,

# An algebraic criterion of the Darboux integrability of differential-
difference equations and systems

I T Habibullin, M N Kuznetsova  Institute of Mathematics, Ufa Federal Research
Centre, Russian Academy of Sciences, 112 Chernyshevsky Street, Ufa 450008,
Russian Federation<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The article investigates systems of differential-difference equations of
hyperbolic type, integrable in sense of Darboux. The concept of a complete set
of independent characteristic integrals underlying Darboux integrability is
discussed. A close connection is found between integrals and characteristic
Lie-Rinehart algebras of the system. It is proved that a system of equations
is Darboux integrable if and only if its characteristic algebras in both
directions are finite-dimensional.

###### pacs:

02.30.Ik

††: J. Phys. A: Math. Theor.

##  1 Introduction

The article deals with systems of differential-difference equations of the
following form:

$u^{j}_{n+1,x}=F^{j}(x,n,u^{j}_{n,x},u_{n},u_{n+1}),\quad j=1,2,\ldots,N,$
(1.1)

where $u_{n}=(u^{1}_{n},u^{2}_{n},\ldots,u^{N}_{n})$ is the sought function,
its components $u^{j}_{n}=u^{j}_{n}(x)$ depend on two variables: real $x$ and
integer $n$. It is assumed that $F^{j}$ analytically depends on the dynamical
variables $u^{j}_{n,x}$, $u_{n}$, $u_{n+1}$ in some domain of the space
$\mathbb{C}^{2N+1}$. We study the system from the point of view of Darboux
integrability. Below we will briefly explain the essence of this concept.
Fundamental ideas in the study of the problem of constructing in closed form
general solution of partial differential equations of hyperbolic type go back
to the classical works of Laplace, Liouville, Lie, Darboux, Goursat, Vessio,
and others. One of the basic approaches to the problem is worked out by
Darboux. The first step in Darboux’s method is to find integrals over each
characteristic direction, which gives two ordinary differential equations. In
the second one a formula is looked for parameterizing general solutions of
these ODEs. Let us explain Darboux’s method using the example of the well
known Liouville equation $u_{xy}=e^{u}$ (see, for instance, [1]). It has
integrals of the form

$I=u_{xx}-\frac{1}{2}u_{x}^{2},\qquad J=u_{yy}-\frac{1}{2}u_{y}^{2}.$

Functions $I$ and $J$ are such that for arbitrary solution $u(x,y)$ of the
Liouville equation the equalities are fulfilled: $\frac{d}{dy}I=0$ and
$\frac{d}{dx}J=0$. That immediately implies that each solution of the equation
satisfies a pair of the ordinary differential equations

$u_{xx}-\frac{1}{2}u_{x}^{2}=p(x),\qquad u_{yy}-\frac{1}{2}u_{y}^{2}=q(y).$
(1.2)

It is noteworthy that the representation (1.2) of an arbitrary solution makes
it possible to derive a well-known formula that parametrizes the general
solution of the Liouville equation

$u(x,y)=\log\left(\frac{2\phi^{\prime}(x)\psi^{\prime}(y)}{(\phi(x)+\psi(y))^{2}}\right).$
(1.3)

It is not surprising that Liouville-type equations admit an explicit formula
for the general solution. Actually existence of a nontrivial integral makes it
possible to find a differential substitution that reduces the equation to the
d’Alembert equation. In the case if the Liouville equation it is done easily.
Indeed, by taking $v_{x}=u_{xx}-\frac{1}{2}u_{x}^{2}$ we obtain that $v(x,y)$
due to formula (1.2) satisfies equation $v_{xy}=0$.

To find the integrals (as well as to find out the integrability of a given
equation) Darboux used Laplace cascade method. In later studies (see [2], [3],
[4], [5], the direct method using characteristic vector fields becomes the
main tool for searching for integrals (it is within this approach that
apparently, the first lists of equations with integrals in both directions
were obtained, see [2], [3]).

Interest in this topic revived at the end of the twentieth century after the
works [6, 7], where the concept of characteristic algebra was introduced,
which became an effective implement for studying and classifying systems of
partial differential equations of hyperbolic type. The fundamental fact of
this theory is that a system of PDEs is Darboux integrable if and only if its
characteristic algebras in both characteristic directions are of finite
dimension (see [8]). Note that characteristic algebras of the exponential type
systems of differential equations are actually Lie algebras over the field of
constants. But later it was observed that in general characteristic algebra is
really an algebra over the ring of locally analytic functions, such kind
generalization of the characteristic algebra was used for the sake of the
integrable classification of differential-difference equations in [9]. We
thank D.Millionschikov who pointed out that such kind algebra is a well known
object called in literature a Lie-Rinehart algebra [10, 11, 12].

At present, there are various alternative approaches to the problem of
classifying equations of the Liouville type, such as the method of higher
symmetries, the method of Laplace invariants, etc. Interesting classification
results, a description of the current state of research in this area and
references can be found in [1], [13].

Note that interest in discrete equations is steadily growing due to their
important applications in physics, informatics, biology, etc. Various
approaches to constructing solutions of such equations are often discussed in
the literature. One of the purposes of this work is to transfer the main
stages of the Darboux method to differential-difference systems of the form
(1.1). It should be noted that different aspects of the Darboux integrability
of differential-difference equations nowadays are actively discussed (see, for
instance, [14], [15], [16], [17], [18], [19], [9]).

The second purpose of the article is related to a recently found application
of the system (1.1) in the problem of the integrable classification of the
lattices in 3D. In a paper [20] by A. Khakimova and one of the authors of this
article, a method was proposed for the integrable classification of nonlinear
equations in 3D

$u_{n+1,x}^{j}=F(u_{n,x}^{j},u_{n}^{j+1},u_{n+1}^{j},u_{n}^{j},u_{n+1}^{j-1}),$
(1.4)

with two discrete and one continuous independent variables by reducing the
problem to studying higher order differential-difference Darboux integrable
systems with two independent variables. The above-mentioned method of
integrable reductions is also an effective tool for constructing particular
solutions of three-dimensional models of the form (1.4). However, some
important questions remained unexplored in [20], such as the description of
the structure of integrals of Darboux integrable differential-difference
systems, as well as the proof of the algebraic criterion for Darboux
integrability, although the criterion was formulated there as a conjecture.
These gaps are filled in this work.

In [20] it was observed that all known integrable equations with one
continuous and two discrete independent variables, presented in [21] can be
written in the form (1.4), it was checked that these equations admit
reductions being finite field Darboux integrable systems of semi-discrete
hyperbolic type equations (1.1). Class of equations (1.4) defines
differential-difference versions of the famous Hirota type difference
equations and is studied by many authors, the references can be found in [21].

A classification method based on the Darboux integrable reductions has earlier
been used in our articles [22, 23, 24] for studying differential-difference
equations of the form

$u_{n,xy}=f(u_{n+1},u_{n},u_{n-1},u_{n,x},u_{n,y}),$ (1.5)

having one discrete and two continuous independent variable $n$, $x$ and $y$.
Classification results concerned to the equations (1.5) can be found in [24,
25, 26, 23].

It worth mentioning that the integrability theory for the Liouville type
systems of hyperbolic PDE has been developed earlier by many authors. In the
works [8], [27] such important aspects of the theory, as searching of the
complete set of integrals by using characteristic algebras, integrability
criterion, finding general solutions has been studied in detail. Below we
discuss the similar problems for systems of the differential-difference
equations. Some properties of these two objects are very resembling, in those
cases we use the technique worked out in [8], [27]. However, there are aspects
where new ideas are needed. For instance, the structure of the characteristic
algebra for the second case is essentially more complex.

The main result of this paper is the proof of the statement that for the
Darboux integrability of the system of differential-difference equations (1.1)
it is necessary and sufficient that the characteristic Lie-Rinehart algebras
corresponding to both characteristic directions be of finite dimension.

Let’s briefly comment on the content of the article. In §2 we give definitions
of integrals of the system (1.1) for both characteristic directions $x$ and
$n$, explain the essence of Darboux integrability. To describe the structure
of the variety of integrals, we introduce the concept of a set of independent
integrals of minimal orders. In fact, the set thus defined constitutes the
basis of the integrals of the system (1.1). In §3, we recall the definition of
the characteristic Lie-Rinehart algebras in directions of $x$ and $n$, discuss
some of their properties, and derive an algebraic criterion of existence of a
complete set of $n$-integrals (see Theorems 3.2 and 3.3 ). The proof of
Theorem 3.3 contains a detailed description of the method for constructing the
set of integrals of minimal orders. Illustrative examples are given in §4.

##  2 Basis of integrals

In this section we discuss the problem of constructing bases of the integrals
for the main object of study of the present article, the system of
differential-difference hyperbolic type equations (1.1).

Let us recall the necessary definitions and explain the basic notations.
Variables $u^{j}_{n},u^{j}_{n\pm 1},u^{j}_{n\pm 2},\ldots$ and
$u^{j}_{n,x},u^{j}_{n,xx},u^{j}_{n,xxx},\ldots$ are called dynamical variables
and we consider them independent. We denote by $D_{x}$ the operator of the
total derivative with respect to $x$; $D_{n}$ stands for the operator shifting
the discrete argument $n$ such that $D_{n}y(n)=y(n+1)$. For higher-order
derivatives with respect to $x$, we use the notation:

$u^{j}_{n,[m]}=\frac{\partial^{m}u^{j}_{n}}{\partial x^{m}}.$

Definition 2.1. A function

$I=I(x,n,u_{n},u_{n,x},u_{n,xx},\ldots,u_{n,[m]})$

is called an $n$-integral of the order $m$, if at least one of the derivatives
$\frac{\partial I}{\partial u^{j}_{n,[m]}}$ where $j=1,2,\ldots,N$ does not
vanish identically and the condition $D_{n}I=I$ is satisfied by virtue of
system (1.1).

Set of $n$-integrals $I^{1},\ldots,I^{N}$ of the orders
$n_{1},n_{2},\ldots,n_{N}$ is called independent if the set of the functions
$D^{j}_{x}I^{i}$, where $i=1,\ldots,N$, $j=0,1,\ldots,s-n_{i}$ and
$s=max\\{n_{i}\\}$ is functionally independent.

