On Reductions of the Hirota–Miwa Equation

On Reductions of the Hirota–Miwa Equation††This paper is a contribution to the
Special Issue on Symmetries and Integrability of Difference Equations. The
full collection is available at http://www.emis.de/journals/SIGMA/SIDE12.html

Andrew N.W. HONE, Theodoros E. KOULOUKAS and Chloe WARD A.N.W. Hone, T.E.
Kouloukas and C. Ward School of Mathematics, Statistics and Actuarial Science,
University of Kent,
Canterbury CT2 7NF, UK<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>https://www.kent.ac.uk/smsas/our-
people/profiles/hone_andrew.html

Received May 02, 2017, in final form July 17, 2017; Published online July 23,
2017

The Hirota–Miwa equation (also known as the discrete KP equation, or the
octahedron recurrence) is a bilinear partial difference equation in three
independent variables. It is integrable in the sense that it arises as the
compatibility condition of a linear system (Lax pair). The Hirota–Miwa
equation has infinitely many reductions of plane wave type (including a
quadratic exponential gauge transformation), defined by a triple of integers
or half-integers, which produce bilinear ordinary difference equations of
Somos/Gale–Robinson type. Here it is explained how to obtain Lax pairs and
presymplectic structures for these reductions, in order to demonstrate
Liouville integrability of some associated maps, certain of which are related
to reductions of discrete Toda and discrete KdV equations.

Hirota–Miwa equation; Liouville integrable maps; Somos sequences; cluster
algebras

70H06; 37K10; 39A20; 39A14; 13F60

##  1 Introduction

The Hirota–Miwa equation, or discrete Kadomtsev–Petviashvili equation
(discrete KP), is the bilinear partial difference equation

$\displaystyle T_{1}T_{-1}=T_{2}T_{-2}+T_{3}T_{-3},$ (1.1)

which serves as a generating equation for the full KP hierarchy of partial
differential equations [11, 21]. In the above, the tau function $T$ is a
function of three independent variables, $T=T(m_{1},m_{2},m_{3})$, and
subscripts denote shifts, i.e. $T_{\pm i}=T|_{m_{i}\rightarrow m_{i}\pm 1}$,
for $i=1,2,3$. The equation (1.1) is integrable in the sense that it arises as
the compatibility condition of the following linear system (Lax pair) [17]:

$\displaystyle T_{-1,3}\Psi_{1,2}+T\Psi_{2,3}=T_{2,3}\Psi,$ $\displaystyle
T\Psi_{-1,2}+T_{-1,3}\Psi_{2,-3}=T_{-1,2}\Psi$ (1.2)

(for a comparison of alternative Lax pairs see [29]). From another viewpoint
of discrete integrability, the equation (1.1) can be extended to a system with
an arbitrary number of independent variables, in which case it satisfies the
multidimensional consistency property, and it can be further extended to a
system with self-consistent sources (see [3] and references).

In this paper we consider reductions of (1.1) to bilinear ordinary difference
equations, in order to show that they can be interpreted as discrete
integrable systems in an appropriate sense, namely that Liouville’s theorem
for symplectic maps can be applied [18, 27]. Certainly it should be no
surprise that reductions of a discrete integrable system to a lower number of
independent variables produce integrable maps. Indeed, from a different point
of view, one can consider the algebraic entropy of such reductions, and show
that all one-dimensional reductions of (1.1) have quadratic degree growth [20,
Theorem 6.8], so their algebraic entropy is zero. However, in general the
connections between algebraic entropy, Lax integrability and Liouville
integrability are rather subtle, and the bilinear ordinary difference
equations that arise as plane wave reductions of (1.1) are not symplectic as
they stand. In order to interpret them correctly, we make use of the
properties of recurrences derived from cluster algebras [7], which provide an
appropriate presymplectic structure for bilinear difference equations [6] (see
also [4, 8, 15] and references), and a further reduction procedure leads to
symplectic maps of a certain kind, referred to in [12] as U-systems. By
applying the reduction process to the linear equations (1.2), we find a Lax
pair for the corresponding bilinear ordinary difference equations, leading to
the construction of conserved quantities, and then in each case the problem
remains to check that this provides a sufficient number of first integrals in
involution for the associated U-system.

General properties of plane wave reductions of the Hirota–Miwa equation are
described in Section 2, and one particular example is considered in full
detail. This example turns out to be a member of a two-parameter family of
bilinear equations that arise from reductions of a lattice equation of
discrete Toda type, considered in Section 3. In Section 4 we consider another
two-parameter family, consisting of pairs of bilinear equations related to
discrete KdV reductions, and we consider another example in detail, before
making some conclusions in Section 5.

##  2 Plane wave reductions

The Hirota–Miwa equation (1.1) admits the plane wave reduction

$\displaystyle
T(m_{1},m_{2},m_{3})=a_{1}^{{m_{1}}^{2}}a_{2}^{{m_{2}}^{2}}a_{3}^{{m_{3}}^{2}}\tau_{m},\qquad
m=m_{0}+\delta_{1}m_{1}+\delta_{2}m_{2}+\delta_{3}m_{3},$ (2.1)

where $m_{0}$ is arbitrary and the distinct values of the parameters
$\delta_{1}$, $\delta_{2}$, $\delta_{3}$ are all chosen to be integers or
half-integers (the case when any of the $\delta_{j}$ coincide is not
interesting). The above expression includes a quadratic exponential gauge
transformation, which allows the insertion of coefficients into the right-hand
side of the parameter-free equation (1.1). By substituting (2.1) into (1.1),
it follows that $\tau_{m}$ satisfies the autonomous bilinear ordinary
difference equation

$\displaystyle\tau_{m+\delta_{1}}\tau_{m-\delta_{1}}=\alpha\tau_{m+\delta_{2}}\tau_{m-\delta_{2}}+\beta\tau_{m+\delta_{3}}\tau_{m-\delta_{3}},$

where the ratios of the arbitrary parameters $a_{j}$ in (2.1) yield the
coefficients

$\displaystyle\alpha=\frac{a_{2}^{2}}{a_{1}^{2}},\qquad\beta=\frac{a_{3}^{2}}{a_{1}^{2}}.$

We can shift the last equation by $\delta_{1}$ and rewrite it as

$\displaystyle\tau_{m+N}\tau_{m}=\alpha\tau_{m+\delta_{1}+\delta_{2}}\tau_{m+\delta_{1}-\delta_{2}}+\beta\tau_{m+\delta_{1}+\delta_{3}}\tau_{m+\delta_{1}-\delta_{3}}\qquad\mathrm{with}\quad
N=2\delta_{1},$ (2.2)

and without loss of generality we can assume that

$\displaystyle\delta_{1}>\max(\delta_{2},\delta_{3}),$

so that the overall order of the recurrence (2.2) is $N$.

The iteration of the recurrence (2.2) is equivalent to iteration of the
birational map $\varphi\colon$ $\mathbb{C}^{N}\rightarrow\mathbb{C}^{N}$,
given by

$\displaystyle\varphi(\tau_{0},\tau_{1},\ldots,\tau_{N-2},\tau_{N-1})=\left(\tau_{1},\tau_{2},\ldots,\tau_{N-1},\frac{\alpha\tau_{\delta_{1}+\delta_{2}}\tau_{\delta_{1}-\delta_{2}}+\beta\tau_{\delta_{1}+\delta_{3}}\tau_{\delta_{1}-\delta_{3}}}{\tau_{0}}\right),$
(2.3)

which is an example of a cluster map: it arises from a sequence of mutations
in a cluster algebra, constructed from a quiver that is cluster mutation-
periodic with period 1, with the coefficients $\alpha$, $\beta$ corresponding
to frozen variables [7]. By making gauge transformations one can also consider
the original equation (1.1) with non-autonomous coefficients, and the same is
true for its reductions (see [20]). In due course we will also find it useful
to allow the coefficients $\alpha$, $\beta$ to depend on the index $m$.

###  2.1 Reduction of Lax pairs

We can apply the plane wave reduction (2.1) at the level of the Lax pair.
However, before doing so it is important to note that the compatibility
condition of the linear system (1.2) is not exactly (1.1), but rather

$\displaystyle R_{1,-3}=R,\qquad
R=\frac{T_{1}T_{-1}-T_{3}T_{-3}}{T_{2}T_{-2}}.$ (2.4)

In order to get precisely (1.1), corresponding to $R\equiv 1$, it is necessary
to augment the pair (1.2) with a third equation, to get a Lax triad. However,
for our purposes it will be convenient to leave this slight ambiguity.

