Darboux and Binary Darboux Transformations for Discrete Integrable Systems. II

Darboux and Binary Darboux Transformations
for Discrete Integrable Systems.
II. Discrete Potential mKdV Equation

Ying SHI †, Jonathan NIMMO ‡ and Junxiao ZHAO §

Y. Shi, J. Nimmo and J.X. Zhao

† School of Science, Zhejiang University of Science and Technology,
† Hangzhou 310023, P.R. China<EMAIL_ADDRESS>
‡ Department of Mathematics, University of Glasgow, Glasgow G12 8QQ, UK
<EMAIL_ADDRESS>
§ School of Mathematics, University of Chinese Academy of Sciences,
§ Beijing 100190, P.R. China<EMAIL_ADDRESS>
Received December 17, 2016, in final form May 16, 2017; Published online May
28, 2017

The paper presents two results. First it is shown how the discrete potential
modified KdV equation and its Lax pairs in matrix form arise from the
Hirota–Miwa equation by a 2-periodic reduction. Then Darboux transformations
and binary Darboux transformations are derived for the discrete potential
modified KdV equation and it is shown how these may be used to construct exact
solutions.

partial difference equations; integrability; reduction; Darboux transformation

39A14; 37K10; 35C08

##  1 Introduction

The discrete version of the potential modified KdV equation that we want to
investigate in this paper is the nonlinear partial difference equation

$\displaystyle Q(v,v_{1},v_{2},v_{12};a_{1},a_{2})\equiv
a_{1}(vv_{2}-v_{1}v_{12})=a_{2}(vv_{1}-v_{2}v_{12}).$ (1.1)

The notation we adopted here and later is as follows, with forward shift
operators $T_{n_{1}}$, $T_{n_{2}}$:

$\displaystyle v:=v(n_{1},n_{2}),\qquad$ $\displaystyle
v_{1}:=T_{n_{1}}(v)=v(n_{1}+1,n_{2}),$ $\displaystyle
v_{2}:=T_{n_{2}}(v)=v(n_{1},n_{2}+1),\qquad$ $\displaystyle
v_{12}:=T_{n_{1}}T_{n_{2}}(v)=v(n_{1}+1,n_{2}+1),$

and $a_{1}$, $a_{2}$ denote lattice parameters associated with the directions
$n_{1}$, $n_{2}$ respectively. Equation (1.1) was derived in [14] from the
Cauchy matrix approach, and was originally found in [16, 20] through the
direct linearization approach. Up to a gauge transformation $v\rightarrow
i^{n_{1}+n_{2}}v$ and changing the lattice parameters as their reciprocals,
equation (1.1) is equivalent to the equation H3δ=0 in the Adler–Bobenko–Suris
(ABS) classification [1],

$\displaystyle H3_{\delta}\equiv
a_{1}(vv_{1}+v_{2}v_{12})-a_{2}(vv_{2}+v_{1}v_{12})=\delta\big{(}a_{2}^{2}-a_{1}^{2}\big{)}.$
(1.2)

There are several papers dedicated to closed-form $N$-soliton solutions of the
‘ABS list’ [5, 6, 8, 14]. So, in [14], based on a Cauchy matrix structure, the
closed-form $N$-soliton solution of equation (1.1) was derived, in [8],
following Hirota’s method, the authors derive bilinear difference equations of
equation (1.2) and its $N$-soliton solutions in terms of Casoratian
determinants, and in [5], by the discrete inverse scattering transform, the
authors point out that the soliton solutions of equation (1.2) derived from
the Cauchy matrix approach are exactly the solutions obtained from
reflectionless potentials.

The Hirota–Miwa equation [9, 13] is the three-dimensional discrete integrable
system

$\displaystyle(a_{1}-a_{2})\tau_{12}\tau_{3}+(a_{2}-a_{3})\tau_{23}\tau_{1}+(a_{3}-a_{1})\tau_{13}\tau_{2}=0,$
(1.3)

where lattice parameters $a_{k}$ are constants, $k=1,2,3$, and for
$\tau=\tau(n_{1},n_{2},n_{3})$ each subscript $i$ denotes a forward shift in
the corresponding discrete variable $n_{i}$. It was discovered by Hirota [9]
as a fully discrete analogue of the two-dimensional Toda equation and later
Miwa [13] showed that it was intimately related to the KP
(Kadomtsev–Petviashvili) hierarchy. In paper [10], Hirota gives the
discretization of the potential modified KdV equation, which can be
transformed into the form (1.1), and shows that it is a 4-reduction of the
Hirota–Miwa equation (which Hirota named as the discrete analogue of a
generalized Toda equation).

In this paper, we discuss in detail the Darboux and binary Darboux
transformations and how these may be used to obtain exact solutions of the
discrete potential modified Korteweg–de Vries (d-p-mKdV) equation (1.1). In
contrast to the approaches presented in [4, 14, 16], we get (1.1) and its Lax
pairs by reducing the Hirota–Miwa equation (1.3) and its Lax pairs. In fact,
the 2-periodic reduction method studied here has already been investigated in
[2] where authors present a multidimensionally consistent hierarchy of
discrete systems whose first member is the equation (1.1). Otherwise, this was
refined and extended to the non-commutative case in [7]. In [2, 3, 4, 7, 17],
as we see that the integrability is understood in the sense of the
multidimensional consistency property, which gives a Lax pair directly. We
here, through a 2-periodic reduction of the linear systems of the Hirota–Miwa
equation (1.3), obtain the Lax pairs of the equation (1.1) which allows the
application of the classical Darboux transformations [11, 12]. However, up to
gauge transformations, these Lax pairs are coincident with the ones given by
the multidimensional consistency property [4]. This paper is part of the work
which will explore the equations in the ABS list, their Lax pairs and Darboux
transformations as reductions of the Hirota–Miwa equation.

The outline of this paper is as follows. In Section 2, we recall important
results on Darboux transformations and binary Darboux transformations of the
Hirota–Miwa equation. In particular, in a departure from the results in [18,
19, 21], we write the linear system of Hirota–Miwa equation in a different
form, by the gauge transformation
$\phi\rightarrow\prod\limits_{i=1}^{3}a_{i}^{-n_{i}}\phi$, which is suitable
for making the reduction. In Section 3, we show that how the d-p-mKdV equation
and its Lax pairs in matrix form arise from the Hirota–Miwa equation by a
$2$-periodic reduction. Then its Darboux transformations and binary Darboux
transformations are derived and it is shown how these may be used to construct
exact solutions.

##  2 Hirota–Miwa equation

The Hirota–Miwa equation (1.3) arises as the compatibility conditions of the
linear system

$\displaystyle\phi_{i}-\phi_{j}=(a_{i}-a_{j})u^{ij}\phi,\qquad 1\leq i<j\leq
3,$ (2.1)

where for $\phi=\phi(n_{1},n_{2},n_{3})$ each subscript $i$ denotes a forward
shift in the corresponding discrete variable $n_{i}$, for example,
$\phi_{1}=T_{{}_{n_{1}}}(\phi)=\phi(n_{1}+1,n_{2},n_{3})$. This linear system
(2.1) is compatible if and only if

$\displaystyle(a_{1}-a_{2})u^{12}+(a_{2}-a_{3})u^{23}+(a_{3}-a_{1})u^{13}=0,$
(2.2a)
$\displaystyle\big{(}u^{ij}\big{)}_{{}_{k}}u^{ik}=\big{(}u^{ik}\big{)}_{{}_{j}}u^{ij}.$
(2.2b)

Note that when one uses the formula $u^{ij}=\tau_{ij}\tau/\tau_{i}\tau_{j}$,
(2.2a) gives (1.3) and (2.2b) is satisfied identically. A second way is to
suppose $u^{ij}=(v_{j}-v_{i}+(a_{i}-a_{j}))/(a_{i}-a_{j})$. This ansatz solves
(2.2a) exactly and (2.2b) becomes the discrete potential KP (d-p-KP) equation
[15].

In this paper, in particular we deal with the Hirota–Miwa equation (1.3)
together with its linear system in the form (2.1). Using the reversal-
invariance property of the Hirota–Miwa equation, i.e., it is invariant with
respect to the reversal of all lattice directions $n_{i}\rightarrow-n_{i}$, we
have the linear system in formal adjoint form [18]

$\displaystyle\psi_{\overline{i}}-\psi_{\overline{j}}=(a_{i}-a_{j})\frac{\tau_{\overline{i}\overline{j}}\tau}{\tau_{\overline{i}}\tau_{\overline{j}}}\psi,\qquad
1\leq i<j\leq 3.$ (2.3)

The subscript $\overline{i}$ denotes a backward shift with respect to $n_{i}$,
for example,
$\psi_{\overline{1}}=T_{{}_{n_{1}}}^{-1}(\psi)=\psi(n_{1}-1,n_{2},n_{3})$.

###  2.1 Darboux and binary Darboux transformations

The basic Darboux transformation for the Hirota–Miwa equation is stated in the
following proposition.

######  Proposition 2.1.

Let $\theta$ be a non-zero solution of the linear system (2.1) for some
$\tau$. Then the transformation

$\displaystyle\mathrm{DT}^{\theta}\colon\
\phi\rightarrow\frac{C_{{}_{[i]}}(\theta,\phi)}{\theta},\qquad\tau\rightarrow\theta\tau,$

leaves (2.1) invariant, where
$C_{{}_{[i]}}(\theta,\phi)=\theta\phi_{i}-\theta_{i}\phi$, $i=1,2,3$, using
the subscript $[i]$ to designate that the forward shifts of the determinant
$C_{{}_{[i]}}(\theta,\phi)$ is with respect to the variable $n_{i}$.

Next we write down the closed form expression for the result of $N$
applications of the above Darboux transformation, which give solutions in
Casoratian determinant form. To do this we need to define the Casoratian of
$N$ solutions. Let
$\bm{\theta}=({\theta^{1}}(n_{1},n_{2},n_{3}),{\theta^{2}}(n_{1},n_{2},n_{3}),\dots$,
${\theta^{N}}(n_{1},n_{2},n_{3}))^{T}$ be an $N$-vector solution of (2.1). The
Casoratian determinant (with forward-shifts) can be written as

$\displaystyle
C\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}=\big{|}\bm{\theta},T_{{}_{n_{i}}}(\bm{\theta}),T_{{}_{n_{i}}}^{2}(\bm{\theta})\dots,T_{{}_{n_{i}}}^{N-1}(\bm{\theta})\big{|},\qquad
1\leq i\leq 3,$

which may also be unambiguously defined in the following notation as

$\displaystyle
C_{{}_{[i]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}=\big{|}\bm{\theta}(0),\bm{\theta}(1),\bm{\theta}(2),\dots,\bm{\theta}(N-1)\big{|},\qquad
1\leq i\leq 3,$

where $\bm{\theta}(k)$ denotes the $N$-vector
$\big{(}{\theta^{1}}(n_{1},n_{2},n_{3}),{\theta^{2}}(n_{1},n_{2},n_{3}),\dots,{\theta^{N}}(n_{1},n_{2},n_{3})\big{)}^{T}$
subject to the $k$ times shift $T_{{}_{n_{i}}}^{k}$ on $n_{i}$ which gives
$n_{i}\rightarrow n_{i}+k$, $0\leq k\leq N-1$, and $i=1,2$ or $3$, the same
value being taken for $i$ in each column in the determinant. Then we have the
following.

######  Proposition 2.2.

Let $\theta^{1},\theta^{2},\dots,\theta^{N}$ be non-zero, independent
solutions of the linear system (2.1) for some $\tau$, such that
$C_{{}_{[i]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}\neq 0$. Then
the $N$-fold Darboux transformation

$\displaystyle\phi\rightarrow\frac{C_{{}_{[i]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N},\phi\big{)}}{C_{{}_{[i]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}},\qquad\tau\rightarrow
C_{{}_{[i]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}\tau,$

leaves (2.1) invariant.

