# Squared eigenfunction symmetry of the D$\Delta$mKP hierarchy
and its constraint

Kui Chen, Cheng Zhang, Da-jun Zhang111Corresponding author. Email:
<EMAIL_ADDRESS>
Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China

###### Abstract

In this paper squared eigenfunction symmetry of the differential-difference
modified Kadomtsev-Petviashvili (D$\Delta$mKP) hierarchy and its constraint
are considered. Under the constraint, the Lax triplets of the D$\Delta$mKP
hierarchy, together with their adjoint forms, give rise to the positive
relativistic Toda (R-Toda) hierarchy. An invertible transformation is given to
connect the positive and negative R-Toda hierarchies. The positive R-Toda
hierarchy is reduced to the differential-difference Burgers hierarchy. We also
consider another D$\Delta$mKP hierarchy and show that its squared
eigenfunction symmetry constraint gives rise to the Volterra hierarchy. In
addition, we revisit the Ragnisco-Tu hierarchy which is a squared
eigenfunction symmetry constraint of the differential-difference Kadomtsev-
Petviashvili (D$\Delta$KP) system. It was thought the Ragnisco-Tu hierarchy
does not exist one-field reduction, but here we find an one-field reduction to
reduce the hierarchy to the Volterra hierarchy. Besides, the differential-
difference Burgers hierarchy are also investigated in Appendix. A multi-
dimensionally consistent 3-point discrete Burgers equation is given.

PACS numbers:

02.30.Ik, 02.30.Ks, 05.45.Yv

Keywords:

D$\Delta$mKP, squared eigenfunction symmetry constraint, relativistic Toda,
one-field reduction, Burgers

##  1 Introduction

It is common that an integrable system with a Lax pair usually has a squared
eigenfunction symmetry composed by the wave functions governed by the Lax pair
and its adjoint form. Such a symmetry is deeply related to $N$-soliton
expression [19], gradients of eigenvalues and nonlinearization of Lax pairs
[6, 7], Mel’nikov-type integrable systems with self-consistent sources [38]
(also see [55]), etc. The squared eigenfunction symmetries and their
constraints have received intensively attention in the early 1990s and many
remarkable results were obtained, such as, by the symmetry constraint bridging
a gap between continuous (2+1)-dimensional and (1+1)-dimensional integrable
systems [13, 29, 30], interpreting the squared eigenfunction symmetries as an
assemble of isospectral flow symmetries [37], understanding and solving the
constrainted (2+1)-dimensional systems [11, 14, 12, 15, 17, 35, 36, 41, 43,
44, 52, 53, 58, 57, 56], and so on.

The research has also been extended in 1990s to differential-difference case
with one independent discrete variable [23, 31, 42] but the understanding did
not go as far as the continuous case. This is because at that time the
discrete integrable systems were less understood than the continuous ones.
Recently, in [10] it is shown that by the squared eigenfunction symmetry
constraint the differential-difference Kadomtsev-Petviashvili (D$\Delta$KP)
system is related to the (1+1)-dimensional differential-difference system, the
Ragnisco-Tu hierarchy [39, 46], which is a second discretization of the
Ablowitz-Kaup-Newell-Segur(AKNS) system but different from the Ablowitz-Ladik
system [1] by a different discretization of wave functions [10]. The
D$\Delta$KP hierarchy is related to the pseudo-difference operator $M$ (2.5)
and the differential-difference modified Kadomtsev-Petviashvili (D$\Delta$mKP)
hierarchy is related to $L$ (2.6), respectively. In this paper we focus on the
squared eigenfunction symmetry of the D$\Delta$mKP system, and also revisit
the D$\Delta$KP, because the gauge connection between $M$ and $L$ will play a
backstage role in the research. As new results of the present paper, we obtain
the following.

  * •

We reduce the Ragnisco-Tu hierarchy to the Volterra hierarchy, which breaks
the statement in [39] that the hierarchy “are essentially given by the non-
existence of one-field reduction”.

  * •

We show that the squared eigenfunction symmetry constraint of the D$\Delta$mKP
system gives rise to the positive relativistic Toda (R-Toda) hierarchy in
(1+1)-dimension.

  * •

The R-Toda($\pm$) hierarchies are unified by an invertible transformation.

  * •

The differential-difference Burgers hierarchy is obtained as reductions of the
R-Toda($+$) hierarchy.

  * •

The D$\Delta$mKP(E) system that is related to $\bar{L}$ (5.2) and its squared
eigenfunction symmetry constraint not only gives rise to a decomposition of
the Volterra hierarchy, but also can be reduced to the later.

This paper is organized as follows. In Sec.2 we introduce basic notations and
derive the scalar D$\Delta$KP and D$\Delta$mKP hierarchies. In Sec.3 we
revisit the squared eigenfunction symmetry constraint of the D$\Delta$KP
hierarchy and reduce the Ragnisco-Tu hierarchy to the Volterra hierarchy. Then
in Sec.4 we deal with the D$\Delta$mKP system and show that its symmetry
constraint gives rise to the R-Toda hierarchy. And in Sec.5 we consider the
D$\Delta$mKP(E) hierarchy and its squared eigenfunction symmetry constraint.
Finally, conclusions are given in Sec.6. There is an Appendix in which we will
have a close look at the continuous and discrete Burgers equations.

##  2 Preliminary

###  2.1 Notations

Suppose that $u=u(n,x,t)$ and $v=v(n,x,t)$ are smooth functions of
$(n,x,t)\in\mathbb{Z}\times\mathbb{R}^{2}$ and $C^{\infty}$ w.r.t.
$(x,t)\in\mathbb{R}^{2}$. Let $S[u]$ be a Schwartz space composed by all
$f(u)$ that are $C^{\infty}$ differentiable w.r.t. $u$. Here for two functions
$f,g\in S$, the G$\hat{\mathrm{a}}$teaux derivative of $f$ w.r.t. $u$ in
direction $g$ is

$\displaystyle f^{\prime}[g]=\frac{d}{d\varepsilon}f(u+\varepsilon
g)\Big{|}_{\varepsilon=0},$ (2.1)

and a Lie product $\llbracket\cdot,\,\cdot\rrbracket$ is defined as

$\llbracket f,g\rrbracket=f^{\prime}[g]-g^{\prime}[f].$ (2.2)

Without confusion usually we write $f(u)=f_{n}$. We denote a shift by
$Ef_{n}=f_{n+1}$ and a difference by $\Delta f_{n}=(E-1)f_{n}=f_{n+1}-f_{n}$.
Note that there is an extended Leibniz rule for $\Delta$,

$\displaystyle\Delta^{s}g_{n}=\sum^{\infty}_{i=0}\mathrm{C}_{s}^{i}\,(\Delta^{i}g_{n+s-i})\Delta^{s-i},\quad
s\in\mathbb{Z},$ (2.3)

where $\mathrm{C}_{s}^{i}$ is defined as

$\displaystyle\mathrm{C}_{s}^{i}=\frac{s(s-1)(s-2)\cdots(s-i+1)}{i!}$ (2.4)

and $\mathrm{C}_{0}^{0}=1$.

In this paper we are interested in the following pseudo-difference operators

$M=\Delta+u+u_{1}\Delta^{-1}+u_{2}\Delta^{-2}+\cdots,$ (2.5)

and

$L=v\Delta+v_{0}+v_{1}\Delta^{-1}+\cdots,$ (2.6)

where $u,u_{i}\in S[u]$ and $v-1,v_{i}\in S[v]$. For two functions
$f_{n},g_{n}\in S[u]\cup S[v]$, their inner product is defined as

$<f_{n},g_{n}>=\sum^{+\infty}_{n=-\infty}f_{n}g_{n},$ (2.7)

by which we can define an adjoint operator of an operator $T$, denoted by
$T^{*}$, via

$<Tf_{n},g_{n}>=<f_{n},T^{*}g_{n}>.$ (2.8)

$M^{*}$ and $L^{*}$ can be defined in this way.

###  2.2 The scalar D$\Delta$KP hierarchy[18]

Let us briefly recall the scalar D$\Delta$KP hierarchy derived from the Lax
triplet composed by222Although in isospectral case $x\equiv t_{1}$, when
considering master symmetry which is in non-isospectral case, $x$ and $t_{1}$
must be considered different [18].

$\displaystyle M\Psi=\lambda\Psi,$ (2.9a) $\displaystyle\Psi_{x}=B_{1}\Psi,$
(2.9b) $\displaystyle\Psi_{t_{s}}=B_{s}\Psi,~{}~{}s=1,2,\cdots,$ (2.9c)

where $\lambda$ is a spectral parameter, $B_{s}=(M^{s})_{\geq 0}$ contains
only non-negative $\Delta^{j}$ terms (i.e. $j\geq 0$) of $M^{s}$, e.g.,

$\displaystyle B_{1}$ $\displaystyle=\Delta+u,$ (2.10a) $\displaystyle B_{2}$
$\displaystyle=\Delta^{2}+(Eu+u)\Delta+(\Delta u)+u^{2}+(Eu_{1}+u_{1}).$
(2.10b)

The compatibility of (2.9) leads to

$\displaystyle M_{t_{s}}=[B_{s},M],$ (2.11a) $\displaystyle M_{x}=[B_{1},M],$
(2.11b) $\displaystyle
B_{1,t_{s}}-B_{s,x}+[B_{1},B_{s}]=0,~{}~{}s=1,2,\cdots,$ (2.11c)

where $[A,B]=AB-BA$, and among which (2.11b) serves to express $u_{i}$ via $u$
by

$\displaystyle\Delta u_{1}=u_{x},$ (2.12a) $\displaystyle\Delta
u_{s+1}=u_{s,x}-\Delta
u_{s}-uu_{s}+\sum_{j=0}^{s-1}(-1)^{j}\mathrm{C}_{s-1}^{j}u_{s-j}\Delta^{j}E^{-s}u,~{}s=1,2,\cdots,$
(2.12b)

and (2.11c) provides a zero curvature representation for the scalar
D$\Delta$KP hierarchy

$u_{t_{s}}=B_{s,x}-[B_{1},B_{s}],~{}~{}s=1,2,\cdots.$ (2.13)

This hierarchy can also be expressed as

$u_{t_{s}}=\Delta\underset{\Delta}{\textrm{Res}}(M^{s}),~{}~{}s=1,2,\cdots.$
(2.14)

Denote the hierarchy by

$u_{t_{s}}=G_{s},$ (2.15)

in which the D$\Delta$KP equation is [16, 24]

$u_{t_{2}}=G_{2}=(1+2\Delta^{-1})u_{xx}+2uu_{x}-2u_{x}.$ (2.16)

It can be proved that

$\llbracket G_{j},G_{s}\rrbracket=0,$

which means the flow $G_{j}$ is a symmetry of the whole D$\Delta$KP hierarchy
(2.13).

Note that one may also alternatively rewrite $M$ in terms of $E$, denoted by

$\bar{M}=E+\bar{u}+\bar{u}_{1}E^{-1}+\cdots,$ (2.17)

where

$\bar{u}=u-1.$ (2.18)

In that case, $\bar{u}+1,\bar{u}_{i}\in S[u]$, and the Lax triplet

$\displaystyle\bar{M}\Psi=\lambda\Psi,$ (2.19a)
$\displaystyle\Psi_{x}=\bar{B}_{1}\Psi,$ (2.19b)
$\displaystyle\Psi_{t_{s}}=\bar{B}_{s}\Psi,~{}~{}s=1,2,\cdots,$ (2.19c)

leads to another scalar D$\Delta$KP hierarchy

$\bar{u}_{t_{s}}=\bar{G}_{s},~{}~{}s=1,2,\cdots,$ (2.20)

which is the same as (2.13) under (2.18).

