# Symmetries of the D$\Delta$mKP hierarchy and their continuum limits

Jin Liu1, Da-jun Zhang1111Corresponding author. Email:
<EMAIL_ADDRESS>Xuehui Zhao2,3
1Department of Mathematics, Shanghai University, Shanghai 200444, China
2College of Mathematical Science, Inner Mongolia Normal University, Hohhot
010022, China
3Center for Applied Mathematical Science, Inner Mongolia Normal University,
Hohhot 010022, China

###### Abstract

In the recent paper [Stud. App. Math. 147 (2021) 752], squared eigenfunction
symmetry constraint of the differential-difference modified Kadomtsev-
Petviashvili (D$\Delta$mKP) hierarchy converts the D$\Delta$mKP system to the
relativistic Toda spectral problem and its hierarchy. In this paper we
introduce a new formulation of independent variables in the squared
eigenfunction symmetry constraint, under which the D$\Delta$mKP system gives
rise to the discrete spectral problem and a hierarchy of the differential-
difference derivative nonlinear Schrödinger equation of the Chen-Lee-Liu type.
In addition, by introducing nonisospectral flows, two sets of symmetries of
the D$\Delta$mKP hierarchy and their algebraic structure are obtained. We then
present a unified continuum limit scheme, by which we achieve the
correspondence of the mKP and the D$\Delta$mKP hierarchies and their
integrable structures.

Keywords:

D$\Delta$mKP hierarchy, eigenfunction, symmetry, constraint, derivative
nonlinear Schrödinger, continuum limit

PACS numbers:

02.30.Ik, 02.30.Ks, 05.45.Yv

##  1 Introduction

The Kadomtsev-Petviashvili (KP) hierarchy and modified KP (mKP) hierarchy are
elementary continuous (2+1)-dimensional integrable systems. They are
respectively formulated as the compatible conditions of

$\displaystyle\mathcal{L}\psi=\lambda\psi,~{}~{}\mathcal{L}=\partial_{x}+u_{2}\partial^{-1}_{x}+u_{3}\partial^{-2}_{x}+\cdots,$
(1.1a)
$\displaystyle\psi_{t_{m}}=\mathcal{A}_{m}\psi,~{}~{}\mathcal{A}_{m}=(\mathcal{L}^{m})_{\geq
0},$ (1.1b)

and

$\displaystyle
L\phi=\lambda\phi,~{}~{}L=\partial_{x}+v_{0}+v_{1}\partial^{-1}_{x}+v_{2}\partial^{-2}_{x}+\cdots,$
(1.2a) $\displaystyle\phi_{t_{m}}=A_{m}\phi,~{}~{}A_{m}=(L^{m})_{\geq 1}.$
(1.2b)

Note that most of the involved notations can be referred from Sec.2 of this
paper. It is well known that the squared eigenfunction symmetry constraints
bridge a gap between (2+1)-dimensional and (1+1)-dimensional integrable
systems, which allows us to study (2+1)-dimensional equations using
(1+1)-dimensional ones, and vice versa. By squared eigenfunction symmetry
constraints, (1.1) together with the adjoint form
$\psi^{*}_{t_{m}}=-\mathcal{A}^{*}_{m}\psi^{*}$, namely, the KP system, gives
rise to the Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem and the AKNS
hierarchy [3, 10, 18, 19]; while for the mKP system composed by (1.2) together
with the adjoint form $\phi^{*}_{t_{m}}=-A^{*}_{m}\phi^{*}$, gives rise to the
Kaup-Newell (KN) spectral problem and the derivative nonlinear Schrödinger
(DNLS) hierarchy of the Chen-Lee-Liu (CLL) type [4]. More details about the
above links can also be found in [5].

For the KP system, such links have been extended to the differential-
difference case, namely, the differential-difference KP (D$\Delta$KP) system,
composed by

$\displaystyle\bar{\mathcal{L}}\bar{\psi}_{n}=\lambda\bar{\psi}_{n},~{}~{}\bar{\mathcal{L}}=\Delta+\bar{u}_{0}+\bar{u}_{1}\Delta^{-1}+\bar{u}_{2}\Delta^{-2}+\cdots,$
(1.3a)
$\displaystyle\bar{\psi}_{n,t_{m}}=\bar{\mathcal{A}}_{m}\bar{\psi}_{n},~{}~{}\bar{\psi}_{n,t_{m}}^{*}=\bar{\mathcal{A}}_{m}^{*}\bar{\psi}_{n}^{*},~{}~{}\bar{\mathcal{A}}_{m}=(\bar{\mathcal{L}}^{m})_{\geq
0}.$ (1.3b)

It turns out that, by the squared eigenfunction symmetry constraint, the
D$\Delta$KP system is converted to the Merola-Ragnisco-Tu spectral problem and
a semi-discrete AKNS hierarchy [8]. Note that the Merola-Ragnisco-Tu spectral
problem [22] (see also [11]) is a second discretization of the AKNS spectral
problem (see [8]), which is also known as the Darboux transformation of the
AKNS spectral problem [2], but is different from the familiar discretization,
namely, the Ablowitz-Ladik spectral problem [1].

In a recent paper [9] the squared eigenfunction symmetry constraint for the
differential-difference modified KP (D$\Delta$mKP) hierarchy was investigated.
The later is related to the spectral problem

$\bar{L}\bar{\phi}_{n}=\lambda\bar{\phi}_{n},~{}~{}\bar{L}=\bar{v}\Delta+\bar{v}_{0}+\bar{v}_{1}\Delta^{-1}+\bar{v}_{2}\Delta^{-2}+\cdots.$
(1.4)

It is surprising as well as interesting that with the constraint converts the
D$\Delta$mKP system to the spectral problem and hierarchy of the relativistic
Toda, not as we expected to the semi-discrete DNLS system of the CLL type.

In the present paper we will give a new formulation of independent variables
in the squared eigenfunction symmetry constraint such that the constrained
D$\Delta$mKP system can yield a semi-discrete system that matches the DNLS
system in a unified continuum limit. Here by “ _unified_ ” we want to
emphasize that one should use a same continuum limit scheme to recover
continuous objects from their semi-discrete counterparts, including equation
hierarchy, their integrable characteristics and their algebraic structures,
e.g.[28, 15] for the semi-discrete AKNS and [14, 8] for the D$\Delta$KP. Note
that sometimes combinations of objects are needed, e.g. [15, 23]. In this
paper, apart from reinvestigating the squared eigenfunction symmetry
constraint, we will also derive nonisospectral D$\Delta$mKP flows which not
only provide a master symmetry but also lead to a set of symmetries for the
D$\Delta$mKP hierarchy. Finally, we will provide a unified continuum limit
scheme, with which we will show the correspondence between all the above semi-
discrete objects and their continuous counterparts.

The paper is organized as follows. Section 2 serves as the preliminary to
introduce some notions and notations. In Section 3, we quickly review the mKP
hierarchy, their symmetries and the squared eigenfunction symmetry constraint
of the mKP system. Then in Section 4, we derive nonisospectral flows and
symmetries for the D$\Delta$mKP heirarchy and present their algebraic
structures. In addition, as the main result of the paper, we give a new
formulation of the squared eigenfunction symmetry constraint and derive the
differential-difference CLL spectral problem and the differential-difference
DNLS hierarchy from the constrained D$\Delta$mKP system. After that, in
Section 5, we present a unified scheme of the continuum limits and show that
the obtained results and their algebraic structures can well match their
continuous counterparts given in Section 3. Finally, in Section 6, conclusions
are given. There are two appendices. One provides a detailed proof of formula
(4.32), and in the another, we discuss discrete CLL spectral problem and the
related discrete integrable systems.

##  2 Preliminary

We briefly list some notions and notations used in our paper. One can also
refer to [16] and [14]. Let $v=v(x,y,t)$ be a $C^{\infty}$ function defined on
$\mathbb{R}^{3}$ and decrease rapidly as $|x|,|y|\to\infty$. By $S[v]$ we
denote a Schwartz space composed by all functions $f(v)$ that are $C^{\infty}$
differentiable with respect to $v$ and its derivatives. For two functions
$f,g\in S[v]$, the G$\hat{\mathrm{a}}$teaux derivative of $f$ with respect to
$v$ in direction $g$ is defined as

$f^{\prime}[g]\doteq f^{\prime}(v)[g]=\frac{\mathrm{d}\,f(v+\varepsilon
g)}{\mathrm{d}\,\varepsilon}\Big{|}_{\varepsilon=0},$

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

$\llbracket f,g\rrbracket=f^{\prime}[g]-g^{\prime}[f].$ (2.1)

For an evolution equation

$v_{t}=K(v)$ (2.2)

where $K(v)\in S[v]$, we say $\omega=\omega(v)$ is its symmetry if for all
solutions $v$ of equation (2.2) there is

$\omega_{t}=K^{\prime}[\omega].$

In the pseudo-differential operator $\mathcal{L}$ given in (1.1a),
$\partial_{x}=\frac{\partial}{\partial x}$ and
$\partial^{-1}_{x}\partial_{x}=\partial_{x}\partial^{-1}_{x}=1$. For two
functions $f,g\in S[v]$, their inner product is defined as

$\langle
f,g\rangle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}fg\,\mathrm{d}x\mathrm{d}y.$

For an operator $T$ living on $S[v]$, its adjoint operator $T^{*}$ is defined
through

$\langle f,Tg\rangle=\langle T^{*}f,g\rangle.$ (2.3)

In the semi-discrete (differential-difference) case, suppose that
$\tilde{v}=\tilde{v}(n,\bar{x},\bar{t})$ is a function of
$(n,\bar{x},\bar{t})\in\mathbb{Z}\times\mathbb{R}^{2}$, $C^{\infty}$ with
respect to $(\bar{x},\bar{t})$, and decreases rapidly as
$|n|,|\bar{x}|\to\infty$. Similar to the continuous case, one can introduce a
Schwartz function space $\bar{S}[\tilde{v}]$ and define
G$\hat{\mathrm{a}}$teaux derivative, Lie product, and symmetry for an
evolution equation $\tilde{v}_{\bar{t}}=K(\tilde{v})$. Without confusion we
write $f(\tilde{v})=f_{n}$. By $\Delta$ and $E$ we denote the difference
operator and shift operator with respect to $n$, i.e. $\Delta
f_{n}=(E-1)f_{n}=f_{n+1}-f_{n}$. There is an extended Leibniz rule for
$\Delta$ [13],

$\Delta^{s}g_{n}=\sum_{i=0}^{\infty}\mathrm{C}_{s}^{i}(\Delta^{i}g_{n+s-i})\Delta^{s-i},~{}~{}~{}s\in\mathbb{Z},$
(2.4)

where

$\mathrm{C}_{0}^{0}=1,~{}~{}\mathrm{C}_{s}^{i}=\frac{s(s-1)(s-2)\cdots(s-i+1)}{i!}.$

An inner product of two functions in $\bar{S}[\tilde{v}]$ is defined as

$\langle
f_{n},g_{n}\rangle=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}f_{n}g_{n}\,\mathrm{d}\bar{x},$
(2.5)

and for an operator $T$ its adjoint operator $T^{*}$ is defined as $\langle
f_{n},\bar{T}g_{n}\rangle=\langle\bar{T}^{*}f_{n},g_{n}\rangle$.

##  3 Results of the mKP

In the following we quickly review the mKP hierarchy, their symmetries and
algebraic structures (see also [12]), and the squared eigenfunction symmetry
constraint of the mKP system (see also [4, 5]). These will be finally
recovered from the results of the D$\Delta$mKP using a unified continuum
limit.