Definition 2.2. A function

$J=J(x,n,u_{n-k},u_{n-k+1},\ldots,u_{n-1},u_{n},u_{n+1},\ldots,u_{n+m-1},u_{n+m}),$

$k,m=0,1,2,\ldots$ is called an $x$-integral of the order $m+k$, if at least
for one pair of the numbers $j,s=1,\ldots,N$ the product $\frac{\partial
J}{\partial u^{j}_{n+m}}\dot{\frac{\partial J}{\partial u^{s}_{n-k}}}$ does
not vanish identically and the following condition $D_{x}J=0$ holds by virtue
of system (1.1).

Since the shift transformation $J\rightarrow D_{n}J$ maps an $x$-integral into
$x$-integral we can set here $k=0$. Let us given a set of $x$-integrals
$J^{i}(x,n,u_{n},\ldots,u_{n+m_{i}})$ of the orders $m_{i}$. We call the set
$J^{1},\ldots,J^{N}$ a set of independent $x$-integrals if a system of
functions $D^{j}_{n}J^{i}$, where $i=1,\ldots,N$, $j=0,1,\ldots,s-s_{i}$ and
$s=max\\{s_{i}\\}$ is functionally independent.

For the convenience of further work, we introduce the concept of a set of
$n$-integrals of minimal orders: $n_{1}<n_{2}<\ldots<n_{s}$. Below we will
give the definition of the notion of set of integrals of minimal orders. We
will discuss a practical way of constructing this set by using the
characteristic algebra in the proof of the Theorem 3.3, and also illustrate it
with concrete examples (see § 4). Let

$\omega=\omega(x,n,u_{n},u_{n,x},\ldots,u_{n,[k]})$ (2.1)

be a function satisfying the conditions

$D_{n}\omega=\omega,\quad\frac{\partial\omega}{\partial u_{n,[k]}}\neq 0.$
(2.2)

Minimality of the order $n_{1}$ means that a function $\omega$ of the form
(2.1) satisfying (2.2), is a trivial integral as soon as $k<n_{1}$. Moreover,
for $k=n_{1}$, $\omega$ is not a trivial integral. We denote
$\omega^{1,1},\ldots,\omega^{1,m_{1}}$ the complete set of functionally
independent integrals of order $n_{1}$. Then we look for a number $n_{2}$ such
that there is no integral of the order $n<n_{2}$, which cannot be represented
as a function of $x$, the already found integrals
$\omega^{1,1},\ldots,\omega^{1,m_{1}}$ and their derivatives. Such an
essentially new integral exists for $n=n_{2}$. In other words we look for a
number $n_{2}$, assuming that $\omega$ is a function only on $x$, functions
$\omega^{1,1},\ldots,\omega^{1,m_{1}}$ and on the derivatives of these
functions with respect to $x$, as soon as $k\in[n_{1},n_{2})$. We denote by
$\omega^{2,1},\ldots,\omega^{2,m_{2}}$ a functionally independent set of
integrals of order $n_{2}$, which are not expressed in terms of
$x,\omega^{1,1},\ldots,\omega^{1,m_{1}}$ and their derivatives. Continuing
this process, we obtain a set of $n$-integrals of minimal orders
$n_{1}<n_{2}<\ldots<n_{s}$:

$\omega^{1,1},\ldots,\omega^{1,m_{1}},\omega^{2,1},\ldots,\omega^{2,m_{2}},\ldots,\omega^{s,1},\ldots,\omega^{s,m_{s}},$
(2.3)

where $m_{1}+m_{2}+\cdots+m_{s}=N$.

In a similar way we determine a set of $x$-integrals of minimal orders.

Let system (1.1) have a set of $n$-integrals $I^{1},I^{2},\ldots I^{N}$ of the
minimal orders $s_{1},s_{2},\ldots s_{N}$ correspondingly. Let us denote
$s=\mathrm{max}_{j}(s_{j})$, $j=1,2,\ldots,N$ and determine a new set of
integrals by taking $W^{i}=D^{s-s_{i}}_{x}I^{i}$. It is clear that functions
$W^{1},W^{2},\ldots W^{N}$ are integrals of system (1.1) of one and the same
order $s$.

Definition 2.3. We say that $n$-integrals $I^{1},I^{2},\ldots I^{N}$ of system
(1.1) constitute a set of independent in principal integrals if the following
condition holds

$\mathrm{det}\,\left(\frac{\partial W^{k}}{\partial u^{i}_{[s]}}\right)\neq
0,\quad k=1,2,\ldots N.$ (2.4)

Let system (1.1) have a set of $x$-integrals $J^{1},J^{2},\ldots J^{N}$ of the
form
$J^{i}=J^{i}(x,n,u_{n},\ldots,u_{n+m_{i}})$ of the minimal orders $r_{1}\leq
r_{2},\ldots,\leq r_{N}$ correspondingly. We determine a new set of integrals
by taking $Z^{i}=D^{r_{N}-r_{i}}_{n}J^{i}$.

Definition 2.4. We say that $x$-integrals of the minimal orders
$J^{1},J^{2},\ldots J^{N}$ of system (1.1) constitute a set of independent in
principal integrals if the following condition holds

$\mathrm{det}\,\left(\frac{\partial Z^{k}}{\partial
u^{i}_{n+r_{N}}}\right)\neq 0,\quad k=1,2,\ldots,N.$ (2.5)

The notion of the independence in principal allows one to deduce an effective
criterion of the independent set of integrals.

Let us consider a set of $n$-integrals

$\omega^{1},\omega^{2},\ldots,\omega^{N}$ (2.6)

of the minimal orders $k_{1}\leq k_{2}\leq\ldots\leq k_{N}$. We assume that
(2.6) are analytic functions of the dynamical variables
$u_{n},u_{n,x},u_{n,xx},\ldots$ defined on a domain $D$.

Lemma 2.1. Set of $n$-integrals (2.6) is independent if and only if it is
independent in principal.

Proof. Assume that integrals (2.6) are independent and the determinant
vanishes

$\mathrm{det}\,\left(\frac{\partial W^{k}}{\partial
u^{i}_{n,[k_{N}]}}\right)=\left|\begin{array}[]{cccc}\frac{\partial
W^{1}}{\partial u^{1}_{[n,k_{N}]}}&\frac{\partial W^{1}}{\partial
u^{2}_{[n,k_{N}]}}&...&\frac{\partial W^{1}}{\partial u^{N}_{[n,k_{N}]}}\\\
...&...&...&...\\\ \frac{\partial W^{N}}{\partial
u^{1}_{[n,k_{N}]}}&\frac{\partial W^{N}}{\partial
u^{2}_{[n,k_{N}]}}&...&\frac{\partial W^{N}}{\partial
u^{N}_{[n,k_{N}]}}\end{array}\right|=0.$ (2.7)

The due to the theorem on implicit functions one of the functions
$W^{1},W^{2},\ldots,W^{N}$ is expressed in terms of the others and the
variables $x,n,u_{n},u_{n,x},\ldots,u_{n,[k_{N}-1]}$. For the definiteness we
set

$W^{N}=F(W^{1},W^{2},\ldots,W^{N-1},x,n,u_{n},u_{n,x},\ldots,u_{n,[k_{N}-1]}).$
(2.8)

Let us expand the function (2.8) in the Taylor series around some fixed point
$(W^{1}_{0},W^{2}_{0},\ldots,W^{N-1}_{0})\in D$:

$\displaystyle
F=\sum^{\infty}_{i_{1},i_{2},\ldots,i_{N-1}=0}\alpha_{i_{1}i_{2}\ldots
i_{N-1}}(x,n,u_{n},u_{n,x},\ldots,u_{n,[k_{N}-1]})\times$
$\displaystyle\times(W^{1}-W^{1}_{0})^{i_{1}}\cdots(W^{N-1}-W^{N-1}_{0})^{i_{N-1}}.$
(2.9)

Apply the operator $D_{n}-1$ to both sides of (2.9) and get

$(D_{n}-1)\alpha_{i_{1}\ldots
i_{N-1}}(x,n,u_{n},u_{n,x},\ldots,u_{n,[k_{N}-1]})=0.$

Therefore the coefficients of the series (2.9) are integrals of the orders
less than $k_{N}$. Due to the definition of the set of integrals of the
minimal orders they are expressed in terms of
$x,\omega^{1},\omega^{2},\ldots,\omega^{N}$. Hence by virtue of the
coincidence $W^{N}=\omega^{N}$ we obtain that integrals (2.6) are dependent,
that contradicts the assumption above.

We prove now the converse statement. Assume that condition

$\left|\begin{array}[]{cccc}\frac{\partial W^{1}}{\partial
u^{1}_{n,[k_{N}]}}&\frac{\partial W^{1}}{\partial
u^{2}_{n,[k_{N}]}}&...&\frac{\partial W^{1}}{\partial u^{N}_{n,[k_{N}]}}\\\
...&...&...&...\\\ \frac{\partial W^{N}}{\partial
u^{1}_{n,[k_{N}]}}&\frac{\partial W^{N}}{\partial
u^{2}_{n,[k_{N}]}}&...&\frac{\partial W^{N}}{\partial
u^{N}_{n,[k_{N}]}}\end{array}\right|\neq 0$

holds and integrals
$\omega^{1},D_{x}\omega^{1},\ldots,D^{k_{n}-k_{1}}_{x}\omega^{1},\omega^{2},D_{x}\omega^{2},\ldots,D^{k_{N}-k_{2}}_{x}\omega^{2},\ldots,\omega^{N}$
are dependent. The latter means that there exists a function $\Phi$ such that

$\Phi(\omega^{1},\ldots,\omega^{N},D_{x}\omega^{1},\ldots,D_{x}\omega^{N},\ldots,W^{1},\ldots,W^{N})=0.$
(2.10)

We assume that function $\Phi$ satisfies the relation

$\sum^{N}_{k=1}\left|\frac{\partial\Phi}{\partial W^{k}}\right|^{2}\neq 0,$
(2.11)

i.e. at least one of the derivatives $\frac{\partial\Phi}{\partial W^{k}}$
doesn’t vanish identically. If it is not the case and $\Phi$ doesn’t depend on
any of $W^{j}$ we apply to (2.10) operator $D_{x}$ and get a new relation of
the form (2.10), that would satisfy (2.11). If necessary we perform this trik
several times to get equation (2.10) satisfying (2.11). Then we differentiate
the relation (2.10) with respect to each of the variables $u^{i}_{s}$, where
$s=n_{N}$ and $i=1,2,\ldots,N$. As a result we get an equation

$\frac{\partial\Phi}{\partial W^{1}}\frac{\partial W^{1}}{\partial
u^{i}_{s}}+\frac{\partial\Phi}{\partial W^{2}}\frac{\partial W^{2}}{\partial
u^{i}_{s}}+\cdots+\frac{\partial\Phi}{\partial W^{N}}\frac{\partial
W^{N}}{\partial u^{i}_{s}}=0$

which implies that

$\mathrm{det}\left(\frac{\partial W^{k}}{\partial u^{i}_{s}}\right)=0.$

Lemma is proved.

