical characteristics of the quasi-periodic solutions (wave
numbers, phase velocities, amplitudes of the interacting modes) are defined by
a compact Riemann surface. There are numerous applications of the finite-gap
integration theory in condensed matter physics, state physics and fluid
mechanics. For example, in peierls state, phonon produce a finite-gap
potential for electrons, and the peierls state is a lattice of solutions at
low densities of electrons [7]. A most famous mechanical system, the
Kowalewski top, was the focus of interest in the 19th century. The equation of
motion of the top can be solved through finite-gap theory [7]. A problem of
fundamental interest in fluid mechanics is to provide an accurate description
of waves on a water surface. The Kadomtsev-Petviashvili equation is known to
describe the evolution of waves in shallow water and admits a large family of
quasi-periodic solutions. Each solution has $N$ independent phases.
Experiments demonstrate the existence of genuinely two-dimensional shallow
water waves that are full periodic in two spatial directions and time. The
comparisons with experiments showed that the two-periodic wave solutions of
the KP equation describe shallow water waves with much accuracy [18, 19].

The algebro-geometric theory, however, needs Lax pairs and involves
complicated calculus on the Riemann surfaces. It is rather difficult to
directly determine the characteristic parameters of waves such as frequencies
and phase shifts for a function of given wave-numbers and amplitudes. On the
other hand, the bilinear derivative method developed by Hirota is a powerful
approach for constructing exact solution of nonlinear equations. Once a
nonlinear equation is written in bilinear forms by a dependent variable
transformation, then multi-soliton solutions are usually obtained [20]–[26].
It was based on Hirota forms that Nakamura proposed a convenient way to
construct a kind of quasi-periodic solutions of nonlinear equations [27, 28,
29], where the periodic wave solutions of the KdV equation and the Boussinesq
equation were obtained. Such a method indeed exhibits some advantages over
algebro-gometric methods. For example, it does not need any Lax pairs and
Riemann surface for the considered equation, allows the explicit construction
of multi-periodic wave solutions, only relies on the existence of the Hirota’s
bilinear form, as well as all parameters appearing in Riemann matrix are
arbitrary. Recently, further development was made to investigate the discrete
Toda lattice, (2+1)-dimensional Kadomtsev-Petviashvili equation and
Bogoyavlenskii’s breaking soliton equation [30]-[35]. Indeed there are some
differences between quasi-periodic solutions and algebro-geometric solutions.
A quasi-periodic solution needs not be an algebro-geometric one. Sometimes a
quasi-periodic solution may not correspond to any Riemann surface and is
generically associated with infinite bands, not just finitely-many, for
instance with a Riemann surface of infinite genus.

The concept of supersymmetry was originally introduced and developed for
applications in elementary particle physics thirty years ago [36]–[38]. It is
found that supersymmetry can be applied to a variety of problems such as
relativistic, non-relativistic physics and nuclear physics. In recent years,
supersymmetry has been a subject of considerable interest both in physics and
mathematics. The mathematical formulation of the supersymmetry is based on the
introduction of Grassmann variables along with the standard ones [39]. In a
such way, a number of well known mathematical physical equations have been
generalized into the supersymmetric analogues, such as supersymmetric versions
of sine-Gordon, KdV, KP hierarchy, Boussinesq, MKdV etc. [40]–[50]. It has
been shown that these supersymmetric integrable systems possess bi-Hamiltonian
structure, Painleve property, infinite many symmetries, Darboux
transformation, Backlund transformation, bilinear form and multi-soliton
solutions. The systematic bilinear transcription of supersymmetric equations
was introduced by Carstea [43, 44]. This required an extension of the Hirota’s
bilinear operator to supersymmetric case. Despite this bilinearization of
supersymmetric equations, the standard construction did not lead to malti-
soliton solutions. In recent years, Carsta, Liu, Ghosh et al. have done much
on the construction of soliton solutions of supersymmetric equations [43]–[50]
. However, the quasi-periodic solutions of the supersymmetric systems, which
can be considered as a generalization of the soliton solutions, are still not
available (both by algebro-geometric method and by bilinear methods or others)
to the knowledge of the author.

The motivation of this paper is to show how the quasi-periodic wave solutions
of nonlinear supersymmetric equations can be constructed with Hirota’s
bilinear method in superspace. To achieve this aim, we devise a Riemannn theta
function formula, which actually provides us a lucid and straightforward way
for applying in a class of nonlinear 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. This method considerably improves the key steps of the existing
methods, where repetitive recursion and computation must be preformed for each
equation [30]-[35]. As illustrative example, we shall construct quasi-periodic
wave solutions to the $\mathcal{N}=1$ supersymmetric Sawada-Kotera-Ramani
equation and $\mathcal{N}=2$ supersymmetric KdV equation.

The organization of this paper is as follows. In section 2, we briefly give
some properties on superspace and super-Hirota bilinear operators. In section
3, we introduce a super Riemann theta function and discuss its quasi-
periodicity. In particular, we provide a key formula for constructing periodic
wave solutions of supersymmetric equations. As applications of our method, in
section 4 and section 5, we construct one- and two-periodic wave solutions to
the $\mathcal{N}=1$ supersymmetric Sawada-Kotera-Ramani equation and
$\mathcal{N}=2$ supersymmetric KdV equation, respectively. The propagation of
the quasi-periodic waves are displayed with help of software Mathematica. In
addition, we further present a simple and effective limiting procedure to
analyze asymptotic behavior of the periodic wave solutions. It is rigorously
shown that the quasi-periodic wave solutions tend to the soliton solutions
under small amplitude limits. At last, we briefly discuss the conditions on
the construction of multi-periodic wave solutions of supersymmetric equations
in section 6.
2\. Super space and super-Hirota bilinear form

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 [51, 52].

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

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

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

A function $f(\boldsymbol{x}):\mathbb{R}_{\Lambda}^{m,n}\rightarrow\Lambda$ is
said to be superdifferentiable at the point $x\in\mathbb{R}_{\Lambda}^{m,n}$,
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

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

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:

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

In this paper, we consider functions with two ordinary even variables $x,t$
and a Grassmann odd variable $\theta$. The associated space
$\mathbb{R}_{\Lambda}^{2,1}=\Lambda_{0}^{2}\times\Lambda_{1}$ (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$G_{1}(\sigma)=2$.
Therefore, any $\mu\in G_{1,1}$ have the form $\mu=\beta\sigma,\
\beta\in\mathbb{C}$. Under traveling wave frame in space
$\mathbb{R}_{\Lambda}^{2,1}$, the phase variable should have the form

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

For the functions
$f(x,t,\theta),g(x,t,\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda$,
the Hirota bilinear differential operators $D_{x}$ and $D_{t}$ about ordinary
variables $x,t$ are defined by

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

The super-Hirota bilinear operator is defined as [43]

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

where the differential operator
$\mathfrak{D}=\partial_{\theta}+\theta\partial_{x}$ is the super derivative,
and 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 $k$ ($[k]\leq k\leq[k]+1$).

We point out here that throughout this paper the natural number $N$ (which
will denote powers, the number of phase variables, number of terms etc.) is
different form $\mathcal{N}$ which is related to supersymmetry or superspace.

Proposition 1. Suppose that functions
$f(x,t,\theta),g(x,t,\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\Lambda$,
then Hirota bilinear operators $D_{x},D_{t}$ and super-Hirota bilinear
operator $S_{x}$ have properties [43]

$\displaystyle S_{x}^{2N}f\cdot g=D_{x}^{N}f\cdot g,$ $\displaystyle
D_{x}^{m}D_{t}^{n}e^{\xi_{1}}\cdot
e^{\xi_{2}}=(\alpha_{1}-\alpha_{2})^{m}(\omega_{1}-\omega_{2})^{n}e^{\xi_{1}+\xi_{2}},$
$\displaystyle S_{x}e^{\xi_{1}}\cdot
e^{\xi_{2}}=[\sigma(\beta_{1}-\beta_{2})+\theta(\alpha_{1}-\alpha_{2})]e^{\xi_{1}+\xi_{2}},$

where $\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j}$, $\
\alpha_{j},\omega_{j},\sigma_{j},\delta_{j}\in\Lambda_{0}$ are parameters,
$j=1,2$. In fact, the third formula above is defined with an accuracy to
within an addition constant of the $c\sigma\in^{\perp}\Lambda_{1}$. More
generally, we have

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

where $F(S_{t},D_{x},D_{t})$ is a polynomial about operators $S_{t},D_{x}$ and
$D_{t}$. This properties are useful in deriving Hirota’s bilinear form and
constructing the quasi-periodic wave solutions of the supersymmetric
equations.
3\. Super Riemann theta function and addition formulae

In the following, we introduce a multi-dimensional super Riemann theta
function on superspace $\mathbb{R}_{\Lambda}^{2,1}$ and discuss its quasi-
periodicity, which plays a central role in the construction of quasi-periodic
solutions of supersymmetric equations. The multi-dimensional Riemann theta
function reads

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

Here the integer value vector
$\boldsymbol{n}=(n_{1},\cdots,n_{N})^{T}\in\mathbb{Z}^{N}$, complex parameter
vectors
$\boldsymbol{s}=(s_{1},\cdots,s_{N})^{T},\boldsymbol{\varepsilon}=(\varepsilon_{1},\cdots,\varepsilon_{N})^{T}\in\mathbb{{C}}^{N}$.
The complex phase variables $\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})^{T},\
\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j}$, $\
\alpha_{j},\omega_{j},\beta_{j},\delta_{j}\in\Lambda_{0}$, $j=1,2,\cdots,N$,
where $x,t$ are ordinary variables and $\theta$ is Grassmann variable.
Moreover, for two vectors $\boldsymbol{f}=(f_{1},\cdots,f_{N})^{T}$ and
$\boldsymbol{g}=(g_{1},\cdots,g_{N})^{T}$, their inner product is defined by

$\langle\boldsymbol{f},\boldsymbol{g}\rangle=f_{1}g_{1}+f_{2}g_{2}+\cdots+f_{N}g_{N}.$

The $\boldsymbol{\tau}=(\tau_{ij})$ is a positive definite and real-valued
symmetric $N\times N$ matrix, which is independent of $\theta$ and $\sigma$ in
superspace $\mathbb{R}_{\Lambda}^{2,1}$. The entries $\tau_{ij}$ of the period
matrix $\boldsymbol{\tau}$ can be considered as free parameters of the theta
function (3.1).

In this paper, we take the $\tau$ to be pure imaginary matrix to make the
theta function (3.1) real-valued. In the definition of the theta function
(3.1), for the case $\boldsymbol{s}=\boldsymbol{\varepsilon}=\boldsymbol{0}$,
hereafter we use
$\vartheta(\boldsymbol{\xi},{\boldsymbol{\tau}})=\vartheta(\boldsymbol{\xi},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})$
for simplicity. Moreover, we have
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})=\vartheta(\boldsymbol{\xi}+\boldsymbol{\varepsilon},\boldsymbol{\tau})$.
It is obvious that the Riemann theta function (3.1) converges absolutely and
superdifferentiable on superspace $\mathbb{R}_{\Lambda}^{2,1}$.

Remark 1. The period matrix $\boldsymbol{\tau}$ here is different form
algebro-geometric theory discussed in [2]-[34], where it is usually
constructed via a compact Riemann surface $\Gamma$ of genus $N\in\mathbb{N}$.
We take two sets of regular cycle paths: $a_{1},a_{2},\cdots,a_{N}$;
$b_{1},b_{2},\cdots,b_{N}$ on $\Gamma$ in such a way that the intersection
numbers of cycles satisfies

$a_{k}\circ a_{j}=b_{k}\circ b_{j}=0,a_{k}\circ b_{j}=\delta_{kj},\ \
k,j=1,\cdots,N.$

We choose the normalized holomorphic differentials $\omega_{j},j=1,\cdots,N$
on $\Gamma$ and let

$a_{jk}=\int_{a_{k}}{{\omega}}_{j},\ \ b_{jk}=\int_{b_{k}}{{\omega}}_{j},$

then $N\times N$ matrices $\boldsymbol{A}=(a_{jk})$ and
$\boldsymbol{B}=(b_{jk})$ are invertible. Define matrices $\boldsymbol{C}$ and
$\boldsymbol{\tau}$ by

$\boldsymbol{C}=(c_{jk})=\boldsymbol{A}^{-1},\ \
\boldsymbol{\tau}=(\tau_{jk})=\boldsymbol{A}^{-1}\boldsymbol{B}.$

It is can be shown that the matrix $\boldsymbol{\tau}$ is symmetric and has
positive definite imaginary part. However, we see that the entries in such a
matrix $\boldsymbol{\tau}$ are not free and difficult to be explicitly given.
$\square$

Definition 2. A function $g(\boldsymbol{x},t)$ on
$\mathbb{C}^{N}\times\mathbb{C}$ is said to be quasi-periodic in $t$ with
fundamental periods $T_{1},\cdots,T_{k}\in\mathbb{C}$ if $T_{1},\cdots,T_{k}$
are linearly dependent over $\mathbb{Z}$ and there exists a function
$G(\boldsymbol{x},t)\in\mathbb{C}^{N}\times\mathbb{C}^{k}$ such that

$G(\boldsymbol{x},y_{1},\cdots,y_{j}+T_{j},\cdots,y_{k})=G(\boldsymbol{x},y_{1},\cdots,y_{j},\cdots,y_{k}),\
\ {\rm for\ all}\ y_{j}\in\mathbb{C},\ j=1,\cdots,k.$
$G(\boldsymbol{x},t,\cdots,t,\cdots,t)=g(x,t).$

In particular, $g(\boldsymbol{x},t)$ becomes periodic with $T$ if and only if
$T_{j}=m_{j}T$. $\square$

Let’s first see the periodicity of the theta function
$\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$.

