Multi-Component Extension of CAC Systems

Multi-Component Extension of CAC Systems

Dan-Da ZHANG †, Peter H. VAN DER KAMP ‡ and Da-Jun ZHANG §

D.-D. Zhang, P.H. van der Kamp and D.-J. Zhang

† School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R.
China<EMAIL_ADDRESS>
‡ Department of Mathematics and Statistics, La Trobe University, Victoria
3086, Australia<EMAIL_ADDRESS>
§ Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
<EMAIL_ADDRESS>
Received December 07, 2019, in final form June 14, 2020; Published online July
01, 2020

In this paper an approach to generate multi-dimensionally consistent
$N$-component systems is proposed. The approach starts from scalar multi-
dimensionally consistent quadrilateral systems and makes use of the cyclic
group. The obtained $N$-component systems inherit integrable features such as
Bäcklund transformations and Lax pairs, and exhibit interesting aspects, such
as nonlocal reductions. Higher order single component lattice equations (on
larger stencils) and multi-component discrete Painlevé equations can also be
derived in the context, and the approach extends to $N$-component
generalizations of higher dimensional lattice equations.

lattice equations; consistency around the cube; cyclic group; multi-component;
Lax pair; Bäcklund transformation; nonlocal

37K60

##  1 Introduction

Integrability of nonlinear partial differential equations is of sovereign
importance in the study of soliton theory. For discrete equations, especially
quadrilateral equations, consistency around the cube (CAC) [12, 43, 48]
provides an interpretation of integrability. A classification of discrete
integrable equations with scalar-valued fields was obtained by
Adler–Bobenko–Suris (ABS) in [4] and more general classes of scalar equations
where classified in [5, 13]. The CAC property is also applicable to higher
order lattice equations by introducing suitable multi-component forms [24, 52,
57]. Further examples of multi-component integrable systems can be found in
the literature [34, 35, 36, 50]. However, general classification results have
not been obtained yet.

For CAC equations, the equations on the side faces of the consistent cube can
be interpreted as an auto-Bäcklund transformation (BT). CAC also enables one
to construct Lax pairs [14, 43], as well as to find soliton solutions [27, 28,
29]. Whereas the classification in [4] requires the equations on the six faces
of the cube to be the same, in [7] alternative auto-BTs were given for several
ABS equations, giving rise to consistent systems where the equations on the
side faces are different from the equation on the top and bottom of the cube.
Moreover, (non-auto) BTs between distinct equations were also provided,
corresponding to consistent systems with different equations on the top and
the bottom faces of the cube. Other classifications of CAC systems with
asymmetrical properties, other relaxations, and 3D affine linear lattice
equations with 4D consistency have been also considered [5, 6, 13, 25, 53]. We
will refer to a system of equations which is consistent on a cube, as a cube
system.

In Appendix B of the PhD. Thesis of J. Atkinson [8], multi-component versions
of CAC scalar systems were introduced under the name “The trivial Toeplitz
extension”. Atkinson trivially extends a scalar equation for a field $u$ to an
$N$-component system for fields $u_{1},\ldots,u_{N}$, and then applies the
transformation

$u_{i}(l,m)\rightarrow u_{i+l+m\ {\rm mod}\ N}(l,m).$

He remarks that the resulting system is trivial (indeed the inverse
transformation decouples it), and that extensions of multi-dimensional
consistent scalar equations are multi-dimensional consistent (and hence would
emerge in classifications of multi-component discrete integrable systems).

In [22], $(N=2)$-component ABS equations resulting from such an extension were
investigated. Here the CAC property (with affine linearity, D4 symmetry and
the tetrahedron property) of these coupled systems was established, and
solutions provided.

In this paper, we consider more general (but still trivial) extensions of not
only scalar equations, but also of cube systems. For scalar equations these
extension take the (non-Toeplitz) form

$u_{i}(l,m)\rightarrow u_{i+al+bm\ {\rm mod}\ N}(l,m),\qquad\text{with}\quad
a,b\in\mathbb{Z}.$

We will provide Lax pairs, BTs, solutions and reductions for multi-component
extensions of CAC lattice equations and cube systems, as well as multi-
component extensions of 3D lattice equations with 4D consistency.

The paper is organized as follows. In Section 2 we construct multi-component
extensions of two distinguished kinds of systems, CAC cube systems (Section
2.1) and CAC lattice systems (quadrilateral equations in Section 2.2, and
higher dimensional equations in Section 2.3). CAC lattice systems can be
consistently posed on a lattice, whereas CAC cube systems need to be
accompanied by reflected cube systems and posed on lattices similar to so-
called black and white lattices [5, 58]. Examples include multi-component
extensions of a Boll cube system [13] (Section 2.1), an equation from the ABS
list [4], and (auto and non-auto) Bäcklund transformations [7] (Section
2.2.1). We show that multi-component extensions admit Lax-pairs in Section
2.2.2. Assuming D4 symmetry we count the number of different non-decoupled
$N$-component extensions in Section 2.2.3. In Section 3 we provide several
kinds of reductions. Nonlocal reductions are given in Section 3.1, reductions
to higher order scalar equations are given in Section 3.2, and a reduction to
a multi-component Painlevé lattice equation is considered in Section 3.3. In
Section 4.1 some particular solutions are given. These can be constructed from
$N$ solutions of the scalar equation. A solution for a nonlocal equation is
provided in Section 4.2. In Section 5 we summarize and discuss the results,
and we point out that particular examples of $N$-component generalised systems
have appeared in the literature in different contexts.

##  2 Multi-component extension of CAC systems

In this section we construct multi-component systems that are consistent
around the cube, a.k.a. CAC. We first focus on a single 3D cube with
consistent face equations.

###  2.1 Multi-component CAC cube systems

We will be concerned with quadrilateral equations of the form

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=0,$ (2.1)

where we use $\widetilde{u}$ and $\widehat{u}$ to denote shifts of $u$ in two
different directions. Posing six such equations on a cube yields a general
type of system of the form

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
Q^{*}\big{(}\overline{u},\overline{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.2a) $\displaystyle
A\big{(}u,\widetilde{u},\overline{u},\overline{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
A^{*}\big{(}\widehat{u},\widehat{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.2b) $\displaystyle
B\big{(}u,\widehat{u},\overline{u},{\overline{\widehat{u}}}\big{)}=0,\qquad$
$\displaystyle
B^{*}\big{(}\widetilde{u},\widehat{\widetilde{u}},\overline{\widetilde{u}},\overline{\widehat{\widetilde{u}}}\big{)}=0.$
(2.2c)

We assume the functions $Q$, $A$, $B$, $Q^{*}$, $A^{*}$, $B^{*}$ are affine
linear with respect to each variable, and the symbols
$\widetilde{u},\widehat{u},\overline{u},\dots,\overline{\widehat{\widetilde{u}}}$
represent the values of the field at the vertices of the cube, see Fig. 1(a).
Each equation may depend on additional (edge) parameters but we omit these.

The system (2.2) is called CAC if the three values for
$\overline{\widehat{\widetilde{u}}}$ calculated from the three starred
equations coincide for arbitrary initial data $u$, $\widetilde{u}$,
$\widehat{u}$, $\overline{u}$, i.e.,

$\displaystyle\overline{\widehat{\widetilde{u}}}=F\big{(}u,\widetilde{u},\widehat{u},\overline{u}\big{)}.$
(2.3)

In order to get a multi-component extension of the system (2.2), we consider
the vertex symbols $u$, $\widetilde{u}$, $\widehat{u}$,
$\widehat{\widetilde{u}}$, $\overline{\widetilde{u}}$,
${\overline{\widehat{u}}}$, $\overline{\widehat{\widetilde{u}}}$ to be
$N\times N$ diagonal matrices, e.g.,

$\displaystyle
u=\operatorname{Diag}(u_{1},u_{2},\dots,u_{N}),\qquad\widetilde{u}=\operatorname{Diag}(\widetilde{u}_{1},\widetilde{u}_{2},\dots,\widetilde{u}_{N}).$
(2.4)

We introduce a cyclic group using the generator $\sigma=\sigma_{N}$, defined
as the $N\times N$ matrix with elements given by

$\displaystyle(\sigma_{N})_{i,j}=\begin{cases}1,&i+1\equiv j\ \text{mod}\
N,\\\ 0,&\text{otherwise}.\end{cases}$ (2.5)

Thus, a cyclic transformation of $u$ (a permutation of the components on the
diagonal) can be denoted by

$\displaystyle u\mapsto T_{k}u=\sigma^{k}u\sigma^{-k},\qquad k\in\mathbb{Z}\
({\rm mod}\ N).$ (2.6)

Note that $\sigma^{N}=I_{N}$ which is the $N\times N$ identity matrix.

######  Lemma 2.1.

If the scalar cube system (2.2) is CAC, the following multi-component cube
system

$\displaystyle
Q\big{(}u,T_{k_{1}}\widetilde{u},T_{k_{2}}\widehat{u},T_{k_{4}}\widehat{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
Q^{*}\big{(}T_{k_{3}}\overline{u},T_{k_{5}}\overline{\widetilde{u}},T_{k_{6}}{\overline{\widehat{u}}},T_{k_{7}}\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.7a) $\displaystyle
A\big{(}u,T_{k_{1}}\widetilde{u},T_{k_{3}}\overline{u},T_{k_{5}}\overline{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
A^{*}\big{(}T_{k_{2}}\widehat{u},T_{k_{4}}\widehat{\widetilde{u}},T_{k_{6}}{\overline{\widehat{u}}},T_{k_{7}}\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.7b) $\displaystyle
B\big{(}u,T_{k_{3}}\overline{u},T_{k_{2}}\widehat{u},T_{k_{6}}{\overline{\widehat{u}}}\big{)}=0,\qquad$
$\displaystyle
B^{*}\big{(}T_{k_{1}}\widetilde{u},T_{k_{5}}\overline{\widetilde{u}},T_{k_{4}}\widehat{\widetilde{u}},T_{k_{7}}\overline{\widehat{\widetilde{u}}}\big{)}=0$
(2.7c)

is CAC as well, where the variables $u$, $\widetilde{u}$, $\widehat{u}$,
$\widehat{\widetilde{u}}$, $\overline{\widetilde{u}}$,
${\overline{\widehat{u}}}$, $\overline{\widehat{\widetilde{u}}}$ are diagonal
matrices as in (2.4), and the $T_{k_{i}}$ are cyclic transformations as
defined in (2.6), with $k_{i}\in\mathbb{Z}$ $({\rm mod}~{}N)$.

###### Proof.

Note that the equations in the system (2.2) are affine linear and the
variables denote the values of fields at the vertices of the cube. Replacing
these variables by diagonal matrices they remain commutative. If the system
(2.2) is CAC and $\overline{\widehat{\widetilde{u}}}$ has a unique expression
(2.3), it then follows that the system (2.7) is CAC in the sense that
$\overline{\widehat{\widetilde{u}}}$ has a unique expression

$\displaystyle
T_{k_{7}}\overline{\widehat{\widetilde{u}}}=F\big{(}u,T_{k_{1}}\widetilde{u},T_{k_{2}}\widehat{u},T_{k_{3}}\overline{u}\big{)}$

in which the function $F$ is the same as in equation (2.3), and the fields at
the vertices are relabelled. ∎

3(50,30)(0,0) 3(50,30)(150,30) 3(50,30)(50,130) 2(25, 65)(125, 65) 2(75,
15)(75, 115) 2(50, 50)(100, 80)
$B$$B^{*}$$Q$$Q^{*}$$A$$A^{*}$$u$$\widetilde{u}$$\widehat{u}$$\overline{u}$$\widehat{\widetilde{u}}$${\overline{\widehat{u}}}$$\overline{\widetilde{u}}$$\overline{\widehat{\widetilde{u}}}$(a)
CAC scalar cube system (2.2)

3(50,30)(0,0) 3(50,30)(150,30) 3(50,30)(50,130) 2(25, 65)(125, 65) 2(75,
15)(75, 115) 2(50, 50)(100, 80)
$B$$B^{*}$$Q$$Q^{*}$$A$$A^{*}$$u$$T_{k_{1}}\widetilde{u}$$T_{k_{2}}\widehat{u}$$T_{k_{3}}\overline{u}$$T_{k_{4}}\widehat{\widetilde{u}}$$T_{k_{6}}{\overline{\widehat{u}}}$$T_{k_{5}}\overline{\widetilde{u}}$$T_{k_{7}}\overline{\widehat{\widetilde{u}}}$(b)
CAC multi-component cube system (2.7)

Figure 1: Single cubes with equations defined on their faces.

In Fig. 1 the cube on the right has equation
$A\big{(}u,T_{k_{1}}\widetilde{u},T_{k_{3}}\overline{u},T_{k_{5}}\overline{\widetilde{u}}\big{)}=0$
on its front face, which can be thought of in two distinct ways: as a
relabeling of the variable names (which we do in the proof), or, as
introducing coupling between different components of the fields at the
vertices (which yield multi-component coupled systems of equations).

As an example we consider a cube system of Boll [13, equations (3.31),
(3.32)]. With $N=2$, denoting the field components by $u$, $v$ (instead of
$u_{1}$, $u_{2}$), and taking $k_{i}=\frac{1}{2}(1-(-1)^{i})$ we find the
following 2-component cube system (written as vector system instead of as a
matrix system):

$\displaystyle
Q=\begin{pmatrix}\widehat{u}\widehat{\widetilde{u}}\delta_{{1}}+\widehat{u}\widetilde{v}\delta_{{2}}+u\widehat{u}+\widetilde{v}\widehat{\widetilde{u}}\\\
\widetilde{u}\widehat{v}\delta_{{2}}+\widehat{v}\widehat{\widetilde{v}}\delta_{{1}}+\widetilde{u}\widehat{\widetilde{v}}+v\widehat{v}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},$ $\displaystyle
A=\begin{pmatrix}\alpha\overline{v}\overline{\widetilde{v}}\delta_{{1}}+\overline{v}\widetilde{v}\delta_{{2}}+u\overline{v}+\widetilde{v}\overline{\widetilde{v}}\\\
\alpha\overline{u}\overline{\widetilde{u}}\delta_{{1}}+\overline{u}\widetilde{u}\delta_{{2}}+v\overline{u}+\widetilde{u}\overline{\widetilde{u}}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},$ $\displaystyle
B=\begin{pmatrix}u\overline{v}+\widehat{u}{\overline{\widehat{u}}}-\alpha\big{(}u\widehat{u}+{\overline{\widehat{u}}}\overline{v}\big{)}+\delta_{{1}}\delta_{{2}}\big{(}{\alpha}^{2}-1\big{)}\widehat{u}\overline{v}\\\
v\overline{u}+\widehat{v}{\overline{\widehat{v}}}-\alpha\big{(}\overline{u}{\overline{\widehat{v}}}+v\widehat{v}\big{)}+\delta_{{1}}\delta_{{2}}\big{(}{\alpha}^{2}-1\big{)}\widehat{v}\overline{u}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},$ $\displaystyle
Q^{*}=\begin{pmatrix}\delta_{{1}}\overline{v}\overline{\widetilde{v}}+\overline{v}\overline{\widehat{\widetilde{v}}}\delta_{{2}}+{\overline{\widehat{u}}}\overline{v}+\overline{\widehat{\widetilde{v}}}\overline{\widetilde{v}}\\\
\delta_{{1}}\overline{u}\overline{\widetilde{u}}+\overline{u}\overline{\widehat{\widetilde{u}}}\delta_{{2}}+\overline{u}{\overline{\widehat{v}}}+\overline{\widehat{\widetilde{u}}}\overline{\widetilde{u}}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},$ $\displaystyle
A^{*}=\begin{pmatrix}\delta_{{1}}\alpha\widehat{u}\widehat{\widetilde{u}}+\widehat{u}\overline{\widehat{\widetilde{v}}}\delta_{{2}}+\widehat{u}{\overline{\widehat{u}}}+\overline{\widehat{\widetilde{v}}}\widehat{\widetilde{u}}\\\
\delta_{{1}}\alpha\widehat{v}\widehat{\widetilde{v}}+\widehat{v}\overline{\widehat{\widetilde{u}}}\delta_{{2}}+\overline{\widehat{\widetilde{u}}}\widehat{\widetilde{v}}+\widehat{v}{\overline{\widehat{v}}}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},$ $\displaystyle
B^{*}=\begin{pmatrix}\widetilde{v}\overline{\widetilde{v}}+\overline{\widehat{\widetilde{v}}}\widehat{\widetilde{u}}-\alpha\big{(}\widetilde{v}\widehat{\widetilde{u}}+\overline{\widehat{\widetilde{v}}}\overline{\widetilde{v}}\big{)}\\\
\widetilde{u}\overline{\widetilde{u}}+\overline{\widehat{\widetilde{u}}}\widehat{\widetilde{v}}-\alpha\big{(}\widetilde{u}\widehat{\widetilde{v}}+\overline{\widehat{\widetilde{u}}}\overline{\widetilde{u}}\big{)}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},$

where $\alpha$, $\delta_{1}$, $\delta_{2}$ are parameters. It is consistent
around the cube.