To make the reduction, we require that the tau function $T$ is given by (2.1),
and take the wave function $\Psi$ in the form

$\displaystyle\Psi(m_{1},m_{2},m_{3})=\lambda_{1}^{m_{1}}\lambda_{2}^{m_{2}}\lambda_{3}^{m_{3}}T(m_{1},m_{2},m_{3})\phi_{m},$
(2.5)

where ${\lambda}_{j}$ correspond to spectral parameters, and upon setting

$\displaystyle\zeta=\lambda_{2}\lambda_{1}^{-1},\qquad\xi=\big{(}a_{1}^{2}\lambda_{1}\lambda_{2}\big{)}^{-1},\qquad\lambda_{3}=a_{3}^{2}\lambda_{2}$

we can derive a corresponding linear system for the reductions (2.2) of (1.1).

######  Proposition 2.1.

The plane wave reduction of the linear system (1.2) is

$\displaystyle
Y_{m}\phi_{m+\delta_{1}+\delta_{2}}+\beta\zeta\phi_{m+\delta_{2}+\delta_{3}}=\xi\phi_{m},$
$\displaystyle\phi_{m+\delta_{1}-\delta_{2}}-X_{m}\phi_{m+\delta_{1}-\delta_{3}}=\zeta\phi_{m},$
(2.6)

where

$\displaystyle
X_{m}=\frac{\tau_{m+2\delta_{1}-2\delta_{3}}\tau_{m+\delta_{1}-\delta_{2}}}{\tau_{m+2\delta_{1}-\delta_{2}-\delta_{3}}\tau_{m+\delta_{1}-\delta_{3}}},\qquad
Y_{m}=\frac{\tau_{m+2\delta_{1}+\delta_{2}-\delta_{3}}\tau_{m}}{\tau_{m+\delta_{1}+\delta_{2}}\tau_{m+\delta_{1}-\delta_{3}}}.$

The compatibility condition of (2.6) is a Somos-$N$ recurrence with a periodic
coefficient, given by

$\displaystyle\tau_{m+N}\tau_{m}=\alpha_{m}\tau_{m+\delta_{1}+\delta_{2}}\tau_{m+\delta_{1}-\delta_{2}}+\beta\tau_{m+\delta_{1}+\delta_{3}}\tau_{m+\delta_{1}-\delta_{3}},\qquad\alpha_{m+\delta_{1}-\delta_{3}}=\alpha_{m},$
(2.7)

with $N=2\delta_{1}$.

###### Proof.

This follows by substituting (2.1) and (2.5) into (1.2): the first linear
equation produces the top equation in (2.6), and the second equation in (1.2)
almost immediately yields the bottom equation in (2.6) after shifting the
indices on each $\phi_{j}$ to $\phi_{j+\delta_{1}-\delta_{2}}$, except that we
have also made an overall shift $m\to m+\delta_{1}-\delta_{3}$ in the
dependent variables which appear in the combinations $X_{m}$, $Y_{m}$ (which
we are free to do, since the compatibility of the linear system gives
autonomous relations between these quantities). Now observe that the parameter
$\alpha$ in (2.2) does not appear anywhere in the linear system (2.6), and the
compatibility condition of the latter is obtained by substituting (2.1) into
(2.4), which implies that $\tau_{m}$ satisfies the non-autonomous bilinear
recurrence (2.7) with the coefficient $\alpha_{m}$ of period
$\delta_{1}-\delta_{3}$. ∎

Each iteration of (2.7) is equivalent to an iteration of a map
$\varphi_{m}\colon{\mathbb{C}}^{N}\to{\mathbb{C}}^{N}$, which is of the same
form as (2.3), except that the coefficient ${\alpha}$ is replaced by
$\alpha_{m}$, depending on $m\bmod\delta_{1}-\delta_{3}$. The iterates of this
map have a matrix Lax representation, and the dynamics preserves an associated
spectral curve.

######  Corollary 2.2.

The scalar Lax pair (2.6) is equivalent to a matrix linear system of size
$K=\max(\delta_{1}-\delta_{2},\delta_{1}-\delta_{3})$, of the form

$\displaystyle{\bf
L}_{m}(\zeta)\mathbf{\Phi}_{m}=\xi\mathbf{\Phi}_{m},\qquad\mathbf{\Phi}_{m+1}={\bf
M}_{m}(\zeta)\mathbf{\Phi}_{m}.$ (2.8)

The compatibility condition of the latter system is the discrete Lax equation

$\displaystyle{\bf L}_{m+1}{\bf M}_{m}={\bf M}_{m}{\bf L}_{m},$ (2.9)

which preserves the spectral curve in the $(\zeta,\xi)$ plane given by

$\displaystyle{\cal P}(\zeta,\xi)\equiv\det({\bf
L}_{m}(\zeta)-\xi\mathbf{1})=0.$ (2.10)

###### Proof.

Upon introducing the vector
$\mathbf{\Phi}_{m}=(\phi_{m},\phi_{m+1},\ldots,\phi_{m+K-1})^{T}$, the top
equation in (2.6) becomes an eigenvalue equation for a matrix ${\bf L}_{m}$,
while the bottom equation can be rewritten in matrix form with another matrix
${\bf M}_{m}$, where ${\bf L}_{m}$, ${\bf M}_{m}$ in (2.8) are $K\times K$
matrix functions of $\zeta$ and the dynamical variables. The compatibility
condition (2.9) shows that this is an isospectral evolution, preserving the
coefficients of the polynomial ${\cal P}(\zeta,\xi)$. ∎

######  Example 2.3.

Choosing $(\delta_{1},\delta_{2},\delta_{3})=(4,3,0)$ in (2.7) gives rise to
the recurrence

$\displaystyle\tau_{m+8}\tau_{m}=\alpha_{m}\tau_{m+7}\tau_{m+1}+\beta\tau_{m+4}^{2},\qquad\alpha_{m+4}=\alpha_{m}.$
(2.11)

As we shall see below, the latter recurrence also arises as a reduction of a
5-point lattice equation of discrete Toda type appearing in [9]. In this case,
the scalar linear system (2.6) consists of two equations, of orders 7 and 4
respectively, with coefficients

$\displaystyle X_{m}=\frac{\tau_{m+8}\tau_{m+1}}{\tau_{m+5}\tau_{m+4}},\qquad
Y_{m}=\frac{\tau_{m+11}\tau_{m}}{\tau_{m+7}\tau_{m+4}}.$

After using the equation of fourth order to eliminate higher shifts of
$\phi_{m}$ from the equation of order 7, this gives the system

$\displaystyle\phi_{m+4}=\frac{1}{X_{m}}(\phi_{m+1}-\zeta\phi_{m}),$
$\displaystyle\xi\phi_{m}=\zeta\left(\beta-\frac{Y_{m}}{X_{m+3}}\right)\phi_{m+3}+\frac{Y_{m}}{X_{m}X_{m+3}}(\phi_{m+1}-\zeta\phi_{m}).$

The matrix entries in the Lax pair of Corollary 2.2 can then be written in a
fairly compact form in terms of the quantities $X_{m+j}$ for $0\leq j\leq 6$,
$Y_{m+k}$ for $0\leq k\leq 3$ and $\beta$, $\zeta$, in which case the
compatibility conditions are

$\displaystyle X_{m}Y_{m+4}=X_{m+7}Y_{m+1},\qquad
X_{m}Y_{m+4}-X_{m+7}Y_{m}=\beta X_{m+7}(X_{m}-X_{m+3}).$ (2.12)

When these equations are rewritten in terms of the tau function $\tau_{m}$,
the first of them is just a tautology, while the second one says that

$\displaystyle\frac{\tau_{m+12}\tau_{m+4}-\beta\tau_{m+8}^{2}}{\tau_{m+11}\tau_{m+5}}=\frac{\tau_{m+8}\tau_{m}-\beta\tau_{m+4}^{2}}{\tau_{m+7}\tau_{m+1}},$

which is equivalent to (2.11). We can further use the equation (2.11) to
rewrite the entries of ${\bf L}_{m}$, ${\bf M}_{m}$ in terms of the tau
function and $\alpha_{m}$ as well as $\beta$, $\zeta$, but rather than doing
this here we follow [28] and introduce the quantity

$\displaystyle z_{m}=\frac{\tau_{m+5}\tau_{m}}{\tau_{m+4}\tau_{m+1}},$ (2.13)

so that $X_{m}=z_{m+1}z_{m+2}z_{m+3}$, $Y_{m}=z_{m}X_{m}X_{m+3}$, to find
(setting $m\to 0$ for convenience)