###  3.1 The scalar mKP hierarchy and symmetries

Consider the Lax triad222We should consider $y$ and $t_{2}$ independently,
especially in the nonisospectral case, see [14]. [12]

$\displaystyle L\phi=\lambda\phi,~{}~{}~{}\lambda_{t_{m}}=0,$ (3.1a)
$\displaystyle\phi_{y}=A_{2}\phi,$ (3.1b)
$\displaystyle\phi_{t_{m}}=A_{m}\phi,~{}~{}(m=1,2,\cdots),$ (3.1c)

where $L$ is the pseudo-differential operator given in (1.2a) with
$v_{0}\equiv v,~{}v_{j}\in S[v]$, $A_{m}=(L^{m})_{\geq 1}$ which contains the
pure differential part of $L^{m}$, e.g.,

$\displaystyle A_{1}=\partial_{x},$ (3.2a) $\displaystyle
A_{2}=\partial^{2}_{x}+2v_{0}\partial_{x},$ (3.2b) $\displaystyle
A_{3}=\partial^{3}_{x}+3v_{0}\partial^{2}_{x}+3(v_{1}+v_{0,x}+v_{0}^{2})\partial_{x}.$
(3.2c)

Compatibility of the triplet (3.1) gives rise to

$\displaystyle L_{y}=[A_{2},L],$ (3.3a) $\displaystyle L_{t_{m}}=[A_{m},L],$
(3.3b) $\displaystyle A_{2,t_{m}}-A_{m,y}+[A_{2},A_{m}]=0,$ (3.3c)

where $[A,B]=AB-BA$. Among them Eq.(3.3a) plays a role to express $v_{j}$ in
terms of $v_{0}\equiv v$ for $j=1,2,\cdots$, Eq.(3.3b) agrees with
$A_{m}=(L^{m})_{\geq 1}$ with the boundary condition
$A_{m}|_{\bm{v=0}}=\partial^{m}_{x}$ where $\bm{v}=(v_{0},v_{1},v_{2},\cdots)$
(see the following Remark 3.1), and the third equation (3.3c) gives rises to
the zero curvature representation of the scalar isospectral mKP hierarchy

$v_{t_{m}}=K_{m}=\frac{1}{2}(A_{m,y}+[A_{m},A_{2}])\partial^{-1}_{x},~{}~{}~{}m=1,2,\cdots.$
(3.4)

Here we list out the first three equations in the hierarchy:

$\displaystyle v_{t_{1}}=K_{1}=v_{x},$ (3.5a) $\displaystyle
v_{t_{1}}=K_{2}=v_{y},$ (3.5b) $\displaystyle
v_{t_{3}}=K_{3}=\frac{1}{4}v_{xxx}-\frac{3}{2}v^{2}v_{x}+\frac{3}{2}v_{x}\partial^{-1}_{x}v_{y}+\frac{3}{4}\partial^{-1}v_{yy},$
(3.5c)

where the third equation is known as the mKP equation.

######  Remark 3.1.

If we assume $A_{m}$ has a form

$A_{m}=a_{0}\partial_{x}^{m}+a_{1}\partial_{x}^{m-1}+\cdots+a_{m-1}\partial_{x},~{}~{}A_{m}|_{\bm{v=0}}=\partial^{m}_{x},$

then $\\{a_{j}\\}$ can be uniquely determined from (3.3b) and it turns out
that $A_{m}=(L^{m})_{\geq 1}$. See also [27].

The nonisospectral mKP flows are formulated from the triad [12]

$\displaystyle L\phi=\lambda\phi,~{}~{}~{}\lambda_{t_{m}}=\lambda^{m-1},$
(3.6a) $\displaystyle\phi_{y}=A_{2}\phi,$ (3.6b)
$\displaystyle\phi_{t_{m}}=B_{m}\phi,~{}~{}(m=1,2,\cdots),$ (3.6c)

where $L$ and $A_{2}$ are the same as the isospectral case, while $B_{m}$ is a
differential operator

$B_{m}=b_{0}\partial^{m}_{x}+b_{1}\partial^{m-1}_{x}+\cdots+b_{m-1}\partial_{x},$

with assumption

$B_{m}|_{\bm{v}=0}=\begin{cases}2y\partial_{x},&m=1,\\\
2y\partial^{m}_{x}+x\partial^{m-1}_{x},&m\geq 2.\end{cases}$ (3.7)

The compatibility of (3.6) leads to

$\displaystyle L_{y}=[A_{2},L],$ (3.8a) $\displaystyle
L_{t_{m}}=[B_{m},L]+L^{m-1},$ (3.8b) $\displaystyle
A_{2,t_{m}}-B_{m,y}+[A_{2},B_{m}]=0,$ (3.8c)

among which, (3.8a) is the same as (3.3a) which means $v_{j}$ are expressed in
terms of $v_{0}\equiv v$ as same as the isospectral case; by (3.8b) one can
uniquely determine $B_{m}$ with the boundary condition (3.7), and we have

$\displaystyle B_{1}=2y\partial_{x},$ $\displaystyle B_{2}=2yA_{2}+xA_{1},$
$\displaystyle B_{3}=2yA_{3}+xA_{2},$

etc, where $A_{j}$ are given in (3.2); Eq.(3.8c) provides the zero curvature
representation of the scalar nonisospectral mKP hierarchy

$u_{t_{m}}=\sigma_{m}=\frac{1}{2}(B_{m,y}+[B_{m},A_{2}])\partial^{-1}_{x},$
(3.9)

where the first few equations are

$\displaystyle v_{t_{1}}=\sigma_{1}=2yK_{1}+1,$ (3.10a) $\displaystyle
v_{t_{2}}=\sigma_{2}=2yK_{2}+xK_{1}+v,$ (3.10b) $\displaystyle
v_{t_{3}}=\sigma_{3}=2yK_{3}+xK_{2}-\frac{1}{2}v^{2}+\frac{3}{2}\partial^{-1}_{x}v_{y}-\frac{1}{2}v_{x},$
(3.10c)

with $K_{j}$ given in (3.5). Note that asymptotically we have

$K_{m}|_{v=0}=0,~{}~{}~{}\sigma_{m}|_{v=0}=\begin{cases}1,&m=1,\\\ 0,&m\geq
2.\end{cases}$ (3.11)

The flows $\\{K_{m},\sigma_{s}\\}$ compose a Lie algebra with respect to the
product $\llbracket\cdot,\cdot\rrbracket$ given in (2.1) [12].

######  Theorem 3.1.

The mKP flows $\\{K_{m},\sigma_{s}\\}_{m\geq 1,n\geq 1}$ compose a Lie algebra
with the following structure

$\displaystyle[\\![K_{m},K_{s}]\\!]=0,$ (3.12a)
$\displaystyle[\\![K_{m},\sigma_{s}]\\!]=mK_{m+s-2},$ (3.12b)
$\displaystyle[\\![\sigma_{m},\sigma_{s}]\\!]=(m-s)\sigma_{m+s-2},$ (3.12c)

in which we specially define $K_{0}=\sigma_{0}=0$. Note that $\sigma_{3}$
plays a role of master symmetry.

The above structure immediately yields symmetries of the mKP hierarchy [12].

######  Theorem 3.2.

Each equation

$v_{t_{m}}=K_{m}$ (3.13)

in the isospectral mKP hierarchy (3.4) has two sets of symmetries:

$\\{K_{s}\\}_{s\geq
1},\qquad\\{\tau_{k}^{m}=m\,t_{m}K_{m+k-2}+\sigma_{k}\\}_{k\geq 1},$ (3.14)

and they compose a Lie algebra with structure

$\displaystyle[\\![K_{s},K_{k}]\\!]=0,$ (3.15a)
$\displaystyle[\\![K_{s},\tau_{k}^{m}]\\!]=s\,K_{s+k-2},$ (3.15b)
$\displaystyle[\\![\tau_{s}^{m},\tau_{k}^{m}]\\!]=(s-k)\tau^{m}_{s+k-2},$
(3.15c)

where we define $K_{0}=\tau^{m}_{0}=0$.

###  3.2 Squared eigenfunction symmetry constraint

In addition to the Lax triad (3.1), we consider the following adjoint system

$\displaystyle L^{*}\phi^{*}=\lambda\phi^{*},~{}~{}~{}\lambda_{t_{m}}=0,$
(3.16a) $\displaystyle\phi_{y}^{*}=-A_{2}^{*}\phi^{*},$ (3.16b)
$\displaystyle\phi_{t_{m}}^{*}=-A_{m}^{*}\phi^{*},~{}~{}(m=1,2,\cdots),$
(3.16c)

where $\phi^{*}$ stands for the eigenfunction, $L^{*}$ and $A_{m}^{*}$ are the
adjoint operators of $L$ and $A_{m}$ defined through (2.3). One can prove that
the above system yields the iosospectral mKP hierarchy $v_{t_{m}}=K_{m}$ as
well. In addition, it can be proved that

$\omega=(\phi\phi^{*})_{x}$ (3.17)

is a (squared eigenfunction) symmetry of the iosospectral mKP hierarchy (3.4),
provided $\phi$ and $\phi^{*}$ are solutions to the triplets (3.1) and (3.16),
respectively.

Since $v_{x}$ is also a symmetry of the iosospectral mKP hierarchy, by
imposing $v_{x}=(\phi\phi^{*})_{x}$ we have a squared eigenfunction symmetry
constraint [4]

$v=\phi\phi^{*}=qr,~{}~{}(\phi=q,~{}\phi^{*}=r).$ (3.18)

With such a constraint, the pseudo-differential operator $L$ can be
consistently written as

$L=\partial_{x}+q\partial_{x}^{-1}r\partial_{x},$ (3.19)

and consequently the spectral problem $L\varphi=\lambda\varphi$ gives rise to

$\left(\begin{array}[]{c}\varphi_{1}\\\
\varphi_{2}\end{array}\right)_{x}=\left(\begin{array}[]{cc}-\eta^{2}&\eta q\\\
\eta r&-qr\end{array}\right)\left(\begin{array}[]{c}\varphi_{1}\\\
\varphi_{2}\end{array}\right),$ (3.20)

where we have taken $\varphi_{1}=\varphi$ and $\lambda=-\eta^{2}$. Introducing
a gauge transformation

$\varphi_{i}=e^{-\frac{1}{2}(\eta^{2}x+\partial^{-1}_{x}qr)}\psi_{i},~{}~{}(i=1,2),$

the spectral problem (3.20) yields a SL(2) version

$\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)_{x}=\left(\begin{array}[]{cc}-\frac{1}{2}(\eta^{2}-qr)&\eta
q\\\ \eta
r&\frac{1}{2}(\eta^{2}-qr)\end{array}\right)\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right),$ (3.21)

which is the KN spectral problem to generate the DNLS hierarchy of the Chen-
Lee-Liu type [25]. In the following we call it the CLL spectral problem for
short. In addition, the coupled system
$\phi_{t_{m}}=A_{m}\phi,~{}\phi_{t_{m}}^{*}=-A_{m}^{*}\phi^{*}$ gives rise to
the DNLS hierarchy of the Chen-Lee-Liu type (i.e. the CLL hierarchy)

$\left(\begin{array}[]{c}q\\\
r\end{array}\right)_{t_{m+1}}=T\left(\begin{array}[]{c}q\\\
r\end{array}\right)_{t_{m}},~{}~{}~{}\left(\begin{array}[]{c}q\\\
r\end{array}\right)_{t_{1}}=\left(\begin{array}[]{c}q\\\
r\end{array}\right)_{x},~{}~{}(m=1,2,\cdots),$ (3.22)

where the recursion operator $T$ is

$T=\left(\begin{array}[]{cc}\partial_{x}+qr+q_{x}\partial_{x}^{-1}r-q\partial_{x}^{-1}r_{x}&q_{x}\partial_{x}^{-1}q+q\partial_{x}^{-1}q_{x}\\\
r_{x}\partial_{x}^{-1}r+r\partial_{x}^{-1}r_{x}&-\partial_{x}+qr+r_{x}\partial_{x}^{-1}q-r\partial_{x}^{-1}q_{x}\end{array}\right).$
(3.23)

For more details of the above squared eigenfunction symmetry constraint, one
can refer to [4, 5].