Now we can apply the reflections $n_{i}\rightarrow-n_{i}$, $i=1,2,3$, to the
above results to deduce adjoint Darboux transformation for the second linear
system (2.3).

######  Proposition 2.3.

Let $\rho$ be a non-zero solution of the linear system (2.3) for some $\tau$.
Then the transformation

$\displaystyle\mathrm{DT}^{\rho}\colon\
\psi\rightarrow\frac{C_{{}_{[\overline{i}]}}(\rho,\psi)}{\rho},\qquad\tau\rightarrow\rho\tau,$

leaves (2.3) invariant, where
$C_{{}_{[\overline{i}]}}(\rho,\psi)=\rho_{{}_{\overline{i}}}\psi-\rho\psi_{{}_{\overline{i}}}$,
$i=1,2,3$, using the subscript $[\overline{i}]$ to designate that the backward
shifts of the determinant $C_{{}_{[\overline{i}]}}(\rho,\psi)$ is with respect
to the variable $n_{i}$.

The $N$-fold adjoint Darboux transformation is expressed in terms of the
Casoratian

$\displaystyle
C_{{}_{[\overline{i}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}=|\bm{\rho}(0),\bm{\rho}(-1),\bm{\rho}(-2),\dots,\bm{\rho}(-N+1)|,\qquad
1\leq i\leq 3,$

where $\bm{\rho}=\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}^{T}$ and
$\bm{\rho}(-k)=T_{{}_{n_{i}}}^{-k}(\bm{\rho})=\bm{\rho}|_{{}_{n_{i}\rightarrow
n_{i}-k}}$, $0\leq k\leq N-1$, the same $i=1,2$ or $3$ be taken in all
columns.

######  Proposition 2.4.

Let ${\rho^{1}},{\rho^{2}},\dots,{\rho^{N}}$ be $N$ non-zero independent
solutions of the linear system (2.3) for some $\tau$, such that
$C_{{}_{[\overline{i}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}\neq 0$.
Then the $N$-fold adjoint Darboux transformation

$\displaystyle\psi\rightarrow\frac{C_{{}_{[\overline{i}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N},\psi\big{)}}{C_{{}_{[\overline{i}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}},\qquad\tau\rightarrow
C_{{}_{[\overline{i}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}\tau,$

leaves (2.3) invariant.

To construct a binary Darboux transformation, we introduce the potential
$\omega=\omega(\phi,\psi)$, defined by the relations

$\displaystyle\Delta_{i}\omega(\phi,\psi)=\phi\psi_{i},\qquad i=1,2,3,$ (2.4)

where $\Delta_{i}=T_{n_{i}}-1$ is the forward-difference operator in discrete
variable $n_{i}$. If $\phi$ and $\psi$ satisfy the linear systems (2.1) and
(2.3) for some $\tau$, respectively, then (2.4) are compatible in the sense as
$\Delta_{i}(\phi\psi_{j})=\Delta_{j}(\phi\psi_{i})$, for $i<j$. So the
potential $\omega$ is well-defined.

The following proposition gives the binary Darboux transformation of the
Hirota–Miwa equation (1.3).

######  Proposition 2.5.

For some $\tau$, let $\theta$ and $\phi$ be two non-zero solutions of the
linear system (2.1), $\rho$ and $\psi$ be two non-zero solutions of the linear
system (2.3), then

$\displaystyle\mathrm{BDT^{\theta,\rho}}\colon\quad$
$\displaystyle\phi\rightarrow\phi-\theta\omega(\theta,\rho)^{-1}\omega(\phi,\rho),\qquad$
$\displaystyle\tau\rightarrow\omega(\theta,\rho)\tau,$
$\displaystyle\mathrm{aBDT^{\theta,\rho}}\colon\quad$
$\displaystyle\psi\rightarrow\psi-\rho\omega(\theta,\rho)^{-1}\omega(\theta,\psi),\qquad$
$\displaystyle\tau\rightarrow\omega(\theta,\rho)\tau,$

leave (2.1) and (2.3) respectively invariant.

The $N$-fold iteration of these binary Darboux transformations are given
below.

######  Proposition 2.6.

Let $\bm{\theta}=\big{(}\theta^{1},\dots,\theta^{N}\big{)}^{T}$ and
$\bm{\rho}=\big{(}\rho^{1},\dots,\rho^{N}\big{)}^{T}$ satisfy linear systems
(2.1) and (2.3) for some $\tau$ respectively. Then

$\displaystyle\phi\rightarrow\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\bm{\theta}\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\phi\\\
\end{vmatrix}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}^{-1},\qquad$
$\displaystyle\tau\rightarrow\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}\tau,$
$\displaystyle\psi\rightarrow\begin{vmatrix}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}&\bm{\rho}\\\
\omega\big{(}\bm{\theta}^{T},\psi\big{)}&\psi\\\
\end{vmatrix}\big{|}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}\big{|}^{-1},\qquad$
$\displaystyle\tau\rightarrow\big{|}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}\big{|}\tau,$

leave (2.1) and (2.3) respectively invariant.

Here
$\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}=\big{(}\omega\big{(}\theta^{(i)},\rho^{(j)}\big{)}\big{)}_{i,j=1,\dots,N}$,
$\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}=\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}^{T}$
are $N\times N$ matrix,
$\omega\big{(}\phi,\bm{\rho}^{T}\big{)}=\big{(}\omega\big{(}\phi,\rho^{(j)}\big{)}\big{)}_{j=1,\dots,N}$
and
$\omega\big{(}\bm{\theta}^{T},\psi\big{)}=\big{(}\omega\big{(}\theta^{(i)},\psi\big{)}\big{)}_{i=1,\dots,N}$
are $N$-row vectors.

The proofs of those above propositions are straightforward computation, so we
do not give the details. The reader is also referred to the papers [18, 19].

###  2.2 Explicit solutions obtained by Darboux transformations

Here we present explicit examples of the classes of solutions that may be
obtained by means of the Darboux transformations derived above. We choose the
seed solution of the Hirota–Miwa equation (1.3) as $\tau=\tau_{0}=1$. With
this choice, the first linear system (2.1) reads

$\displaystyle\phi_{i}-\phi_{j}=(a_{i}-a_{j})\phi,\qquad 1\leq i<j\leq 3,$

and the basic eigenfunctions, depending on a single parameter $p$ are found to
be

$\displaystyle\phi(n_{1},n_{2},n_{3};p)=\prod_{i=1}^{3}(a_{i}+p)^{n_{i}}.$
(2.5)

In a similar way the basic eigenfunctions of the adjoint linear system (2.3),
depending on a single parameter $q$, are

$\displaystyle\psi(n_{1},n_{2},n_{3};q)=\prod_{i=1}^{3}(a_{i}+q)^{-n_{i}}.$
(2.6)

For these eigenfunctions above we may integrate (2.4) and obtain the potential

$\displaystyle\omega(\phi,\psi)=\frac{1}{p-q}\prod_{i=1}^{3}\left(\frac{a_{i}+p}{a_{i}+q}\right)^{n_{i}}+c.$
(2.7)

Given the above expression it is straightforward to write down the following
explicit solution for the Hirota–Miwa equation (1.3)

$\displaystyle\tau(n_{1},n_{2},n_{3})=C_{{{}_{[i]}}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}\tau_{0},$
(2.8)

where
$\theta^{k}=\alpha_{k}\theta(n_{1},n_{2},n_{3};p_{k})+\theta(n_{1},n_{2},n_{3};p^{\prime}_{k})$
where $\theta(n_{1},n_{2},n_{3};p_{k})$ and
$\theta(n_{1},n_{2},n_{3};p_{k}^{\prime})$ are given by (2.5) and $p_{k}$,
$p^{\prime}_{k}$, $p_{k}\neq p^{\prime}_{k}$ and $\alpha_{k}$ are arbitrary
constants;

$\displaystyle\tau(n_{1},n_{2},n_{3})=C_{{{}_{[\overline{i}]}}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}\tau_{0},$

where
$\rho^{k}=\beta_{k}\rho(n_{1},n_{2},n_{3};q_{k})+\rho(n_{1},n_{2},n_{3};q^{\prime}_{k})$
where $\rho(n_{1},n_{2},n_{3};q_{k})$ and
$\rho(n_{1},n_{2},n_{3};q_{k}^{\prime})$ are given by (2.6) and $q_{k}$,
$q^{\prime}_{k}$, $q_{k}\neq q^{\prime}_{k}$ and $\beta_{k}$ are arbitrary
constants;

$\displaystyle\tau(n_{1},n_{2},n_{3})=\det(\omega_{k,l})\tau_{0},\qquad
k,l=1,2,\dots,N,$

where $\omega_{k,l}$ is given by (2.7) with and $p=p_{k}$, $q=q_{l}$ and
$c=c_{kl}$.

##  3 Discrete potential modified KdV equation

###  3.1 From Hirota–Miwa equation to d-p-mKdV equation:
2-periodic reductions

Here, we explain the way to obtain the d-p-mKdV equation (1.1) from the
Hirota–Miwa equation (1.3) through a 2-periodic reduction technique.

For the Hirota–Miwa equation (1.3), from (2.8), it is easy to get its one
soliton solution in discrete exponential function form

$\displaystyle\tau(n_{1},n_{2},n_{3})=1+\alpha\prod_{i=1}^{3}\left(\frac{a_{i}+p}{a_{i}+p^{\prime}}\right)^{n_{i}}.$
(3.1)

Introduce $f(n_{1},n_{2},n_{3})$ and $\bar{f}(n_{1},n_{2},n_{3})$ and impose a
2-periodic property on the $\tau$ function (3.1) as below

$\displaystyle\tau=f=T_{n_{3}}^{2}(f),\qquad\overline{f}=T_{n_{3}}(f).$ (3.2)

Note here that the reduction condition (3.2) gives $a_{3}=0$, $p^{\prime}=-p$,
and

$\displaystyle\overline{f}=T_{n_{3}}^{2}(\overline{f}),\quad
f=T_{n_{3}}(\overline{f}).$ (3.3)

Moreover (3.2) and (3.3) indicate the symmetric property between $f$ and
$\overline{f}$, with respect to $n_{3}$.

By applying the reduction condition (3.2) to the Hirota–Miwa equation (1.3),
together with parameter reduction $a_{3}=0$, we get

$\displaystyle(a_{1}-a_{2})f_{12}\overline{f}=a_{1}f_{2}\overline{f}_{1}-a_{2}f_{1}\overline{f}_{2},$
(3.4a)
$\displaystyle(a_{1}-a_{2})\overline{f}_{12}f=a_{1}\overline{f}_{2}f_{1}-a_{2}\overline{f}_{1}f_{2}.$
(3.4b)

There are two ways to obtain the equation (3.4b), one way is applying the
symmetric property between $f$ and $\overline{f}$ to the equation (3.4a), the
another one is taking the shift operator $T_{n_{3}}$ on the Hirota–Miwa
equation (1.3), and using the reduction condition (3.3).

Define two functions (potentials)

$\displaystyle v(n_{1},n_{2},n_{3})=\frac{\overline{f}}{f},\qquad
u(n_{1},n_{2},n_{3})=\frac{f_{12}f}{f_{1}f_{2}}.$ (3.5)

By substituting (3.5) into (3.4), we get

$\displaystyle(a_{1}-a_{2})vu=a_{1}v_{1}-a_{2}v_{2},$ (3.6a)
$\displaystyle(a_{1}-a_{2})v_{12}u=a_{1}v_{2}-a_{2}v_{1}.$ (3.6b)

Eliminating $u$ in (3.6) gives

$\displaystyle Q(v,v_{1},v_{2},v_{12};a_{1},a_{2})\equiv
v_{12}(a_{1}v_{1}-a_{2}v_{2})-v(a_{1}v_{2}-a_{2}v_{1})=0,$ (3.7)

which is the d-p-mKdV equation (1.1) and is exactly same as the one first
given by Nijhoff, cf. [14], through the Cauchy matrix approach. Moreover, the
relation (3.6) serves as the discrete Miura transformation between the d-KdV
equation

$\displaystyle\frac{1}{u_{1}}-\frac{1}{u_{2}}=\frac{a_{1}-a_{2}}{a_{1}+a_{2}}\left(u_{12}-u\right),$

in potential $u$ (or more specifically, say $u_{\overline{2}}$ [21]) and the
d-p-mKdV equation (3.7) in potential $v$.