######  Remark 2.2.

As in the scalar case, one can not straightforwardly impose the cube system
(2.7) on the $\mathbb{Z}^{3}$ lattice. It needs to be accompanied by 7 other
cube systems which are obtained from the original one by reflections. If
$R_{i}$ denotes a reflection in the $i$th direction, e.g., application of
$R_{1}$ gives the cube system depicted in Fig. 2, then on the cube with center
$\big{(}n+\frac{1}{2},m+\frac{1}{2},l+\frac{1}{2}\big{)}$ one should impose
the cube system reflected by $R_{1}^{n}R_{2}^{m}R_{3}^{l}$.

3(50,30)(0,0) 3(50,30)(150,30) 3(50,30)(50,130) 2(25, 65)(125, 65) 2(75,
15)(75, 115) 2(50, 50)(100, 80)
$B^{*}$$B$$Q$$Q^{*}$$A$$A^{*}$$T_{k_{1}}u$$\widetilde{u}$$T_{k_{4}}\widehat{u}$$T_{k_{5}}\overline{u}$$T_{k_{2}}\widehat{\widetilde{u}}$$T_{k_{7}}{\overline{\widehat{u}}}$$T_{k_{3}}\overline{\widetilde{u}}$$T_{k_{6}}\overline{\widehat{\widetilde{u}}}$
Figure 2: CAC system (2.7) reflected in $\sim$ direction.

###  2.2 Multi-component CAC lattice systems

In this section we consider CAC lattice systems. The difference with the
previous section is that we now require that the lattice equation $Q=0$ can be
consistently imposed on the entire $\mathbb{Z}^{2}$ lattice, together with the
cubes they are part of. The consequence of this requirement is two-fold:

  * •

we have to restrict ourselves to cube systems with $A=A^{*}$ and $B=B^{*}$,

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
Q^{*}\big{(}\overline{u},\overline{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.8a) $\displaystyle
A\big{(}u,\widetilde{u},\overline{u},\overline{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
A\big{(}\widehat{u},\widehat{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.8b) $\displaystyle
B\big{(}u,\widehat{u},\overline{u},{\overline{\widehat{u}}}\big{)}=0,\qquad$
$\displaystyle
B\big{(}\widetilde{u},\widehat{\widetilde{u}},\overline{\widetilde{u}},\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.8c)

because consistent cubes with $Q=0$ on the bottom face need to be glued
together so that their common faces carry same equation, $A=0$ or $B=0$. Note
that we want to allow for the possibility that $Q\neq Q^{*}$, so that (non-
auto) Bäcklund transformations are included in the same framework.

  * •

we need to restrict the values the parameters of the extension, $k_{i}$, can
acquire.

######  Theorem 2.3.

Suppose that the system (2.8) is CAC in the sense
$\overline{\widehat{\widetilde{u}}}$ is uniquely determined by (2.3) in terms
of initial values $u$, $\widetilde{u}$, $\widehat{u}$, $\overline{u}$.
Extending $u$ to be a diagonal matrix (2.4), the system

$\displaystyle
Q\big{(}u,T_{a}\widetilde{u},T_{b}\widehat{u},T_{a+b}\widehat{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
Q^{*}\big{(}T_{c}\overline{u},T_{a+c}\overline{\widetilde{u}},T_{b+c}{\overline{\widehat{u}}},T_{a+b+c}\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.9a) $\displaystyle
A\big{(}u,T_{a}\widetilde{u},T_{c}\overline{u},T_{a+c}\overline{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
A\big{(}T_{b}\widehat{u},T_{a+b}\widehat{\widetilde{u}},T_{b+c}{\overline{\widehat{u}}},T_{a+b+c}\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.9b) $\displaystyle
B\big{(}u,T_{c}\overline{u},T_{b}\widehat{u},T_{b+c}{\overline{\widehat{u}}}\big{)}=0,\qquad$
$\displaystyle
B\big{(}T_{a}\widetilde{u},T_{a+c}\overline{\widetilde{u}},T_{a+b}\widehat{\widetilde{u}},T_{a+b+c}\overline{\widehat{\widetilde{u}}}\big{)}=0$
(2.9c)

is CAC as well, where $a,b,c\in\mathbb{Z}$ $({\rm mod}~{}N)$, and can be
consistently defined on $\mathbb{Z}^{2}\otimes\\{0,1\\}$.

###### Proof.

The CAC property follows directly from Lemma 2.1. We have to show that we can
consistently define the same cube system on neighboring cubes.

3(50,30)(0,0) 3(50,30)(150,30) 3(50,30)(50,130) 2(25, 65)(125, 65) 2(75,
15)(75, 115) 2(50, 50)(100, 80)
$B$$B$$Q$$Q^{*}$$A$$A$$u$$T_{a}\widetilde{u}$$T_{b}\widehat{u}$$T_{c}\overline{u}$$T_{a+b}\widehat{\widetilde{u}}$$T_{b+c}{\overline{\widehat{u}}}$$T_{a+c}\overline{\widetilde{u}}$$T_{a+b+c}\overline{\widehat{\widetilde{u}}}$

3(50,30)(0,0) 3(50,30)(150,30) 3(50,30)(50,130) 2(25, 65)(125, 65) 2(75,
15)(75, 115) 2(50, 50)(100, 80)
$B$$B$$Q$$Q^{*}$$A$$A$$\widetilde{u}$$T_{a}\widetilde{\widetilde{u}}$$T_{b}\widehat{\widetilde{u}}$$T_{c}\overline{\widetilde{u}}$$T_{a+b}\widehat{\widetilde{\widetilde{u}}}$$T_{b+c}\overline{\widehat{\widetilde{u}}}$$T_{a+c}\bar{\widetilde{\widetilde{u}}}$$T_{a+b+c}\overline{\widehat{\widetilde{\widetilde{u}}}}$

Figure 3: Neighboring cubes supporting the same cube system.

Consider two neighboring cubes, as in Fig. 3. Before we can glue them together
we need to apply $T_{a}$ to every vertex of the cube on the right. But then we
have to establish, e.g., that the shifted system of equations

$\displaystyle{\cal
S}_{n}Q\big{(}u,T_{a}\widetilde{u},T_{b}\widehat{u},T_{a+b}\widehat{\widetilde{u}}\big{)}=Q\big{(}\widetilde{u},T_{a}\widetilde{\widetilde{u}},T_{b}\widehat{\widetilde{u}},T_{a+b}\widehat{\widetilde{\widetilde{u}}}\big{)}=0,$
(2.10)

where ${\cal S}_{n}f(n,m)=f(n+1,m)$, is the same system of equations as the
system

$\displaystyle
Q\big{(}T_{a}\widetilde{u},T_{2a}\widetilde{\widetilde{u}},T_{a+b}\widehat{\widetilde{u}},T_{2a+b}\widehat{\widetilde{\widetilde{u}}}\big{)}=0.$
(2.11)

Indeed, we have the identity

$T_{a}Q\big{(}\widetilde{u},T_{a}\widetilde{\widetilde{u}},T_{b}\widehat{\widetilde{u}},T_{a+b}\widehat{\widetilde{\widetilde{u}}}\big{)}=Q\big{(}T_{a}\widetilde{u},T_{2a}\widetilde{\widetilde{u}},T_{a+b}\widehat{\widetilde{u}},T_{2a+b}\widehat{\widetilde{\widetilde{u}}}\big{)},$

which shows that the system (2.11) is just a rearrangement of the components
of the system (2.10). In fact, it can be shown that for any fractional affine
linear function $g$ of diagonal $N\times N$ matrices $(m_{1},\ldots,m_{n})$,
we have

$\displaystyle\theta g(m_{1},\ldots,m_{n})\theta^{-1}=g\big{(}\theta
m_{1}\theta^{-1},\dots,\theta m_{n}\theta^{-1}\big{)},$

for any invertible $N\times N$ matrix $\theta$. ∎

It follows from the proof that the equations in (2.9) on the right may be
simplified, i.e.,

$\displaystyle
Q^{*}\big{(}\overline{u},T_{a}\overline{\widetilde{u}},T_{b}{\overline{\widehat{u}}},T_{a+b}\overline{\widehat{\widetilde{u}}}\big{)}=0,\qquad
A\big{(}\widehat{u},T_{a}\widehat{\widetilde{u}},T_{c}{\overline{\widehat{u}}},T_{a+c}\overline{\widehat{\widetilde{u}}}\big{)}=0,\qquad
B\big{(}\widetilde{u},T_{c}\overline{\widetilde{u}},T_{b}\widehat{\widetilde{u}},T_{b+c}\overline{\widehat{\widetilde{u}}}\big{)}=0.$

It also follows that in the case where $Q^{*}=Q$, the cube system can be
imposed on the entire $\mathbb{Z}^{3}$-lattice. In the case where $Q^{*}\neq
Q$ one needs a second cube system obtained from the first by the reflection
$R_{3}$, and impose the reflected system on cubes with center
$\big{(}n+\frac{1}{2},m+\frac{1}{2},l+\frac{1}{2}\big{)}$ with $l$ odd.

######  Remark 2.4.

The cubes in Figs. 1(b), 2, and 3 are useful to define the equations which
live on the faces of the cubes. However, one should be aware that the field
$u$ (which provides the support for equations (2.7) and (2.9) is defined in
the usual way, namely $\widetilde{u}(n,m,l)=u(n+1,m,l)$, as in Fig. 1. We do
not have $\widetilde{u}(n,m,l)=T_{a}u(n+1,m,l)$.

####  2.2.1 Examples

The $N$-component equation

$\displaystyle
Q\big{(}u,T_{a}\widetilde{u},T_{b}\widehat{u},T_{a+b}\widehat{\widetilde{u}}\big{)}=0,$
(2.12)

where $u$ is an $N\times N$ diagonal matrix (2.4)), will be referred to as the
$N[a,b]$ extension of the scalar equation (2.1). Similarly, the multi-
component cube system (2.9) will be referred to as the $N[a,b,c]$ extension of
the cube system (2.8). In this terminology, the trivial Toeplitz extension
introduced in Appendix B of [8] corresponds to the $N[1,1]$ extension.

#### Discrete Burgers

A simple example of a CAC scalar equation is the 3-point discrete Burgers
equation [16, 39]

$\displaystyle\widehat{\widetilde{u}}(p-q+\widehat{u}-\widetilde{u})=p\widehat{u}-q\widetilde{u}.$
(2.13)

The parameters in this equation, $p$, $q$, are called lattice parameters, $p$
corresponds to the tilde-direction, $q$ corresponds to the hat-direction, and
there is a third parameter, $r$, which corresponds to the bar-direction. The
equations on the faces of the corresponding consistent cube are each of the
form (2.13) with different dependence on the lattice parameters, i.e., setting

$\displaystyle
Q\big{(}\widetilde{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=Q\big{(}\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q\big{)}:=\widehat{\widetilde{u}}(p-q+\widehat{u}-\widetilde{u})-p\widehat{u}+q\widetilde{u},$
$\displaystyle
A\big{(}\widetilde{u},\overline{u},\overline{\widetilde{u}}\big{)}=Q\big{(}\widetilde{u},\overline{u},\overline{\widetilde{u}};p,r\big{)},\qquad
B\big{(}\widehat{u},\overline{u},{\overline{\widehat{u}}}\big{)}=Q\big{(}\widehat{u},\overline{u},{\overline{\widehat{u}}};q,r\big{)},$

the cube system

$\displaystyle
Q\big{(}\widetilde{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
Q\big{(}\overline{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}}\big{)}=0,$
$\displaystyle
A\big{(}\widetilde{u},\overline{u},\overline{\widetilde{u}}\big{)}=0,\qquad$
$\displaystyle
A\big{(}\widehat{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}}\big{)}=0,$
$\displaystyle
B\big{(}\widehat{u},\overline{u},{\overline{\widehat{u}}}\big{)}=0,\qquad$
$\displaystyle
B\big{(}\widehat{\widetilde{u}},\overline{\widetilde{u}},\overline{\widehat{\widetilde{u}}}\big{)}=0,$

is CAC (with no dependence on $u$).

The $2[0,1]$ extension of equation (2.13) is

$\displaystyle\widehat{\widetilde{u}}(p-q+\widehat{v}-\widetilde{u})=p\widehat{v}-q\widetilde{u},\qquad\widehat{\widetilde{v}}(p-q+\widehat{u}-\widetilde{v})=p\widehat{u}-q\widetilde{v},$
(2.14)

and its $2[1,1]$ extension is

$\displaystyle\widehat{\widetilde{u}}(p-q+\widehat{v}-\widetilde{v})=p\widehat{v}-q\widetilde{v},\qquad\widehat{\widetilde{v}}(p-q+\widehat{u}-\widetilde{u})=p\widehat{u}-q\widetilde{u}.$

The $2[1,0]$ extension is the same as (2.14) (after interchanging the two
equations).

#### ABS equations

The ABS equations also depend on lattice parameters. They are scalar equations
of the form

$Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q\big{)}=0$

and can be embedded in a CAC system as follows

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q\big{)}=0,\qquad$
$\displaystyle
Q\big{(}\overline{u},\overline{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}};p,q\big{)}=0,$
$\displaystyle
Q\big{(}u,\widetilde{u},\overline{u},\overline{\widetilde{u}};p,r\big{)}=0,\qquad$
$\displaystyle
Q\big{(}\widehat{u},\widehat{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}};p,r\big{)}=0,$
$\displaystyle
Q\big{(}u,\widehat{u},\overline{u},{\overline{\widehat{u}}};q,r\big{)}=0,\qquad$
$\displaystyle
Q\big{(}\widetilde{u},\widehat{\widetilde{u}},\overline{\widetilde{u}},\overline{\widehat{\widetilde{u}}};q,r\big{)}=0.$

Similar to the discrete Burgers equation, each ABS equation has two types of
2-component generalizations, $2[0,1]$ and $2[1,1]$, which are respectively
given by

$\displaystyle
Q\big{(}u,\widetilde{u},T\widehat{u},T\widehat{\widetilde{u}};p,q\big{)}=0,$
(2.15)

and

$\displaystyle
Q\big{(}u,T\widetilde{u},T\widehat{u},\widehat{\widetilde{u}};p,q\big{)}=0.$
(2.16)

The latter form has been investigated in [22]. Explicitly, for the H1
equation, also known as the lattice potential KdV equation (lpKdv),

$\displaystyle\big{(}u-\widehat{\widetilde{u}}\big{)}\big{(}\widetilde{u}-\widehat{u}\big{)}+q-p=0,$
(2.17)

the $2[0,1]$ extension is

$\displaystyle\big{(}u-\widehat{\widetilde{v}}\big{)}\big{(}\widetilde{u}-\widehat{v}\big{)}+q-p=0,\qquad\big{(}v-\widehat{\widetilde{u}}\big{)}\big{(}\widetilde{v}-\widehat{u}\big{)}+q-p=0,$
(2.18)

and the $2[1,1]$ extension is

$\displaystyle\big{(}u-\widehat{\widetilde{u}}\big{)}\big{(}\widetilde{v}-\widehat{v}\big{)}+q-p=0,\qquad\big{(}v-\widehat{\widetilde{v}}\big{)}\big{(}\widetilde{u}-\widehat{u}\big{)}+q-p=0.$

The latter appeared in [14], where the CAC property was used to construct its
Lax pair. The $2[1,1]$ extension of the lattice Schwarzian KdV equation was
given in [8, equation (B.6)].

When $N=3$ one can have $3[0,1]$, $3[0,2]$, $3[1,1]$, and $3[1,2]$
generalizations. For example, the $3[1,1]$ extension of H1 is

$\displaystyle\big{(}u-\widehat{\widetilde{w}}\big{)}\big{(}\widetilde{v}-\widehat{v}\big{)}+q-p=0,\qquad\big{(}v-\widehat{\widetilde{u}}\big{)}\big{(}\widetilde{w}-\widehat{w}\big{)}+q-p=0,$
$\displaystyle\big{(}w-\widehat{\widetilde{v}}\big{)}\big{(}\widetilde{u}-\widehat{u}\big{)}+q-p=0.$
(2.19)

######  Remark 2.5.

While each ABS-equation is part of a CAC system which comprises copies of the
same equation (with appropriate dependence on the lattice parameters), this is
not necessarily the case for their multi-component extensions. For example,
the 2-component ($Q$) equation (2.18) sits in the cube system $2[0,1,0]$
together with

$\displaystyle A\colon\
\begin{array}[]{l}(u-\overline{\widetilde{u}})(\widetilde{u}-\overline{u})+r-p=0,\\\
(v-\overline{\widetilde{v}})(\widetilde{v}-\overline{v})+r-p=0,\end{array}$
(2.22) $\displaystyle B\colon\
\begin{array}[]{l}(u-{\overline{\widehat{v}}})(\widehat{v}-\overline{u})+r-q=0,\\\
(v-{\overline{\widehat{u}}})(\widehat{u}-\overline{v})+r-q=0.\end{array}$
(2.25)

Here the $B$ equation has the same form as the $Q$ equation, but the $A$
equation is decoupled.

