数理解析研究所講究録
第 1700 巻 2010 年 118-131

118

Darboux transformations for twisted derivations
C.X. Li1* and J.J.C. Nimmo2\dagger
1School of Mathematical Sciences,
Capital Normal University, Beijing 100048, CHINA
2Department of Mathematics,
University of Glasgow, Glasgow G12 8QW, UK

Abstract

This paper is concemed with a generalized type of Darboux transformations defined in
is a homoterms of a twisted derivation $D$ satisfying $D(AB)=D(A)+\sigma(A)B$ where
q-difference
derivations,difference andmorphism. Such twisted derivations include regular
operators and superderivatives as special cases. Remarkably, the formulae for the iteration ofDarboux transformations are identical with those in the standard case of a regular derivation
and are expressed in terms of quasideterminants. As an example, we revisit the Darboux
transformations for the Manin-Radul super $KdV$ equation.
$\sigma$

Introduction

1

Recently noncommutative versions of integrable systems have received much attention [1-14].
It has been shown that such systems often have solutions expressed in terms of quasideterminants [15]. The prototypical example of this is the class of solutions of the noncommutative KP
equation found using Darboux transformations [16]. In [12] also, a second type of quasideterminant solutions for this equation were found using binary Darboux transformations.
Supersymmetric integrable systems are a particular noncommutative extension of integrable
systems. Among these, the Manin-Radul super $KdV$ equation [17] is perhaps the best known
example. Motivated in part by the properties of superderivatives, we consider a generalized
derivation which has regular derivations, difference operators, q-difference operators and superderivatives as some of its special cases. We call this a twisted derivation, following theterminology used in [18, 19]. We show that one can formulate Darboux transformations for such
twisted derivations and the iteration formulae are expressed in terms of quasideterminants inwhich one simply replaces the derivative with the twisted derivation.
In [20, 21] solutions for the Manin-Radul super $KdV$ equation were constructed by means ofDarboux transformations and binary Darboux transformations. In this paper, we use an alternative approach to such Darboux transformations using quasideterminants. This is successful
in obtaining simple unified formulae for the solutions. From these quasideterminant solutions,
we recover the superdeterminant solutions given in [20, 21] and also get a superdeterminant
representation in the cases not considered in the earlier work.
The paper is organized as follows. In Section 2, we give a brief review of relevant propertiesof quasideterminants. In Section 3, in order to introduce the basic ideas, we discuss Darboux
transformations for the noncommutative KP equation. Then, in Section 4, the main results are*trishaJi2001@yahoo.com$\dagger$j.nimmo@maths.gla.ac.uk

119described. Twisted derivation and related Darboux transformation are defined and a quasideterminant iteration formula for twisted Darboux transformation is obtained. Section 5 contains
some basic facts about supersymmetric objects. In Section 6 the Darboux and binary Darboux
transformations to the Manin-Radul super $KdV$ system are discussed. Finally, in Section 7, thesolutions obtained using iterated Darboux and binary Darboux transformations are reexpressedin terms of superdeterminants. Proofs of the results stated in this paper are given in [22].

2Properties of quasideterminants

In this section, we record some basic facts about quasideterminants [15, 16, 23]. The reader isreferred to the above mentioned literature for more details.
quasideAn $n\cross n$ matrix $M=(m_{i)j})$ over a ring(noncommutative, in general) has, which are also elements of . They are definedterminants written as $|M|_{i,j}$ for $i,j=1,$
recursively by$n^{2}$$\mathcal{R}$$\ldots,$$\mathcal{R}$$n$$|M|_{i,j}=m_{i_{1}j}-r_{i}^{j}(M^{i,j})^{-1}c_{j}^{i}$,$M^{-1}=(|M|_{j_{l}i}^{-1})_{i,j=1,\ldots,n}$.

(1)

the jth column
In the above
represents the ith row of $M$ with the jth element removed,with the ith element removed andthe submatrix obtained by removing the ith row andthe jth column from $M$ . Quasideterminants can be also denoted as shown below by boxing theentry about which the expansion is made$r_{i}^{j}$$c_{j}^{i}$$M^{i,j}$$|M|_{i,j}=|_{r_{i}^{j}}^{M^{i,j}}$$m_{i_{l}j}c_{j}^{i}|$.

Note that if the entries in $M$ commute then$|M|_{i,j}=(-1)^{i+j} \frac{\det(M)}{\det(M^{i,j})}$(2).

Noncommutative Jacobi Identity There is a quasideterminant version of the Jacobi identity for determinants [15]. The simplest version of this identity is given by

where $f,$ $g,$ $h,$over$\mathcal{R}$$i\in \mathcal{R},$

A

B

$C$

$D$

$f$

$g$

$E$

$h$

$i$

$A$

.

is an

$n\cross n$$=|_{E}^{A}$$\underline{\prod iC}|-|\begin{array}{ll}A BE h\end{array}||\begin{array}{ll}A BD f\end{array}||\begin{array}{ll}A CD \underline{\Pi}\end{array}|$,(3)

matrix and $B,$ $C$ (resp. $D,$ $E$ ) are column (resp. row) n-vectors

Quasi-Pl\"ucker coordinates Given an $(n+k)\cross n$ matrix $A$ , denote the ith row of $A$ by, the submatrix of $A$ having rows with indices in a subset of $\{1, 2, \ldots, n+k\}$ by
and
, one
by
. Given $i,j\in\{1,2, \ldots, n+k\}$ and such that $\# I=n-1$ anddefines the (right) quasi-Plucker coordinates$I$$A_{i}$$A_{\{1,\ldots,n+k\}\backslash \{i\}}$$A_{\hat{l}}$$A_{I}$$I$$r_{ij}^{I}=r_{ij}^{I}(A):=|\begin{array}{l}A_{I}A_{i}\end{array}||\begin{array}{l}A_{I}A_{j}\end{array}|=-|\begin{array}{ll}A_{I} 0A_{i} 0A_{j} 1\end{array}|$$j\not\in I$,(4)

for any column index $s\in\{1, \ldots, n\}$ . The final equality in (4) comes from an identity of theform (3) and proves that the definition is independent of the choice of .
$s$

120
Derivatives of quasideterminants

Consider the derivative of an arbitrary quasideterminant(5)$|\begin{array}{ll}A BC d\end{array}|=d’-C’ A^{-1}B+CA^{-1}A’ A^{-1}B-CA^{-1}B’$

where is an $n\cross n$ matrix, $C$ is a row vector and $B$ a column vector. Let denote the $n\cross n$
identity matrix and let
denote the kth row and the kth column of a matrix $Z$ ,andrespectively. Then
$A$

$I$

$Z^{k}$$Z_{k}$$|\begin{array}{ll}A BC d\end{array}|’=|\begin{array}{ll}A BC d\end{array}|+ \sum_{k=1}^{n}|\begin{array}{ll}A I_{k}C 0\end{array}| |\begin{array}{ll}A B(A^{k})’ (B^{k})’\end{array}|$

3

.