For the $x$-integrals we can prove the analogue of the Lemma 1.1, claiming
that set of $x$-integrals is independent if and only if it is independent in
principal.

Now we will prove that a set of $n$-integrals independent in principal
constitutes in a sense a basis of integrals by showing that any integral of
the system is expressed through the members of this set.

Theorem 2.1. Assume that system (1.1) has an independent in principal set of
$n$-integrals $I^{1},\ldots,I^{N}$ analytically depending on the dynamical
variables in a domain $D$, having the minimal orders $n_{1}\leq
n_{2}\leq...\leq n_{N}$. Then any other $n$-integral $G$ of (1.1) analytic in
$D$ is a function of

$x,I^{1},I^{2},\ldots,I^{N},D_{x}I^{1},D_{x}I^{2},\ldots
D_{x}I^{N},D^{2}_{x}I^{1},D^{2}_{x}I^{2},\ldots,D^{2}_{x}I^{N},\ldots$

Proof. Denote by $s$ the order of the integral $G$. If $s\leq n_{N}$ then the
statement of the theorem follows from the definition of the set of the minimal
orders. For the simplicity we take the case $s=n_{N}+1$. Then by applying
operator $D_{x}$ we get a new set of the integrals
$W^{1}_{1},W^{2}_{1},\ldots,W^{N}_{1}$ where $W^{i}_{1}:=D_{x}W^{i}$. Since
the relations hold

$\mathrm{det}\left|\frac{\partial W^{i}_{1}}{\partial
u^{i}_{n,[s]}}\right|=\mathrm{det}\left|\frac{\partial W^{i}}{\partial
u^{i}_{n,[s-1]}}\right|\neq 0$ (2.12)

function $G$ can be represented as

$G=F(W^{1}_{1},\ldots,W^{N}_{1},u_{n,[s]},u_{n,[s-1]},\ldots,u_{n},x,n).$
(2.13)

Moreover we can get rid $u_{n,[s]}$ due to (2.12) and rewrite (2.13) as

$G=\bar{F}(W^{1}_{1},\ldots,W^{N}_{1},W^{1},\ldots,W^{N},u_{n,[s-1]},\ldots,u_{n},x,n).$
(2.14)

Then we expand $\bar{F}$ into a power series around a point
$(W^{1}_{0},\ldots,W^{N}_{0};W^{1}_{01},\ldots,W^{N}_{01})\in D:$

$\displaystyle\bar{F}=\sum\alpha_{i_{1}\ldots
i_{N},j_{1},\ldots,j_{N}}(x,n,u_{n},u_{n,x},\ldots,u_{n,[s-1]})\times$
$\displaystyle\times(W^{1}-W^{1}_{0})^{i_{1}}\cdots(W^{N}-W^{N}_{0})^{i_{N}}(W^{1}_{1}-W^{1}_{01})^{j_{1}}\cdots(W^{N}_{1}-W^{N}_{01})^{j_{N}}.$

Now we apply operator $D_{n}-1$ to $\bar{F}$ and prove that coefficients of
the series are $n$-integrals of the order less than or equal to $s-1=n_{N}$,
therefore they are expressed through $x$, integrals $I^{1},I^{2},\ldots,I^{N}$
and their derivatives with respect to $x$. This completes the proof.

A version of the Theorem 2.1 concerned the $x$-integrals is also true. It can
be proved in a similar way.

Theorem 2.2. Assume that system (1.1) has an independent in principal set of
$x$-integrals $J^{1},\ldots,J^{N}$ analytically depending on the dynamical
variables in a domain $\bar{D}$, having the minimal orders $n_{1}\leq
n_{2}\leq...\leq n_{N}$. Here for any $j$ integral $J^{j}$ is of the form
$J^{j}=J^{j}(x,n,u_{n},\ldots,u_{n+n_{j}})$. Then any other $x$-integral
$\bar{G}=\bar{G}(x,n,u_{n},\ldots,u_{n+s})$ of (1.1) analytic in $\bar{D}$ is
a function of

$n,J^{1},J^{2},\ldots,J^{N},D_{n}J^{1},D_{n}J^{2},\ldots
D_{n}J^{N},D^{2}_{n}J^{1},D^{2}_{n}J^{2},\ldots,D^{2}_{n}J^{N},\ldots$

Definition 2.5. System (1.1) is called Darboux integrable if it has sets of
integrals independent in principal on both characteristic directions of $x$
and $n$.

##  3 Algebraic criterion of Darboux integrability

In this section we introduce characteristic Lie-Rinehart algebras for the
system of differential-difference equations of the form (1.1). These objects
are very important attributes of the Darboux integrability since the finite
dimensionality property of the characteristic algebras is necessary and
sufficient for the Darboux integrability. For the scalar discrete equation the
notion of the characteristic algebra was introduced for the first time in
[28]. For the scalar differential-difference equation it was studied in [9].
The notion was generalized to the systems of differential-difference equations
in [20]. Below we use the definitions given in [20].

Suppose that system (1.1) has an integral in the direction of $x$, we denote
it by $J$. It is easy to check that function $J$ can depend only on the
variables $x,n,u_{n},u_{n\pm 1},u_{n\pm 2},\ldots$. According to Definition
2.2, we have

$D_{x}J=K_{0}J=\left(\frac{\partial}{\partial
x}+\sum_{j=1}^{N}u_{n,x}^{j}\frac{\partial}{\partial
u_{n}^{j}}+u_{n+1,x}^{j}\frac{\partial}{\partial
u_{n+1}^{j}}+u_{n-1,x}^{j}\frac{\partial}{\partial
u^{j}_{n-1}}+\ldots\right)J=0.$ (3.1)

Taking into account system (1.1), the operator can be easily rewritten in the
form

$K_{0}=\frac{\partial}{\partial
x}+\sum_{j=1}^{N}(u_{n,x}^{j}\frac{\partial}{\partial
u_{n}^{j}}+F_{n}^{j}\frac{\partial}{\partial
u_{n+1}^{j}}+G_{n}^{j}\frac{\partial}{\partial
u^{j}_{n-1}}+F_{n+1}^{j}\frac{\partial}{\partial
u_{n+2}^{j}}+G_{n-1}^{j}\frac{\partial}{\partial u^{j}_{n-2}}+\ldots)$ (3.2)

where $F^{j}_{n+i}=F^{j}(x,n+i,u_{n+i,x}^{j},u_{n+i},u_{n+i+1})$ and
$G^{j}_{n-i}=G^{j}(x,n-i,u_{n-i,x}^{j},u_{n-i},u_{n-i-1})$. We call $K_{0}$
the characteristic operator in the $x$ direction.

Consequently, the $x$ -integral is a solution to the first order linear
differential equation

$K_{0}J=0.$ (3.3)

The peculiarity of the equation is that the coefficients of the equation
depend on the derivatives of the function $u$ with respect to $x$, while the
solution $J$ cannot depend on these variables. This actually means that
equation (3.3) is strongly overdetermined. Indeed, the sought function, in
addition to (3.3), satisfies several more equations

$X_{k}J=0,\quad k=1,2,\ldots,N,$ (3.4)

where $X_{k}=\frac{\partial}{\partial u^{k}_{n,x}}$. Thus any $x$-integral is
nullified by the operators

$X_{1},X_{2},\ldots,X_{N},K_{0}$ (3.5)

their multiple commutators as well as by linear combinations of the obtained
operators with variable coefficients, depending on the dynamical variables.
Therefore the $x$-integral belongs to the kernel of the operators which lie in
some generalization of the Lie algebra, where the operators can be multiplied
not only by constants but also by elements of the ring of locally analytic
functions depending on the dynamical variables, we denote the ring by $A$.

In other words, any $x$-integral belongs to the kernel of the operators in the
Lie-Rinehart algebra $L_{x}$, generated by the operators (3.5) over the ring
$A$. We call it characteristic algebra in the direction of $x$. Note that the
operation of multiplication by a function and taking the commutator of two
operators are connected by conditions:

  * 1)

$[W_{1},aW_{2}]=W_{1}(a)W_{2}+a[W_{1},W_{2}]$,

  * 2)

$(aW_{1})b=aW_{1}(b)$,

that are valid for any $W_{1},W_{2}\in L_{x}$ and $a,b\in A$. This means that,
if $W_{1}\in L_{x}$ and $a\in A$ then $aW_{1}\in L_{x}$ (see [10]).

The algebra $L_{x}$ is finite dimensional if there exists a basis having a
finite number of the elements $W_{1},W_{2},\ldots,W_{k}\in L_{x}$ such that an
arbitrary element $W\in L_{x}$ can be represented as a linear combination of
the form

$W=a_{1}W_{1}+a_{2}W_{2}+\dots+a_{k}W_{k},$ (3.6)

where the coefficients are functions $a_{1},a_{2},\ldots,a_{k}\in A$. If in
(3.6) $W=0$ then we have $a_{1}=0$, $a_{2}=0$, $\ldots$, $a_{k}=0$.

The requirement of local analyticity of the elements of the ring A is
important from the point of view of the existence of a basis.

Let us briefly discuss the difference between Lie and Lie-Rinehart algebras.

Example 3.1. Obviously, the Lie algebra generated by the operators
$Z_{1}=x^{2}\frac{\partial}{\partial x}$ and
$Z_{2}=x^{3}\frac{\partial}{\partial x}$ is infinite-dimensional. For example,
the commutator $[Z_{1},Z_{2}]=x^{4}\frac{\partial}{\partial x}$ is not a
linear combination of $Z_{1}$ and $Z_{2}$ with constant coefficients. At the
same time the Lie-Rinehart algebra corresponding to the ring $A$ of functions
analytic in the domain $x\neq 0$ of the complex plane generated by the same
operators is one-dimensional, since any element $Z$ in the algebra can be
represented as $Z=f(x)Z_{0}$, where $Z_{0}=\frac{\partial}{\partial x}$, since
here linear combinations with variable coefficients are allowed.