##  4 The D$\Delta$mKP: symmetries and constraint

Since in the recent paper [9] the squared eigenfunction symmetry constraint
leads the D$\Delta$mKP system to the spectral problem and the positive
hierarchy of the relativistic Toda, not those of the DNLS as we expected, our
particular interest in the current paper is to find a new formulation of the
squared eigenfunction symmetry constraint so that the constrained the
D$\Delta$mKP system matches the continuous results described in Sec.3. In
addition, the match of the algebraic structures in the continuum limit is also
important because this means the discretization keeps well the integrable
structures. In the following we will first derive nonisospectral D$\Delta$mKP
flows, from which we will obtain symmetries of the D$\Delta$mKP hierarchy and
their Lie algebraic structure. After that, we will look at the squared
eigenfunction symmetry constraint.

###  4.1 Symmetries and algebraic structures

In the following, to consider continuum limit, we introduce the spacing
parameter $h$ in the $n$-direction. Thus, instead of the pseudo-difference
operator $\bar{L}$ given in (1.4), we use the following form,

$\bar{L}=h^{-1}\bar{v}\Delta+\bar{v}_{0}+h\bar{v}_{1}\Delta^{-1}+\cdots+h^{j}\bar{v}_{j}\Delta^{-j}+\cdots,$
(4.1)

where

$\bar{v}=1+h\tilde{v}$ (4.2)

and we assume $\tilde{v}$ and $\bar{v}_{i}$ belong to $\bar{S}[\tilde{v}]$.

Consider the isospectral Lax triad (cf.[9])

$\displaystyle\bar{L}\bar{\phi}=\lambda\bar{\phi},~{}~{}~{}\lambda_{t_{m}}=0,$
(4.3a)
$\displaystyle\bar{\phi}_{\bar{x}}=\bar{A}_{1}\bar{\phi},~{}~{}~{}\bar{A}_{1}=h^{-1}\bar{v}\Delta,$
(4.3b)
$\displaystyle\bar{\phi}_{\bar{t}_{m}}=\bar{A}_{m}\bar{\phi},~{}~{}~{}m=1,2,\cdots,$
(4.3c)

where $\bar{A}_{m}=(\bar{L}^{m})_{\geq 1}$ contains the pure difference part
(in terms of $\Delta$) of $\bar{L}^{m}$, and the first three are

$\displaystyle\bar{A}_{1}=h^{-1}\bar{v}\Delta,$ (4.4a)
$\displaystyle\bar{A}_{2}=h^{-2}\bar{v}(E\bar{v})\Delta^{2}+h^{-2}\bar{v}(\Delta\bar{v})\Delta+h^{-1}\bar{v}(E\bar{v}_{0})\Delta+h^{-1}\bar{v}\bar{v}_{0}\Delta,$
(4.4b)
$\displaystyle\bar{A}_{3}=h^{-3}\bar{a}_{0}\Delta^{3}+h^{-2}\bar{a}_{1}\Delta^{2}+h^{-1}\bar{a}_{2}\Delta,$
(4.4c)

in which

$\displaystyle\bar{a}_{0}$ $\displaystyle=\bar{v}(E\bar{v})(E^{2}\bar{v}),$
$\displaystyle\bar{a}_{1}$
$\displaystyle=2h^{-1}\bar{v}(E\bar{v})(E\Delta\bar{v})+\bar{v}(E\bar{v})(E^{2}\bar{v}_{0})+h^{-1}\bar{v}(\Delta\bar{v})(E\bar{v})+\bar{v}(E\bar{v}_{0})(E\bar{v})+\bar{v}\bar{v}_{0}(E\bar{v}),$
$\displaystyle\bar{a}_{2}$
$\displaystyle=h^{-2}\bar{v}(E\bar{v})(\Delta^{2}\bar{v})+2h^{-1}\bar{v}(E\bar{v})(E\Delta\bar{v}_{0})+\bar{v}(E\bar{v})(E^{2}\bar{v}_{1})+h^{-2}\bar{v}(\Delta\bar{v})^{2}+2h^{-1}\bar{v}(\Delta\bar{v})(E\bar{v}_{0})$
$\displaystyle~{}~{}~{}+\bar{v}(E\bar{v}_{0})^{2}+h^{-1}\bar{v}\bar{v}_{0}(\Delta\bar{v})+\bar{v}\bar{v}_{0}(E\bar{v}_{0})+h^{-1}\bar{v}^{2}(\Delta\bar{v}_{0})+\bar{v}^{2}(E\bar{v}_{1})+\bar{v}\bar{v}_{1}(E^{-1}\bar{v})+\bar{v}_{0}^{2}\bar{v},$

where $\bar{v}=1+h\tilde{v}$. Obviously, $\bar{A}_{m}$ obeys the asymptotic
condition

$\bar{A}_{m}|_{\tilde{v}=0}=h^{-m}\Delta^{m}.$ (4.5)

The compatibility of (4.3) gives rise to

$\displaystyle\bar{L}_{\bar{x}}=[\bar{A}_{1},\bar{L}],$ (4.6a)
$\displaystyle\bar{L}_{\bar{t}_{m}}=[\bar{A}_{m},\bar{L}],$ (4.6b)
$\displaystyle\bar{A}_{1,\bar{t}_{m}}-\bar{A}_{m,\bar{x}}+[\bar{A}_{1},\bar{A}_{m}]=0,$
(4.6c)

Among them, Eq.(4.6a) yields

$\displaystyle\bar{v}\Delta\bar{v}_{0}=\bar{v}_{x},$ (4.7a) $\displaystyle
h\bar{v}(E\bar{v}_{1})-h(E^{-1}\bar{v})\bar{v}_{1}=h\bar{v}_{0,x}-\bar{v}(\Delta\bar{v}_{0}),$
(4.7b) $\displaystyle
h\bar{v}(E\bar{v}_{s+1})-h(E^{-1-s}\bar{v})\bar{v}_{s+1}=h\bar{v}_{s,x}-\bar{v}(\Delta\bar{v}_{s})+\sum_{i=1}^{s}\mathrm{C}_{s}^{i-1}(-1)^{s-i+1}h^{i-s}\bar{v}_{i}E^{-s-1}\Delta^{s-i+1}\bar{v}$
(4.7c)

for $s\geq 1$, from which one can ‘integrate’ $\bar{v}_{s}$ with boundary
condition $\bar{v}_{s}|_{\tilde{v}=0}=0$, and then one has

$\displaystyle\bar{v}_{0}=\Delta^{-1}(\ln\bar{v})_{\bar{x}},~{}~{}~{}\bar{v}_{1}=\frac{\Delta^{-2}(\ln\bar{v})_{\bar{x}\bar{x}}-h^{-1}\Delta^{-1}\bar{v}_{\bar{x}}}{(E^{-1}\bar{v})},~{}~{}~{}\cdots.$
(4.8)

Eq.(4.6c) provides the zero curvature representation of the isopectral
D$\Delta$mKP hierarchy,

$\tilde{v}_{\bar{t}_{m}}=\bar{K}_{m}(\tilde{v})=(\bar{A}_{m,\bar{x}}-[\bar{A}_{1},\bar{A}_{m}])\Delta^{-1},~{}~{}~{}m=1,2,\cdots,$
(4.9)

where the first three of them are

$\displaystyle\tilde{v}_{\bar{t}_{1}}=\bar{K}_{1}(\tilde{v})$
$\displaystyle=\tilde{v}_{\bar{x}},$ (4.10a)
$\displaystyle\tilde{v}_{\bar{t}_{2}}=\bar{K}_{2}(\tilde{v})$
$\displaystyle=h^{-1}[\bar{v}(1+2\Delta^{-1})(\ln\bar{v})_{\bar{x}\bar{x}}+\bar{v}_{\bar{x}}(1+2\Delta^{-1})(\ln\bar{v})_{\bar{x}}-2h^{-1}\bar{v}\bar{v}_{\bar{x}}],$
(4.10b) $\displaystyle\tilde{v}_{\bar{t}_{3}}=\bar{K}_{3}(\tilde{v})$
$\displaystyle=h^{-1}[\bar{v}(3\Delta^{-2}+3\Delta^{-1}+1)(\ln\bar{v})_{\bar{x}\bar{x}\bar{x}}+\bar{v}_{\bar{x}}(3\Delta^{-2}+3\Delta^{-1}+1)(\ln\bar{v})_{\bar{x}\bar{x}}$
$\displaystyle~{}~{}~{}+\bar{v}_{\bar{x}}H_{1}^{2}+2\bar{v}H_{1}H_{2}-(\bar{v}(\Delta^{-1}(\ln\bar{v})_{\bar{x}}))_{\bar{x}}H_{1}-\bar{v}(\Delta^{-1}(\ln\bar{v})_{\bar{x}})H_{2}$
$\displaystyle~{}~{}~{}+3h^{-2}\bar{v}^{2}\bar{v}_{\bar{x}}-3h^{-1}(\bar{v}(1+\Delta^{-1})\bar{v}_{\bar{x}})_{\bar{x}}-3h^{-1}(\bar{v}^{2}\Delta^{-1}(\ln\bar{v})_{\bar{x}})_{\bar{x}}],$
(4.10c)

where

$H_{1}=(1+2\Delta^{-1})(\ln\bar{v})_{\bar{x}},~{}~{}H_{2}=(1+2\Delta^{-1})(\ln\bar{v})_{\bar{x}\bar{x}}.$

######  Remark 4.1.

If we assume $\bar{A}_{m}$ has a form

$\bar{A}_{m}=\bar{a}_{0}\Delta^{m}+\bar{a}_{1}\Delta^{m-1}+\cdots+\bar{a}_{m-1}\Delta,~{}~{}\bar{A}_{m}|_{\tilde{v}=0}=\Delta^{m},$

then $\\{\bar{a}_{j}\\}$ can be uniquely determined from (4.6b) and it gives
rise to $\bar{A}_{m}=(\bar{L}^{m})_{\geq 1}$.

To find a master symmetry, we consider the nonisospectral Lax triad

$\displaystyle\bar{L}\bar{\phi}=\lambda\bar{\phi},~{}~{}~{}\lambda_{t_{m}}=h\lambda^{m}+\lambda^{m-1},$
(4.11a) $\displaystyle\bar{\phi}_{\bar{x}}=\bar{A}_{1}\bar{\phi},$ (4.11b)
$\displaystyle\bar{\phi}_{t_{m}}=\bar{B}_{m}\bar{\phi},~{}~{}~{}m=2,3,\cdots,$
(4.11c)

where $\bar{B}_{m}$ is assumed to be a difference operator with the form

$\bar{B}_{m}=\sum_{j=0}^{m-1}h^{-(m-j)}\bar{b}_{j}\Delta^{m-j},$

with the asymoptotic condition

$\displaystyle{\bar{B}_{m}}|_{\tilde{v}=0}=h^{-(m-1)}\bar{x}\Delta^{m}+h^{-(m-1)}(\bar{x}+nh)\Delta^{m-1},~{}~{}~{}m\geq
2.$ (4.12)

The compatibility of (4.11) gives rise to

$\displaystyle\bar{L}_{\bar{x}}=[\bar{A}_{1},\bar{L}],$ (4.13a)
$\displaystyle\bar{L}_{\bar{t}_{m}}=[\bar{B}_{m},\bar{L}]+h\bar{L}^{m}+\bar{L}^{m-1},$
(4.13b)
$\displaystyle\bar{A}_{1,\bar{t}_{m}}-\bar{B}_{m,\bar{x}}+[\bar{A}_{1},\bar{B}_{m}]=0.$
(4.13c)

Again, (4.13a) yields (4.7) which expresses $\bar{v}_{j}$ in terms of
$\tilde{v}$. $\\{\bar{B}_{m}\\}$ are uniquely defined from (4.13b) but no
$\bar{B}_{1}$ can be obtained in that approach, therefore there is no the
nonisospectral D$\Delta$mKP flow $\bar{\sigma}_{1}$ defined from (4.13). The
first two $\bar{B}_{m}$ are