(6)

Darboux transformations for the $ncKP$ equation

To introduce the key aspects of Darboux transformations we consider the standard example ofthe noncommutative KP $(ncKP)$ equation [1-9, 12, 16]$(v_{t}+v_{xxx}+3v_{x}v_{x})_{x}+3v_{yy}-3[v_{x}, v_{y}]=0$.

(7)

Its Lax pair is$L=\partial_{x}^{2}+v_{x}-\partial_{y}$,(8)$M=4\partial_{x}^{3}+6v_{x}\partial_{x}+3v_{xx}+3v_{y}+\partial_{t}$Let$\theta$

be such that $L(\theta)=M(\theta)=0$ , and we call$\theta$.

(10)in the sense that

$G_{\theta}$$\tilde{L}=G_{\theta}LG_{\theta}^{-1}$(9)an eigenfunction. Define the operator$G_{\theta}=\theta\partial_{x}\theta^{-1}=\partial_{x}-\theta_{x}\theta^{-1}$

The Lax pair is covariant with respect to.

,$\overline{M}=G_{\theta}MG_{\theta}^{-1}$,and $M$ with changed to cir $=v+2\theta_{x}\theta^{-1}$ . This transformation ishave the same form as
called a Darboux transformation. Since the form of and $M$ is preserved, it induces a B\"acklundtransformation for the $ncKP$ equation.
be a generic eigenfunction andThis transformation may be iterated as follows. Let. Thenbe invertible eigenfunctions of $(L[0], M[0])=(L, M)$ . Define
let$:=G_{\theta[0]}(\phi[0])$ andforare eigenfunctions
$L$

$v$

$L$

$\phi_{[0]}=\phi$$\theta_{0},$$\ldots,$$\theta[0]=\theta_{0}$$\theta_{n-1}$$\phi[1]$$\theta[1]=\phi[1]|_{\phiarrow\theta_{1}}$$(L[1], M[1])=(G_{\theta[0]}L[0]G_{\theta[0]}^{-1}, G_{\theta[0]}M[0]G_{\theta[0]}^{-1})$

In general, for $n\geq 0$ define the nth Darboux transform ofby$\phi$$\phi[n+1]=\phi[n]^{(1)}-\theta[n]^{(1)}\theta[n]^{-1}\phi[n]$in which$\theta[k]=\phi[k]|_{\phiarrow\theta_{k}}$

After

$n$

.

,

.

Darboux transformations the change of the Lax pair is that$v[n]=v+2 \sum_{i=0}^{n-1}\theta[i]_{x}\theta[i]^{-1}$

.

(11)

121
Further, it may be proved by induction that$\sum_{i=0}^{n-1}\theta[i]_{x}\theta[i]^{-1}=-|\begin{array}{ll}\Theta 0\vdots \vdots\Theta^{(n-2)} 0\Theta^{(n-1)} 1\Theta^{(n)} 0\end{array}|$

(12)

,

where
and
is its kth derivative with respect to .
To define a binary Darboux transformation one needs to consider the adjoint Lax pair$\Theta^{(k)}$$\Theta=(\theta_{0}, \ldots, \theta_{n-1})$$x$$L^{\uparrow}=\partial_{x}^{2}+v_{x}^{\uparrow}+\partial_{y}$

,

(13)

$M^{\uparrow}=-4\partial_{x}^{3}-6v_{x}^{\uparrow}\partial_{x}-3v_{xx}^{\uparrow}+3v_{y}^{\uparrow}-\partial_{t}$

.

(14)

Following the standard construction of a binary Darboux transformation (see [24, 25]) one
introduces a potential
satisfying$\Omega(\phi, \psi)$$\Omega(\phi, \psi)_{x}=\psi^{\uparrow\emptyset}$,,$\Omega(\emptyset, \psi)_{t}=-4(\psi\uparrow\phi_{xx}-\psi_{x}^{1}\phi_{x}+\psi_{xx}^{\uparrow}\phi)-6\psi\dagger_{v_{x}\phi}$$\Omega(\phi, \psi)_{y}=\psi^{\uparrow}\phi_{x}-\psi_{x}^{\uparrow\emptyset}$.

(15)$L^{\uparrow}(\psi)=M^{\uparrow}(\psi)=0$
The definition is consistent whenever $L(\phi)=M(\phi)=0$ and. Moregenerally, we can define
for any row vectors andsuch that $L(\Phi)=M(\Phi)=0$ and$L^{\uparrow}(\Psi)=M^{\uparrow}(\Psi)=0$ . If
is an n-vector and is an m-vector thenis an $m\cross n$ matrix.
A binary Darboux transformation is then defined by$\Psi$$\Phi$$\Omega(\Phi, \Psi)$$\Phi$$\Psi$$\Omega(\Phi, \Psi)$$\phi_{[n+1]}=\phi_{[n]}-\theta_{[n]}\Omega(\theta_{[n]}, \rho_{[n]})^{-1}\Omega(\phi_{[n]}, \rho_{[n]})$and$\psi_{[n+1]}=\psi_{[n]}-\rho_{[n]}\Omega(\theta_{[n]}, \rho_{[n]})^{-\dagger}\Omega(\theta_{[n]}, \psi_{[n]})^{\dagger}$

,

where
$\theta_{[n]}=\phi_{[n]}|_{\phiarrow\theta_{n}}$Using the notation
for $n\geq 1$ ,$\Theta=(\theta_{0}, \ldots, \theta_{n-1})$,$\rho_{[n]}=\psi_{[n]}|_{\psiarrow\rho_{n}}$(as above) and $P=(\rho_{0}, \ldots, \rho_{n-1})$ it is can be shown that$\phi_{[n]}=|\begin{array}{llll}\Omega(\Theta P) \Omega(\phi P)\Theta \phi \end{array}|$$\Omega(\Theta, P)^{\uparrow}$$\Omega(\Theta, \psi)^{\uparrow}$$P$$\psi$,(16),$\psi_{[n]}=$

(17)

and
$\Omega(\phi_{[n]}, \psi_{[n]})=|\begin{array}{ll}\Omega(\Theta,P) \Omega(\phi,P)\Omega(\Theta,\psi) \Omega(\phi,\psi)\end{array}|$

.