The algebraic background of the Darboux integrability theory is based on the
following important fact.

Theorem 3.1. System (1.1) admits an independent set of $x$-integrals if and
only if its characteristic Lie-Rinehart algebra $L_{x}$ has a finite
dimension.

The proof of Theorem 3.1. for the case $N=2$ is given in the recent article
[29] by A.V.Zhiber and one of the authors. The proof, presented in [29],
carries over without special difficulty to the general case, therefore we do
not present it here.

Let us turn to the consideration of $n$-integrals. Suppose that $I$ is a
nontrivial $n$-integral of system (1.1). It is easy to check that $I$ can
depend only on $n$, $x$ and dynamical variables
$u_{n},u_{n,x},u_{n,xx},\ldots$. According to Definition 2.1, the equality
holds

$I(x,n+1,u_{n+1},u_{n+1,x},u_{n+1,xx},...)=I(x,n,u_{n},u_{n,x},u_{n,xx},\ldots),$

which, by virtue of equation (1.1), can be rewritten in the following form

$I(x,n+1,u_{n+1},F,D_{x}F,D_{x}^{2}F,\ldots)=I(x,n,u_{n},u_{n,x},u_{n,xx},\ldots).$
(3.7)

Hence, it is clear that to find the $n$-integral it is necessary to solve a
functional equation, since the equality (3.7) connects the values of the
required function at two different points. Note that (3.7) admits a large
number of differential consequences. Indeed, since the right-hand side of
(3.7) does not contain dependence on the variable $u_{n+1}$, then for any $j$
the equality $\frac{\partial}{\partial u^{j}_{n+1}}D_{n}I=0$ holds, from which
it easily follows that

$Y_{j,1}I:=D_{n}^{-1}\frac{\partial}{\partial
u^{j}_{n+1}}D_{n}I=0\quad\mbox{for}\quad j=1,2,\ldots,N.$ (3.8)

In expanded form, (3.7) is represented as a vector field

$Y_{j,1}=\frac{\partial}{\partial
u^{j}_{n}}+\sum_{i=1}^{N}D_{n}^{-1}\left(\frac{\partial F_{n}^{i}}{\partial
u^{j}_{n+1}}\right)\frac{\partial}{\partial
u^{i}_{n,x}}+D_{n}^{-1}\left(\frac{\partial F_{n,x}^{i}}{\partial
u^{j}_{n+1}}\right)\frac{\partial}{\partial u^{i}_{n,xx}}+\dots,$ (3.9)

where $F_{n,x}^{i}:=D_{x}F_{n}^{i}$. We set $Y_{j,0}=\frac{\partial}{\partial
u^{j}_{n+1}}$ and then rewrite representation as follows

$Y_{j,1}=\frac{\partial}{\partial
u^{j}_{n}}+\sum_{i=1}^{N}D_{n}^{-1}\left(Y_{j,0}\left(F_{n}^{i}\right)\right)\frac{\partial}{\partial
u^{i}_{n,x}}+D_{n}^{-1}\left(Y_{j,0}\left(F_{n,x}^{i}\right)\right)\frac{\partial}{\partial
u^{i}_{n,xx}}+\dots.$

Repeatedly applying the operator $D_{n}$ to the equality $D_{n}I=I$, we obtain
$D^{k}_{n}I=I$, applying the operator $\frac{\partial}{\partial u^{j}_{n+1}}$
to the latter, we derive the equality

$Y_{j,k}I:=D_{n}^{-k}\frac{\partial}{\partial
u^{j}_{n+1}}D_{n}^{k}I=0\quad\mbox{for}\quad j=1,2,\ldots,N.$ (3.10)

Operators $Y_{j,k}$ allow coordinate representation

$Y_{j,k}=\sum_{i=1}^{N}D_{n}^{-1}\left(Y_{j,k-1}\left(F_{n}^{i}\right)\right)\frac{\partial}{\partial
u^{i}_{n,x}}+D_{n}^{-1}\left(Y_{j,k-1}\left(F_{n,x}^{i}\right)\right)\frac{\partial}{\partial
u^{i}_{n,xx}}+\dots.$ (3.11)

Since the coefficients of the equations (3.10) depend on the variables
$u^{j}_{n-s}$ for $1\leq s\leq k$ and $1\leq j\leq N$, while the solution to
the equation does not depend on them, then such equations

$\bar{X}_{i,s}I=0,$

where $\bar{X}_{i,s}=\frac{\partial}{\partial u^{i}_{n-s}}$ should hold for
these values of $i$ and $s$.

In what follows $A_{k}$ denotes a ring of locally analytic functions of the
variables $u_{n-k},u_{n-k+1},\ldots,u_{n}$;
$u_{nx},u_{n,xx},u_{n,xxx},\ldots$. In the next theorem, we define the
characteristic Lie-Rinehart algebra $L_{n}$ and derive a necessary condition
for the existence of $n$-integrals.

Theorem 3.2. If system (1.1) admits an independent set of $n$-integrals then
the following two conditions hold:

1) The linear space $V$ spanned by the operators $\\{Y_{i,s}\\}$ has finite
dimension, which we denote by $N_{1}$. Assume that
$Z_{1},Z_{2},\ldots,Z_{N_{1}}$ constitute a basis in $V$, such that for any
$Z\in V$ we have an expansion

$Z=\lambda_{1}Z_{1}+\lambda_{2}Z_{2}+\ldots+\lambda_{N_{1}}Z_{N_{1}},$

We emphasize that the coefficients in this expansion, as a rule, are not
constant, they are analytic functions of dynamic variables. Then due to
construction of the space $V$ there exists a number $N_{2}$, such that
$[Z_{j},\bar{X}_{i,s}]=0$ for all $j=1,2,\ldots,N_{1}$, $i=1,2,\ldots,N$ and
$s>N_{2}$.

2) The Lie-Rinehart algebra $L_{n}$ generated by the operators
$\\{Z_{j}\\}_{j=1}^{N_{1}}$ and $\\{\bar{X}_{i,s}\\}_{s=1,i=1}^{N_{2},N}$ over
the ring $A_{N_{2}}$ has a finite dimension.

Proof of the theorem. Assume that system (1.1) admits an essentially
independent set of $n$-integrals $I^{(1)}$, $I^{(2)}$,… , $I^{(N)}$ of the
order $m$. Then the relation holds

$\det\left(\frac{\partial I}{\partial u_{[m]}}\right)\neq 0$ (3.12)

where $I=(I^{(1)},I^{(2)},...,I^{(N)})$. Our goal is now to prove that the
corresponding characteristic algebra is of finite dimension.

Let us change the variables in the characteristic vector fields $Y_{j,k}$ in
the following way. We pass from the subset of the dynamical variables

$u_{n},u_{n,x},u_{n,xx},...$

to the subset

$u_{n},u_{n,x},u_{n,xx},...,u_{n,[m-1]},I,D_{x}(I),D^{2}_{x}(I),\ldots.$

In terms of these new variables characteristic operators take the form

$Y_{j,k}=\sum_{i}a^{(0)}_{j,k,i}\frac{\partial}{\partial
u^{i}_{n}}+a^{(1)}_{j,k,i}\frac{\partial}{\partial
u^{i}_{n,x}}+...+a^{(m)}_{j,k,i}\frac{\partial}{\partial
I^{i}}+a^{(m+1)}_{j,k,i}\frac{\partial}{\partial I^{i}_{[1]}}+...$

where $I^{i}_{[s]}=D_{x}^{s}(I^{i})$. However, since the variables $I^{i}$,
$I^{i}_{[1]}$,… are n-integrals we have
$a^{(m+s)}_{j,k,i}=Y_{j,k}(I^{i}_{[s]})=0$ for all $s\geq 0$. Therefore the
characteristic algebra $L_{n}$ is generated by the differential operators

$Y_{j,k}=\sum_{i}a^{(0)}_{j,k,i}\frac{\partial}{\partial
u^{i}_{n}}+a^{(1)}_{j,k,i}\frac{\partial}{\partial
u^{i}_{n,x}}+...+a^{(m-1)}_{j,k,i}\frac{\partial}{\partial u^{i}_{n,[m-1]}}$

and a finite set of the operators $X_{j,k}$. Obviously this algebra is of a
finite dimension. Consequently both conditions 1) and 2) of the Theorem 3.2.
are satisfied.

The following theorem gives a sufficient condition for the existence of the
$n$-integrals. Theorems 3.2, 3.3 were given without proofs in work [20]. In
the course of the proof, we clarified the essence the concept of a complete
set of integrals.

Theorem 3.3. If algebra $L_{n}$ for the system (1.1) is of a finite dimension
then the system admits a set of independent $n$-integrals.

Proof. The proof consists of two essentially different parts. In the first
part we construct preliminary set of functions which are independent in
principal solutions to the characteristic system of equations (see below
(3.15)). In part 2 we construct the desired set of integrals by using the
preliminary set found in the first part. This complexity is due to the
specificity of discrete equations.

Part 1. Let us assume that algebra $L_{n}$ is of a finite dimension. We denote
by $L$ a linear span of $\bar{X_{i,s}}$. We first decompose algebra $L_{n}$
into a direct sum of two subalgebras $L_{n}=L\oplus B$. Then we focus on the
subalgebra $B$, consisting of elements of the following form:

$\displaystyle Z_{i}=\frac{\partial}{\partial
u^{i}_{n}}+\sum^{N}_{i=1}\left(\alpha^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,x}}+\beta^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,xx}}+\cdots\right),\quad i=1,2,\ldots,N,$ (3.13) $\displaystyle
Z_{N+k}=\sum^{N}_{i=1}\left(\gamma^{i}_{(N+k)}\frac{\partial}{\partial
u^{i}_{n,x}}+\delta^{i}_{(N+k)}\frac{\partial}{\partial
u^{i}_{n,xx}}\right),\quad k\geq 1.$ (3.14)

The aim of this part is to construct an independent in principal set of
solutions of the system

$Z_{j}(n)G(n)=0$ (3.15)

where the operators $Z_{j}$ are from the list (3.14).