$\displaystyle\mathbf{L}=\begin{pmatrix}-\zeta z_{0}&z_{0}&0&\zeta(\beta-
z_{0}z_{1}z_{2}z_{3})\\\
\\!\zeta^{2}\left(z_{4}-\frac{\beta}{z_{1}z_{2}z_{3}}\right)&\\!\zeta\left(\frac{\beta}{z_{1}z_{2}z_{3}}-z_{1}-z_{4}\right)&z_{1}&0\\\
0&\zeta^{2}\left(z_{5}-\frac{\beta}{z_{2}z_{3}z_{4}}\right)&\\!\zeta\left(\frac{\beta}{z_{2}z_{3}z_{4}}-z_{2}-z_{5}\right)&z_{2}\\\
-\frac{\zeta}{z_{1}z_{2}}&\frac{1}{z_{1}z_{2}}&\zeta^{2}\left(z_{6}-\frac{\beta}{z_{3}z_{4}z_{5}}\right)&\\!\zeta\left(\frac{\beta}{z_{3}z_{4}z_{5}}-z_{3}-z_{6}\right)\\!\end{pmatrix},$
$\displaystyle\mathbf{M}=\begin{pmatrix}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
\frac{-\zeta}{z_{1}z_{2}z_{3}}&\frac{1}{z_{1}z_{2}z_{3}}&0&0\end{pmatrix},$
(2.14)

with ${\bf L}={\bf L}_{0}$, ${\bf M}={\bf M}_{0}$, in which case the second
equation in (2.12) becomes a recurrence of order 7 for $z_{m}$, namely

$\displaystyle
z_{m+7}-z_{m}=\beta\left(\frac{1}{z_{m+4}z_{m+5}z_{m+6}}-\frac{1}{z_{m+1}z_{m+2}z_{m+3}}\right).$
(2.15)

The corresponding spectral curve (2.10) is of genus 9, and takes the form

$\displaystyle\Gamma\colon\
\xi^{4}+H_{1}\zeta\xi^{3}+H_{2}\zeta^{2}\xi^{2}+\big{(}H_{3}\zeta^{3}-1\big{)}\xi+H_{4}\zeta^{7}+\beta\zeta^{4}=0,$
(2.16)

where

$\displaystyle
H_{1}=\sum_{k=0}^{6}z_{k}-\beta\sum_{k=0}^{2}\frac{1}{z_{k+1}z_{k+2}z_{k+3}},$
$\displaystyle H_{2}=\sum_{k=0}^{6}z_{k}z_{k+1}+z_{k}z_{k+2}$
$\displaystyle\hphantom{H_{2}=}{}-\beta\left(\frac{1}{z_{1}z_{3}}+\frac{1}{z_{2}z_{3}}+\frac{1}{z_{2}z_{4}}+\frac{1}{z_{3}z_{4}}+\frac{1}{z_{3}z_{5}}+\frac{z_{0}+z_{1}}{z_{3}z_{4}z_{5}}+\frac{z_{0}+z_{6}}{z_{2}z_{3}z_{4}}+\frac{z_{5}+z_{6}}{z_{1}z_{2}z_{3}}\right)$
$\displaystyle\hphantom{H_{2}=}{}+\beta^{2}\left(\frac{1}{z_{1}z_{2}^{2}z_{3}^{2}z_{4}}+\frac{1}{z_{1}z_{2}z_{3}^{2}z_{4}z_{5}}+\frac{1}{z_{2}z_{3}^{2}z_{4}^{2}z_{5}}\right),$
$\displaystyle
H_{3}=\sum_{k=0}^{6}z_{k}z_{k+1}z_{k+2}-\beta\left(\frac{1}{z_{3}}+\frac{z_{0}}{z_{3}z_{4}}+\frac{z_{6}}{z_{2}z_{3}}+\frac{z_{0}z_{1}}{z_{3}z_{4}z_{5}}+\frac{z_{0}z_{6}}{z_{2}z_{3}z_{4}}+\frac{z_{5}z_{6}}{z_{1}z_{2}z_{3}}\right)$
$\displaystyle\hphantom{H_{3}=}{}+\beta^{2}\left(\frac{1}{z_{1}z_{2}z_{3}^{2}z_{4}}+\frac{1}{z_{2}z_{3}^{2}z_{4}z_{5}}+\frac{z_{0}}{z_{2}z_{3}^{2}z_{4}^{2}z_{5}}+\frac{z_{6}}{z_{1}z_{2}^{2}z_{3}^{2}z_{4}}\right)-\frac{\beta^{3}}{z_{1}z_{2}^{2}z_{3}^{3}z_{4}^{2}z_{5}},$
$\displaystyle
H_{4}=\frac{1}{z_{3}}\prod_{k=0}^{3}\left(z_{k}z_{k+3}-\frac{\beta}{z_{k+1}z_{k+2}}\right),$
(2.17)

with indices read $\bmod$ 6 where necessary. These coefficients of the
spectral curve provide 4 functionally independent integrals for the map in 7
dimensions defined by (2.15). We shall consider the question of Liouville
integrability when we return to this example in Section 2.3 below.

######  Remark 2.4.

The replacement
$(\delta_{1},\delta_{2},\delta_{3})\to(\delta_{1},\delta_{3},\delta_{2})$
leads to a recurrence with the same terms as (2.7), but with the roles of the
coefficients $\alpha,\beta$ reversed. For instance, in the previous example,
choosing the parameters $(\delta_{1},\delta_{2},\delta_{3})=(4,0,3)$ gives
$\delta_{1}-\delta_{3}=1$, so with both $\alpha$ and $\beta$ as constant
coefficients, the recurrence is just

$\displaystyle\tau_{m+8}\tau_{m}=\alpha\tau_{m+7}\tau_{m+1}+\beta\tau_{m+4}^{2}.$

Another method for finding first integrals of discrete KP reductions was given
in [19], based on reduction of conservation laws.

###  2.2 Somos recurrences and cluster maps

The plane wave reductions of the discrete KP equation (1.1) are recurrence
relations of Somos type. They are also particular examples of cluster maps,
which arise from cluster mutations of quivers which are periodic with period 1
[7], and this provides an appropriate presymplectic structure (i.e. a closed
2-form of constant rank) that is preserved by the dynamics.

The general Somos-$N$ recurrence is a quadratic recurrence relation of the
form

$\displaystyle
x_{m+N}x_{m}=\sum_{j=1}^{\left\lfloor{\frac{N}{2}}\right\rfloor}a_{j}x_{m+N-j}x_{m+j},$
(2.18)

where $a_{j}$ are coefficients. The case when there are only two non-zero
coefficients $a_{j}$, $a_{k}$ on the right-hand side corresponds to the
reductions (2.2) of the discrete KP equation (1.1), while the case of three
non-zero coefficients $a_{j}$, $a_{k}$, $a_{\ell}$ with $j+k+\ell=0$
$(\bmod\,N)$ arises from reductions of the discrete BKP equation. These
particular cases are also sometimes referred to as 3-term or 4-term
Gale–Robinson recurrences [16], respectively (where the total number of terms
in the equation is counted), and are the only cases which display the Laurent
phenomenon [5]. In general, if there are more than three terms on the right-
hand side of (2.18) then the recurrence does not appear to be integrable: the
growth of degrees of the terms, or the growth of logarithmic heights of
generic iterates in ${\mathbb{Q}}$, is exponential. However, for certain
choices of coefficients and initial data such recurrences with more terms can
still be produced by identities for abelian functions [1].