(18)

The effect of this transformations on the Lax pair is to give new coefficients defined in terms of$\hat{v}=v+2\theta\Omega(\theta, \rho)^{-1}\rho^{\uparrow}$

Thus after$n$.

binary Darboux transformations we obtain$v_{[n]}=v+2 \sum_{k=0}^{n-1}\theta_{[k]}\Omega(\theta_{[k]}, \rho_{[k]})^{-1}\rho_{[k]}^{\dagger}$

,

(19)

122
and this may be reexpressed in terms of a single quasideterminant as$v_{[n|}=v-2|\begin{array}{lll}\Omega(\Theta P) P^{\uparrow}\Theta 0\end{array}|$

.

(20)

In this way one obtains a second expression for solutions of the $ncKP$ equation in terms ofquasideterminants.

4

Darboux transformations for twisted derivations

Suppose thatis an associative, unital algebra over ring $K$ . Suppose that there is a homo$\sigma(a+b)=\sigma(a)+\sigma(b)$(i.e. for all $\alpha\in K,$ $a,$morphismsatisfying
and $\sigma(ab)=\sigma(a)\sigma(b))$ and a twisted derivation or -derivation [18, 19]$D(K)=0$ and $D(ab)=D(a)b+\sigma(a)D(b)$ .
depend on a
Important particular examples of such a set-up arise when elements
variable , say.
$\mathcal{A}$$\sigma:\mathcal{A}arrow \mathcal{A}$$b\in \mathcal{A},$$\sigma(\alpha a)=\alpha\sigma(a),$$D:\mathcal{A}arrow \mathcal{A}$$\sigma$$a\in \mathcal{A}$$x$

Derivative Here

$D=\partial/\partial x$

satisfies $D(ab)=D(a)b+aD(b)$ and

Forward difference The homomorphism is the shift operator
and the twisted derivation is$\sigma$$T$

is the identity mapping.

, where $T(a(x))=a(x+1)$$\Delta(a(x))=\frac{a(x+h)-a(x)}{h}$ ,satisfying $\triangle(ab)=D(a)b+T(a)D(b)$ .

Jackson derivative The homomorphism is a q-shift operator defined by $S_{q}(a(x))=a(qx)$ andthe twisted derivation is$D_{q}(a(x))= \frac{a(qx)-a(x)}{(q-1)x}$ .

satisfying $D_{q}(ab)=D_{q}(a)b+S_{q}(a)D_{q}(b)$ .
Superderivative As described in Section 5, for $a,$
where is the grade involution.

$b\in \mathcal{A}$, a superalgebra, $D(ab)=D(a)b+\hat{a}D(b)$$\sim$

4.1

Darboux transformations

Here we consider a more abstract situation modelled on the Darboux transformation for the KP
equation. Let
be a sequence in . Consider the sequencein ,generated from the first sequence by Darboux transformations of the form$\theta_{0},$$\theta_{1},$$\theta_{2},$$\mathcal{A}$$\theta[0],$ $\theta[1],$$\ldots$$G_{\theta}=\sigma(\theta)D\theta^{-1}=D-D(\theta)\theta^{-1}$$\theta[2],$$\mathcal{A}$$\ldots$,(21)

where $D$ andare the twisted derivation and homomorphism defined above. To be specific,
and $G[0]=G_{\theta[0]}$ , then let$\sigma$$\theta[0]=\theta_{0}$(22)$\theta[1]=G[0](\theta_{1})=D(\theta_{1})-D(\theta_{0})\theta_{0}^{-1}\theta_{1}$and $G[1]=G_{\theta[1]},$ $\theta[2]=G[1]\circ G[0](\theta_{2})$ and $G[2]=G_{\theta[2]}$ and so on. In general, for$\theta[k]=G[k-1]\circ G[k-2]0\cdots oG[0](\theta_{k})$ ,$G[k]=\sigma(\theta[k])D\theta[k]^{-1}$,$k\in \mathbb{N}$

,

(23)

123
in which we require that each
is invertible.
In the standard case of a derivation, $D=\partial$ andId, it is well known that the terms
in the sequence of Darboux transformations have closed form expressions in terms of the original sequence. In the case thatis commutative, they are expressed as ratios of wronskian
determinants [26],$\theta[k]$$\sigma=$$\mathcal{A}$$\theta[n]=\frac{1_{\theta_{0}^{(n-1)}\ldots\theta_{n-1}^{(n-1)}\theta_{n}^{(n-1)1}}^{\theta_{0}^{(.\cdot.1)}.\cdot.\cdot.\cdot\theta_{n-1}^{(1)}\theta_{n}^{(1)}}\theta_{0}^{(n)}\theta_{n-1}^{(n)}\theta_{n}^{(n)}\theta_{0}\ldots\theta_{n.-1}\theta_{n}}{|\begin{array}{lll}\theta_{0} \cdots \theta_{n-1}\theta_{0}^{(1)} \cdots \theta_{n-1}^{(1)}| |\theta_{0}^{(n-1)} \cdots \theta_{n-1}^{(n-1)}\end{array}|}$

,

where
denotes. In the case that
expressed as quasideterminants [16],$\theta_{j}^{(i)}$$\partial^{i}(\theta_{j})$$\theta_{0}$$\theta_{0}^{(1)}$$\mathcal{A}$$\theta_{n-1}$$\theta_{n}$...

$\theta_{n-1}^{(1)}$$\theta_{n}^{(1)}$:$\theta_{0}^{(n-1)}$$\theta_{0}^{(n)}$

...
...

,

(24)

is not commutative, the terms in the sequence are...

:$\theta[n]=$$n\in \mathbb{N}$$n\in \mathbb{N}$$\theta_{n-1}^{(n-1)}$$\theta_{n}^{(n-1)}$$\theta_{n-1}^{(n)}$$\theta_{n}^{(n)}$

.

(25)

The following theorem gives a generalisation of this formula to the case of general $D$ and . Notein particular that the expressions do not depend on and are obtained simply by replacingwith $D$ . It is proved by induction.
$\sigma$$\partial$$\sigma$

Theorem 1. Let $\phi[0]=\phi$ and for$n\in \mathbb{N}$let$\phi[n]=D(\phi[n-1])-D(\theta[n-1])\theta[n-1]^{-1}\phi[n-1]$ ,where$\theta[n]=\phi[n]|_{\phiarrow\theta_{n}}$. Then, for$\phi[n]=$$n\in \mathbb{N}$,$\theta_{0}$$\theta_{n-1}$$\phi$$D(\theta_{0})$$D(\theta_{n-1})$$D(\phi)$:$D^{n-1}(\theta_{0})$$D^{n}(\theta_{0})$

...

:

:

$D^{n-1}(\theta_{n-1})$$D^{n-1}(\phi)$...