###  2.3 The scalar D$\Delta$mKP hierarchy

Here we employ the Lax triplet approach to derive the scalar D$\Delta$mKP
hierarchy from the pseudo-difference operator (2.6) that has been considered
in [49]. Let us start from the triplet

$\displaystyle L\Phi=\lambda\Phi,$ (2.21a) $\displaystyle\Phi_{x}=A_{1}\Phi,$
(2.21b) $\displaystyle\Phi_{t_{s}}=A_{s}\Phi,~{}~{}s=1,2,\cdots,$ (2.21c)

where $A_{s}=(L^{s})_{\geq 1}$, e.g.,

$\displaystyle A_{1}=v\Delta,$ (2.22a) $\displaystyle
A_{2}=v(Ev)\Delta^{2}+v(Ev_{0}+v_{0}+\Delta v)\Delta,$ (2.22b)
$\displaystyle\cdots\cdots.$

The compatibility of (2.21) gives rise to

$\displaystyle L_{t_{s}}=[A_{s},L],$ (2.23a) $\displaystyle L_{x}=[A_{1},L],$
(2.23b) $\displaystyle
A_{1,t_{s}}-A_{s,x}+[A_{1},A_{s}]=0,~{}~{}s=1,2,\cdots.$ (2.23c)

From (2.23b) one can find

$\displaystyle v\Delta v_{0}=v_{x},$ (2.24a) $\displaystyle
vEv_{1}-(E^{-1}v)v_{1}=v_{0,x}-v\Delta v_{0},$ (2.24b) $\displaystyle
vEv_{s+1}-(E^{-1-s}v)v_{s+1}=v_{s,x}-v\Delta
v_{s}+\sum_{i=1}^{s}(-1)^{s+1-i}v_{i}E^{-s-1}\Delta^{s+1-i}v,$ (2.24c)

which further yields expressions for $v_{s}$ in terms of $v$, i.e.

$\displaystyle v_{0}=\Delta^{-1}(\ln v)_{x},$ (2.25a) $\displaystyle
v_{1}=\frac{\Delta^{-2}(\ln v)_{xx}-\Delta^{-1}v_{x}}{(E^{-1}v)},$ (2.25b)
$\displaystyle\cdots\cdots.$

(2.23c) serves as a zero curvature representation for the scalar D$\Delta$mKP
hierarchy

$v_{t_{s}}=K_{s}=(A_{s,x}-[A_{1},A_{s}])\Delta^{-1},~{}~{}s=1,2,\cdots.$
(2.26)

The first nonlinear equation of this hierarchy reads [49]

$(\ln v)_{t_{2}}=(1+2\Delta^{-1})(\ln v)_{xx}+(\ln v)_{x}(1+2\Delta^{-1})(\ln
v)_{x}-2v_{x},$ (2.27)

which is known as the D$\Delta$mKP equation and gives rise to the mKP equation
in continuum limit. Similar to [18], one can prove that

$(\ln
v)_{t_{s}}=\Delta\underset{\Delta}{\mathrm{Res}}(L^{s}\Delta^{-1}),~{}~{}s=1,2,\cdots,$
(2.28)

and $K_{j}$ is a symmetry of the whole D$\Delta$mKP hierarchy due to

$\llbracket K_{j},K_{s}\rrbracket=0.$

###  2.4 Gauge equivalence

Motivated by [42], Ref.[49] considered gauge transformation of the pseudo-
difference operators (2.5) and (2.6) by introducing an undetermined function
$f_{n}$ such that

$f_{n}L=Mf_{n}.$ (2.29)

As a result one has

$\displaystyle v=\frac{f_{n+1}}{f_{n}},~{}~{}v_{0}=\partial_{x}\ln f_{n},$
(2.30a) $\displaystyle u=\Delta^{-1}\partial_{x}\ln v+1-v,$ (2.30b)

where the later is considered as a Miura transformation to connect the
D$\Delta$KP hierarchy $u_{t_{s}}=G_{s}$ and the D$\Delta$mKP hierarchy
$v_{t_{s}}=K_{s}$, i.e. (2.13) and (2.26). In addition, comparing (2.9a) and
(2.21a) we have

$\Psi=f_{n}\Phi.$ (2.31)

Corresponding to the Lax triplets (2.9) and (2.21), their adjoint forms are

$\displaystyle M^{*}\Psi^{*}=\lambda\Psi^{*},$ (2.32a)
$\displaystyle\Psi^{*}_{x}=-B_{1}^{*}\Psi^{*},$ (2.32b)
$\displaystyle\Psi^{*}_{t_{s}}=-B_{s}^{*}\Psi^{*},~{}~{}s=1,2,\cdots,$ (2.32c)

and

$\displaystyle L^{*}\Phi^{*}=\lambda\Phi^{*},$ (2.33a)
$\displaystyle\Phi^{*}_{x}=-A_{1}^{*}\Phi^{*},$ (2.33b)
$\displaystyle\Phi^{*}_{t_{s}}=-A_{s}^{*}\Phi^{*},~{}~{}s=1,2,\cdots,$ (2.33c)

where $A^{*}$ and $B^{*}$ are adjoint operators of $A_{s}$ and $B_{s}$, and
$\Psi^{*}$ and $\Phi^{*}$ stands for solutions of (2.32) and (2.33). Note that
(2.32) and (2.33) generate the D$\Delta$KP hierarchy $u_{t_{s}}=G_{s}$ and the
D$\Delta$mKP hierarchy $v_{t_{s}}=K_{s}$ as well. Since $f_{n}$ is a scalar
function, from the way to define adjoint operators, it is easy to get

$L^{*}f_{n}=f_{n}M^{*}$ (2.34)

and consequently

$\Phi^{*}=f_{n}\Psi^{*}.$ (2.35)

##  3 Squared eigenfunction symmetry of the D$\Delta$KP

In this section we first briefly revisit the squared eigenfunction symmetry of
the D$\Delta$KP hierarchy and its constraint. Then we present a new one-field
reduction of the Ragnisco-Tu hierarchy.

###  3.1 Squared eigenfunction symmetry

It has been proved by use of additional symmetry in [10, 34] that

######  Proposition 1.

$(\Psi\Psi^{*})_{x}$ is a symmetry of the whole D$\Delta$KP hierarchy (2.13),
provided $\Psi$ and $\Psi^{*}$ satisfy the Lax triplets (2.9) and (2.32).

In the following we revisit this result from the view point of $\tau$ function
and present the following relation,

$\sigma=(\Psi\Psi^{*})_{x}=\sum^{+\infty}_{s=1}G_{s}\lambda^{-s},$ (3.1)

where $G_{s}$ are the flows in the D$\Delta$KP hierarchy (2.13). To obtain the
above relation, let us employ the results in [2, 23], which includes some more
explicit formulas of the 2DTL hierarchy that was first systematically studied
by the Sato’s approach in the pioneer paper [50]. In fact, the 2DTL hierarchy,
which involves both pseudo-difference operators $\bar{M}$ and $\bar{L}^{*}$
(see (5.2)), amounts to two differential-difference KP hierarchies. By the
rational transformation (cf.[23])

$u-1=\partial_{x}\ln\frac{\tau_{n+1}}{\tau_{n}}$ (3.2)

where $\tau_{n}=\tau(n,x,t_{1},t_{2},\cdots)$, the scalar D$\Delta$KP
hierarchy (2.14) can be cast into bilinear forms [50, 23, 2]

$(D_{t_{s}}-p_{s}(\widetilde{\mathbf{D}}))\tau_{n+1}\cdot\tau_{n}=0,~{}~{}t_{1}=x,~{}~{}s=2,3,\cdots$
(3.3)

where the $p_{s}(\mathbf{x})$ with $\mathbf{x}=(x_{1},x_{2},\cdots)$ are
elementary Schur polynomials defined through

$\exp\,\biggl{(}\sum^{+\infty}_{j=1}x_{j}k^{j}\biggr{)}=\sum_{s=0}^{\infty}p_{s}(\mathbf{x})k^{s},$
(3.4)

$\widetilde{\mathbf{D}}=(D_{t_{1}},\frac{1}{2}D_{t_{2}},\frac{1}{3}D_{t_{3}},\cdots)$
and $D_{x}$ is the Hirota bilinear operator defined as [21]

$e^{hD_{x}}f(x)\cdot g(x)=f(x+h)g(x-h).$ (3.5)

The hierarchy (3.3) can be alternatively written as

$1+\sum_{s=1}^{\infty}\lambda^{-s}\partial_{t_{s}}\ln\frac{\tau_{n+1}}{\tau_{n}}=\frac{\tau_{n+1}(\mathbf{t}+[\lambda^{-1}])\tau_{n}(\mathbf{t}-[\lambda^{-1}])}{\tau_{n+1}\tau_{n}}$
(3.6)

with $\mathbf{t}=(t_{1},t_{2},\cdots)$ and
$[\lambda]=(\lambda,\lambda^{2}/2,\lambda^{3}/3,\cdots)$. Then, taking
derivative w.r.t. $x$ on (3.6) and making use of (3.2), it follows that

$\biggl{(}\frac{\tau_{n+1}(\mathbf{t}+[\lambda^{-1}])\tau_{n}(\mathbf{t}-[\lambda^{-1}])}{\tau_{n+1}\tau_{n}}\biggr{)}_{x}=\sum_{s=1}^{\infty}\lambda^{-s}u_{t_{s}}.$
(3.7)

On the other hand, the eigenfunctions $\Psi$ and $\Psi^{*}$ can be expressed
in terms of $\tau_{n}$ as the following [23, 2],

$\displaystyle\Psi=\frac{\tau_{n}(\mathbf{t}-[\lambda^{-1}])}{\tau_{n}}e^{\sum_{s=1}^{\infty}t_{s}\lambda^{s}}\lambda^{n},$
(3.8a)
$\displaystyle\Psi^{*}=\frac{\tau_{n+1}(\mathbf{t}+[\lambda^{-1}])}{\tau_{n+1}}e^{-\sum_{s=1}^{\infty}t_{s}\lambda^{s}}\lambda^{-n}.$
(3.8b)

Combining (3.7), (3.8a) and (3.8b) together, we immediately reach

$(\Psi\Psi^{*})_{x}=\sum_{s=1}^{\infty}\lambda^{-s}u_{t_{s}},$ (3.9)

i.e. (3.1). This means $\sigma=(\Psi\Psi^{*})_{x}$ is not only a symmetry of
the whole D$\Delta$KP hierarchy (2.13), but also assembles all flows
(isospectral symmetries) of the hierarchy.

###  3.2 Symmetry constraint and the Ragnisco-Tu hierarchy

Noting that $u_{x}$ is also a symmetry of the D$\Delta$KP hierarchy (2.13), we
consider the symmetry constraint $\sigma=u_{x}+(\Psi\Psi^{*})_{x}=0$, which
leads to333In principle there is an integration constant $c$. Here we take
$c=1$, which is slightly different from [10] where $c$ is taken to be zero.