On the other hand, (with the inclusion of coefficients $\alpha,\beta$) the
reductions (2.2) of the discrete KP equation are particular cases of
recurrences of the form

$\displaystyle
x_{m+N}x_{m}=\prod_{j=1}^{N-1}x_{m+j}^{[b_{1,j+1}]_{+}}+\prod_{j=1}^{N-1}x_{m+j}^{[-b_{1,j+1}]_{+}},$
(2.19)

which arise from sequences of mutations in a coefficient-free cluster algebra.
In the above, $[b]_{+}=\max(b,0)$, and the exponents appearing on the right-
hand side belong to the first row of the exchange matrix $B=(b_{ij})$, an
$N\times N$ skew-symmetric integer matrix which defines a quiver (directed
graph) consisting of $N$ nodes without 1- or 2-cycles: the rule is that
$[b_{ij}]_{+}$ is the number of arrows from node $i$ to node $j$. It was shown
by Fordy and Marsh [7] that the quiver has cluster mutation-periodicity with
period 1, meaning that a single mutation of the quiver is equivalent to a
cyclic permutation of the nodes, iff the matrix entries of $B$ satisfy

$\displaystyle b_{j,N}=b_{1,j+1},\qquad j=1,\ldots,N-1,$ $\displaystyle
b_{j+1,k+1}=b_{j,k}+b_{1,j+1}[-b_{1,k+1}]_{+}-b_{1,k+1}[-b_{1,j+1}]_{+},\qquad
1\leq j,k\leq N-1.$ (2.20)

In this case, the above conditions mean that matrix $B$ is completely
determined by the elements of its first row, which are the exponents appearing
in (2.19). The cluster map defined by (2.19) is the birational map

$\displaystyle\begin{array}[]{@{}lrcl}\varphi\colon&{\mathbb{C}}^{N}&\rightarrow&{\mathbb{C}}^{N},\\\
&(x_{1},\ldots,x_{N-1},x_{N})&\mapsto&\displaystyle\bigg{(}x_{2},\ldots,x_{N},x_{1}^{-1}\bigg{(}\prod_{j}x_{j+1}^{[b_{1,j+1}]_{+}}+\prod_{j}x_{j+1}^{[-b_{1,j+1}]_{+}}\bigg{)}\bigg{)}.\end{array}$
(2.23)

Furthermore, the conditions (2.20) are also necessary and sufficient for the
2-form

$\displaystyle\omega=\sum_{i<j}\frac{b_{ij}}{x_{i}x_{j}}{\mathrm{d}}x_{i}\wedge{\mathrm{d}}x_{j},$
(2.24)

to be an invariant presymplectic form for the map, i.e.
$\varphi^{*}\omega=\omega$ [6]. The form (2.24) is log-canonical, i.e. it is
constant in the coordinates $\log x_{i}$.

For the conditions (2.20) to hold, the non-zero entries in the first row of
$B$ must be palindromic. Hence, in the case of the discrete KP reductions,
with a sum of two quadratic terms on the right-hand side, the pattern of non-
zero terms is either of the form $1,\ldots,-2,\ldots,1$ or
$1,\ldots,-1,\ldots,-1,\ldots,1$ (up to an overall choice of sign), and the
other entries of $B$ are completely fixed by this pattern.

######  Example 2.5.

Up to sending $B\to-B$, the exchange matrix for the recurrence in Example 2.3
is

$\displaystyle B=\left(\begin{matrix}0&1&0&0&-2&0&0&1\\\ -1&0&1&0&2&-2&0&0\\\
0&-1&0&1&0&2&-2&0\\\ 0&0&-1&0&1&0&2&-2\\\ 2&-2&0&-1&0&1&0&0\\\
0&2&-2&0&-1&0&1&0\\\ 0&0&2&-2&0&-1&0&1\\\
-1&0&0&2&0&0&-1&0\end{matrix}\right).$

###  2.3 Reduction to symplectic maps

In order to investigate Liouville integrability in the context of cluster
maps, it is convenient to make a reduction to a symplectic map on a space of
even dimension $r=\operatorname{rank}B$, that is the space of leaves of the
null foliation for $\omega$. This is achieved by choosing a
${\mathbb{Z}}$-basis ${\bf v}_{1},\ldots,{\bf v}_{r}$ for
$\operatorname{im}B\cap{\mathbb{Z}}^{N}$, and then mapping to a corresponding
set of Laurent monomials ${\bf u}=(u_{1},\ldots,u_{r})$ in the coordinates
${\bf x}=(x_{j})$, via

$\displaystyle\begin{array}[]{@{}lrcl}\pi\colon&{\mathbb{C}}^{N}&\rightarrow&{\mathbb{C}}^{r},\\\
&{\bf x}&\mapsto&{\bf u}=\big{(}{\bf x}^{{\bf v}_{1}},\ldots,{\bf x}^{{\bf
v}_{r}}\big{)}.\end{array}$ (2.27)

Theorem 2.6 in [6] says that this produces a symplectic birational map
$\hat{\varphi}$ in the reduced coordinates ${\bf u}$. By a further refinement
of this result [12, Proposition 3.9] one can choose a special
${\mathbb{Z}}$-basis consisting of palindromic vectors, such that
$\hat{\varphi}$ is equivalent to iteration of a single recurrence relation
called the U-system.

######  Theorem 2.6.

For any cluster map $\varphi$ given by (2.23) with associated exchange matrix
$B$ satisfying (2.20), there is a palindromic ${\mathbb{Z}}$-basis for
$\operatorname{im}B\cap{\mathbb{Z}}^{N}$ $($unique up to an overall sign$)$,
such that under the reduction (2.27), the map $\hat{\varphi}$ with
$\hat{\varphi}\circ\pi=\pi\circ\varphi$ is equivalent to the iteration of the
corresponding U-system, of the form

$\displaystyle u_{m+r}u_{m}=\mathcal{F}(u_{m+1},\ldots,u_{m+r-1}),$

for a certain rational function $\cal F$. Moreover, $\hat{\varphi}$ preserves
the symplectic form $\hat{\omega}$ such that $\pi^{*}\hat{\omega}=\omega$,
which is log-canonical in the coordinates ${\bf u}=(u_{1},\ldots,u_{r})$.

The above theorem immediately applies to the plane wave reductions (2.2) of
discrete KP, with coordinates $x_{j}\to\tau_{j}$, and the symplectic structure
obtained by reduction is unaltered even when non-autonomous coefficients are
included, in particular when ${\alpha}$ and/or $\beta$ are allowed to be
periodic.

######  Example 2.7.

For the matrix $B$ in Example 2.5, $\operatorname{rank}B=6$, and the vector
${\bf v}_{1}=(1,-2,1,0,0$, $0,0,0)^{T}$, together with ${\bf
v}_{2},\ldots,{\bf v}_{6}$ obtained by shifting the non-zero block to the
right, provides a palindromic basis for $\operatorname{im}B$, so the reduction
(2.27) gives

$\displaystyle u_{m}=\frac{\tau_{m}\tau_{m+2}}{\tau_{m+1}^{2}}.$ (2.28)

Then the U-system corresponding to (2.11) is the non-autonomous recurrence

$\displaystyle
u_{m+6}u_{m}=\frac{{\alpha}_{m}u_{m+5}u_{m+4}^{2}u_{m+3}^{3}u_{m+2}^{2}u_{m+1}+\beta}{u_{m+5}^{2}u_{m+4}^{3}u_{m+3}^{4}u_{m+2}^{3}u_{m+1}^{2}},\qquad{\alpha}_{m+4}={\alpha}_{m}.$
(2.29)

Upon applying Theorem 2.6, we see that (2.29) preserves the symplectic form

$\displaystyle\hat{\omega}=\sum_{i<j}\frac{\hat{b}_{ij}}{u_{i}u_{j}}{\mathrm{d}}u_{i}\wedge{\mathrm{d}}u_{j},\qquad\hat{B}=\big{(}\hat{b}_{ij}\big{)}=\left(\begin{matrix}0&1&2&3&2&1\\\
-1&0&2&4&4&2\\\ -2&-2&0&3&4&3\\\ -3&-4&-3&0&2&2\\\ -2&-4&-4&-2&0&1\\\
-1&-2&-3&-2&-1&0\end{matrix}\right).$

Equivalently, each iteration of the corresponding six-dimensional map
$\hat{\varphi}_{m}$, which depends on $m\bmod 4$, preserves the nondegenerate
log-canonical Poisson bracket (obtained by inverting the matrix $\hat{B}$)
given by

$\displaystyle\\{u_{i},u_{j}\\}=c_{j-i}u_{i}u_{j},\qquad 0\leq i<j\leq 5,$
(2.30)

with $c_{1}=1$, $c_{2}=c_{5}=0$, $c_{3}=-c_{4}=-2$.