$D^{n}(\theta_{n-1})$$D^{n}(\phi)$(26)

As an application of this theorem, we will apply it to the super $KdV$ equation in which thetwisted derivation is a superderivative. Before that, we will recall the definition of a superalgebra
and related concepts.

5Superalgebras and superderivatives

In this section, we collect together some basic facts about supersymmetric objects such as
superderivatives, supermatrices, supertranspose and superdeterminants [27, 28] and about therelationship between superdeterminants and quasideterminants [29].

124
be a supercommutative, associative, unital superalgebra over a (commutative) ringLet
$K$ . There is a standard. Elements of thatsuch that-grading
are called even and those inare called homogeneous; those inor
belong to eitherof a homogeneous element is if it is even and 1 if it isare called odd. The parity$\mathcal{A}$$\mathcal{A}=\mathcal{A}_{0}\oplus \mathcal{A}_{1}$$\mathbb{Z}_{2}$$\mathcal{A}_{0}$$\mathcal{A}_{1}$$\mathcal{A}_{0}$$\mathcal{A}$$\mathcal{A}_{i}\mathcal{A}_{j}\subseteq \mathcal{A}_{i+j}$$|a|$$\mathcal{A}_{1}$

$0$

$a$

$thatallodd.Ithomogeneouse1ementsa,bsatisfyba=(-1)|_{a||b|}ab,ie.evene1ementscommutewithf_{0}11owsthatifa,barehomogeneousthen|ab$

all elements, and odd elements anticommute. In particular, this implies that $a_{1}^{2}=0$ , for all$a_{1}\in \mathcal{A}_{1}$.

satisfying
Grade involution and superderivative The homomorphism$(-1)^{i}a_{i}$ for, expressed as $a=a_{0}+a_{1}$is called the grade involution. For general$\hat{a}=a_{0}-a_{1}$$M=(m_{ij})$. It isany
over.
matrixfor,Also
we havewhereeasy to see that.
$D$
such that $D(K)=0$ andis a linear mapping
A superde rivativeand satisfying $D(ab)=D(a)b+\hat{a}D(b)$ . One way to obtain a superderivative is as
where is an even variable and is an odd (Grassmann) variable. For such a superderivative.
, it follows that $D(\hat{a})=D(a_{0})-D(a_{1})=-\overline{D(a)}$andNote that sinceand so grade involution and superderivatives anticommute.
$\wedge:\mathcal{A}arrow \mathcal{A}$$\hat{a}_{i}=$$a\in \mathcal{A}$$a_{i}\in \mathcal{A}_{i}$$\mathcal{A},\hat{M}:=(\hat{m}_{ij})$$a_{i}\in \mathcal{A}_{i}$$\hat{\hat{a}}=a$$D:\mathcal{A}arrow \mathcal{A}$$D(\mathcal{A}_{i})\subseteq \mathcal{A}_{i+1}$$D=\partial_{\theta}+\theta\partial_{x}$$\theta$$x$$D^{2}=\partial_{x}$$D(\mathcal{A}_{1})\subseteq \mathcal{A}_{0}$$D(\mathcal{A}_{0})\subseteq \mathcal{A}_{1}$A block matrix

Even and odd supermatrices

over$\mathcal{M}=(\begin{array}{ll}X YZ T\end{array})$$\mathcal{A}$where $X$ is $r\cross m,$

$Y$

$Z$
for integers $r,$ $m$ and with $m\geq 1$ and $n\geq 0$ is calledisis $s\cross m$ and $T$ is
an $(r|s)\cross(m|n)$ supermatrix. It is said to be even, and has parity , if $X$ and $T$ (if not empty)
have even entries and $Y$ and $Z$ (if non-empty) have odd entries. One the other hand, if $X$ andis said to be odd, and has parity 1. It
have odd entries and $Y,$ $Z$ have even entries thenis said to be homogeneous if it is either even or odd.
$s\cross n$$r\cross n,$

$n$

$s,$

$s,$

$r,$

$0$

$T$

$\mathcal{M}$Supertranspose

The supertranspose of a homogeneous supermatrix$\mathcal{M}^{st}=(\begin{array}{ll}X^{t} (-1)^{|\mathcal{M}|}Z^{t}-(-1)^{|\mathcal{M}|}Y^{t} T^{t}\end{array})$$\mathcal{M}$, is defined to be

,

(27)

where denotes the normal matrix transpose. In particular, an even $(m|n)$ -row vector has theform $(a_{01}, a_{02}, \ldots , a_{0m}, a_{11}, a_{12}, \ldots, a_{1n}))$ where $a_{ij}\in A$ , and its supertranspose is$t$$(a_{01}, a_{02}, \ldots, a_{0m}, a_{11}, a_{12}, \ldots, a_{1n})^{st}=(a_{01}, a_{02}, \ldots, a_{0m}, -a_{11}, -a_{12}, \ldots, -a_{1n})^{t}$

.

(28)

On the other hand, an odd $(m|n)$ -row vector has the form $(a_{11}, a_{12}, \ldots, a_{1m}, a_{01}, a_{02}, \ldots , a_{0n})$ ,and the supertranspose$(a_{11}, a_{12}, \ldots, a_{1m}, a_{01}, a_{02}, \ldots, a_{0n})^{st}=(a_{11}, a_{12}, \ldots, a_{1m}, a_{01}, a_{02}, \ldots, a_{0n})^{t}$

For homogenous supermatrices$\mathcal{L},$$\mathcal{M}$and$\mathcal{N}$

.

(29)

, it is known that$(\mathcal{M}\mathcal{N})^{st}=(-1)^{|\mathcal{M}||\mathcal{N}|}\mathcal{N}^{st}\mathcal{M}^{st}$$(\mathcal{M}^{st})^{st}=(-1)^{1\mathcal{M}}$I$\hat{\mathcal{M}}$

,

(30)
(31)

.

Supertranspose commutes with the grade involution but not with a superderivative; for a ho,mogeneous matrix$\mathcal{M}$$(\mathcal{M}st_{=\overline{(\mathcal{M})^{st}}}$,$(D(\mathcal{M}))^{st}=(-1)^{|\Lambda 4|}D(\hat{\mathcal{M}}^{st})$

.

(32)

125
Superdeterminants$X$

Consider an even $(m|n)\cross(m|n)$ supermatrix

and $T$ are non-singular. The superdeterminant, or Berezinian, of$\mathcal{M}=(\begin{array}{ll}X YZ T\end{array})$$\mathcal{M}$in which

is defined to be

Ber $( \mathcal{M})=\frac{\det(X-YT^{-1}Z)}{\det(T)}=\frac{\det(X)}{\det(T-ZX^{-1}Y)}$ .