$\displaystyle\bar{B}_{2}=h\bar{x}\bar{A}_{2}+(\bar{x}+nh)\bar{A}_{1},$
$\displaystyle\bar{B}_{3}=h\bar{x}\bar{A}_{3}+(\bar{x}+nh)\bar{A}_{2}+\bar{v}(\Delta^{-1}\bar{v}_{0})\Delta-h^{-1}\bar{v}(\Delta^{-1}\bar{v})\Delta+h^{-1}n\bar{v}\Delta.$

where $\bar{A}_{j}$ are given in (4.4). Then, (4.13c) defines the
nonisopectral D$\Delta$mKP hierarchy

$\tilde{v}_{\bar{t}_{m}}=\bar{\sigma}_{m}(\tilde{v})=(\bar{B}_{m,\bar{x}}+[\bar{B}_{m},\bar{A}_{1}])\Delta^{-1},~{}~{}m=2,3,\cdots,$
(4.14)

in which the first two equations are

$\displaystyle\tilde{v}_{\bar{t}_{2}}$
$\displaystyle=\bar{\sigma}_{2}(\tilde{v})=h\bar{x}\bar{K}_{2}+(\bar{x}+nh)\bar{K}_{1}+\bar{v}(1+2\Delta^{-1})(\ln\bar{v})_{\bar{x}}+h^{-1}\bar{v}-h^{-1}{\bar{v}}^{2}.$
$\displaystyle\tilde{v}_{\bar{t}_{3}}$
$\displaystyle=\bar{\sigma}_{3}(\tilde{v})$
$\displaystyle=h\bar{x}\bar{K}_{3}+(\bar{x}+nh)\bar{K}_{2}+h^{-2}\bar{v}(E\bar{v})(\Delta^{2}\bar{v})+2h^{-1}\bar{v}(E\bar{v})(E\Delta\bar{v}_{0})+\bar{v}(E\bar{v})(E^{2}\bar{v}_{1})$
$\displaystyle~{}~{}~{}+h^{-2}\bar{v}(\Delta\bar{v})^{2}+2h^{-1}\bar{v}(\Delta\bar{v})(E\bar{v}_{0})+\bar{v}(E\bar{v}_{0})^{2}+h^{-1}\bar{v}\bar{v}_{0}(\Delta\bar{v})+\bar{v}\bar{v}_{0}(E\bar{v}_{0})$
$\displaystyle~{}~{}~{}+h^{-1}\bar{v}^{2}(\Delta\bar{v}_{0})+\bar{v}^{2}(E\bar{v}_{1})+\bar{v}\bar{v}_{1}(E^{-1}\bar{v})+\bar{v}^{2}_{0}\bar{v}+h^{-2}\bar{v}(\Delta\bar{v})+h^{-1}\bar{v}(E\bar{v}_{0})$
$\displaystyle~{}~{}~{}+h^{-1}\bar{v}\bar{v}_{0}+(\bar{v}(\Delta^{-1}\bar{v}_{0}))_{\bar{x}}-h^{-1}(\bar{v}(\Delta^{-1}\bar{v}))_{\bar{x}}+h^{-1}n\bar{v}_{\bar{x}}-h^{-1}\bar{v}(E\bar{v})\bar{v}_{0},$

where $\bar{v}_{0}$ and $\bar{v}_{1}$ are given in (4.8) and $\bar{K}_{j}$ are
given in (4.10). In addition, one can find that

$\bar{K}_{m}|_{\tilde{v}=0}=0,~{}~{}~{}\bar{\sigma}_{s}|_{\tilde{v}=0}=0,~{}~{}~{}(m=1,2,\cdots,~{}~{}s=2,3,\cdots).$
(4.15)

Eqs. (4.9) and (4.14) are the zero curvature representations of the
D$\Delta$mKP flows $\\{\bar{K}_{m},\bar{\sigma}_{s}\\}_{m\geq 1,s\geq 2}$,
which can be used to determine algebraic structures of these flows (e.g. [6,
7, 14, 21]). To proceed, one should have a zero curvature equation that allows
unique solutions.

######  Lemma 4.1.

Suppose $\bar{X}=\bar{X}(\tilde{v})\in\bar{S}[\tilde{v}]$ and $\bar{N}$ is a
difference operator living on $\bar{S}[\tilde{v}]$, with a form

$\bar{N}=\bar{a}_{0}\Delta^{m}+\bar{a}_{1}\Delta^{m-1}+\cdots+\bar{a}_{m-1}\Delta,$

where $\bar{a}_{j}=\bar{a}_{j}(\tilde{v})\in\bar{S}[\tilde{v}]$ and it follows
that $\bar{N}|_{\tilde{v}=0}=0$. Then, the following equation

$\bar{X}-\bar{N}_{\bar{x}}\Delta^{-1}+[\bar{A}_{1},\bar{N}]\Delta^{-1}=0$
(4.16)

has only zero solutions $\bar{X}=0$ and $\bar{N}=0$, where $\bar{A}_{1}$ is
given as (4.4a).

In fact, since $\bar{v}=1+h\tilde{v}\to 1$ as $|n|\to\infty$, substituting
$\bar{N}$ into (4.16) one can successfully find $\bar{a}_{0}=0$,
$\bar{a}_{1}=0$, $\cdots$, $\bar{a}_{m-1}=0$ and finally $\bar{N}=0$ and
$\bar{X}=0$.

Then we have the following.

######  Theorem 4.1.

The D$\Delta$mKP flows $\\{\bar{K}_{m},\bar{\sigma}_{s}\\}_{m\geq 1,s\geq 2}$
compose a Lie algebra with the following structure

$\displaystyle\llbracket\bar{K}_{m},\bar{K}_{s}\rrbracket=0,$ (4.17a)
$\displaystyle\llbracket\bar{K}_{m},\bar{\sigma}_{s}\rrbracket=m(h\bar{K}_{m+s-1}+\bar{K}_{m+s-2}),$
(4.17b)
$\displaystyle\llbracket\bar{\sigma}_{m},\bar{\sigma}_{s}\rrbracket=(m-s)(h\bar{\sigma}_{m+s-1}+\bar{\sigma}_{m+s-2}).$
(4.17c)

###### Proof.

First, for the D$\Delta$mKP flows $\\{\bar{K}_{m},\bar{\sigma}_{n}\\}_{m\geq
1,n\geq 2}$, using their zero curvature representations, we can find that

$\displaystyle\llbracket\bar{K}_{m},\bar{K}_{s}\rrbracket=(\langle\bar{A}_{m},\bar{A}_{s}\rangle_{y}+[\langle\bar{A}_{m},\bar{A}_{s}\rangle,\bar{A}_{2}])\Delta^{-1},$
(4.18a)
$\displaystyle\llbracket\bar{K}_{m},\bar{\sigma}_{s}\rrbracket=(\langle\bar{A}_{m},\bar{B}_{s}\rangle_{y}+[\langle\bar{A}_{m},\bar{B}_{s}\rangle,\bar{A}_{2}])\Delta^{-1},$
(4.18b)
$\displaystyle\llbracket\bar{\sigma}_{m},\bar{\sigma}_{s}\rrbracket=(\langle\bar{B}_{m},\bar{B}_{s}\rangle_{y}+[\langle\bar{B}_{m},\bar{B}_{s}\rangle,\bar{A}_{2}])\Delta^{-1},$
(4.18c)

where

$\displaystyle\langle\bar{A}_{m},\bar{A}_{s}\rangle=\bar{A}_{m}^{\prime}[\bar{K}_{s}]-\bar{A}_{s}^{\prime}[\bar{K}_{m}]+[\bar{A}_{m},\bar{A}_{s}],$
$\displaystyle\langle\bar{A}_{m},\bar{B}_{s}\rangle=\bar{A}_{m}^{\prime}[\bar{\sigma}_{s}]-\bar{B}_{s}^{\prime}[\bar{K}_{m}]+[\bar{A}_{m},\bar{B}_{s}],$
$\displaystyle\langle\bar{B}_{m},\bar{B}_{s}\rangle=\bar{B}_{m}^{\prime}[\bar{\sigma}_{s}]-\bar{B}_{s}^{\prime}[\bar{\sigma}_{m}]+[\bar{B}_{m},\bar{B}_{s}].$

Let us take (4.18b) as an example to explain the above relations. From the
zero curvature representation (4.9) we have

$\displaystyle\bar{K}_{m}^{\prime}(\tilde{v})[\bar{\sigma}_{s}]$
$\displaystyle=((\bar{A}_{m,\bar{x}}+[\bar{A}_{m},\bar{A}_{1}])\Delta^{-1})^{\prime}[\bar{\sigma}_{s}]$
$\displaystyle=\bigl{(}\bar{A}_{m}^{\prime}[\bar{\sigma}_{s}])_{\bar{x}}+[\bar{A}_{m}^{\prime}[\bar{\sigma}_{s}],\bar{A}_{1}]+[\bar{A}_{m},\bar{A}_{1}^{\prime}[\bar{\sigma}_{s}]]\bigr{)}\Delta^{-1}$
$\displaystyle=\bigl{(}(\bar{A}_{m}^{\prime}[\bar{\sigma}_{s}])_{\bar{x}}+[\bar{A}_{m}^{\prime}[\bar{\sigma}_{s}],\bar{A}_{1}]+[\bar{A}_{m},h^{-1}\bar{\sigma}_{s}\Delta]]\bigr{)}\Delta^{-1}$
$\displaystyle=\bigl{(}(\bar{A}_{m}^{\prime}[\bar{\sigma}_{s}])_{\bar{x}}+[\bar{A}_{m}^{\prime}[\bar{\sigma}_{s}],\bar{A}_{1}]+[\bar{A}_{m},\bar{B}_{s,\bar{x}}]+[\bar{A}_{m},[\bar{B}_{s},\bar{A}_{1}]]\bigr{)}\Delta^{-1}.$

Similarly, from (4.14) we have

$\bar{\sigma}_{s}^{\prime}(\tilde{v})[\bar{K}_{m}]=\bigl{(}(\bar{B}_{s}^{\prime}[\bar{K}_{m}])_{\bar{x}}+[\bar{B}_{s}^{\prime}[\bar{K}_{m}],\bar{A}_{1}]+[\bar{B}_{s},\bar{A}_{m,\bar{x}}]+[\bar{B}_{s},[\bar{A}_{m},\bar{A}_{1}]]\bigr{)}\Delta^{-1}.$

Subtracting $\bar{\sigma}_{s}^{\prime}[\bar{K}_{m}]$ from
$\bar{K}_{m}^{\prime}[\bar{\sigma}_{s}]$ and making use of the Jacobi identity
of $[\cdot,\cdot]$, we obtain (4.18b).

Next, noticing that the following asymptotic conditions

$\displaystyle\langle\bar{A}_{m},\bar{A}_{s}\rangle|_{\bar{v}=1}=0,$
$\displaystyle\langle\bar{A}_{m},\bar{B}_{s}\rangle|_{\bar{v}=1}=h^{-(m+s-2)}m(\Delta^{m+s-1}+\Delta^{m+s-2}),$
$\displaystyle\langle\bar{B}_{m},\bar{B}_{s}\rangle|_{\bar{v}=1}=h^{-(m+s-3)}(m-s)(\bar{x}\Delta^{m+s-1}+(2\bar{x}+hn)\Delta^{m+s-2}+(\bar{x}+hn)\Delta^{m+s-3})$

hold, comparing them with (4.5) and (4.12), and using Lemma 4.1, we obtain the
algebraic relations (4.17).

∎

With these structures we have the following.