Upon comparing (2.28) with (2.13), we obtain the formula

$\displaystyle z_{m}=u_{m}u_{m+1}u_{m+2}u_{m+3}$ (2.31)

for all $m$. We can use this to write $z_{j}$ for $0\leq j\leq 2$ in terms of
a set of initial coordinates $u_{0},u_{1},\ldots,u_{5}$ for the U-system
(2.29). However, for higher shifts of $z_{j}$ we note the identity

$\displaystyle z_{m+3}z_{m}=\alpha_{m}u_{m+3}+\frac{\beta}{z_{m+1}z_{m+2}},$

which follows from (2.31) and (2.29), so after substituting for $z_{3}$,
$z_{4}$, $z_{5}$, $z_{6}$ in (2.14) it is clear that, in addition to $u_{j}$
for $0\leq j\leq 5$, the parameters ${\alpha}_{0}$, ${\alpha}_{1}$,
${\alpha}_{2}$, ${\alpha}_{3}$ will also appear in ${\bf L}={\bf L}_{0}$ and
${\bf M}={\bf M}_{0}$ (and similarly for ${\bf L}_{m}$, ${\bf M}_{m}$,
replacing the index $j$ on each variable in ${\bf L}$, ${\bf M}$ by $m+j$). In
this way we obtain the Lax representation for the U-system itself, while from
the spectral curve (2.16), the coefficients $H_{1}$, $H_{2}$, $H_{3}$ in
(2.17) can be pulled back by the formula (2.31), to provide 3 independent
functions of $u_{j}$, in involution with respect to the bracket (2.30) in 6
dimensions (see [28] for explicit formulae in the case
${\alpha}_{m}={\alpha}={\rm const}$). Pulling back the fourth coefficient we
find

$\displaystyle H_{4}={\alpha}_{0}{\alpha}_{1}{\alpha}_{2}{\alpha}_{3},$ (2.32)

which (like any cyclically symmetric function of ${\alpha}_{0}$,
${\alpha}_{1}$, ${\alpha}_{2}$, ${\alpha}_{3}$) is a trivial first integral of
the U-system. (By an abuse of notation, in (2.32) and elsewhere we use the
same symbol to denote a function and its pullback.) Thus we see that in the
autonomous case we have a 6-dimensional symplectic map $\hat{\varphi}$ given
by

$\displaystyle\hat{\varphi}(u_{0},u_{1},u_{2},u_{3},u_{4},u_{5})=\left(u_{1},u_{2},u_{3},u_{4},u_{5},\frac{\beta+\alpha
u_{1}u_{2}^{2}u_{3}^{3}u_{4}^{2}u_{5}}{u_{0}u_{1}^{2}u_{2}^{3}u_{3}^{4}u_{4}^{3}u_{5}^{2}}\right),$

obtained from (2.29) by fixing ${\alpha}_{m}={\alpha}={\rm const}$, with 3
commuting first integrals, so this is an integrable system in the Liouville
sense. In the non-autonomous case, we have instead a family of symplectic maps
$\hat{\varphi}_{m}$ cycling with $m\bmod 4$, but we can interpret the fourfold
composition
$\hat{\varphi}_{3}\circ\hat{\varphi}_{2}\circ\hat{\varphi}_{1}\circ\hat{\varphi}_{0}$
as an autonomous system with the three commuting invariants $H_{1}$, $H_{2}$,
$H_{3}$ described above, so again this is a Liouville integrable system.

These results also lead to an interpretation of the 7-dimensional map defined
by (2.15) as an integrable system. Indeed, the expression (2.31) means that
the bracket (2.30) can be lifted to the coordinates $z_{j}$, $0\leq j\leq 6$,
to give a Poisson bracket of rank 6 specified by

$\displaystyle\\{z_{0},z_{1}\\}=0=\\{z_{0},z_{2}\\},\qquad\\{z_{0},z_{3}\\}=-z_{0}z_{3}+\beta(z_{1}z_{2})^{-1},$
$\displaystyle\\{z_{0},z_{4}\\}=z_{0}z_{4}-\beta^{2}\big{(}z_{1}^{2}z_{2}^{2}z_{3}^{2}\big{)}^{-1},$
$\displaystyle\\{z_{0},z_{5}\\}=-\beta^{2}\big{(}z_{1}z_{2}^{2}z_{3}^{2}z_{4}\big{)}^{-1}+\beta^{3}\big{(}z_{1}^{2}z_{2}^{3}z_{3}^{3}z_{4}^{2}\big{)}^{-1},$
$\displaystyle\\{z_{0},z_{6}\\}=-\beta^{2}\big{(}z_{1}z_{2}z_{3}^{2}z_{4}z_{5}\big{)}^{-1}+\beta^{3}\big{(}z_{1}^{2}z_{2}^{2}z_{3}^{3}z_{4}^{2}z_{5}^{2}\big{)}^{-1}(z_{1}+z_{5})-\beta^{4}\big{(}z_{1}^{2}z_{2}^{3}z_{3}^{4}z_{4}^{3}z_{5}^{2}\big{)}^{-1};$

all other brackets follow by shifting indices. The preceding results imply
that, as functions of $z_{j}$, the quantities $H_{1}$, $H_{2}$, $H_{3}$ in
(2.17) are in involution with respect to the above bracket, while from the
expression (2.32) it follows that $H_{4}$ is a Casimir for this bracket; this
can also be verified directly from (2.17).

######  Remark 2.8.

In the previous example, the formula (2.14) is a “big Lax pair” for the
system: the spectral curve $\Gamma$ in (2.16) has genus 9, while the Liouville
tori (level sets of first integrals) are only 3-dimensional. One way to
understand this is by noting that $\Gamma$ is invariant under the action of
$C_{3}$ (the cyclic group of order 3), generated by
$(\zeta,\xi)\to(e^{2\pi{\mathrm{i}}/3}\zeta,e^{2\pi{\mathrm{i}}/3}\xi)$, so
that $\Gamma$ is a threefold cover of the curve $\tilde{\Gamma}=\Gamma/C_{3}$,
ramified at $(0,0)$ and $(\infty,\infty)$, and $\tilde{\Gamma}$ has genus 3. A
more direct way to obtain the curve $\tilde{\Gamma}$ is to note that the
system is one of a family of reductions of a lattice equation of discrete Toda
type, considered in the next section (see (3.1) below). This reduction
procedure yields $2\times 2$ Lax pairs, which will be described in future work
[13].

##  3 Reductions of discrete Toda type

In this section we briefly consider the two-parameter family of plane wave
reductions of discrete KP with $(\delta_{1},\delta_{2},\delta_{3})=(P,P-Q,0)$,
for integers $P>Q$, which includes Example 2.3 when $P=4$, $Q=1$. These
examples arise from travelling wave solutions of the five-point lattice
equation

$\displaystyle\frac{V_{k,l}}{V_{k+1,l}}-\frac{V_{k-1,l}}{V_{k,l}}+\alpha\left(\frac{V_{k+1,l-1}}{V_{k,l}}-\frac{V_{k,l}}{V_{k-1,l+1}}\right)=0,$
(3.1)

which for $\alpha=pq$ appears in [9], and can be considered as a discrete time
Toda equation [2]. If we impose the periodicity

$\displaystyle V_{k+Q,l-P}=V_{k,l},$

which is called the $(Q,-P)$ periodic reduction of equation (3.1), then we may
write

$\displaystyle V_{k,l}=v_{m},\qquad\mathrm{where}\quad m=kP+lQ$

is a travelling wave variable, and $v_{m}$ satisfies the ordinary difference
equation

$\displaystyle\frac{v_{m}}{v_{m+P}}-\frac{v_{m-P}}{v_{m}}+\alpha\left(\frac{v_{m+P-Q}}{v_{m}}-\frac{v_{m}}{v_{m+Q-P}}\right)=0.$
(3.2)

######  Proposition 3.1.

Suppose that

$\displaystyle v_{m}=\frac{\tau_{m}}{\tau_{m+Q}}$

is a solution of (3.2). Then $\tau_{m}$ is a solution of

$\displaystyle\tau_{m+2P}\tau_{m}=\alpha\tau_{m+2P-Q}\tau_{m+Q}+\beta_{m}\tau_{m+P}^{2},\qquad\beta_{m}=\beta_{m+Q},$
(3.3)

and the converse is also true.

###### Proof.

By setting $v_{m}=\frac{\tau_{m}}{\tau_{m+Q}}$ in (3.2), up to an overall
shift we find that

$\displaystyle\frac{\tau_{m+2P}\tau_{m}-\alpha\tau_{m+2P-Q}\tau_{m+Q}}{\tau_{m+P}^{2}}=\frac{\tau_{m+2P+Q}\tau_{m+Q}-\alpha\tau_{m+2P}\tau_{m+2Q}}{\tau_{m+P+Q}^{2}},$

and denoting the left-hand side above by $\beta_{m}$ we see that this quantity
is periodic with period $Q$, which yields the bilinear equation in (3.3).
Conversely, any solution of (3.3) with a $Q$-periodic coefficient $\beta_{m}$
provides a solution of (3.2). ∎

######  Example 3.2.

In the running example above, with
$(\delta_{1},\delta_{2},\delta_{3})=(4,3,0)$, we have

$\displaystyle v_{m}=\frac{\tau_{m}}{\tau_{m+1}},\qquad\mathrm{with}\quad
u_{m}=\frac{\tau_{m}\tau_{m+2}}{\tau_{m+1}^{2}}=\frac{v_{m}}{v_{m+1}}$

being a solution of the autonomous version of the U-system (2.29), where
${\alpha}_{m}={\alpha}=\mathrm{const}$. After shifting once to eliminate
$\beta$ from the U-system, an equation of order 7 for $u_{m}$ arises, and this
is equivalent to a relation of order 8 for $v_{m}$, namely

$\displaystyle\frac{v_{m+4}}{v_{m+8}}-\frac{v_{m}}{v_{m+4}}+\alpha\left(\frac{v_{m+7}}{v_{m+4}}-\frac{v_{m+4}}{v_{m+1}}\right)=0,$

which is the $(1,-4)$ periodic reduction of the discrete Toda equation (3.1).