It is also convenient to define$Ber^{*}(\mathcal{M})=\frac{1}{Ber(\mathcal{M})}$.

Relationship between quasideterminants and superdeterminants The basic formulaeconnecting quasideterminants of even supermatrices with their Berezinians are given in [29].

Theorem 2. Let$\mathcal{M}$be an $(m|n)\cross(m|n)$ -supermatrix. Then

(33)$|\mathcal{M}|_{ij})=\{\begin{array}{ll}(-1)^{i+j}\frac{Ber(\mathcal{M},)}{Ber(\mathcal{M}^{ij})} 1\leq i, j\leq m,(-1)^{i+j}\frac{Ber^{*}(\mathcal{M},)}{Ber^{*}(\mathcal{M}^{ij})} m+1\leq i,j\leq m+n,\end{array}$(cf. (2).)
is given
Roughly speaking, a quasideterminant with indices in one of the even blocks ofas a ratio of Berezinians. A quasideterminant with its indices in the one of the odd blocks is
not well-defined.
$\mathcal{M}$

6

The Manin-Radul super $KdV$ equation

The Manin-Radul supersymmetric $KdV$ (MRSKdV) system [17] is$\alpha_{t}=\frac{1}{4}(\alpha_{xx}+3\alpha D(\alpha)+6\alpha u)_{x}$,$u_{t}= \frac{1}{4}(u_{xx}+3u^{2}+3\alpha D(u))_{x}$

,

(34)

where andare even and odd dependent variables respectively, $x,$ are even independent, where is a Grassmann odd
variables and $D$ is the superderivative defined byvariable, satisfying. This system has the Lax pair
$u$

$t$

$\alpha$$D=\partial_{\theta}+\theta\partial_{x}$$\theta$$D^{2}=\partial_{x}$$L=\partial_{x}^{2}+\alpha D+u$

,

(35)

,

(36)

$M= \partial_{x}^{3}+\frac{3}{4}((\alpha\partial_{x}+\partial_{x}\alpha)D+u\partial_{x}+\partial_{x}u)$

in the sense that $L_{t}+[L, M]=0$ implies (34). Eigenfunctions satisfy$L(\phi)=\lambda\phi$for eigenvalue

6.1$\lambda$,$\phi_{t}=M(\phi)$

,

(37)

.

Darboux transformations

A Darboux transformation for this system [21] is$\phiarrow D(\phi)-D(\theta)\theta^{-1}\phi$

(38)

,

$\alphaarrow-\alpha+2(D(\theta)\theta^{-1})_{x}$,$uarrow u+D(\alpha)-2D(\theta)\theta^{-1}(\alpha-(D(\theta)\theta^{-1})_{x})$

(39)

,

(40)

126
where is an invertible, and hence necessarily even, solution of (37). Note that it is an exampleof the general type of Darboux transformation discussed in Section 4.1. As discussed there, thisof (37) to obtaintransformation may be iterated by taking solutions$\theta$$\theta_{0},$$\theta_{1},$$\theta_{2},$$\ldots$$\phi[k+1]=D(\phi[k])-D(\theta[k])\theta[k]^{-1}\phi[k]$ ,

(41)

.

(42)

$\theta[k]=\phi[k]|_{\phiarrow\theta_{k}}$is invertible means that it must be even and consequently that
The requirement that each$u[0]=u$ andmust have parity . The corresponding solutions of MRSKdV are$\theta[k]$$\alpha[0]=\alpha,$$i$$\theta_{i}$$\alpha[k+1]=-\alpha[k]+2(D(\theta[k])\theta[k]^{-1})_{x}$

(43)

,

$u[k+1]=u[k]+D(\alpha[k])-2D(\theta[k])\theta[k]^{-1}(\alpha[k]-(D(\theta[k])\theta[k]^{-1})_{x})$

.

(44)

as a quasideterminant andFrom Theorem 1, we have a closed-form expression (26) for
and $u[n]$ may also be found. For $j\geq 0$ define thethe corresponding expressions for$\phi[n]$$i,$$\alpha[n]$quasideterminants$\theta_{0}$$D(\theta_{0})$

...
...

$\theta_{n-1}$$0$$D(\theta_{n-1})$

$0$

.

:

$D^{n-j-2}(\theta_{n-1})$$0$$D^{n-j-1}(\theta_{n-1})$$1$$D^{n-j}(\theta_{n-1})$

$0$

:

:

$D^{n-1}(\theta_{n-1})$$0$$D^{n+i}(\theta_{n-1})$

$0$

.

.

$D^{n-j-2}(\theta_{0})$$Q_{n}(i, j)=D^{n-j-1}(\theta_{0})$$D^{n-j}(\theta_{0})$

...
...
...

:

.
...

(45)

$.\cdot.\cdot$$D^{n-1}(\theta_{0})$$D^{n+i}(\theta_{0})$$\theta_{0}$.
.
$D^{n-j-2}(\theta_{0})$

$=-$

.

...
...

$\theta_{n-1}$.
$\theta_{0}$$D^{n-j-2}(\theta_{n-1})$$D^{n-j}(\theta_{n-1}):$for any $s=1,$$\ldots,$

.

. . .

$D^{n-1}(\theta_{n-1})$$D^{n+i}(\theta_{0})$...

$D^{n+i}(\theta_{n-1})$

$n$

’

(46)

$.\cdot.\cdot$$..\cdot.\cdot$$D^{n-1}(\theta_{0})$$-1$$\theta_{n-1}$:$D^{n-j}(\theta_{0})$:...

$D^{n-1}(\theta_{0})$.

$D^{n-1}(\theta_{n-1})_{n-j,s}:$$n,s$(see (4)).

Theorem 3. After repeated Darboux transformations, the MRSKdV system has new solutions
and $u[n]$ expressed in terms of $Q_{n}(0,0)$ and $Q_{n}(0,1)$ .
$n$$\alpha[n]$$\alpha[n]=(-1)^{n}\alpha-2Q_{n}(0,0)_{x}$$u[n]=u-2Q_{n}(0,1)_{x}-2Q_{n}(0,0)((-1)^{n} \alpha-Q_{n}(0,0)_{x})+\frac{1-(-1)^{n}}{2}D(\alpha)$

6.2

(47)

,

.