Proposition 2. [53] Let $\boldsymbol{e_{j}}$ be the $j-$th column of $N\times
N$ identity matrix $I_{N}$; ${\tau_{j}}$ be the $j-$th column of
$\boldsymbol{\tau}$, and $\tau_{jj}$ the $(j,j)$-entry of $\boldsymbol{\tau}$.
Then the theta function $\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$ has
the periodic properties

$None$
$\displaystyle\vartheta(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}},\boldsymbol{\tau})=\exp(-2\pi
i\xi_{j}+\pi\tau_{jj})\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}).$

The theta function $\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$ which
satisfies the condition (4.4) is called a multiplicative function. We regard
the vectors $\\{\boldsymbol{e_{j}},\ \ j=1,\cdots,N\\}$ and
$\\{i\boldsymbol{\tau_{j}},\ \ j=1,\cdots,N\\}$ as periods of the theta
function $\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$ with multipliers $1$
and $\exp({-2\pi i\xi_{j}+\pi\tau_{jj}})$, respectively. Here, only the first
$N$ vectors are actually periods of the theta function
$\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$, but the last $N$ vectors are
the periods of the functions
$\partial^{2}_{\xi_{k},\xi_{l}}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$
and
$\partial_{\xi_{k}}\ln[\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau})/\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau})],\
k,l=1,\cdots,N$.

Proposition 3. Let $\boldsymbol{e_{j}}$ and $\boldsymbol{\tau_{j}}$ be defined
as above proposition 2. The meromorphic functions $f(\boldsymbol{\xi})$ on
$\mathbb{R}_{\Lambda}^{2,1}$ are as follow

$\displaystyle(i)\ \ \ \ \
f(\boldsymbol{\xi})=\partial_{\xi_{k}\xi_{l}}^{2}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}),\
\ \boldsymbol{\xi}\in C^{N},\ \ \ k,l=1,\cdots,N,$ $\displaystyle(ii)\ \ \ \
f(\boldsymbol{\xi})=\partial_{\xi_{k}}\ln\frac{\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau})},\
\ \boldsymbol{\xi},\ \boldsymbol{e},\ \boldsymbol{h}\in C^{N},\ \
j=1,\cdots,N.$

then in all two cases (i) and (ii), it holds that

$None$ $\displaystyle
f(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}})=f(\boldsymbol{\xi}),\
\ \ \boldsymbol{\xi}\in C^{N},\ \ \ j=1,\cdots,N.$

Proof. By using (3.2), it is easy to see that

$\displaystyle\frac{\vartheta^{\prime}_{\xi_{k}}(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}},\boldsymbol{\tau})}=-2\pi
i\delta_{jk}+\frac{\vartheta^{\prime}_{\xi_{k}}(\boldsymbol{\xi},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})},$

or equivalently

$None$
$\displaystyle\partial_{\xi_{k}}\ln\vartheta(\boldsymbol{\xi}+\boldsymbol{e_{j}}+i\boldsymbol{\tau_{j}},\boldsymbol{\tau})=-2\pi
i\delta_{jk}+\partial_{\xi_{k}}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}).$

Differentiating (3.4) with respective to $\xi_{l}$ again immediately proves
the formula (3.3) for the case (i). The formula (3.4) can be proved for the
case (ii) in a similar manner. $\square$

Theorem 1. Suppose that
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})$
and
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
are two Riemann theta functions on $\mathbb{R}_{\Lambda}^{2,1}$, in which
$\boldsymbol{\varepsilon}=(\varepsilon_{1},\dots,\varepsilon_{N})$,
$\boldsymbol{\varepsilon^{\prime}}=(\varepsilon_{1}^{\prime},\dots,\varepsilon_{N}^{\prime})$,
and $\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})$,
$\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j},\ \
j=1,2,\cdots,N$. Then Hirota bilinear operators $D_{x},D_{t}$ and super-Hirota
bilinear operator $S_{x}$ exhibit the following perfect properties when they
act on a pair of theta functions

$None$ $\displaystyle
D_{x}\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
$\displaystyle=\left[\sum_{\boldsymbol{\mu}}\partial_{x}\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon},-\boldsymbol{\mu}/2|2\boldsymbol{\tau})|_{\boldsymbol{\xi}=\boldsymbol{0}}\right]\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}+\boldsymbol{\varepsilon},\boldsymbol{\mu}/2|2\boldsymbol{\tau}),$
$None$ $\displaystyle
S_{x}\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
$\displaystyle=\left[\sum_{\boldsymbol{\mu}}\mathfrak{D}_{x}\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon},-\boldsymbol{\mu}/2|2\boldsymbol{\tau})|_{\boldsymbol{\xi}=\boldsymbol{0}}\right]\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}+\boldsymbol{\varepsilon},\boldsymbol{\mu}/2|2\boldsymbol{\tau}),$

where $\boldsymbol{\mu}=(\mu_{1},\cdots,\mu_{N})$, and the notation
$\sum_{\boldsymbol{\mu}}$ represents $2^{N}$ different transformations
corresponding to all possible combinations $\mu_{1}=0,1;\cdots;\mu_{N}=0,1$.

In general, for a polynomial operator $F(S_{x},D_{x},D_{t})$ with respect to
$S_{x},D_{x}$ and $D_{t}$, we have the following useful formula

$None$ $\displaystyle
F(S_{x},D_{x},D_{t})\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})=\left[\sum_{\boldsymbol{\mu}}C(\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\varepsilon},\boldsymbol{\mu})\right]\vartheta(2\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}}+\boldsymbol{\varepsilon},\boldsymbol{\mu}/2|2\boldsymbol{\tau}),$

in which, explicitly

$None$ $\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}F(\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle-2\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right].$

where we denote $\boldsymbol{\mathcal{M}}=(4\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\alpha}\rangle,\ 4\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\omega}\rangle,\ 4\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\sigma}+\theta\boldsymbol{\alpha}\rangle).$

Proof. For simplicity we prove the formula (3.6) for one-dimensional case. The
proof for $N$-dimensional case can be performed simply by replacing one-
dimensional vectors by $N$-dimensional ones.

Making use of Proposition 1, we obtain the relation

$\displaystyle\Delta\equiv
S_{x}\vartheta(\xi,\varepsilon^{\prime},0|{\tau})\cdot\vartheta(\xi,\varepsilon,0|{\tau})$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}\mathfrak{D}_{x}\exp\\{2\pi
im^{\prime}(\xi+\varepsilon^{\prime})-\pi m^{\prime
2}{\tau}\\}\cdot\exp\\{2\pi im(\xi+\varepsilon)-\pi m^{2}{\tau}\\},$
$\displaystyle=\sum_{m^{\prime},m\in\mathbb{Z}}2\pi
i(\beta\sigma+\theta\alpha)(m^{\prime}-m)\exp\left\\{2\pi
i(m^{\prime}+m)\xi-2\pi
i(m^{\prime}\varepsilon^{\prime}+m\varepsilon)-\pi{\tau}[m^{\prime
2}+m^{2}]\right\\}$

By shifting sum index as $m=l^{\prime}-m^{\prime}$, then

$\displaystyle\Delta=\sum_{l^{\prime},m^{\prime}\in\mathbb{Z}}2\pi
i(\sigma+\theta\alpha)(2m^{\prime}-l^{\prime})\exp\left\\{2\pi
il^{\prime}\xi-2\pi
i[m^{\prime}\varepsilon^{\prime}+(l^{\prime}-m^{\prime})\varepsilon]-\pi{\tau}[m^{\prime
2}+(l^{\prime}-m^{\prime})^{2}]\right\\}$
$\displaystyle\stackrel{{\scriptstyle l^{\prime}=2l+\mu}}{{=}}\sum_{\mu=0,1}\
\ \sum_{l,m^{\prime}\in\mathbb{Z}}2\pi
i(\beta\sigma+\theta\alpha)(2m^{\prime}-2l-\mu)\exp\\{4\pi i\xi(l+\mu/2)$
$\displaystyle\ \ \ \ \ -2\pi
i[m^{\prime}\varepsilon^{\prime}-(m^{\prime}-2l-\mu)\varepsilon]-\pi[m^{\prime
2}+(m^{\prime}-2l-\mu)^{2}]{\tau}\\}$

Finally letting $m^{\prime}=n+l$, we conclude that

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

by using the following relations

$\displaystyle n+l=(n-\mu/2)+(l+\mu/2),\ \ n-l-\mu=(n-\mu/2)-(l+\mu/2).$

In a similar way, we can prove the formula (3.5). The formula (3.7) follows
from (3.5) and (3.6). $\Box$

Remark 2. The formulae (3.7) and (3.8) show that if the following equations
are satisfied

$None$
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=0,$

for all possible combinations $\mu_{1}=0,1;\mu_{2}=0,1;\cdots;\mu_{N}=0,1$, in
other word, all such combinations are solutions of equation (3.9), then
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})$
and
$\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})$
are $N$-periodic wave solutions of the bilinear equation

$F(S_{t},D_{x},D_{t})\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{0}|\boldsymbol{\tau})\cdot\vartheta(\boldsymbol{\xi},\boldsymbol{\varepsilon},\boldsymbol{0}|\boldsymbol{\tau})=0.$

We call the formula (3.9) constraint equations, whose number is $2^{N}$. This
formula actually provides us an unified approach to construct multi-periodic
wave solutions for supersymmetric equations. Once a supersymmetric equation is
written bilinear forms, then its multi-periodic wave solutions can be directly
obtained by solving system (3.9).

Theorem 2. Let
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
and $F(S_{x},D_{x},D_{t})$ be given in Theorem 1, and make a choice such that
$\varepsilon_{j}^{\prime}-\varepsilon_{j}=\pm 1/2,\ j=1,\cdots,N$. Then

(i) If $F(S_{x},D_{x},D_{t})$ is an even function in the form

$F(-S_{x},-D_{x},-D_{t})=F(S_{x},D_{x},D_{t}),$

then
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
vanishes automatically for the case when $\sum_{j=1}^{N}\mu_{j}$ is an odd
number, namely

$None$
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})|_{\boldsymbol{\mu}}=0,\
\ {\rm for}\ \ \ \sum_{j=1}^{N}\mu_{j}=1,\ {\rm mod}\ 2.$

(ii) If $F(S_{x},D_{x},D_{t})$ is an odd function in the form

$F(-S_{x},-D_{x},-D_{t})=-F(S_{x},D_{x},D_{t}),$

then
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
vanishes automatically for the case when $\sum_{j=1}^{N}\mu_{j}$ is an even
number, namely

$None$
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})|_{\boldsymbol{\mu}}=0,\
\ {\rm for}\ \sum_{j=1}^{N}\mu_{j}=0,\ {\rm mod}\ 2.$

Proof. We are going to consider the case where $F(S_{x},D_{x},D_{t})$ is an
even function and prove the formula (3.9). The formula (3.11) is analogous.
Making transformation $\boldsymbol{n}=-\boldsymbol{\bar{n}}+\boldsymbol{\mu}\
(\boldsymbol{\bar{n}}=(\bar{n}_{1},\cdots,\bar{n}_{N}),\
\bar{n}_{j}\in\mathbb{Z},\ \ j=1,\cdots,N.$ ), and noting
$F(S_{x},D_{x},D_{t})$ is even, we then deduce that

$\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{\bar{n}}\in\mathbb{Z}^{N}}F(-\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{\bar{n}}-\boldsymbol{\mu}/2),\boldsymbol{\bar{n}}-\boldsymbol{\mu}/2\rangle+2\pi
i\langle\boldsymbol{\bar{n}}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\exp\left(4\pi
i\langle\boldsymbol{\bar{n}}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon}\rangle\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\exp\left(\pm
2\pi i\sum_{j=1}^{N}\bar{n}_{j}\right)\exp\left(\pm\pi
i\sum_{j=1}^{N}\mu_{j}\right)$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\exp(\pm\pi
i)=-C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu}),$

which proves the formula (3.10). $\square$

Corollary 1.  Let $\varepsilon_{j}^{\prime}-\varepsilon_{j}=\pm 1/2,\
j=1,\cdots,N$. Assume $F(S_{x},D_{x},D_{t})$ is a linear combination of even
and odd functions

$F(S_{x},D_{x},D_{t})=F_{1}(S_{x},D_{x},D_{t})+F_{2}(S_{x},D_{x},D_{t}),$

where $F_{1}(S_{x},D_{x},D_{t})$ is even and $F_{2}(S_{x},D_{x},D_{t})$ is
odd. In addition,
$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
corresponding (3.8) is given by

$C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})+C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu}),$

where

$C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}F_{1}(\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle-2\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right],$
$C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{N}}F_{2}(\boldsymbol{\mathcal{M}})\exp\left[-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle-2\pi
i\langle\boldsymbol{n}-\boldsymbol{\mu}/2,\boldsymbol{\varepsilon^{\prime}}-\boldsymbol{\varepsilon})\right].$

Then

$None$ $\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\
\ {\rm for}\ \ \ \sum_{j=1}^{N}\mu_{j}=1,\ {\rm mod}\ 2,$ $None$
$\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu}),\
\ {\rm for}\ \sum_{j=1}^{N}\mu_{j}=0,\ {\rm mod}\ 2.$

Proof. In a similar to the proof of Theorem 2, shifting sum index as
$\boldsymbol{n}=-\boldsymbol{\bar{n}}+\boldsymbol{\mu}$, and using
$F_{1}(S_{x},D_{x},D_{t})$ even and $F_{2}(S_{x},D_{x},D_{t})$ odd, we have

$None$ $\displaystyle
C(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})+C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
=\left[C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})-C_{2}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})\right]\exp\left(\pm\pi
i\sum_{j=1}^{N}\mu_{j}\right).$

Then for $\sum_{j=1}^{N}\mu_{j}=1,\ {\rm mod}\ 2$, the equation (3.15) gives

$C_{1}(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon^{\prime}},\boldsymbol{\mu})=0,$

which implies the formula (3.12). The formula (3.13) is analogous. $\square$

The theorem 2 and corollary 1 are very useful to deal with coupled super-
Hirota’s bilinear equations, which will be seen in the following section 5.