In the general $(Q,-P)$ periodic reduction of (3.1), one can consider a
Poisson structure for the variables $v_{m}$ in (3.2), and a “small” ($2\times
2$) Lax representation for this reduction and the associated U-system. We
propose to treat these details elsewhere [13].

##  4 Reductions of discrete KdV type

In this section, for a pair of integers $L>M$, we consider two different plane
wave reductions of discrete KP, corresponding to the choices

$\displaystyle(\delta_{1},\delta_{2},\delta_{3})=\left(L+\frac{M}{2},\frac{M}{2},L-\frac{M}{2}\right)\qquad\mathrm{or}\qquad\left(M+\frac{L}{2},\frac{L}{2},\left|M-\frac{L}{2}\right|\right).$

It turns out that these two different reductions are very closely related to
each other: they both correspond to the $(L,M)$ periodic reduction of the
lattice KdV equation [10]

$\displaystyle
V_{k+1,l}-V_{k,l+1}=\alpha\left(\frac{1}{V_{k,l}}-\frac{1}{V_{k+1,l+1}}\right).$
(4.1)

This connection leads to an alternative $2\times 2$ Lax pair for these
discrete KP reductions and their associated U-systems, as well as linking the
Liouville integrability of the latter with that of the corresponding discrete
KdV reduction.

Observe that if the $(L,M)$ reduction is imposed on (4.1), then we have

$\displaystyle V_{k+L,l+M}=V_{k,l}\implies V_{k,l}=v_{m},\qquad m=lL-kM,$

and we can write the following ordinary difference equation in terms of the
travelling wave variable $m$:

$\displaystyle
v_{m+{L}+{M}}-v_{m}=\alpha\left(\frac{1}{v_{m+L}}-\frac{1}{v_{m+M}}\right).$
(4.2)

######  Remark 4.1.

Up to sending $v_{m}\to 1/v_{m}$ and redefining ${\alpha}$, the $(L,M)$
reduction and the $(L,-M)$ reduction of (4.1) are equivalent. The parameter
${\alpha}$ can be removed by scaling.

######  Proposition 4.2.

Suppose that

$\displaystyle v_{m}=\frac{\tau_{m}\tau_{m+L+M}}{\tau_{m+M}\tau_{m+L}}$ (4.3)

is a solution of (4.2). Then $\tau_{m}$ satisfies the following two bilinear
equations

$\displaystyle\tau_{m+2L+M}\tau_{m}=\beta_{m}\tau_{m+L+M}\tau_{m+L}-\alpha\tau_{m+2L}\tau_{m+M},\qquad\beta_{m+M}=\beta_{m},$
(4.4)
$\displaystyle\tau_{m+2M+L}\tau_{m}=\beta^{\prime}_{m}\tau_{m+L+M}\tau_{m+M}+\alpha\tau_{m+2M}\tau_{m+L},\qquad\beta^{\prime}_{m+L}=\beta^{\prime}_{m}.$
(4.5)

Conversely, if $\tau_{m}$ is a solution of either (4.4) or (4.5), then $v_{m}$
given by (4.3) satisfies (4.2).

###### Proof.

With (4.3), (4.2) is equivalent to either of the two equalities

$\displaystyle\frac{\tau_{m+2L+M}\tau_{m}+\alpha\tau_{m+2L}\tau_{m+M}}{\tau_{m+L}\tau_{m+L+M}}=\frac{\tau_{m+2L+2M}\tau_{m+M}+\alpha\tau_{m+2L+M}\tau_{m+2M}}{\tau_{m+L+M}\tau_{m+L+2M}},$
$\displaystyle\frac{\tau_{m+2M+L}\tau_{m}-\alpha\tau_{m+2M}\tau_{m+L}}{\tau_{m+M}\tau_{m+M+L}}=\frac{\tau_{m+2M+2L}\tau_{m+L}-\alpha\tau_{m+2M+L}\tau_{m+2L}}{\tau_{m+M+L}\tau_{m+M+2L}},$

from which the result follows. ∎

######  Remark 4.3.

In [14, Proposition 8], it was proved that the solutions of $(d-1,-1)$
reductions of the lattice KdV equation (4.1) are Liouville integrable, using
the observation that (with ${\alpha}=-1$) these reductions are given in terms
of a tau function that satisfies the bilinear recurrence relation

$\displaystyle\tau_{m+d+1}\tau_{m}=\beta_{m}\tau_{m+d}\tau_{m+1}+\tau_{m+d-1}\tau_{m+2},$

with the coefficient $\beta_{m}$ having period $d-1$. The above result extends
this observation to the general $(L,M)$ reduction, and shows that in each case
there are actually two different bilinear equations involved.

###  4.1 $\boldsymbol{2\times 2}$ Lax pairs

Without loss of generality, we can assume from now on that $L$, $M$ are
coprime (since otherwise the equations split into copies of systems in lower
dimension). According to Corollary 2.2, the reductions (4.4) and (4.5) of
discrete KP each admit a Lax representation, with $L\times L$ and
$\min{(L,2M)}\times\min{(L,2M)}$ Lax matrices respectively. However, in these
cases there is also a $2\times 2$ Lax representation, derived from the Lax
representation of the lattice KdV equation.

The lattice KdV equation (4.1) admits a zero curvature representation with a
$2\times 2$ Lax pair. Specifically, equation (4.1) is equivalent to

$\displaystyle{\bf L}(V_{k,l+1},V_{k+1,l+1},{\lambda}){\bf
M}(V_{k,l},{\lambda})={\bf M}(V_{k+1,l},{\lambda}){\bf
L}(V_{k,l},V_{k+1,l},{\lambda}),$ (4.6)

where ${\lambda}$ is a spectral parameter and

$\displaystyle{\bf
L}(V,W,{\lambda})=\left(\begin{matrix}V-\frac{\alpha}{W}&{\lambda}\\\
1&0\end{matrix}\right),\qquad{\bf
M}(V,{\lambda})=\left(\begin{matrix}V&{\lambda}\\\
1&\frac{{\alpha}}{V}\end{matrix}\right).$ (4.7)

It is well known that the Lax representation of quadrilateral lattice
equations gives rise to Lax representations of their periodic reductions (see,
e.g., [25] and references). Hence, a $2\times 2$ Lax representation can be
obtained for the $(L,M)$ periodic reduction (4.2) and consequently for the
corresponding discrete KP reductions, as well as their associated U-systems.
First integrals of these systems are derived from the spectrum of their
corresponding monodromy matrix.

###  4.2 Example: a discrete KdV reduction of order 5

The Liouville integrability of (4.2) in the case $L=4$, $M=1$ follows from the
results of [14], so here we consider a different example of order 5, namely
the case $L=3$, $M=2$. In the latter case, the recurrences (4.4) and (4.5)
become

$\displaystyle\tau_{m+8}\tau_{m}=-\alpha\tau_{m+6}\tau_{m+2}+\beta_{m}\tau_{m+3}\tau_{m+5},\qquad\beta_{m+2}=\beta_{m},$
(4.8)
$\displaystyle\tau_{m+7}\tau_{m}=\alpha\tau_{m+4}\tau_{m+3}+\beta^{\prime}_{m}\tau_{m+2}\tau_{m+5},\qquad\beta^{\prime}_{m+3}=\beta^{\prime}_{m},$
(4.9)

respectively.