(48)

Binary Darboux transformations

Binary Darboux transformations for the MRSKdV system were discussed in [20, 30]. In thesearticles, solutions expressed in terms of determinants were obtained. As discussed in connection
with Darboux transformations it is to be expected that solutions for this supersymmetric system

127
should be superdeterminants in general. In this section, we will construct a more general type ofbinary Darboux transformation which will be shown to give these superdeterminants solutions
and includes the solutions found in 20, 30 as a special case.
First we recall the definition of the adjoint for supersymmetric linear operators. For a linear
operator $P,$ $|P|$ denotes its parity. For example, $|D|=1$ and $|\partial|=0$ , where
denotes any
derivative with respect to an even variable, and the parity of multiplication by a homogeneous
element is the parity of that element (in the usual sense). The rules defining the superadjoint
$]$

$[$

$\partial$are$D^{\uparrow}=-D$

where$\mathcal{M}$denotes any matrix over$\mathcal{A}$,$\partial^{\uparrow}=-\partial$,$\mathcal{M}^{\dagger}=\mathcal{M}^{st}$

(49)

,

, together with the product rule$(PQ)^{\dagger}=(-1)^{|P||Q|}Q^{\dagger}P^{\dagger}$

,

(50)

where $P$ and $Q$ are operators (cf. (30) for the case of matrices). In particular, this gives$(D^{n})^{\uparrow}=(-1)^{n(n+1)/2}D^{n}$ and, consistently,. For any
.
(35),
pair
(36) has the adjoint form
The Lax$(\partial^{n})^{\uparrow}=(-1)^{n}\partial^{n}$$L^{\uparrow}=\partial_{x}^{2}+D\alpha+u$$a\in \mathcal{A},$$a^{\uparrow}=a$

,

(51)

$M^{\dagger}=- \partial_{x}^{3}-\frac{3}{4}(D(\alpha\partial_{x}+\partial_{x}\alpha)+u\partial_{x}+\partial_{x}u)$

,

(52)

and adjoint eigenfunctions satisfy$L^{\uparrow}(\psi)=\xi\psi$,$-\psi_{t}=M^{\dagger}(\psi)$

,

(53)

for eigenvalue . Given an (eigenfunction, adjoint) eigenfunction pairtransformation [20, 30] is given by$\xi$$\phiarrow\phi-\theta\Omega(\theta, \rho)^{-1}\Omega(\phi, \rho)$$(\theta, \rho)$, the binary Darboux,$\psiarrow\psi-\rho\Omega(\theta, \rho)^{-1}\Omega(\theta, \psi)$

(54)
,

(55)
(56)

$\alphaarrow\alpha+2(\theta\Omega(\theta, \rho)^{-1}\hat{\rho})_{x}$$uarrow u-2(\alpha+(\theta\Omega(\theta, \rho)^{-1}\hat{\rho})_{x})\theta\Omega(\theta, \rho)^{-1}\hat{\rho}+2(\theta\Omega(\theta, \rho)^{-1}D(\rho))_{x}$

,

(57)

where eigenfunction and adjoint eigenfunction have opposite parities. Since $D(\Omega(\phi, \psi)=\psi\phi$ ,is even and assumed to be invertible. When iterating this transformation, both previous
papers [20, 30] on this topic considered the case that all eigenfunctions are even and all adjoint
eigenfunctions are odd. We will show that this is not the most general possibility however.
Consider an even $(m|n)$ -row vector eigenfunction
and an odd $(m|n)$ -row
vector adjoint eigenfunction $\mathcal{O}=(\rho 0, \ldots, \rho_{m+n-1})$ , where
for $i=0,$$m-1$ and
for
even
and
$n-1$ are$j=0,$
for $i=0,$
for $j=0,$$m-1$ and$n-1$ are odd.
These row vectors satisfy$\theta$$\rho$$\Omega$$\mathcal{E}=(\theta_{0}, \ldots\theta_{m+n-1})$$\theta_{i}$$\rho_{m+j}$$\ldots,$$\rho_{i}$$\ldots,$$\ldots,$$L(\mathcal{E})=\mathcal{E}\Lambda$$L^{\dagger}(\mathcal{O})=\mathcal{O}\Xi$

where
Then

$\Lambda$

and

$\Xi$

$\theta_{m+j}$,,$\mathcal{E}_{t}=M(\mathcal{E})$$\ldots,$,$-\mathcal{O}_{t}=M^{\uparrow}(\mathcal{O})$

(58)

,

(59)

are constant $(m+n)\cross(m+n)$ diagonal matrices containing the eigenvalues.
is an even $(m|n)\cross(m|n)$ -supermatrix defined up to a constant by$\Omega=\Omega(\mathcal{E}, \mathcal{O})$$D(\Omega)=\mathcal{O}^{\uparrow}\mathcal{E}$,$\Omega\Lambda-\Xi\Omega=D(\mathcal{O}^{\uparrow\dagger}\mathcal{E}_{x}-\mathcal{O}_{x}\mathcal{E})-\hat{\mathcal{O}}^{\uparrow}\alpha \mathcal{E}$

,

(60)

. (61)$\Omega_{t}=D(\mathcal{O}_{xx}^{\uparrow \mathcal{E}-\mathcal{O}_{x}^{\uparrow \mathcal{E}_{x}+\mathcal{O}^{\uparrow \mathcal{E}_{xx})+\frac{3}{2}\hat{\mathcal{O}}_{x}^{\dagger}\alpha \mathcal{E}+\frac{3}{4}\hat{\mathcal{O}}^{\dagger}\alpha_{x}\mathcal{E}+\frac{3}{2}D(\mathcal{O}^{\dagger}\alpha D(\mathcal{E}))+\frac{3}{2}D(\mathcal{O}u\mathcal{E})}}}\dagger$

The closed form expressions for the results of iterated binary Darboux transformations arestated in the following theorems.

128
Theorem 4. Iterating the binary Darboux $transfom\iota ations(54),$ (55) for $m+n\geq 1$ , one obtains$\phi[m+n]=\Omega(\mathcal{E}, \mathcal{O})\mathcal{E}$$\psi[m+n]=\Omega(\mathcal{E}, \mathcal{O})^{\uparrow}\mathcal{O}$$\Omega(\phi, \mathcal{O})\phi$$\Omega(\mathcal{E}, \psi)^{\uparrow}\psi|=|_{\Omega^{\frac{\overline(\mathcal{E},\mathcal{O}}{(\mathcal{E},\psi}})}^{\Omega)}$’$\mathcal{O}^{1}\psi|$

(62)

with
$\Omega(\phi[m+n], \psi[m+n])=|\begin{array}{ll}\Omega(\mathcal{E},\mathcal{O}) \Omega(\phi,\mathcal{O})\Omega(\mathcal{E},\psi) \Omega(\phi,\psi)\end{array}|$

.