In the proof, we consider various possible situations corresponding to the
structure of the subalgebra $B$ and specify how to choose the solutions of the
system (3.15) in order to obtain an independent in principal set of solutions.
It is clear that the dimension of the subalgebra $B$ satisfies the inequality
${\rm dim}\,B\geq N+1$. Let us first assume that ${\rm dim}\,B=N+1$. Then the
basis can be taken as the union of the vector fields $Z_{i}$, $i=1,2,\ldots N$
and

$Z_{N+1}=\sum^{N}_{i=1}\left(\varphi^{i}_{N+1,m}\frac{\partial}{\partial
u^{i}_{n,[m]}}+\varphi^{i}_{N+1,m+1}\frac{\partial}{\partial
u^{i}_{n,[m+1]}}+\cdots\right),\quad m\geq 1.$ (3.16)

If $m=1$ then we consider the system

$\left(\frac{\partial}{\partial
u^{i}_{n}}+\sum^{N}_{i=1}\alpha^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,x}}\right)\omega=0,\quad\sum^{N}_{k=1}\varphi^{k}_{N+1,1}\frac{\partial\omega}{\partial
u^{k}_{n,x}}=0,\quad i=1,2,\ldots N.$ (3.17)

This system consists of $N+1$ equations for a function of $2N$ variables
$u^{i}_{n},u^{i}_{n,x}$, $i=1,\ldots,N$. Thus it has $N-1$ functionally
independent solutions $\omega^{i}(x,n,u_{n},u_{n,x})$, $i=1,2,\ldots,N-1$.

Now let us consider the system:

$\displaystyle\left(\frac{\partial}{\partial
u^{i}_{n}}+\sum^{N}_{i=1}\left(\alpha^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,x}}+\beta^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,xx}}\right)\right)W=0,\quad i=1,2,\ldots,N,$
$\displaystyle\sum^{N}_{k=1}\left(\varphi^{k}_{N+1,1}\frac{\partial}{\partial
u^{k}_{n,x}}+\varphi^{k}_{N+1,2}\frac{\partial}{\partial
u^{k}_{n,xx}}\right)W=0.$ (3.18)

It consists of $N+1$ equations on a function of $3N$ variables. Thus we have
$2N-1$ functionally independent solutions

$\omega^{i}(x,n,u_{n},u_{n,x}),\quad D_{x}\omega^{i}(x,n,u_{n},u_{n,x}),\quad
W(x,n,u_{n},u_{n,x},u_{n,xx}),\quad i=1,2,\ldots,N-1.$

Thus system (3.15) possesses a complete set of independent in principal
solutions $\omega^{i}$, $W$, $i=1,2,\ldots,N-1$.

Let us consider case $m\geq 2$. Then to find integrals of the first order
$\omega(x,n,u_{n},u_{n,x})$ we have the following system:

$\left(\frac{\partial}{\partial
u^{i}_{n}}+\sum^{N}_{i=1}\alpha^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,x}}\right)\omega=0,\quad i=1,2,\ldots,N$

with $N$ equations and $2N$ independent variables. Thus the system admits $N$
functionally independent solutions $\omega^{i}(x,n,u_{n},u_{n,x})$,
$i=1,2,\ldots,N$.

Now we consider the case when ${\rm dim}\,B>N+1$. Assume that ${\rm
dim}\,B=N+2$, then the basis is defined by the vector fields $Z_{i}$,
$i=1,2,\ldots,N$ and

$\displaystyle
Z_{N+1}=\sum^{N}_{i=1}\left(\varphi^{1}_{N+1,m}\frac{\partial}{\partial
u^{i}_{n,[m]}}+\varphi^{i}_{N+1,m+1}\frac{\partial}{\partial
u^{i}_{n,[m+1]}}+\cdots\right),$ $\displaystyle
Z_{N+2}=\sum^{N}_{i=1}\left(\psi^{i}_{N+2,s}\frac{\partial}{\partial
u^{i}_{n,[s]}}+\psi^{i}_{N+2,s+1}\frac{\partial}{\partial
u^{i}_{n,[s+1]}}+\cdots\right),\quad m,s\geq 1,\quad s\geq m.$

If $m\geq 2$ then the system of equations

$\left(\frac{\partial}{\partial
u^{i}_{n}}+\sum^{N}_{i=1}\alpha^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,x}}\right)\omega=0,\quad i=1,2,\ldots,N$

has $N$ functionally independent solutions
$\omega^{i}=\omega^{i}(x,n,u_{n},u_{n,x})$ because here we have $N$ equations
and $2N$ variables. These solutions also satisfy equations
$Z_{N+1}\omega^{i}=0$, $Z_{N+2}\omega^{i}=0$.

Let us consider the case when $m=1$. If $s\geq 3$ then in addition to system
(3.18) we have also the following equation

$\tilde{Z}_{N+2}=\psi^{1}_{N+2,3}\frac{\partial}{\partial
u^{1}_{n,xxx}}+\ldots+\psi^{N}_{N+2,3}\frac{\partial}{\partial
u^{N}_{n,xxx}}+\sum^{\infty}_{k=4}\left(\mu^{1}_{k}\frac{\partial}{\partial
u^{1}_{n,[k]}}+\ldots+\mu^{N}_{k}\frac{\partial}{\partial
u^{N}_{n,[k]}}\right).$

As it was shown earlier, the entire family of solutions of system (3.18) is
given by the integrals

$\omega^{i}(x,n,u_{n},u_{n,x}),\quad W(x,n,u_{n},u_{n,x},u_{n,xx}),\quad
i=1,2,\ldots,N-1.$

In addition the integrals satisfy also equations
$\tilde{Z}_{N+2}\omega^{i}=0$, $\tilde{Z}_{N+2}W=0$, $i=1,2,\ldots,N_{1}$.
Thus these integrals define the whole family of solutions of system (3.18).

Let us assume that $s=2$ (if $m=1$). Then we obtain system (3.17) for unknown
$\omega(x,n,u_{n},u_{n,x})$. So this system has $N-1$ functionally independent
integrals. To find solutions of the form $W(x,n,u_{n},u_{n,x},u_{n,xx})$ we
obtain system (3.18) and additionally the equation

$\sum^{N}_{i=1}\psi^{i}_{N+2,2}\frac{\partial W}{\partial u^{i}_{n,xx}}=0.$

Thus we deal with the system of $N+2$ equations on a function on $3N$
variables. In this case the system has $2N-2$ functionally independent
solutions $\omega^{i}(x,n,u_{n},u_{n,x})$,
$D_{x}\omega^{i}(x,n,u_{n},u_{n,x})$. To find solutions of the form
$I(x,n,u_{n},u_{n,x},u_{n,xx},u_{n,xxx})$ we consider the system:

$\displaystyle\left(\frac{\partial}{\partial
u^{i}_{n}}+\sum^{N}_{i=1}\left(\alpha^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,x}}+\beta^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,xx}}+\gamma^{i}_{(1)}\frac{\partial}{\partial
u^{i}_{n,xxx}}\right)\right)I=0,\quad i=1,2,\ldots,N,$
$\displaystyle\sum^{N}_{k=1}\left(\varphi^{k}_{N+1,1}\frac{\partial}{\partial
u^{i}_{n,x}}+\varphi^{k}_{N+1,2}\frac{\partial}{\partial
u^{i}_{n,xx}}+\varphi^{k}_{N+1,3}\frac{\partial}{\partial
u^{i}_{n,xxx}}\right)I=0,$
$\displaystyle\sum^{N}_{i=1}\left(\psi^{i}_{N+2,2}\frac{\partial}{\partial
u^{i}_{n,xx}}+\psi^{i}_{N+2,3}\frac{\partial}{\partial
u^{i}_{n,xxx}}\right)I=0.$

This is a system of $N+2$ equations on a function on $4N$ variables. Thus, it
has $3N-2$ solutions:

$\omega^{i}(x,n,u_{n},u_{n,x}),\,D_{x}\omega^{i}(x,n,u_{n},u_{n,x}),\,D^{2}_{x}\omega^{i}(x,n,u_{n},u_{n,x}),\,I(x,n,u_{n},u_{n,x},u_{n,xx},u_{n,xxx}),$

$i=1,2,\ldots,N-1$.

It remains to consider the case $s=1$ (if $m=1$). We can assume that vector
fields $Z_{N+1}$, $Z_{N+2}$ have the forms:

$\displaystyle\tilde{Z}_{N+1}=\frac{\partial}{\partial
u^{1}_{n,x}}+\varphi^{2}_{1}\frac{\partial}{\partial
u^{2}_{n,x}}+\cdots+\varphi^{N}_{1}\frac{\partial}{\partial
u^{N}_{n,x}}+\sum^{\infty}_{l=2}\left(d^{1}_{l}\frac{\partial}{\partial
u^{1}_{n,[l]}}+\cdots+d^{N}_{l}\frac{\partial}{\partial
u^{N}_{n,[l]}}\right),$ (3.19)
$\displaystyle\tilde{Z}_{N+2}=s^{1}_{1}\frac{\partial}{\partial
u^{1}_{n,x}}+\cdots+s^{N}_{1}\frac{\partial}{\partial
u^{N}_{n,x}}+\sum^{\infty}_{l=2}\left(\mu^{1}_{l}\frac{\partial}{\partial
u^{1}_{n,[l]}}+\cdots+\mu^{N}_{l}\frac{\partial}{\partial
u^{N}_{n,[l]}}\right).$

If

$\frac{s^{1}_{1}}{1}=\frac{s^{2}_{1}}{\varphi^{2}_{1}}=\ldots=\frac{s^{N}_{1}}{\varphi^{N}_{1}}$
(3.20)

then we replace $\tilde{Z}_{N+2}$ by

$\tilde{\tilde{Z}}_{N+2}=\tilde{Z}_{N+2}-s^{1}_{1}\tilde{Z}_{N+1}.$

This vector field has the following form:

$\tilde{\tilde{Z}}_{N+2}=\eta^{1}_{k}\frac{\partial}{\partial
u^{1}_{n,[k]}}+\cdots+\eta^{N}_{k}\frac{\partial}{\partial
u^{1}_{n,[k]}}+\sum^{\infty}_{s=k+1}\left(\nu^{1}_{s}\frac{\partial}{\partial
u^{1}_{n,[s]}}+\cdots+\nu^{N}_{s}\frac{\partial}{\partial
u^{1}_{n,[s]}}\right),\quad k\geq 2.$

This case has already been investigated above.