Another interesting result is that with the periodic property of $f$ and
$\overline{f}$, we have the following formulae on the potentials $u$ and $v$
as follows

$\displaystyle T_{n_{3}}(u)=uv_{12}vv_{1}^{-1}v_{2}^{-1},\qquad
T_{n_{3}}^{2}(u)=u,\qquad T_{n_{3}}(v)=v^{-1},\qquad T_{n_{3}}^{2}(v)=v.$

So the potentials $u$ and $v$ also satisfy the 2-periodic property in the
virtual variable $n_{3}$. We observe that if $v$ is a solution of the d-p-mKdV
equation then, as in the continuous case, $-v$ is a solution, but in the
discrete case, $v^{-1}$ is a yet another solution.

Under the reduction condition (3.2), from the $\tau$ function (3.1), we easily
get the exact solution of (3.4)

$\displaystyle
f(n_{1},n_{2},n_{3})=1+\alpha(-1)^{n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}+p}{a_{1}-p}\right)^{n_{i}},$
$\displaystyle\overline{f}(n_{1},n_{2},n_{3})=1-\alpha(-1)^{n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}+p}{a_{1}-p}\right)^{n_{i}},$

which directly gives the one soliton solution of the d-p-mKdV equation

$\displaystyle
v(n_{1},n_{2})=\frac{\overline{f}}{f}=\frac{1-\alpha(-1)^{n_{3}}\prod\limits_{i=1}^{2}\left(\frac{a_{i}+p}{a_{1}-p}\right)^{n_{i}}}{1+\alpha(-1)^{n_{3}}\prod\limits_{i=1}^{2}\left(\frac{a_{i}+p}{a_{1}-p}\right)^{n_{i}}}.$

Note here that in the equation (3.7), there is no shift depends on the
discrete variable $n_{3}$. So treating the $n_{3}$ as a virtual variable for
the d-p-mKdV equation is allowable.

Next, we show the way of discovering the linear system in matrix form of the
d-p-mKdV equation (3.7) from the linear system of the Hirota–Miwa equation
(2.1) through the $2$-periodic reduction technique.

Introduce eigenfunctions $\phi(n_{1},n_{2},n_{3})$ and
$\overline{\phi}(n_{1},n_{2},n_{3})$ and impose a 2-periodic condition on the
eigenfunction $\phi(n_{1},n_{2},n_{3})$ in the linear system (2.1) as below

$\displaystyle\phi=\lambda^{-2}T_{n_{3}}^{2}(\phi),\qquad\overline{\phi}=\lambda^{-1}T_{n_{3}}(\phi),$
(3.8)

where the parameter $\lambda$ serves as the spectral parameter. From (3.8), we
have

$\displaystyle\overline{\phi}=\lambda^{-2}T_{n_{3}}^{2}(\overline{\phi}),\qquad\phi=\lambda^{-1}T_{n_{3}}(\overline{\phi}).$
(3.9)

So (3.8) and (3.9) mean the symmetric property between $\phi$ and
$\overline{\phi}$, with respect to $n_{3}$.

By applying the reduction conditions (3.2) and (3.8), together with $a_{3}=0$,
to the linear system (2.1), we get

$\displaystyle\phi_{1}-\phi_{2}=(a_{1}-a_{2})\frac{f_{12}f}{f_{1}f_{2}}\phi,$
(3.10a)
$\displaystyle\phi_{2}-\lambda\overline{\phi}=a_{2}\frac{\overline{f}_{2}f}{f_{2}\overline{f}}\phi,$
(3.10b)
$\displaystyle\lambda\overline{\phi}-\phi_{1}=-a_{1}\frac{\overline{f}_{1}f}{f_{1}\overline{f}}\phi.$
(3.10c)

Then by using the symmetric property (3.3) and (3.9) respectively between $f$
and $\overline{f}$, $\phi$ and $\overline{\phi}$, we get

$\displaystyle\overline{\phi}_{1}-\overline{\phi}_{2}=(a_{1}-a_{2})\frac{\overline{f}_{12}\overline{f}}{\overline{f}_{1}\overline{f}_{2}}\overline{\phi},$
(3.11a)
$\displaystyle\overline{\phi}_{2}-\lambda\phi=a_{2}\frac{f_{2}\overline{f}}{\overline{f}_{2}f}\overline{\phi},$
(3.11b)
$\displaystyle\lambda\phi-\overline{\phi}_{1}=-a_{1}\frac{f_{1}\overline{f}}{\overline{f}_{1}f}\overline{\phi}.$
(3.11c)

Substituting (3.5) into (3.10) and (3.11) gives

$\displaystyle\phi_{1}-\phi_{2}=(a_{1}-a_{2})u\phi,$ (3.12a)
$\displaystyle\phi_{2}-\lambda\overline{\phi}=a_{2}v_{2}v^{-1}\phi,$ (3.12b)
$\displaystyle\lambda\overline{\phi}-\phi_{1}=-a_{1}v_{1}v^{-1}\phi,$ (3.12c)

and

$\displaystyle\overline{\phi}_{1}-\overline{\phi}_{2}=(a_{1}-a_{2})uv_{12}vv_{1}^{-1}v_{2}^{-1}\overline{\phi},$
(3.13a)
$\displaystyle\overline{\phi}_{2}-\lambda\phi=a_{2}v_{2}^{-1}v\overline{\phi},$
(3.13b)
$\displaystyle\lambda\phi-\overline{\phi}_{1}=-a_{1}v_{1}^{-1}v\overline{\phi}.$
(3.13c)

Through the discrete Miura transformation (3.6), the equations (3.12a) and
(3.13a) can be derived by (3.12b) and (3.12c), (3.13b) and (3.13c),
respectively.

Defining a vector eigenfunction $\bm{\Phi}=(\phi,\overline{\phi})^{T}$, which
satisfies the condition (3.8), then (3.12c) and (3.13c), (3.12b) and (3.13b),
can be respectively written in matrix form as below

$\displaystyle\bm{\Phi}_{1}=\bm{L}\bm{\Phi},$ (3.14a)
$\displaystyle\bm{\Phi}_{2}=\bm{M}\bm{\Phi},$ (3.14b)

where

$\displaystyle\bm{L}=\left(\begin{matrix}a_{1}v_{1}v^{-1}&\lambda\\\
\lambda&a_{1}v_{1}^{-1}v\end{matrix}\right),\qquad\bm{M}=\left(\begin{matrix}a_{2}v_{2}v^{-1}&\lambda\\\
\lambda&a_{2}v_{2}^{-1}v\end{matrix}\right).$

One then finds that

$\displaystyle
0=\bm{\Phi}_{12}-\bm{\Phi}_{21}=(\bm{L}_{2}\bm{M}-\bm{M}_{1}\bm{L})\bm{\Phi}=\lambda
Q(v,v_{1},v_{2},v_{{12}};a_{1},a_{2})\left(\begin{matrix}0&-v_{1}^{-1}v_{2}^{-1}\\\
v^{-1}v_{12}^{-1}&0\end{matrix}\right)\bm{\Phi}.$

So the compatibility condition of the above linear system (3.14) in
eigenfunction $\bm{\Phi}$ is that $v$ obeys the d-p-mKdV equation (3.7).

###  3.2 Darboux and binary Darboux transformations

In this section, we will see that through the reduction conditions (3.2) and
(3.8), it is easy to investigate the Darboux and binary Darboux
transformations of d-p-mKdV equation.

Let $v$ be a solution of the d-p-mKdV equation (3.7) and
$\bm{\Phi}=(\phi,\overline{\phi})^{T}$ be a vector solution of its Lax pair
(3.14). The fundamental Darboux transformation of the d-p-mKdV equation is
given as below.

######  Proposition 3.1.

Suppose $(\theta,\overline{\theta})^{T}$, which holds the $2$-periodic
property $\theta=\mu^{-2}T_{{}_{n_{3}}}^{2}(\theta)$,
$\overline{\theta}=\mu^{-1}T_{{}_{n_{3}}}(\theta)$, is a vector solution of
the linear system (3.14) by taking $\lambda=\mu$ for some $v$, then

$\displaystyle\mathrm{DT}^{\theta,\overline{\theta}}\colon\
\phi\rightarrow\frac{C_{{}_{[3]}}(\theta,\phi)}{\theta},\qquad\overline{\phi}\rightarrow\frac{C_{{}_{[3]}}(\overline{\theta},\overline{\phi})}{\overline{\theta}},\qquad
v\rightarrow\frac{T_{{}_{n_{3}}}(\theta)}{\theta}v=\mu^{2}\frac{\overline{\theta}}{T_{{}_{n_{3}}}(\overline{\theta})}v$
(3.15)

leaves (3.14) invariant. Otherwise,

$\displaystyle\frac{C_{{}_{[3]}}(\overline{\theta},\overline{\phi})}{\overline{\theta}}=\lambda^{-1}T_{{}_{n_{3}}}\left(\frac{C_{{}_{[3]}}(\theta,\phi)}{\theta}\right),\qquad\frac{C_{{}_{[3]}}(\theta,\phi)}{\theta}=\lambda^{-1}T_{{}_{n_{3}}}\left(\frac{C_{{}_{[3]}}(\overline{\theta},\overline{\phi})}{\overline{\theta}}\right).$

We remark that may also write the gauge transformation of
$\bm{\Phi}=(\phi,\overline{\phi})^{T}$ in (3.15) in matrix form as follows

$\displaystyle\mathrm{DT}^{\theta,\overline{\theta}}\colon\
\bm{\Phi}\rightarrow\left(\begin{matrix}-\mu\theta^{-1}\overline{\theta}&\lambda\\\
\lambda&-\mu\theta{\overline{\theta}}^{-1}\end{matrix}\right)\bm{\Phi}.$

But for later convenience of the construction of the binary Darboux
transformation, we here write in scalar form shown in (3.15).

Next we write down the closed form expression for the result of $N$
applications of the above Darboux transformation, which give solutions in
Casoratian determinant form.