By introducing differential operators

$\displaystyle\nabla=(\partial_{\xi_{1}},\partial_{\xi_{2}},\cdots,\partial_{\xi_{N}}),$
$\displaystyle\partial_{x}=\alpha_{1}\partial_{\xi_{1}}+\alpha_{2}\partial_{\xi_{2}}+\cdots+\alpha_{N}\partial_{\xi_{N}}=\boldsymbol{\alpha}\cdot\nabla,$
$\displaystyle\partial_{t}=\omega_{1}\partial_{\xi_{1}}+\omega_{2}\partial_{\xi_{2}}+\cdots+\omega_{N}\partial_{\xi_{N}}=\boldsymbol{\omega}\cdot\nabla,$
$\displaystyle\mathfrak{D}=(\sigma_{1}+\theta\alpha_{1})\partial_{\xi_{1}}+(\sigma_{2}+\theta\alpha_{2})\partial_{\xi_{2}}+\cdots+(\sigma_{N}+\theta\alpha_{N})\partial_{\xi_{N}}=(\boldsymbol{\sigma}+\theta\boldsymbol{\alpha})\cdot\nabla,$

then we have

$None$
$\displaystyle\mathfrak{D}\partial_{x}^{k}\partial_{t}^{j}\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=[(\boldsymbol{\sigma}+\theta\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\alpha}\cdot\nabla)^{k}(\boldsymbol{\omega}\cdot\nabla)^{j}\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$
$\displaystyle=(\boldsymbol{\sigma}\cdot\nabla)(\boldsymbol{\alpha}\cdot\nabla)^{k}(\boldsymbol{\omega}\cdot\nabla)^{j}\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})+\theta(\boldsymbol{\alpha}\cdot\nabla)^{k+1}(\boldsymbol{\omega}\cdot\nabla)^{j}\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}),$
$\displaystyle j,k=0,1,\cdots.$

4\. $\mathcal{N}=1$ supersymmetric Sawada-Kotera-Ramani equation

The supersymmetric Sawada-Kotera-Ramani equation takes the form

$None$
$\displaystyle\phi_{t}+\mathfrak{D}^{2}\left[10(\mathfrak{D}\phi)\mathfrak{D}^{4}\phi+5(\mathfrak{D}^{5}\phi)\phi+15(\mathfrak{D}\phi)^{2}\phi\right]+\mathfrak{D}^{10}\phi=0,$

where
$\phi=\phi(x,t,\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is fermionic superfield depending on usual independent variable $x$, $t$ and
Grassmann variable $\theta$. This equation was first proposed by Carstea [43].
The soliton solutions, Lax representation and infinite conserved quantities of
the equation have been further obtained recently [56, 57]. Here we are
interested in quasi-periodic wave solutions to the supersymmetric equation
(4.1). We will show that the soliton solutions can be obtained as limiting
case of these quasi-periodic solutions.

To apply the Hirota bilinear method in superspace for constructing multi-
periodic wave solutions of the equation (4.1), we hope more fee variables and
consider a general variable transformation

$None$ $\phi=\phi_{0}+2\mathfrak{D}^{3}\ln f(x,t,\theta),$

where
$f(x,t,\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{1,0}$
is an even superfield and
$\phi_{0}=\phi_{0}(\theta):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is an odd special solution of the equation (4.1). Substituting (4.2) into
(4.1) and integrating with respect to $x$, we then get the following super
Hirota’s bilinear form

$None$ $\displaystyle F(S_{x},D_{x},D_{t})f\cdot
f=(S_{x}D_{t}+S_{x}D_{x}^{5}+15\phi_{0}^{2}D_{x}^{2}+5\phi_{0}D_{x}^{4}+c)f\cdot
f=0,$

where
$c=c(\theta,t):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is an odd integration constant. In the special case when $\phi_{0}=c=0$,
starting from the bilinear equation (4.3), it is easy to find that the
equation (4.1) admits one-soliton solution (also called one-supersoliton
solution) in superspace $\mathbb{R}_{\Lambda}^{2,1}$ over two-dimensional
Grassmann algebra $G_{1}(\sigma)$

$None$ $\phi_{1}=2\mathfrak{D}^{3}\ln(1+e^{\eta}),$

with phase variable $\eta=px-p^{5}t+q\theta\sigma+r$ with
$p,q,r\in\Lambda_{0}$. While two-soliton solution (super two-soliton solution)
reads

$None$
$\phi_{2}=2\mathfrak{D}^{3}\ln(1+e^{\eta_{1}}+e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}}),$

with $\eta_{j}=p_{j}x-p_{j}^{3}t+q_{j}\theta\sigma+r_{j},\ \ j=1,2$ and

$\displaystyle
e^{A_{12}}=\frac{(p_{1}-p_{2})^{2}(p_{1}^{2}-p_{1}p_{2}+p_{2}^{2})}{(p_{1}+p_{2})^{2}(p_{1}^{2}+p_{1}p_{2}+p_{2}^{2})}\left(1+2\theta\sigma\frac{p_{1}q_{2}-p_{2}q_{1}}{p_{1}-p_{2}}\right),$

and here $p_{j},q_{j},r_{j}\in\Lambda_{0},j=1,2$ are free constants.

Next, we turn to see the periodicity of the solution (4.2), the function $f$
is chosen to be a Riemann theta function, namely,

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

where phase variable $\xi$ is taken as the form
$\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})^{T}$,
$\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j},\ \
j=1,2,\cdots,N.$ With Proposition 3, we refer to

$\phi=\phi_{0}+2\sum_{k,l=1}^{N}\alpha_{k}(\beta_{l}\sigma+\theta\alpha_{l})\partial_{\xi_{k}\xi_{l}}^{2}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}),$

which shows that the solution $\phi$ is indeed a quasi-periodic function with
$2N$ fundamental periods $\\{\boldsymbol{e_{j}},\ \ j=1,\cdots,N\\}$ and
$\\{i\boldsymbol{\tau}_{j},\ \ j=1,\cdots,N\\}$. The quasi-periodic means that
$\phi$ is periodic in each of the $N$ phases $\\{\xi_{j},\ \ j=1,\cdots,N\\}$,
if the other $N-1$ phases are fixed.
3.1. One-periodic waves and asymptotic analysis

We first consider one-periodic wave solutions of the equation (4.1). As a
simple case of the theta function (3.1) when $N=1,s=\varepsilon=0$, we take
$f$ as

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

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

Next, we let the Riemann theta function (4.6) be a solution of the bilinear
equation (4.3). By using Theorem 1 and the formula (3.9), the following two
equations ( corresponding to $\mu=0$ and $1$ respectively) should be satisfied

$None$
$\displaystyle\sum_{n\in\mathbb{Z}}\\{-(4\pi(n-\mu/2))^{2}(\beta\sigma+\theta\alpha)\omega-(4\pi(n-\mu/2))^{6}(\beta\sigma+\theta\alpha)\alpha^{5}-15(4\pi(n-\mu/2))^{2}\alpha^{2}\phi_{0}^{2}$
$\displaystyle+5(4\pi(n-\mu/2))^{4}\alpha^{4}\phi_{0}+c\\}\exp(-2\pi\tau(n-\mu/2)^{2})=0,\
\ \mu=0,1.$

We introduce the notations by

$\begin{aligned}
&\lambda=e^{-\pi\tau/2},\quad\vartheta_{1}(\xi,\lambda)=\vartheta(2\mathbf{\xi},0,0|2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{4n^{2}}\exp(4i\pi
n\xi),\\\
&\vartheta_{2}(\xi,\lambda)=\vartheta(2\xi,0,-1/2|2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{(2n-1)^{2}}\exp[2i\pi(2n-1)\xi],\end{aligned})$

the equation (4.7) can be written as a linear system about $\omega,c$

$None$
$\displaystyle(\beta\sigma+\theta\alpha)\vartheta_{j}^{\prime\prime}\omega+(\beta\sigma+\theta\alpha)\alpha^{5}\vartheta_{j}^{(6)}+3\alpha^{2}\vartheta_{j}^{\prime\prime}\phi_{0}^{2}+5\alpha^{4}\vartheta_{j}^{(4)}\phi_{0}+\vartheta_{j}c=0,\
\ j=1,\ 2,$

where $\omega\in\Lambda_{0}$ is even and $c,\phi_{0}\in\Lambda_{1}$ are odd,
and we have denoted the derivative value of $\vartheta_{j}(\xi,\lambda)$ at
$\xi=0$ by simple notations

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

Moreover, we see that the functions $\vartheta_{j}$ and their derivatives are
independent of Grassmann variable $\theta$ and anticommuting number $\sigma$.

We take $\phi_{0}=0$ for the simplicity. It is obvious that the coefficient
determinant of the system (4.8) is nonzero and
$(\alpha^{5}\vartheta_{1}^{(6)},\alpha^{5}\vartheta_{2}^{(6)})^{T}\not=0$,
therefore the system (4.8) admits a solution

$None$
$\displaystyle\omega=\frac{\alpha^{5}(\vartheta_{2}^{(6)}\vartheta_{1}-\vartheta_{1}^{(6)}\vartheta_{2})}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}},\
\ \
b_{1}=\frac{\alpha^{5}(\beta\sigma+\theta\alpha)(\vartheta_{2}^{(6)}\vartheta_{1}^{\prime\prime}-\vartheta_{1}^{(6)}\vartheta_{2}^{\prime\prime})}{\vartheta_{1}^{\prime\prime}\vartheta_{2}-\vartheta_{2}^{\prime\prime}\vartheta_{1}},$

where $\omega$ is independent of Grassmann variable $\theta$ and auticommuting
number $\sigma$, and parameter $\alpha$ is free.

In this way, a one-periodic wave solution of the equation (4.1) is explicitly
obtained by

$None$ $\phi=2\mathfrak{D}^{3}\ln\vartheta(\xi,\tau),$

with the theta function $\vartheta(\xi,\tau)$ given by (4.6) and parameters
$\omega$, $c$ by (4.9), while other parameters
$\alpha,\beta,\tau,\delta\in\Lambda_{0}$ are free. Among them, the three
parameters $\alpha$ and $\tau$ completely dominate a one-periodic wave.

In summary, one-periodic wave (4.10) possesses the following features:

(i) It is one-dimensional, i.e. there is a single phase variable $\xi$.
Moreover, it has two fundamental periods $1$ and $i\tau$ in phase variable
$\xi$, but it need not to be periodic in $x$, $t$ and $\theta$ directions.

(ii) It can be viewed as a parallel superposition of overlapping one-soliton
waves, placed one period apart ( see $(a)$ and $(b)$ in Figure 1 ).

(iii) Different form the purely bosonic case, it is observed shows that there
is an influencing band among the one-periodic waves under the presence of the
Grassmann variable (in contour plot, the bright hexagons are crests and the
dark hexagons are troughs). The one-periodic waves are symmetric about the
band but collapse along with the band. Furthermore, the amplitudes of the
quasi-periodic waves increase as the waves move away from the band ( see $(a)$
and $(b)$ in Figure 1 ).

(iv) The quasi-periodic wave will degenerate to pure bosonic quasi-periodic
wave when $\theta$ becomes small ( see Figure 2 ).