#### Auto-Bäcklund transformations

There also exist CAC systems containing two different equations. For example,
one can compose a CAC system using the lattice potential modified KdV (lpmKdV,
or H$3^{0}$) equation

$\displaystyle
B\big{(}u,\widehat{u},\overline{u},{\overline{\widehat{u}}};q,r\big{)}=q\big{(}u\widehat{u}-\overline{u}{\overline{\widehat{u}}}\big{)}-r\big{(}u\overline{u}-\widehat{u}{\overline{\widehat{u}}}\big{)},$
(2.26)

on the side faces and the discrete sine-Gordon (dsG) equation

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q\big{)}=p\big{(}u\widehat{\widetilde{u}}-\widetilde{u}\widehat{u}\big{)}-q\big{(}u\widetilde{u}\widehat{u}\widehat{\widetilde{u}}-1\big{)},$

for the other four faces of the cube $(Q,A)$ [13, 26]. Multi-component
extension yields the CAC system

$\displaystyle
Q\big{(}u,T_{a}\widetilde{u},T_{b}\widehat{u},T_{a+b}\widehat{\widetilde{u}};p,q\big{)}=0,$
(2.27a) $\displaystyle
Q\big{(}u,T_{a}\widetilde{u},T_{c}\overline{u},T_{a+c}\overline{\widetilde{u}};p,r\big{)}=0,$
(2.27b) $\displaystyle
B\big{(}u,T_{b}\widehat{u},T_{c}\overline{u},T_{b+c}{\overline{\widehat{u}}};q,r\big{)}=0,$
(2.27c)

with their shifts. In particular, we mention the $2[1,1]$ extension of the dsG
equation

$\displaystyle
p\big{(}u\widehat{\widetilde{u}}-\widetilde{v}\widehat{v}\big{)}-q\big{(}u\widetilde{v}\widehat{v}\widehat{\widetilde{u}}-1\big{)}=0,\qquad
p\big{(}v\widehat{\widetilde{v}}-\widetilde{u}\widehat{u}\big{)}-q\big{(}v\widetilde{u}\widehat{u}\widehat{\widetilde{v}}-1\big{)}=0,$
(2.28)

whose auto-BT consists of the dsG equation and the lpmKdV equation, which
gives rise to an asymmetric Lax-pair, given in Section 2.2.2.

The auto-Bäcklund transformations given in Table 2 of [7] provide other
instances of the same situation. For example, one can take the ABS equation
called Q$1^{1}$ as the $Q$-equation, that is

$Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q\big{)}:=p\big{(}u-\widehat{u}\big{)}\big{(}\widetilde{u}-\widehat{\widetilde{u}}\big{)}-q\big{(}u-\widetilde{u}\big{)}\big{(}\widehat{u}-\widehat{\widetilde{u}}\big{)}+pq(p-q)=0.$

at the bottom face, and
$Q\big{(}\overline{u},\overline{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}};p,q\big{)}=0$
on the top face. A CAC system is obtained by placing the auto-BT, where the
Bäcklund parameter $r$ plays the role of the lattice parameter,

$\displaystyle
A\big{(}u,\widetilde{u},\overline{u},\overline{\widetilde{u}},p,r\big{)}:=\big{(}u-\widetilde{u}\big{)}\big{(}\overline{u}-\overline{\widetilde{u}}\big{)}+p\big{(}u+\widetilde{u}+\overline{u}+\overline{\widetilde{u}}+p+2r\big{)}=0,$
(2.29)

on the front face,
$A\big{(}\widehat{u},\widehat{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}},p,r\big{)}=0$
on the back face,
$A\big{(}u,\widehat{u},\overline{u},{\overline{\widehat{u}}},q,r\big{)}=0$ on
the left face and
$A\big{(}\widetilde{u},\widehat{\widetilde{u}},\overline{\widetilde{u}},\overline{\widehat{\widetilde{u}}},q,r\big{)}=0$
on the right face. Such CAC lattice systems can be consistently extended to
multi-component CAC lattice systems by virtue of Theorem 2.3.

Thus, when $Q^{*}=Q$ the multi-component equations

$\displaystyle
A\big{(}u,T_{a}\widetilde{u},T_{c}\overline{u},T_{a+c}\widetilde{\overline{u}};p,r\big{)}=0,$
(2.30a) $\displaystyle
B\big{(}u,T_{c}\overline{u},T_{b}\widehat{u},T_{b+c}{\widehat{\overline{u}}};r,q\big{)}=0,$
(2.30b)

can be interpreted as an auto-BT, mapping one solution $u$ to another solution
$\overline{u}$. This is because the top equation in (2.9) can be rewritten as

$T_{c}Q\big{(}\overline{u},T_{a}\widetilde{\overline{u}},T_{b}{\widehat{\overline{u}}},T_{a+b}\widehat{\widetilde{\overline{u}}};p,q\big{)}=0.$

We remark that the equation (2.29) (in fact, any auto-BT) is an integrable
equation on the $\mathbb{Z}^{2}$ lattice. The equation (2.29) is not in the
ABS list and neither is the sine-Gordon equation, because they are not CAC
with copies of themselves. However, they do posses an auto-BT and hence (non-
symmetric) Lax pairs (where the Bäcklund parameter provides the so called
spectral parameter) can be constructed, see Section 2.2.2.

#### Bäcklund transformations

For (non-auto) Bäcklund transformations, such as the ones given in Table 3 in
[7], the equations on the bottom face, $Q$, and on the top face $Q^{*}$ are
different. For example, taking $Q=$H2,

$\displaystyle\big{(}{u}-\widehat{\widetilde{u}}\big{)}\big{(}\widetilde{{u}}-\widehat{{u}}\big{)}-(p-q)\big{(}{u}+\widetilde{{u}}+\widehat{{u}}+\widehat{\widetilde{u}}+p+q\big{)}=0,$

and posing

$\displaystyle A\colon\
u+\widetilde{u}+p=2{\overline{u}}\widetilde{\overline{u}},\qquad B\colon\
u+\widehat{u}+q=2{\overline{u}}\widehat{\overline{u}},$

and their shifted versions, $\widehat{A}$ and $\widetilde{B}$ on the side
faces, one finds that on the top face the variable ${\overline{u}}$ satisfies
the $Q^{*}=$H1 equation (2.17).

In the general $N$-component cube system, denoted $N[a,b,c]$, the side system

$\displaystyle
u+T_{a}\widetilde{u}+p=2(T_{c}{\overline{u}})\big{(}T_{a+c}\widetilde{\overline{u}}\big{)},\qquad
u+T_{b}\widehat{u}+q=2(T_{c}{\overline{u}})\big{(}T_{b+c}\widehat{\overline{u}}\big{)}$

provides a BT between the $N$-component H2 system

$\displaystyle\big{(}{u}-T_{a+b}\widehat{\widetilde{u}}\big{)}\big{(}T_{a}\widetilde{{u}}-T_{b}\widehat{{u}}\big{)}-(p-q)\big{(}{u}+T_{a}\widetilde{{u}}+T_{b}\widehat{u}+T_{a+b}\widehat{\widetilde{u}}+p+q\big{)}=0,$

and the $N$-component H1 system

$\displaystyle\big{(}T{\overline{u}}-T_{a+b}\widehat{\widetilde{\overline{u}}}\big{)}\big{(}T_{a}\widetilde{{\overline{u}}}-T_{b}\widehat{{\overline{u}}}\big{)}-p+q=0$

There are other examples of CAC lattice systems such as the ones in [25].
These all allow multi-component extension.

####  2.2.2 Lax pairs

In a consistent system of the form (2.9) the Lax pair of equation (2.12) can
be constructed through the BT (2.30) following the standard procedure, cf. [4,
14, 43]. Here one would introduce $\overline{u}=gf^{-1}$ with

$\displaystyle f=\operatorname{diag}(f_{1},f_{2},\dots,f_{N}),\qquad
g=\operatorname{diag}(g_{1},g_{2},\dots,g_{N}),$ (2.31)

and $\Psi=(f_{1},f_{2},\dots,f_{N},g_{1},g_{2},\dots,g_{N})^{\rm T}$, and then
equation (2.30) can be cast into the form
$\widetilde{\Psi}=L(u,\widetilde{u})\Psi$,
$\widehat{\Psi}=M(u,\widehat{u})\Psi$. With the correct scaling factors, the
pair of matrices $L$, $M$ form a $2N\times 2N$ Lax pair of (2.12). In Appendix
A we show how this approach yields (2.34).

On the other hand, one can directly write down a Lax pair of the $N$-component
system (2.12) in terms of Lax matrices of the scalar equation. Suppose that
(2.8) is a scalar 3D consistent lattice system, and the bottom equation

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=0$ (2.32)

has a $2\times 2$ Lax pair (e.g., the one obtained from the BT at hand)

$\displaystyle\widetilde{\psi}=\mathcal{L}(u,\widetilde{u})\psi,\qquad\widehat{\psi}=\mathcal{M}(u,\widehat{u})\psi,$
(2.33)

where $\psi=(\psi_{1},\psi_{2})^{\rm T}$ and $\mathcal{L}$ and $\mathcal{M}$
are $2\times 2$ matrices. Considering $N$ copies of the equation, each scalar
equation
$Q\big{(}u_{i},\widetilde{u}_{i},\widehat{u}_{i},\widehat{\widetilde{u}}_{i}\big{)}=0$
has a $2\times 2$ Lax pair $\mathcal{L}(u_{i},\widetilde{u}_{i})$,
$\mathcal{M}(u_{i},\widehat{u}_{i})$. Then, we have the following Lax pair for
the multi-component case.

######  Theorem 2.6.

Suppose that the scalar equation (2.32) has a Lax pair (2.33). Then the
$N$-component extension (2.12) has a $2N\times 2N$ Lax pair

$\displaystyle\widetilde{\Phi}=\theta^{-a-c}L(u,T_{a}\widetilde{u})\theta^{c}\Phi,\qquad\widehat{\Phi}=\theta^{-b-c}M(u,T_{b}\widehat{u})\theta^{c}\Phi.$
(2.34)

where, with $u$ the diagonal matrix (2.4) and
$v=\operatorname{diag}(v_{1},v_{2},\dots,v_{N})$,

$\displaystyle
L(u,v)=\operatorname{diag}(\mathcal{L}(u_{1},v_{1}),\mathcal{L}(u_{2},v_{2}),\dots,\mathcal{L}(u_{N},v_{N})),$
(2.35a) $\displaystyle
M(u,v)=\operatorname{diag}(\mathcal{M}(u_{1},v_{1}),\mathcal{M}(u_{2},v_{2}),\dots,\mathcal{M}(u_{N},v_{N}))$
(2.35b)

in which $\mathcal{L}$ and $\mathcal{M}$ are the Lax matrices given in (2.33),
$\theta=\sigma_{2N}^{2}$ and $\sigma_{2N}$ is a $2N\times 2N$ cyclic matrix
defined by (2.5).

Note that the gauge transformation $\Phi^{\prime}=\theta^{c}\Phi$, transforms
the Lax pair (2.34) into

$\displaystyle\widetilde{\Phi}^{\prime}=\theta^{-a}L(u,T_{a}\widetilde{u})\Phi^{\prime},\qquad\widehat{\Phi}^{\prime}=\theta^{-b}M(u,T_{b}\widehat{u})\Phi^{\prime},$
(2.36)

which does not depend on $c$.

###### Proof.

The compatibility of the linear system
$\widetilde{\Phi}=L(u,\widetilde{u})\Phi$,
$\widehat{\Phi}=M(u,\widehat{u})\Phi$ equals

$\displaystyle
L\big{(}\widehat{u},\widehat{\widetilde{u}}\big{)}M(u,\widehat{u})=M\big{(}\widetilde{u},\widehat{\widetilde{u}}\big{)}L(u,\widetilde{u}),$
(2.37)

which is equivalent to equation (2.32) with diagonal matrix $u$ given by
(2.4). From the compatibility of (2.34) we find

$\displaystyle\theta^{-a-c}L\big{(}\widehat{u},T_{a}\widehat{\widetilde{u}}\big{)}\theta^{c}\theta^{-b-c}M(u,T_{b}\widehat{u})\theta^{c}=\theta^{-b-c}M\big{(}\widetilde{u},T_{b}\widehat{\widetilde{u}}\big{)}\theta^{c}\theta^{-a-c}L(u,T_{a}\widetilde{u})\theta^{c}$
$\displaystyle\qquad{}\Longleftrightarrow\theta^{b}L\big{(}\widehat{u},T_{a}\widehat{\widetilde{u}}\big{)}\theta^{-b}M(u,T_{b}\widehat{u})=\theta^{a}M\big{(}\widetilde{u},T_{b}\widehat{\widetilde{u}}\big{)}\theta^{-a}L(u,T_{a}\widetilde{u})$
$\displaystyle\qquad{}\Longleftrightarrow
L\big{(}T_{b}\widehat{u},T_{a+b}\widehat{\widetilde{u}}\big{)}M(u,T_{b}\widehat{u})=M\big{(}T_{a}\widetilde{u},T_{a+b}\widehat{\widetilde{u}}\big{)}L(u,T_{a}\widetilde{u}),$

which gives rise to (2.12), as (2.32) arises from (2.37). ∎

As an example, consider the H1 equation (2.17) which admits the Lax pair

$\displaystyle\mathcal{L}(u,\widetilde{u})=\mathcal{H}(u,\widetilde{u},p),\qquad\mathcal{M}(u,\widehat{u})=\mathcal{H}(u,\widehat{u},q),$

with

$\displaystyle\mathcal{H}(u,\widetilde{u},p)=\left(\begin{matrix}-u&u\widetilde{u}+p-r\\\
-1&\widetilde{u}\end{matrix}\right).$

According to (2.36), the $3[1,1]$ extension of H1 (2.19) admits the Lax pair

$\displaystyle\widetilde{\Phi}=\left(\begin{matrix}0&0&\mathcal{H}(w,\widetilde{u},p)\\\
\mathcal{H}(u,\widetilde{v},p)&0&0\\\
0&\mathcal{H}(v,\widetilde{w},p)&0\end{matrix}\right)\Phi,$
$\displaystyle\widehat{\Phi}=\left(\begin{matrix}0&0&\mathcal{H}(w,\widehat{u},q)\\\
\mathcal{H}(u,\widehat{v},q)&0&0\\\
0&\mathcal{H}(v,\widehat{w},q)&0\end{matrix}\right)\Phi.$

#### Asymmetrical Lax pairs

When equation $A$ is not related to equation $B$ by a standard change in
lattice parameters, the corresponding Lax pair for $Q$ is asymmetrical.
Similarly, one also finds asymmetry if one constructs a Lax pair for an auto-
BT ($A$), using its auto-BT given by $B$, $Q$. For example, the auto-BT
(2.27b)–(2.27c) provides an asymmetrical Lax pair for the 2-component dsG
equation (2.28),

$\widetilde{\Phi}=\left(\begin{matrix}0&\mathcal{L}(u,\widetilde{v})\\\
\mathcal{L}(v,\widetilde{u})&0\end{matrix}\right)\Phi,\qquad\widehat{\Phi}=\left(\begin{matrix}0&\mathcal{M}(u,\widehat{v})\\\
\mathcal{M}(v,\widehat{u})&0\end{matrix}\right)\Phi,$

where $a=b=c=1$ and

$\mathcal{L}(u,\widetilde{u})=\left(\begin{matrix}p&-r\widetilde{u}\\\
-\dfrac{r}{u}&\dfrac{p\widetilde{u}}{u}\\\
\end{matrix}\right),\qquad\mathcal{M}(u,\widehat{u})=\left(\begin{matrix}\dfrac{r\widehat{u}}{u}&-\frac{q}{u}\\\
-q\widehat{u}&r\end{matrix}\right).$

For all the auto-BTs gives in [7, Table 2] a superposition principle emerges
for solutions of the equation that are related by the auto-BT. This gives rise
to a different asymmetrical Lax-pair for the auto-BT. The superposition
principle for solutions of the lmpKdV equation related by the dsG equation is

$\displaystyle
Q\big{(}u,\widetilde{u},\dot{u},\dot{\widehat{u}};p,s\big{)}=s\big{(}u\widetilde{u}-\dot{u}\dot{\widetilde{u}}\big{)}-p\big{(}u\dot{u}-\widetilde{u}\dot{\widetilde{u}}\big{)},$
(2.38)

which is an lmpKdV equation with $p$ and $s$ interchanged. The cube system
with the superposition principle (2.38) on the bottom and top faces and the
dsG equation on the side faces, cf. Fig. 4, is consistent, and admits multi-
component extension.

Figure 4: On the inner cube we have the lmpKdV ($B$) on the left face and on
the right face. They are connected by the auto-BT (with parameter $p$) which
is the dsG equation ($Q$) on the front, back, bottom and top faces. On the
left cube the auto-BT (in dot-direction with parameter $s$) yields an lmpKdV
equation on the left face. On the right cube the auto-BT with parameter $s$
yields a fourth lmpKdV equation on the right face. The four solutions to these
four lmpKdV equations are related by the superposition principle. On the front
cube the superposition principle ($SP$), which is the lmpKdV equation with $p$
and $s$ interchanged, is on the top and the bottom face, and the dsG equation
is on the four side faces. This provides another CAC cube system.