$u=-\Psi\Psi^{*}+1.$ (3.10)

Taking

$\Psi=Q_{n},~{}~{}\Psi^{*}=R_{n},$ (3.11)

the pseudo-difference operator (2.5) is written as (cf. [10])

$M=E-Q_{n}R_{n}-Q_{n}\Delta^{-1}R_{n},$ (3.12)

and the spectral problem (2.9a) is cast into a matrix form

$\left(\begin{array}[]{c}\psi_{1,n+1}\\\ \psi_{2,n+1}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\lambda+Q_{n}R_{n}&Q_{n}\\\
R_{n}&1\\\ \end{array}\right)\left(\begin{array}[]{c}\psi_{1,n}\\\
\psi_{2,n}\\\ \end{array}\right).$ (3.13)

Meanwhile, (2.9b) and (2.32b) turn out to be

$\left(\begin{array}[]{c}Q_{n}\\\ R_{n}\\\
\end{array}\right)_{x}=H_{1}=\left(\begin{array}[]{l}Q_{n+1}-Q_{n}^{2}R_{n}\\\
-R_{n-1}+Q_{n}R_{n}^{2}\end{array}\right).$ (3.14)

Note that the Ragnisco-Tu spectral problem (3.13) together with (3.14), after
replacing $R_{n}$ with $R_{n+1}$ and $\lambda$ with $2(\lambda-\beta)$, is
actually the Darboux transformation of the AKNS spectral problem (cf. [4,
26]), and (3.13) is gauge equivalent to the form [10]

$\left(\begin{array}[]{c}\psi_{1,n+1}\\\ -\psi_{2,n-1}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\eta&Q_{n}\\\ R_{n}&-\eta\\\
\end{array}\right)\left(\begin{array}[]{c}\psi_{1,n}\\\ \psi_{2,n}\\\
\end{array}\right),$ (3.15)

which is a discretisation of the AKNS spectral problem in a way different from
the Ablowitz-Ladik spectral problem [1]

$\left(\begin{array}[]{c}\phi_{1,n+1}\\\ \phi_{2,n+1}\\\
\end{array}\right)=\left(\begin{array}[]{cc}z&U_{n}\\\ V_{n}&1/z\\\
\end{array}\right)\left(\begin{array}[]{c}\phi_{1,n}\\\ \phi_{2,n}\\\
\end{array}\right).$ (3.16)

The Ragnisco-Tu hierarchy is composed by (2.9c) and (2.32c).

######  Proposition 2.

Under the symmetry constraint (3.10) together with (3.11), the system (2.9c)
and (2.32c) give rise to the recursive structure of the Ragnisco-Tu hierarchy

$\left(\begin{array}[]{c}Q_{n}\\\ R_{n}\\\
\end{array}\right)_{t_{s+1}}=H_{s+1}=M_{R}\left(\begin{array}[]{c}Q_{n}\\\
R_{n}\\\
\end{array}\right)_{t_{s}},~{}~{}H_{1}=\left(\begin{array}[]{l}Q_{n+1}-Q_{n}^{2}R_{n}\\\
-R_{n-1}+Q_{n}R_{n}^{2}\end{array}\right),~{}~{}s=1,2,\cdots,$ (3.17)

where the recursion operator $M_{R}$ is

$M_{R}=\left(\begin{array}[]{cc}1&0\\\ 0&E^{-1}\\\
\end{array}\right)\bigg{(}\mu_{n}I-\left(\begin{array}[]{c}Q_{n}\\\
-R_{n+1}\\\
\end{array}\right)(E+1)\Delta^{-1}(R_{n+1},Q_{n})\bigg{)}\left(\begin{array}[]{cc}E&0\\\
0&1\\\ \end{array}\right)$ (3.18)

with $\mu_{n}=1+Q_{n}R_{n+1}$ and $I$ being the $2\times 2$ identity matrix.

###### Proof.

Introduce $\mathcal{L}=M-1$. Based on the results in [10], one has the
relation

$\left(\begin{array}[]{l}(\mathcal{L}^{s})_{\geq 0}\Psi\\\
((\mathcal{L}^{s})_{\geq 0})^{*}(-\Psi^{*})\\\
\end{array}\right)=(M_{R}-1)^{s}\left(\begin{array}[]{c}\Psi\\\ -\Psi^{*}\\\
\end{array}\right),~{}~{}s=1,2,\cdots$ (3.19)

which leads to

$\displaystyle\left(\begin{array}[]{l}Q_{n}\\\ R_{n}\\\
\end{array}\right)_{t_{s}}$
$\displaystyle=\left(\begin{array}[]{l}(M^{s})_{\geq 0}Q_{n}\\\
-[(M^{s})_{\geq 0}]^{*}R_{n}\\\
\end{array}\right)=\left(\begin{array}[]{l}[(\mathcal{L}+1)^{s}]_{\geq
0}Q_{n}\\\ -[[(\mathcal{L}+1)^{s}]_{\geq 0}]^{*}R_{n}\end{array}\right)$
$\displaystyle=\sum_{i=0}^{s}\mathrm{C}_{s}^{i}\left(\begin{array}[]{l}(\mathcal{L}^{i})_{\geq
0}Q_{n}\\\ -[(\mathcal{L}^{i})_{\geq 0}]^{*}R_{n}\\\
\end{array}\right)=\sum_{i=0}^{s}\mathrm{C}_{s}^{i}(M_{R}-1)^{i}\left(\begin{array}[]{c}Q_{n}\\\
-R_{n}\\\ \end{array}\right)$
$\displaystyle=(M_{R})^{s}\left(\begin{array}[]{c}Q_{n}\\\ -R_{n}\\\
\end{array}\right).$

∎

###  3.3 The Ragnisco-Tu hierarchy: Refined with asymptotic condition

In the previous subsection, to derive the recursive structure (3.17) of the
Ragnisco-Tu hierarchy, we employed the results in [10]. Note that the proof
given in [10] has nothing with the asymptotic condition of $(Q_{n},R_{n})$
(i.e. $(\Psi,\Psi^{*})$).

In the following, in order to meet the relation (3.10), we refine the
Ragnisco-Tu hierarchy under the asymptotic condition

$Q_{n+j}R_{n+i}\to-1,~{}~{}(|n|\to+\infty),$ (3.21)

where $i,j$ are finite integers. This will be helpful in discussing one-field
reduction later. Let us also introduce

$H^{\prime}_{0}=H_{0}=(Q_{n},-R_{n})^{T}$ (3.22)

and define

$H^{\prime}_{s}=M_{R}^{s}H^{\prime}_{0},$ (3.23)

where $M_{R}$ is given as (3.18). With the asymptotic condition (3.21) it is
easy to obtain

$H^{\prime}_{1}=M_{R}H^{\prime}_{0}=H_{1}+2H^{\prime}_{0}=\left(\begin{array}[]{l}Q_{n+1}-2Q_{n}-Q_{n}^{2}R_{n}\\\
-R_{n-1}+2R_{n}+Q_{n}R_{n}^{2}\end{array}\right).$ (3.24)

Write $H^{\prime}_{s}=(H^{\prime}_{s,1},H^{\prime}_{s,2})^{T}$, and by
$[f_{n}]$ we denote the residue terms of $f_{n}$ after modular
$Q_{n+j}R_{n+i}$. For example, $[Q_{n+1}R_{n-1}]=0$, $[Q_{n}^{2}R_{n}]=Q_{n}$
and $[Q_{n+j}]=Q_{n+j}$. It is then easy to get

$[H^{\prime}_{0,1}]=Q_{n},~{}~{}[H^{\prime}_{1,1}]=Q_{n+1}-2Q_{n}-Q_{n}.$

By observation of $[H^{\prime}_{0,i}],~{}[H^{\prime}_{1,i}]$ and the structure
of the “integration” part of $M_{R}$, we can find that under the asymptotic
condition (3.21) $[H^{\prime}_{s,1}]$ is always a linear combination of
$\\{Q_{n+j}\\}$ and $[H^{\prime}_{s,2}]$ is always a linear combination of
$\\{R_{n+j}\\}$. This means, when we take into account of the asymptotic
condition (3.21) in deriving the refined Ragnisco-Tu hierarchy

$(Q_{n},R_{n})^{T}_{t_{s}}=H^{\prime}_{s},~{}~{}s=0,1,\cdots,$ (3.25)

the flow $H^{\prime}_{s}$ is actually certain linear combination of the flows
$\\{H_{j}\\}$ with a form

$H^{\prime}_{s}=\sum^{s}_{j=0}c_{j}H_{j},~{}~{}c_{s}\equiv
1,~{}c_{j}\in\mathbb{Z}.$ (3.26)

###  3.4 Reduced to the Volterra hierarchy

In [39] it is addressed that the Ragnisco-Tu hierarchy “are essentially given
by the non-existence of one-field reduction”. In the following let us show how
the Volterra hierarchy arises as a reduction from the refined Ragnisco-Tu
hierarchy (3.25).

Consider reduction

$R_{n+1}=-\frac{1}{Q_{n}}$ (3.27)

and introduce

$q_{n}=\ln\frac{Q_{n}}{Q_{n-1}},$ (3.28)

which indicates that

$e^{q_{n}}=-Q_{n}R_{n}=\frac{Q_{n}}{Q_{n-1}}.$ (3.29)

By (3.27) the two equations in $(Q_{n},R_{n})^{T}_{t_{1}}=H^{\prime}_{1}$ with
(3.24) are reduced to the same one,

$(\ln Q_{n})_{t_{1}}=\frac{Q_{n+1}}{Q_{n}}+\frac{Q_{n}}{Q_{n-1}}-2,$ (3.30)

which gives rise to the well known Volterra equation

$q_{n,t_{1}}=V_{1}=e^{q_{n+1}}-e^{q_{n-1}}.$ (3.31)

Next, for the recursive Ragnisco-Tu hierarchy (3.17), under the reduction
(3.27), it is reduced to a scaler relation

$(\ln
Q_{n})_{t_{s+1}}=(1+2\Delta^{-1})\Bigl{(}\frac{Q_{n+1}}{Q_{n}}E-E^{-1}\frac{Q_{n+1}}{Q_{n}})(\ln
Q_{n})_{t_{s}},~{}~{}s=1,2,\cdots.$ (3.32)

Applying $E^{-1}\Delta$ on both sides, we immediately arrive at

$q_{n,t_{s+1}}=V_{s+1}=L_{V}q_{n,t_{s}},~{}~{}L_{V}=(E+1)(e^{q_{n}}E-E^{-1}e^{q_{n}})\Delta^{-1},$
(3.33)

which is the recursive structure of the Volterra hierarchy (cf.[47]).

Let us sum up the reduction results as follows.

######  Theorem 3.1.

Consider the relation (3.26), the refined Ragnisco-Tu hierarchy (3.25) is
reduced under the reduction (3.27) to the Volterra hierarchy

$q_{n,t_{s}}=V^{\prime}_{s},~{}~{}s=1,2,\cdots,$ (3.34)

where $q_{n}$ is defined as (3.28), $V^{\prime}_{1}=V_{1}$ as defined in
(3.31),

$V^{\prime}_{s}=\sum^{s}_{j=1}c_{j}V_{j},~{}~{}c_{s}\equiv
1,~{}c_{j}\in\mathbb{Z},$ (3.35)

and $V_{j}=L^{j-1}_{V}V_{1}$ with $L_{V}$ defined in (3.33).

##  4 Squared eigenfunction symmetry of the D$\Delta$mKP

In this section we consider squared eigenfunction symmetry constraint of the
D$\Delta$mKP system. As a result, we will obtain the R-Toda hierarchy, which
further is reduced to two differential-difference Burgers hierarchies.

###  4.1 Squared eigenfunction symmetry

As for the squared eigenfunction symmetry of the D$\Delta$mKP hierarchy
(2.26), we have the following.

######  Theorem 4.1.

The following relation holds

$(\Phi E\Delta^{-1}\Phi^{*})_{x}=-\sum_{s=1}^{\infty}\lambda^{-s}v_{t_{s}},$
(4.1)

where $\Phi$ and $\Phi^{*}$ satisfy the Lax triplets (2.21) and (2.33).