$(b)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$

Figure 1. A one-periodic wave for $\mathcal{N}=1$ supersymmetric Sawada-
Kotera-Ramani equation with parameters: $\alpha=0.1,$ $\tau=32,\sigma=0.013$.
(a) Perspective view of wave. (b) Overhead view of wave, with contour plot
shown.

$(b)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$

Figure 2. A purely bosonic case of one-periodic wave to $\mathcal{N}=1$
supersymmetric Sawada-Kotera-Ramani equation with parameters: $\alpha=0.1,$
$\tau=32,\ \sigma=0.013,\ \theta=0.00000001$. (a) Perspective view of wave.
(b) Overhead view of wave, with contour plot shown.

In the following, we further consider asymptotic properties of the one-
periodic wave solution. Interestingly, the relation between the one-periodic
wave solution (4.10) and the one-soliton solution (4.4) can be established as
follows.

Theorem 3. Suppose that $\omega$ and $c$ are given by (4.9), and for the one-
periodic wave solution (4.10), we let

$None$ $\alpha=\frac{p}{2\pi i},\ \ \beta=\frac{q}{2\pi i},\ \
\delta=\frac{r+\pi\tau}{2\pi i},$

where the $p,q$ and $r$ are given in (4.4). Then we have the following
asymptotic properties

$\displaystyle c\longrightarrow 0,\ \ 2\pi
i\xi-\pi\tau\longrightarrow\eta=px-p^{5}t+q\theta\sigma+r,$
$\displaystyle\vartheta(\xi,\tau)\longrightarrow 1+e^{\eta},\ \ {\rm as}\ \
\lambda\rightarrow 0.$

In other words, the one-periodic solution (4.10) tends to the soliton solution
(4.4) under a small amplitude limit, that is,

$None$ $\phi\longrightarrow\phi_{1},\ \ {\rm as}\ \ \lambda\rightarrow 0.$

Proof. Here we will directly use the system (4.8) to analyze asymptotic
properties of one-periodic solution (4.10), which is more simple and effective
than our original method by solving the system [30]-[34]. Since the
coefficients of system (4.8) are power series about $\lambda$, its solution
$(\omega,c)$ also should be a series about $\lambda$. We explicitly expand the
coefficients of system (4.8) as follows

$None$ $\displaystyle\vartheta_{1}=1+2\lambda^{4}+\cdots,\quad\
\vartheta_{1}^{\prime\prime}=-32\pi^{2}\lambda^{4}+\cdots,$
$\displaystyle\vartheta_{1}^{(6)}=-8192\pi^{6}\lambda^{4}+\cdots,\ \
\vartheta_{2}=2\lambda+2\lambda^{9}+\cdots$
$\displaystyle\vartheta_{2}^{\prime\prime}=-8\pi^{2}\lambda+\cdots,\ \ \
\vartheta_{1}^{(6)}=-128\pi^{6}\lambda+\cdots.$

Let the solution of the system (4.8) be of the form

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

Substituting the expansions (4.13) and (4.14) into the system (4.8) (the
second equation is divided by $\lambda$ ) and letting $\lambda\longrightarrow
0$, we immediately obtain the following relations

$\displaystyle c_{0}=0,\ \
-8\pi^{2}(\beta\sigma+\theta\alpha)\omega_{0}+2c_{0}-128\pi^{6}(\beta\sigma+\theta\alpha)\alpha^{5}=0,$

which has a solution

$None$ $c_{0}=0,\ \ w_{0}=-16\pi^{4}\alpha^{5}.$

Combining (4.14) and (4.15) then yields

$c\longrightarrow 0,\ \ 2\pi
i\omega\longrightarrow-32i\pi^{5}\alpha^{5}=-p^{5},\ \ {\rm as}\ \
\lambda\rightarrow 0.$

Hence we conclude

$None$ $\displaystyle\hat{\xi}=2\pi i\xi-\pi\tau=px+2\pi i\omega
t+q\theta\sigma+r$ $\displaystyle\quad\longrightarrow
px-p^{5}t+q\theta\sigma+r=\eta,\ \ {\rm as}\ \ \lambda\rightarrow 0.$

It remains to consider asymptotic properties of the one-periodic wave solution
(4.10) under the limit $\lambda\rightarrow 0$. By expanding the Riemann theta
function $\vartheta(\xi,\tau)$ and by using (4.16), it follows that

$\displaystyle\vartheta(\xi,\tau)=1+\lambda^{2}(e^{2\pi i\xi}+e^{-2\pi
i\xi})+\lambda^{8}(e^{4\pi i\xi}+e^{-4\pi i\xi})+\cdots$ $\displaystyle\ \ \ \
\
=1+e^{\hat{\xi}}+\lambda^{4}(e^{-\hat{\xi}}+e^{2\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}+e^{3\hat{\xi}})+\cdots$
$\displaystyle\ \ \ \ \ \quad\longrightarrow 1+e^{\hat{\xi}}\longrightarrow
1+e^{\eta},\ \ {\rm as}\ \ \lambda\rightarrow 0,$

which together with (4.10) lead to (4.12). Therefore we conclude that the one-
periodic solution (4.10) just goes to the one-soliton solution (4.4) as the
amplitude $\lambda\rightarrow 0$. $\square$
3.2. Two-periodic wave solutions and asymptotic analysis

We proceed to the construction of the two-periodic wave solutions to the
supersymmetric Sawada-Kotera-Ramani equation (4.1), which are a two-
dimensional generalization of one-periodic wave solutions. The two-periodic
waves of interest here have three-dimensional velocity fields and two-
dimensional surface patterns.

For the case when $N=2,\boldsymbol{s}=\boldsymbol{\varepsilon}=\boldsymbol{0}$
in the Riemann theta function (3.1), we takes $f$ as

$None$
$f=\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}\exp\\{2\pi
i\langle\boldsymbol{\xi},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\},$

where $\boldsymbol{n}=(n_{1},n_{2})^{T}\in\mathbb{Z}^{2},\ \
\boldsymbol{\xi}=(\xi_{1},\xi_{2})^{T}\in\mathbb{C}^{2},\ \
\xi_{i}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta\sigma+\delta_{j},\ \ j=1,2$;
The matrix $\boldsymbol{\tau}$ is a positive definite and real-valued
symmetric ${2\times 2}$ matrix, which can takes of the form

$\boldsymbol{\tau}=(\tau_{ij})_{2\times 2},\ \ \tau_{12}=\tau_{21},\ \
\tau_{11}>0,\ \ \tau_{22}>0,\ \ \tau_{11}\tau_{22}-\tau_{12}^{2}>0.$

Next, we explore the conditions to make the Riemann theta function (4.17)
satisfy the bilinear equation (4.3). Theorem 1 and the formula (3.9) give rise
to the following four constraint equations

$None$ $\displaystyle\sum_{n_{1},n_{2}\in\mathbb{Z}}\left[-4\pi^{2}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\sigma}+\theta\boldsymbol{\alpha}\rangle\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\omega}\rangle-64\pi^{6}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{5}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\sigma}+\theta\boldsymbol{\alpha}\rangle\right.$
$\displaystyle-60\pi^{2}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{2}\phi_{0}^{2}+80\pi^{4}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{4}\phi_{0}+c]\exp\\{-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle\\}=0,$

where $\boldsymbol{\mu}=(\mu_{1},\mu_{2})$ takes all possible combinations of
$\mu_{1},\mu_{2}=0,1$.

By introducing the notations

$\displaystyle\lambda_{kl}=e^{-\pi\tau_{kl}/2},k,l=1,2,\boldsymbol{\lambda}=(\lambda_{11},\lambda_{12},\lambda_{22})$
$\displaystyle\vartheta_{j}(\boldsymbol{\xi},\boldsymbol{\lambda})=\vartheta(2\xi,\boldsymbol{0},-\boldsymbol{s}_{j}/2|2\tau)=\sum_{n_{1},n_{2}\in
Z}\exp\\{4\pi
i\langle\boldsymbol{\xi},\boldsymbol{n}-\boldsymbol{s_{j}}/2\rangle\\}\prod_{k,l=1}^{2}\lambda_{kl}^{(2n_{k}-s_{j,k})(2n_{j}-s_{j,l})},$
$\displaystyle\boldsymbol{s_{j}}=(s_{j,1},s_{j,2}),\quad j=1,2,3,4,\ \
\boldsymbol{s_{1}}=(0,0),\ \ \boldsymbol{s_{2}}=(1,0),\ \
\boldsymbol{s_{3}}=(0,1),\ \ \boldsymbol{s_{4}}=(1,1),$

then by using (3.15), the system (4.18) can be written as a linear system

$None$
$\displaystyle[(\boldsymbol{\beta}\sigma+\theta\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\omega}\cdot\nabla)\vartheta_{j}(0,\boldsymbol{\lambda})+[(\boldsymbol{\beta}\sigma+\theta\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\alpha}\cdot\nabla)^{5}\vartheta_{j}(0,\boldsymbol{\lambda})$
$\displaystyle+15(\boldsymbol{\alpha}\cdot\nabla)^{2}\vartheta_{j}(0,\boldsymbol{\lambda})\phi_{0}^{2}+5(\boldsymbol{\alpha}\cdot\nabla)^{4}\vartheta_{j}(0,\boldsymbol{\lambda})\phi_{0}+\vartheta_{j}(0,\boldsymbol{\lambda})c=0.$

This system is easy to be solved in such a way: $\phi_{0}$ by solving a
quadratic equation with one unknown; $\omega_{1},\omega_{2}$ and $c$ by
solving a linear system. With such a solution
$(\omega_{1},\omega_{2},\phi_{0},c)$, we then get an exact two-periodic wave
solution

$None$
$\phi=\phi_{0}+2\mathfrak{D}^{3}\ln\vartheta(\boldsymbol{\xi},\boldsymbol{\tau}),$

with $\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})$ and
$\omega_{1},\omega_{2},\phi_{0},c$ given by (4.17) and (4.19), respectively,
while other parameters $\beta_{1},\beta_{2},\alpha_{1},$
$\alpha_{2},\tau_{11},\tau_{22},\tau_{12},\delta_{1},\delta_{2}\in\Lambda_{0}$
are free.

In summary, two-periodic wave (4.20), which is a direct generalization of one-
periodic wave, has the following features:

(i) The two-periodic wave solution is genuinely two-dimensional. Its surface
pattern is two-dimensional, namely, there are two phase variables $\xi_{1}$
and $\xi_{2}$.

(ii) It has two independent spatial periods in two independent horizontal
directions. It has $4$ fundamental periods $\\{e_{1},e_{2}\\}$ and
$\\{i\tau_{1},i\tau_{2}\\}$ in $(\xi_{1},\xi_{2})$. It is spatially periodic
in two directions $\xi_{1},\xi_{2}$, but it does not need periodic in the all
$x$-, $t$\- and $\theta$-directions.

(iii) As in the case of on-periodic waves, there is an influencing band among
the two-periodic waves under the presence of the Grassmann variable. ( see
Figure 3 ).

$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$

Figure 3. A two-super periodic wave for $\mathcal{N}=1$ supersymmetric Sawada-
Kotera-Ramani equation with parameters: $\alpha_{1}=0.1,\alpha_{2}=-0.1,$
$\tau_{11}=2,\tau_{12}=0.2,\tau_{22}=2,\sigma_{1}=0.1,\sigma_{2}=0.1$. (a)
Perspective view of wave. (b) Overhead view of wave, with contour plot shown.

At last, we consider the asymptotic properties of the two-periodic solution
(4.20). In a similar way to Theorem 3, we can establish the relation between
the two-periodic solution (4.20) and the two-soliton solution as follows.

Theorem 4. Assume that $(\omega_{1},\omega_{2},\phi_{0},c)^{T}$ is a solution
of the system (4.19). We choose the parameters in the two-periodic wave
solution (4.20) as follows

$None$ $\displaystyle\alpha_{j}=\frac{p_{j}}{2\pi i},\ \
\beta_{j}=\frac{q_{j}}{2\pi i},\ \ \delta_{j}=\frac{r_{j}+\pi\tau_{jj}}{2\pi
i},\ \ \tau_{12}=-\frac{A_{12}}{2\pi},\ \ j=1,2,$

with the $p_{j},q_{j},r_{j},j=1,2$ as those given in (4.5). Then under
constraint $\alpha_{1}\beta_{2}=\alpha_{2}\beta_{1}$, we have the following
asymptotic relations

$None$ $\displaystyle\phi_{0}\longrightarrow 0,\ \ \ c\longrightarrow 0,\ \
2\pi i\xi_{j}-\pi\tau_{jj}\longrightarrow
p_{j}x-p_{j}^{5}t+q_{j}\theta\sigma+r_{j}=\eta_{j},\ \ j=1,2,$
$\displaystyle\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})\longrightarrow
1+e^{\eta_{1}}+e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},\ \ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$

So the two-periodic wave solution (5.20) tends to the two-soliton solution
(4.5), namely,

$\phi\longrightarrow\phi_{2},\ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$

Proof. From (4.21), the constraint $\alpha_{1}\beta_{2}=\alpha_{2}\beta_{1}$
leads to $p_{1}q_{2}-p_{2}q_{1}=0$, which implies that
$\tau_{12}=-A_{12}/2\pi$ is independent of Grassmann variable $\theta$
according to (4.5).