One can also construct Lax-pairs from non-auto BTs [7, Table 3], however,
these will not contain a spectral parameter.

####  2.2.3 Counting $\boldsymbol{N}$-component extensions of ABS lattice
equations

#### D4 symmetry

The ABS lattice equations are D4 symmetric, i.e.,

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q\big{)}=\pm
Q\big{(}u,\widehat{u},\widetilde{u},\widehat{\widetilde{u}};q,p\big{)}=\pm
Q\big{(}\widehat{u},\widehat{\widetilde{u}},u,\widetilde{u};p,q\big{)}.$
(2.39)

In (2.39) we can consider $u$ and its shifts to be diagonal matrices (2.4). By
relabelling we also have

$\displaystyle
Q\big{(}u,T_{a}\widetilde{u},T_{b}\widehat{u},T_{a+b}\widehat{\widetilde{u}};p,q\big{)}=$
$\displaystyle\pm
Q\big{(}u,T_{b}\widehat{u},T_{a}\widetilde{u},T_{a+b}\widehat{\widetilde{u}};q,p\big{)}$
$\displaystyle=$ $\displaystyle\pm
Q\big{(}T_{a}\widetilde{u},u,T_{a+b}\widehat{\widetilde{u}},T_{b}\widehat{u};p,q\big{)}=\pm
Q\big{(}T_{b}\widehat{u},T_{a+b}\widehat{\widetilde{u}},u,T_{a}\widetilde{u};p,q\big{)}.$

Now we introduce $v(n,m)=u(-n,m)$, and in terms of $v$ we write (2.12) as

$\displaystyle
Q\big{(}v,T_{a}\underaccent{\tilde}{v},T_{b}\widehat{v},T_{a+b}\widehat{\underaccent{\tilde}{v}};p,q\big{)}=0.$

Applying a tilde-shift, by virtue of D4 symmetry we obtain

$\displaystyle
Q\big{(}\widetilde{v},T_{a}v,T_{b}\widehat{\widetilde{v}},T_{a+b}\widehat{v};p,q\big{)}=0$
$\displaystyle\Rightarrow
Q\big{(}T_{a}v,\widetilde{v},T_{a+b}\widehat{v},T_{b}\widehat{\widetilde{v}};p,q\big{)}=0$
$\displaystyle\Rightarrow
T_{a}Q\big{(}v,T_{-a}\widetilde{v},T_{b}\widehat{v},T_{b-a}\widehat{\widetilde{v}};p,q\big{)}=0$
$\displaystyle\Rightarrow
Q\big{(}v,T_{-a}\widetilde{v},T_{b}\widehat{v},T_{b-a}\widehat{\widetilde{v}};p,q\big{)}=0.$

Since $T_{-a}=T_{N-a}$, the above relation indicates that $N[a,b]$ and
$N[N-a,b]$ generate same $N$-component system up to reflection $R_{1}\colon
n\leftrightarrow-n$. This leads to the following proposition.

######  Proposition 2.7.

For the ABS equation (2.1), due to D4 symmetry, the cases

$N[a,b],\quad N[b,a],\quad N[a,N-b],\quad N[N-a,b],\quad N[N-a,N-b]$

are all equivalent up to coordinate refections. Consequently, we can assume
$0\leq a\leq b\leq c\leq N/2$ without loss of generality.

#### Decoupling

Let us consider the $4[2,2]$ extension of (2.1):

$\displaystyle
Q\big{(}u_{1},\widetilde{u}_{3},\widehat{u}_{3},\widehat{\widetilde{u}}_{1};p,q\big{)}=0,$
(2.40a) $\displaystyle
Q\big{(}u_{2},\widetilde{u}_{4},\widehat{u}_{4},\widehat{\widetilde{u}}_{2};p,q\big{)}=0,$
(2.40b) $\displaystyle
Q\big{(}u_{3},\widetilde{u}_{1},\widehat{u}_{1},\widehat{\widetilde{u}}_{3};p,q\big{)}=0,$
(2.40c) $\displaystyle
Q\big{(}u_{4},\widetilde{u}_{2},\widehat{u}_{2},\widehat{\widetilde{u}}_{4};p,q\big{)}=0.$
(2.40d)

This system is decoupled into two $2[1,1]$ systems, namely (2.40a), (2.40c)
and (2.40b), (2.40d).

In the next theorem, for which we include a proof in Appendix B, we give
conditions which decide when a system is decoupled or non-decoupled. The
greatest common divisor between integers $a,b,\ldots,c$ will be denoted
$\operatorname{gcd}(a,b,\ldots,c)$.

######  Theorem 2.8.

Let $d=\operatorname{gcd}(a,b,N)$. If $d>1$ the $N[a,b]$ extension (2.12) can
be decomposed into $d$ sets of $s[a/d,b/d]$ form of the equation (2.1), where
$s=N/d$. If $d=1$ the system is non-decoupled.

It follows that if $N$ is prime the only decoupled case is $N[0,0]$, which
corresponds to the trivial multi-component extension. For $N=6$, we have five
extensions which decouple,

$6[0,0],\quad 6[0,2],\quad 6[0,3],\quad 6[2,2],\quad 6[3,3]$

and five that do not decouple,

$6[0,1],\quad 6[1,1],\quad 6[1,2],\quad 6[1,3],\quad 6[2,3].$

We let $\alpha_{N}$ denote the number of $N$-component extensions (2.12) that
decouple, and we let $\beta_{N}$ denote the number of $N$-component extensions
(2.12) that do not decouple. Thus, $\alpha_{6}=\beta_{6}=5$.

In the following theorem we give formulas for the functions $\alpha_{N}$ and
$\beta_{N}$. We use notation as follows. Let $s$ be a set. By $\mathcal{P}(s)$
we denote the powerset of $s$, $\\#s$ we denote the number of elements in $s$,
and $\Pi s$ denotes the product of the elements in $s$, e.g., with
$s=\\{1,2,3\\}$ we have

$\mathcal{P}(s)=\\{\varnothing,\\{1\\},\\{2\\},\\{3\\},\\{1,2\\},\\{1,3\\},\\{2,3\\},\\{1,2,3\\}\\},\qquad\\#s=3,\qquad\Pi
s=6,$

and $\\#\varnothing=0$, $\Pi\varnothing=1$. Furthermore, for $n\in\mathbb{N}$
we denote the set of prime divisors of $n$ by $\mathbb{P}_{n}$, i.e., if $n$
has prime decomposition $n=\prod\limits_{i=1}^{l}p_{i}^{m_{i}}$, then
$\mathbb{P}_{n}=\\{p_{1},p_{2},\ldots,p_{l}\\}$.

######  Theorem 2.9.

For any given positive integer $N>1$, the numbers of decoupled and non-
decoupled $N$-component ABS systems (2.12) are respectively given by

$\displaystyle\alpha_{N}=\sum_{s\in\mathcal{P}(\mathbb{P}_{N})\backslash\varnothing}(-1)^{\\#s+1}\begin{pmatrix}\left\lfloor\frac{N}{2\Pi
s}\right\rfloor+2\\\ 2\end{pmatrix}$ (2.41)

and

$\displaystyle\beta_{N}=\sum_{s\in\mathcal{P}(\mathbb{P}_{N})}(-1)^{\\#s}\begin{pmatrix}\left\lfloor\frac{N}{2\Pi
s}\right\rfloor+2\\\ 2\end{pmatrix}$ (2.42)

where $\binom{n}{m}=\frac{n!}{m!(n-m)!}$ and
${\left\lfloor{\cdot}\right\rfloor}$ represents the floor function.

Formulas (2.41) and (2.42) yield

$\displaystyle\alpha_{2,3,\ldots,20}=1,\,1,\,3,\,1,\,5,\,1,\,6,\,3,\,8,\,1,\,13,\,1,\,12,\,8,\,15,\,1,\,22,\,1,\,24,$
$\displaystyle\beta_{2,3,\ldots,20}=2,\,2,\,3,\,5,\,5,\,9,\,9,\,12,\,13,\,20,\,15,\,27,\,24,\,28,\,30,\,44,\,33,\,54,\,42,$

and we note that
$\alpha_{2n}+\beta_{2n}=\alpha_{2n+1}+\beta_{2n+1}=\binom{n+2}{2}$, and
$\alpha_{p}=1$ when $p$ is prime.

###### Proof.

Due to $0\leq a\leq b\leq N/2$, the total number of $N[a,b]$ extensions is

$\displaystyle\alpha_{N}+\beta_{N}=\binom{{\left\lfloor{N/2}\right\rfloor}+1}{2}+\binom{{\left\lfloor{N/2}\right\rfloor}+1}{1}=\binom{{\left\lfloor{N/2}\right\rfloor}+2}{2}.$
(2.43)

First consider the decoupled case. For any $p\in\mathbb{P}_{N}$, if

$a,b\in\\{0,p,2p,3p,\dots,{\left\lfloor{N/(2p)}\right\rfloor}p\\}$

then $p$ is a divisor of $\operatorname{gcd}(a,b,N)$. By Theorem 2.8 this
gives rise to $\binom{{\left\lfloor{N/(2p)}\right\rfloor}+2}{2}$ decoupled
$N$-component systems. Similarly, for $q\in\mathbb{P}_{N}$ with $q\neq p$ we
find $\binom{{\left\lfloor{N/(2q)}\right\rfloor}+2}{2}$ decoupled
$N$-component systems. However, a number of these systems we would have
already counted, namely the
$\binom{{\left\lfloor{N/(2pq)}\right\rfloor}+2}{2}$ systems where

$a,b\in\\{0,pq,2pq,3pq,\dots,{\left\lfloor{N/(2pq)}\right\rfloor}pq\\}.$

Running through all the primes in $\mathbb{P}_{N}$ by the inclusion-exclusion
principle one finds the formula (2.41) for $\alpha_{N}$. Due to (2.43) the
formula for $\beta_{N}$ is then given by (2.42). ∎

###  2.3 Multi-component CAC 3D lattice equations

3D lattice equations defined on a 3D cube can be consistent around a 4D cube.
In affine linear case, certain 8-point and 6-point lattice equations, see Fig.
5, have been verified to be CAC in this sense [4, 6]. We mention in particular
the lattice AKP equation (a.k.a. the Hirota equation [30])

$\displaystyle\alpha_{1}\widetilde{u}{\widehat{\overline{u}}}+\alpha_{2}\widehat{u}\widetilde{\overline{u}}+\alpha_{3}\overline{u}\widehat{\widetilde{u}}=0$
(2.44)

and the lattice BKP equation (a.k.a. the Miwa equation [41])

$\displaystyle\alpha_{1}\widetilde{u}{\widehat{\overline{u}}}+\alpha_{2}\widehat{u}\widetilde{\overline{u}}+\alpha_{3}\overline{u}\widehat{\widetilde{u}}+\alpha_{4}u\widehat{\widetilde{\overline{u}}}=0.$
(2.45)

One can include arbitrary coefficients $\\{\alpha_{j}\\}$ in the equations,
which can be gauged to any nonzero value [49]. The stencils of these two
equations are depicted in Fig. 5.

${\widetilde{\overline{u}}}$$\overline{u}$${{\widehat{\overline{u}}}}$${\widehat{\widetilde{u}}}$$\widetilde{u}$$\widehat{u}$(1275,2708)(3075,2708)
(1275,2708)(450,1883) (450,1883)(450,83) (3075,2708)(450,1883)
(3075,2708)(2250,83) (450,1883)(2250,83) (3075,2708)(3075,908)
60.000(1275,2708)(450,83) 60.000(1275,2708)(3075,908) (450,83)(2250,83)
(3075,908)(2250,83) 70.000(3075,908)(450,83) (a) 6-point stencil of lattice
AKP (2.44).

${\widetilde{\overline{u}}}$$\overline{u}$${{\widehat{\overline{u}}}}$${\widehat{\widetilde{u}}}$$u$${\widehat{\widetilde{\overline{u}}}}$$\widetilde{u}$$\widehat{u}$(1275,2708)(3075,2708)
(1275,2708)(450,1883) (450,1883)(450,83) (3075,2708)(2250,1883)
(450,1883)(2250,1883) (3075,2633)(3075,908) 60.000(1275,908)(450,83)
60.000(1275,908)(3075,908) (2250,1883)(2250,83) (450,83)(2250,83)
(3075,908)(2250,83) 60.000(1275,2633)(1275,908) (b) 8-point stencil of lattice
BKP (2.45).

Figure 5: Stencils of 3D lattice equations.

Any 3D CAC lattice equation can be generalised to a multi-component 3D
equation which inherits the CAC property. We have the following result.

######  Theorem 2.10.

Suppose that the scalar lattice system

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\overline{u},\widehat{\widetilde{u}},\overline{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}}\big{)}=0,\qquad$
$\displaystyle
Q\big{(}\dot{u},\dot{\widetilde{u}},\dot{\widehat{u}},\dot{\overline{u}},\dot{\widehat{\widetilde{u}}},\dot{\overline{\widetilde{u}}},\dot{{\overline{\widehat{u}}}},\dot{\overline{\widehat{\widetilde{u}}}}\big{)}=0,$
$\displaystyle
A\big{(}u,\widetilde{u},\widehat{u},\dot{u},\widehat{\widetilde{u}},\dot{\widetilde{u}},\dot{\widehat{u}},\dot{\widehat{\widetilde{u}}}\big{)}=0,\qquad$
$\displaystyle
A\big{(}\overline{u},\overline{\widetilde{u}},{\overline{\widehat{u}}},\dot{\overline{u}},\overline{\widehat{\widetilde{u}}},\dot{\overline{\widetilde{u}}},\dot{{\overline{\widehat{u}}}},\dot{\overline{\widehat{\widetilde{u}}}}\big{)}=0,$
$\displaystyle
B\big{(}u,\widetilde{u},\dot{u},\overline{u},\dot{\widetilde{u}},\overline{\widetilde{u}},\dot{\overline{u}},\dot{\overline{\widetilde{u}}}\big{)}=0,\qquad$
$\displaystyle
B\big{(}\widehat{u},\widehat{\widetilde{u}},\dot{u},{\overline{\widehat{u}}},\dot{\widehat{\widetilde{u}}},\overline{\widehat{\widetilde{u}}},\dot{{\overline{\widehat{u}}}},\dot{\overline{\widehat{\widetilde{u}}}}\big{)}=0,$
$\displaystyle
C\big{(}u,\dot{u},\widehat{u},\overline{u},\dot{\widehat{u}},\dot{\overline{u}},{\overline{\widehat{u}}},\dot{{\overline{\widehat{u}}}}\big{)}=0,\qquad$
$\displaystyle
C\big{(}\widetilde{u},\dot{\widetilde{u}},\widehat{\widetilde{u}},\overline{\widetilde{u}},\dot{\widehat{\widetilde{u}}},\dot{\overline{\widetilde{u}}},\overline{\widehat{\widetilde{u}}},\dot{\overline{\widehat{\widetilde{u}}}}\big{)}=0$

is consistent around the 4D cube in Fig. 6, i.e., the value of
$\dot{\overline{\widehat{\widetilde{u}}}}$ is uniquely determined by suitably
given initial values. Then, after replacing $u$ with diagonal form (2.4), the
following system

$\displaystyle
Q\big{(}u,T_{a}\widetilde{u},T_{b}\widehat{u},T_{c}\overline{u},T_{a+b}\widehat{\widetilde{u}},T_{a+c}\overline{\widetilde{u}},T_{b+c}{\overline{\widehat{u}}},T_{a+b+c}\overline{\widehat{\widetilde{u}}}\big{)}=0,$
(2.46a) $\displaystyle
Q\big{(}T_{d}\dot{u},T_{a+d}\dot{\widetilde{u}},T_{b+d}\dot{\widehat{u}},T_{c+d}\dot{\overline{u}},T_{a+b+d}\dot{\widehat{\widetilde{u}}},T_{a+c+d}\dot{\overline{\widetilde{u}}},T_{b+c+d}\dot{{\overline{\widehat{u}}}},T_{a+b+c+d}\dot{\overline{\widehat{\widetilde{u}}}}\big{)}=0,$
(2.46b) $\displaystyle
A\big{(}u,T_{a}\widetilde{u},T_{b}\widehat{u},T_{d}\dot{u},T_{a+b}\widehat{\widetilde{u}},T_{a+c}\dot{\widetilde{u}},T_{b+d}\dot{\widehat{u}},T_{a+b+d}\dot{\widehat{\widetilde{u}}}\big{)}=0,$
(2.46c) $\displaystyle
A\big{(}T_{c}\overline{u},T_{a+c}\overline{\widetilde{u}},T_{b+c}{\overline{\widehat{u}}},T_{c+d}\dot{\overline{u}},T_{a+b+c}\overline{\widehat{\widetilde{u}}},T_{a+b+d}\dot{\overline{\widetilde{u}}},T_{b+c+d}\dot{{\overline{\widehat{u}}}},T_{a+b+c+d}\dot{\overline{\widehat{\widetilde{u}}}}\big{)}=0,$
(2.46d) $\displaystyle
B\big{(}u,T_{a}\widetilde{u},T_{d}\dot{u},T_{c}\overline{u},T_{a+d}\dot{\widetilde{u}},T_{a+c}\overline{\widetilde{u}},T_{c+d}\dot{\overline{u}},T_{a+c+d}\dot{\overline{\widetilde{u}}}\big{)}=0,$
(2.46e) $\displaystyle
B\big{(}T_{b}\widehat{u},T_{a+b}\widehat{\widetilde{u}},T_{d}\dot{u},T_{b+c}{\overline{\widehat{u}}},T_{a+b+d}\dot{\widehat{\widetilde{u}}},T_{a+b+c}\overline{\widehat{\widetilde{u}}},T_{b+c+d}\dot{{\overline{\widehat{u}}}},T_{a+b+c+d}\dot{\overline{\widehat{\widetilde{u}}}}\big{)}=0,$
(2.46f) $\displaystyle
C\big{(}u,T_{d}\dot{u},T_{b}\widehat{u},T_{c}\overline{u},T_{b+d}\dot{\widehat{u}},T_{c+d}\dot{\overline{u}},T_{b+c}{\overline{\widehat{u}}},T_{b+c+d}\dot{{\overline{\widehat{u}}}}\big{)}=0,$
(2.46g) $\displaystyle
C\big{(}T_{a}\widetilde{u},T_{a+d}\dot{\widetilde{u}},T_{a+b}\widehat{\widetilde{u}},T_{a+c}\overline{\widetilde{u}},T_{a+b+d}\dot{\widehat{\widetilde{u}}},T_{a+c+d}\dot{\overline{\widetilde{u}}},T_{a+b+c}\overline{\widehat{\widetilde{u}}},T_{a+b+c+d}\dot{\overline{\widehat{\widetilde{u}}}}\big{)}=0$
(2.46h)

is also consistent around the 4D cube.