######  Theorem 4.2.

Each equation

$\tilde{v}_{t_{k}}=\bar{K}_{k}$ (4.20)

in the isospectral D$\Delta$mKP hierarchy (4.9) has two sets of symmetries:

$\\{\bar{K}_{s}\\}_{s\geq 1},\qquad\\{\bar{\tau}_{m}^{k}=k\
\bar{t}_{k}(h\bar{K}_{m+k-1}+\bar{K}_{m+k-2})+\bar{\sigma}_{m}\\}_{m\geq 2},$
(4.21)

and they compose a Lie algebra with structure

$\displaystyle[\\![\bar{K}_{s},\bar{K}_{m}]\\!]=0,$ (4.22a)
$\displaystyle[\\![\bar{K}_{s},\bar{\tau}_{m}^{k}]\\!]=s(h\bar{K}_{m+s-1}+\bar{K}_{m+s-2}),$
(4.22b)
$\displaystyle[\\![\bar{\tau}_{s}^{k},\bar{\tau}_{m}^{k}]\\!]=(s-m)(h\bar{\tau}^{k}_{m+s-1}+\bar{\tau}^{k}_{m+s-2}).$
(4.22c)

###  4.2 Squared eigenfunction symmetry constraint

####  4.2.1 Symmetry and constraint

Consider the triplet (4.3) and its adjoint form with eigenfunctions $\Phi$ and
$\Phi^{*}$, respectively, we introduce the adjoint form

$\displaystyle\bar{L}\Phi=\lambda\Phi,~{}~{}~{}~{}~{}\bar{L}^{*}\Phi^{*}=\lambda\Phi^{*},$
(4.23a)
$\displaystyle\Phi_{\bar{x}}=\bar{A}_{1}\Phi,~{}~{}~{}~{}\Phi^{*}_{\bar{x}}=-\bar{A}^{*}_{1}\Phi^{*},$
(4.23b)
$\displaystyle\Phi_{\bar{t}_{s}}=\bar{A}_{s}\Phi,~{}~{}~{}\Phi^{*}_{\bar{t}_{s}}=-\bar{A}^{*}_{s}\Phi^{*},~{}~{}s=1,2,\cdots,$
(4.23c)

where $\bar{L}^{*}$ and $\bar{A}^{*}_{s}$ are the adjoint operators of
$\bar{L}$ and $\bar{A}_{s}$, respectively, defined through the inner product
(2.5). It can be proved that [9]

$\omega=h(\Phi
E\Delta^{-1}\Phi^{*})_{\bar{x}}=-\sum^{\infty}_{s=1}\bar{K}_{s}(\tilde{v}),$
(4.24)

i.e., $\omega$ is the squared eigenfunction symmetry of the D$\Delta$mKP
hierarchy. Note that in [9], by the constraint

$\bar{v}=\Phi E\Delta^{-1}\Phi^{*},$ (4.25)

the D$\Delta$mKP system composed by (with $\bar{x}=\bar{t}_{1}$)

$\displaystyle\bar{L}\Phi=\lambda\Phi,$ (4.26a)
$\displaystyle\Phi_{\bar{t}_{s}}=-\bar{A}_{s}\Phi,~{}~{}\Phi^{*}_{\bar{t}_{s}}=-\bar{A}^{*}_{s}\Phi^{*},~{}~{}s=1,2,\cdots$
(4.26b)

gives rise to the the spectral problem and positive hierarchy of the
relativistic Toda, not those of the DNLS as we expect (cf. in the continuous
case [4]).

####  4.2.2 The discrete CLL spectral problem

In the following, instead of (4.25), we introduce another constraint (note
that $\bar{v}=1+h\tilde{v}$)

$\tilde{v}=h\Phi
E\Delta^{-1}\Phi^{*}=a_{n}b_{n},~{}~{}~{}a_{n}=\Phi,~{}~{}~{}b_{n}=hE\Delta^{-1}\Phi^{*},$
(4.27)

under which, first, one can rewrite $\bar{L}$ in terms of $a_{n}$ and $b_{n}$.

######  Proposition 4.1.

For the two triplets in (4.23), with the constraint (4.27), the pseudo-
difference operator $\bar{L}$ (4.1) can be written into the form

$\bar{L}=h^{-1}(1+ha_{n}b_{n})\Delta+a_{n}\Delta^{-1}b_{n}\Delta.$ (4.28)

###### Proof.

In light of formula (2.4) with $s=-1$, we only need to prove that all the
$\\{\bar{v}_{s}\\}^{\infty}_{s=0}$ defined by (4.7c) can be expressed in terms
of $a_{n}$ and $b_{n}$ as the following:

$\bar{v}_{s}=(-1)^{s}h^{-s}a_{n}E^{-s-1}\Delta^{s}b_{n},~{}~{}~{}s=0,1,2,\cdots.$
(4.29)

Then (4.28) follows.

The proof of (4.29) is similar to Proposition 3 in [9]. Below we sketch main
steps and leave the details in Appendix A. First, it follows from (4.23b) and
(4.27) that

$a_{n,\bar{x}}=h^{-1}(1+ha_{n}b_{n})(a_{n+1}-a_{n}),~{}~{}b_{n,\bar{x}}=h^{-1}(1+ha_{n}b_{n})(b_{n}-b_{n-1}),$
(4.30)

which will be used to eliminate derivatives in (4.7), in particular, from
(4.7a) and (4.7b) we have

$\bar{v}_{0}=a_{n}b_{n-1},~{}~{}\bar{v}_{1}=-h^{-1}a_{n}E^{-2}\Delta b_{n}.$

Then, to make use of mathematical induction, we assume the following
expression holds for $i=0,1,2,\cdots,s$, i.e.

$\bar{v}_{i}=(-1)^{i}h^{-i}a_{n}E^{-i-1}\Delta^{i}b_{n},~{}~{}~{}i=0,1,2,\cdots,s.$
(4.31)

Then, for $s+1$, the right-hand side of (4.7c) gives rise to (refer to
Appendix A for details)

$(-1)^{s+1}h^{-s}[(1+ha_{n}b_{n})E(a_{n}E^{-s-2}\Delta^{s+1}b_{n})-(E^{-s-1}(1+ha_{n}b_{n}))(a_{n}E^{-s-2}\Delta^{s+1}b_{n})].$
(4.32)

Compared with the left-hand side of (4.7c), we can find

$\bar{v}_{s+1}=(-1)^{s+1}h^{-s-1}a_{n}E^{-s-2}\Delta^{s+1}b_{n}.$

The proof is completed.

∎

######  Theorem 4.3.

With the constraint (4.27), the spectral problem (4.11a) gives rise to the
discrete CLL spectral problem

$\left(\begin{array}[]{c}\bar{\psi}_{1,n+1}\\\
\bar{\psi}_{2,n+1}\end{array}\right)=\left(\begin{array}[]{cc}-h\eta^{2}+1+ha_{n}b_{n}&h\eta
a_{n}\\\ h\eta
b_{n}&1\end{array}\right)\left(\begin{array}[]{c}\bar{\psi}_{1,n}\\\
\bar{\psi}_{2,n}\end{array}\right).$ (4.33)

###### Proof.

In light of Proposition 4.1, the spectral problem (4.11a) is written as

$\bar{L}\bar{\phi}=[h^{-1}(1+ha_{n}b_{n})\Delta+a_{n}\Delta^{-1}b_{n}\Delta]\bar{\phi}=\lambda\bar{\phi}.$
(4.34)

Introducing $\varphi_{1,n}$ and $\varphi_{2,n}$ by

$\lambda=-\eta^{2},~{}~{}\bar{\varphi}_{1,n}=\bar{\phi},~{}~{}\eta\Delta\bar{\varphi}_{2,n}=-b_{n}\Delta\bar{\varphi}_{1,n},$
(4.35)

(4.34) yields

$\displaystyle(1+ha_{n}b_{n})\Delta\bar{\varphi}_{1,n}=-h\eta^{2}\bar{\varphi}_{1,n}+h\eta
a_{n}\bar{\varphi}_{2,n},$ (4.36a)
$\displaystyle(1+ha_{n}b_{n})\Delta\bar{\varphi}_{2,n}=h\eta
b_{n}\bar{\varphi}_{1,n}-ha_{n}b_{n}\bar{\varphi}_{2,n},$ (4.36b)

i.e.

$\displaystyle(1+ha_{n}b_{n})\bar{\varphi}_{1,n+1}=(-h\eta^{2}+1+ha_{n}b_{n})\bar{\varphi}_{1,n}+h\eta
a_{n}\bar{\varphi}_{2,n},$
$\displaystyle(1+ha_{n}b_{n})\bar{\varphi}_{2,n+1}=h\eta
b_{n}\bar{\varphi}_{1,n}+\bar{\varphi}_{2,n}.$

Next, we introduce a gauge transformation

$\left(\begin{array}[]{c}\bar{\psi}_{1,n}\\\
\bar{\psi}_{2,n}\end{array}\right)=\left(\begin{array}[]{cc}(1+ha_{n}b_{n})^{n}&0\\\
0&(1+ha_{n}b_{n})^{n}\end{array}\right)\left(\begin{array}[]{c}\bar{\varphi}_{1,n}\\\
\bar{\varphi}_{2,n}\end{array}\right),$

and immediately we arrive at the spectral problem (4.33). If we further
introduce

$\left(\begin{array}[]{c}\bar{\psi}_{1,n}\\\
\bar{\psi}_{2,n}\end{array}\right)=\left(\begin{array}[]{cc}1&0\\\
0&1/\eta\end{array}\right)\left(\begin{array}[]{c}\psi_{1,n}\\\
\psi_{2,n}\end{array}\right),$

(4.33) gives rise to

$\left(\begin{array}[]{c}\psi_{1,n+1}\\\
\psi_{2,n+1}\end{array}\right)=\left(\begin{array}[]{cc}-h\eta^{2}+1+ha_{n}b_{n}&ha_{n}\\\
h\eta^{2}b_{n}&1\end{array}\right)\left(\begin{array}[]{c}\psi_{1,n}\\\
\psi_{2,n}\end{array}\right),$ (4.37)

which is known as the discrete CLL spectral problem [11], see also [24, 17].

∎

We will introduce more about the discrete CLL spectral problem (4.33) and
(4.37) in Appendix B.

####  4.2.3 The differential-difference CLL hiararchy

Next, let us look at (4.26b) and reveal their explicit forms under the
constraint (4.27) and the $(a_{n},b_{n})$ formulation. The procedure is
similar to [9] but we need examine all details. First, we have the following.

######  Proposition 4.2.

For $\bar{L}$ given in (4.28), $A_{s}$ obeys the following two recursive
relations:

$\displaystyle\bar{A}_{s+1}=\bar{L}\bar{A}_{s}+h^{-1}(ha_{n}b_{n}+1)(E(\bar{L}^{s})_{0})\Delta-
a_{n}\Delta^{-1}(E\Delta^{-1}\bar{A}_{s}^{*}E^{-1}\Delta b_{n})\Delta,$
(4.38a)
$\displaystyle\bar{A}_{s+1}=\bar{A}_{s}\bar{L}+h^{-1}(\bar{L}^{s})_{0}(ha_{n}b_{n}+1)\Delta-(\bar{A}_{s}a_{n})\Delta^{-1}b_{n}\Delta,$
(4.38b)

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

###### Proof.

Using the identities [9]

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

and noticing that $\bar{A}_{s+1}=(\bar{L}\bar{L}^{s})_{\geq 1}$ and
$\bar{A}_{s+1}=(\bar{L}^{s}\bar{L})_{\geq 1}$, one can obtain (4.38a) and
(4.38b) respectively. ∎

Next, we have the following.