In the same way as the proof of Theorem 3, we expand the Riemann function
$\vartheta(\xi_{1},\xi_{2},\boldsymbol{\tau})$ in the following form

$\displaystyle\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=1+(e^{2\pi
i\xi_{1}}+e^{-2\pi i\xi_{1}})e^{-\pi{\tau}_{11}}+(e^{2\pi i\xi_{2}}+e^{-2\pi
i\xi_{2}})e^{-\pi\tau_{22}}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ +(e^{2\pi
i(\xi_{1}+\xi_{2})}+e^{-2\pi
i(\xi_{1}+\xi_{2})})e^{-\pi(\tau_{11}+2\tau_{12}+\tau_{22})}+\cdots$

Further by using (4.21) and making a transformation $\hat{\omega}_{j}=2\pi
i\omega_{j},j=1,2$, we get

$\displaystyle\vartheta(\boldsymbol{\xi},\boldsymbol{\tau})=1+e^{\hat{\xi}_{1}}+e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}-2\pi\tau_{12}}+\lambda_{11}^{4}e^{-\hat{\xi}_{1}}+\lambda_{22}^{4}e^{-\hat{\xi}_{2}}+\lambda_{11}^{4}\lambda_{22}^{4}e^{-\hat{\xi}_{1}-\hat{\xi}_{2}-2\pi\tau_{12}}+\cdots$
$\displaystyle\ \ \ \ \ \longrightarrow
1+e^{\hat{\xi}_{1}}+e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}+A_{12}},\
\ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0,$

where $\hat{\xi}_{j}=p_{j}x+\hat{\omega}_{j}t+p_{j}\theta\sigma+r_{j},\ \
j=1,2.$

It remains to prove that

$None$ $\displaystyle c\longrightarrow 0,\ \ \
\hat{\omega}_{j}\longrightarrow-p_{j}^{5},\ \
\hat{\xi}_{j}\longrightarrow\eta_{j},\ \ j=1,2,\ \ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$

As in the case when $N=1$, we let the solution of the system (4.19) be the
form

$None$
$\displaystyle\omega_{1}=\omega_{1,0}+\omega_{1,1}\lambda_{11}+\omega_{1,2}\lambda_{22}+\cdots=\omega_{1,0}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle\omega_{2}=\omega_{2,0}+\omega_{2,1}\lambda_{11}+\omega_{2,2}\lambda_{22}+\cdots=\omega_{2,0}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle\phi_{0}=\phi_{0,0}+\phi_{0,1}\lambda_{11}+\phi_{0,2}\lambda_{22}+\cdots=\phi_{0,0}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle
c=c_{0}+c_{1}\lambda_{11}+c_{2}\lambda_{22}+\cdots=c_{0}+o(\lambda_{11},\lambda_{22}).$

Expanding functions $\vartheta_{j},j=1,2,3,4$ in the system (4.19), together
with substitution of assumption (4.24), the second and third equation is
divided by $\lambda_{11}$ and $\lambda_{22}$, respectively; the fourth
equation is divided by $\lambda_{11}\lambda_{22}$, and letting
$\lambda_{11},\lambda_{22}\longrightarrow 0$ , we then obtain

$\displaystyle c_{0}=0,\
-8\pi(\beta_{1}\sigma+\theta\alpha_{1})\omega_{1}-128\pi^{6}\alpha_{1}^{5}(\beta_{1}\sigma+\theta\alpha_{1})-120\pi\alpha_{1}^{2}\phi_{0,0}^{2}+2\phi_{0,0}=0,$
$\displaystyle-8\pi(\beta_{2}\sigma+\theta\alpha_{2})\omega_{2}-128\pi^{6}\alpha_{2}^{5}(\beta_{2}\sigma+\theta\alpha_{2})-120\pi\alpha_{2}^{2}\phi_{0,0}^{2}+2\phi_{0,0}=0,$

which has solution

$None$ $\displaystyle c_{0}=0,\ \ \phi_{0,0}=0,\ \ \
\omega_{1,0}=-16\pi^{4}\alpha_{1}^{5},\ \
\omega_{2,0}=-16\pi^{4}\alpha_{2}^{5}.$

The expressions (4.24) and (4.25) implies that

$\displaystyle\phi_{0}=o(\lambda_{11},\lambda_{22})\longrightarrow 0,\ \
c=o(\lambda_{11},\lambda_{22})\longrightarrow 0,\ \
\omega_{1}=-16\pi^{4}\alpha_{1}^{5}+o(\lambda_{11},\lambda_{22})\longrightarrow-16\pi^{4}\alpha_{1}^{5},$
$\displaystyle\omega_{2}=-16\pi^{4}\alpha_{2}^{5}+o(\lambda_{11},\lambda_{22})\longrightarrow-16\pi^{4}\alpha_{2}^{5},\
\ \ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0,$

thus proving (4.23). We conclude that the two-periodic wave solution (4.20)
tends to the two-soliton solution (4.5) as
$\lambda_{11},\lambda_{22}\rightarrow 0$. $\square$
4\. $\mathcal{N}=2$ supersymmetric KdV equation

We consider $\mathcal{N}=2$ supersymmetric KdV equation

$None$
$\displaystyle\phi_{t}=-\phi_{xxx}+3(\phi\mathfrak{D}_{1}\mathfrak{D}_{2}\phi)_{x}+\frac{1}{2}(a-1)(\mathfrak{D}_{1}\mathfrak{D}_{2}\phi^{2})_{x}+3a\phi^{2}\phi_{x}^{2},$

which was originally introduced by Laberge and Mathieu [58, 59]. In the
equation (5.1),
$\phi=\phi(x,t,\theta_{1},\theta_{2}):\mathbb{R}_{\Lambda}^{2,2}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is a superboson function depending on temporal variable $t$, spatial variable
$x$ and its fermionic counterparts $\theta_{1},\theta_{2}$. The operators
$\mathfrak{D}_{1}$ and $\mathfrak{D}_{2}$ are the super derivatives defined by
$\mathfrak{D}_{1}=\partial_{\theta_{1}}+\theta_{1}\partial_{x},\ \
\mathfrak{D}_{2}=\partial_{\theta_{2}}+\theta_{2}\partial_{x}$ and $a$ is a
parameter. The equation (5.1) is called supersymmetric KdVa equation [60]. For
the cases when $a=1$ and $a=4$, the Lax representation, Hamiltonian structure,
Painleve analysis and soliton solutions of the equation (5.1) can refer to,
for instance, papers [58]–[60].

Here we are interested in quasi-periodic wave solutions to the supersymmetric
equation (5.1) by using Theorem 1 and 5. We only consider the case when $a=1$,
so the equation (5.1) reduces to

$None$
$\displaystyle\phi_{t}=-\phi_{xxx}+3(\phi\mathfrak{D}_{1}\mathfrak{D}_{2}\phi)_{x}+3\phi^{2}\phi_{x}^{2}.$

To apply the Hirota bilinear method for constructing multi-periodic wave
solutions of the equation (5.2), we add two variables and consider a general
variable transformation

$None$ $\displaystyle\phi=u+\theta_{2}v,\ \ u=i\partial_{x}\ln\frac{f}{g},\ \
v=v_{0}-\partial_{x}\mathfrak{D}\ln(fg),$

where
$u(x,t,\theta_{1}),f(x,t,\theta_{1}),g(x,t,\theta_{1}):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{1,0}$,
and
$v(x,t,\theta_{1}),v_{0}=v_{0}(\theta_{1}):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is a special solution of the equation (5.2). Hereafter we use
$\mathfrak{D}=\mathfrak{D}_{1}$ for simplicity, Substituting (5.3) into (5.2),
we then get the following bilinear form

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

where
$c=c(\theta_{1},t):\mathbb{R}_{\Lambda}^{2,1}\rightarrow\mathbb{R}_{\Lambda}^{0,1}$
is an odd integration constant to variable $x$; The equation (5.4) is a type
of coupled bilinear equations, which is more difficult to be dealt with than
the bilinear equation (4.3) due to appearance of two functions and two
equations. We will take full advantages of Theorem 2 to reduce the number of
constraint equations.

Now we take into account the periodicity of the solution (5.3), in which we
take $f$ and $g$ as

$f=\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau}),\ \
g=\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau}),\ \
\boldsymbol{e},\boldsymbol{h}\in\mathbb{Z}^{N},$

where phase variable $\xi$ is taken as the form
$\boldsymbol{\xi}=(\xi_{1},\cdots,\xi_{N})^{T}$,
$\xi_{j}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta_{1}\sigma+\delta_{j},\ \
j=1,2,\cdots,N.$ By means of Proposition 3, we deduce that

$\displaystyle\phi=u+\theta_{2}v,\ \
u=i\sum_{k=1}^{N}\alpha_{k}\partial_{\xi_{k}}\ln\frac{\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau})},$
$\displaystyle
v=v_{0}-2\sum_{k,l=1}^{N}\alpha_{k}(\beta_{l}\sigma+\theta\alpha_{l})\partial_{\xi_{k}\xi_{l}}^{2}\ln[{\vartheta(\boldsymbol{\xi}+\boldsymbol{e},\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi}+\boldsymbol{h},\boldsymbol{\tau})}],$

which indicates that the solution $\phi$ is a quasi-periodic function with
$2N$ fundamental periods $\\{\boldsymbol{e_{j}},\ \ j=1,\cdots,N\\}$ and
$\\{i\boldsymbol{\tau_{j}},\ \ j=1,\cdots,N\\}$.

In the special case of $v_{0}=c=0$, starting from the bilinear equation (5.4),
Zhang et al. found that the equation (5.2) admits one-soliton solution [60]

$None$
$\phi_{1}=i\partial_{x}\ln\frac{f_{1}}{g_{1}}+\theta_{2}[v_{0}-\partial_{x}\mathfrak{D}\ln(f_{1}g_{1})],$

with

$f_{1}=1+e^{\eta},\ \ g_{1}=1-e^{\eta}$

and phase variable $\eta=px-p^{3}t+q\theta_{1}\sigma+r$ with
$p,q,r\in\Lambda_{0}$. While two-soliton solution takes the form

$None$
$\phi_{1}=i\partial_{x}\ln\frac{f_{2}}{g_{2}}+\theta_{2}[v_{0}-\partial_{x}\mathfrak{D}\ln(f_{2}g_{2})],$

with

$None$ $\displaystyle
f_{2}=1+e^{\eta_{1}}+e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},$
$\displaystyle
g_{2}=1-+e^{\eta_{1}}-e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},$

and $\eta_{j}=p_{j}x-p_{j}^{3}t+q_{j}\theta_{1}\sigma+r_{j},\ \ j=1,2,$

$\displaystyle
e^{A_{12}}=\frac{(p_{1}-p_{2})^{2}}{(p_{1}+p_{2})^{2}}+2\theta_{1}\sigma\frac{(p_{1}-p_{2})(p_{1}q_{2}-p_{2}q_{1})}{(p_{1}+p_{2})^{2}},$

here $p_{j},q_{j},r_{j}\in\Lambda_{0},j=1,2$ are free constants.
4.1. One-periodic waves and asymptotic analysis

We first construct one-periodic wave solutions of the equation (5.2). As a
simple case of the theta function (3.2) when $N=1,s=0$, we take $f$ and $g$ as

$None$ $\displaystyle
f=\vartheta(\xi,0,0|\tau)=\sum_{n\in\mathbb{Z}}\exp({2\pi in\xi-\pi
n^{2}\tau}),$ $\displaystyle
g=\vartheta(\xi,1/2,0|\tau)=\sum_{n\in\mathbb{Z}}\exp({2\pi in(\xi+1/2)-\pi
n^{2}\tau})$ $\displaystyle\ \ \ =\sum_{n\in\mathbb{Z}}(-1)^{n}\exp({2\pi
in\xi-\pi n^{2}\tau}),$

where the phase variable $\xi=\alpha x+\omega t+\beta\theta_{1}\sigma+\delta$,
and the parameter $\tau>0$.