###### Proof.

We prove this Theorem by using the results of the D$\Delta$KP hierarchy and
the gauge equivalence relations. Making use of relations (2.30b), (2.31) and
(2.35), from (3.9) we can find

$\displaystyle(\Phi\Phi^{*})_{x}=$
$\displaystyle(\Psi\Psi^{*})_{x}=\sum_{s=1}^{\infty}\lambda^{-s}\partial_{t_{s}}u$
$\displaystyle=$
$\displaystyle\sum_{s=1}^{\infty}\lambda^{-s}(\Delta^{-1}\partial_{x}\ln
v-v)_{t_{s}}$ $\displaystyle=$
$\displaystyle\Bigl{(}\Delta^{-1}\partial_{x}\frac{1}{v}-1\Bigr{)}\sum_{s=1}^{\infty}\lambda^{-s}v_{t_{s}}.$
(4.2)

Note that from (2.21b) and (2.33b) we can derive a relation

$(\Phi E\Delta^{-1}\Phi^{*})_{x}=v\Delta(\Phi
E\Delta^{-1}\Phi^{*}-\Phi\Phi^{*}),$ (4.3)

from which we can find

$(\Delta^{-1}\partial_{x}\frac{1}{v}-1)(\Phi
E\Delta^{-1}\Phi^{*})_{x}=-(\Phi\Phi^{*})_{x}.$ (4.4)

Thus, combining (4.2) and (4.4) we immediately reach (4.1) and complete the
proof. ∎

Note that (4.1) indicates $\sigma=(\Phi E\Delta^{-1}\Phi^{*})_{x}$ provides a
symmetry for the whole D$\Delta$mKP hierarchy (2.26). Such a symmetry can also
be constructed using the additional symmetry approach [41] (also see [22]).

###  4.2 Symmetry constraint and the R-Toda hierarchy

####  4.2.1 Spectral problem

Since both $v_{x}$ and $(\Phi E\Delta^{-1}\Phi^{*})_{x}$ are symmetries of the
D$\Delta$mKP hierarchy, we consider the following symmetry constraint

$v=\Phi E\Delta^{-1}\Phi^{*}.$ (4.5)

For convenience, introduce

$a_{n}=\Phi,~{}~{}b_{n}=E\Delta^{-1}\Phi^{*},$ (4.6)

under which the pseudo-difference operator $L$ is written as

$L=a_{n}b_{n}\Delta+a_{n}\Delta^{-1}b_{n}\Delta.$ (4.7)

Note that (4.5) and (4.6) indicate

$v=a_{n}b_{n}.$ (4.8)

To prove the form (4.7), we need to express all $v_{s}$ in terms of $a_{n}$
and $b_{n}$.

######  Proposition 3.

Under the symmetry constraint (4.5), all the $\\{v_{s}\\}_{s=0}^{\infty}$
defined by (2.24) can be expressed in terms of $a_{n}$ and $b_{n}$ as the
following,

$v_{s}=(-1)^{s}a_{n}E^{-1-s}\Delta^{s}b_{n},~{}~{}s=0,1,2,\cdots,$ (4.9)

where $a_{n}$ and $b_{n}$ are defined in (4.6).

###### Proof.

Under the constraint (4.5), the coupled system (2.21b) and (2.33b) turns out
to be

$a_{n,x}=a_{n}b_{n}(a_{n+1}-a_{n}),~{}~{}b_{n,x}=a_{n}b_{n}(b_{n}-b_{n-1}).$
(4.10)

This will be used to eliminate derivatives of $a_{n}$ and $b_{n}$ w.r.t. $x$
in (2.24). Inserting (4.8) and (4.10) into (2.24a) and (2.24b) respectively,
$v_{0}$ and $v_{1}$ are written as

$v_{0}=a_{n}b_{n-1},~{}~{}v_{1}=-a_{n}E^{-2}\Delta b_{n}.$ (4.11)

Then we assume

$v_{i}=(-1)^{i}a_{n}E^{-1-i}\Delta^{i}b_{n},~{}~{}i=0,1,\cdots,m,$ (4.12)

by which the right hand side of (2.24c) with $s=m$ is expressed as

$a_{n}b_{n}E\big{(}(-1)^{m+1}a_{n}E^{-2-m}\Delta^{m}b_{n}\big{)}-(E^{-1-s}a_{n}b_{n})(-1)^{m+1}a_{n}E^{-2-m}\Delta^{m}b_{n}.$
(4.13)

By comparison we immediately from (2.24c) find

$v_{m+1}=(-1)^{m+1}a_{n}E^{-2-m}\Delta^{m}b_{n}.$ (4.14)

This means on basis of mathematical induction the expression (4.9) is valid
for all $s=0,1,\cdots$.

∎

Substituting (4.9) into (2.6) and make use of formula (2.3) with $s=-1$, one
can arrive at (4.7). This indicates that the D$\Delta$mKP system is closed
under the constraint (4.5).

Now, with (4.7) in hand, the spectral problem (2.21a) turns out to be

$\big{(}a_{n}b_{n}\Delta+a_{n}\Delta^{-1}b_{n}\Delta\big{)}\Phi=\lambda\Phi,$
(4.15)

which can be written as

$\left(\begin{array}[]{c}\phi_{1,n+1}\\\ \phi_{2,n+1}\\\
\end{array}\right)=\left(\begin{array}[]{cc}{(\eta^{2}-v_{n})}/{r_{n}}&-\eta/r_{n}\\\
\eta&0\\\ \end{array}\right)\left(\begin{array}[]{c}\phi_{1,n}\\\
\phi_{2,n}\\\ \end{array}\right),$ (4.16)

where $\phi_{1,n}=\Phi/a_{n},~{}\lambda=\eta^{2}$, and

$v_{n}=a_{n}b_{n},~{}~{}r_{n}=-a_{n+1}b_{n}.$ (4.17)

Here we specially note that we employ $v_{n}$ to denote the function $v$ of
$n$, without making any confusion.

The spectral problem (4.16) can be further gauge transformed into

$\left(\begin{array}[]{c}\phi^{\prime}_{1,n+1}\\\ \phi^{\prime}_{2,n+1}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\eta^{2}-v_{n}&-\eta\\\ \eta
r_{n}&0\\\ \end{array}\right)\left(\begin{array}[]{c}\phi^{\prime}_{1,n}\\\
\phi^{\prime}_{2,n}\\\ \end{array}\right)$ (4.18)

by taking

$(\phi_{1,n},\phi_{2,n})=e^{-\Delta^{-1}\ln
r_{n}}(\phi^{\prime}_{1,n},\phi^{\prime}_{2,n}).$ (4.19)

After a further gauge transformation [9]

$\left(\begin{array}[]{c}\phi^{\prime}_{1,n}\\\ \phi^{\prime}_{2,n}\\\
\end{array}\right)=(-\eta)^{n}\left(\begin{array}[]{cc}1&0\\\
0&\frac{r_{n-1}}{\alpha}\\\
\end{array}\right)\Omega_{n},~{}\eta=\frac{1}{\zeta},~{}v_{n}=1+\alpha
R^{\prime}_{n},~{}r_{n}=\alpha^{2}Q^{\prime}_{n},$ (4.20)

the spectral problem (4.18) goes to the R-Toda spectral problem that reads
[47]

$\Omega_{n+1}=\left(\begin{array}[]{cc}\zeta(1+\alpha
R^{\prime}_{n})-\zeta^{-1}&\zeta Q^{\prime}_{n-1}\\\ -\alpha&0\\\
\end{array}\right)\Omega_{n},$ (4.21)

where $\alpha$ is an arbitrary constant.

Note that the one-component form of (4.18), which reads

$\phi^{\prime}_{1,n+1}+v_{n}\phi^{\prime}_{1,n}=\eta^{2}(\phi^{\prime}_{1,n}-r_{n-1}\phi^{\prime}_{1,n-1}),$
(4.22)

has been used as an unusual spectral problem (see Eq.(2.1) in [27]) to study
the R-Toda lattice; (4.18) was also restudied in [51] without mentioning the
connection with the R-Toda lattice.

####  4.2.2 The R-Toda hierarchy

Under the constraint (4.5), the two equations (2.21b) and (2.33b) in the Lax
triplets (2.21) and (2.33) can be cast into an evolution equation in terms of
$(a_{n},b_{n})$ like (4.10), i.e.

$(\ln a_{n})_{x}=b_{n}(a_{n+1}-a_{n}),~{}~{}(\ln
b_{n})_{x}=a_{n}(b_{n}-b_{n-1}),$ (4.23)

where $a_{n}$ and $b_{n}$ are given in (4.6). To look at explicit forms of
(2.21c) and (2.33c) in terms of $(a_{n},b_{n})$, we need to use some recursive
structures.

######  Proposition 4.

With the compact form (4.7) of $L$, $A_{s}$ allows the following two recursive
relations

$\displaystyle A_{s+1}=LA_{s}+a_{n}b_{n}(E(L^{s})_{0})\Delta-
a_{n}\Delta^{-1}(E\Delta^{-1}(A_{s})^{*}E^{-1}\Delta b_{n})\Delta,$ (4.24a)
$\displaystyle
A_{s+1}=A_{s}L+(L^{s})_{0}a_{n}b_{n}\Delta-(A_{s}a_{n})\Delta^{-1}b_{n}\Delta,$
(4.24b)

where $(L^{s})_{0}$ stands for the constant term of the operator $L^{s}$ with
respect to $\Delta$.

###### Proof.

By means of the identities

$(A_{s}a_{n}\Delta^{-1}b_{n}\Delta)_{\leq
0}=(A_{s}a_{n})\Delta^{-1}b_{n}\Delta,~{}~{}~{}\Delta^{-1}b_{n}\Delta
A_{s}=\Delta^{-1}(E\Delta^{-1}A_{s}^{*}E^{-1}\Delta b_{n})\Delta,$ (4.25)

(4.24a) and (4.24b) can be derived from assuming $A_{s+1}=(L^{s}L)_{\geq 1}$
and $A_{s+1}=(LL^{s})_{\geq 1}$, respectively. The detailed procedures are
similar to the D$\Delta$KP case in [10]. ∎

Now let us come to (2.21c) and (2.33c).

######  Theorem 4.2.

(2.21c) and (2.33c) give rise to the recursive hierarchy

$\left(\begin{array}[]{c}\ln a_{n}\\\ \ln b_{n}\\\
\end{array}\right)_{t_{s+1}}=L_{R}\left(\begin{array}[]{c}\ln a_{n}\\\ \ln
b_{n}\\\ \end{array}\right)_{t_{s}},~{}~{}s=1,2,\cdots,$ (4.26)

where the initial member reads

$(\ln a_{n})_{t_{1}}=a_{n+1}b_{n}-a_{n}b_{n},~{}~{}(\ln
b_{n})_{t_{1}}=a_{n}b_{n}-a_{n}b_{n-1},$ (4.27)

and the recursion operator $L_{R}$ is

$L_{R}=\left(\begin{array}[]{cc}L_{R}^{11}&L_{R}^{12}\\\
L_{R}^{21}&L_{R}^{22}\\\ \end{array}\right)$ (4.28)

with elements

$\displaystyle L_{R}^{11}=b_{n}\Delta a_{n}+\Delta^{-1}b_{n}\Delta
a_{n}+(\Delta a_{n})b_{n}E\Delta^{-1},$ $\displaystyle
L_{R}^{12}=\Delta^{-1}b_{n}(\Delta a_{n})+(\Delta a_{n})b_{n}E\Delta^{-1},$
$\displaystyle L_{R}^{21}=E\Delta^{-1}a_{n}(E^{-1}\Delta
b_{n})+a_{n}(E^{-1}\Delta b_{n})\Delta^{-1},$ $\displaystyle
L_{R}^{22}=-a_{n}E^{-1}\Delta b_{n}+E\Delta^{-1}a_{n}E^{-1}\Delta
b_{n}+a_{n}(E^{-1}\Delta b_{n})\Delta^{-1}.$