######  Proposition 3.2.

Let
$\big{(}\theta^{1},\overline{\theta}^{1}\big{)}^{T},\big{(}\theta^{2},\overline{\theta}^{2}\big{)}^{T},\dots,\big{(}\theta^{N},\overline{\theta}^{N}\big{)}^{T}$,
satisfying $\theta^{k}=\lambda_{k}^{-2}T_{{}_{n_{3}}}^{2}(\theta^{k})$,
$\overline{\theta}^{k}=\lambda_{k}^{-1}T_{{}_{n_{3}}}(\theta^{k})$, be $N$
non-zero independent vector solutions of the linear system (3.14) by taking
$\lambda=\lambda_{k}$, $k=1,2,\dots,N$, for some $v$, such that
$C_{{}_{[3]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}\neq 0$. Then

$\displaystyle\phi\rightarrow\widetilde{\phi}=\frac{C_{{}_{[3]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N},\phi\big{)}}{C_{{}_{[3]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}},\qquad\overline{\phi}\rightarrow\widetilde{\overline{\phi}}=\frac{C_{{}_{[3]}}\big{(}\overline{\theta}^{1},\overline{\theta}^{2},\dots,\overline{\theta}^{N},\overline{\phi}\big{)}}{C_{{}_{[3]}}\big{(}\overline{\theta}^{1},\overline{\theta}^{2},\dots,\overline{\theta}^{N}\big{)}},$
(3.16a) $\displaystyle
v\rightarrow\widetilde{v}=\frac{T_{{}_{n_{3}}}\big{(}C_{{}_{[3]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}\big{)}}{C_{{}_{[3]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}}v=\prod_{i=1}^{N}\lambda_{i}^{2}\frac{C_{{}_{[3]}}\big{(}\overline{\theta}^{1},\overline{\theta}^{2},\dots,\overline{\theta}^{N}\big{)}}{T_{{}_{n_{3}}}\big{(}C_{{}_{[3]}}\big{(}\overline{\theta}^{1},\overline{\theta}^{2},\dots,\overline{\theta}^{N}\big{)}\big{)}}v,$
(3.16b)

leaves (3.14) invariant. Otherwise,
$\widetilde{\overline{\phi}}=\lambda^{-1}T_{{}_{n_{3}}}(\widetilde{\phi})$,
$\widetilde{\phi}=\lambda^{-1}T_{{}_{n_{3}}}\big{(}\widetilde{\overline{\phi}}\big{)}$.

The d-p-mKdV equation (3.7) is invariant with respect to the reversal of all
lattice directions $n_{i}\rightarrow-n_{i}$, $i=1,2$. But its linear system
(3.14) does not have such invariance and so the reflections $n_{i}\rightarrow-
n_{i}$, $i=1,2$, acting on (3.14) give a second linear system on the vector
eigenfunction $\bm{\Psi}=(\psi,\overline{\psi})^{T}$, which also satisfy
2-periodic reduction condition

$\displaystyle\psi=\lambda^{-2}T_{{}_{n_{3}}}^{-2}(\psi),\qquad\overline{\psi}=\lambda^{-1}T_{{n_{3}}}^{-1}(\psi),$
(3.17)

as follows

$\displaystyle\bm{\Psi}_{\overline{1}}=\bm{U}\bm{\Psi},$ (3.18a)
$\displaystyle\bm{\Psi}_{\overline{2}}=\bm{V}\bm{\Psi},$ (3.18b)

where

$\displaystyle\bm{U}=\left(\begin{matrix}a_{1}v_{\overline{1}}v^{-1}&\lambda\\\
\lambda&a_{1}v_{\overline{1}}^{-1}v\end{matrix}\right),\qquad\bm{V}=\left(\begin{matrix}a_{2}v_{\overline{2}}v^{-1}&\lambda\\\
\lambda&a_{2}v_{\overline{2}}^{-1}v\end{matrix}\right).$

One then finds that

$\displaystyle
0=\bm{\Psi}_{\overline{1}\overline{2}}-\bm{\Psi}_{\overline{2}\overline{1}}=(\bm{U}_{\overline{2}}\bm{V}-\bm{V}_{\overline{1}}\bm{U})\bm{\Psi}=\lambda
Q\big{(}v,v_{\overline{1}},v_{\overline{2}},v_{\overline{12}};a_{1},a_{2}\big{)}\left(\begin{matrix}0&-v_{\overline{1}}^{-1}v_{\overline{2}}^{-1}\\\
v^{-1}v_{\overline{1}\overline{2}}^{-1}&0\end{matrix}\right)\bm{\Psi}.$

Now we apply the reflections $n_{i}\rightarrow-n_{i}$, $i=1,2$, in order to
deduce Darboux transformation for the second linear system as below.

######  Proposition 3.3.

Suppose $(\rho,\overline{\rho})^{T}$, which holds the $2$-periodic property
$\rho=\mu^{-2}T_{{}_{n_{3}}}^{-2}(\rho)$,
$\overline{\rho}=\mu^{-1}T_{{}_{n_{3}}}^{-1}(\rho)$, is a vector solution of
the linear system (3.18) by taking $\lambda=\mu$ for some $v$, then

$\displaystyle\mathrm{DT}^{\rho,\overline{\rho}}\colon\
\psi\rightarrow\frac{C_{{}_{[\overline{3}]}}(\rho,\psi)}{\rho},\qquad\overline{\psi}\rightarrow\frac{C_{{}_{[\overline{3}]}}(\overline{\rho},\overline{\psi})}{\overline{\rho}},\qquad
v\rightarrow\frac{T_{{}_{n_{3}}}^{-1}(\rho)}{\rho}v=\mu^{2}\frac{\overline{\rho}}{T_{{}_{n_{3}}}^{-1}(\overline{\rho})}v$

leaves (3.18) invariant. Otherwise,

$\displaystyle\frac{C_{{}_{[\overline{3}]}}(\overline{\rho},\overline{\psi})}{\overline{\rho}}=\lambda^{-1}T_{{}_{n_{3}}}^{-1}\left(\frac{C_{{}_{[\overline{3}]}}(\rho,\psi)}{\rho}\right),\qquad\frac{C_{{}_{[\overline{3}]}}(\rho,\psi)}{\rho}=\lambda^{-1}T_{{}_{n_{3}}}^{-1}\left(\frac{C_{{}_{[\overline{3}]}}(\overline{\rho},\overline{\psi})}{\overline{\rho}}\right).$

Next we write down the closed form expression for the result of $N$
applications of the above Darboux transformation, which give solutions in
Casoratian determinant form.

######  Proposition 3.4.

Let
$\big{(}\rho^{1},\overline{\rho}^{1}\big{)}^{T},\big{(}\rho^{2},\overline{\rho}^{2}\big{)}^{T},\dots,\big{(}\rho^{N},\overline{\rho}^{N}\big{)}^{T}$,
satisfying $\rho^{k}=\lambda_{k}^{-2}T_{{}_{n_{3}}}^{-2}(\rho^{k})$,
$\overline{\rho}^{k}=\lambda_{k}^{-1}T_{{}_{n_{3}}}^{-1}(\rho^{k})$, be $N$
non-zero independent vector solutions of the linear system (3.18) by taking
$\lambda=\lambda_{k}$, $k=1,2,\dots,N$, for some $v$, such that
$C_{{}_{[\overline{3}]}}(\rho^{1},\rho^{2},\dots,\rho^{N})\neq 0$. Then

$\displaystyle\psi\rightarrow\widetilde{\psi}=\frac{C_{{}_{[\overline{3}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N},\psi\big{)}}{C_{{}_{[\overline{3}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}},\qquad\overline{\psi}\rightarrow\widetilde{\overline{\psi}}=\frac{C_{{}_{[\overline{3}]}}\big{(}\overline{\rho}^{1},\overline{\rho}^{2},\dots,\overline{\rho}^{N},\overline{\psi}\big{)}}{C_{{}_{[\overline{3}]}}\big{(}\overline{\rho}^{1},\overline{\rho}^{2},\dots,\overline{\rho}^{N}\big{)}},$
$\displaystyle
v\rightarrow\widetilde{v}=\frac{T_{{}_{n_{3}}}^{-1}\big{(}C_{{}_{[\overline{3}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}\big{)}}{C_{{}_{[\overline{3}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}}v=\prod_{i=1}^{N}\lambda_{i}^{2}\frac{C_{{}_{[\overline{3}]}}\big{(}\overline{\rho}^{1},\overline{\rho}^{2},\dots,\overline{\rho}^{N}\big{)}}{T_{{}_{n_{3}}}^{-1}\big{(}C_{{}_{[\overline{3}]}}\big{(}\overline{\rho}^{1},\overline{\rho}^{2},\dots,\overline{\rho}^{N}\big{)}\big{)}}v,$

leaves (3.18) invariant. Otherwise,
$\widetilde{\overline{\psi}}=\lambda^{-1}T_{{}_{n_{3}}}^{-1}(\widetilde{\psi})$,
$\widetilde{\psi}=\lambda^{-1}T_{{}_{n_{3}}}^{-1}\big{(}\widetilde{\overline{\psi}}\big{)}$.

To construct a binary Darboux transformation, we introduce the potentials
$\omega=\omega(\phi,\psi)$ and
$\overline{\omega}=\omega(\overline{\phi},\overline{\psi})$, defined by the
relations

$\displaystyle\Delta_{3}(\omega(\phi,\psi))=\phi T_{{}_{n_{3}}}(\psi),$
(3.19a)
$\displaystyle\Delta_{3}(\omega(\overline{\phi},\overline{\psi}))=\overline{\phi}T_{{}_{n_{3}}}(\overline{\psi}).$
(3.19b)

If $(\phi,\overline{\phi})$ and $(\psi,\overline{\psi})$ satisfy the linear
systems (3.14) and (3.18) for some $v$, respectively. Otherwise, together with
the reductions (3.8) and (3.17), we have the reduction condition for
$(\omega,\overline{\omega})^{T}$ as follows

$\displaystyle
T_{{}_{n_{3}}}(\omega(\phi,\psi))=\omega(\overline{\phi},\overline{\psi}),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\phi,\psi))=\omega(\phi,\psi),$ $\displaystyle
T_{{}_{n_{3}}}(\omega(\overline{\phi},\overline{\psi}))=\omega(\phi,\psi),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\overline{\phi},\overline{\psi}))=\omega(\overline{\phi},\overline{\psi}).$

Especially,

$\displaystyle
T_{{}_{n_{3}}}(\omega(\phi,\rho))=\lambda\mu^{-1}\omega(\overline{\phi},\overline{\rho}),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\phi,\rho))=\lambda^{2}\mu^{-2}\omega(\phi,\rho),$
$\displaystyle
T_{{}_{n_{3}}}(\omega(\overline{\phi},\overline{\rho}))=\lambda\mu^{-1}\omega(\phi,\rho),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\overline{\phi},\overline{\rho}))=\lambda^{2}\mu^{-2}\omega(\overline{\phi},\overline{\rho});$
$\displaystyle
T_{{}_{n_{3}}}(\omega(\theta,\psi))=\lambda^{-1}\mu~{}\omega(\overline{\theta},\overline{\psi}),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\theta,\psi))=\lambda^{-2}\mu^{2}\omega(\theta,\psi),$
$\displaystyle
T_{{}_{n_{3}}}(\omega(\overline{\theta},\overline{\psi}))=\lambda^{-1}\mu~{}\omega(\theta,\psi),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\overline{\theta},\overline{\psi}))=\lambda^{-2}\mu^{2}\omega(\overline{\theta},\overline{\psi});$
$\displaystyle
T_{{}_{n_{3}}}(\omega(\theta,\rho))=\omega(\overline{\theta},\overline{\rho}),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\theta,\rho))=\omega(\theta,\rho),$ $\displaystyle
T_{{}_{n_{3}}}(\omega(\overline{\theta},\overline{\rho}))=\omega(\theta,\rho),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\overline{\theta},\overline{\rho}))=\omega(\overline{\theta},\overline{\rho});$

and

$\displaystyle
T_{{}_{n_{3}}}(\omega(\theta^{k},\rho^{l}))=\left(\frac{\lambda_{k}}{\lambda_{l}}\right)\omega(\overline{\theta}^{k},\overline{\rho}^{l}),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\theta^{k},\rho^{l}))=\left(\frac{\lambda_{k}}{\lambda_{l}}\right)^{2}\omega(\theta^{k},\rho^{l}),$
$\displaystyle
T_{{}_{n_{3}}}(\omega(\overline{\theta}^{k},\overline{\rho}^{l}))=\left(\frac{\lambda_{k}}{\lambda_{l}}\right)\omega(\theta^{k},\rho^{l}),\qquad
T_{{}_{n_{3}}}^{2}(\omega(\overline{\theta}^{k},\overline{\rho}^{l}))=\left(\frac{\lambda_{k}}{\lambda_{l}}\right)^{2}\omega(\overline{\theta}^{k},\overline{\rho}^{l}).$

The following proposition gives the binary Darboux transformation of the d-p-
mKdV equation.