Due to the fact that $F(D_{t},D_{x})$ is an odd function, its constraint
equations in the formula (3.10) vanish automatically for $\mu=0$. Similarly
the constraint equations associated with $G(S_{x},D_{t},D_{x})$ also vanish
automatically for $\mu=1$. Therefore, the Riemann theta function (5.8) is a
solution of the bilinear equation (5.4), provided the following equations

$None$ $\displaystyle\sum_{n\in\mathbb{Z}}\\{2\pi
i(2n-\mu)\omega-i(2\pi\alpha)^{3}(2n-\mu)^{3}\\}\exp[-2\pi\tau(n-\mu/2)^{2}+\pi
i(n-\mu/2)]|_{\mu=1}=0,$
$\displaystyle\sum_{n\in\mathbb{Z}}\\{-[2\pi(2n-\mu)]^{2}(\beta\sigma+\theta_{1}\alpha)\omega+(2\pi(2n-\mu)]^{4}(\beta\sigma+\theta_{1}\alpha)\alpha^{3}-(2\pi(2n-\mu)\alpha]^{2}v_{0}+c\\}$
$\displaystyle\times\exp(-2\pi\tau(n-\mu/2)^{2}+\pi i(n-\mu/2))|_{\mu=0}=0.$

We introduce the notations by

$\displaystyle\lambda=e^{-\pi\tau/2},$
$\displaystyle\vartheta_{1}(\xi,\lambda)=\vartheta(2\mathbf{\xi},1/4,-1/2|2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{(2n-1)^{2}}\exp[4i\pi(n-1/2)(\xi+1/4)],$
$\displaystyle\vartheta_{2}(\xi,\lambda)=\vartheta(2\xi,1/4,0|2\tau)=\sum_{n\in\mathbb{Z}}\lambda^{4n^{2}}\exp[4i\pi
n(\xi+1/4)],$

the equation (5.9) can be written as a linear system about $\omega,c$

$None$
$\displaystyle\vartheta_{1}^{\prime}\omega+\alpha^{3}\vartheta_{1}^{\prime\prime\prime}=0,$
$\displaystyle(\beta\sigma+\theta_{1}\alpha)\vartheta_{2}^{\prime\prime}\omega+\vartheta_{2}c+(\beta\sigma+\theta_{1}\alpha)\alpha^{3}\vartheta_{2}^{(4)}+\alpha^{2}\vartheta_{2}^{\prime\prime}v_{0}=0.$

where $\omega\in\Lambda_{0}$ is even and $c,v_{0}\in\Lambda_{1}$ are odd, and
we have denoted the derivative value of $\vartheta_{j}(\xi,\lambda)$ at
$\xi=0$ by simple notations

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

Moreover, we see that the functions $\vartheta_{j}$ and their derivatives are
independent of Grassmann variable $\theta$ and anticommuting number $\sigma$.

We take $v_{0}=\gamma\alpha(\sigma+\theta\alpha),\ \gamma\in\Lambda_{0}$ for
the simplicity, then the system (5.10) admits a solution

$None$
$\displaystyle\omega=-\frac{\alpha^{3}\vartheta_{1}^{\prime\prime\prime}}{\vartheta_{1}^{\prime}},\
\ \
c=\frac{\alpha^{3}(\beta\sigma+\theta_{1}\alpha)}{\vartheta_{1}^{\prime}\vartheta_{2}}(\vartheta_{1}^{\prime\prime\prime}\vartheta_{2}^{\prime\prime}-\vartheta_{1}^{\prime}\vartheta_{2}^{(4)}-\gamma\vartheta_{1}^{\prime}\vartheta_{2}^{\prime\prime}),$

where $\omega$ is independent of Grassmann variable $\theta$ and auticommuting
number $\sigma$. In this way, a one-periodic wave solution reads

$None$
$\phi=i\partial_{x}\ln\frac{\vartheta(\xi,0,0|\tau)}{\vartheta(\xi,1/2,0|\tau)}+\theta_{2}\\{v_{0}-\partial_{x}\mathfrak{D}\ln[\vartheta(\xi,0,0|\tau)\vartheta(\xi,1/2,0|\tau)]\\},$

where parameters $\omega$ and $c$ are given by (5.11), while other parameters
$\alpha,\beta,\tau,\delta\in\Lambda_{0}$ are free. Among them, the three
parameters $\alpha$ and $\tau$ completely dominate a one-periodic wave.

In summary, one-periodic wave (5.12) has the following features:

(i) It is one-dimensional and has two fundamental periods $1$ and $i\tau$ in
phase variable $\xi$. It can be viewed as a parallel superposition of
overlapping one-soliton waves, placed one period apart (see Figure 5-7).

(ii) As in the case of the supersymmetric Sawada-Kotera-Ramani equation, there
is also an influencing band among the real part of one-periodic waves for the
supersymmetric KdV equation under the presence of the Grassmann variable (see
Figure 4).

(iii) It was not observed that influencing band appears among the imaginary
part and modulus of the one-periodic wave. Moreover, they seem to have the
same shape from the observation of their plots (see Figures 5 and 6).

$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$

Figure 4. Real part of one-periodic wave for $\mathcal{N}=2$ supersymmetric
KdV equation with parameters: $\alpha=0.1,$ $\tau=3,\sigma_{1}=0.01$. (a)
Perspective view of wave. (b) Overhead view of wave, with contour plot shown.

$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$

Figure 5. Imaginary part of one-periodic wave for $\mathcal{N}=2$
supersymmetric KdV equation with parameters: $\alpha=0.1,$
$\tau=3,\sigma_{1}=0.01$. (a) Perspective view of wave. (b) Overhead view of
wave, with contour plot shown.

$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$

Figure 6. Modulus of one-periodic wave for $\mathcal{N}=2$ supersymmetric KdV
equation with parameters: $\alpha=0.1,$ $\tau=3,\sigma_{1}=0.01$. (a)
Perspective view of wave. (b) Overhead view of wave, with contour plot shown.

In the following, we further consider asymptotic properties of the one-
periodic wave solution. The relation between the one-periodic wave solution
(5.12) and the one-soliton solution (5.5) can be established as follows.

Theorem 5. Suppose that $\omega$ and $c$ are given by (5.11). In the one-
periodic wave solution (5.12), we choose parameters as

$None$ $\gamma=0,\ \ \alpha=\frac{p}{2\pi i},\ \ \beta=\frac{q}{2\pi i},\ \
\delta=\frac{r+\pi\tau}{2\pi i},$

where the $p,q$ and $r$ are the same as those in (5.5). Then we have the
following asymptotic properties

$c\longrightarrow 0,\ \ \xi\longrightarrow\frac{\eta+\pi\tau}{2\pi i},\ \
f\longrightarrow 1+e^{\eta},\ \ g\longrightarrow 1-e^{\eta},\ \ {\rm as}\ \
\lambda\rightarrow 0.$

In other words, the one-periodic solution (5.12) tends to the one-soliton
solution (5.5) under a small amplitude limit , that is,

$None$ $\phi\longrightarrow\phi_{1},\ \ {\rm as}\ \ \lambda\rightarrow 0.$

Proof. Here we will directly use the system (5.10) to analyze asymptotic
properties of one-periodic solution (5.12). We explicitly expand the
coefficients of system (5.10) as follows

$None$
$\displaystyle\vartheta_{1}^{\prime}=-4\pi\lambda+12\pi\lambda^{9}+\cdots,\quad\vartheta_{1}^{\prime\prime\prime}=16\pi^{3}\lambda+432\pi^{3}\lambda^{9}+\cdots,$
$\displaystyle\vartheta_{2}=-2+2\lambda^{4}+\cdots,\quad\vartheta_{2}^{\prime\prime}=32\pi^{2}\lambda^{4}+\cdots,$
$\displaystyle\vartheta_{2}^{(4)}=512\pi^{4}\lambda^{4}+\cdots$

Suppose that the solution of the system (5.10) is of the form

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

Substituting the expansions (5.15) and (5.16) into the system (5.11) and
letting $\lambda\longrightarrow 0$, we immediately obtain the following
relations

$\displaystyle-4\pi\omega_{0}+16\pi^{3}\alpha^{3}=0,\ \ c_{0}=0,$

which has a solution

$None$ $c_{0}=0,\ \ w_{0}=4\pi^{2}\alpha^{3}.$

Combining (5.13) and (5.17) leads to

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

or equivalently

$None$ $\displaystyle\hat{\xi}=2\pi i\xi-\pi\tau=px+2\pi i\omega
t+q\theta_{1}\sigma+r$ $\displaystyle\quad\longrightarrow
px-p^{3}t+q\theta_{1}\sigma+r=\eta,\ \ {\rm as}\ \ \lambda\rightarrow 0.$

It remains to identify that the one-periodic wave (5.12) possesses the same
form with the one-soliton solution (5.5) under the limit $\lambda\rightarrow
0$. For this purpose, we start to expand the functions $f$ and $g$ in the form

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

By using (5.13) and (5.17), it follows that

$None$ $\displaystyle
f=1+e^{\hat{\xi}}+\lambda^{4}(e^{-\hat{\xi}}+e^{2\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}+e^{3\hat{\xi}})+\cdots$
$\displaystyle\quad\longrightarrow 1+e^{\hat{\xi}}\longrightarrow 1+e^{\eta},\
\ {\rm as}\ \ \lambda\rightarrow 0;$ $\displaystyle
g=1-e^{\hat{\xi}}+\lambda^{4}(e^{2\hat{\xi}}-e^{-\hat{\xi}})+\lambda^{12}(e^{-2\hat{\xi}}-e^{3\hat{\xi}})+\cdots$
$\displaystyle\quad\longrightarrow 1-e^{\hat{\xi}}\longrightarrow 1-e^{\eta},\
\ {\rm as}\ \ \lambda\rightarrow 0.$

The expression (5.13) follows from (5.19), and thus we conclude that the one-
periodic solution (5.12) just goes to the one-soliton solution (5.5) as the
amplitude $\lambda\rightarrow 0$. $\square$
4.2. Two-periodic waves and asymptotic properties

We now consider two-periodic wave solutions to the supersymmetric KdV equation
(5.2). For the case when $N=2,\ \boldsymbol{s}=\boldsymbol{0},\
\boldsymbol{\varepsilon}=\boldsymbol{1/2}=(1/2,1/2)$ in the Riemann theta
function (3.2), we choose $f$ and $g$ to be

$None$ $\displaystyle
f=\vartheta(\boldsymbol{\xi},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}\exp\\{2\pi
i\langle\boldsymbol{\xi},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\},$
$\displaystyle
g=\vartheta(\boldsymbol{\xi},\boldsymbol{1/2},\boldsymbol{0}|\boldsymbol{\tau})=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}\exp\\{2\pi
i\langle\boldsymbol{\xi}+\boldsymbol{1/2},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\}$
$\displaystyle\ \
=\sum_{\boldsymbol{n}\in\mathbb{Z}^{2}}(-1)^{n_{1}+n_{2}}\exp\\{2\pi
i\langle\boldsymbol{\xi},\boldsymbol{n}\rangle-\pi\langle\boldsymbol{\tau}\boldsymbol{n},\boldsymbol{n}\rangle\\}$

where we denote $\boldsymbol{n}=(n_{1},n_{2})\in Z^{2},\ \
\boldsymbol{\xi}=(\xi_{1},\xi_{2})\in\mathcal{C}^{2},\ \
\xi_{i}=\alpha_{j}x+\omega_{j}t+\beta_{j}\theta_{1}\sigma+\delta_{j},\ \
j=1,2$, and $\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2}),\
\boldsymbol{\omega}=(\omega_{1},\omega_{2}),\
\boldsymbol{\beta}=(\beta_{1},\beta_{2})\in\mathcal{C}^{2},$; The matrix
$\boldsymbol{\tau}$ is a positive definite and real-valued symmetric ${2\times
2}$ matrix, which can take the form

$\boldsymbol{\tau}=(\tau_{ij})_{2\times 2},\ \ \tau_{12}=\tau_{21},\ \
\tau_{11}>0,\ \ \tau_{22}>0,\ \ \tau_{11}\tau_{22}-\tau_{12}^{2}>0.$

According to Theorem 5, constraint equations associated with $F(D_{t},D_{x})$
vanish automatically for $(\mu_{1},\mu_{2})=(0,0),(1,1)$, and the constraint
equations associated with $G(S_{x},D_{t},D_{x})$ vanish automatically for
$(\mu_{1},\mu_{2})=(1,0),(0,1)$. Hence, making the theta functions $f$ and $g$
satisfy the bilinear equation (5.4) gives to the following constraint
equations

$None$ $\displaystyle\sum_{n_{1},n_{2}\in\mathbb{Z}}\left[2\pi i\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\omega}\rangle-8\pi^{3}i\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{3}\right]\exp\\{-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle$
$\displaystyle+\pi
i\sum_{j=1}^{2}(n_{j}-\mu_{j}/2)\\}|_{\boldsymbol{\mu}=(\mu_{1},\mu_{2})}=0,\
\ {\rm for}\ \ (\mu_{1},\mu_{2})=(0,1),\ (1,0).$

and

$None$ $\displaystyle\sum_{n_{1},n_{2}\in\mathbb{Z}}[-4\pi^{2}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\sigma\boldsymbol{\beta}+\theta_{1}\boldsymbol{\alpha}\rangle\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\omega}\rangle+16\pi^{4}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\sigma\boldsymbol{\beta}+\theta_{1}\boldsymbol{\alpha}\rangle\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{3}$
$\displaystyle\ \ \ \ -4\pi^{2}\langle
2\boldsymbol{n}-\boldsymbol{\mu},\boldsymbol{\alpha}\rangle^{2}v_{0}+c]\exp\\{-2\pi\langle\boldsymbol{\tau}(\boldsymbol{n}-\boldsymbol{\mu}/2),\boldsymbol{n}-\boldsymbol{\mu}/2\rangle$
$\displaystyle\ \ \ \ +\pi
i\sum_{j=1}^{2}(n_{j}-\mu_{j}/2)\\}|_{\boldsymbol{\mu}=(\mu_{1},\mu_{2})}=0,\
\ {\rm for}\ \ (\mu_{1},\mu_{2})=(0,0),\ (1,1).$