###### Proof.

Based on (2.28) and (4.8), one has

$(L^{s})_{0}=\Delta^{-1}(\ln v)_{t_{s}}=\Delta^{-1}(\ln
a_{n}b_{n})_{t_{s}}=\Delta^{-1}\frac{1}{a_{n}}a_{n,t_{s}}+\Delta^{-1}\frac{1}{b_{n}}b_{n,t_{s}}.$
(4.29)

Then, using the recursive form (4.24a) and expression (4.7), one has

$a_{n,t_{s+1}}=A_{s+1}a_{n}=(a_{n}b_{n}\Delta+a_{n}\Delta^{-1}b_{n}\Delta)a_{n,t_{s}}+a_{n}b_{n}(E(L^{s})_{0})\Delta
a_{n}+a_{n}\Delta^{-1}b_{n,t_{s}}\Delta a_{n},$

which, coupled with (4.29), gives rise to

$(\ln a_{n})_{t_{s+1}}=L_{R}^{11}(\ln a_{n})_{t_{s}}+L_{R}^{12}(\ln
b_{n})_{t_{s}}.$

Similarly, using (4.24b) and (4.29) one can find

$(\ln b_{n})_{t_{s+1}}=L_{R}^{21}(\ln a_{n})_{t_{s}}+L_{R}^{22}(\ln
b_{n})_{t_{s}}.$

∎

Further, after some calculations we can find that

######  Theorem 4.3.

Under transformation (4.17), the hierarchy (4.26) gives rise to the R-Toda(+)
hierarchy

$\left(\begin{array}[]{c}\ln v_{n}\\\ \ln r_{n}\\\
\end{array}\right)_{t_{s+1}}=L_{R^{+}}\left(\begin{array}[]{c}\ln v_{n}\\\ \ln
r_{n}\\\ \end{array}\right)_{t_{s}},~{}~{}s=1,2,\cdots,$ (4.30)

where the first member is the R-Toda lattice (denoted by R-Toda$(+1)$)

$\displaystyle v_{n,t_{1}}=v_{n}(r_{n}-r_{n-1}),$ (4.31a) $\displaystyle
r_{n,t_{1}}=r_{n}(r_{n+1}+v_{n+1}-r_{n-1}-v_{n}),$ (4.31b)

and the recursion operator $L_{R^{+}}$ reads

$L_{R^{+}}=\left(\begin{array}[]{cc}v_{n}&(r_{n}E-E^{-1}r_{n})\Delta^{-1}\\\
(E+1)v_{n}&(Er_{n}E-E^{-1}r_{n})\Delta^{-1}+r_{n}+\Delta v_{n}\Delta^{-1}\\\
\end{array}\right).$ (4.32)

###  4.3 Further discussion on the R-Toda hierarchy

####  4.3.1 The R-Toda$(-)$ hierarchy

The R-Toda(+) hierarchy (4.30) can be also derived from the spectral problem
(4.18) by considering the compatible condition with the time part

$\left(\begin{array}[]{c}\phi^{\prime}_{1,n}\\\ \phi^{\prime}_{2,n}\\\
\end{array}\right)_{t_{s}}=\left(\begin{array}[]{cc}A^{\prime}_{s}&B_{s}^{\prime}\\\
C_{s}^{\prime}&D_{s}^{\prime}\\\
\end{array}\right)\left(\begin{array}[]{c}\phi^{\prime}_{1,n}\\\
\phi^{\prime}_{2,n}\\\ \end{array}\right),~{}~{}s=1,2,\cdots,$ (4.33)

where $A_{s}^{\prime},B^{\prime}_{s},C^{\prime}_{s}$ and $D_{s}^{\prime}$ are
polynomials of $\eta$. On the other side, if expanding
$A_{s}^{\prime},B^{\prime}_{s},C^{\prime}_{s}$ and $D_{s}^{\prime}$ into
polynomials of $1/\eta$, one can derive the so-called R-Toda$(-)$ hierarchy

$\left(\begin{array}[]{c}\ln v^{\prime}_{n}\\\ \ln r^{\prime}_{n}\\\
\end{array}\right)_{t_{-(s+1)}}=L_{R^{-}}\left(\begin{array}[]{c}\ln
v^{\prime}_{n}\\\ \ln r^{\prime}_{n}\\\
\end{array}\right)_{t_{-s}},~{}~{}s=1,2,\cdots,$ (4.34)

where the recursion operator $L_{R^{-}}$ is

$L_{R^{-}}=L_{R^{+}}^{-1}=\left(\begin{array}[]{cc}\frac{1}{v^{\prime}_{n}}+\frac{1}{v^{\prime}_{n}}(r^{\prime}_{n}E-E^{-1}r^{\prime}_{n})\frac{1}{v^{\prime}_{n}}(E+1)\Delta^{-1}&-\frac{1}{v^{\prime}_{n}}(r^{\prime}_{n}E-E^{-1}r^{\prime}_{n})\frac{1}{v^{\prime}_{n}}\Delta^{-1}\\\
-\Delta\frac{1}{v^{\prime}_{n}}(E+1)\Delta^{-1}&\Delta\frac{1}{v^{\prime}_{n}}\Delta^{-1}\\\
\end{array}\right).$ (4.35)

Here for the purpose of identification, we have used
$(v^{\prime}_{n},r^{\prime}_{n})$ in stead of $(v_{n},r_{n})$ in the
R-Toda$(-)$ hierarchy. The first equation in the R-Toda(–) hierarchy is,
denoted by the R-Toda(–1),

$\left(\begin{array}[]{c}v^{\prime}_{n,t_{-1}}\\\ r^{\prime}_{n,t_{-1}}\\\
\end{array}\right)=\left(\begin{array}[]{c}r^{\prime}_{n}/v^{\prime}_{n+1}-r^{\prime}_{n-1}/v^{\prime}_{n-1}\\\
r^{\prime}_{n}/v^{\prime}_{n}-r^{\prime}_{n}/v^{\prime}_{n+1}\\\
\end{array}\right).$ (4.36)

Ref.[27] used to introduce a transformation

$v_{n}\rightarrow\frac{1}{v^{\prime}_{n}},~{}~{}r_{n}\rightarrow\frac{r^{\prime}_{n}}{v^{\prime}_{n}v^{\prime}_{n+1}},~{}~{}t_{1}\rightarrow-
t_{-1},$ (4.37)

by which the R-Toda(+1) equation (4.31) and the R-Toda(–1) equation (4.36) can
be transformed to each other. Next, we will show that the same transformation
can be extended to the whole hierarchy of the R-Toda lattice.

######  Theorem 4.4.

The R-Toda(+) hierarchy (4.30) and the R-Toda(–) hierarchy (4.34) are
equivalent to each other up to the transformation

$v_{n}=\frac{1}{v^{\prime}_{n}},~{}~{}r_{n}=\frac{r^{\prime}_{n}}{v^{\prime}_{n}v^{\prime}_{n+1}},~{}~{}t_{s}=-t_{-s},~{}~{}s=1,2,\cdots.$
(4.38)

###### Proof.

The relation (4.38) indicates

$\left(\begin{array}[]{c}\ln v_{n}\\\ \ln r_{n}\\\
\end{array}\right)=T\left(\begin{array}[]{c}\ln v^{\prime}_{n}\\\ \ln
r^{\prime}_{n}\\\ \end{array}\right),~{}~{}T=\left(\begin{array}[]{cc}-1&0\\\
-E-1&1\\\ \end{array}\right).$ (4.39)

Noting that $T^{-1}=T$, and under (4.38) there is

$L_{R^{-}}=TL_{R^{+}}T,$

we can unify the R-Toda$(\pm)$ hierarchies and then complete the proof. ∎

####  4.3.2 The differential-difference Burgers hierarchy

In the following we will see that one-field reduction of the R-Toda$(+)$
hierarchy can give rise to the differential-difference Burgers hierarchy.

Imposing reduction

$r_{n}=-v_{n}$ (4.40)

on the R-Toda$(+1)$ equation (4.31), we have

$v_{n,t_{1}}=W_{1}=v_{n}(v_{n-1}-v_{n}),$ (4.41)

and on the R-Toda$(+)$ hierarchy (4.30) we get

$(\ln v_{n})_{t_{s+1}}=T_{1}(\ln v_{n})_{t_{s}}=-T^{s}_{1}\Delta
v_{n-1}=W_{s},$ (4.42a) where the recursion operator is $T_{1}=-\Delta
E^{-1}v_{n}\Delta^{-1}.$ (4.42b)

This hierarchy can be explicitly written as

$(\ln
v_{n})_{t_{s}}=W_{s}=(-1)^{s}\Delta\prod^{s}_{j=1}v_{n-j},~{}~{}s=1,2,\cdots.$
(4.43)

Note that by a discrete Cole-Hopf transformation

$v_{n}=\frac{\alpha_{n}}{\alpha_{n+1}},$ (4.44)

the hierarchy (4.43) can be linearized as

$\alpha_{n,t_{s}}=\alpha_{n-s}-c_{s}(t)\alpha_{n},$ (4.45)

where $c_{s}(t)$ is an arbitrary function of $t$ but independent of $n$.

Equation (4.41) is known as the differential-difference Burgers equation. In
fact, taking

$v_{n}=1+\varepsilon\gamma_{n},~{}~{}\partial_{t^{\prime}_{1}}=\frac{2}{\varepsilon}(\partial_{t_{1}}+\partial_{x})$
(4.46)

and letting $n\to\infty$, $\varepsilon\to 0$ while $n\varepsilon=x$, (4.41)
gives rises to

$\gamma_{t^{\prime}_{1}}+2\gamma\gamma_{x}-\gamma_{xx}=0$ (4.47)

in its leading term, which is the Burgers equation.

To consider the continuum limit of the whole hierarchy (4.42a), we introduce

$v_{n}=e^{\varepsilon\gamma},~{}~{}x=\varepsilon n.$ (4.48)

Noticing that formally

$\displaystyle
v_{n}=e^{\varepsilon\gamma}=1+\varepsilon\gamma+O(\varepsilon^{2}),$ (4.49a)
$\displaystyle
E=e^{\varepsilon\partial_{x}}=1+\varepsilon\partial_{x}+O(\varepsilon^{2}),$
(4.49b) $\displaystyle
E^{-1}=e^{-\varepsilon\partial_{x}}=1-\varepsilon\partial_{x}+O(\varepsilon^{2}),$
(4.49c)
$\displaystyle\Delta=(E-1)=\varepsilon\Bigl{(}\partial_{x}+\frac{\varepsilon}{2}\partial_{x}^{2}+O(\varepsilon^{2})\Bigr{)},$
(4.49d)
$\displaystyle\Delta^{-1}=\frac{1}{\varepsilon}\Bigl{(}\partial^{-1}_{x}-\frac{\varepsilon}{2}+O(\varepsilon^{2})\Bigr{)},$
(4.49e)

from (4.42b) we have

$T_{1}=-1+\varepsilon\partial_{x}(\partial_{x}-\gamma)\partial^{-1}_{x}+O(\varepsilon^{2}).$
(4.50)

This means, after a combination of the flows $W_{j}$,

$\gamma_{n,t_{s}}=W^{\prime}_{s}=(T_{1}+1)^{s-1}W_{1}=(T_{1}+1)^{s-1}\gamma_{n},~{}~{}s=1,2,\cdots,$
(4.51)

can be considered as the differential-difference Burgers hierarchy, as with
the continuum limit scheme (4.49) it gives rise to the Burgers hierarchy
(cf.[54])

$\gamma_{t^{\prime}_{s}}=\partial_{x}(\partial_{x}-\gamma)^{s-1}\gamma,$
(4.52)

where $\partial_{t_{s}}$ has been replaced with
$\varepsilon^{s}\partial_{t^{\prime}_{s}}$.