######  Proposition 3.5.

For some $v$, let $(\theta,\overline{\theta})^{T}$ and
$(\phi,\overline{\phi})^{T}$ be two non-zero vector solutions of the linear
system (3.14), respectively corresponding to spectrum parameters $\mu$ and
$\lambda$; $(\rho,\overline{\rho})^{T}$ and $(\psi,\overline{\psi})^{T}$ be
two non-zero vector solutions of the linear system (3.18), respectively
corresponding to spectrum parameters $\mu$ and $\lambda$, then

$\displaystyle\mathrm{BDT}\colon\quad$
$\displaystyle\phi\rightarrow\phi-\theta\omega(\theta,\rho)^{-1}\omega(\phi,\rho),~{}\overline{\phi}\rightarrow\overline{\phi}-\overline{\theta}\omega(\overline{\theta},\overline{\rho})^{-1}\omega(\overline{\phi},\overline{\rho}),$
$\displaystyle
v\rightarrow\frac{T_{{}_{n_{3}}}(\omega(\theta,\rho))}{\omega(\theta,\rho)}v=\frac{\omega(\overline{\theta},\overline{\rho})}{T_{{}_{n_{3}}}(\omega(\overline{\theta},\overline{\rho}))}v,$
(3.20) $\displaystyle\mathrm{aBDT}\colon\quad$
$\displaystyle\psi\rightarrow\psi-\rho\omega(\theta,\rho)^{-1}\omega(\theta,\psi),~{}\overline{\psi}\rightarrow\overline{\psi}-\overline{\rho}\omega(\overline{\theta},\overline{\rho})^{-1}\omega(\overline{\theta},\overline{\psi}),$
$\displaystyle
v\rightarrow\frac{T_{{}_{n_{3}}}(\omega(\theta,\rho))}{\omega(\theta,\rho)}v=\frac{\omega(\overline{\theta},\overline{\rho})}{T_{{}_{n_{3}}}(\omega(\overline{\theta},\overline{\rho}))}v,$
(3.21)

leave (3.14) and (3.18) respectively invariant. Otherwise,

$\displaystyle\overline{\phi}-\overline{\theta}\omega(\overline{\theta},\overline{\rho})^{-1}\omega(\overline{\phi},\overline{\rho})=\lambda^{-1}T_{{}_{n_{3}}}\big{(}\phi-\theta\omega(\theta,\rho)^{-1}\omega(\phi,\rho)\big{)},$
$\displaystyle\phi-\theta\omega(\theta,\rho)^{-1}\omega(\phi,\rho)=\lambda^{-1}T_{{}_{n_{3}}}\big{(}\overline{\phi}-\overline{\theta}\omega(\overline{\theta},\overline{\rho})^{-1}\omega(\overline{\phi},\overline{\rho})\big{)},$
$\displaystyle\overline{\psi}-\overline{\rho}\omega(\overline{\theta},\overline{\rho})^{-1}\omega(\overline{\theta},\overline{\psi})=\lambda^{-1}T_{{}_{n_{3}}}^{-1}\big{(}\psi-\rho\omega(\theta,\rho)^{-1}\omega(\theta,\psi)\big{)},$
$\displaystyle\psi-\rho\omega(\theta,\rho)^{-1}\omega(\theta,\psi)=\lambda^{-1}T_{{}_{n_{3}}}^{-1}\big{(}\overline{\psi}-\overline{\rho}\omega(\overline{\theta},\overline{\rho})^{-1}\omega(\overline{\theta},\overline{\psi})\big{)}.$

The $N$-fold iteration of these binary Darboux transformations are given
below.

######  Proposition 3.6.

Let
$\big{(}\theta^{1},\overline{\theta}^{1}\big{)}^{T},\big{(}\theta^{2},\overline{\theta}^{2}\big{)}^{T}\\!,\dots,\big{(}\theta^{N},\overline{\theta}^{N}\big{)}^{T}$
and
$\big{(}\rho^{1},\overline{\rho}^{1}\big{)}^{T},\big{(}\rho^{2},\overline{\rho}^{2}\big{)}^{T}\\!,\dots,\big{(}\rho^{N},\overline{\rho}^{N}\big{)}^{T}$
be $N$ independent vector solutions, holding
$\theta^{k}=\lambda_{k}^{-2}T_{{}_{n_{3}}}^{2}(\theta^{k})$,
$\overline{\theta}^{k}=\lambda_{k}^{-1}T_{{}_{n_{3}}}(\theta^{k})$, and
$\rho^{k}=\lambda_{k}^{-2}T_{{}_{n_{3}}}^{-2}(\rho^{k})$,
$\overline{\rho}^{k}=\lambda_{k}^{-1}T_{{}_{n_{3}}}^{-1}(\rho^{k})$, by taking
$\lambda=\lambda_{k}$, $k=1,2,\dots,N$, satisfy linear systems (3.14) and
(3.18) for some $v$ respectively. Then

$\displaystyle\phi\rightarrow\widehat{\phi}=\frac{\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\bm{\theta}\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\phi\end{vmatrix}}{\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}},\qquad\overline{\phi}\rightarrow\widehat{\overline{\phi}}=\frac{\begin{vmatrix}\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}}^{T}\big{)}&\bm{\overline{\theta}}\\\
\omega\big{(}\overline{\phi},\bm{\overline{\rho}}^{T}\big{)}&\overline{\phi}\end{vmatrix}}{\big{|}\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}^{T}}\big{)}\big{|}},$
$\displaystyle
v\rightarrow\widehat{v}=\frac{\big{|}T_{{}_{n_{3}}}\big{(}\omega(\bm{\theta},\bm{\rho}^{T})\big{)}\big{|}}{\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}}v=\frac{\big{|}\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}}^{T}\big{)}\big{|}}{\big{|}T_{{}_{n_{3}}}\big{(}\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}}^{T}\big{)}\big{)}\big{|}}v,$

and

$\displaystyle\psi\rightarrow\widehat{\psi}=\frac{\begin{vmatrix}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}&\bm{\rho}\\\
\omega\big{(}\bm{\theta}^{T},\psi\big{)}&\psi\end{vmatrix}}{\big{|}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}\big{|}},\qquad\overline{\psi}\rightarrow\widehat{\overline{\psi}}=\frac{\begin{vmatrix}\omega\big{(}\bm{\overline{\theta}}^{T},\bm{\overline{\rho}}\big{)}&\bm{\overline{\rho}}\\\
\omega\big{(}\bm{\overline{\theta}}^{T},\overline{\psi}\big{)}&\overline{\psi}\end{vmatrix}}{\big{|}\omega\big{(}\bm{\overline{\theta}}^{T},\bm{\overline{\rho}}\big{)}\big{|}},$
$\displaystyle
v\rightarrow\widehat{v}=\frac{\big{|}T_{{}_{n_{3}}}\big{(}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}\big{)}\big{|}}{\big{|}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}\big{|}}v=\frac{\big{|}\omega\big{(}\bm{\overline{\theta}}^{T},\bm{\overline{\rho}}\big{)}\big{|}}{\big{|}T_{{}_{n_{3}}}\big{(}\omega\big{(}\bm{\overline{\theta}}^{T},\bm{\overline{\rho}}\big{)}\big{)}\big{|}}v,$

leave (3.14) and (3.18) respectively invariant, where
$\bm{\theta}=\big{(}\theta^{1},\dots,\theta^{N}\big{)}^{T}$ and
$\bm{\rho}=\big{(}\rho^{1},\dots,\rho^{N}\big{)}^{T}$. Otherwise,
$\widehat{\overline{\phi}}=\lambda^{-1}T_{{}_{n_{3}}}(\widehat{\phi})$,
$\widehat{\phi}=\lambda^{-1}T_{{}_{n_{3}}}(\widehat{\overline{\phi}})$,
$\widehat{\overline{\psi}}=\lambda^{-1}T_{{}_{n_{3}}}^{-1}(\widehat{\psi})$,
$\widehat{\psi}=\lambda^{-1}T_{{}_{n_{3}}}^{-1}(\widehat{\overline{\psi}})$.

###  3.3 Explicit solutions obtained by Darboux transformations

Here we present explicit examples of the classes of solutions that may be
obtained by means of the Darboux transformations derived above. We choose the
seed solution of the d-p-mKdV equation (3.7) as $v=v_{0}=1$. With this choice,
the first linear system (3.14) reads

$\displaystyle\phi_{1}=a_{1}\phi+\lambda\overline{\phi},\qquad\overline{\phi}_{1}=a_{1}\overline{\phi}+\lambda\phi,$
$\displaystyle\phi_{2}=a_{2}\phi+\lambda\overline{\phi},\qquad\overline{\phi}_{2}=a_{2}\overline{\phi}+\lambda\phi,$

and the eigenfunctions are found to be

$\displaystyle\phi(n_{1},n_{2},n_{3};\lambda)=\lambda^{n_{3}}\prod_{i=1}^{2}(a_{i}+\lambda)^{n_{i}}+(-\lambda)^{n_{3}}\prod_{i=1}^{2}(a_{i}-\lambda)^{n_{i}},$
(3.22a)
$\displaystyle\overline{\phi}(n_{1},n_{2},n_{3};\lambda)=\lambda^{n_{3}}\prod_{i=1}^{2}(a_{i}+\lambda)^{n_{i}}-(-\lambda)^{n_{3}}\prod_{i=1}^{2}(a_{i}-\lambda)^{n_{i}},$
(3.22b)

which hold $\phi=\lambda^{-2}T_{{}_{n_{3}}}(\phi)$,
$\overline{\phi}=\lambda^{-1}T_{{}_{n_{3}}}(\phi)$.

In a similar way the eigenfunctions of the second linear system (3.18) are

$\displaystyle\psi(n_{1},n_{2},n_{3};\lambda)=\lambda^{-n_{3}}\prod_{i=1}^{2}(a_{i}+\lambda)^{-n_{i}}+(-\lambda)^{-n_{3}}\prod_{i=1}^{2}(a_{i}-\lambda)^{-n_{i}},$
(3.23a)
$\displaystyle\overline{\psi}(n_{1},n_{2},n_{3};\lambda)=\lambda^{-n_{3}}\prod_{i=1}^{2}(a_{i}+\lambda)^{-n_{i}}-(-\lambda)^{-n_{3}}\prod_{i=1}^{2}(a_{i}-\lambda)^{-n_{i}},$
(3.23b)

which hold $\psi=\lambda^{-2}T_{{}_{n_{3}}}^{-1}(\psi)$,
$\overline{\psi}=\lambda^{-1}T_{{}_{n_{3}}}^{-1}(\psi)$.

For these eigenfunctions (3.22) and (3.23) above we may integrate (3.19) and
obtain the potentials

$\displaystyle\omega(\phi,\psi)=\frac{1}{2\lambda}\left[(-1)^{n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}+\lambda}{a_{i}-\lambda}\right)^{n_{i}}-(-1)^{-n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}+\lambda}{a_{i}-\lambda}\right)^{-n_{i}}\right],$
(3.24a)
$\displaystyle\omega(\overline{\phi},\overline{\psi})=\frac{1}{2\lambda}\left[(-1)^{-n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}+\lambda}{a_{i}-\lambda}\right)^{-n_{i}}-(-1)^{n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}+\lambda}{a_{i}-\lambda}\right)^{n_{i}}\right],$
(3.24b)

which hold $\omega(\phi,\psi)=T_{{}_{n_{3}}}^{2}(\omega(\phi,\psi))$,
$\omega(\overline{\phi},\overline{\psi})=T_{{}_{n_{3}}}(\omega(\phi,\psi))$.