Next, introducing the following notations

$\displaystyle\lambda_{kl}=e^{-\pi\tau_{kl}/2},k,l=1,2,\boldsymbol{\lambda}=(\lambda_{11},\lambda_{12},\lambda_{22})$
$\displaystyle\vartheta_{j}(\boldsymbol{\xi},\boldsymbol{\lambda})=\vartheta(2\xi,\boldsymbol{1/4},-\boldsymbol{s}_{j}/2|2\tau)=\sum_{n_{1},n_{2}\in
Z}\exp\\{4\pi
i\langle\boldsymbol{\xi}+\boldsymbol{1/4},\boldsymbol{n}-\boldsymbol{s_{j}}/2\rangle\\}\prod_{k,l=1}^{2}\lambda_{kl}^{(2n_{k}-s_{j,k})(2n_{j}-s_{j,l})},$
$\displaystyle\boldsymbol{s_{j}}=(s_{j,1},s_{j,2}),\quad j=1,2,3,4,\ \
\boldsymbol{s_{1}}=(0,1),\ \ \boldsymbol{s_{2}}=(1,0),\ \
\boldsymbol{s_{3}}=(0,0),\ \ \boldsymbol{s_{4}}=(1,1),$

then by using (3.15), the system (5.21) and (5.22) can be written as a linear
system

$None$
$\displaystyle(\boldsymbol{\omega}\cdot\nabla)\vartheta_{j}(0,\boldsymbol{\lambda})+(\boldsymbol{\alpha}\cdot\nabla)^{3}\vartheta_{j}(0,\boldsymbol{\lambda})=0,\
j=1,2,$
$\displaystyle[(\sigma\boldsymbol{\beta}+\theta_{1}\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\omega}\cdot\nabla)\vartheta_{j}(0,\boldsymbol{\lambda})+[(\sigma\boldsymbol{\beta}+\theta_{1}\boldsymbol{\alpha})\cdot\nabla](\boldsymbol{\alpha}\cdot\nabla)^{3}\vartheta_{j}(0,\boldsymbol{\lambda})$
$\displaystyle+(\boldsymbol{\alpha}\cdot\nabla)^{2}\vartheta_{j}(0,\boldsymbol{\lambda})v_{0}+\vartheta_{j}(0,\boldsymbol{\lambda})c=0,\
j=3,4.$

This system can be solved in such a way: After we obtain
$\omega,_{1},\omega_{2}$ form the first two equations, We substitute them into
last two equations to get $v_{0},c$. With the solution
$(\omega_{1},\omega_{2},v_{0},c)$, we get a two-periodic wave solution to the
supersymmetric equation (5.2)

$None$
$\phi=i\partial_{x}\ln\frac{\vartheta(\boldsymbol{\xi},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})}{\vartheta(\boldsymbol{\xi},\boldsymbol{1/2},\boldsymbol{0}|\boldsymbol{\tau})}+\theta_{2}\\{v_{0}-\partial_{x}\mathfrak{D}\ln[\vartheta(\boldsymbol{\xi},\boldsymbol{0},\boldsymbol{0}|\boldsymbol{\tau})\vartheta(\boldsymbol{\xi},\boldsymbol{1/2},\boldsymbol{0}|\boldsymbol{\tau})]\\},$

where parameters $\omega_{1},\omega_{2},v_{0}$ and $c$ are given by (5.22),
while other parameters $\sigma_{1},\sigma_{2},\alpha_{1},$
$\alpha_{2},\tau_{11}$, $\tau_{22},\tau_{12},\delta_{1}$ and $\delta_{2}$ are
free.

In summary, the two-periodic wave (5.24), which is a direct generalization of
one-periodic waves, has the following features:

(i) Its surface pattern is two-dimensional, namely, there are two phase
variables $\xi_{1}$ and $\xi_{2}$. It has $4$ fundamental periods
$\\{e_{1},e_{2}\\}$ and $\\{i\tau_{1},i\tau_{2}\\}$ in $(\xi_{1},\xi_{2})$,
and is spatially periodic in two directions $\xi_{1},\xi_{2}$. Its real part
is not periodic in $\theta$ direction, while its real part, imaginary part and
modulus are all periodic in $x$ and $t$ directions.

(iii) There is also an influencing band among the Real part of two-periodic
waves for the supersymmetric KdV equation under the presence of the Grassmann
variable ( see Figure 7 ).

(iv) It was not found that influencing band appears among the imaginary part
and modulus of two-periodic waves to the supersymmetric KdV equation ( see
Figures 8 and 9 ).

$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$

Figure 7. Real part of two-periodic wave for $\mathcal{N}=2$ supersymmetric
KdV equation with parameters: $\alpha_{1}=0.1,\ \alpha_{2}=0.2$ $\tau_{11}=3,\
\tau_{12}=0.2,\tau_{22}=3,\sigma_{1}=0.2,\ \sigma_{2}=-0.1$. (a) Perspective
view of wave. (b) Overhead view of wave, with contour plot shown.

$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$

Figure 8. Imaginary part of two-periodic wave for $\mathcal{N}=2$
supersymmetric KdV equation with parameters: $\alpha_{1}=0.1,\ \alpha_{2}=0.2$
$\tau_{11}=3,\ \tau_{12}=0.2,\tau_{22}=3,\sigma_{1}=0.2,\ \sigma_{2}=-0.1$.
(a) Perspective view of wave. (b) Overhead view of wave, with contour plot
shown.

$(a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $(b)$

Figure 9. Modulus of two-periodic wave for $\mathcal{N}=2$ supersymmetric KdV
equation with parameters: $\alpha_{1}=0.1,\ \alpha_{2}=0.2$ $\tau_{11}=3,\
\tau_{12}=0.2,\tau_{22}=3,\sigma_{1}=0.2,\ \sigma_{2}=-0.1$. (a) Perspective
view of wave. (b) Overhead view of wave, with contour plot shown.

Finally, we consider the asymptotic properties of the two-periodic solution
(5.24). In a similar way to Theorem 5, we can establish the relation between
the two-periodic solution (5.24) and the two-soliton solution (5.6) as
follows.

Theorem 6. Assume that $(\omega_{1},\omega_{2},v_{0},c)$ is a solution of the
system (5.23). In the two-periodic wave solution (5.24), a choice of
parameters is given by

$None$ $\displaystyle\alpha_{j}=\frac{p_{j}}{2\pi i},\ \
\beta_{j}=\frac{q_{j}}{2\pi i},\ \ \delta_{j}=\frac{r_{j}+\pi\tau_{jj}}{2\pi
i},\ \ \tau_{12}=-\frac{A_{12}}{2\pi},\ \ j=1,2,$

with the $p_{j},q_{j},r_{j}\in\Lambda_{0},j=1,2$ and $A_{12}$ as those given
in (5.6). Then under the constraint $\alpha_{1}\beta_{2}=\alpha_{2}\beta_{1}$,
we have the following asymptotic relations

$None$ $\displaystyle v_{0}\longrightarrow 0,\ \ \ c\longrightarrow 0,\ \
\xi_{j}\longrightarrow\frac{\eta_{j}+\pi\tau_{jj}}{2\pi i},\ \ j=1,2,$
$\displaystyle f\longrightarrow
1+e^{\eta_{1}}+e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},\ \ g\longrightarrow
1-e^{\eta_{1}}-e^{\eta_{2}}+e^{\eta_{1}+\eta_{2}+A_{12}},$ $\displaystyle\ \ \
{\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0.$

So the two-periodic wave solution (5.24) just tends to the two-soliton
solution (5.6) under a certain limit

$\phi\longrightarrow\phi_{2},\ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$

Proof. From (5.25), the constraint $\alpha_{1}\beta_{2}=\alpha_{2}\beta_{1}$
leads to $p_{1}q_{2}-p_{2}q_{1}=0$, which implies that
$\tau_{12}=-A_{12}/2\pi$ is independent of Grassmann variable $\theta$
according to (5.7).

Using (5.20), we explicitly expand the functions $f$ and $g$ in the following
form

$\displaystyle f=1+(e^{2\pi i\xi_{1}}+e^{-2\pi
i\xi_{1}})e^{-\pi\tau_{11}}+(e^{2\pi i\xi_{2}}+e^{-2\pi
i\xi_{2}})e^{-\pi\tau_{22}}$ $\displaystyle\ \ \ \ \ \ +(e^{2\pi
i(\xi_{1}+\xi_{2})}+e^{-2\pi
i(\xi_{1}+\xi_{2})})e^{-\pi(\tau_{11}+2\tau_{12}+\tau_{22})}+\cdots$
$\displaystyle g=1-(e^{2\pi i\xi_{1}}+e^{-2\pi
i\xi_{1}})e^{-\pi\tau_{11}}-(e^{2\pi i\xi_{2}}+e^{-2\pi
i\xi_{2}})e^{-\pi\tau_{22}}$ $\displaystyle\ \ \ \ \ \ +(e^{2\pi
i(\xi_{1}+\xi_{2})}+e^{-2\pi
i(\xi_{1}+\xi_{2})})e^{-\pi(\tau_{11}+2\tau_{12}+\tau_{22})}+\cdots$

Further using (4.5) and making a transformation $\hat{\omega}_{j}=2\pi
i\omega_{j},j=1,2$, we infer that

$\displaystyle
f=1+e^{\hat{\xi}_{1}}+e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}-2\pi\tau_{12}}+\lambda_{11}^{4}e^{-\hat{\xi}_{1}}+\lambda_{22}^{4}e^{-\hat{\xi}_{2}}+\lambda_{11}^{4}\lambda_{22}^{4}e^{-\hat{\xi}_{1}-\hat{\xi}_{2}-2\pi\tau_{12}}+\cdots$
$\displaystyle\ \ \ \ \ \longrightarrow
1+e^{\hat{\xi}_{1}}+e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}+A_{12}},\
\ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0,$ $\displaystyle
g=1-e^{\hat{\xi}_{1}}-e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}-2\pi\tau_{12}}-\lambda_{11}^{4}e^{-\hat{\xi}_{1}}-\lambda_{22}^{4}e^{-\hat{\xi}_{2}}+\lambda_{11}^{4}\lambda_{22}^{4}e^{-\hat{\xi}_{1}-\hat{\xi}_{2}-2\pi\tau_{12}}+\cdots$
$\displaystyle\ \ \ \ \ \longrightarrow
1-e^{\hat{\xi}_{1}}-e^{\hat{\xi}_{2}}+e^{\hat{\xi}_{1}+\hat{\xi}_{2}+A_{12}},\
\ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow 0,$

where $\hat{\xi}_{j}=p_{j}x+\hat{\omega}_{j}t+q_{j}\theta_{1}\sigma+r_{j},\ \
j=1,2.$

It remains to prove that

$None$ $\displaystyle c\longrightarrow 0,\ \ \
\hat{\omega}_{j}\longrightarrow-p_{j}^{3},\ \
\hat{\xi}_{j}\longrightarrow\eta_{j},\ \ j=1,2,\ \ \ {\rm as}\ \
\lambda_{11},\lambda_{22}\rightarrow 0.$

As in the case when $N=1$, we let the solution of the system (5.23) be the
form

$None$
$\displaystyle\omega_{1}=\omega_{1,0}+\omega_{1,1}\lambda_{11}+\omega_{2,2}\lambda_{22}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle\omega_{2}=\omega_{2,0}+\omega_{2,1}\lambda_{11}+\omega_{2,2}\lambda_{22}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle
v_{0}=v_{0,0}+v_{0,1}\lambda_{11}+v_{0,2}\lambda_{22}+o(\lambda_{11},\lambda_{22}),$
$\displaystyle
c=c_{0}+c_{1}\lambda_{11}+c_{2}\lambda_{22}+o(\lambda_{11},\lambda_{22}).$

Expanding functions $\vartheta_{j},j=1,2,3,4$ in the system (5.23), together
with substitution of assumption (5.28), the second and third equation is
divided by $\lambda_{11}$ and $\lambda_{22}$, respectively; the fourth
equation is divided by $\lambda_{11}\lambda_{22}$, and letting $v_{0,0}=0,\
\lambda_{11},\lambda_{22}\longrightarrow 0$ , we then obtain