There is another reduction

$r_{n}=-v_{n+1},$ (4.53)

which leads the R-Toda$(+)$ hierarchy (4.30) to

$(\ln v_{n})_{t_{s+1}}=T_{2}(\ln v_{n})_{t_{s}}=-T^{s}_{2}\Delta v_{n}=Y_{s},$
(4.54a) where the recursion operator is $T_{2}=-\Delta v_{n}E\Delta^{-1}.$
(4.54b)

Note that the two hierarchies (4.42) and (4.54) are simply related by
$v_{n+j}\to v_{n-j}$ and $t_{s}\to-t_{s}$. The first equation in the hierarchy
in (4.54) reads

$v_{n,t_{1}}=Y_{1}=v_{n}(v_{n}-v_{n+1}).$ (4.55)

which, under (4.46), gives the Burgers equation

$\gamma_{t^{\prime}_{1}}+2\gamma\gamma_{x}+\gamma_{xx}=0.$ (4.56)

The whole hierarchy can also be explicitly written as

$(\ln
v_{n})_{t_{s}}=Y_{s}=(-1)^{s}\Delta\prod^{s}_{j=1}v_{n+j},~{}~{}s=1,2,\cdots,$
(4.57)

which, by a discrete Cole-Hopf transformation,

$v_{n}=\frac{\beta_{n+1}}{\beta_{n}},$ (4.58)

are linearized as

$\beta_{n,t_{s}}=\beta_{n+s}-c_{s}(t)\beta_{n},$ (4.59)

where $c_{s}(t)$ is an arbitrary function of $t$ but independent of $n$. The
combined hierarchy

$\gamma_{n,t_{s}}=Y^{\prime}_{s}=(T_{2}+1)^{s-1}Y_{1}=(T_{2}+1)^{s-1}\gamma_{n},~{}~{}s=1,2,\cdots$
(4.60)

can be considered as a second differential-difference Burgers hierarchy with
continuum limit (cf.[8, 45, 54])

$\gamma_{t^{\prime}_{s}}=\partial_{x}(\partial_{x}+\gamma)^{s-1}\gamma.$
(4.61)

Note that in the continuum limit scheme (4.49) the equation

$(\ln v_{n})_{t_{2}}=\Delta v_{n}v_{n+1}$ (4.62)

goes to the Burgers equation (4.56) when $n\to+\infty$, and (4.62) was derived
in [33] as a discrete Burgers equation.

The derivation of the Burgers hierarchy in [33] indicates that the spectral
problem for the continuous Burgers hierarchy is (also see Eq.(5.2) in [28])

$\psi_{x}-\gamma\psi=\lambda\psi.$ (4.63)

For the differential-difference Burgers hierarchy (4.42), following from
(4.22) and (4.40), its spectral problem reads

$\phi^{\prime}_{1,n+1}+v_{n}\phi^{\prime}_{1,n}=\eta^{2}(\phi^{\prime}_{1,n}+v_{n-1}\phi^{\prime}_{1,n-1}),$
(4.64)

which is gauge equivalent to

$\varphi_{n+1}=\zeta v_{n}\varphi_{n},$ (4.65)

where

$\varphi_{n}=(-\zeta)^{-n}\phi^{\prime}_{n},~{}~{}\zeta=\eta^{-2}.$ (4.66)

Now we take

$\varphi_{n}=e^{\varepsilon\psi_{n}},~{}~{}\zeta=e^{\varepsilon\lambda},$
(4.67)

by which, together with (4.49), the discrete spectral problem (4.65) gives
rise to the continuous spectral problem (4.63).

The reduction (4.53) leads to a second discrete spectral problem from (4.53),
which is

$\phi^{\prime}_{1,n+1}+v_{n}(\phi^{\prime}_{1,n}-\eta^{2}\phi^{\prime}_{1,n-1})=\eta^{2}\phi^{\prime}_{1,n}.$
(4.68)

Again, employing the transformation (4.66), together with defining
$\varphi^{\prime}_{n}=\Delta\varphi_{n-1}$, we arrive at
$\varphi^{\prime}_{n+1}=\zeta v_{n}\varphi^{\prime}_{n}$, i.e. (4.65), which
leads to the spectral problem (4.63) in continuum limit.

We will have a closer look at the differential-difference Burgers hierarchies
in Appendix A.

##  5 The D$\Delta$mKP(E) and constraint

###  5.1 The D$\Delta$mKP(E) hierarchy

Note that the negative powers of $\Delta$ can be expressed in terms of
backward shifts like

$\Delta^{-s}=\sum_{i=s}^{\infty}\mathrm{C}^{s-1}_{i-1}E^{-i},~{}~{}s=1,2,\cdots,$
(5.1)

by which we can rewrite the pseudo-difference operator (2.6) as the following

$\bar{L}=wE+w_{0}+w_{1}E^{-1}+\cdots,$ (5.2)

where the new valuables $\\{w,w_{s}\\}$ are related to $\\{v,v_{s}\\}$ through

$w=v,~{}~{}w_{0}=v_{0}-v,~{}~{}w_{s}=\sum_{j=1}^{s}\mathrm{C}_{s-1}^{j-1}\,v_{j},~{}~{}s=1,2,\cdots.$
(5.3)

Note also that asymptotically

$w\to 1,~{}~{}w_{0}\to-1,~{}~{}w_{s}\to 0,~{}(s=1,2,\cdots)$

as $|n|\to+\infty$.

The hierarchy resulted from (5.2) is named as the D$\Delta$mKP(E) hierarchy,
which is generated from the Lax triplet

$\displaystyle\bar{L}\Theta=\lambda\Theta,$ (5.4a)
$\displaystyle\Theta_{x}=\bar{A}_{1}\Theta,$ (5.4b)
$\displaystyle\Theta_{t_{s}}=\bar{A}_{s}\Theta,~{}~{}s=1,2,\cdots,$ (5.4c)

where $\bar{A}_{s}=(\bar{L}^{s})_{\geq 1}$, of which the first two of
$\bar{A}_{s}$ are

$\displaystyle\bar{A}_{1}=wE,$ (5.5a)
$\displaystyle\bar{A}_{2}=w(Ew)E^{2}+(w(Ew_{0})+ww_{0})E.$ (5.5b)

The compatibility of (5.4) is

$\displaystyle\bar{L}_{t_{s}}=[\bar{A}_{s},\bar{L}],$ (5.6a)
$\displaystyle\bar{L}_{x}=[\bar{A}_{1},\bar{L}],$ (5.6b)
$\displaystyle\bar{A}_{1,t_{s}}-\bar{A}_{s,x}+[\bar{A}_{1},\bar{A}_{s}]=0,~{}~{}s=1,2,\cdots.$
(5.6c)

From (5.6b) one can express $w_{s}$ in terms of $w$ as the following,

$w_{0}=\Delta^{-1}(\ln w)_{x}-1,~{}~{}w_{1}=\frac{\Delta^{-2}(\ln
w)_{xx}}{E^{-1}w},~{}~{}w_{s+1}=\pi_{s+1}^{-1}\Delta^{-1}\pi_{s}\,w_{s,x},~{}~{}s=1,2,\cdots,$
(5.7)

where $\\{\pi_{s}\\}$ is defined by $\pi_{s}=\prod_{i=1}^{s}(E^{-i}w)$, for
$s=1,2,\cdots$; and from (5.6c) we have

$w_{t_{s}}=\bar{K}_{s}=(\bar{A}_{s,x}-[\bar{A}_{1},\bar{A}_{s}])E^{-1},~{}~{}s=1,2,\cdots,$
(5.8)

which provides a zero curvature expression of the scalar D$\Delta$mKP(E)
hierarchy. An alternative expression of (5.8) is

$w_{t_{s}}=\bar{K}_{s}=w\Delta\underset{E}{\mathrm{Res}}(\bar{L}^{s}E^{-1}),~{}~{}s=1,2,\cdots.$
(5.9)

The first two equations are

$w_{t_{1}}=\bar{K}_{1}=w_{x}$ (5.10)

and

$(\ln w)_{t_{2}}=(1+2\Delta^{-1})(\ln w)_{xx}+(\ln w)_{x}(1+2\Delta^{-1})(\ln
w)_{x}-2(\ln w)_{x},$ (5.11)

or in the form

$\hat{w}_{t_{2}}=(1+2\Delta^{-1})\hat{w}_{xx}+\hat{w}_{x}(1+2\Delta^{-1})\hat{w}_{x}-2\hat{w}_{x}$
(5.12)

with $w=e^{\hat{w}}$.

In addition, similar to [18], we can prove that

######  Theorem 5.1.

The $\\{S_{s}\\}$ defined by (5.9) are the infinitely many symmetries of the
scalar D$\Delta$mKP(E) hierarchy (5.9), i.e. $[\\![S_{i},S_{j}]\\!]=0$.

###  5.2 Squared eigenfunction symmetry

Consider the Lax triplet (5.4) and its adjoint form

$\displaystyle\bar{L}^{*}\Theta^{*}=\lambda\Theta^{*},$ (5.13a)
$\displaystyle\Theta^{*}_{x}=-\bar{A}^{*}_{1}\Theta^{*},$ (5.13b)
$\displaystyle\Theta^{*}_{t_{s}}=-\bar{A}^{*}_{s}\Theta^{*},~{}~{}s=1,2,\cdots.$
(5.13c)

One can verify that if

$\bar{L}_{z}=[-\Theta E\Delta^{-1}\Theta^{*},\bar{L}],$ (5.14)

then $[\partial_{z},\partial_{t_{s}}]\bar{L}=0$. (5.14) indicates that

$w_{z}=w(\Delta\Theta\Theta^{*}),$ (5.15)

which means $\sigma=w(\Delta\Theta\Theta^{*})$ is a (squared eigenfunction)
symmetry of the whole D$\Delta$mKP(E) hierarchy (5.8).

###  5.3 Symmetry constraint

####  5.3.1 Spectral problem

Now let us consider a constraint

$w_{x}=w(\Delta\Theta\Theta^{*}).$ (5.16)

For convenience we take $\Theta=c_{n},~{}\Theta^{*}=d_{n}$. Then, first,
compared with $w_{0}$ in (5.7) we find

$w_{0}=c_{n}d_{n}.$ (5.17)

Next, noting that $\Theta$ and $\Theta^{*}$ satisfy (5.4b) and (5.13b), by
calculation from (5.16) we can find

$w=c_{n}d_{n+1},$ (5.18)

which gives an explicit form of the symmetry constraint (5.16). Further, those
$w_{s}$ defined in (5.7) can be explicitly expressed as

$w_{s}=c_{n}d_{n-s},~{}~{}s=1,2,\cdots.$ (5.19)

Then, making use of the formula of $\Delta^{-1}$ in (5.1), we can rewrite
$\bar{L}$ in terms of $(c_{n},d_{n})$ as the compact form

$\bar{L}=c_{n}E^{2}\Delta^{-1}d_{n}.$ (5.20)

Thus, the spectral problem (5.4a) reads

$c_{n}E^{2}\Delta^{-1}d_{n}\Theta=\lambda\Theta,$ (5.21)

which is gauge equivalent to

$\Theta^{\prime}_{n+1}-e^{q_{n}}\Theta^{\prime}_{n-1}=\xi\Theta^{\prime}_{n}$
(5.22)

by the transformation

$\Theta=c_{n}\lambda^{n/2}\left(\prod^{n}_{k=-\infty}\frac{-1}{c_{k}d_{k}}\right)\Theta^{\prime}_{n},~{}~{}\lambda=\xi^{2},$
(5.23)

and

$e^{q_{n}}=-c_{n}d_{n}.$ (5.24)

(5.22) is nothing but the spectral problem of the Volterra lattice hierarchy.