If at least one of the equalities (3.20) is not satisfied then the operator
$\tilde{Z}_{N+2}$ can be replaced by

$\tilde{\tilde{Z}}_{N+2}=\eta^{2}_{1}\frac{\partial}{\partial
u^{2}_{n,x}}+\cdots+\eta^{N}_{1}\frac{\partial}{\partial
u^{N}_{n,x}}+\sum^{\infty}_{s=2}\left(\nu^{1}_{s}\frac{\partial}{\partial
u^{1}_{n,[s]}}+\cdots+\nu^{N}_{s}\frac{\partial}{\partial
u^{N}_{n,[s]}}\right).$ (3.21)

So, we have the following system:

$Z_{i}W=0,\quad\tilde{Z}_{N+1}W=0,\quad\tilde{\tilde{Z}}_{N+2}W=0,\quad
i=1,2,\ldots,N,$

where $\tilde{Z}_{N+1}$ is defined by (3.19), $\tilde{\tilde{Z}}_{N+2}$ is
defined by (3.21). The system has $N-2$ solutions
$\omega^{i}(x,n,u_{n},u_{n,x})$, $i=1,\ldots,N-2$, because in this case we
have $N+2$ equations and $2N$ variables. The system has $2N-2$ solutions:

$\displaystyle\omega^{i}(x,n,u_{n},u_{n,xx}),\quad
D_{x}\omega^{i}(x,n,u_{n},u_{n,x}),\quad i=1,\ldots N-2,$ $\displaystyle
W^{1}(x,n,u_{n},u_{n,x},u_{n,xx}),\quad W^{2}(x,n,u_{n},u_{n,x},u_{n,xx}).$

Thus we obtain a complete set of functions from the kernel of the operators in
the algebra $L_{n}$ containing functions
$\omega^{i}(x,n,u_{n},u_{n,x},u_{n,xx})$, $i=1,\ldots,N-2$,
$W^{1}(x,n,u_{n},u_{n,x})$, $W^{2}(x,n,u_{n},u_{n,x},u_{n,xx})$. The case
${\rm dim}B>N+2$ is considered in a similar way. That completes the first part
of the proof, where we constructed a set of functions defining an independent
in principal set of solutions to the system (3.15). However these solutions
aren’t $n$-integrals to the system of differential-difference equations (1.1)
because they do not satisfy the shift condition (see Definition 2.1).

Part 2. Now we have to construct required integrals by using the obtained in
the first part set of solutions to the first order linear equations of the
form

$Z_{j}(n)G(n)=0,\quad j=1,2,\ldots,M,\quad M=N+k,$ (3.22)

where the operators $Z_{j}=Z_{j}(n)$ are given in (3.14). Denote these
solutions as $G^{(1)}$, $G^{(2)}$,…, $G^{(N)}$, recall that they constitute a
set of independent in principal functions. Introduce a notation for the vector
valued solution of the system (3.22) by setting

$G_{n}=(G^{(1)}_{n},G^{(2)}_{n},\ldots,G^{(N)}_{n}).$

We notice that for $\forall i$ function $G^{i}_{n}$ depends on
$u_{n},u_{n,x},\ldots,u_{n,[m]}$ and satisfy the inequality

$\displaystyle\frac{\partial G_{n}}{\partial u_{n,[m]}}={\rm
det}\left(\begin{array}[]{cccc}\frac{\partial G^{(1)}}{\partial
u^{1}_{n,[m]}}&\frac{\partial G^{(1)}}{\partial
u^{2}_{n,[m]}}&\ldots&\frac{\partial G^{(1)}}{\partial u^{N}_{n,[m]}}\\\
\ldots&\ldots&\ldots&\ldots\\\ \ \frac{\partial G^{(N)}}{\partial
u^{1}_{n,[m]}}&\frac{\partial G^{(N)}}{\partial
u^{2}_{n,[m]}}&\ldots&\frac{\partial G^{(N)}}{\partial
u^{N}_{n,[m]}}\end{array}\right)\neq 0.$ (3.26)

Lemma 3.1. Functions $\left\\{G_{n}\right\\}^{n=+\infty}_{n=-\infty}$ solve
one and the same system of the first order linear equations, i.e. the
relations hold

$Z_{j}(0)G_{n}=0,\quad j=1,2,\ldots,M.$ (3.27)

Proof of the Lemma. Due to the definition following relations hold

$\displaystyle D^{-1}Y_{j,1}D=Y_{j,2}+X_{j,1},$ $\displaystyle
D^{-1}Y_{j,k}D=Y_{j,k+1},\quad k\geq 2,$ (3.28) $\displaystyle
D^{-1}X_{j,k}D=X_{j,k+1}.$

Relations (3.28) imply that subalgebra $B$ of the Lie-Rinehart algebra $L_{n}$
generated by the operators $\left\\{Z_{j}(n)\right\\}^{N}_{j=1}$ is invariant
under the action of the automorphism defined due to the rule

$Z\rightarrow D^{-1}_{n}ZD_{n}.$ (3.29)

Therefore if $G_{n}$ is a solution of the system (3.27) then function
$G_{n+1}=D_{n}G_{n}$ is again a solution. This completes the proof of the
Lemma.

Now turn back to the proof of the theorem. Since for arbitrary integer $k$ the
set $G^{(1)}_{k},G^{(2)}_{k},\ldots,G^{(N)}_{k}$ constitutes a set of
independent in principal solutions of the system (3.27) of the order $m$ then
any other solution of the order $m$ is a function of these basic solutions. In
other words there exists a function
$H_{n}=\left\\{H^{(1)}_{n},H^{(2)}_{n},\ldots,H^{(N)}_{n}\right\\}$ depending
also on $n$ such that

$G_{n+k}=H_{n+k}(G_{k}).$ (3.30)

Due to the chain rule we can easily derive a relation for the Jacobians

$\frac{\partial G_{n+k}}{\partial u_{n,[m]}}=\frac{\partial H_{n+k}}{\partial
G_{n}}\frac{\partial G_{n}}{\partial u_{n,[m]}}$ (3.31)

that obviously implies that Jacobian $\frac{\partial H_{n+k}}{\partial G_{k}}$
doesn’t vanish. Indeed due to the property of the independence in principal we
have that $\frac{\partial G_{n+k}}{\partial u_{n,[m]}}\neq 0$ and
$\frac{\partial G_{n}}{\partial u_{n,[m]}}\neq 0$. Let us take $k=0$ and find
by inversion of the relation (3.30) that $G_{0}=H^{-1}_{n}(G_{n})$ and
$G_{0}=H^{-1}_{n+1}(G_{n+1})=DH^{-1}_{n}(G_{n})$. It follows from these two
relations that function $I=H^{-1}_{n}(G_{n})$ is an integral of the system
(1.1). Indeed we have $DI=I$.

Theorem 3.3. is proved.

Thus we are ready to formulate the main result of the article.

Theorem 3.4 (Algebraic criterion of Darboux integrability). System (1.1) is
integrable in the sense of Darboux if and only if both its characteristic
algebras $L_{x}$ and $L_{n}$ have finite dimension.

##  4 Construction of a set of independent integrals of minimal orders.
Illustrative Examples.

In this section, we show two examples demonstrating an algorithm for defining
the characteristic Lie-Rinehart algebra. We then discuss the subsequent
application of algebra to the computation of the set of integrals of minimal
orders. The reader can find a large number of examples in [20].

Example 4.1. Let us consider a scalar equation [30]

$u_{n+1,x}=u_{n,x}+u^{2}_{n+1}+u_{n+1}-u^{2}_{n}-u_{n}.$ (4.1)

We apply the method of characteristic algebras to find its $x$-integral. First
we write down characteristic operator. In this case it can be presented as

$K_{0}=u_{n,x}\frac{\partial}{\partial
u_{n}}+(u_{n,x}+f_{n})\frac{\partial}{\partial
u_{n+1}}+(u_{n,x}+g_{n})\frac{\partial}{\partial u_{n-1}}+\cdots,$ (4.2)

where $f_{n}=u^{2}_{n+1}+u_{n+1}-u^{2}_{n}-u_{n}$,
$g_{n}=u^{2}_{n-1}+u_{n-1}-u^{2}_{n}-u_{n}$. According to the scheme above we
have to describe the Lie-Rinehart algebra $L_{x}$ generated by $K_{0}$ and
$X=\frac{\partial}{\partial u_{n,x}}$. We rewrite (4.2) in a more convenient
form $K_{0}=u_{n,x}\tilde{X}+K$, where

$\tilde{X}=\sum^{+\infty}_{-\infty}\frac{\partial}{\partial u_{k}},\quad
K=\sum^{+\infty}_{-\infty}(u^{2}_{n+k}+u_{n+k}-u^{2}_{n}-u_{n})\frac{\partial}{\partial
u_{n+k}},$

and then find
$K_{1}=\left[\tilde{X},K\right]=\sum^{+\infty}_{-\infty}(u_{n+k}-u_{n})\frac{\partial}{\partial
u_{n+k}}$. Since $x$-integral does not depend on $u_{n,x}$ we concentrate on
the subalgebra $L_{0}$ generated by the operators $\tilde{X}$ and $K$.
Subalgebra is of dimension three, operators $\tilde{X},K,K_{1}$ constitute its
basis, since we have obviously $[\tilde{X},K_{1}]=0$,
$[K_{1},K]=2K-2u_{n,x}\tilde{X}-(2u_{n}+1)K_{1}$. Finalizing the reasoning
above we can conclude that $x$-integral $J$ of the equation (4.1) should
satisfy a system of three equations

$K_{1}J=0,\quad KJ=0,\quad\tilde{X}J=0.$ (4.3)

Without loosing generality we can set $J=J(u_{n},\ldots,u_{n+m})$. Note that
the order $m$ of a nontrivial integral is greater than two. Indeed, for
$J(u_{n},u_{n+1},u_{n+2})$ we have from (4.3) a system of linear equations

$\displaystyle J_{u}+J_{u_{1}}+J_{u_{2}}=0,$
$\displaystyle(u^{2}_{n+1}+u_{n+1}-u^{2}_{n}-u_{n})J_{u_{1}}+(u^{2}_{n+2}+u_{n+2}-u^{2}_{n+1}-u_{n+1})J_{u_{2}}=0,$
$\displaystyle(u_{n+1}-u_{n})J_{u_{1}}+(u_{n+2}-u_{n})J_{u_{2}}=0$

that implies $J_{u}=J_{u_{1}}=J_{u_{2}}=0$. Therefore $J$ is a trivial
integral depending only on $n$. We can find a nontrivial integral by taking
$m=3$. In this case we get

$J=\frac{(u_{n+3}-u_{n+1})(u_{n+2}-u_{n})}{(u_{n+3}-u_{n+2})(u_{n+1}-u_{n})}$

which is evidently an integral of the minimal order. For this equation the
$n$-integral is easily found, since the equation (4.1) is rewritten as
$u_{n+1,x}-u^{2}_{n+1}-u_{n+1}=u_{n,x}-u^{2}_{n}-u_{n}$. Therefore
$I=u_{n,x}-u^{2}_{n}-u_{n}$ is an $n$-integral of the first order.