We will call (abusing notation) equation (2.46a) the $N[a,b,c]$ extension of
the scalar equation

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\overline{u},\widehat{\widetilde{u}},\overline{\widetilde{u}},{\overline{\widehat{u}}},\overline{\widehat{\widetilde{u}}}\big{)}=0.$
(2.47)

( 0,1800)(700,2200) (1800,1800)(2500, 2200) (1800, 0)(2500, 400) 50(0, 0)(700,
400) 50(700,400)(0,0) 50(700,400)(2500,400) 50(700,400)(700,2200) (850,
750)(1450 ,750) (1650, 850)(1450 ,750) (850, 750)(850,1350) (1450,
1350)(850,1350) (1450, 1350)(1450,750) (1050, 1450)(850,1350) (1050,
1450)(1650,1450) (1450, 1350)(1650,1450) (1650, 850)(1650,1450) 50(850,
750)(1050 ,850) 50(1650, 850)(1050 ,850) 50(1050, 1450)(1050 ,850) 18(1800,
0)(1450, 750) 18(0, 0)(850, 750) 18(700, 400)(1050, 850) 18(2500,400)(1650,
850) 18(2500,2200)(1650, 1450) 18(700, 2200)(1050, 1450) 18(1800, 1800)(1450,
1350) 18(0, 1800)(850, 1350)
$\dot{u}$$\dot{\widetilde{u}}$$\widetilde{u}$$u$$\dot{\widehat{u}}$$\widehat{u}$$\dot{\widehat{\widetilde{u}}}$$\widehat{\widetilde{u}}$$\dot{\overline{\widehat{\widetilde{u}}}}$$\overline{\widehat{\widetilde{u}}}$$\dot{{\overline{\widehat{u}}}}$${\overline{\widehat{u}}}$$\dot{\overline{\widetilde{u}}}$$\overline{\widetilde{u}}$$\dot{\overline{u}}$$\overline{u}$
Figure 6: 4D consistency around a hypercube.

As examples, the $2[1,1,1]$ extension of the AKP equation (2.44) is

$\displaystyle\alpha_{1}\widetilde{v}{\widehat{\overline{u}}}+\alpha_{2}\widehat{v}\widetilde{\overline{u}}+\alpha_{3}\overline{v}\widehat{\widetilde{u}}=0,$
(2.48a)
$\displaystyle\alpha_{1}\widetilde{u}{\widehat{\overline{v}}}+\alpha_{2}\widehat{u}\widetilde{\overline{v}}+\alpha_{3}\overline{u}\widehat{\widetilde{v}}=0,$
(2.48b)

and the $2[1,1,1]$ extension of the BKP equation (2.45) is

$\displaystyle\alpha_{1}\widetilde{v}{\widehat{\overline{u}}}+\alpha_{2}\widehat{v}\widetilde{\overline{u}}+\alpha_{3}\overline{v}\widehat{\widetilde{u}}+\alpha_{4}u\widehat{\widetilde{\overline{v}}}=0,$
(2.49a)
$\displaystyle\alpha_{1}\widetilde{u}{\widehat{\overline{v}}}+\alpha_{2}\widehat{u}\widetilde{\overline{v}}+\alpha_{3}\overline{u}\widehat{\widetilde{v}}+\alpha_{4}v\widehat{\widetilde{\overline{u}}}=0.$
(2.49b)

Both of them are 4D consistent by virtue of Theorem 2.10.

##  3 Reductions

###  3.1 Nonlocal systems

A few years ago, nonlocal integrable systems were introduced by Ablowitz and
Musslimani [1]. They studied the nonlocal nonlinear Schrödinger equation

$iq_{t}(x,t)+q_{xx}(x,t)+2q^{2}(x,t)q^{*}(-x,t)=0,$

where $i$ is the imaginary unit and $q^{*}$ denotes the complex conjugate of
$q$. The equation is called nonlocal as it involves functions which depend on
points $-x$, $x$ which are far apart. Most nonlocal integrable systems are
continuous or semi-discrete. In this section we show that nonlocal fully
discrete integrable equations can be constructed as reductions of 2-component
ABS systems.

For a $2[0,1]$ ABS system, (2.15), that is a system of the form

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{v},\widehat{\widetilde{v}};p,q\big{)}=0,$
(3.1a) $\displaystyle
Q\big{(}v,\widetilde{v},\widehat{u},\widehat{\widetilde{u}};p,q\big{)}=0,$
(3.1b)

where $u$ and $v$ are scalar functions of $(n,m)$. Introducing the relation

$\displaystyle v(n,m)=u(-n,m),$ (3.2)

(3.1a) reduces to

$\displaystyle Q(u(n,m),u(n+1,m),u(-n,m+1),u(-n-1,m+1);p,q)=0.$ (3.3)

Equation (3.3) is a nonlocal ABS equation. In nonlocal reduction, usually the
coupled system does not collapse to one single equation, but solutions of the
second equation can be obtained from the first one. Here, if $u(n,m)$ is a
solution of (3.3), then $v(n,m)$ given by (3.2) is solution of the equation
reduced from (3.1b). This is similar to the continuous case [1].

A $2[1,1]$ ABS system (2.16), i.e.,

$\displaystyle
Q\big{(}u,\widetilde{v},\widehat{v},\widehat{\widetilde{u}};p,q\big{)}=0,$
(3.4a) $\displaystyle
Q\big{(}v,\widetilde{u},\widehat{u},\widehat{\widetilde{v}};p,q\big{)}=0,$
(3.4b)

admits the reduction

$\displaystyle v(n,m)=u(-n,-m),$

which reduces (3.4a) to the nonlocal equation

$\displaystyle Q(u(n,m),u(-n-1,-m),u(-n,-m-1),u(n+1,m+1);p,q)=0.$ (3.5)

As examples, the nonlocal H1 equation of type (3.3) reads

$\displaystyle(u(n,m)-u(-n-1,m+1))(u(n+1,m)-u(-n,m+1))+q-p=0,$ (3.6)

and the nonlocal H1 equation of type (3.5) is

$\displaystyle(u(n,m)-u(n+1,m+1))(u(-n-1,-m)-u(-n,-m-1))+q-p=0.$ (3.7)

We note that for any ABS equation which is invariant under the transformation
$u\to-u$, its 2-component extension (3.1) allows nonlocal reduction by
introducing

$\displaystyle v(n,m)=\epsilon u(-n,m),\qquad\epsilon=\pm 1,$

and its 2-component extension (3.4) allows nonlocal reduction

$\displaystyle v(n,m)=\epsilon u(-n,-m),\qquad\epsilon=\pm 1.$

Thus, besides (3.6) and (3.7), the H1 equation also has nonlocal forms

$\displaystyle(u(n,m)+u(-n-1,m+1))(u(n+1,m)+u(-n,m+1))+q-p=0,$

and

$\displaystyle(u(n,m)-u(n+1,m+1))(u(-n-1,-m)-u(-n,-m-1))-q+p=0.$ (3.8)

###  3.2 Higher order equations from eliminations

Multi-component extensions can be reduced to higher order lattice equations by
elimination of field components. Examples of higher order lattice equations
can be found in [12, 17, 21, 23, 33, 38, 47, 51]. The first instance of such
an elimination procedure appeared in the theory of integrable discretisation
of holomorphic and harmonic functions [18, 40]. More recently such techniques
were applied in discrete integrable systems, e.g., in [33], where multi-
quadratic relations were obtained, and related to Yang-Baxter difference
systems.

#### The discrete Burgers equation

An example where the elimination can be done by a single substitution, is
provided by the discrete Burgers equation. Eliminating the variable $v$ in the
2[0,1] extension (2.14) yields the four point equation

$\frac{\widehat{\widetilde{\widetilde{u}}}\widetilde{\widetilde{u}}-q\widetilde{\widetilde{u}}-\widehat{\widetilde{\widetilde{u}}}(p-q)}{\widehat{\widetilde{\widetilde{u}}}-p}\left(p-q+\widehat{u}-\frac{\widetilde{\widetilde{u}}\widetilde{\widetilde{\underaccent{\hat}{u}}}-q\widetilde{\widetilde{\underaccent{\hat}{u}}}-\widetilde{\widetilde{u}}(p-q)}{\widetilde{\widetilde{u}}-p}\right)=p\widehat{u}-q\frac{\widetilde{\widetilde{u}}\widetilde{\widetilde{\underaccent{\hat}{u}}}-q\widetilde{\widetilde{\underaccent{\hat}{u}}}-\widetilde{\widetilde{u}}(p-q)}{\widetilde{\widetilde{u}}-p}.$

Equations on similar but larger stencils are easily obtained from
$N$-component extensions.

#### ABS equations

For ABS-equations, the generic form of the function $Q$ is, cf. [56],

$\displaystyle
Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=k_{1}u\widetilde{u}\widehat{u}\widehat{\widetilde{u}}+k_{2}\big{(}u\widetilde{u}\widehat{u}+u\widehat{u}\widehat{\widetilde{u}}+u\widetilde{u}\widehat{\widetilde{u}}+\widetilde{u}\widehat{u}\widehat{\widetilde{u}}\big{)}+k_{3}\big{(}\widetilde{u}\widehat{u}+u\widehat{\widetilde{u}}\big{)}$
$\displaystyle\hphantom{Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=}{}+k_{4}\big{(}u\widetilde{u}+\widehat{u}\widehat{\widetilde{u}}\big{)}+k_{5}\big{(}u\widehat{u}+\widetilde{u}\widehat{\widetilde{u}}\big{)}+k_{6}\big{(}u+\widetilde{u}+\widehat{u}+\widehat{\widetilde{u}}\big{)}+k_{7},$
(3.9)

where the coefficients depend on the lattice parameters $p$, $q$. Due the D4
symmetry property

$Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q\big{)}=\pm
Q\big{(}\widehat{u},\widehat{\widetilde{u}},u,\widetilde{u};p,q\big{)},$

there exists a function $G$ such that

$\displaystyle\widehat{\widetilde{u}}=G\big{(}\widehat{u},u,\widetilde{u}\big{)},\qquad\widetilde{u}=G\big{(}u,\widehat{u},\widehat{\widetilde{u}}\big{)}.$

#### From $\boldsymbol{2[0,1]}$ ABS equations to scalar six-point equations

For the $2[0,1]$ ABS equation (3.1), we have

$\displaystyle\widetilde{v}=G\big{(}v,\underaccent{\hat}{u},\underaccent{\hat}{\t
u}\big{)},\qquad\widetilde{v}=G\big{(}v,\widehat{u},\widehat{\widetilde{u}}\big{)}.$

The equation $G\big{(}v,\underaccent{\hat}{u},\underaccent{\hat}{\t
u}\big{)}=G\big{(}v,\widehat{u},\widehat{\widetilde{u}}\big{)}$ is quadratic
in $v$, so there exists a rational function $F$, linear in $\omega$, such that

$\displaystyle v=F\big{(}\underaccent{\hat}{u},\underaccent{\hat}{\t
u},\widehat{u},\widehat{\widetilde{u}};\omega\big{)},$ (3.10)

with

$\displaystyle\omega^{2}=\varphi\big{(}\underaccent{\hat}{u},\underaccent{\hat}{\t
u},\widehat{u},\widehat{\widetilde{u}}\big{)}.$ (3.11)

Substituting (3.10) into (3.1a) gives rise to

$\displaystyle Q\big{(}\underaccent{\hat}{\dt
u},\underaccent{\hat}{u},F\big{(}\underaccent{\hat}{\dt
u},\underaccent{\hat}{u},\underaccent{\tilde}{\h
u},\widehat{u};\underaccent{\tilde}{\omega}\big{)},F\big{(}\underaccent{\hat}{u},\underaccent{\hat}{\t
u},\widehat{u},\widehat{\widetilde{u}};\omega\big{)}\big{)}=0,$

which is multi-linear in the radicals $\omega$ and
$\underaccent{\tilde}{\omega}$, i.e., of the form
$m_{\omega,\underaccent{\tilde}{\omega}}:=c_{1}+c_{2}\omega+c_{3}\underaccent{\tilde}{\omega}+c_{4}\omega\underaccent{\tilde}{\omega}=0$.
Combining the different roots, using the identity

$m_{\omega,\underaccent{\tilde}{\omega}}m_{-\omega,\underaccent{\tilde}{\omega}}m_{\omega,-\underaccent{\tilde}{\omega}}m_{-\omega,-\underaccent{\tilde}{\omega}}=\big{(}{c_{1}}^{2}{\omega}^{2}{{\underaccent{\tilde}{\omega}}}^{2}-{c_{2}}^{2}{\omega}^{2}-{c_{3}}^{2}{{\underaccent{\tilde}{\omega}}}^{2}+{c_{4}}^{2}\big{)}^{2}-(2c_{1}c_{4}-2c_{2}c_{3})^{2}{\omega}^{2}{{\underaccent{\tilde}{\omega}}}^{2}$

and substituting in the expressions for $\omega^{2}$,
${\underaccent{\tilde}{\omega}}^{2}$, (3.11), one obtains a six-point equation
of the form (see Fig. 7(a))

$\displaystyle H\big{(}\underaccent{\hat}{\dt u},\underaccent{\tilde}{\h
u},\underaccent{\hat}{\t
u},\underaccent{\hat}{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=0,$ (3.12)

which is multi-quadratic in each variable. Multi-quadratic CAC equations
related to the ABS-equations have been studied in the literature, cf. [9, 33].
The multi-quadratic equations in [9, 33] have been written in quadrilateral
form. However, considering, e.g., d$Q1^{\ast}$ [33, Table 5], in the lpKdV
variable $x$, which is related to $u$ by $u=\tilde{x}-x$, the equation lives
on a six-point stencil.

For our $2[0,1]$H1 equation, we have

$\omega^{2}=\big{(}\widehat{u}-\underaccent{\hat}{u}\big{)}\big{(}\widehat{\widetilde{u}}-\widetilde{\underaccent{\hat}{u}}\big{)}\big{(}\big{(}\widehat{u}-\underaccent{\hat}{u}\big{)}\big{(}\widehat{\widetilde{u}}-\widetilde{\underaccent{\hat}{u}}\big{)}+4q-4p\big{)},$

and equation (3.12) yields the six-point equation

$\displaystyle\big{(}\widehat{\widetilde{u}}-\widehat{\underaccent{\tilde}{u}}\big{)}\big{(}\widehat{\widetilde{u}}-\underaccent{\hat}{\dt
u}\big{)}\big{(}\widetilde{\underaccent{\hat}{u}}-\underaccent{\hat}{\dt
u}\big{)}\big{(}\widetilde{\underaccent{\hat}{u}}-\widehat{\underaccent{\tilde}{u}}\big{)}\big{(}\widehat{u}-\underaccent{\hat}{u}\big{)}^{2}+2(p-q)\big{(}\widehat{\widetilde{u}}-\widehat{\underaccent{\tilde}{u}}\big{)}\big{(}\widetilde{\underaccent{\hat}{u}}-\underaccent{\hat}{\dt
u}\big{)}\big{(}\widehat{u}-\underaccent{\hat}{u}\big{)}\big{(}\widehat{\widetilde{u}}-\widetilde{\underaccent{\hat}{u}}-\underaccent{\tilde}{\dh
u}+\widehat{\underaccent{\tilde}{u}}\big{)}$
$\displaystyle\qquad{}+(p-q)^{2}\big{(}\widehat{\widetilde{u}}-\widetilde{\underaccent{\hat}{u}}+\underaccent{\tilde}{\dh
u}-\widehat{\underaccent{\tilde}{u}}\big{)}^{2}=0.$ (3.13)

To our knowledge this is a new equation. It can be written as a quadrilateral
system by introducing variables $x=\tilde{u}-u$, $y=\hat{\hat{u}}-u$.