####  5.3.2 Decomposition of the Volterra hierarchy

With new valuables $c_{n}$ and $d_{n}$, (5.4c) and (5.13c) read

$c_{n,t_{s}}=\bar{A}_{s}c_{n},~{}~{}d_{n,t_{s}}=-\bar{A}^{*}_{s}d_{n},~{}~{}s=1,2,\cdots,$
(5.25)

where $\bar{A}_{s}=((c_{n}E^{2}\Delta^{-1}d_{n})^{s})_{\geq 1}$. Let us see
what the known integrable hierarchy is related to the above system.

First, (5.4b) and (5.13b) are

$c_{n,x}=c_{n}c_{n+1}d_{n+1},~{}~{}d_{n,x}=-d_{n}d_{n-1}c_{n-1},$ (5.26)

which gives rise to the Volterra equation

$q_{n,x}=-e^{q_{n+1}}+e^{q_{n-1}}$ (5.27)

provided $q_{n}$ is defined through (5.24). In order to understand the
recursive structure behind (5.25), we need the following relations

$\displaystyle\bar{A}_{s}c_{n}E\Delta^{-1}d_{n}=(\bar{A}_{s}c_{n}E\Delta^{-1}d_{n})_{\geq
1}+(\bar{A}_{s}c_{n})E\Delta^{-1}d_{n},$ (5.28a) $\displaystyle
c_{n}E\Delta^{-1}d_{n}\bar{A}_{s}=(c_{n}E\Delta^{-1}d_{n}\bar{A}_{s})_{\geq
1}+c_{n}E\Delta^{-1}(\bar{A}^{*}_{s}d_{n}),$ (5.28b)

which holds by considering the definition of $\bar{A}_{s}$ and the formula of
$\Delta^{-1}$ in (5.1). Then we have the recursive structure for $\bar{A}_{s}$
and $\bar{A}^{*}_{s}$:

$\displaystyle\bar{A}_{s+1}=c_{n}Ed_{n}(\bar{A}_{s}+(\bar{L}^{s})_{0})+c_{n}E\Delta^{-1}d_{n}\bar{A}_{s}-c_{n}E\Delta^{-1}(\bar{A}^{*}_{s}d_{n}),$
(5.29a)
$\displaystyle\bar{A}^{*}_{s+1}=d_{n}E^{-1}c_{n}(\bar{A}^{*}_{s}+(\bar{L}^{s})_{0})-d_{n}\Delta^{-1}c_{n}\bar{A}^{*}_{s}+d_{n}\Delta^{-1}(\bar{A}_{s}c_{n}),$
(5.29b)

in which the term $(\bar{L}^{s})_{0}$ is written as

$(\bar{L}^{s})_{0}=\Delta^{-1}(\ln c_{n})_{t_{s}}+\Delta^{-1}E(\ln
d_{n})_{t_{s}},~{}~{}s=1,2,\cdots$ (5.30)

by noting that (5.18) and (5.9). Then, with the help of the above relations,
it is not difficult to prove that

######  Theorem 5.2.

(5.25) can be written as the recursive form

$\left(\begin{array}[]{c}\ln c_{n}\\\ \ln d_{n}\\\
\end{array}\right)_{t_{s}}=T^{s-1}\left(\begin{array}[]{c}c_{n+1}d_{n+1}\\\
-d_{n-1}c_{n-1}\\\ \end{array}\right),~{}~{}s=1,2,\cdots,$ (5.31)

where the recursion operator $T$ is defined as

$T=\left(\begin{array}[]{cc}E&0\\\ 0&-E^{-1}\\\
\end{array}\right)\Delta^{-1}(Ec_{n}d_{n}E-c_{n}d_{n})\Delta^{-1}\left(\begin{array}[]{cc}1&1\\\
1&1\\\ \end{array}\right).$ (5.32)

Further we have

######  Theorem 5.3.

By defining $q_{n}$ as in (5.24), the hierarchy (5.31) gives rise to the
Volterra hierarchy

$q_{n,t_{s}}=(-1)^{s}L_{V}^{s-1}(e^{q_{n+1}}-e^{q_{n-1}}),~{}~{}s=1,2,\cdots,$
(5.33)

where the recursion operator $L_{V}$ is given in (3.33). In this context, we
say that (5.31) provides a decomposition of the Volterra hierarchy (5.33).

Note that if we look for $c_{n}d_{n+j}=-1$ type of reduction, we find the only
available case is $j=-2$, i.e.

$d_{n}=\frac{1}{d_{n-2}}.$ (5.34)

This leads (5.26) to a single equation

$(\ln c_{n})_{x}=-\frac{c_{n}}{c_{n-2}},$ (5.35)

which is the the Volterra equation (5.27) if we take

$q_{n}=\ln\frac{c_{n}}{c_{n-2}},~{}~{}(\mathrm{or}~{}e^{q_{n}}=-c_{n}d_{n},~{}d_{n}={-1}/{c_{n-2}}).$
(5.36)

It is interesting that the reduction (5.34) is valid as well for the whole
hierarchy (5.31). As a result, we obtain a scalar hierarchy

$(\ln
c_{n})_{t_{s+1}}=-E\Delta^{-1}\Bigl{(}E\frac{c_{n}}{c_{n-2}}E-\frac{c_{n}}{c_{n-2}}\Bigr{)}(E^{-1}+E^{-2})(\ln
c_{n})_{t_{s}},~{}~{}s=1,2,\cdots,$

which is, by acting $(1-E^{-2})$ on both sides,

$q_{n,t_{s+1}}=-L_{V}q_{n,t_{s}},~{}~{}s=1,2,\cdots,$ (5.37)

i.e., the Volterra hierarchy where $L_{V}$ is given in (3.33).

####  5.3.3 Correlation of the D$\Delta$KP and D$\Delta$KP(E)

Note that both the D$\Delta$KP and D$\Delta$KP(E) give rise to the Volterra
hierarchy via there squared eigenfunction symmetry constraints, there should
have some correlations behind the fact.

Let us consider the gauge transformation

$h_{n}\bar{L}=\bar{M}h_{n}$ (5.38)

with an unfixed scalar function $h_{n}$, which gives rise to

$h_{n}w=h_{n+1},~{}~{}w_{0}=\bar{u},~{}~{}h_{n}w_{s}=\bar{u}_{s}h_{n-s},~{}~{}s=1,2,\cdots.$
(5.39)

From (5.9) we know that

$(\ln h_{n})_{t_{s}}=(\bar{L}^{s})_{0}.$ (5.40)

Then, under (5.38), (5.40) and the Lax equation (5.6a) we find

$\displaystyle\bar{M}_{t_{s}}$
$\displaystyle=(h_{n}\bar{L}h_{n}^{-1})_{t_{s}}=h_{n,t_{s}}\bar{L}h_{n}^{-1}+h_{n}\bar{L}_{t_{s}}h_{n}^{-1}-h_{n}\bar{L}h_{n}^{-1}h_{n,t_{s}}h_{n}^{-1}$
$\displaystyle=h_{n,t_{s}}h_{n}^{-1}\bar{M}+h_{n}[\bar{A}_{s},\bar{L}]h_{n}^{-1}-\bar{M}h_{n,t_{s}}h_{n}^{-1}$
$\displaystyle=[h_{n,t_{s}}h_{n}^{-1}+h_{n}\bar{A}_{s}h_{n}^{-1},\bar{M}]$
$\displaystyle=[h_{n}(\bar{L}^{s})_{0}h_{n}^{-1}+h_{n}(h_{n}^{-1}\bar{M}^{s}h_{n})_{\geq
1}h_{n}^{-1},\bar{M}]$ $\displaystyle=[\bar{B}_{s},\bar{M}].$

This indicates that the Lax equation $\bar{M}_{t_{s}}=[\bar{B}_{s},\bar{M}]$
is a consequence of (5.6a) if $h_{n}$ provides the gauge transformation
(5.38). Note that $\bar{M}_{t_{s}}=[\bar{B}_{s},\bar{M}]$ and (2.11a) generate
the same D$\Delta$KP hierarchy, and with (5.38) we have

$\Psi=h_{n}\Theta_{n},~{}~{}\Psi^{*}=\Theta^{*}/h_{n}.$

By the replacement we employed previously, i.e.
$\Psi=Q_{n},~{}\Psi^{*}=R_{n},~{}\Theta=c_{n},~{}\Theta^{*}=d_{n}$, we
formally have

$Q_{n}=h_{n}c_{n},~{}R_{n}=d_{n}/h_{n},$

which gives rise to $Q_{n}R_{n}=c_{n}d_{n}$. Thus (3.29) and (5.24) coincide,
and it is then not surprised that both the D$\Delta$KP and D$\Delta$mKP(E) are
related to the Volterra hierarchy by squared eigenfunction symmetry
constraints.

##  6 Concluding remarks

We have mainly considered squared eigenfunction symmetry constraint of the
D$\Delta$mKP system together with the D$\Delta$KP systems. These two systems
are entangled each other, as we can see in the paper. As the new results, we
have achieved the following. The D$\Delta$KP gives rise to the Ragnisco-Tu
hierarchy which is a discretization of the AKNS system and is reduced to the
Volterra hierarchy. Note that it used to be thought that there does not exist
one-field reduction for the Ragnisco-Tu hierarchy. The D$\Delta$mKP gives rise
to the R-Toda hierarchy which is reduced to the differential-difference
Burgers hierarchy. The D$\Delta$mKP(E) leads to the Volterra hierarchy as
well, by either decomposition or reduction. In some reductions, we have taken
into account of nonzero asymptotic conditions of the wave functions so that
reductions can be implemented reasonably.

The D$\Delta$KP and D$\Delta$mKP systems can be formally considered as
subsystems of the 2DTL system, since the 2DTL system employs both pseudo-
difference operators $\bar{M}$ and $\bar{L}^{*}$ [50]. There exists a
constraint imposed on $\bar{M}$ and $\bar{L}^{*}$ such that the time part in
the Lax triplet of the 2DTL together with its adjoint form gives rise to the
Ablowitz-Ladik hierarchy (cf. [48]); but that is different from the constraint
of squared eigenfunction symmetry. Besides, there exists a local
transformation to bring the Ablowitz-Ladik hierarchy to the R-Toda hierarchy
(4.26), which is given by [27], (also see §18.11 of [47]),

$a_{n}=-\frac{Q_{n}}{R_{n-1}},~{}~{}b_{n}=-\frac{Q_{n}}{R_{n-1}}+Q_{n}R_{n-1}.$
(6.1)

In addition, $(a_{n},b_{n})$ can also be locally expressed by $(u_{n},v_{n})$
of the Ragnisco-Tu hierarchy via [27]

$a_{n}=u_{n}v_{n}+\frac{u_{n}}{v_{n-1}},~{}~{}b_{n}=-\frac{u_{n}}{v_{n-1}}.$
(6.2)

However, the connection between the Ragnisco-Tu and Ablowitz-Ladik potentials,

$u_{n}=Q_{n}h_{n},~{}~{}v_{n}=\frac{R_{n-1}}{h_{n}},~{}~{}h_{n}=\prod^{n-1}_{j=-\infty}(1-Q_{j}R_{j}),$
(6.3)

are not local. Although the above three hierarchies are connected via
transformations, they are not essentially equivalent, for example, they
exhibit quite different one-field reduction features.