Example 4.2.  Let us consider the following system

$u^{0}_{n+1,x}=u^{0}_{n,x}+e^{u^{0}_{n+1}-u^{1}_{n}},\quad
u^{1}_{n+1,x}=u^{1}_{n,x}-e^{u^{0}_{n+1}-u^{1}_{n}}.$ (4.4)

Assume that function $H(u^{0}_{n},u^{1}_{n},u^{0}_{n+1},u^{1}_{n+1},\ldots)$
is an $x$-integral for the system (4.4). Then by definition we get
$D_{x}H(u^{0}_{n},u^{1}_{n},u^{0}_{n+1},u^{1}_{n+1},\ldots)=0$. Let us apply
$D_{x}$ to $H$, then

$\displaystyle D_{x}H=K_{0}H=\left(u^{0}_{n,x}\frac{\partial}{\partial
u^{0}_{n}}+u^{1}_{n,x}\frac{\partial}{\partial u^{1}_{n}}+\right.$
$\displaystyle\left.+(u^{0}_{n,x}+e^{u^{0}_{n+1}-u^{1}_{n}})\frac{\partial}{\partial
u^{0}_{n+1}}+(u^{1}_{n,x}-e^{u^{0}_{n+1}-u^{1}_{n}})\frac{\partial}{\partial
u^{1}_{n+1}}+\cdots\right),$

In addition we also have two equations:

$X_{1}H=0,\quad X_{2}H=0,$

where $X_{1}=\frac{\partial}{\partial u^{0}_{n,x}}$,
$X_{2}=\frac{\partial}{\partial u^{1}_{n,x}}$. Operator $K_{0}$ is represented
as a linear combination of the vector fields:

$K_{0}=u^{0}_{n,x}Y_{1}+u^{1}_{n,x}Y_{2}+W,$

where

$\displaystyle Y_{1}=\frac{\partial}{\partial
u^{0}_{n}}+\frac{\partial}{\partial u^{0}_{n+1}}+\frac{\partial}{\partial
u^{0}_{n+2}}+\cdots,$ $\displaystyle Y_{2}=\frac{\partial}{\partial
u^{1}_{n}}+\frac{\partial}{\partial u^{1}_{n+1}}+\frac{\partial}{\partial
u^{1}_{n+2}}+\cdots,$ $\displaystyle
W=e^{u^{0}_{n+1}-u^{1}_{n}}\frac{\partial}{\partial
u^{0}_{n+1}}-e^{u^{0}_{n+1}-u^{1}_{n}}\frac{\partial}{\partial u^{1}_{n+1}}+$
$\displaystyle+\left(e^{u^{0}_{n+1}-u^{1}_{n}}+e^{u^{0}_{n+2}-u^{1}_{n+1}}\right)\frac{\partial}{\partial
u^{0}_{n+2}}-\left(e^{u^{0}_{n+1}-u^{1}_{n}}+e^{u^{0}_{n+2}-u^{1}_{n+1}}\right)\frac{\partial}{\partial
u^{1}_{n+2}}.$

Thus we have

$Y_{1}H=0,\quad Y_{2}H=0,\quad WH=0.$ (4.5)

Commutators of operators $Y_{1},Y_{2},W$ satisfy the following relations:

$[Y_{1},Y_{2}]=0,\quad[Y_{1},W]=W,\quad[Y_{2},W]=-W.$

At the first step we are looking for solutions of system (4.5) of the form
$H=H(u^{0}_{n},u^{1}_{n},u^{0}_{n+1},u^{1}_{n+1})$. Then $H$ must satisfy the
system:

$\frac{\partial H}{\partial u^{0}_{n}}+\frac{\partial H}{\partial
u^{0}_{n+1}}=0,\quad\frac{\partial H}{\partial u^{1}_{n}}+\frac{\partial
H}{\partial u^{1}_{n+1}}=0,\quad\frac{\partial H}{\partial
u^{0}_{n+1}}-\frac{\partial H}{\partial u^{1}_{n+1}}=0.$

The system has a solution of the form

$H=P(u^{1}_{n+1}+u^{0}_{n+1}-u^{0}_{n}-u^{1}_{n}).$

Therefore we can chose the integral

$J_{1}=u^{1}_{n+1}+u^{0}_{n+1}-u^{0}_{n}-u^{1}_{n}.$ (4.6)

In order to find two functionally independent solution of system (4.5) we
should consider the function $H$ of the form
$\bar{H}=\bar{H}(u^{0}_{n},u^{1}_{n},u^{0}_{n+1},u^{1}_{n+1},u^{0}_{n+2})$. To
find $H$ we have the following system:

$\displaystyle\frac{\partial\bar{H}}{\partial
u^{0}_{n}}+\frac{\partial\bar{H}}{\partial
u^{0}_{n+1}}+\frac{\partial\bar{H}}{\partial u^{0}_{n+2}}=0,$
$\displaystyle\frac{\partial\bar{H}}{\partial
u^{1}_{n}}+\frac{\partial\bar{H}}{\partial u^{1}_{n+1}}=0,$ $\displaystyle
e^{u^{0}_{n+1}-u^{1}_{n}}\left(\frac{\partial\bar{H}}{\partial
u^{0}_{n+1}}-\frac{\partial\bar{H}}{\partial
u^{1}_{n+1}}\right)+(e^{u^{0}_{n+1}-u^{1}_{n}}+e^{u^{0}_{n+2}-u^{1}_{n+1}})\frac{\partial\bar{H}}{\partial
u^{0}_{n+2}}=0.$

This system has a solution of the form:

$\bar{H}=\bar{H}(u^{1}_{n+1}-u^{1}_{n}+u^{0}_{n+1}-u^{0}_{n},e^{u^{0}_{n+1}-u^{0}_{n}}+e^{2u^{0}_{n+1}-u^{0}_{n}-u^{0}_{n+2}+u^{1}_{n+1}-u^{1}_{n}}).$

We can chose the following integral:

$J_{2}=e^{-(u^{1}_{n+1}-u^{1}_{n}+u^{0}_{n+1}-u^{0}_{n})}\left(e^{u^{0}_{n+1}-u^{0}_{n}}+e^{2u^{0}_{n+1}-u^{0}_{n}-u^{0}_{n+2}+u^{1}_{n+1}-u^{1}_{n}}\right)=e^{u^{1}_{n}-u^{1}_{n+1}}+e^{u^{0}_{n+1}-u^{0}_{n+2}}.$

Thus we obtain two independent $x$-integrals of the minimal orders $n_{1}=1$
and $n_{2}=2$

$J_{1}=u^{1}_{n+1}+u^{0}_{n+1}-u^{0}_{n}-u^{1}_{n},\quad\mbox{and}\quad
J_{2}=e^{u^{1}_{n}-u^{1}_{n+1}}+e^{u^{0}_{n+1}-u^{0}_{n+2}}$

of system (4.4).

##  5 Conclusion

It was observed in [20] that every known three-dimensional integrable equation
of the form (1.4) possesses an infinite set of the Darboux integrable
reductions that are systems of differential-difference equations with two
independent variables $x$ and $n$. This circumstance is very important from
the integrable classification viewpoint, since the Darboux integrable systems
are closely related to the Lie-Rinehart algebras. In the present article we
proved that a finite system of differential-difference hyperbolic type
equations is integrable in the sense of Darboux if and only if both its
characteristic Lie-Rinehart algebras are of finite dimension. This fact
creates a good basis for the development of an algebraic algorithm for the
classification of equations of the form (1.4) with three independent
variables.

## Acknowledgements

The research was supported by a grant from the Russian Science Foundation
No. 21-11-00006, https://rscf.ru/project/21-11-00006/.