$None$ $\displaystyle c_{0}=0,$
$\displaystyle-8\pi\omega_{1,0}+32\pi^{4}\alpha_{1}^{3}=0,$
$\displaystyle-8\pi\omega_{2,0}+32\pi^{4}\alpha_{2}^{3}=0,$
$\displaystyle[-8\pi^{2}(\omega_{1,0}+\omega_{2,0})+32\pi^{4}(\alpha_{1}+\alpha_{2})^{3}]\lambda_{12}$
$\displaystyle\ \ \ \
+[-8\pi^{2}(\omega_{1,0}-\omega_{2,0})+32\pi^{4}(\alpha_{1}-\alpha_{2})^{3}]\lambda_{12}^{-1}=0.$

The first three equations in the system (5.9) have a solution

$None$ $\displaystyle c_{0}=0,\ \ v_{0,0}=0,\ \
\omega_{1,0}=4\pi^{2}\alpha_{1}^{3},\ \ \omega_{2,0}=4\pi^{2}\alpha_{2}^{3},$

The fourth equation in the system (5.29) satisfied automatically by using
(5.25) and (5.30), thus we have

$None$ $\displaystyle c_{0}=c_{1}=c_{2}=0,\ \ v_{0,0}=0,\ \
\omega_{1,0}=4\pi^{2}\alpha_{1}^{3},\ \ \omega_{2,0}=4\pi^{2}\alpha_{2}^{3}.$

Using (5.28) and (5.31), we conclude that

$\displaystyle v_{0}=o(\lambda_{11},\lambda_{22})\longrightarrow 0,\ \
c=o(\lambda_{11},\lambda_{22})\longrightarrow 0,\ \
\omega_{1}=4\pi^{2}\alpha_{1}^{3}+o(\lambda_{11},\lambda_{22})\longrightarrow
4\pi^{2}\alpha_{1}^{3},$
$\displaystyle\omega_{2}=4\pi^{2}\alpha_{2}^{3}+o(\lambda_{11},\lambda_{22})\longrightarrow
4\pi^{2}\alpha_{2}^{3},\ \ \ {\rm as}\ \ \lambda_{11},\lambda_{22}\rightarrow
0,$

and therefore we have (5.26). So the two-periodic wave solution (5.23) tends
to the two-supersoliton solution (5.6) as
$\lambda_{11},\lambda_{22}\rightarrow 0$. $\square$
6\. Conclusion and future work

Following the procedure described in this paper, we are able to construct
quasi-periodic wave solutions for other supersymmetric equations also can be
dealt with by the same way. For instance,

(1) Supersymmetric KdV equation [40, 42, 43]

$\displaystyle\Phi_{t}+3\left(\Phi\mathfrak{D}\Phi\right)_{x}+\Phi_{xxx}=0,$

(2) Supersymmetric MKdV equation [41, 49, 65]

$\displaystyle\phi_{t}-3\phi\mathfrak{D}(\phi_{x})\mathfrak{D}\phi-3(\mathfrak{D}\phi)^{2}\phi_{x}+\phi_{xxx}=0,$

(3) Supersymmetric It’s equation [66]

$\displaystyle\mathfrak{D}_{t}F_{t}+6(F_{x}(\mathfrak{D}_{t}F))_{x}+\mathfrak{D}_{t}F_{xxx}=0,$

(4) Supersymmetric two-boson equation [67, 68]

$\displaystyle\phi_{1,t}=\mathfrak{D}((\mathfrak{D}\phi_{1})^{2})+2\phi_{2,x}-\phi_{1,xx},$
$\displaystyle\phi_{2,t}=2((\mathfrak{D}\phi_{1})\phi_{2})_{x}+\phi_{2,xx}.$

The system (3.10) indicates that constructing multi-periodic wave solutions
depends on the solvability of the system. We consider the number of constraint
equations and some unknown parameters. Obviously, the number of constraint
equations of the type (3.10) is $2^{N}$. On the other hand we have parameters
$\tau_{ii},\tau_{ij},\alpha_{i},\omega_{i},\phi_{0},c$, whose total number is
$\frac{1}{2}N(N+1)+3N+2$. Among them, $2N$ parameters $\tau_{ii},\omega_{i}$
are taken to be the given parameters related to the amplitudes and wave
numbers of $N$-periodic waves. Hence, the number of the unknown parameters is
$\frac{1}{2}N(N+1)+N+2$. while $\frac{1}{2}N(N+1)$ 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 $N+2$.
The number of equations is larger than the unknown parameters in the case when
$N>2$. This fact means that if equation (3.10) is satisfied by the unknowns,
we have at least $N$-periodic wave solutions ($N\leq 2$). It is still possible
to construct multi-periodic wave solutions for $N\geq 3$ by adding the number
of parameters (for example, using constant solution and integration constant)
or decreasing the number of equations (for example, using odd and even
properties of considered equations). In this paper, we consider one-periodic
wave solution of the equation (1.2), which belongs to the cases when $N=1$ and
$N=2$ in the Riemann theta function (3.1). There are still certain
difficulties in the calculation for the case $N\geq 3$.

We expect the proposed method to be extended to $\mathcal{N}=1$ supersymmetric
sine-Gordon equation and $\mathcal{N}=1$ supersymmetric KP equation, as well
as other discrete supersymmetric equations (like supersymmetric Toda lattice).
For the $\mathcal{N}=2$ supersymmetric equations with ordinary variables $x,t$
and Grassmann variables $\theta_{1},\theta_{2}$, their corresponding
superspace is
$\mathbb{R}_{\Lambda}^{2,2}=\Lambda_{0}^{2}\times\Lambda_{1}^{2}$. In this
case, the Grassmann algebra $G_{2}(\sigma_{1},\sigma_{2})$ whose dimension is
four. The $\tau$ matrix will be dependent on the odd parameters
$\sigma_{1},\sigma_{2}$. In superspace $\mathbb{R}_{\Lambda}^{2,2}$, the super
bilinear forms of $\mathcal{N}=2$ supersymmetric equations, their quasi-
periodic solutions and asymptotic properties remain open. We intend to return
to these question in some future publications.
Acknowledgment

The work described in this paper was supported by grants from 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]
  * [2] S. P. Novikov, Funct. Anal. Appl. 8(1974): 236-246.
  * [3] B. A. Dubrovin, Funct. Anal. Appl. 9(1975): 265-277.
  * [4] A. Its and V. Matveev, Funct. Anal. Appl. 9(1975): 65-66.
  * [5] P. D. Lax, Comm. Pure Appl. 28(1975): 141-188.
  * [6] H. P. Mckean and P. Moerbeke, Invent. Math. 30(1975): 217-274.
  * [7] E. Belokolos, A. Bobenko, V. Enol’skij, A. Its and V. Matveev, Algebro-Geometrical Approach to Nonlinear Integrable Equations ( Springer, Berlin, 1994).
  * [8] F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions, (Cambridge University Press, 2003).
  * [9] F. Gesztesy and H Holden. _Phil Trans R Soc A_ 366  (2008): 1025-1054.
  * [10] Z. J. Qiao, _Comm Math Phys_ , 239 (2003): 309-341.
  * [11] L. Zampogni, _Advanced Nonl Studies_ , 3 (2007): 345-380.
  * [12] R. G. Zhou, J. Math. Phys. 38(1997): 2535-2346.
  * [13] C. W. Cao, Y. T. Wu and X. G. Geng. J. Math. Phys. 40(1999): 3948-3970.
  * [14] X. G. Geng, Y. T. Wu and C. W. Cao. J. Phys. A 32 (1999): 3733-3742.
  * [15] X. G. Geng, and C. W. Cao. Nonlinearity, 14 (2001): 1433-1452.
  * [16] X. G. Geng, H. H. Dai, J. Y. Zhu and H. Y. Wang. Stud. Appl. Math. 118 (2007): 281-312.
  * [17] Y. C. Hon and E. G. Fan, J. Math.Phys. 46 (2005): 032701-21.
  * [18] J. Hammack, N. Scheffner and H. Segur, J. Fluid Mech. 209 567-589 (1989)
  * [19] J. Hammack, D. McCallister, N. Scheffner and H. Segur, J. Fluid Mech. 285 95-112 (1995)
  * [20] R. Hirota and J. Satsuma: Prog. Theor. Phys. 57 (1977) 797.
  * [21] R. Hirota: Direct methods in soliton theory (Springer-verlag, Berlin, 2004).
  * [22] X. B. Hu and P. A. Clarkson, J. Phys. A 28 (1995) 5009.
  * [23] X. B. Hu, C X Li, J. J. C. Nimmo and G. F. Yu, J. Phys. A, 38 (2005) 195.
  * [24] R Hirota and Y. Ohta: J. Phys. Soc. Jpn. 60 (1991) 798\.
  * [25] D. J. Zhang, J. Phys. Soc. Jpn. 71 (2002) 2649.
  * [26] K. Sawada and T Kotera: Prog. Theor. Phys. 51 (1974) 1355.
  * [27] A. Nakamura, J. Phys. Soc. Jpn. 47, 1701-1705 (1979).
  * [28] A. Nakamura, J. Phys. Soc. Jpn. 48(1980), 1365.
  * [29] R. Hirota and M. Ito, J. Phys. Soc. Jpn. 48(1980), 1365.
  * [30] H. H. Dai, E. G. Fan and X. G. Geng, arxiv.org/pdf/nlin/0602015
  * [31] Y. Zhang, L. Y. Ye, Y. N. Lv and H. Q. Zhao, J. Phys A, 40 (2007), 5539.
  * [32] Y. C. Hon, E. G. Fan and Z. Y. Qin, Modern Phys Lett B, 22 (2008), 547.
  * [33] E. G. Fan and Y. C. Hon, Phys Rev E, 78 (2008), 036607.
  * [34] E. G. Fan, J. Phys A, 42 (2009), 095206.
  * [35] W. X. Ma, R. G. Zhou, J. Math. Phys, 24 (2009), 1677.
  * [36] P. Ramond, Phys. Rev. D, 3 (1971), 2415.
  * [37] A. Neveu and J. H. Schwarz, Nucl. Phys. B, 31 (1971), 86.
  * [38] J. Wess and B. Zumino, Nucl. Phys. B, 70 (1974), 39.
  * [39] F. A. Berezin, Introduction to super-Analysi, Dordrecht: Reidel, 1987.
  * [40] Yu. I. Manin and A. O. Radul, Commun. Math. Phys. 98 (1985), 65.
  * [41] P. Mathieu, J. Math. Phys. 29 (1988), 2499.
  * [42] W. Oevel and Z. Popowicz, Commun. Math. Phys. 139 (1991), 441.
  * [43] A. S. Carstea, Nonlinearity 13 (2000), 1645.
  * [44] A. S. Carstea, A. Ramani and B Grammaticos, Nonlinearity 14 (2000), 1419.
  * [45] D. Sarma, Nucl. Phys. B 681 (2004), 351.
  * [46] Q. P. Liu, Lett Math. Phys. 35 (1995), 115.
  * [47] Q. P. Liu and Y. F. Xie, Phys. Lett. A 325 (2004), 139.
  * [48] Q. P. Liu and X. B. Hu, J. Phys. A 38 (2005), 6371.
  * [49] Q. P. Liu, X. B. Hu and M. X. Zhang, Nonlinearity 18 (2005), 1597.
  * [50] S. Ghosh, J. Nonl. Math. Phys. 10 (2003), 526.
  * [51] V. S. Vladimirov, Theor. Math. Phys. 59 (1984), 3.
  * [52] V. S. Vladimirov, Theor. Math. Phys. 60 (1984), 169\.
  * [53] D. Mumford, Tata Lectures on Theta II Progress in Mathmatics, Vol. 43 ( Boston: Birkhäuser, 1984).
  * [54] R. Hirota and M Ito, J. Phys. Soc. Jpn. 52 (1983) 744
  * [55] K. Konno, J. Phys. Soc. Jpn. 61 (1992) 51
  * [56] Y. X. Yu, Commun. Theor. Phys. 49 (2008), 685.
  * [57] K. Tian and Q. P. Liu, Phys. Lett. A 373 (2009), 169.
  * [58] C. A. Laberge, P. Mathieu, Phys. Lett. B 215 (1988), 718.
  * [59] C. A. Laberge, P. Mathieu, J. Math. Phys. 32 (1991), 923.
  * [60] M. X. Zhang, Q. P. Liu, Y. L. Shen and K. Wu, Sic. in China. 52 (2009), 1973.
  * [61] Z. Popowicz. Phys. Lett. A 174 (1993), 411.
  * [62] S. Bourque and P. Mathieu, J. Math. Phys. 42 (2001), 3517.
  * [63] T. Inami and H. Kanno, Nuclear Phys. B 359 (1991), 201.
  * [64] Q. P. Liu and Y. F. Xie, Phys. Lett. A 235 (1997), 335.
  * [65] S. Ghosh and D. Sarma, Nonlinearity, 16 (2003), 411.
  * [66] S. Q. Liu, X. B. Hu and Q. P. Liu, J. Phys. Soc. Jpn 75 (2006), 064004\.
  * [67] J. C. Brunelli, A. Das, Phys. Lett. B, 337 (1994), 303\.
  * [68] Q. P. Liu, X. X. Yang, Phys. Phys A. 351 (2006), 131.