(63)

respectively be even and odd
be a solution of MRSKdV and let andTheorem 5. Let$(m|n)$ -row vectors satisfying (58) and (59). Then for any integers $m+n\geq 0$$\mathcal{O}$$\mathcal{E}$$(\alpha, u)$$\alpha[m+n]=\alpha-2A[m+n]_{x}$ ,$u[m+n]=u+2(\alpha-A[m+n]_{x})A[m+n]-2U[m+n]_{x}$ , (64)where$A[m+n]=|\begin{array}{lll}\Omega(\mathcal{E} \mathcal{O}) \hat{o}\dagger\mathcal{E} 0\end{array}|$,$U[m+n]=|\begin{array}{lll}\Omega(\mathcal{E} \mathcal{O}) D(\mathcal{O}^{\uparrow})\mathcal{E} 0\end{array}|$

,

(65)

are also solutions of MRSKd .
$V$

6.3

From quasideterminants to superdeterminants

It is usual in a supersymmetric integrable system for the solutions to be expressable in termsof superdeterminants. Indeed, in this section we will show that this can be done here. The
expressions we will obtain coincide with the superdeterminant solutions found in [21] and we
also find the superdeterminant expressions in the case that they did not.
Let us therefore introduce the relabeling of the eigenfunctions used in the Darboux transformations(66), $\theta_{2k+1}=O_{k}$ .
$\theta_{2k}=E_{k}$are odd. Also, we write
are even and all
Recall that has parity so that all$D^{2j+1}(\theta)=D(\theta^{(j)})$ , where
denotes the jth derivative with respect to .
andConsider the matrix$\theta_{i}$$O_{k}$$E_{k}$

$i$

$(j)$

$D^{2j}(\theta)=\theta^{(j)}$$x$$W_{n}=\{\begin{array}{lll}\theta_{0}\ddots \cdots \theta_{n-l}| \ddots |D^{n-1}(\theta_{0}) \cdots D^{n-1}(\theta_{n-1})\end{array}\}$

,

(67)

appearing in the definition (46) of $Q_{n}(i,j)$ . There is a natural reordering of the rows and columns$W_{n}arrow \mathcal{W}_{n}=\{\begin{array}{ll}X_{n} Y_{n}Z_{n} T_{n}\end{array}\}$

,

(68)

. This reordering does not change the value of any associatedwhich gives an even matrix
quasideterminant, as long as the expansion point in each refers to the same element. In the casethat is even,$\mathcal{W}_{n}$$n$$X_{2k}=\{\begin{array}{lll}E_{0} \cdots E_{k-l}| \ddots |E_{0}^{(k-1)} \cdots E_{k-1}^{(k-1)}\end{array}\}$,$Y_{2k}=\{\begin{array}{lll}O_{0} \cdots O_{k-1}| \ddots |O_{0}^{(k-1)} \cdots O_{k-1}^{(k-1)}\end{array}\}$129
and $Z_{2k}=D(X_{2k})$ and $T_{2k}=D(Y_{2k})$ are all
matrices. In the case that$(k+1)\cross(k+1),$ $Y_{2k+1}$ is $(k+1)\cross k,$ $Z_{2k+1}$ is $k\cross(k+1)$ and $T_{2k+1}$ is a
precise form can be easily deduced from the above description.
Similarly, consider the matrix$k\cross k$$n$is odd, $X_{2k+1}$ ismatrix whose$k\cross k$$W_{n}=\{\begin{array}{lll}\theta_{0}\ddots \cdots \theta_{n-1}| \ddots |D^{n-3}(\theta_{0}) \cdots D^{n-3}(\theta_{n-1})D^{n-l}(\theta_{0}) \cdots D^{n-1}(\theta_{n-1})D^{n}(\theta_{0}) \cdots D^{n}(\theta_{n-1})\end{array}\}$

,

(69)

appearing in the definition (46) of $Q_{n}(0,1)$ . A similar reordering of this matrix(70)$W_{n}’arrow \mathcal{W}_{n}=\{\begin{array}{ll}X_{n}’ Y_{n}Z_{n}’ T_{n}’\end{array}\}$gives another even matrix$\mathcal{W}_{n}$where, for example,$X_{2k}=\{\begin{array}{lll}E_{0} \cdots E_{k-1}|\cdots \ddots |E_{0}^{(k-2)}E_{0}^{(k)} \cdots E_{k\frac{k}{k(k}1}^{(-2)}E_{-1})\end{array}\}$.

The solutions obtained by use of Darboux transformations (47) $-(48)$ are expressed in termsof two particular quasideterminants $Q_{n}(0,0)$ and $Q_{n}(0,1)$ , The following theorem given the
superdeterminant expressions for these.

Theorem 6. For$n\in \mathbb{N}$,$Q_{n}(0,0)=D(\log(B(\mathcal{W}_{n})))$

where $B=$ Ber if$n$,$Q_{n}(0,1)=- \frac{B(\mathcal{W}_{n}’)}{B(\mathcal{W}_{n})}$

(71)

,

is even, and $B=$ Ber $*ifn$ is odd.

Next we will show how the quasideterminant solutions $(A[m+n], U[m+n])$ obtained using
binary Darboux transformations can also be expressed in terms of superdeterminants. To dothis, it is necessary to introduce a more detailed notation for row vector eigenfunctions andadjoint eigenfunctions. Recall that for the general transformation we use $(m|n)$ -row vectors
andwhich are even and odd with entries
andrespectively. Here we will also write
andfor the row vectors containing the first entries ofandrespectively, and denote by subscript and 1 the even and odd element parts of andrespectively. Thusand.

$\mathcal{E}$$\mathcal{O}$$\theta_{i}$$\rho_{i}$$\mathcal{E}^{i}=(\theta_{0}, \ldots, \theta_{i-1})$$\mathcal{O}^{i}=(\rho 0, \ldots, \rho_{i-1})$$\mathcal{O}$

$i$

$0$

$\mathcal{O}$$\mathcal{E}=(\mathcal{E}_{0}, \mathcal{E}_{1})$$\mathcal{E}$$\mathcal{E}$$\mathcal{O}=(\mathcal{O}_{1}, \mathcal{O}_{0})$Theorem 7. The expressions $(A[m+n], U[m+n])$ can expressed as$A[m+n]=D$ $(\log$ Ber$(\mathcal{G}_{(m|n)}))$,$U[m+n]= \frac{Ber(\mathcal{G}_{(m+1|n)}’)}{Ber(\mathcal{G}_{(m|n)})}$

where$\mathcal{G}_{(m|n)}=(-.\sim\sim\sim\sim--\sim\sim\sim\sim-;^{\mathfrak{l}}\sim--\sim-\sim\Omega(\overline{\overline{\mathcal{E}}}_{0},\overline{\mathcal{O}}_{0})\mathfrak{l}_{1}\overline{\overline{\Omega}}(\overline{\mathcal{E}}_{1},\overline{\overline{\overline{\mathcal{O}}}}_{0}^{\frac{)}{)})}\Omega(\mathcal{E}_{0},\mathcal{O}_{1})\mathfrak{l}\prime \mathfrak{l}\Omega(\mathcal{E}_{1},.\mathcal{O}_{1}g’$