Note that due to the symmetric positions for $u$ and $v$ in (3.1), variable
$v$ satisfies (3.12) as well, and therefore system (3.1) can serve as either a
Lax pair or an auto-BT for (3.12).

#### From $\boldsymbol{2[1,1]}$ ABS equations to scalar five-point equations

For the $2[1,1]$ ABS equation, the second component (3.4b) can be written
variously as

$\displaystyle\widetilde{v}=G\big{(}u,\underaccent{\hat}{v},\widetilde{\underaccent{\hat}{u}}\big{)},\qquad\widehat{v}=G\big{(}\widehat{\underaccent{\tilde}{u}},\underaccent{\tilde}{v},u\big{)},$
(3.14)

and the first component (3.4a) is equivalent to

$\displaystyle\underaccent{\hat}{v}=G\big{(}\underaccent{\tilde}{\dh
u},\underaccent{\tilde}{v},u\big{)}.$ (3.15)

Substituting the formulas (3.14) and (3.15) into (3.4a) gives

$\displaystyle Q\big{(}u,G\big{(}u,G(\underaccent{\tilde}{\dh
u},\underaccent{\tilde}{v},u),\widetilde{\underaccent{\hat}{u}}\big{)},G\big{(}\widehat{\underaccent{\tilde}{u}},\underaccent{\tilde}{v},u\big{)},\widehat{\widetilde{u}}\big{)}=0.$
(3.16)

Miraculously, when $Q$ has the form (3.9) the equation (3.16) factorises. One
factor is biquadratic in $u$, $\underaccent{\tilde}{v}$ and the other is

$\displaystyle H\big{(}\underaccent{\tilde}{\dh
u},\widehat{\underaccent{\tilde}{u}},\widetilde{\underaccent{\hat}{u}},u,\widehat{\widetilde{u}}\big{)}=\big{(}l_{1}u^{2}\\!+l_{2}u+l_{3}\big{)}\big{(}\underaccent{\tilde}{\dh
u}\widetilde{\underaccent{\hat}{u}}\widehat{\underaccent{\tilde}{u}}-\underaccent{\tilde}{\dh
u}\widetilde{\underaccent{\hat}{u}}\widehat{\widetilde{u}}-\underaccent{\tilde}{\dh
u}\widehat{\widetilde{u}}\widehat{\underaccent{\tilde}{u}}+\widehat{\widetilde{u}}\widehat{\underaccent{\tilde}{u}}\widetilde{\underaccent{\hat}{u}}\big{)}+\big{(}l_{2}u^{2}\\!+l_{4}u+l_{5}\big{)}\big{(}\widehat{\underaccent{\tilde}{u}}\widetilde{\underaccent{\hat}{u}}-\widehat{\widetilde{u}}\underaccent{\tilde}{\dh
u}\big{)}$ $\displaystyle\hphantom{H\big{(}\underaccent{\tilde}{\dh
u},\widehat{\underaccent{\tilde}{u}},\widetilde{\underaccent{\hat}{u}},u,\widehat{\widetilde{u}}\big{)}=}{}+\big{(}l_{3}u^{2}+l_{5}u+l_{6}\big{)}\big{(}\widehat{\underaccent{\tilde}{u}}+\widetilde{\underaccent{\hat}{u}}-\widehat{\widetilde{u}}-\underaccent{\tilde}{\dh
u}\big{)}=0$ (3.17)

with parameters defined by

$\displaystyle l_{1}=k_{1}k_{3}-k_{2}^{2},\qquad
l_{2}=-k_{5}k_{2}-k_{2}k_{4}+k_{1}k_{6}+k_{3}k_{2},\qquad
l_{3}=-k_{4}k_{5}+k_{6}k_{2}$ $\displaystyle
l_{4}=-k_{4}^{2}+k_{3}^{2}-k_{5}^{2}+k_{1}k_{7},\qquad
l_{5}=k_{7}k_{2}-k_{6}k_{4}-k_{5}k_{6}+k_{6}k_{3},\qquad
l_{6}=k_{7}k_{3}-k_{6}^{2},$

quadratic in $u$ and multi-linear in $\underaccent{\tilde}{\dh u}$,
$\widetilde{\underaccent{\hat}{u}}$, $\widehat{\underaccent{\tilde}{u}}$,
$\widehat{\widetilde{u}}$, see Fig. 7(b). We note that (3.4) provides a Lax
pair as well as an auto-BT for (3.17).

(0,20)(140,140) (0,140)(140,140) (0,20)(140,20) (140,20)(0,140)
$\underaccent{\tilde}{\dh
u}$$\underaccent{\hat}{u}$$\widehat{\underaccent{\tilde}{u}}$$\widehat{u}$$\widehat{\widetilde{u}}$$\widetilde{\underaccent{\hat}{u}}$(a)
Stencil of equaton (3.12)

(0,20)(0,160) (0,160)(140,20) (0,20)(140,160) (140,20)(140,160)
$\underaccent{\tilde}{\dh
u}$$\widetilde{\underaccent{\hat}{u}}$$\widehat{\underaccent{\tilde}{u}}$$\widehat{\widetilde{u}}$$u$(b)
Stencil of equation (3.17)

Figure 7: Equations on five- and six-point stencils are obtained from
2-component ABS equations.

The 5-point equation (3.17) is not new; it can also be obtained by elimination
of the single shifts from 4 copies of (3.9) on a $2\times 2$ stencil, and it
coincides with the discrete Toda-type equations classified by Adler in [2, 3].
This can be seen as follows. The ABS equations admit a three-leg form [4,
Section 5], either an additive form,

$\displaystyle\psi(u,\widetilde{u};p)-\psi(u,\widehat{u};q)=\phi\big{(}u,\widehat{\widetilde{u}};p,q\big{)},$

or an multiplicative form,

$\displaystyle\Psi(u,\widetilde{u};p)/\Psi(u,\widehat{u};q)=\Phi\big{(}u,\widehat{\widetilde{u}};p,q\big{)},$

with certain functions $\psi$, $\phi$, $\Psi$ and $\Phi$. In the additive
case, by means of $D4$ symmetry, the $2[1,1]$ equations we have used to derive
(3.17) can be expressed in terms of the three-leg form as

$\displaystyle
Q\big{(}u,\widetilde{v},\widehat{v},\widehat{\widetilde{u}};p,q\big{)}=0\Leftrightarrow\psi(u,\widetilde{v};p)-\psi(u,\widehat{v};q)=\phi\big{(}u,\widehat{\widetilde{u}};p,q\big{)},$
$\displaystyle
Q(\underaccent{\tilde}{v},u,\widehat{\underaccent{\tilde}{u}},\widehat{v};p,q)=0\Leftrightarrow
Q(u,\underaccent{\tilde}{v},\widehat{v},\widehat{\underaccent{\tilde}{u}};p,q)=0\Leftrightarrow\psi(u,\underaccent{\tilde}{v};p)-\psi(u,\widehat{v};q)=\phi(u,\widehat{\underaccent{\tilde}{u}};p,q),$
$\displaystyle
Q(\underaccent{\hat}{v},\widetilde{\underaccent{\hat}{u}},u,\widetilde{v};p,q)=0\Leftrightarrow
Q(u,\widetilde{v},\underaccent{\hat}{v},\widetilde{\underaccent{\hat}{u}};p,q)=0\Leftrightarrow\psi(u,\widetilde{v};p)-\psi(u,\underaccent{\hat}{v};q)=\phi(u,\widetilde{\underaccent{\hat}{u}};p,q),$
$\displaystyle Q(\underaccent{\tilde}{\dh
u},\underaccent{\hat}{v},\underaccent{\tilde}{v},u;p,q)=0\Leftrightarrow
Q(u,\underaccent{\tilde}{v},\underaccent{\hat}{v},\underaccent{\tilde}{\dh
u};p,q)=0\Leftrightarrow\psi(u,\underaccent{\tilde}{v};p)-\psi(u,\underaccent{\hat}{v};q)=\phi(u,\underaccent{\tilde}{\dh
u};p,q).$

Eliminating $\psi$ we arrive at a five-point lattice equation

$\displaystyle\phi\big{(}u,\widehat{\widetilde{u}};p,q\big{)}+\phi(u,\widehat{\underaccent{\tilde}{u}};p,q)=\phi(u,\widetilde{\underaccent{\hat}{u}};p,q)+\phi(u,\underaccent{\tilde}{\dh
u};p,q),$

which is called a discrete Toda-type equation in [2, 3]. In the multiplicative
case, we have

$\displaystyle\Phi\big{(}u,\widehat{\widetilde{u}};p,q\big{)}\Phi(u,\underaccent{\tilde}{\dh
u};p,q)=\Phi(u,\widehat{\underaccent{\tilde}{u}};p,q)\Phi(u,\widetilde{\underaccent{\hat}{u}};p,q),$

which is again a discrete Toda-type equation after a transformation $\Phi={\rm
e}^{\phi}$.

Explicitly, the 5-point equation derived from $2[1,1]$H1 or
$2[1,1]\text{Q1}^{0}$ can be written as

$\displaystyle\frac{1}{u-\underaccent{\tilde}{\dh
u}}+\frac{1}{u-\widehat{\widetilde{u}}}=\frac{1}{u-\widetilde{\underaccent{\hat}{u}}}+\frac{1}{u-\widehat{\underaccent{\tilde}{u}}}.$

This equation can also be found by elimination of the single shifts in the
7-point equation [46, equation (3)]. The systems $2[1,1]$H2,
$2[1,1]\text{Q1}^{1}$ and $2[1,1]\text{A1}^{1}$ give rise to

$\displaystyle\frac{\big{(}u-\widehat{\widetilde{u}}+k\big{)}(u-\underaccent{\tilde}{\dh
u}+k)}{\big{(}u-\widehat{\widetilde{u}}-k\big{)}(u-\underaccent{\tilde}{\dh
u}-k)}=\frac{(u-\widehat{\underaccent{\tilde}{u}}+k)(u-\widetilde{\underaccent{\hat}{u}}+k)}{(u-\widehat{\underaccent{\tilde}{u}}-k)(u-\widetilde{\underaccent{\hat}{u}}-k)},\qquad
k=p-q,$

and $2[1,1]\text{H3}^{\delta}$, $2[1,1]$A2 yield

$\displaystyle\frac{\big{(}ku-\widehat{\widetilde{u}}\big{)}(ku-\underaccent{\tilde}{\dh
u})}{\big{(}u-k\widehat{\widetilde{u}}\big{)}(u-k\underaccent{\tilde}{\dh
u})}=\frac{(ku-\widehat{\underaccent{\tilde}{u}})(ku-\widetilde{\underaccent{\hat}{u}})}{(u-k\widehat{\underaccent{\tilde}{u}})(u-k\widetilde{\underaccent{\hat}{u}})},\qquad
k=p/q.$

In [35] certain $a$\- and $m$-bond systems lead to double H-type vertex
systems, cf. [35, Proposition 6.1]. As mentioned in [35, Remark 6.2], two of
these are two-component extensions of the type discussed here, the third is a
mixed 2-component system, H1$\times$H2, and they provide Bäcklund
transformations for the above 5-point schemes.

The other $2[1,1]$ABS equations relate to the following 5-point equations:

$\displaystyle 2[1,1]\text{Q2}\colon\quad$
$\displaystyle\underaccent{\tilde}{\dh
u}\widetilde{\underaccent{\hat}{u}}\widehat{\underaccent{\tilde}{u}}-\underaccent{\tilde}{\dh
u}\widetilde{\underaccent{\hat}{u}}\widehat{\widetilde{u}}-\underaccent{\tilde}{\dh
u}\widehat{\widetilde{u}}\widehat{\underaccent{\tilde}{u}}+\widehat{\widetilde{u}}\widehat{\underaccent{\tilde}{u}}\widetilde{\underaccent{\hat}{u}}-2\big{(}u+k^{2}\big{)}\big{(}\widehat{\underaccent{\tilde}{u}}\widetilde{\underaccent{\hat}{u}}-\widehat{\widetilde{u}}\underaccent{\tilde}{\dh
u}\big{)}$
$\displaystyle{}+\big{(}u-k^{2}\big{)}^{2}\big{(}\widehat{\underaccent{\tilde}{u}}+\widetilde{\underaccent{\hat}{u}}-\widehat{\widetilde{u}}-\underaccent{\tilde}{\dh
u}\big{)}=0,\qquad k=p-q,$ $\displaystyle 2[1,1]\text{Q3}\colon\quad$
$\displaystyle\underaccent{\tilde}{\dh
u}\widetilde{\underaccent{\hat}{u}}\widehat{\underaccent{\tilde}{u}}-\underaccent{\tilde}{\dh
u}\widetilde{\underaccent{\hat}{u}}\widehat{\widetilde{u}}-\underaccent{\tilde}{\dh
u}\widehat{\widetilde{u}}\widehat{\underaccent{\tilde}{u}}+\widehat{\widetilde{u}}\widehat{\underaccent{\tilde}{u}}\widetilde{\underaccent{\hat}{u}}-\big{(}k+k^{-1}\big{)}u\big{(}\widehat{\underaccent{\tilde}{u}}\widetilde{\underaccent{\hat}{u}}-\widehat{\widetilde{u}}\underaccent{\tilde}{\dh
u}\big{)}$
$\displaystyle{}+\left(u^{2}+\frac{\delta^{2}\big{(}k-k^{-1}\big{)}^{2}}{4}\right)\big{(}\widehat{\underaccent{\tilde}{u}}+\widetilde{\underaccent{\hat}{u}}-\widehat{\widetilde{u}}-\underaccent{\tilde}{\dh
u}\big{)}=0,\qquad k=p/q,$ $\displaystyle 2[1,1]\text{Q4}\colon\quad$
$\displaystyle\big{(}1-k^{2}u^{2}\big{)}\big{(}\underaccent{\tilde}{\dh
u}\widetilde{\underaccent{\hat}{u}}\widehat{\underaccent{\tilde}{u}}-\underaccent{\tilde}{\dh
u}\widetilde{\underaccent{\hat}{u}}\widehat{\widetilde{u}}-\underaccent{\tilde}{\dh
u}\widehat{\widetilde{u}}\widehat{\underaccent{\tilde}{u}}+\widehat{\widetilde{u}}\widehat{\underaccent{\tilde}{u}}\widetilde{\underaccent{\hat}{u}}\big{)}-2\big{(}1-k^{2}\big{)}u\big{(}\widehat{\underaccent{\tilde}{u}}\widetilde{\underaccent{\hat}{u}}-\widehat{\widetilde{u}}\underaccent{\tilde}{\dh
u}\big{)}$
$\displaystyle{}+\big{(}u^{2}-k^{2}\big{)}\big{(}\widehat{\underaccent{\tilde}{u}}+\widetilde{\underaccent{\hat}{u}}-\widehat{\widetilde{u}}-\underaccent{\tilde}{\dh
u}\big{)}=0,\qquad k=\mathrm{sn}(p-q).$

#### Higher order scalar equations from $\boldsymbol{2[1,1,1]}$ AKP and BKP

Here we show that higher order equations can also be obtained from multi-
component extensions of 3D lattice equations.