In each case, the associated exchange matrix $B$ has rank 4. The corresponding
U-systems are obtained by setting

$\displaystyle u_{m}=\frac{\tau_{m}\tau_{m+4}}{\tau_{m+1}\tau_{m+3}},\qquad
u^{\prime}_{m}=\frac{\tau_{m}\tau_{m+3}}{\tau_{m+1}\tau_{m+2}},$ (4.10)

to get

$\displaystyle u_{m}u_{m+1}u_{m+2}u_{m+3}u_{m+4}=\beta_{m}-{\alpha}u_{m+2},$
(4.11) $\displaystyle
u_{m}^{\prime}u_{m+1}^{\prime}({u}_{m+2}^{\prime})^{2}u_{m+3}^{\prime}u_{m+4}^{\prime}=\beta^{\prime}_{m}u_{m+2}^{\prime}+{\alpha},$
(4.12)

respectively. By Theorem [6], the latter are equivalent to iteration of
4-dimensional birational symplectic maps $\hat{\varphi}_{m}^{(j)}$, $j=1,2$,
where

$\displaystyle\hat{\varphi}_{0}^{(1)}(u_{0},u_{1},u_{2},u_{3})=\left(u_{1},u_{2},u_{3},\frac{\beta_{0}-\alpha
u_{2}}{u_{0}u_{1}u_{2}u_{3}}\right),$
$\displaystyle\hat{\varphi}_{0}^{(2)}(u_{0}^{\prime},u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})=\left(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime},\frac{\alpha+\beta^{\prime}_{0}u_{2}^{\prime}}{u_{0}^{\prime}u_{1}^{\prime}(u_{2}^{\prime})^{2}u_{3}^{\prime}}\right),$

respectively (we have just written the case $m=0$), with the nondegenerate
Poisson bracket being specified by

$\displaystyle\\{u_{j},u_{j+1}\\}_{1}=0,\qquad\\{u_{j},u_{j+2}\\}_{1}=u_{j}u_{j+2},\qquad\\{u_{j},u_{j+3}\\}_{1}=-u_{j}u_{j+3}$
(4.13)

for the first one and

$\displaystyle\\{u_{j}^{\prime},u_{j+1}^{\prime}\\}_{2}=0,\qquad\\{u_{j}^{\prime},u_{j+2}^{\prime}\\}_{2}=u_{j}^{\prime}u_{j+2}^{\prime},\qquad\\{u_{j}^{\prime},u_{j+3}^{\prime}\\}_{2}=-u_{j}^{\prime}u_{j+3}^{\prime}$
(4.14)

for the second. Observe that the two brackets (4.13) and (4.14) are identical.

On the other hand, from Proposition 4.2, by setting

$\displaystyle v_{j}=\frac{\tau_{j}\tau_{j+5}}{\tau_{j+2}\tau_{j+3}},$ (4.15)

the discrete KP reductions (4.8) and (4.9) both yield the $(3,2)$ periodic
reduction of the lattice KdV equation, which is equivalent to the
5-dimensional birational map

$\displaystyle(v_{0},v_{1},v_{2},v_{3},v_{4})\mapsto\left(v_{1},v_{2},v_{3},v_{4},v_{0}+\alpha\left(\frac{1}{v_{3}}-\frac{1}{v_{2}}\right)\right).$
(4.16)

Lax representation and first integrals. Corollary 2.2 produces a $3\times 3$
Lax representation for both (4.8) and (4.9), and for their corresponding
U-systems (4.11) and (4.12), with a trigonal spectral curve. However, it is
more straightforward to apply the (3,2) periodic reduction to the discrete KdV
Lax pair (4.6), directly giving a Lax pair for (4.16) in terms of the
coordinates $v_{j}$, which can then be rewritten in terms of the $u_{j}$ or
$u_{j}^{\prime}$ as desired.

The monodromy matrix of the $(3,2)$ KdV periodic reduction, obtained by the
staircase method [25], is

$\displaystyle\mathcal{M}(v_{0},v_{1},v_{2},v_{3},v_{4};{\lambda})={\bf
M}(v_{3}){\bf L}(v_{1},v_{3}){\bf M}(v_{4}){\bf L}(v_{2},v_{4}){\bf
L}(v_{0},v_{2}),$

where the $2\times 2$ matrices ${\bf M}$ and ${\bf L}$ are given in (4.7). The
map (4.16) satisfies the equation

$\displaystyle\mathcal{M}(v_{0},v_{1},v_{2},v_{3},v_{4})\mathcal{L}=\mathcal{L}\mathcal{M}(v_{1},v_{2},v_{3},v_{4},v_{5}),$

where

$\displaystyle\mathcal{L}={\bf L}(v_{0},v_{2})^{-1}{\bf M}(v_{5}){\bf
L}(v_{3},v_{5}){\bf L}(v_{1},v_{3})\qquad\text{and}\qquad
v_{5}=v_{0}+\alpha\left(\frac{1}{v_{3}}-\frac{1}{v_{2}}\right).$

The associated hyperelliptic spectral curve in the $({\lambda},\mu)$ plane is
of genus 2, being given by

$\displaystyle\det({\cal
M}({\lambda})-\mu\mathbf{I})\equiv\mu^{2}-\Pi({\lambda})\mu-{\lambda}^{3}({\lambda}-{\alpha})^{2}=0,$

where the trace of the monodromy matrix ${\cal
M}({\lambda})=\mathcal{M}(v_{0},v_{1},v_{2},v_{3},v_{4};{\lambda})$ has the
form

$\displaystyle\Pi({\lambda})=H_{2}{\lambda}^{2}+H_{1}{\lambda}+H_{0}.$

From the trace of the monodromy matrix we find three functionally independent
first integrals for the map (4.16), which are conveniently chosen as

$\displaystyle I_{1}=\Pi(0)=H_{0},\qquad I_{2}=H_{2},\qquad
I_{3}=\Pi({\alpha})=H_{2}{\alpha}^{2}+H_{1}{\alpha}+H_{0},$

so that they have the explicit form

$\displaystyle I_{1}=-\frac{1}{v_{2}}(\alpha-v_{0}v_{2})(\alpha-
v_{1}v_{3})(\alpha-v_{2}v_{4}),$ $\displaystyle
I_{2}=v_{0}+v_{1}+v_{2}+v_{3}+v_{4}-\frac{\alpha}{v_{2}},$ $\displaystyle
I_{3}=v_{2}(\alpha+v_{0}v_{3})(\alpha+v_{1}v_{4}).$

In order to obtain the corresponding first integrals in terms of the variables
for the two different U-systems, we compare (4.10) with (4.15) to see that we
can write the variables for (4.16) in two different ways, as

$\displaystyle
v_{m}=u_{m}u_{m+1}=u_{m}^{\prime}u_{m+1}^{\prime}u_{m+2}^{\prime}.$

Now to pull back $I_{1}$, $I_{2}$, $I_{3}$ to the first U-system, we must
iterate the recurrence (4.11) to get

$\displaystyle v_{0}=u_{0}u_{1},\qquad v_{1}=u_{1}u_{2},\qquad
v_{2}=u_{2}u_{3},\qquad v_{3}=\frac{\beta_{0}-\alpha
u_{2}}{u_{0}u_{1}u_{2}},\qquad v_{4}=\frac{\beta_{1}-\alpha
u_{3}}{u_{1}u_{2}u_{3}}.$ (4.17)

In that case, we find that $I_{1}$, $I_{2}$ are two independent functions of
the $u_{j}$, while pulling back the third integral yields

$\displaystyle I_{3}=\beta_{0}\beta_{1},$ (4.18)

which is a trivial first integral for the symplectic map
$\hat{\varphi}_{m}^{(1)}$. Similarly, by iterating the second U-system (4.11),
we obtain

$\displaystyle v_{0}=u_{0}^{\prime}u_{1}^{\prime}u_{2}^{\prime},\qquad
v_{1}=u_{1}^{\prime}u_{2}^{\prime}u_{3}^{\prime},\qquad
v_{2}=\frac{\alpha+\beta^{\prime}_{0}u_{2}^{\prime}}{u_{0}^{\prime}u_{1}^{\prime}u_{2}^{\prime}},$
(4.19) $\displaystyle
v_{3}=\frac{\alpha+\beta^{\prime}_{1}u_{3}^{\prime}}{u_{1}^{\prime}u_{2}^{\prime}u_{3}^{\prime}},\qquad
v_{4}=\frac{\alpha\beta^{\prime}_{2}+\beta^{\prime}_{0}\beta^{\prime}_{2}u_{2}^{\prime}+{\alpha}u_{0}^{\prime}u_{1}^{\prime}(u_{2}^{\prime})^{2}u_{3}^{\prime}}{{\alpha}u_{2}^{\prime}u_{3}^{\prime}+\beta_{0}^{\prime}(u_{2}^{\prime})^{2}u_{3}^{\prime}}.$

The quantities $I_{2}$, $I_{3}$ pull back to two independent functions of the
$u_{j}^{\prime}$, while for the first quantity we find a trivial first
integral of the symplectic map $\hat{\varphi}_{m}^{(2)}$, namely

$\displaystyle I_{1}=\beta_{0}^{\prime}\beta_{1}^{\prime}\beta_{2}^{\prime}.$
(4.20)

Bi-Hamiltonian structure and Liouville integrability. From (the pullbacks of)
the formulae for $I_{1}$, $I_{2}$ one can verify directly that, as functions
of $u_{j}$, they are in involution with respect to the bracket (4.13), which
implies that the symplectic map $\hat{\varphi}_{m}^{(1)}$ is Liouville
integrable. Similarly, one can check that the same conclusion holds for
$\hat{\varphi}_{m}^{(2)}$, by using (4.14) to verify that
$\\{I_{2},I_{3}\\}_{2}=0$. However, there is another way to obtain this
result, by lifting the brackets for both U-systems to obtain two different
Poisson structures for (4.16).