######  Theorem 4.4.

Equation (4.26b) gives rise to the recursive hierarchy

$\left(\begin{array}[]{c}a_{n}\\\
b_{n}\end{array}\right)_{\bar{t}_{s+1}}=\bar{T}\left(\begin{array}[]{c}a_{n}\\\
b_{n}\end{array}\right)_{\bar{t}_{s}},$ (4.39)

where the initial equation reads

$a_{n,\bar{t}_{1}}=h^{-1}(1+ha_{n}b_{n})(a_{n+1}-a_{n}),~{}~{}b_{n,\bar{t}_{1}}=h^{-1}(1+ha_{n}b_{n})(b_{n}-b_{n-1}),$
(4.40)

and the recursion operator is

$\bar{T}=\left(\begin{array}[]{cc}\bar{T}_{11}&\bar{T}_{12}\\\
\bar{T}_{21}&\bar{T}_{22}\end{array}\right)$ (4.41)

with entries

$\begin{array}[]{l}\bar{T}_{11}=h^{-1}(ha_{n}b_{n}\Delta+\Delta+ha_{n}\Delta^{-1}b_{n}\Delta)+h^{-1}(ha_{n}b_{n}+1)(\Delta
a_{n})E\Delta^{-1}\frac{hb_{n}}{ha_{n}b_{n}+1},\\\
\bar{T}_{12}=h^{-1}(ha_{n}b_{n}+1)(\Delta
a_{n})E\Delta^{-1}\frac{ha_{n}}{ha_{n}b_{n}+1}-a_{n}\Delta^{-1}(\Delta
a_{n}),\\\ \bar{T}_{21}=h^{-1}(ha_{n}b_{n}+1)(E^{-1}\Delta
b_{n})\Delta^{-1}\frac{hb_{n}}{ha_{n}b_{n}+1}+b_{n}E\Delta^{-1}(E\Delta
b_{n}),\\\
\bar{T}_{22}=h^{-1}(-ha_{n}b_{n}E^{-1}\Delta-E^{-1}\Delta+hb_{n}E\Delta^{-1}a_{n}E^{-1}\Delta)\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}+h^{-1}(ha_{n}b_{n}+1)(E^{-1}\Delta
b_{n})\Delta^{-1}\frac{ha_{n}}{ha_{n}b_{n}+1}.\end{array}$ (4.42)

###### Proof.

The initial equation (4.40) is obvious. Next, recalling the residue form of
the D$\Delta$mKP hierarchy [9],

$\tilde{v}_{\bar{t}_{m}}=\bar{K}_{m}=h^{-1}\bar{v}\Delta\mathop{\mathrm{Res}}\limits_{\Delta}(L^{m}\Delta^{-1}),$
(4.43)

one has

$(\bar{L}^{s})_{0}=\Delta^{-1}(\ln\bar{v})_{\bar{t}_{s}}=\Delta^{-1}(\ln(ha_{n}b_{n}+1))_{\bar{t}_{s}}=\Delta^{-1}\left(\frac{hb_{n}}{ha_{n}b_{n}+1}a_{n,\bar{t}_{s}}+\frac{ha_{n}}{ha_{n}b_{n}+1}b_{n,\bar{t}_{s}}\right).$

Then using (4.38a) we have

$\displaystyle a_{n,\bar{t}_{s+1}}$ $\displaystyle=\bar{A}_{s+1}a_{n}$
$\displaystyle=\bar{L}a_{n,\bar{t}_{s}}+h^{-1}(ha_{n}b_{n}+1)(E(\bar{L}^{s})_{0})\Delta
a_{n}+a_{n}\Delta^{-1}(\Delta a_{n})b_{n,t_{s}},$ (4.44)

where we have used

$a_{n,\bar{t}_{s}}=\bar{A}_{s}a_{n},~{}~{}b_{n,t_{s}}=-E\Delta^{-1}\bar{A}_{s}^{*}E^{-1}\Delta
b_{n}.$

Then, substituting $(\bar{L}^{s})_{0}$ into (4.44) we have

$a_{n,\bar{t}_{s+1}}=\bar{T}_{11}a_{n,\bar{t}_{s}}+\bar{T}_{12}b_{n,\bar{t}_{s}},$

where $\bar{T}_{11}$ and $\bar{T}_{12}$ are given in (4.42). Similarly, one
can prove that

$b_{n,\bar{t}_{s+1}}=\bar{T}_{21}a_{n,\bar{t}_{s}}+\bar{T}_{22}b_{n,\bar{t}_{s}}.$

Thus, the proof is completed.

∎

In the next section, we will show a continuum limit in which (4.39) recovers
the CLL hierarchy.

##  5 Continuum limits of the D$\Delta$mKP

Our purpose of this section is to implement continuum limits on the integrable
structures of the D$\Delta$mKP and match them with those of the continuous
case. The technique of implementing the continuum limits is similar to that we
have used in [14] for investigating the D$\Delta$KP, where we introduced a
notion called “degree” as a practical and effective tool to figure out the
leading terms in the continuum limits. In the following we will first present
a unified scheme of the continuum limits and calculate degree of each object
of the D$\Delta$mKP. After that the continuum limits will be investigated. We
will skip some details, for which one can refer to the analogues in [14]. We
will see that all the results in Section 3 for the mKP will be recovered from
those of the D$\Delta$mKP in the continuum limits.

###  5.1 Scheme of the continuum limits

Our plan for the continuum limit is as the following (cf.[14] for the
D$\Delta$KP case).

  * •

Let $n\rightarrow\infty$ and $h\rightarrow 0$ simultaneously such that $nh$ is
finite.

  * •

Introduce an auxiliary continuous variable

$\tau=nh,$ (5.1)

which maps $\bar{f}(n+j)$ to $f(\tau+jh)$.

  * •

Define coordinates relation

$x=\bar{x}+\tau,~{}~{}~{}y=-\frac{1}{2}h\tau,~{}~{}~{}t_{m}=\bar{t}_{m},$
(5.2)

under which one has

$\partial_{\bar{x}}=\partial_{x},~{}~{}~{}\partial_{\tau}=\partial_{x}-\frac{1}{2}h\partial_{y},~{}~{}~{}\partial_{\bar{t}_{m}}=\partial_{t_{m}}.$
(5.3)

  * •

Define functions relation

$\displaystyle\bar{v}(n,\bar{x},\bar{t}_{m})=1+h\tilde{v}(n,\bar{x},\bar{t}_{m})=1+hv(x,y,t_{m}),$
(5.4a) $\displaystyle\bar{\phi}(n,\bar{x},\bar{t}_{m})=\phi(x,y,t_{m}).$
(5.4b)

Base on the above scheme one can quickly find

$\displaystyle\Delta$
$\displaystyle=h\partial_{\tau}+\frac{1}{2!}h^{2}\partial_{\tau}^{2}+\frac{1}{3!}h^{3}\partial_{\tau}^{3}+O(h^{4})$
$\displaystyle=h\partial_{x}+\frac{h^{2}}{2}(\partial_{x}^{2}-\partial_{y})+\frac{h^{3}}{6}(\partial_{x}^{3}-3\partial_{x}\partial_{y})+O(h^{4}).$
(5.5)

Consequently,

$\displaystyle\Delta^{-1}$
$\displaystyle=h^{-1}\partial_{x}^{-1}+\frac{1}{2}(\partial_{x}^{-2}\partial_{y}^{-1}-1)+O(h),$
(5.6)

and in general,

$\Delta^{j}=h^{j}\partial^{j}_{x}+O(h^{j+1}),~{}~{}j\in\mathbb{Z}.$ (5.7)

###  5.2 Degrees

By degree we mean (see Definition 5.1 in [14]) in the scheme presented in
Section 5.1 the order of the leading term of a function
$\bar{f}(n,\bar{x},\bar{t}_{m})$ (or operator $\bar{P}(\tilde{v},\Delta)$)
after expanded in a series of $h$, denoted by deg $\bar{f}$ (or deg
$\bar{P}$).

Let us first examine the degrees of $\\{\bar{v}_{j}\\}$. From (4.7) we find

$\bar{v}_{0}=\Delta^{-1}(\ln(1+h\tilde{v}))_{\bar{x}}=v+O(h),$

and

$\bar{v}_{1}=-\frac{1}{2}\tilde{v}^{2}+\frac{1}{2}(\partial_{x}^{-1}\partial_{y}-\partial_{x})\tilde{v}+O(h)=v_{1}+O(h).$

For general $j$, if we assume $\mathrm{deg}\,\bar{v}_{j}=0$ for
$j=1,2,\cdots,s$, then, by analyzing both sides of (4.7c), we can find the
degree of $\bar{v}_{s+1}$ must be zero too. Thus, we have

$\mathrm{deg}\,\bar{v}_{j}=0$

and then we may assume

$\bar{v}_{j}(n,\bar{x},\bar{t}_{m})=v_{j}(x,y,t_{m})+O(h),~{}~{}j=1,2,\cdots.$

Thus, for the pseudo-difference operator $\bar{L}$ given in (4.1) we have

$\mathrm{deg}\,\bar{L}=0,~{}~{}\bar{L}=L+O(h),$ (5.8)

and consequently,

$\bar{L}\bar{\phi}-\lambda\bar{\phi}=(L\phi-\lambda\phi)+O(h).$ (5.9)

In addition, for $\bar{A}_{1}$, using (5.5), we find

$\bar{A}_{1}=A_{1}+\frac{h}{2}(A_{2}-\partial_{y})+O(h^{2}),$ (5.10)

and it then follows that

$\displaystyle\bar{\phi}_{\bar{x}}-\bar{A}_{1}\bar{\phi}=\frac{h}{2}(\phi_{y}-A_{2}\phi)+O(h^{2})$
(5.11)

and

$\bar{L}_{\bar{x}}-[\bar{A}_{1},\bar{L}]=\frac{h}{2}(L_{y}-[A_{2},L])+O(h^{2}).$

Besides, for the first few flows of $\\{\bar{K}_{j}\\}$ and
$\\{\bar{\sigma}_{j}\\}$, one can also check that

$\displaystyle\mathrm{deg}\,\bar{K}_{j}=0,~{}~{}\bar{K}_{j}=K_{j}+O(h),~{}~{}(j=1,2,3),$
$\displaystyle\mathrm{deg}\,\bar{\sigma}_{j}=0,~{}~{}\bar{\sigma}_{j}=\sigma_{j}+O(h),~{}~{}(j=2,3).$

In the following, in order to obtain degrees of more elements, we recall some
rules for calculating degrees developed in [14]. Let us list them below.

######  Proposition 5.1.

For functions (or operators) $\bar{f}(\tilde{v})$ and $\bar{g}(\tilde{v})$, it
holds that

$\displaystyle\mathrm{deg\,}\bar{f}\cdot\bar{g}=\mathrm{deg\,}\bar{f}+\mathrm{deg\,}\bar{g},$
(5.12a)
$\displaystyle\mathrm{deg\,}(\bar{f}+\bar{g})\geq\min\\{\mathrm{deg\,}\bar{f},\mathrm{deg\,}\bar{g}\\}.$
(5.12b)

######  Proposition 5.2.

For functions $\bar{f}(\tilde{v}),\bar{g}(\tilde{v})\in\bar{S}[\tilde{v}]$ and
$f(v),g(v)\in S[v]$, suppose that there are relations in the continuum limit:

$\displaystyle\bar{f}(\tilde{v})=f(v)h^{i}+O(h^{i+1}),\quad\bar{g}(\tilde{v})=g(v)h^{j}+O(h^{j+1}).$

It then holds that

$\displaystyle\llbracket\bar{f}(\tilde{v}),\bar{g}(\tilde{v})\rrbracket_{\tilde{v}}=\llbracket
f(v),g(v)\rrbracket_{v}\,h^{i+j-1}+O(h^{i+j}),$ (5.13)
$\displaystyle\mathrm{deg\,}\llbracket\bar{f}(\tilde{v}),\bar{g}(\tilde{v})\rrbracket_{\tilde{v}}\geq\mathrm{deg\,}\bar{f}(\bar{u})+\mathrm{deg\,}\bar{g}(\bar{u})-1,$
(5.14)

where the subscripts $\tilde{v}$ and $v$ indicate the Gâteaux derivatives in
$\llbracket\cdot,\,\cdot\rrbracket$ are defined with respect to $\tilde{v}$
and $v$, respectively.

In addition, similar to Lemma 5.1, Lemma 5.2 and Proposition 5.6 in [14], for
the D$\Delta$mKP system, we have the following.