One can eliminate $u$ from the $2[1,1,1]$ AKP equation (2.48) as follows.
Denote the left hand side from (2.48a) by $E$ and the left hand side from
(2.48b) by $F$. First we solve $\widehat{\underaccent{\tilde}{u}}$ from
$\underaccent{\tilde}{E}=0$, $\widehat{\underaccent{\bar}{u}}$ from
$\underaccent{\bar}{E}=0$, $\widetilde{\underaccent{\hat}{u}}$ from
$\underaccent{\bar}{\dt F}=0$ and $\underaccent{\bar}{\dh u}$ from
$\underaccent{\hat}{\dt F}=0$. Then we substitute
$\widehat{\underaccent{\tilde}{u}},\widehat{\underaccent{\bar}{u}}$ into
$\underaccent{\bar}{\dt
F}/(\underaccent{\tilde}{v}\underaccent{\bar}{v}\widehat{v})$ and
$\widetilde{\underaccent{\hat}{u}}$, $\overline{\underaccent{\hat}{u}}$ into
$\underaccent{\hat}{E}/(\underaccent{\hat}{v}\widetilde{v}\overline{v})$. The
difference of the results gives rise to the 12-point equation

$\displaystyle a_{1}^{2}\left[\frac{\underaccent{\bar}{\underaccent{\hat}{\t
v}}}{\underaccent{\hat}{v}\underaccent{\bar}{v}\widetilde{v}}-\frac{{\widehat{\overline{\underaccent{\tilde}{v}}}}}{\underaccent{\tilde}{v}\widehat{v}\overline{v}}\right]+a_{2}^{2}\left[\frac{\underaccent{\bar}{\underaccent{\tilde}{\h
v}}}{\underaccent{\tilde}{v}\underaccent{\bar}{v}\widehat{v}}-\frac{\widetilde{\overline{\underaccent{\hat}{v}}}}{\underaccent{\hat}{v}\widetilde{v}\overline{v}}\right]+a_{3}^{2}\left[\frac{\underaccent{\tilde}{\underaccent{\hat}{\b
v}}}{\underaccent{\tilde}{v}\underaccent{\hat}{v}\overline{v}}-\frac{\widehat{\widetilde{\underaccent{\bar}{v}}}}{\underaccent{\bar}{v}\widetilde{v}\widehat{v}}\right]=0.$
(3.18)

Similar to the 2D case, the coupled system (2.48) can be considered as either
a BT or a Lax pair of (3.18).

From the 2-component BKP equation (2.49), using a similar elimination scheme,
one obtains the 14-point King–Schief equation

$\displaystyle a_{1}^{2}\left[\frac{\underaccent{\bar}{\underaccent{\hat}{\t
v}}}{\underaccent{\hat}{v}\underaccent{\bar}{v}\widetilde{v}}-\frac{{\widehat{\overline{\underaccent{\tilde}{v}}}}}{\underaccent{\tilde}{v}\widehat{v}\overline{v}}\right]+a_{2}^{2}\left[\frac{\underaccent{\bar}{\underaccent{\tilde}{\h
v}}}{\underaccent{\tilde}{v}\underaccent{\bar}{v}\widehat{v}}-\frac{\widetilde{\overline{\underaccent{\hat}{v}}}}{\underaccent{\hat}{v}\widetilde{v}\overline{v}}\right]+a_{3}^{2}\left[\frac{\underaccent{\tilde}{\underaccent{\hat}{\b
v}}}{\underaccent{\tilde}{v}\underaccent{\hat}{v}\overline{v}}-\frac{\widehat{\widetilde{\underaccent{\bar}{v}}}}{\underaccent{\bar}{v}\widetilde{v}\widehat{v}}\right]+a_{4}^{2}\left[\frac{\widehat{\widetilde{\overline{v}}}}{\widetilde{v}\widehat{v}\overline{v}}-\frac{\underaccent{\bar}{\dth
v}}{\underaccent{\bar}{v}\underaccent{\tilde}{v}\underaccent{\hat}{v}}\right]=0,$
(3.19)

which arose in the study of nondegenerate Cox lattices [37]. Its BT/Lax pair
is provided by (2.49), with $u$ acting as an eigenvalue function, cf. [37,
equation (30)], and the equation degenerates to equation (3.18) when
$a_{4}=0$. The equations (3.18) and (3.19) are satisfied by solutions of the
AKP equation and the BKP equation respectively.

The analysis in [37, Section 5] reveals that Cox–Menelaus lattices are
intimately related to the AKP equation. We expect that (3.18) will play a
similar role in that context as (3.19) plays in the context of nondegenerate
Cox lattices. We further note that equation (3.19) relates, by a simple
coordinate transformation [55], to an equation that appeared in a completely
different setting, namely through the notion of duality, employing
conservation laws of the lattice AKP equation [54].

###  3.3 $\boldsymbol{N}$-component $\boldsymbol{\mathfrak{q}}$-Painlevé III
equation

The results in Section 2 remain true for non-autonomous multi-dimensionally
consistent systems, extending spacing parameters $p\to p(n)$, $q\to q(m)$ and
$r\to r(l)$. There are close relations, cf. [32, 42], between non-autonomous
ABS lattice equations and discrete Painlevé equations exploing the affine Weyl
group. A particular example is provided by a non-autonomous version of the
lpmKdV equation (2.26) which can be reduced to a $\mathfrak{q}$-Painlevé III
equation, by performing a periodic reduction. We use this example to
illustrate that such a link extends to the multi-component case.

For the non-autonomous lpmKdV equation

$Q\big{(}u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}}\big{)}=p\big{(}u\widetilde{u}-\widehat{u}\widehat{\widetilde{u}}\big{)}-q\big{(}u\widehat{u}-\widetilde{u}\widehat{\widetilde{u}}\big{)}=0,\qquad
p=p_{0}\mathfrak{q}^{n},\qquad q=q_{0}\mathfrak{q}^{m},$

we use the bottom and front equations on its multi-component consistent cube,
i.e.,

$\displaystyle
p\big{(}uT_{a}\widetilde{u}-(T_{b}\widehat{u})\big{(}T_{a+b}\widehat{\widetilde{u}}\big{)}\big{)}-q\big{(}uT_{b}\widehat{u}-(T_{a}\widetilde{u})\big{(}T_{a+b}\widehat{\widetilde{u}}\big{)}\big{)}=0,$
(3.20a) $\displaystyle
p\big{(}uT_{a}\widetilde{u}-(T_{c}\overline{u})\big{(}T_{a+c}\overline{\widetilde{u}}\big{)}\big{)}-r\big{(}uT_{c}{\overline{u}}-(T_{a}\widetilde{u})\big{(}T_{a+c}\overline{\widetilde{u}}\big{)}\big{)}=0.$
(3.20b)

Imposing the periodic reduction (cf. [32])

$\displaystyle\overline{\widehat{\widetilde{u}}}=u,\qquad a+b+c=0$ (3.21)

on (3.20), and replacing $p$, $q$ by $p\mathfrak{q}$, $q\mathfrak{q}$, and the
first equation (3.20a) is unchanged but we rewrite it as

$\displaystyle
T_{a}\widetilde{u}\big{(}pu+qT_{a+b}\widehat{\widetilde{u}}\big{)}=T_{b}\widehat{u}\big{(}qu+pT_{a+b}\widehat{\widetilde{u}}\big{)}$
(3.22a) for convenience. For (3.20b), after a tilde/hat-shift and making use
of the reduction (3.21), we have $\displaystyle
T_{a}\widehat{\widetilde{\widetilde{u}}}\big{(}p\mathfrak{q}\widehat{\widetilde{u}}+rT_{-b}\widetilde{u}\big{)}=T_{-a-b}u\big{(}r\widehat{\widetilde{u}}+p\mathfrak{q}T_{-b}\widetilde{u}\big{)}.$
(3.22b)

Then, introducing diagonal matrices $f$ and $g$ by

$f=(T_{a}\widetilde{u})\big{(}T_{a+b}\widehat{\widetilde{u}}\big{)}^{-1},\qquad
g=\big{(}T_{a+b}\widehat{\widetilde{u}}\big{)}u^{-1},\qquad
t=\frac{\mathfrak{q}p}{r},\qquad k=\frac{r}{q\mathfrak{q}},$

from (3.22a) and (3.22b) we find

$\displaystyle T_{-a}\undertilde{f}=\frac{1+ktg}{fg(kt+g)},\qquad
T_{a}\widetilde{g}=\frac{1+tf}{fg(t+f)},$

which is an $N$-component $\mathfrak{q}$-Painlevé III equation.

Taking $N=3$ and $a=1$, this yield the 3-component $\mathfrak{q}$-Painlevé III
system

$\displaystyle\undertilde{f}_{3}=\frac{1+ktg_{1}}{f_{1}g_{1}(kt+g_{1})},\qquad$
$\displaystyle\widetilde{g}_{2}=\frac{1+tf_{1}}{f_{1}g_{1}(t+f_{1})},$
$\displaystyle\undertilde{f}_{1}=\frac{1+ktg_{2}}{f_{2}g_{2}(kt+g_{2})},\qquad$
$\displaystyle\widetilde{g}_{3}=\frac{1+tf_{2}}{f_{2}g_{2}(t+f_{2})},$
$\displaystyle\undertilde{f}_{2}=\frac{1+ktg_{3}}{f_{3}g_{3}(kt+g_{3})},\qquad$
$\displaystyle\widetilde{g}_{1}=\frac{1+tf_{3}}{f_{3}g_{3}(t+f_{3})}.$

##  4 Exact solutions

###  4.1 Solutions with jumping property

Using $N$ solutions to the scalar equation, one can construct a solution for
the $N$-component equation (2.12). For $j=1,2,\dots,N$, let $w_{j}(n,m)$ be a
solution to the scalar equation (2.32), i.e.,

$\displaystyle
Q\big{(}w_{j},\widetilde{w}_{j},\widehat{w}_{j},\widehat{\widetilde{w}}_{j};p,q\big{)}=0.$
(4.1)

If we set

$\displaystyle u_{k}(n,m)=w_{k-an-bm}(n,m),$ (4.2)

where the sub-index is taken modulo $N$, the system of equations (2.12)
comprises $N$ copies of the scalar equation. Thus, (4.2) provides a solution
to (2.12). If (2.12) is non-decoupled, i.e., when
$\operatorname{gcd}(a,b,N)=1$, then $u_{k}$ will run over $\\{w_{j}\colon
j=1,2,\dots N\\}$ by virtue of Lemma B.1 in Appendix B. The pattern of $u_{k}$
is depicted in Fig. 8.

(0,20)(80,20) (0,50)(80,50) (0,80)(80,80) (0,110)(80,110) (0,140)(80,140)
3.000(110,20)(80,20) 3.000(110,50)(80,50) 3.000(110,80)(80,80)
3.000(110,110)(80,110) 3.000(110,140)(80,140) (110,20)(160,20)
(110,50)(160,50) (110,80)(160,80) (110,110)(160,110) (110,140)(160,140)
(20,0)(20,80) (50,0)(50,80) (80,0)(80,80) (110,0)(110,80) (140,0)(140,80)
3.000(20,110)(20,80) 3.000(50,110)(50,80) 3.000(80,110)(80,80)
3.000(110,110)(110,80) 3.000(140,110)(140,80) (20,110)(20,160)
(50,110)(50,160) (80,110)(80,160) (110,110)(110,160) (140,110)(140,160)
$w_{k}$$w_{k+a}$$w_{k+2a}$$w_{k-a}$$w_{k}$$w_{k}$$w_{k+a}$$w_{k+2a}$$w_{k-a}$$w_{k}$$w_{k+b}$$w_{k+2b}$$w_{k-b}$$w_{k+b}$$w_{k+2b}$$w_{k-b}$
Figure 8: $u_{k}$ defined by (4.2) on $(n,m)$ lattice.

For the $2[1,1]$ ABS equation (2.16), according to (4.2) its solution can be
given by

$\displaystyle u_{1}=\begin{cases}w_{1},&n+m\equiv 0\ (\text{mod}\ 2),\\\
w_{2},&n+m\equiv 1,\end{cases}\qquad u_{2}=\begin{cases}w_{2},&n+m\equiv 0,\\\
w_{1},&n+m\equiv 1,\end{cases}$

where each $w_{i}$ satisfies scalar equation (4.1). This coincides with the
result in [22]. For the $3[1,1]$ ABS equation (2.12), solutions can be
presented by

$u_{k}=\left\\{\begin{array}[]{ll}w_{k},&n+m\equiv 0\ (\text{mod}\ 3),\\\
w_{k-1},&n+m\equiv 1,\\\ w_{k-2},&n+m\equiv 2,\end{array}\right.$ (4.3)

provided each $w_{i}$ solves the scalar equation (4.1). Note that (4.2) has
the so-called jumping property (cf. [22]) and for (4.3) this property is
illustrated by Fig. 9.

$w_{3}$:$n$$n+1$$n+2$$n+3$$n+4$$n+5$(875,2708)(6675,2708)
(875,1708)(6675,1708) (875,708)(6675,708)
$w_{2}$:$n$$n+1$$n+2$$n+3$$n+4$$n+5$$w_{1}$:$n$$n+1$$n+2$$n+3$$n+4$$n+5$50.000(1275,2708)(3275,708)
50.000(4275,2708)(3275,708) 50.000(4275,2708)(6275,708)
150.000(1275,1708)(2275,708) 150.000(3275,2708)(2275,708)
150.000(3275,2708)(5275,708) 150.000(6275,2708)(5275,708)
300.000(1275,708)(2275,2708) 300.000(4275,708)(2275,2708)
300.000(4275,708)(5275,2708) 300.000(6275,1708)(5275,2708)
$u_{1}$:300.000(1075,100)(2075,100) $u_{2}$:150.000(2875,100)(3875,100)
$u_{3}$:60.000(4675,100)(5675,100)  Figure 9: Jumping property of $u_{i}$ in
(4.3) in the tilde-direction.

Solutions for multi-component extensions of 3D equations, given in Section
2.3, can be given in a similar fashion as for 2D equations. If $\\{w_{j}\\}$
are $N$ solutions of the 3D scalar equation (2.47), then

$\displaystyle u_{k}(n,m,l)=w_{k-an-bm-cl}(n,m,l),$

where the sub-index is taken modulo $N$, provides a solution of the $N[a,b,c]$
extension (2.46a).

#### Bilinear equations

Many equations in the ABS list have been bilinearized [27]. If a scalar ABS
equation (2.32) has a bilinear form111Some equations need more than two
functions to get bilinear forms. Here we just employ (4.4) as a generic form.

$\displaystyle
H\big{(}\mathfrak{f},\mathfrak{g},\widetilde{\mathfrak{f}},\widehat{\mathfrak{f}},\widehat{\mathfrak{g}},\widehat{\mathfrak{g}},\widehat{\widetilde{\mathfrak{f}}},\widehat{\widetilde{\mathfrak{g}}}\big{)}=0,$
(4.4)

with transformation $u=F(\mathfrak{f},\mathfrak{g})$, for example, H1 equation
(2.17) has bilinear form

$\displaystyle\widehat{g}\widetilde{f}-\widetilde{g}\widehat{f}+(\alpha-\beta)\big{(}\widetilde{f}\widehat{f}-f\widehat{\widetilde{f}}\big{)}=0,\qquad
g\widehat{\widetilde{f}}-\widehat{\widetilde{g}}f+(\alpha+\beta)\big{(}f\widehat{\widetilde{f}}-\widetilde{f}\widehat{f}\big{)}=0,$

where $p=-\alpha^{2}$, $q=-\beta^{2}$, then for the $N[a,b]$ system (2.12),
its bilinear form can be given by

$\displaystyle
H\big{(}f,g,T_{a}\widetilde{f},T_{a}\widetilde{g},T_{b}\widehat{f},T_{b}\widehat{g},T_{a+b}\widehat{\widetilde{f}},T_{a+b}\widehat{\widetilde{g}}\big{)}=0,$
(4.5)

through the transformation $u=F(f,g)$ where $f$, $g$ are diagonal forms in
(2.31).

With respect to solutions, suppose $(\mathfrak{f}_{j},\mathfrak{g}_{j})$ are
any arbitrary solutions of (4.4). Using them we define

$\displaystyle f_{k}(n,m)=\mathfrak{f}_{k-an-bm}(n,m),\qquad
g_{k}(n,m)=\mathfrak{g}_{k-an-bm}(n,m).$

Then, $(f,g)$ composed by such components will be a solution to (4.5).

As an example, the $2[0,1]$ H1 equation (2.18) has a bilinear form

$\displaystyle\widehat{g}_{2}\widetilde{f}_{1}-\widetilde{g}_{1}\widehat{f}_{2}+(\alpha-\beta)\big{(}\widetilde{f}_{1}\widehat{f}_{2}-f_{1}\widehat{\widetilde{f}}_{2}\big{)}=0,$
$\displaystyle\widehat{g}_{1}\widetilde{f}_{2}-\widetilde{g}_{2}\widehat{f}_{1}+(\alpha-\beta)\big{(}\widetilde{f}_{2}\widehat{f}_{1}-f_{1}\widehat{\widetilde{f}}_{1}\big{)}=0,$
$\displaystyle
g_{1}\widehat{\widetilde{f}}_{2}-\widehat{\widetilde{g}}_{2}f_{1}+(\alpha+\beta)\big{(}f_{1}\widehat{\widetilde{f}}_{2}-\widetilde{f}_{1}\widehat{f}_{2}\big{)}=0,$
$\displaystyle
g_{2}\widehat{\widetilde{f}}_{1}-\widehat{\widetilde{g}}_{1}f_{2}+(\alpha+\beta)\big{(}f_{2}\widehat{\widetilde{f}}_{1}-\widetilde{f}_{2}\widehat{f}_{1}\big{)}=0$

with transformation $u_{i}=\alpha n+\beta m+r-{g_{i}}/{f_{i}}$, and (with $m$
taken modulo 2)

$\displaystyle f_{1}=\begin{cases}\mathfrak{f}_{1},&m\equiv 0,\\\
\mathfrak{f}_{2},&m\equiv
1,\end{cases}\qquad\\!f_{2}=\begin{cases}\mathfrak{f}_{2},&m\equiv 0,\\\
\mathfrak{f}_{1},&m\equiv
1,\end{cases}\qquad\\!g_{1}=\begin{cases}\mathfrak{g}_{1},&m\equiv 0,\\\
\mathfrak{g}_{2},&m\equiv
1,\end{cases}\qquad\\!g_{2}=\begin{cases}\mathfrak{g}_{2},&m\equiv 0,\\\
\mathfrak{g}_{1},&m\equiv 1,\end{cases}\\!$

where $(\mathfrak{f}_{i},\mathfrak{g}_{i})$ are solutions of (4.4).