,

,

(72)

130
in an even $(m|n)\cross(m|n)$ -supe $7matrix$ and$\mathcal{G}_{(m+1|n)}’(\begin{array}{lll}\Omega(\mathcal{E}_{0},\mathcal{O}_{1}) D(\mathcal{O}_{1}\dagger^{\acute{l}})’-\prime- \Omega(\mathcal{E}_{1},\mathcal{O}_{1})\text{コ}\sim\sim-\sim\sim----\sim\sim-\sim\Omega(\mathcal{E}_{0},\mathcal{O}_{0})\mathcal{E}_{0} ----\sim\sim--\sim-\prime\prime\sim- \text{コ}\sim-\vee\sim,--\sim-\sim-D(\mathcal{O}_{o-}^{\uparrow-})--\Omega(\mathcal{E}_{1}\mathcal{O}_{0})0’\prime \mathcal{E}_{1}\mathfrak{l}| \end{array})$,in an even $(m+1|n)\cross(m+1|n)$ -supermatrix.

Remark 1. The earlier papers on this topic [20,30] deal with the case $n=0$ only. In this case,and the solutions can be expressed in terms of determinants rather thanandthe more general superdeterminants$\mathcal{E}=\mathcal{E}_{0}$$\mathcal{O}=\mathcal{O}_{1}$$A[m]=D$ (log det$\Omega(\mathcal{E}_{0},$$\mathcal{O}_{1})$),and$U[m]= \frac{\det(\begin{array}{lll}\Omega(\mathcal{E}_{0} \mathcal{O}_{1}) D(\mathcal{O}_{1}^{t})\mathcal{E}_{0} 0\end{array})}{\det(\Omega(\mathcal{E}_{0},\mathcal{O}_{1}))}$.

7

Conclusions

In this paper, we considered a twisted derivation which includes normal derivative, forward
difference operator, q-difference operator and superderivatives as special cases. Darboux transformations defined in terms of twisted derivations have an quasideterminant iteration formulavery similar to the known one for the untwisted case. This result gives a framework for a
unified approach to Darboux transformations for differential, superdifferential, difference andq-difference operators. As an example we showed how this is achieved for superderivatives in
the Manin-Radul super $KdV$ equation.

Acknowledgement
Much of the work presented in this paper was carried out during the authors’ visit to the Newton
Institute for Mathematical Sciences as part of the Programme on Discrete Integrable Systems(January-July 2009). The authors are grateful to the organizers for their invitation and to the
Isaac Newton Institute for financial support and hospitality. This work was supported by the
National Natural Science Foundation of China (grant No. 10601028) and the Project-sponsoredby SRF for ROCS, SEM.

References[1] Kupershmidt, B. A. KP or mKP, vol. 78 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2000). Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems.

[2] Paniak, L. D. Exact Noncommutative KP and KdV Multi-solitons (2001). hep-th/0105185.

[3] Sakakibara, M. J. Phys. A 37 (2004) L599-L604.
[4] Wang, N. and Wadati, M. J. Phys. Soc. Japan 72 (2003) 1366-1373.

131[5] Wang, N. and Wadati, M. J. Phys. Soc. Japan 72 (2003) 1881-1888.
[6] Wang, N. and Wadati, M. J. Phys. Soc. Japan 73 (2004) 1689-1698.
[7] Hamanaka,$th/0303256$ .

M. Noncommutative solitons and D-branes. Ph.D. thesis (2003). hep-

[8] Hamanaka, M. and Toda, K. Phys. Lett. A 316 (2003) 77-83.

[9] Dimakis, A. and M\"uller-Hoissen, F. J. Phys. A 38 (2005) 5453-5505.

[10] Nimmo, J. J. C. J. Phys. A 39 (2006) 5053-5065.
[11] Li, C. X., Nimmo, J. J. C. and Tamizhmani, K. M. Proc. R. Soc. A 465 (2009) 1441-1451.
[12] Gilson C.R. and Nimmo J.J.C. J. Phys. A: Math. Theor. 40(2007) 3839-3850.
[13] Gilson, C. R., Nimmo, J. J. C. and Ohta, Y. J. Phys. A: Math. Theor. 40 (2007) 12607-

12617.

[14] Li, C. X. and Nimmo, J. J. C., Proc. R. Soc. A 464(2008) 951-966.
[15] Gelfand I.M. and Retakh V.S., Determinants of matrices over noncommutative rings. Funct.
Anal. Appl. 25(1991) 13-25.

[16] Etingof P., Gelfand I. and Retakh V., Math. Res. Lett. 4 (1997) 413-425.
[17] Manin Y.I. and Radul A.O., Comm. Math. Phys. 98 (1985) 65-77.
[18] Hartwig J. T., Larsson D. and Silvestrov S. D., J. Algebra 295 (2006) 314-361.
[19] De Concini S. and Procesi C., Quantum Groups, Lecture Notes in Mathematics 1565 (1993)

31-140.
[20] Liu Q.P. and Manas M., Phys. Lett. B394 (1997) 337-342.
[21] Liu Q.P. and Manas M., Phys. Lett. B396 (1997) 133-140.
[22] Li C.X. and Nimmo J.J.C., arXiv: solv-int/0911.1413.

[23] Gelfand I., Gelfand S., Retakh V. and Wilson R.L., Adv. in Math. 193 (2005) 56-141.

[24] Matveev, V.B. and Salle, M.A. Darboux transformations and solitons. Springer-Verlag,
Berlin, (1991).

[25] Oevel, W. and Schief, W. “Darboux theorems and the KP hierarchy” in Applications ofanalytic and geometric methods to nonlinear differential equations (Exeter, 1992), 193-206,
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 413, Kluwer Acad. Publ., Dordrecht, (1993).
[26] Crum M.M., Q, J. Math. 6 (1955) 121-127.
[27] Berezin F.A., Introduction to superanalysis (D. Reidel Publishing Company, Dordrecht,1987).

[28] DeWitt B., Supermanifolds (Cambridge University Press, 1984).

[29] Bergvelt M.J. and Rabin J.M., Super curves, their Jacobians and super KP equations.
arXiv: alg-geom/9601012vl.
[30] Shaw J.C. and Th M.H., J. Math. Phys. 39 (1998) 4773-4784.