######  Lemma 5.1.

For the difference operator

$\bar{W}_{m}=\sum_{j=0}^{m-1}\bar{w}_{j}(\tilde{v})\Delta^{m-j},~{}~{}~{}\mathrm{with}~{}\bar{w}_{j}(\tilde{v})\in\bar{S}[\tilde{v}],~{}~{}j=0,1,m-1,$
(5.15)

if $\bar{W}_{m}$ satisfies $\left[\bar{W}_{m},\bar{L}\right]=0$, then
$\bar{W}_{m}=0$.

###### Proof.

Writing the left-hand side of $\left[\bar{W}_{m},\bar{L}\right]=0$ in terms of
the power of $\Delta$, from the highest order term, we find

$(E\bar{w}_{0})=\frac{(E^{m}\bar{v})}{\bar{v}}\bar{w}_{0}.$

This is a first order ordinary difference equation with respect to
$\bar{w}_{0}$. It allows a general solution

$\bar{w}_{0}=c\prod_{j=0}^{m-1}(E^{j}\bar{v}),$

and since $\bar{w}_{0}\to 0$ and $\bar{v}\to 1$ as $|n|\to\infty$, the
constant $c$ has to be zero, and thus $\bar{w}_{0}=0$. In the same way we can
successfully work out $\bar{w}_{j}=0$ for $j=1,2,\cdots,m-1$.

∎

Similarly, the following holds.

######  Lemma 5.2.

For the differential operator

$W_{m}=\sum_{j=0}^{m-1}w_{j}(v)\partial^{m-j},~{}~{}~{}\mathrm{with}~{}w_{j}(v)\in
S[v],~{}~{}j=0,1,2,\cdots,m-1,$

if $W_{m}$ satisfies $[W,L]=0$, then $W_{m}=0$.

With these two lemmas, for the difference operator $\bar{W}_{m}$ given in
(5.15), we can also prove that (cf. Proposition 5.6 in [14])

$\mathrm{deg\,}[\bar{W}_{m},\bar{L}]=0.$ (5.16)

The above two lemmas and relation (5.16) are important to achieve the
relations between $\bar{A}_{m},\bar{B}_{m}$ and $A_{m},B_{m}$ in the
continuous limit. Below we present these relations, which can be proved via
similar procedures as for Proposition 5.7 and 5.8 in [14].

######  Proposition 5.3.

In the continuum limit, we have

$\displaystyle{\rm{deg}}~{}\bar{A}_{m}=0,~{}~{}\bar{A}_{m}=A_{m}+O(h),~{}~{}m\geq
1,$ (5.17a)
$\displaystyle{\rm{deg}}~{}\bar{B}_{m}=0,~{}~{}\bar{B}_{m}=B_{m}+O(h),~{}~{}m\geq
2.$ (5.17b)

Thus, from the zero curvature representation (4.9) and (4.14) and noticing the
relation (5.10), we have

$\displaystyle\tilde{v}_{\bar{t}_{m}}=\bar{K}_{m}(\tilde{v})$
$\displaystyle=(\bar{A}_{m,\bar{x}}-[\bar{A}_{1},\bar{A}_{m}])\Delta^{-1},$
$\displaystyle=\frac{1}{2}(A_{m,y}-[A_{2},A_{m}])\partial_{x}^{-1}+O(h)$
$\displaystyle=K_{m}+O(h),$

i.e.

$\displaystyle{\rm{deg}}~{}\bar{K}_{m}=0,~{}~{}\bar{K}_{m}=K_{m}+O(h),~{}~{}m=1,2,\cdots.$
(5.18)

Similarly, from (4.14) we have

$\displaystyle{\rm{deg}}~{}\bar{\sigma}_{m}=0,~{}~{}\bar{\sigma}_{m}=\sigma_{m}+O(h),~{}~{}m=2,3,\cdots.$
(5.19)

###  5.3 Integrable structures and the symmetry constraint

####  5.3.1 Lax triads

For the Lax triad (4.3), in addition to (5.9) and (5.11), from (5.17a) we also
have

$\bar{\phi}_{\bar{t}_{m}}-\bar{A}_{m}\bar{\phi}=(\phi_{t_{m}}-A_{m}\phi)+O(h),$

and the compatibility condition (4.6) yields

$\displaystyle\bar{L}_{\bar{x}}-[\bar{A}_{1},\bar{L}]=\frac{h}{2}(L_{y}-[A_{2},L])+O(h^{2}),$
$\displaystyle\bar{L}_{\bar{t}_{m}}-[\bar{A}_{m},\bar{L}]=L_{t_{m}}-[A_{m},L]+O(h),$
$\displaystyle\bar{A}_{1,\bar{t}_{m}}-\bar{A}_{m,\bar{x}}+[\bar{A}_{1},\bar{A}_{m}]=\frac{h}{2}(A_{2,t_{m}}-A_{m,y}+[A_{2},A_{m}])+O(h^{2}).$

For the nonisospectral case, we have

$\displaystyle\bar{L}\bar{\phi}-\lambda\bar{\phi}=(L\phi-\lambda\phi)+O(h),$
$\displaystyle\bar{\phi}_{\bar{x}}-\bar{A}_{1}\bar{\phi}=\frac{h}{2}(\phi_{y}-A_{2}\phi)+O(h^{2}),$
$\displaystyle\bar{\phi}_{\bar{t}_{m}}-\bar{B}_{m}\bar{\phi}=(\phi_{t_{m}}-B_{m}\phi)+O(h),$

and

$\displaystyle\bar{L}_{\bar{x}}-[\bar{A}_{1},\bar{L}]=\frac{h}{2}(L_{y}-[A_{2},L])+O(h^{2}),$
$\displaystyle\bar{L}_{\bar{t}_{m}}-[\bar{B}_{m},\bar{L}]=L_{t_{m}}-[B_{m},L]+L^{m-1}+O(h),$
$\displaystyle\bar{A}_{1,\bar{t}_{m}}-\bar{B}_{m,\bar{x}}+[\bar{A}_{1},\bar{B}_{m}]=\frac{h}{2}(A_{2,t_{m}}-B_{m,y}+[A_{2},B_{m}])+O(h^{2}).$

####  5.3.2 Symmetries and algebra

The algebraic structures in (3.12) and (4.17) are apparently different.
However, they agree with each other in the continuum limits. To see that,
taking (4.17b) as an example, we just need to calculate degrees of its two
sides. Using Proposition 5.2, (5.18) and (5.19), we immediately arrive at

$\displaystyle\llbracket\bar{K}_{m},\bar{\sigma}_{n}\rrbracket_{\tilde{v}}-(m(h\bar{K}_{m+n-1}+\bar{K}_{m+n-2}))=\llbracket{K}_{m},{\sigma}_{n}\rrbracket_{v}-m{K}_{m+n-2}+O(h),$

which means (3.12b) is recovered from (4.17b) in the continuum limit. In the
following we skip presenting details and just list the results.

######  Proposition 5.4.

Except those relations involved with $\sigma_{1}$, in the continuum limits the
algebraic structure (3.12) of flows, symmetries (3.14) and their structure
(3.15) can be obtained from those of the D$\Delta$mKP case, i.e. (4.17),
(4.21) and (4.22).

####  5.3.3 Squared eigenfunction symmetry constraint

We can assume in our scheme of continuum limits, there are

$a_{n}=q(x,y,t_{m}),~{}~{}b_{n}=r(x,y,t_{m})$

and $\mathrm{deg\,}a_{n}=\mathrm{deg\,}b_{n}=0$. It then follows from (4.28)
that

$\bar{L}=h^{-1}(1+ha_{n}b_{n})\Delta+a_{n}\Delta^{-1}b_{n}\Delta=\partial_{x}+q\partial_{x}^{-1}r\partial_{x}+O(h).$

Next, in the spectral problem (4.36), we assume
$\bar{\varphi}_{1,n}=\varphi_{1}(x,y,t_{m}),\,\bar{\varphi}_{2,n}=\varphi_{2}(x,y,t_{m})$.
Then the spectral problem (3.20) can be recovered from (4.36) in the continuum
limit. In addition, for the recursion operator $\bar{T}$ one can check that

$\bar{T}=T+O(h).$

Thus, the spectral problem and CLL hierarchy obtained from the mKP system with
the squared symmetry constraint are also obtained from the continuum limits.

##  6 Conclusions

In this paper we gave a new formulation of the squared eigenfunction symmetry
constraint to convert the D$\Delta$mKP system to the discrete CLL spectral
problem and the differential-difference CLL hierarchy. This enabled us to
introduce a unified continuum limit scheme and build correspondence between
the continuous and the differential-difference mKP hierarchies and their
integrable structures. We also introduced nonisospectral flows and obtained
two sets of symmetries of the D$\Delta$mKP hierarchy and their Lie algebraic
structure, which recover all their continuous counterparts in the continuum
limits. With such correspondence, we have provided a relatively complete
profile of the D$\Delta$mKP.

Note that the two different formulations of independent variables in the
squared eigenfunction symmetry constraint convert the D$\Delta$mKP system to
the relativistic Toda ( cf.[9]) and the differential-difference CLL,
respectively. It would be worthy to understand the link between these two
(1+1)-dimensional differential-difference integrable systems. In addition, the
discrete CLL spectral problem (4.37) is a Darboux transformation of the
continuous CLL spectral problem (see Appendix B). In this context, one can
investigate the connections between the discrete CLL spectral problem and the
fully discrete (potential) mKP equation, which may lead to new algebraic
geometry solution to the later, cf.[26].

### Acknowledgements

This project is supported by the NSF of China (Nos. 12271334, 12126352,
12126343), the Science and technology innovation plan of Shanghai (No.
20590742900) and the Fundamental Research Funds for the Inner Mongolia Normal
University (No. 2022JBBJ008).

##  Appendix A Calculation of formula (4.32)

By substitution of (4.31) the right-hand side of (4.7c) and take $s=i$ yields

$\displaystyle\eqref{dmkp-vs}|_{\mathrm{r.h.s.}}$ $\displaystyle=$
$\displaystyle(-1)^{s}h^{-s}(ha_{n}b_{n}+1)(a_{n+1}-a_{n})E^{-s-1}\Delta^{s}b_{n}+(-1)^{s}h^{-s}a_{n}E^{-s-1}\Delta^{s}(ha_{n}b_{n}+1)(b_{n}-b_{n-1})$
$\displaystyle~{}~{}-(ha_{n}b_{n}+1)\Delta(-1)^{s}h^{-s}a_{n}E^{-s-1}\Delta^{s}b_{n}$
$\displaystyle~{}~{}+\sum_{i=1}^{s}(-1)^{s+1}\mathrm{C}_{s}^{i-1}h^{-s}(a_{n}E^{-i-1}\Delta^{i}b_{n})E^{-s-1}\Delta^{s-i+1}(ha_{n}b_{n}+1)$
$\displaystyle=$ $\displaystyle(-1)^{s}h^{-s}(A+B+C),$

where $\eqref{dmkp-vs}|_{\mathrm{r.h.s.}}$ stands for the right-hand side of
(4.7c), and

$\displaystyle A=$ $\displaystyle(\Delta a_{n})E^{-s-1}\Delta^{s}b_{n}-\Delta
a_{n}E^{-s-1}\Delta^{s}b_{n}+a_{n}E^{-s-2}\Delta^{s+1}b_{n},$ $\displaystyle
B=$ $\displaystyle ha_{n}b_{n}(\Delta
a_{n})E^{-s-1}\Delta^{s}b_{n}-ha_{n}b_{n}\Delta
a_{n}E^{-s-1}\Delta^{s}b_{n}+ha_{n}E^{-s-1}\Delta^{s}a_{n}b_{n}(\Delta
b_{n-1}),$ $\displaystyle C=$
$\displaystyle-h\sum_{i=1}^{s}\mathrm{C}_{s}^{i-1}(a_{n}E^{-i-1}\Delta^{i}b_{n})E^{-s-1}\Delta^{s-i+1}a_{n}b_{n}.$