###  4.2 Nonlocal case

For some nonlocal ABS equations, if the scalar equation (2.32) admits an odd
or even solution, i.e.,

$\displaystyle u(n,m)=\epsilon u(-n,-m),\qquad\epsilon=\pm 1,$

then, such solutions may be used to construct a solution to the nonlocal
equation.

As an example, let us look at the nonlocal H1 equation (3.8). The local H1
equation (2.17) has rational solution [59]

$\displaystyle u=\frac{n}{\mu}+\frac{m}{\nu}-\frac{g_{[N]}}{f_{[N]}},$ (4.6)

where $p=-1/\mu^{2}$, $q=-1/\nu^{2}$, $f_{[N]}$ and $g_{[N]}$ are Casoratians
[59]

$\displaystyle
f_{[N]}=\big{|}\widehat{N-1}\big{|}=|\alpha(n,m,0),\alpha(n,m,1),\dots,\alpha(n,m,N-1)|,$
$\displaystyle g_{[N]}=\big{|}\widehat{N-2},N\big{|}-Nf_{[N]}.$

The Casoratian vector is

$\alpha(n,m,l)=(\alpha_{0},\alpha_{1},\dots,\alpha_{N-1})^{\rm
T},\qquad\alpha_{j}=\frac{1}{(2j+1)!}\partial^{2j+1}_{s_{i}}\psi_{i}|_{s_{i}=0},$

with

$\psi_{i}(n,m,l)=\psi_{i}^{+}(n,m,l)+\psi_{i}^{-}(n,m,l),\qquad\psi_{i}^{\pm}(n,m,l)=(1\pm
s_{i})^{l}(1\pm\mu s_{i})^{n}(1\pm\nu s_{i})^{m}.$

$\psi_{i}^{\pm}$ has the form

$\psi_{i}^{\pm}(n,m,l)=\pm\frac{1}{2}\sum^{\infty}_{h=0}\alpha^{\pm}_{h}s_{i}^{h}=\pm\frac{1}{2}\exp\left[-\sum^{\infty}_{j=1}\frac{(\mp
s_{i})^{j}}{j}\accentset{\circ}{x}_{j}\right],$

where

$\displaystyle\accentset{\circ}{x}_{j}=x_{j}+l,\qquad
x_{j}=\mu^{j}n+\nu^{j}m,\qquad j\in\mathbb{Z}.$

Then, $\alpha_{j}=\alpha^{+}_{2j+1}$ can be expressed in terms of
$\\{x_{j}\\}$, see [59]. The first few $f_{[N]}$ and $g_{[N]}$ are

$\displaystyle f_{[1]}=x_{1},\qquad g_{[1]}=1,\qquad
f_{[2]}=\frac{x_{1}^{3}-x_{3}}{3},\qquad g_{[2]}=x_{1}^{2},$ $\displaystyle
f_{[3]}=\frac{1}{45}x_{1}^{6}-\frac{1}{9}x_{1}^{3}x_{3}+\frac{1}{5}x_{1}x_{5}-\frac{1}{9}x_{3}^{2},\qquad
g_{[3]}=\frac{2}{15}x_{1}^{5}-\frac{1}{3}x_{1}^{2}x_{3}+\frac{1}{5}x_{5}.$

It has been proved in [59] that $f_{[N]}$ and $g_{[N]}$ only depend on
$\\{x_{1},x_{3},\dots,x_{2N-1}\\}$ and are homogeneous with degrees

$\mathcal{D}[f_{N}]=\frac{N(N+1)}{2},\qquad\mathcal{D}[g_{N}]=\frac{N(N+1)}{2}-1,$

defined by the formula $\mathcal{D}\big{[}\prod\limits_{i\geq
1}x_{i}^{k_{i}}\big{]}=\sum\limits_{i\geq 1}ik_{i}$. The function $u$ given by
(4.6) is an odd function, and so it provides a solution to the nonlocal H1
(3.8).

We remark that rational solutions in terms of $\\{x_{j}\\}$ have been obtained
for all the ABS equations except Q4 [59, 61]. This implies rational solutions
for nonlocal ABS equations can be derived, which will be explored elsewhere.

##  5 Conclusion

We have presented a systematic way to generate multi-component lattice
equations which are CAC, by making use of the cyclic group. We note that
cyclic matrices have been used in 3-point differential-difference equations
[10, 15] and fully discrete Lax pairs [20] to generate multi-component
systems.

Although the multi-component extensions we have considered are more general
than “the trivial Toeplitz extension” introduced in Appendix B of the PhD
thesis of J. Atkinson [8], the key idea is the same: starting from a single-
component CAC (2D or 3D) discrete system (2.2), replacing $u$ by a diagonal
matrix (2.4), applying permutations $T_{k}$ on shifted field components, one
obtains a multi-component CAC system (2.9). Posing the equations on lattices,
D4 symmetry, criteria for decoupled and non-decoupled cases, BTs, auto-BTs,
Lax pairs, solutions, nonlocal reductions, elimination of components to get
equations on larger stencils, and reduction to multi-component discrete
Painlevé equations, have been investigated in detail.

Isolated examples of multi-component extensions sporadically appear in the
literature in different contexts. We mention: the two-component potential KdV
system [14, Table 5], cf. [22]; a two-component extension [31, equation
(3.17)] of the (non-potential) lattice mKdV equation [44, equation (2.49)];
and the linear system of tetrahedral equations [37, equation (30)] that
constitutes a two-component version of the BKP equation. We have provided a
basic understanding for all these examples.

What we have not touched upon, is the fact that multi-component extension can
also be applied to systems of equations. For example, the discrete Boussinesq
family contains several multi-component CAC lattice systems [24]. These also
allow $N[a,b]$ extension, cf. the $2[1,1]$ Boussinesq extension given in [22].

Finally, we’d like to point out that the multi-component extensions considered
here are commutative. Non-commutative lattice equations, which are multi-
component generalisations, have also been considered in the literature: 3D
matrix discrete interable systems can be found in [45, equations (1.5)–(1.9)],
a non-commutative version of $\text{Q1}^{0}$ was given in [11, equation (23)],
a matrix version of H1 was obtained in [19, equation (2.1)], cf. [60], where
several matrix discrete integrable equations derived from the Cauchy matrix
approach were presented.

##  Appendix A Multi-component Lax pair from BT

For the scalar consistent lattice system (2.2) with $Q^{*}=Q$, there exist
functions $G_{i}$ such that

$\displaystyle\widehat{\widetilde{u}}=G_{1}(u,\widetilde{u},\widehat{u}),\qquad\overline{\widetilde{u}}=G_{2}(u,\widetilde{u},\overline{u}),\qquad{\overline{\widehat{u}}}=G_{3}(u,\overline{u},\widehat{u}),$
(A.1)

which leads to the Lax pair (2.33) for (2.32). It holds as well if $u$ is
extended to the diagonal form (2.4). Introduce $\overline{u}=gf^{-1}$ where
$f$ and $g$ are given in (2.31). From (A.1) there exist functions $F_{2}$ and
$F_{3}$ such that

$\widetilde{g}\widetilde{f}^{-1}=F_{2}(u,\widetilde{u},f,g),\qquad\widehat{g}\widehat{f}^{-1}=F_{3}(u,\widehat{u},f,g),$

which leads to a Lax pair for (2.1) where $u$ is (2.4):

$\displaystyle\Phi\big{(}\widetilde{f},\widetilde{g}\big{)}=L(u,\widetilde{u})\Phi(f,g),\qquad\Phi\big{(}\widehat{f},\widehat{g}\big{)}=M(u,\widehat{u})\Phi(f,g),$
(A.2)

where

$\displaystyle\Phi(f,g)=(f_{1},g_{1},f_{2},g_{2},\dots,f_{N},g_{N})^{\rm T},$
(A.3)

and $L$ and $M$ are defined by (2.35) through $\mathcal{L}$ and $\mathcal{M}$.

Meanwhile, from (A.1) we have

$\displaystyle
T_{a+b}\widehat{\widetilde{u}}=G_{1}(u,T_{a}\widetilde{u},T_{b}\widehat{u}),\qquad
T_{a+c}\overline{\widetilde{u}}=G_{2}(u,T_{a}\widetilde{u},T_{c}\overline{u}),\qquad
T_{b+c}{\overline{\widehat{u}}}=G_{3}(u,T_{c}\overline{u},T_{b}\widehat{u}),$

and consequently

$\displaystyle
T_{a+c}\big{(}\widetilde{g}\widetilde{f}^{-1}\big{)}=F_{2}(u,T_{a}\widetilde{u},T_{c}f,T_{c}g),\qquad
T_{b+c}\big{(}\widehat{g}\widehat{f}^{-1}\big{)}=F_{3}(u,T_{b}\widehat{u},T_{c}f,T_{c}g).$

From this and the definition (A.3) for $\Phi$, we obtain (with $\theta$
defined in Theorem 2.6)

$\displaystyle\Phi(T_{c}f,T_{c}g)=\theta^{c}\Phi(f,g),\qquad\Phi\big{(}T_{a+c}\widetilde{f},T_{a+c}\widetilde{g}\big{)}=\theta^{a+c}\Phi\big{(}\widetilde{f},\widetilde{g}\big{)},$
$\displaystyle\Phi\big{(}T_{b+c}\widehat{f},T_{b+c}\widehat{g}\big{)}=\theta^{b+c}\Phi\big{(}\widehat{f},\widehat{g}\big{)}.$

Compared with (A.2), it comes to

$\theta^{a+c}\Phi\big{(}\widetilde{f},\widetilde{g}\big{)}=L(u,T_{a}\widetilde{u})\theta^{c}\Phi(f,g),\qquad\theta^{b+c}\Phi\big{(}\widehat{f},\widehat{g}\big{)}=M(u,T_{b}\widehat{u})\theta^{c}\Phi(f,g),$

which gives rise to the Lax pair (2.34) for (2.12).

##  Appendix B Proof of Theorem 2.8

We first prove a useful lemma. Although we believe it is an elementary result
in number theory, we include it for completeness. We denote
$\mathbb{N}=\\{1,2,\ldots\\}$ and $\mathbb{N}^{0}=\\{0,1,2,\ldots\\}$.

######  Lemma B.1.

Let $a,b\in\mathbb{N}^{0}$, $N\in\mathbb{N}$ such that
$\operatorname{gcd}(a,b,N)=1$, and let $\mathbb{A}=\\{ia+jb+kN\colon
i,j,k\in\mathbb{Z}\\}$. There are $i_{0},j_{0}\in\mathbb{N}$ and
$k_{0}\in\mathbb{Z}$ such that

$\displaystyle 1=i_{0}a+j_{0}b+k_{0}N,$ (B.1)

and hence $\mathbb{A}=\mathbb{Z}$.

###### Proof.

Let $s_{0}=i_{0}a+j_{0}b+k_{0}N$ be the smallest positive integer in
$\mathbb{A}$. For all $s>0\in\mathbb{A}$, there are $i,j,k\in\mathbb{Z}$ such
that $s=ia+jb+kN$, and there exist $q,r\in\mathbb{Z}$ such that $s=s_{0}q+r$
where $q>0$ and $0\leq r<s_{0}$. Then we have

$\displaystyle 0\leq
r=s-s_{0}q=(i-i_{0}q)a+(j-j_{0}q)b+(k-k_{0}q)N\in\mathbb{A}.$

Since $s_{0}$ is the smallest positive number in $\mathbb{A}$ and $0\leq
r<s_{0}$, we must have $r=0$, which leads to $s=s_{0}q$. As $s$ was arbitrary
it follows that $s_{0}$ is a divisor of $\operatorname{gcd}(a,b,N)$, which
implies $s_{0}=1$ because $\operatorname{gcd}(a,b,N)=1$. Thus we reach (B.1)
and consequently $\mathbb{A}$ covers $\mathbb{Z}$. If $i_{0}$ and $j_{0}$ are
not positive, there exist $i,j\in\mathbb{N}$ such that $iN+i_{0}>0$ and
$jN+j_{0}>0$, and from (B.1) we have

$1=(iN+i_{0})a+(jN+j_{0})b+(k_{0}-ia-jb)N.$

We now prove Theorem 2.8.

###### Proof.

We consider two cases, $d=\operatorname{gcd}(a,b,N)>1$ and $d=1$.

$d>1$: The $N[a,b]$ system (2.12), viewed as a set which we denote by $S$
here, contains $N$ equations of the form

$\displaystyle Q[k]\colon\
Q\big{(}u_{k},\widetilde{u}_{k+a},\widehat{u}_{k+b},\widehat{\widetilde{u}}_{k+a+b}\big{)}=0,\qquad
k=1,2,\dots,N,$

where the sub-index $i$ on the components $u_{i}$ and hence the ‘argument’ of
$Q[\cdot]$) is taken modulo $N$. These equations depend on $N$ field
components,

$V=\\{u_{i}\colon 1\leq i\leq N\\}.$

For each $1\leq i\leq d$ we define a subset of $M=N/d$ equations

$S_{i}=\\{Q[i],Q[d+i],\ldots,Q[(M-1)d+i]\\},$

so that the disjoint union $\cup_{i=1}^{d}S_{i}$ equals the set $S$. Writing
the set of variables as a disjoint union, $V=\cup_{i=1}^{d}V_{i}$ where
$u_{j}\in V_{i}$ iff $j\equiv i$ mod $N$, we have, for all $i$, that each
equation in $S_{i}$ only depends on the variables in $V_{i}$. Renaming the
variables $u_{i+dj}\in V_{i}$ by $v_{j}$ shows that the system $S_{i}$ is a
$M[a/d,b/d]$ system.

$d=1$: We distinguish three cases: $a=0$, $a=b$, $a<b$.

$a=0$: The generic equation in the system $N[0,b]$, with $b\neq 0$, is

$Q[k]\colon\
Q\big{(}u_{k},\widetilde{u}_{k},\widehat{u}_{k+b},\widehat{\widetilde{u}}_{k+b}\big{)}=0.$

Suppose $Y$ is a subset of equations which depend on a subset of variables
$U\subset V$. If all equations that depend on variables in $U$ are in $Y$ and
$Y$ is a proper subset, then the system is decoupled. Without loss of
generality, suppose $Q[1]\in Y$. As $Q[1]$ depends on $u_{1+b}$, we have
$Q[1+b]\in Y$, which in turn implies that $Q[1+2b]\in Y$. Continuing this
argument

$Y\supset\\{Q[1+ib]\colon 0\leq i<N\\}$ (B.2)

contains $N$ equations. As $1+ib\equiv 1+jb$ mod $N$ implies $i\equiv j$ mod
$N$ when $\operatorname{gcd}(b,N)=1$, they are all distinct. This shows that
$Y$ is not a proper subset and hence the system is non-decoupled.

$a=b$: The generic equation in the system $N[b,b]$, with $b\neq 0$, is

$Q[k]\colon\
Q\big{(}u_{k},\widetilde{u}_{k+b},\widehat{u}_{k+b},\widehat{\widetilde{u}}_{k+2b}\big{)}=0.$

As in the case $a=0$, attempting to construct a proper subset of equations,
$Y$, we find (B.2).

$a<b$: For the system $N[a,b]$, with $0<a<b$, the generic equation has the
form

$Q[k]\colon\
Q\big{(}u_{k},\widetilde{u}_{k+a},\widehat{u}_{k+b},\widehat{\widetilde{u}}_{k+a+b}\big{)}=0.$

Starting from $Q[1]\in Y$, following the dependence on the variables we must
have

$\big{\\{}Q[1+ai+bj]\colon i,j\in\mathbb{N}^{0}\big{\\}}\subset Y.$

It follows from Lemma B.1 that $Y$ contains $N$ distinct equations and
therefore this case is non-decoupled as well. ∎

### Acknowledgments

The authors thank Jarmo Hietarinta for his suggestion to include equations
(2.22), (2.25) and noting they are not of the same form. We thank Pavlos
Kassotakis and Maciej Nieszporski for noting equation (3.13) can be written as
a quadrilateral system. We thank all referees for their comments, especially
the referee who pointed out Appendix B from [8]. DJZ is grateful to Professors
Q.P. Liu and R.G. Zhou for warm discussion. This project is supported by the
NSF of China (grant nos. 11875040, 11631007 and 11801289), the K.C. Wong Magna
Fund in Ningbo University, and a CRSC grant from La Trobe University.

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