Otherwise, for $\lambda=\lambda_{k}$, $v=v_{0}=1$, the first linear system
(3.14) has eigenfunctions

$\displaystyle\theta^{k}(n_{1},n_{2},n_{3};\lambda_{k})=\lambda_{k}^{n_{3}}\prod_{i=1}^{2}(a_{i}+\lambda_{k})^{n_{i}}+(-\lambda_{k})^{n_{3}}\prod_{i=1}^{2}(a_{i}-\lambda_{k})^{n_{i}},$
(3.25a)
$\displaystyle\overline{\theta}^{k}(n_{1},n_{2},n_{3};\lambda_{k})=\lambda_{k}^{n_{3}}\prod_{i=1}^{2}(a_{i}+\lambda_{k})^{n_{i}}-(-\lambda_{k})^{n_{3}}\prod_{i=1}^{2}(a_{i}-\lambda_{k})^{n_{i}},$
(3.25b)

which hold $\theta^{k}=\lambda_{k}^{-2}T_{{}_{n_{3}}}^{2}(\theta^{k})$,
$\overline{\theta}^{k}=\lambda_{k}^{-1}T_{{}_{n_{3}}}(\theta^{k})$.

Similarly, for $\lambda=\lambda_{l}$, $v=v_{0}=1$, the second linear system
(3.18) has eigenfunctions

$\displaystyle\rho^{l}(n_{1},n_{2},n_{3};\lambda_{l})=\lambda_{l}^{-n_{3}}\prod_{i=1}^{2}(a_{i}+\lambda_{l})^{-n_{i}}+(-\lambda_{l})^{-n_{3}}\prod_{i=1}^{2}(a_{i}-\lambda_{l})^{-n_{i}},$
(3.26a)
$\displaystyle\overline{\rho}^{l}(n_{1},n_{2},n_{3};\lambda_{l})=\lambda_{l}^{-n_{3}}\prod_{i=1}^{2}(a_{i}+\lambda_{l})^{-n_{i}}-(-\lambda_{l})^{-n_{3}}\prod_{i=1}^{2}(a_{i}-\lambda_{l})^{-n_{i}},$
(3.26b)

which hold $\rho^{l}=\lambda_{l}^{-2}T_{{}_{n_{3}}}^{-2}(\rho^{l})$,
$\overline{\rho}^{l}=\lambda_{l}^{-1}T_{{}_{n_{3}}}^{-1}(\rho^{l})$.

For these eigenfunctions (3.25) and (3.26) above we may integrate (3.19) and
obtain the potential, for $\lambda_{k}\neq\lambda_{l}$,

$\displaystyle\omega\big{(}\theta^{k},\rho^{l}\big{)}=\lambda_{l}^{-1}\left(\frac{\lambda_{k}}{\lambda_{l}}\right)^{n_{3}}\Bigg{[}\left(\frac{\lambda_{k}}{\lambda_{l}}-1\right)^{-1}\prod_{i=1}^{2}\left(\frac{a_{i}+\lambda_{k}}{a_{i}+\lambda_{l}}\right)^{n_{i}}$
$\displaystyle\hphantom{\omega\big{(}\theta^{k},\rho^{l}\big{)}=}{}-\left(\frac{\lambda_{k}}{\lambda_{l}}+1\right)^{-1}(-1)^{n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}-\lambda_{k}}{a_{i}+\lambda_{l}}\right)^{n_{i}}+\left(\frac{\lambda_{k}}{\lambda_{l}}+1\right)^{-1}(-1)^{n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}+\lambda_{k}}{a_{i}-\lambda_{l}}\right)^{n_{i}}$
$\displaystyle\hphantom{\omega\big{(}\theta^{k},\rho^{l}\big{)}=}{}-\left(\frac{\lambda_{k}}{\lambda_{l}}-1\right)^{-1}\prod_{i=1}^{2}\left(\frac{a_{i}-\lambda_{k}}{a_{i}-\lambda_{l}}\right)^{n_{i}}\Bigg{]},$
(3.27a)
$\displaystyle\omega\big{(}\overline{\theta}^{k},\overline{\rho}^{l}\big{)}=\lambda_{l}^{-1}\left(\frac{\lambda_{k}}{\lambda_{l}}\right)^{n_{3}}\Bigg{[}\left(\frac{\lambda_{k}}{\lambda_{l}}-1\right)^{-1}\prod_{i=1}^{2}\left(\frac{a_{i}+\lambda_{k}}{a_{i}+\lambda_{l}}\right)^{n_{i}}$
$\displaystyle\hphantom{\omega\big{(}\overline{\theta}^{k},\overline{\rho}^{l}\big{)}=}{}+\left(\frac{\lambda_{k}}{\lambda_{l}}+1\right)^{-1}(-1)^{n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}-\lambda_{k}}{a_{i}+\lambda_{l}}\right)^{n_{i}}-\left(\frac{\lambda_{k}}{\lambda_{l}}+1\right)^{-1}(-1)^{n_{3}}\prod_{i=1}^{2}\left(\frac{a_{i}+\lambda_{k}}{a_{i}-\lambda_{l}}\right)^{n_{i}}$
$\displaystyle\hphantom{\omega\big{(}\overline{\theta}^{k},\overline{\rho}^{l}\big{)}=}{}-\left(\frac{\lambda_{k}}{\lambda_{l}}-1\right)^{-1}\prod_{i=1}^{2}\left(\frac{a_{i}-\lambda_{k}}{a_{i}-\lambda_{l}}\right)^{n_{i}}\Bigg{]},$
(3.27b)

which hold

$\displaystyle
T_{{}_{n_{3}}}\big{(}\omega\big{(}\theta^{k},\rho^{l}\big{)}\big{)}=\left(\frac{\lambda_{k}}{\lambda_{l}}\right)\omega\big{(}\overline{\theta}^{k},\overline{\rho}^{l}\big{)},\qquad
T_{{}_{n_{3}}}^{2}\big{(}\omega\big{(}\theta^{k},\rho^{l}\big{)}\big{)}=\left(\frac{\lambda_{k}}{\lambda_{l}}\right)^{2}\omega\big{(}\theta^{k},\rho^{l}\big{)}.$

For $\lambda_{k}=\lambda_{l}$, these eigenfunctions are (3.24) taking
$\lambda=\lambda_{k}=\lambda_{l}$.

Given the above expression it is straightforward to write down the following
explicit solution for the d-p-mKdV equation (3.7)

$\displaystyle
v(n_{1},n_{2})=\frac{T_{{}_{n_{3}}}\big{(}C_{{}_{[3]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}\big{)}}{C_{{}_{[3]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}}v_{0},$

where $\theta^{k}=\theta^{k}(n_{1},n_{2},n_{3};\lambda_{k})$ is given by
(3.25) and $\lambda_{k}$ are arbitrary constants;

$\displaystyle
v(n_{1},n_{2})=\frac{T_{{}_{n_{3}}}^{-1}\big{(}C_{{}_{[\overline{3}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}\big{)}}{C_{{}_{[\overline{3}]}}\big{(}\rho^{1},\rho^{2},\dots,\rho^{N}\big{)}}v_{0},$

where $\rho^{k}=\rho^{k}(n_{1},n_{2},n_{3};\lambda_{k})$ is given by (3.26)
and $\lambda_{k}$ are arbitrary constants;

$\displaystyle
v(n_{1},n_{2},n_{3})=\frac{T_{{}_{n_{3}}}\left(\det(\omega_{k,l})\right)}{\det(\omega_{k,l})}v_{0},\qquad
k,l=1,2,\dots,N,$

where $\omega_{k,l}$ is given by (3.27) with
$\omega_{k,l}=\omega(\theta^{k},\rho^{l})$.

##  4 Conclusions

In this paper, we presents two main results. In the first we show how the d-p-
mKdV equation and its Lax pairs in matrix form arise from the Hirota–Miwa
equation by 2-periodic reduction. The second is that Darboux transformations
and binary Darboux transformations are derived for the d-p-mKdV equation and
we show how these may be used to construct exact solutions. In this paper, we
have revisited the Darboux and binary transformations of the Hirota–Miwa
equation but in a departure from the results in [18, 19, 21], by the gauge
transformation $\phi\rightarrow\prod\limits_{i=1}^{3}a_{i}^{-n_{i}}\phi$, we
write the linear system of Hirota–Miwa equation in a way which is suitable for
obtaining the Lax pair of the d-pmKdV equation naturally by a 2-periodic
reduction. Up to gauge transformations, these Lax pairs, which allow the
application of the classical Darboux transformations, are coincident with the
ones given by the multidimensional consistency property [4]. Hietarinta and
Zhang [8] derived the $N$-soliton solutions to the d-p-mKdV equation using
Hirota’s direct method and the authors mention that the bilinear equations
they get are similar to the Hirota–Miwa equation (1.3). The results in this
paper, in which similar results are obtained by reduction of the Hirota–Miwa
equation, give an explanation of the observations in [8].

##  Appendix A Some proofs

This section contains proofs of some of the propositions in the main text. For
the d-p-mKdV equation, one each of the $N$-fold basic Darboux transformations
and binary Darboux transformations is proved. The omitted proofs are very
similar.

###  A.1 Proof of Proposition 3.2

Let

$\displaystyle
F:=C_{{}_{[3]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}=|\bm{\theta}(0),\bm{\theta}(1),\dots,\bm{\theta}(N-1)|,$
$\displaystyle
G:=C_{{}_{[3]}}\big{(}\theta^{1},\theta^{2},\dots,\theta^{N},\phi\big{)}=\big{|}\bm{\theta}^{{}^{\dagger}}(0),\bm{\theta}^{{}^{\dagger}}(1),\dots,\bm{\theta}^{{}^{\dagger}}(N)\big{|},$
$\displaystyle\overline{F}:=C_{{}_{[3]}}\big{(}\overline{\theta}^{1},\overline{\theta}^{2},\dots,\overline{\theta}^{N}\big{)}=\big{|}\overline{\bm{\theta}}(0),\overline{\bm{\theta}}(1),\dots,\overline{\bm{\theta}}(N-1)\big{|},$
$\displaystyle\overline{G}:=C_{{}_{[3]}}\big{(}\overline{\theta}^{1},\overline{\theta}^{2},\dots,\overline{\theta}^{N},\phi\big{)}=\big{|}\overline{\bm{\theta}}^{{\dagger}}(0),\overline{\bm{\theta}}^{{\dagger}}(1),\dots,\overline{\bm{\theta}}^{{\dagger}}(N)\big{|},$

where $\bm{\theta}=\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\big{)}$,
$\bm{\theta}^{{}^{\dagger}}\\!=\big{(}\theta^{1},\theta^{2},\dots,\theta^{N}\\!,\phi\big{)}$,
$\overline{\bm{\theta}}=\big{(}\overline{\theta}^{1},\overline{\theta}^{2},\dots,\overline{\theta}^{N}\big{)}$,
$\overline{\bm{\theta}}^{{}^{\dagger}}\\!=\big{(}\overline{\theta}^{1},\overline{\theta}^{2},\dots,\overline{\theta}^{N}\\!,\overline{\phi}\big{)}$.
Moreover, by the reduction conditions
$\overline{\phi}=\lambda^{-1}T_{{}_{n_{3}}}(\phi)$,
$\phi=\lambda^{-1}T_{{}_{n_{3}}}(\overline{\phi})$,
$\overline{\theta}^{i}=\lambda_{i}^{-1}T_{{}_{n_{3}}}(\theta^{i})$, and
$\theta^{i}=\lambda_{i}^{-1}T_{{}_{n_{3}}}(\overline{\theta}^{i})$, we easily
get the relation
$\frac{\overline{G}}{\overline{F}}=\lambda^{-1}T_{{}_{n_{3}}}\big{(}\frac{G}{F}\big{)}$,
$\frac{G}{F}=\lambda^{-1}T_{{}_{n_{3}}}\big{(}\frac{\overline{G}}{\overline{F}}\big{)}$.
To verify that (3.14) is invariant under (3.16) we must show that, for
$\kappa=1,2$,