From the first U-system, using the formulae (4.17) we find that the Poisson
bracket $\\{\,,\,\\}_{1}$ in (4.13) pushes forward to a bracket in 5
dimensions (denoted here by the same symbol):

$\displaystyle\\{v_{0},v_{1}\\}_{1}=v_{0}v_{1},\qquad$
$\displaystyle\\{v_{0},v_{2}\\}_{1}=v_{0}v_{2},$
$\displaystyle\\{v_{0},v_{3}\\}_{1}=-v_{0}v_{3}-\alpha,\qquad$
$\displaystyle\\{v_{0},v_{4}\\}_{1}=-v_{0}v_{4}.$ (4.21)

The quantities $I_{1}$, $I_{2}$, $I_{3}$ found from the trace of the monodromy
matrix are in involution with respect to the bracket (4.21). This Poisson
bracket has rank 4, with $I_{3}$ as a Casimir (this follows from the
expression (4.18): a constant function of the $u_{j}$ must lift to a Casimir).

Similarly, pushing forward the second U-system, using the formulae (4.19) we
find that the Poisson bracket in (4.14) lifts to

$\displaystyle\\{v_{0},v_{1}\\}_{2}=v_{0}v_{1},\qquad$
$\displaystyle\\{v_{0},v_{2}\\}_{2}=v_{0}v_{2}-\alpha,$ (4.22)
$\displaystyle\\{v_{0},v_{3}\\}_{2}=-v_{0}v_{3},\qquad$
$\displaystyle\\{v_{0},v_{4}\\}_{2}=-v_{0}v_{4}+\frac{\alpha^{2}}{v_{2}^{2}}.$
(4.23)

Once again, this is a bracket of rank 4, with $I_{1}$ as a Casimir (as follows
from (4.20) above). One can show directly that $I_{2}$, $I_{3}$ provide two
more independent commuting functions of $v_{j}$ with respect to the bracket
$\\{\,,\,\\}_{2}$ given by (4.23).

It turns out that these two brackets for (4.16) are compatible, in the sense
that their sum (and hence any linear combination) also satisfies the Jacobi
identity, so is also a Poisson bracket. In fact, we observe that the
difference of the Poisson brackets

$\displaystyle\\{\cdot,\cdot\\}_{3}=\\{\cdot,\cdot\\}_{1}-\\{\cdot,\cdot\\}_{2}$
(4.24)

coincides (under the transformation $v_{i}\mapsto\frac{1}{v_{i}}$ and by
inserting the parameter $\alpha$) with the one that is derived from the
Lagrangian structure of the lattice KdV equation, recently presented in [24].
Thus we see that the map (4.16) has a bi-Hamiltonian structure, and the
sequence of first integrals $I_{1}$, $I_{2}$, $I_{3}$ gives a finite
Lenard–Magri chain.

###  4.3 Comments on the integrability of the general case

In the case $M=1$, the Liouville integrability of $(L,1)$ periodic reductions
of discrete KdV was proved in [14]. However, the Liouville integrability for
the case of general $(L,M)$ will be the subject of a future publication, and
we only comment on it briefly here. The Liouville integrability of the
corresponding U-systems follows from the Liouville integrability of the
$(L,M)$ periodic reduction of the lattice KdV equation (and vice-versa). For
all $(L,M)$ we find that the two U-systems are of the same dimension and
preserve the same log-canonical symplectic structure.

For $L+M$ odd, we can always find two compatible Poisson structures
$\\{\,,\,\\}_{1,2}$ for the variables $v_{m}$, whose difference (4.24)
coincides with the bracket obtained from a discrete Lagrangian in [24]. Here
we just state the corresponding theorem.

######  Theorem 4.4.

Let $L$, $M$ be coprime with $L>M>1$ and $L+M$ odd. For $0\leq i<j\leq L+M-1$,
the brackets

$\displaystyle\\{v_{i},v_{j}\\}_{1}=\begin{cases}c_{j-i}v_{i}v_{j},&j-i\neq
L,\\\ c_{j-i}v_{i}v_{j}+c_{N}\alpha,&j-i=L,\end{cases}$
$\displaystyle\\{v_{i},v_{j}\\}_{2}=\begin{cases}c_{j-i}v_{i}v_{j},&j-i\neq
kM,\\\ \displaystyle
c_{j-i}v_{i}v_{j}+c_{M}(-\alpha)^{k}\prod\limits_{l=1}^{k-1}v_{i+l\,M}^{-2},&j-i=kM,\end{cases}$

where

$\displaystyle c_{k}=(-1)^{h},\qquad\text{with}\quad h=\frac{k}{M}\bmod(N+M),$

for $k=1,\dots,L+M-1$, define two compatible Poisson structures on
$\mathbb{C}^{L+M}$ of rank $L+M-1$ preserved by the map

$\displaystyle(v_{0},v_{1},\dots,v_{L+M-1})\mapsto\left(v_{1},v_{2},\dots,v_{0}+\alpha\left(\frac{1}{v_{L}}-\frac{1}{v_{M}}\right)\right),$

corresponding to the $(L,M)$ periodic reduction of the lattice KdV equation.

The proof of this theorem, as well as the detailed description of the
U-systems in the general case, will be presented elsewhere.

In the case where $L+M$ is even the situation is slightly different. In this
case the U-systems are $(L+M-2)$-dimensional maps but can be lifted to a space
of one dimension higher. In this $(L+M-1)$-dimensional space there is an
invariant bi-Poisson structure that ensures Liouville integrability. The
integrability of the lifted maps implies the integrability of the initial
U-systems as well as that of the corresponding $(L,M)$ reductions of the
discrete KdV equation.

##  5 Conclusions

We have described the general properties of plane wave reductions of the
Hirota–Miwa (discrete KP) equation (1.1), and have considered two special
families of such reductions that correspond to travelling waves of certain
lattice equations, of discrete Toda and discrete KdV type, respectively. Both
of these families admit a $2\times 2$ Lax representation that gives rise to
hyperelliptic spectral curves, whose coefficients are first integrals of the
corresponding maps. This is in addition to the Lax pairs obtained in Corollary
2.2, which are generally of larger size.

The plane wave reductions of the discrete KP equation yield finite genus
solutions, and in that case the equation (1.1) itself corresponds to the Fay
trisecant identity for the theta function of the spectral curve [26]. The Fay
identity can also be used to derive corresponding solutions of continuous
soliton equations via a limiting process [22, 23], so in a sense the discrete
Hirota equation (1.1) is more fundamental than its continuous counterparts.

It is interesting to note that the discrete Toda and discrete KdV families
exhaust all the discrete KP reductions up to order 7, i.e. all the three-term
Somos recurrences up to Somos-7. If we proceed to higher order discrete KP
reductions, then new families appear. For example, the three-term Somos-8
recurrence

$\displaystyle\tau_{m+8}\tau_{m}={\alpha}\tau_{m+7}\tau_{m+1}+{\beta}\tau_{m+5}\tau_{m+3}$

is neither of Toda nor of KdV type: it belongs to a different family of
recurrences associated with periodic reductions of a Boussinesq type lattice
equation. Concerning three-term Somos-9 and Somos-10 recurrences, all cases
except one are included in the Toda, KdV or Boussinesq families. Further
details of the Liouville integrable maps arising from these and the other
families will be the subject of future work.

### Acknowledgements

Some of these results first appeared in the Ph.D. Thesis [28], which was
supported by EPSRC studentship EP/P50421X/1. ANWH is supported by EPSRC
fellowship EP/M004333/1.

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