For the terms in $A$, using the formula

$\Delta f_{n}g_{n}=(Ef_{n})\Delta g_{n}+(\Delta f_{n})g_{n},$ (A.1)

the first two terms together yield

$-(Ea_{n})E^{-s-1}\Delta^{s+1}b_{n}=-Ea_{n}E^{-s-2}\Delta^{s+1}b_{n}.$

Thus we have

$A=-\Delta a_{n}E^{-s-2}\Delta^{s+1}b_{n}.$

For the terms in $B$, again, by using formula (A.1), the first two terms yield

$-ha_{n}b_{n}(Ea_{n})E^{-s-1}\Delta^{s}b_{n}=-ha_{n}b_{n}Ea_{n}E^{-s-2}\Delta^{s}b_{n}.$

For the third term in $B$, using (2.4) we have

$\displaystyle ha_{n}E^{-s-1}\Delta^{s}a_{n}b_{n}(\Delta
b_{n-1})=ha_{n}\Delta^{s}(\Delta b_{n-s-2})(E^{-s-1}a_{n}b_{n})$
$\displaystyle=$ $\displaystyle
h\sum_{i=0}^{s}\mathrm{C}_{s}^{i}a_{n}(\Delta^{i+1}b_{n-i-2})(\Delta^{s-i}E^{-s-1}a_{n}b_{n})$
$\displaystyle=$ $\displaystyle
h\sum_{i=0}^{s}\mathrm{C}_{s}^{i}a_{n}(\Delta^{i+1}E^{-i-2}b_{n})(\Delta^{s-i}E^{-s-1}a_{n}b_{n}).$

Replacing $i$ with $i-1$ in the above equation we have

$\displaystyle ha_{n}E^{-s-1}\Delta^{s}a_{n}b_{n}(\Delta b_{n-1})$
$\displaystyle=$ $\displaystyle
h\sum_{i=1}^{s+1}\mathrm{C}_{s}^{i-1}a_{n}(\Delta^{i}E^{-i-1}b_{n})(\Delta^{s-i+1}E^{-s-1}a_{n}b_{n})$
$\displaystyle=$ $\displaystyle
h(a_{n}\Delta^{s+1}E^{-s-2}b_{n})(E^{-s-1}a_{n}b_{n})-C.$

Thus,

$\displaystyle\eqref{dmkp-vs}|_{\mathrm{r.h.s.}}$ $\displaystyle=$
$\displaystyle(-1)^{s}h^{-s}(A+B+C)$ $\displaystyle=$
$\displaystyle(-1)^{s+1}h^{-s}[(ha_{n}b_{n}+1)E(a_{n}E^{-s-2}\Delta^{s+1}b_{n})-(E^{-s-1}(ha_{n}b_{n}+1))(a_{n}E^{-s-2}\Delta^{s+1}b_{n})],$

i.e. the equation (4.31).

##  Appendix B On the discrete CLL spectral problem and equation

The discrete CLL spectral problem (4.37), i.e.

$\left(\begin{array}[]{c}\psi_{1,n+1}\\\
\psi_{2,n+1}\end{array}\right)=\left(\begin{array}[]{cc}-h\eta^{2}+1+ha_{n}b_{n+1}&ha_{n}\\\
h\eta^{2}b_{n+1}&1\end{array}\right)\left(\begin{array}[]{c}\psi_{1,n}\\\
\psi_{2,n}\end{array}\right),$ (B.1)

was found as early as in 1983 in [11], where we have replaced $b_{n}$ with
$b_{n+1}$. It is compatible with the continuous spectral problem

$\left(\begin{array}[]{c}\psi_{1,n}\\\
\psi_{2,n}\end{array}\right)_{\bar{x}}=\left(\begin{array}[]{cc}-\eta^{2}+a_{n}b_{n}&\eta
a_{n}\\\ \eta b_{n}&0\end{array}\right)\left(\begin{array}[]{c}\psi_{1,n}\\\
\psi_{2,n}\end{array}\right).$ (B.2)

The later was also given in [11] to lead to the unreduced CLL equations. In
addition, (B.2) is also gauge equivalent to the usual CLL spectral problem
(cf.[25])

$\left(\begin{array}[]{c}\phi_{1,n}\\\
\phi_{2,n}\end{array}\right)_{\bar{x}}=\left(\begin{array}[]{cc}\frac{1}{2}(-\eta^{2}+a_{n}b_{n})&\eta
a_{n}\\\ \eta
b_{n}&-\frac{1}{2}(-\eta^{2}+a_{n}b_{n})\end{array}\right)\left(\begin{array}[]{c}\phi_{1,n}\\\
\phi_{2,n}\end{array}\right).$ (B.3)

by taking
$\phi_{j}=\psi_{j}e^{\frac{1}{2}(\eta^{2}\bar{x}^{2}-\partial_{\bar{x}}^{-1}a_{n}b_{n})}$.
The compatibility between (B.1) and (B.2) gives rise to a differential-
difference CLL equation

$a_{n,\bar{x}}=h^{-1}(1+ha_{n}b_{n})(a_{n+1}-a_{n}),~{}~{}b_{n,\bar{x}}=h^{-1}(1+ha_{n}b_{n})(b_{n}-b_{n-1}),$

i.e. (4.30). Meanwhile, the compatibility allows us to view (B.1) as a Darboux
transformation of the CLL spectral problem (B.2).

It is well known that transformation is one of means of discretization [20].
Based on the above relations, if we denote the Darboux matrix in (B.1) as

$M(\gamma,h,a_{n,m},b_{n+1,m})=\left(\begin{array}[]{cc}-h\gamma+1+ha_{n,m}b_{n+1,m}&ha_{n,m}\\\
h\gamma b_{n+1,m}&1\end{array}\right),$ (B.4)

then the compatibility of

$\Psi_{n+1,m}=M(\gamma,p,a_{n,m},b_{n+1,m})\Psi_{n,m},~{}~{}\Psi_{n,m+1}=M(\gamma,q,a_{n,m},b_{n,m+1})\Psi_{n,m},$

where $\Psi_{n,m}=(\psi_{1}(n,m),\psi_{2}(n,m))^{T}$, yields the discrete CLL
equation [11]

$\displaystyle
pa_{n,m+1}-qa_{n+1,m}-(p-q)a_{n,m}=pq(a_{n+1,m}-a_{n,m+1})a_{n,m}b_{n+1,m+1},$
$\displaystyle
qb_{n,m+1}-qb_{n+1,m}+(p-q)b_{n+1,m+1}=pq(b_{n+1,m}-b_{n,m+1})a_{n,m}b_{n+1,m+1}.$

It is also notable that the discrete CLL spectral problem (4.33) is a direct
discretization of the continuous one (B.2) by just replacing
$(\psi_{i,n})_{x}$ with $(\psi_{i,n+1}-\psi_{i,n})/h$ for $i=1,2$. This is
interesting because the Merola-Ragnisco-Tu spectral problem, which can be
obtained from the squared-eigenfunction-symmetry-constrained D$\Delta$KP
system, can also be viewed as a direct discretization as well as a Darboux
transformation of the AKNS spectral problem, see Eq.(1.5) and Appendix B in
[8].

## References

  * [1] Ablowitz M.J., Ladik J.F., Nonlinear differential-difference equations, J. Math. Phys., 16 (1975) 598-603.
  * [2] Adler V.E., Yamilov R.I., Explicit auto-transformations of integrable chains, J. Phys. A: Math. Gen., 27 (1994) 477-492.
  * [3] Cao C.W., Nonlinearization of the Lax system for AKNS hierarchy, Sci. China A, A33 (1990) 528-536.
  * [4] Chen D.Y., $k$-constraint for the modified Kadomtsev-Petviashvili system, J. Math. Phys., 43 (2002) 1956-1965.
  * [5] Chen D.Y., Introduction to Soliton Theory, Sciences Pub. House, Beijing (2006).
  * [6] Chen D.Y., Xin H.W., Zhang D.J., Lie algebraic structures of some (1+2)-dimensional Lax integrable systems, Chaos, Solitons & Fractals, 15 (2003) 761-770.
  * [7] Chen D.Y., Zhang D.J., Lie algebraic structures of (1+1)-dimensional Lax integrable systems, J. Math. Phys., 37(12), (1996) 5524-5538.
  * [8] Chen K., Deng X., Zhang D.J., Symmetry constraint of the differential-difference KP hierarchy and a second discretization of the ZS-AKNS system, J. Nonl. Math. Phys., Supl. 24 (2017) 18-35.
  * [9] Chen K., Zhang C., Zhang D.J., Squared eigenfunction symmetry of the D$\Delta$mKP hierarchy and its constraint, Stud. Appl. Math., 147 (2021) 752-791.
  * [10] Cheng Y., Li Y.S., The constraint of the Kadomtsev-Petviashvili equation and its special solutions, Phys. Lett. A, 157 (1991) 22-26.
  * [11] Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations: IV, J. Phys. Soc. Japan, 52 (1983) 761-765.
  * [12] Deng X., Chen K., Zhang D.J., Lax triad approach to symmetries of scalar modified Kadomtsev-Petviashvili hierarchy, Commun. Theor. Phys., 67 (2017) 131-136.
  * [13] Elaydi S., An Introduction to Difference Equations, 3rd ed., Springer-Verlag, New York (2005).
  * [14] Fu W., Huang L., Tamizhmani K.M., Zhang D.J., Integrability properties of the differential-difference Kadomtsev-Petviashvili hierarchy and continuum limits, Nonlinearity, 26 (2013) 3197-3229.
  * [15] Fu W., Qiao Z., Sun J., Zhang D.J., Continuous correspondence of conservation laws of the semi-discrete AKNS system, J. Nonlin. Math. Phys., 22(3) (2015) 321-341.
  * [16] Fuchssteiner B., Fokas A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica 4D., 4 (1981) 47-66.
  * [17] Khanizadeh F., Mikhailov A.V., Wang J.P., Darboux transformations and recursion operators for differential-difference equations, Theor. Math. Phys., 177 (2013) 1606-1654.
  * [18] Konopelchenko B., Sidorenko J., Strampp W., (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems, Phys. Lett. A, 157 (1991) 17-21.
  * [19] Konopelchenko B., Strampp W., The AKNS hierarchy as symmetry constraint of the KP hierarchy, Inverse Probl., 7 (1991) L17-24.
  * [20] Levi D., Benguria R., Bäcklund transformations and nonlinear differential-difference equations, Proc. Natl. Acad. Sci. USA, 77 (1980) 5025-5027.
  * [21] Ma W.X., Fuchssteiner B., Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations, J. Math. Phys., 40 (1999) 2400-2418.
  * [22] Merola I., Ragnisco O., Tu G.Z., A novel hierarchy of integrable lattices, Inverse Probl., 10 (1994) 1315-1334.
  * [23] Morosi C., Pizzocchero L., On the continuous limit of integrable lattices I, The Kac–Moerbeke system and KdV theory, Commun. Math. Phys., 180 (1996) 505-528.
  * [24] Tsuchida T., Integrable discretizations of derivative nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002) 7827-7847.
  * [25] Wadati M., Sogo K., Gauge transformations in soliton theory, J. Phys. Soc. Jpn., 52 (1983) 394-398.
  * [26] Xu X.X., Cao C.W., Zhang D.J., Algebro-geometric solutions to the lattice potential modified Kadomtsev-Petviashvili equation, J. Phys. A: Math. Theor., 55 (2022) 375201 (28pp).
  * [27] Zhang D.J., Chen D.Y., Some general formulas in the Sato theory, J. Phys. Soc. Jpn., 72 (2003) 448-449.
  * [28] Zhang D.J., Chen S.T., Symmetries for the Ablowitz-Ladik hierarchy: II. Integrable discrete nonlinear Schrödinger equation and discrete AKNS hierarchy, Stud. Appl. Math., 125 (2010) 419-443.