$\displaystyle
G_{\kappa}T_{{}_{n_{3}}}(F)=a_{\kappa}v_{\kappa}v^{-1}T_{{}_{n_{3}}}(F_{\kappa})G+T_{{}_{n_{3}}}(G)F_{\kappa},$
$\displaystyle\overline{G}_{\kappa}T_{{}_{n_{3}}}(\overline{F})=a_{\kappa}v_{\kappa}^{-1}vT_{{}_{n_{3}}}(\overline{F}_{\kappa})\overline{G}+T_{{}_{n_{3}}}(\overline{G})\overline{F}_{\kappa},$

which is equivalent to, under the backward shift operator
$T_{{}_{n_{3}}}^{-1}T_{{}_{n_{{}_{\kappa}}}}^{-1}$,

$\displaystyle
T_{{}_{n_{3}}}^{-1}(G)F_{{}_{\overline{\kappa}}}=a_{\kappa}vv_{\overline{\kappa}}^{-1}FT_{{}_{n_{3}}}^{-1}(G_{{}_{\overline{\kappa}}})+G_{{}_{\overline{\kappa}}}T_{{}_{n_{3}}}^{-1}(F),$
(A.1a) $\displaystyle
T_{{}_{n_{3}}}^{-1}(\overline{G})\overline{F}_{{}_{\overline{\kappa}}}=a_{\kappa}v^{-1}v_{\overline{\kappa}}\overline{F}T_{{}_{n_{3}}}^{-1}(\overline{G}_{{}_{\overline{\kappa}}})+\overline{G}_{{}_{\overline{\kappa}}}T_{{}_{n_{3}}}^{-1}(\overline{F}).$
(A.1b)

From (3.14), we can deduce the following formulae which are the basic
properties we use in proving

$\displaystyle\bm{\theta}_{\overline{\kappa}}(l)=\bm{\theta}(l-1)+\sum_{i=0}^{l-2}(-\alpha_{\overline{\kappa}})^{i+1}\bm{\theta}(l-2-i)+(-\alpha_{\overline{\kappa}})^{l}\bm{\theta}_{\overline{\kappa}}(0),$
$\displaystyle{\bm{\theta}^{\dagger}}_{\overline{\kappa}}(l)=\bm{\theta}^{\dagger}(l-1)+\sum_{i=0}^{l-2}(-\alpha_{\overline{\kappa}})^{i+1}{\bm{\theta}^{\dagger}}(l-2-i)+(-\alpha_{\overline{\kappa}})^{l}{\bm{\theta}^{\dagger}}_{\overline{\kappa}}(0),$
$\displaystyle\bm{\overline{\theta}}_{\overline{\kappa}}(l)=\bm{\overline{\theta}}(l-1)+\sum_{i=0}^{l-2}(-\beta_{\overline{\kappa}})^{i+1}\bm{\overline{\theta}}(l-2-i)+(-\beta_{\overline{\kappa}})^{l}\bm{\overline{\theta}}{{}_{{}_{\overline{\kappa}}}}(0),$
$\displaystyle{\bm{\overline{\theta}}^{\dagger}}_{\overline{\kappa}}(l)=\bm{\overline{\theta}}^{\dagger}(l-1)+\sum_{i=0}^{l-2}(-\beta_{\kappa})^{i+1}{\bm{\overline{\theta}}^{\dagger}}(l-2-i)+(-\beta_{\overline{\kappa}})^{l}{\bm{\overline{\theta}}^{\dagger}}_{\overline{\kappa}}(0),$

where $\alpha_{\overline{\kappa}}=a_{\kappa}vv_{\overline{\kappa}}^{-1}$ and
$\beta_{\overline{\kappa}}=a_{\kappa}v^{-1}v_{\overline{\kappa}}$ are scalars,
$\kappa=1,2$. Then, for (A.1a), it follows that

$\displaystyle
T_{{}_{n_{3}}}^{-1}(G)=\big{|}\bm{\theta^{\dagger}}(-1),\bm{\theta^{\dagger}}(0),\bm{\theta^{\dagger}}(1),\dots,\bm{\theta^{\dagger}}(N-1)\big{|},$
$\displaystyle
F_{{}_{\overline{\kappa}}}=\big{|}\bm{\theta_{\overline{\kappa}}^{\dagger}}(0),\bm{\theta}(0),\bm{\theta}(1),\dots,\bm{\theta}(N-2)\big{|},$
$\displaystyle\alpha_{\kappa}T_{{}_{n_{3}}}^{-1}(G_{{}_{\overline{\kappa}}})=-\big{|}\bm{\theta^{\dagger}}_{\overline{\kappa}}(0),\bm{\theta^{\dagger}}(-1),\bm{\theta^{\dagger}}(0),\dots,\bm{\theta^{\dagger}}(N-2)\big{|},$
$\displaystyle
F=\big{|}\bm{\theta}(0),\bm{\theta}(1),\bm{\theta}(2),\dots,\bm{\theta}(N-1)\big{|},$
$\displaystyle
G_{{}_{\overline{\kappa}}}=\big{|}\bm{\theta_{\overline{\kappa}}^{\dagger}}(0),\bm{\theta^{\dagger}}(0),\bm{\theta^{\dagger}}(1),\dots,\bm{\theta^{\dagger}}(N-1)\big{|},$
$\displaystyle
T_{{}_{n_{3}}}^{-1}(F)=\big{|}\bm{\theta}(-1),\bm{\theta}(0),\bm{\theta}(1),\dots,\bm{\theta}(N-2)\big{|}.$

Substituting into the left-hand side of (A.1a), and using the Laplace theorem,
we get

$\displaystyle\text{LHS}=\left|\begin{matrix}\bm{\theta^{\dagger}}_{\overline{\kappa}}(0)&\bm{\theta^{\dagger}}(-1)&\bm{\theta^{\dagger}}(0)&\cdots&\bm{\theta^{\dagger}}(N-2)&\bm{0}&\cdots&\bm{0}&\bm{\theta^{\dagger}}(N-1)\\\
\bm{\theta}_{\overline{\kappa}}(0)&\bm{\theta}(-1)&\bm{0}&\cdots&\bm{0}&\bm{\theta}(0)&\cdots&\bm{\theta}(N-2)&\bm{\theta}(N-1)\end{matrix}\right|=0.$

In a similar way, we can prove that (A.1b) is also satisfied.

###  A.2 Proof of Proposition 3.6

The proof is by induction. Let
$\big{(}\theta^{1},\overline{\theta}^{1}\big{)}^{T}=(\theta[0],\overline{\theta}[0])^{T},\big{(}\theta^{2},\overline{\theta}^{2}\big{)}^{T},\dots,\big{(}\theta^{N},\overline{\theta}^{N}\big{)}^{T}$
and
$\big{(}\rho^{1},\overline{\rho}^{1}\big{)}^{T}=(\rho[0],\overline{\rho}[0])^{T},\big{(}\rho^{2},\overline{\rho}^{2}\big{)}^{T},\dots,\big{(}\rho^{N},\overline{\rho}^{N}\big{)}^{T}$
be vector eigenfunctions of ‘seed’ linear system (3.14) and (3.18)
respectively for the ‘seed’ potential $v=v[0]$ and let
$(\phi,\overline{\phi})=(\phi[0],\overline{\phi}[0])$,
$(\psi,\overline{\psi})=(\psi[0],\overline{\psi}[0])$ denote arbitrary
eigenfunctions.

The $N$th iteration of binary Darboux transformations is via the formulae,
$N=1,2,\dots$,

$\displaystyle\phi[N]=\phi[N-1]-\theta[N-1]\omega(\theta[N-1],\rho[N-1])^{-1}\omega(\phi[N-1],\rho[N-1]),$
(A.2a)
$\displaystyle\overline{\phi}[N]=\overline{\phi}[N-1]-\overline{\theta}[N-1]\omega(\overline{\theta}[N-1],\overline{\rho}[N-1])^{-1}\omega(\overline{\phi}[N-1],\overline{\rho}[N-1]),$
(A.2b)
$\displaystyle\psi[N]=\psi[N-1]-\rho[N-1]\omega(\theta[N-1],\rho[N-1])^{-1}\omega(\theta[N-1],\psi[N-1]),$
(A.2c)
$\displaystyle\overline{\psi}[N]=\overline{\psi}[N-1]-\overline{\rho}[N-1]\omega(\overline{\theta}[N-1],\overline{\rho}[N-1])^{-1}\omega(\overline{\theta}[N-1],\overline{\psi}[N-1]),$
(A.2d) $\displaystyle
v[N]=\frac{T_{n_{3}}(\omega(\theta[N-1],\rho[N-1]))}{\omega(\theta[N-1],\rho[N-1])}v[N-1]$
$\displaystyle\hphantom{v[N]}{}=\frac{\omega(\overline{\theta}[N-1],\overline{\rho}[N-1])}{T_{n_{3}}(\omega(\overline{\theta}[N-1],\overline{\rho}[N-1]))}v[N-1],$
(A.2e)

and

$\displaystyle\theta[N]=\phi[N]|_{{}_{\phi\rightarrow\theta^{N+1}}},\qquad\overline{\theta}[N]=\overline{\phi}[N]|_{{}_{\overline{\phi}\rightarrow\overline{\theta}^{N+1}}},$
(A.3a)
$\displaystyle\rho[N]=\psi[N]|_{{}_{\psi\rightarrow\rho^{N+1}}},\qquad\overline{\rho}[N]=\overline{\psi}[N]|_{{}_{\overline{\psi}\rightarrow\overline{\rho}^{N+1}}}.$
(A.3b)

For $N=1$, the iterated binary Darboux transformation is the basic form given
by (3.20) and (3.21).

Suppose for $N=k$, the Proposition 3.6 is right, i.e., we have

$\displaystyle\phi[k]=\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\bm{\theta}\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\phi\end{vmatrix}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}^{-1},\qquad{\overline{\phi}[k]}=\begin{vmatrix}\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}}^{T}\big{)}&\bm{\overline{\theta}}\\\
\omega\big{(}\overline{\phi},\bm{\overline{\rho}}^{T}\big{)}&\overline{\phi}\end{vmatrix}\big{|}\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}^{T}}\big{)}\big{|}^{-1},$
(A.4a)
$\displaystyle\psi[k]=\begin{vmatrix}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}&\bm{\rho}\\\
\omega\big{(}\bm{\theta}^{T},\psi)&\psi\end{vmatrix}\big{|}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}\big{|}^{-1},\qquad{\overline{\psi}[k]}=\begin{vmatrix}\omega\big{(}\bm{\overline{\theta}}^{T},\bm{\overline{\rho}}\big{)}&\bm{\overline{\rho}}\\\
\omega\big{(}\bm{\overline{\theta}}^{T},\overline{\psi}\big{)}&\overline{\psi}\\\
\end{vmatrix}\big{|}\omega\big{(}\bm{\overline{\theta}}^{T},\bm{\overline{\rho}}\big{)}\big{|}^{-1},$
(A.4b) $\displaystyle
v[k]=\frac{\big{|}T_{{}_{n_{3}}}\big{(}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}\big{)}\big{|}}{\big{|}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}\big{|}}v=\frac{\big{|}\omega\big{(}\bm{\overline{\theta}}^{T},\bm{\overline{\rho}}\big{)}\big{|}}{\big{|}T_{{}_{n_{3}}}\big{(}\omega\big{(}\bm{\overline{\theta}}^{T},\bm{\overline{\rho}}\big{)}\big{)}\big{|}}v$
$\displaystyle\hphantom{v[k]}{}=\frac{\big{|}T_{{}_{n_{3}}}\big{(}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{)}\big{|}}{\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}}v=\frac{\big{|}\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}^{T}}\big{)}\big{|}}{\big{|}T_{{}_{n_{3}}}\big{(}\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}^{T}}\big{)}\big{)}\big{|}}v,$
(A.4c)