Note that in continuous case the KP system generates the AKNS stuff [13, 29]
and the mKP system leads to the Kaup-Newell spectral problem [25] and
hierarchy that gives rise to the Burgers hierarchy after a simple reduction
[8]. The result of the present paper implies possible connection between the
squared eigenfunction symmetry constraint of the D$\Delta$mKP and the Kaup-
Newell hierarchy, which is worthy to investigate elsewhere. In addition, the
differential-difference case exhibits richer results and reveals more links
than the continuous case, some of which already emerged in [27, 42], but here
have been much more elaborated. We believe that the results we obtained in the
paper will provide a deeper understanding for the connections of
(1+1)-dimensional and (2+1)-dimensional differential-difference integrable
systems.

In Appendix we will have a close look at the differential-difference Burgers
hierarchy and a fully discrete Burgers equation. It turns out that the former
act as auto Bäcklund transformations for the continuous Burgers hierarchy and
the nonlinear superposition formula of the Bäcklund transformation gives rise
to a fully discrete Burgers equation that is defined on 3 points (not
quadrilateral) but still consistent around cube. However, whether this
discrete Burgers equation is linearisable is left open.

### Acknowledgments

DJZ thanks Prof. Hietarinta for his discussion on the discrete Burgers
equation and Prof. Mikhailov for his comments on the Bäcklund transformation
of the Burgers hierarchy. This work was supported by the NSF of China [grant
numbers 11631007, 11875040, 11601312].

##  Appendix A A close look at the Burgers hierarchies

Since the squared eigenfunction symmetry of the continuous mKP gives rise to
the derivative Schrödinger hierarchy hierarchy that is easily reduced to the
Burgers hierarchy [8], it is not surprising the D$\Delta$mKP generates the
differential-difference Burgers hierarchy. However, the differential-
difference Burgers equation can not only act as a Bäcklund transformation for
the whole continuous Burgers hierarchy, but also generate a fully discrete
Burgers equation from its nonlinear superposition formula. Thus, we may have a
full profile from the continuous to the discrete Burgers equations.

###  A.1 Bäcklund and Galilean transformations of the Burgers hierarchy

Equation (2.21b) in the Lax triplet together with its adjoint form (2.33b)
leads to a differential-difference Burgers equation

$\partial_{x}\ln v_{n}=v_{n-1}-v_{n}.$ (A.1)

Let us look at the relation between (A.1) together with the differential-
difference Burgers hierarchy

$\partial_{t_{s+1}}\ln v_{n}=-\Delta E^{-1}v_{n}\Delta^{-1}\partial_{t_{s}}\ln
v_{n},~{}~{}\partial_{t_{1}}\ln v_{n}=v_{n-1}-v_{n},~{}~{}s=1,2,\cdots,$ (A.2)

and the Burgers hierarchy w.r.t. $v_{n}$

$\partial_{t_{s+1}}v_{n}=-\partial_{x}(\partial_{x}+v_{n})\partial_{x}^{-1}\partial_{t_{s}}v_{n},~{}~{}\partial_{t_{1}}v_{n}=v_{n,x}.$
(A.3)

######  Lemma 1.

Under (A.1) the differential-difference Burgers hierarchy (A.2) gives rise to
the Burgers hierarchy (A.3).

###### Proof.

As a single equation, (A.2) is an $(s+1)$-order difference equation. In
principle, one can repeatedly use (A.1) to express $v_{n-j}$ in terms of
$v_{n}$, e.g.

$v_{n-1}=\frac{v_{n,x}+v_{n}^{2}}{v_{n}},~{}~{}v_{n-2}=\frac{v_{n}^{3}+3v_{n}v_{n,x}+v_{n,xx}}{v_{n}^{2}+v_{n,x}},~{}~{}\cdots,$

to eliminate all $v_{n-j}$ in (A.2). In practice, rewriting (A.2) as

$\partial_{t_{s+1}}E\Delta^{-1}\ln v_{n}=-v_{n}\partial_{t_{s}}\Delta^{-1}\ln
v_{n},$ (A.4)

and replacing the $\Delta^{-1}$ term by using (A.1) which indicates
$\Delta^{-1}\ln v_{n}=-\partial_{x}^{-1}v_{n-1}$, from (A.4) we have

$\partial_{t_{s+1}}v_{n}=-\partial_{x}v_{n}\partial_{t_{s}}\partial_{x}^{-1}v_{n-1}.$

Then, replacing $v_{n-1}$ by (A.1) we have

$\displaystyle\partial_{t_{s+1}}v_{n}$
$\displaystyle=-\partial_{x}v_{n}\partial_{t_{s}}\partial_{x}^{-1}(v_{n}+\partial_{x}\ln
v_{n})$
$\displaystyle=-\partial_{x}(v_{n}+\partial_{x})\partial_{x}^{-1}\partial_{t_{s}}v_{n},$

which is the Burgers hierarchy (A.3). ∎

######  Lemma 2.

If $v_{n}$ satisfies (A.3), then $v_{n-1}$ defined by (A.1) gives rise to the
Burgers hierarchy we well.

###### Proof.

First, making use of (A.1) and (A.3), one has

$\displaystyle\partial_{t_{s+1}}v_{n-1}$
$\displaystyle=\partial_{t_{s+1}}(\partial_{x}\ln v_{n}+v_{n})$
$\displaystyle=(\partial_{x}\frac{1}{v_{n}}+1)\partial_{t_{s+1}}v_{n}$
$\displaystyle=-(\partial_{x}\frac{1}{v_{n}}+1)\partial_{x}(v_{n}+\partial_{x})\partial_{x}^{-1}\partial_{t_{s}}v_{n}.$

Meanwhile,

$\displaystyle\partial_{x}(v_{n-1}+\partial_{x})\partial_{x}^{-1}\partial_{t_{s}}v_{n-1}$
$\displaystyle=\partial_{x}(\frac{v_{n,{x}}}{v_{n}}+v_{n})\partial_{x}^{-1}(\partial_{x}\frac{1}{v_{n}}+1)\partial_{t_{s}}v_{n}$
$\displaystyle=\partial_{x}(\frac{v_{n,x}}{v_{n}}+v_{n})(\frac{1}{v_{n}}\partial_{x}+1)\partial_{x}^{-1}\partial_{t_{s}}v_{n}.$

By direct calculation we can find the above two formulas are same and we then
arrive at

$\partial_{t_{s+1}}v_{n-1}=-\partial_{x}(v_{n-1}+\partial_{x})\partial_{x}^{-1}\partial_{t_{s}}v_{n-1},$
(A.5)

which is the Burgers hierarchy for $v_{n-1}$. ∎

On the basis of the above two Lemmas, we immediately find the following.

######  Theorem A.1.

(A.1) and (A.2) define an auto Bäcklund transformation for the Burgers
hierarchy from (A.5) to (A.3).

Note that (A.1) and (A.2) are parameter free. Let us introduce a parameter
$\lambda$ by replacing $v_{n}$ with $v_{n}+\lambda$. It follows from (A.1) and
(A.2) that

$\displaystyle v_{n,x}=(v_{n}+\lambda)(v_{n-1}-v_{n}),$ (A.6a)
$\displaystyle\partial_{t_{s+1}}\ln(v_{n}+\lambda)=(-1)^{s+1}\Delta[E^{-1}(v_{n}+\lambda)]^{s}v_{n-1},$
(A.6b)

which, by repeating the proof of Lemma 1 and 2, compose a Bäcklund
transformation with parameter $\lambda$ for the Burgers hierarchy

$\partial_{t_{s+1}}v_{n}=(-1)^{s}\partial_{x}(\partial_{x}+v_{n}+\lambda)^{s}v_{n,x}.$
(A.7)

This is a linear combination of the original Burgers flows in (A.3), but also
as a whole it is a Galilean transformation of the $(s+1)$-order Burgers
equation (see Theorem 1 in [54]). Thus we can conclude that

######  Theorem A.2.

(A.6) defines an auto Bäcklund transformation for the Galilean transformed
Burgers hierarchy (A.7).

###  A.2 Nonlinear superposition formula and the discrete Burgers equation

There is a Bianchi identity for the Burgers hierarchy. Rewrite the shifted
Bäcklund transformation (A.6a) into

$\widetilde{v}_{x}=(\widetilde{v}-p)(v-\widetilde{v}),~{}~{}\widehat{v}_{x}=(\widehat{v}-q)(v-\widehat{v}),$

where $\widetilde{v}$ and $\widehat{v}$ denote new solutions by choosing
$\lambda=-p,-q$ respectively, from which Ref.[33] derived a superposition
formula

$\widehat{\widetilde{v}}=\frac{p\widehat{v}-q\widetilde{v}}{p-q+\widehat{v}-\widetilde{v}}.$
(A.8)

Suppose that $v_{n,m}$ is a function defined on $\mathbb{Z}^{2}$ lattice as
depicted in Fig.1 (a), introduce notations

$v\equiv v_{n,m},~{}~{}\widetilde{v}\equiv v_{n+1,m},~{}~{}\widehat{v}\equiv
v_{n,m+1},~{}~{}\widehat{\widetilde{v}}\equiv v_{n+1,m+1},$

and $p,q$ serve as lattice parameters associated with $n$, $m$ directions,
respectively. Then the Bianchi identity (A.8) can be viewed as a discrete
Burgers equation. In fact, introducing $x=m/q$, in the limit $m,q\to\infty$,
(A.8) gives rise to (A.6a) which is a differential-difference Burgers
equation. Note that (A.8) is different from the one derived in [20] by
discretising Lax pair.

$m$$n$$q$$p$$u$$\widehat{u}$$\widetilde{u}$$\widehat{\widetilde{u}}$$l$$m$$n$$u$$\widetilde{u}$$\widehat{u}$$\widehat{\widetilde{u}}$$\overline{u}$$\widetilde{\overline{u}}$$\widehat{\overline{u}}$$\widehat{\widetilde{\overline{u}}}$$p$$r$$q$(a)(b)
Figure 1:  (a). Discrete Burgers equation (A.8) on $(n,m)$-lattice. (b).
Consistency-around-cube of the discrete Burgers equation.

It is remarkable that this 3-point equation (A.8) can be consistently embedded
in a multidimensional lattice (see Fig.1(b) where $r$ is the spacing parameter
of the bar-direction) and the value of $\widehat{\widetilde{\overline{u}}}$ is
uniquely given as $\widehat{\widetilde{\overline{u}}}=\frac{A}{B}$ where

$\displaystyle A$
$\displaystyle=(q-r)(qr\widetilde{u}+p\widehat{u}\overline{u})+(r-p)(rp\widehat{u}+q\overline{u}\widetilde{u})+(p-q)(pq\overline{u}+r\widetilde{u}\widehat{u}),$
$\displaystyle B$
$\displaystyle=p^{2}(q-r)+q^{2}(r-p)+r^{2}(p-q)-(\widetilde{u}-\widehat{u})(pq+r^{2}-r\overline{u})$
$\displaystyle~{}~{}~{}~{}-(\widehat{u}-\overline{u})(qr+p^{2}-p\widetilde{u})-(\overline{u}-\widetilde{u})(pr+q^{2}-q\widehat{u}).$

The consistency-around-cube implies existence of Lax pair (cf.[3, 5, 40]). For
that we have

$\displaystyle\widetilde{\Phi}=U\Phi=\left(\begin{array}[]{cc}p&-r\widetilde{u}\\\
1&p-r-\widetilde{u}\end{array}\right)\Phi,~{}~{}~{}\widehat{\Phi}=V\Phi=\left(\begin{array}[]{cc}q&-r\widehat{u}\\\
1&q-r-\widehat{u}\end{array}\right)\Phi,$ (A.13)

and the compatibility $\widehat{\widetilde{\Phi}}=\widetilde{\widehat{\Phi}}$,
i.e., $\widehat{U}V=\widetilde{V}U$ gives rise to the discrete Burgers
equation (A.8).

Note that (A.8) does not belong to the Adler-Bonenko-Suris classification in
[3] as it is not quadrilateral.

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