where the column vectors are as below

$\displaystyle\bm{\theta}=\big{(}\theta^{1},\theta^{2},\dots,\theta^{k}\big{)}^{T},\qquad\bm{\overline{\theta}}=\big{(}\overline{\theta}^{1},\overline{\theta}^{2},\dots,\overline{\theta}^{k}\big{)}^{T},$
$\displaystyle\bm{\rho}=\big{(}\rho^{1},\rho^{2},\dots,\rho^{k}\big{)}^{T},\qquad\bm{\overline{\rho}}=\big{(}\overline{\rho}^{1},\overline{\rho}^{2},\dots,\overline{\rho}^{k}\big{)}^{T},$

and the $k\times k$ matrices and $1\times k$ row vectors are as follows

$\displaystyle\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}=\left(\begin{matrix}\omega\big{(}\theta^{1},\rho^{1}\big{)}&\omega\big{(}\theta^{1},\rho^{2}\big{)}&\cdots&\omega\big{(}\theta^{1},\rho^{k}\big{)}\\\
\omega\big{(}\theta^{2},\rho^{1}\big{)}&\omega\big{(}\theta^{2},\rho^{2}\big{)}&\cdots&\omega\big{(}\theta^{2},\rho^{k}\big{)}\\\
\vdots&\vdots&\cdots&\vdots\\\
\omega\big{(}\theta^{k},\rho^{1}\big{)}&\omega\big{(}\theta^{k},\rho^{2}\big{)}&\cdots&\omega\big{(}\theta^{k},\rho^{k}\big{)}\end{matrix}\right),\qquad\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}=\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}^{T},$
$\displaystyle\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}}^{T}\big{)}=\left(\begin{matrix}\omega\big{(}\overline{\theta}^{1},\overline{\rho}^{1}\big{)}&\omega\big{(}\overline{\theta}^{1},\overline{\rho}^{2}\big{)}&\cdots&\omega\big{(}\overline{\theta}^{1},\overline{\rho}^{k}\big{)}\\\
\omega\big{(}\overline{\theta}^{2},\overline{\rho}^{1}\big{)}&\omega\big{(}\overline{\theta}^{2},\overline{\rho}^{2}\big{)}&\cdots&\omega\big{(}\overline{\theta}^{2},\overline{\rho}^{k}\big{)}\\\
\vdots&\vdots&\cdots&\vdots\\\
\omega\big{(}\overline{\theta}^{k},\overline{\rho}^{1}\big{)}&\omega\big{(}\overline{\theta}^{k},\overline{\rho}^{2}\big{)}&\cdots&\omega\big{(}\overline{\theta}^{k},\overline{\rho}^{k}\big{)}\end{matrix}\right),\qquad\omega\big{(}\bm{\overline{\theta}}^{T},\bm{\overline{\rho}}\big{)}=\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}}^{T}\big{)}^{T},$
$\displaystyle\omega\big{(}\phi,\bm{\rho}^{T}\big{)}=\left(\begin{matrix}\omega\big{(}\phi,\rho^{1}\big{)}&\omega\big{(}\phi,\rho^{2}\big{)}&\cdots&\omega\big{(}\phi,\rho^{k}\big{)}\end{matrix}\right),$
$\displaystyle\omega\big{(}\overline{\phi},\bm{\overline{\rho}}^{T}\big{)}=\left(\begin{matrix}\omega\big{(}\overline{\phi},\overline{\rho}^{1}\big{)}&\omega\big{(}\overline{\phi},\overline{\rho}^{2}\big{)}&\cdots&\omega\big{(}\overline{\phi},\overline{\rho}^{k}\big{)}\end{matrix}\right),$
$\displaystyle\omega\big{(}\bm{\theta}^{T},\psi\big{)}=\left(\begin{matrix}\omega\big{(}\theta^{1},\psi\big{)}&\omega\big{(}\theta^{2},\psi\big{)}&\cdots&\omega\big{(}\theta^{k},\psi\big{)}\end{matrix}\right),$
$\displaystyle\omega\big{(}\bm{\overline{\theta}}^{T},\overline{\psi}\big{)}=\left(\begin{matrix}\omega\big{(}\overline{\theta}^{1},\overline{\psi}\big{)}&\omega\big{(}\overline{\theta}^{2},\overline{\psi}\big{)}&\cdots&\omega\big{(}\overline{\theta}^{k},\overline{\psi}\big{)}\end{matrix}\right).$

Moerover, we have

$\displaystyle\omega(\phi[k],\psi[k])=\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega(\bm{\theta},\psi)\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\omega(\phi,\psi)\end{vmatrix}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}^{-1},$
(A.5a)
$\displaystyle\omega(\overline{\phi}[k],\overline{\psi}[k])=\begin{vmatrix}\omega\big{(}\bm{\overline{\theta}},\bm{\rho}^{T}\big{)}&\omega(\bm{\overline{\theta}},\overline{\psi})\\\
\omega\big{(}\overline{\phi},\bm{\overline{\rho}}^{T}\big{)}&\omega(\overline{\phi},\overline{\psi})\end{vmatrix}\big{|}\omega\big{(}\bm{\overline{\theta}},\bm{\overline{\rho}}^{T}\big{)}\big{|}^{-1}.$
(A.5b)

The proof of (A.5) is as follows. By the definition of $\omega$ and
$\overline{\omega}$ in (3.19), and together with (A.4a) and (A.4b), we have

$\displaystyle\Delta_{3}(\omega(\phi[k],\psi[k]))=\phi[k]T_{{}_{n_{3}}}(\psi[k])=\Delta_{3}(\omega(\phi,\psi))-T_{{}_{n_{3}}}\big{(}\omega\big{(}\bm{\theta}^{T},\psi\big{)}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}^{-1}\big{)}\Delta_{3}(\omega(\phi,\bm{\rho}))$
$\displaystyle\hphantom{\Delta_{3}(\omega(\phi[k],\psi[k]))=}{}-\Delta_{3}\big{(}\omega\big{(}\bm{\theta}^{T},\psi\big{)}\big{)}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}^{-1}\omega(\phi,\bm{\rho})$
$\displaystyle\hphantom{\Delta_{3}(\omega(\phi[k],\psi[k]))=}{}+T_{{}_{n_{3}}}\big{(}\omega\big{(}\bm{\theta}^{T},\psi\big{)}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}^{-1}\big{)}\Delta_{3}\big{(}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}\big{)}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}^{-1}\omega(\phi,\bm{\rho})$
$\displaystyle\hphantom{\Delta_{3}(\omega(\phi[k],\psi[k]))}{}=\Delta_{3}\big{(}\omega(\phi,\psi)-\omega\big{(}\bm{\theta}^{T},\psi\big{)}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}^{-1}\omega(\phi,\bm{\rho})\big{)}$
$\displaystyle\hphantom{\Delta_{3}(\omega(\phi[k],\psi[k]))}{}=\Delta_{3}\left(\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega(\bm{\theta},\psi)\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\omega(\phi,\psi)\end{vmatrix}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}^{-1}\right),$

which means (A.5a) is right. Similarly, we can get (A.5b) is right as well.

Note here that we use the difference operator property for matrices as follows

$\displaystyle\Delta_{n}\big{(}CA^{-1}B\big{)}=T_{n}(CA^{-1})\Delta_{n}(B)-\Delta_{n}(C)A^{-1}B-C_{n}A_{n}^{-1}\Delta_{n}(A)A^{-1}B,$

where $\Delta_{n}=T_{n}-1$ is the difference operator, $A=A_{N\times N}$,
$B=B_{N\times 1}$ and $C=C_{1\times N}$ are arbitrary function matrix, column
vector and row vector of independent discrete variable $n$ respectively.

Next, let us prove the $(k+1)$-th step is also right. By the iterated formulae
(A.2a), we have

$\displaystyle\phi[k+1]=\phi[k]-\theta[k]\omega(\theta[k],\rho[k])^{-1}\omega(\phi[k],\rho[k]).$
(A.6)

Substitute (A.2a), (A.3a), (A.3b) and (A.5a) into (A.6), we have

$\displaystyle\phi[k+1]=\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\bm{\theta}\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\phi\\\
\end{vmatrix}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}^{-1}-\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\bm{\theta}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\theta^{k+1}\end{vmatrix}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}^{-1}$
$\displaystyle\hphantom{\phi[k+1]=}{}\times\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}\end{vmatrix}^{-1}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}$
$\displaystyle\hphantom{\phi[k+1]=}{}\times\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\omega\big{(}\phi,\rho^{k+1}\big{)}\end{vmatrix}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}^{-1}$
$\displaystyle\hphantom{\phi[k+1]}{}=\left(\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\bm{\theta}\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\phi\end{vmatrix}\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}\end{vmatrix}-\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\bm{\theta}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\theta^{k+1}\end{vmatrix}\right.$
$\displaystyle\left.\hphantom{\phi[k+1]=}{}\times\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\omega\big{(}\phi,\rho^{k+1}\big{)}\end{vmatrix}\right)\left(\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}\\\
\end{vmatrix}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}\right)^{-1}$
$\displaystyle\hphantom{\phi[k+1]}{}=\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}&\bm{\theta}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}&\theta^{k+1}\\\
\omega\big{(}\phi,\bm{\rho}^{T}\big{)}&\omega(\phi,\rho^{k+1})&\phi\end{vmatrix}\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}\end{vmatrix}^{-1}.$

Note here that we use the Jacobi identity.

Similarly, we can have

$\displaystyle\psi[k+1]=\begin{vmatrix}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}&\omega\big{(}\theta^{k+1},\bm{\rho}\big{)}&\bm{\rho}\\\
\omega\big{(}\bm{\theta}^{T},\rho^{k+1}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}&\rho^{k+1}\\\
\omega\big{(}\bm{\theta}^{T},\psi\big{)}&\omega\big{(}\theta^{k+1},\psi\big{)}&\psi\end{vmatrix}\begin{vmatrix}\omega\big{(}\bm{\theta}^{T},\bm{\rho}\big{)}&\omega\big{(}\theta^{k+1},\bm{\rho}\big{)}\\\
\omega\big{(}\bm{\theta}^{T},\rho^{k+1}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}\end{vmatrix}^{-1}.$

Then from (A.2e), for the potential $\widehat{v}$, we get

$\displaystyle
v[k+1]=\frac{T_{n_{3}}(\omega(\theta[k],\rho[k]))}{\omega(\theta[k],\rho[k])}v[k]=T_{n_{3}}\left(\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega(\bm{\theta},\rho^{k+1})\\\
\omega)\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}\end{vmatrix}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}^{-1}\right)$
$\displaystyle\hphantom{v[k+1]=}{}\times\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}\end{vmatrix}^{-1}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}T_{n_{3}}\big{(}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}\big{)}\big{|}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}\big{|}^{-1}v$
$\displaystyle\hphantom{v[k+1]}{}=T_{n_{3}}\left(\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}\end{vmatrix}\right)\begin{vmatrix}\omega\big{(}\bm{\theta},\bm{\rho}^{T}\big{)}&\omega\big{(}\bm{\theta},\rho^{k+1}\big{)}\\\
\omega\big{(}\theta^{k+1},\bm{\rho}^{T}\big{)}&\omega\big{(}\theta^{k+1},\rho^{k+1}\big{)}\end{vmatrix}^{-1}v.$

The proofs of the remaining parts are very similar, we omit them here.

### Acknowledgements

One of the author (YS) would like to acknowledge Professor Jarmo Hietarinta
for his useful suggestions and hospitality when the author visited Turku
University. The authors (YS and JXZ) also thank for the financial support from
NSFC (Grant Numbers 11501510, 11271362, 11271266).

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