## References

## References

  * [1] Zhiber A V, Sokolov V V 2001 Exactly integrable hyperbolic equations of Liouville type Russian Math. Surveys 56:1, 61–101.
  * [2] Goursat E 1899  Recherches sur quelques $\acute{e}$quations aux d$\acute{e}$riv$\acute{e}$es partielles du second ordre, Annales de la facult$\acute{e}$ des Sciences de l’Universit$\acute{e}$ de Toulouse $2^{e}$ s$\acute{e}$rie, tome 1, n0 1, 31–78.
  * [3] Goursat E 1899  Recherches sur quelques $\acute{e}$quations aux d$\acute{e}$riv$\acute{e}$es partielles du second ordre, Annales de la facult$\acute{e}$ des Sciences de l’Universit$\acute{e}$ de Toulouse $2^{e}$ s$\acute{e}$rie, tome 1, n0 1 (1899), 79–163.
  * [4] Vessiot E 1939 Sur les equations aux derivees partialles du second order, $F(x,y,p,q,r,s,t)=0,$ inteqrables par la methode de Darboux, J. Math. Pure Appl. 18, 1–61.
  * [5] Vessiot E 1942 Sur les equations aux derivees partialles du second order, $F(x,y,p,q,r,s,t)=0,$ integrables par la methode de Darboux, J. Math. Pure Appl. 21, 1–68.
  * [6] Shabat A B, Yamilov R I 1981 Exponential systems of type I and the Cartan matrices (in Russian), Preprint, Bashkirian Branch Acad. Sci. USSR, Ufa
  * [7] Leznov A N, Smirnov V G, Shabat A B 1982 The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems Theor. Math. Phys. 51:1, 322–330.
  * [8] Zhiber A V, Murtazina R D, Habibullin I T, Shabat A B 2012 Characteristic Lie Rings and Non-linear Integrable Equations (in Russian), Inst. Computer Studies, Moscow
  * [9] Habibullin I, Zheltukhina N, Pekcan A 2009 Complete list of Darboux integrable chains of the form $t_{1,x}=t_{x}+d(t,t_{1})$. J. Math. Phys. 50, 1–23.
  * [10] Rinehart G 1963 Differential forms for general commutative algebras Trans. Amer. Math. Soc. 108, 195–222.
  * [11] Millionshchikov D 2018 Lie Algebras of Slow Growth and Klein-Gordon PDE Algebr. Represent. Theor. 21, 1037–1069.
  * [12] Millionshikov D V, Smirnov S V 2021 Characteristic algebras and integrable systems of exponential type Ufa Math. J. 13:2, 44–73.
  * [13] Anderson I M, Kamran N 1997 The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane Duke Math. J. 87:2, 265–319.
  * [14] Adler V E, Startsev S Y 1999 Discrete analogues of the Liouville equation Theor. Math. Phys. 121:2, 1484–1495
  * [15] Smirnov S V 2015 Darboux integrability of discrete two-dimensional Toda lattices Theor. Math. Phys. 182:2, 189–210.
  * [16] Garifullin R N, Yamilov R I 2019 On series of Darboux integrable discrete equations on square lattice Ufa Math. J. 11:3, 100–109.
  * [17] Garifullin R N, Habibullin I T 2021 Generalized symmetries and integrability conditions for hyperbolic type semi-discrete equations J. Phys. A: Math. Theor. 54, 205201.
  * [18] Demskoi D K 2010 Integrals of open two-dimensional lattices Theor. Math. Phys. 163, 466–471.
  * [19] Zheltukhin K, Zheltukhina N A 2021 On discretization of Darboux Integrable Systems admitting second-order integrals Ufa Math. J. 13:2, 176–192.
  * [20] Habibullin I T, Khakimova A R 2021 Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D J. Phys. A: Math. Theor. 54:29, 295202
  * [21] Ferapontov E V, Novikov V S, Roustemoglou I 2015 On the classification of discrete Hirota-type equations in 3D Int. Math. Res. Not. IMRN 13 4933–4974
  * [22] Habibullin I 2013 Characteristic Lie rings, finitely-generated modules and integrability conditions for $(2+1)$-dimensional lattices Physica Scripta 87, 065005
  * [23] Habibullin I, Poptsova M 2017 Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings SIGMA 13, 26 pp.
  * [24] Habibullin I T, Kuznetsova M N 2020 A classification algorithm for integrable two-dimensional lattices via Lie–Rinehart algebras Theor. Math. Phys. 203 569–581.
  * [25] Habibullin I T, Kuznetsova M N, Sakieva A U 2020 Integrability conditions for two-dimensional Toda-like equations J. Phys. A: Math. Theor. 53:39, 395203.
  * [26] Ferapontov E V, Habibullin I T, Kuznetsova M N, Novikov V S 2020 On a class of 2D integrable lattice equations J. Math. Phys. 61:7, 073505.
  * [27] Kostrigina O S 2011, Nonlinear hyperbolic systems integrable by Darboux, dissertation.
  * [28] Habibullin I 2005 Characteristic Algebras of Fully Discrete Hyperbolic Type Equations SIGMA 1, 023.
  * [29] Kuznetsova M N, Zhiber A V 2021 Integrals and characteristic Lie rings of semidiscrete systems of equations Ufa Math. J. 13:2, 25–35.
  * [30] Habibullin I, Zheltukhina N, Sakieva A 2010 On Darboux-integrable semi-discrete chains J. Phys. A: Math. Theor. 43:43, 434017.


# Factorizations of Eisenstein Series of Level up to $4$

Henri Cohen,
Université de Bordeaux,
LFANT, IMB, U.M.R. 5251 du C.N.R.S,
351 Cours de la Libération,
33405 TALENCE Cedex, FRANCE

###### Abstract

We show that the most standard Eisenstein series such as $E_{4}(\tau)$ or
$2E_{2}(2\tau)-E_{2}(\tau)$, and also the function $\theta^{2}(\tau)$, are in
a natural way the product of two conjugate Eisenstein series of half their
weight and double their level, as well as a number of similar elementary
identities for $E_{6}$ and Eisenstein series of levels $2$, $3$, and $4$.

##  1 Introduction and Notation

The goal of this paper is to show that the most basic Eisenstein series of
level up to $4$ have natural factorizations into a product of two conjugate
Eisenstein-type series of half their weight and double their level as well as
similar identities, and to study the corresponding functions. No proofs are
explicitly given since in all cases it is simply a matter of working in a
suitable finite-dimensional space of modular forms on some congruence subgroup
of the modular group and checking a few initial Fourier coefficients.

The motivation for this paper comes from the fact that the functions $F_{N,j}$
that we will introduce are essentially normalized Hauptmoduln for several
standard non-cocompact arithmetic triangle groups, see [1].

We use the standard notation of modular forms and functions: we will use in
particular the functions $E_{2}$, $E_{4}$, $E_{6}$, $\eta$, and $\theta$, and
we will also need the Eisenstein series of weight $1$ and levels $3$ and $4$:

$E_{1,-3}(\tau)=1/6+\sum_{n\geq 1}\bigl{(}\sum_{d\mid
n}\mbox{$\left(\frac{-3}{d}\right)$}\bigr{)}q^{n}\text{\quad
and\quad}E_{1,-4}(\tau)=1/4+\sum_{n\geq 1}\bigl{(}\sum_{d\mid
n}\mbox{$\left(\frac{-4}{d}\right)$}\bigr{)}q^{n}\;,$

and we recall that $\theta^{2}=4E_{1,-4}$.

For $N=2$, $3$, and $4$ we also define the forms

$\displaystyle E_{N,2}(\tau)$
$\displaystyle=(NE_{2}(N\tau)-E_{2}(\tau))/(N-1)\in
M_{2}(\Gamma_{0}^{*}(N))\text{\quad and}$ $\displaystyle E_{N,4}(\tau)$
$\displaystyle=(N^{2}E_{4}(N\tau)-E_{4}(\tau))/(N^{2}-1)\in
M_{4}(\Gamma_{0}^{*}(N))\;,$

where as usual $\Gamma_{0}^{*}(N)$ denotes the group generated by
$\Gamma_{0}(N)$ and the Fricke involution $\tau\mapsto-1/(N\tau)$, but we will
not use the function $E_{4,4}$. In levels $3$ and $4$ we will need additional
modular forms which will be introduced when needed.

We could present our results in a more or less uniform manner, but it is more
readable to give them level by level, referring to the above for definitions
of the functions which are used.

##  2 Results in Level $1$

The basic formula in level $1$ is

$E_{4}^{3}-E_{6}^{2}=1728\eta^{24}\;.$

This can be factored in two ways and leads to the following:

###### Definition 2.1

We define

$F_{1,j}(\tau)=E_{6}(\tau)-(-1)^{j}\sqrt{-1728}\eta^{12}(\tau)\text{\quad
and\quad}G_{1,j}(\tau)=E_{4}(\tau)-12\rho^{j}\eta^{8}(\tau)\;,$

where as usual $\rho=(-1+\sqrt{-3})/2$.

###### Proposition 2.2

We have

$\displaystyle F_{1,j}(\tau+1)=F_{1,j+1}(\tau)\;,\quad
F_{1,j}(-1/\tau)=\tau^{6}F_{1,j+1}(\tau)\;,$ $\displaystyle
F_{1,j}(\tau)F_{1,j+1}(\tau)=E_{4}^{3}(\tau)\;,$ $\displaystyle
G_{1,j}(\tau+1)=G_{1,j+1}(\tau)\;,\quad
G_{1,j}(-1/\tau)=\tau^{4}G_{1,j}(\tau)\;,$ $\displaystyle
G_{1,j}(\tau)G_{1,j+1}(\tau)G_{1,j+2}(\tau)=E_{6}^{2}(\tau)\;.$

Proof. Trivial.  $\sqcap$$\sqcup$

######  Theorem 2.3 (Level 1)

  1. (1)

The functions

$\displaystyle f_{1,j}(\tau)=2E_{2}(\tau)$
$\displaystyle-((3+(-1)^{j}\sqrt{-3})/6)E_{2}(\tau/2)$
$\displaystyle-((3-(-1)^{j}\sqrt{-3})/6)E_{2}((\tau+1)/2)$

have the following properties:

$\displaystyle f_{1,j}^{3}(\tau)=F_{1,j}(\tau)\;,$ $\displaystyle
f_{1,j}(\tau)f_{1,j+1}(\tau)=E_{4}(\tau)\;,\quad
f_{1,j}(((-1)^{j}+\sqrt{-3})/2)=0\;,$ $\displaystyle
f_{1,j}(\tau+1)=f_{1,j+1}(\tau)\;,\quad
f_{1,j}(-1/\tau)=((-1-(-1)^{j}\sqrt{-3})/2)\tau^{2}f_{1,j+1}(\tau)\;.$

  2. (2)

The functions

$g_{1,j}(\tau)=(3/2)E_{2}(\tau)-(1/4)\sum_{\begin{subarray}{c}0\leq m\leq 2\\\
m\not\equiv j\allowbreak\ ({\rm{mod}}\,\,3)\end{subarray}}E_{2}((\tau+m)/3)$

have the following properties:

$\displaystyle g_{1,j}^{2}(\tau)=G_{1,j}(\tau)\;,$ $\displaystyle
g_{1,j}(\tau)g_{1,j+1}(\tau)g_{1,j+2}(\tau)=E_{6}(\tau)\;,\quad
g_{1,j}(-j+\sqrt{-1})=0\;,$ $\displaystyle
g_{1,j}(\tau+1)=g_{1,j+1}(\tau)\;,\quad
g_{1,j}(-1/\tau)=\varepsilon_{j}\tau^{2}g_{1,j}(\tau)\;,$

with $\varepsilon_{j}=1$ if $3\mid j$ and $\varepsilon_{j}=-1$ if $3\nmid j$.

Proof. All the functions involved are modular over $\Gamma(2)$ or $\Gamma(3)$,
so the proof is a simple verification of a few coefficients. Evidently this is
not an explanation of the theorem, especially since a similar theorem is valid
in levels $2$, $3$, and $4$.  $\sqcap$$\sqcup$

In particular this shows that $E_{4}$ is the product of two conjugate
Eisenstein series of weight $2$ and level $2$, and that $E_{6}$ is the product
of three conjugate Eisenstein series of weight $2$ and level $3$.

##  3 Results in Level $2$

For simplicity, we set $\eta_{2}(\tau)=\eta(\tau)\eta(2\tau)$, of weight $1$
(not to be confused with $\eta(2\tau)$ itself).

In this level, we have the following eta quotient identities:

$E_{2,2}^{2}(\tau)-E_{2,4}(\tau)=128\eta^{16}(2\tau)/\eta^{8}(\tau)\;,\quad
E_{2,2}^{2}(\tau)+E_{2,4}(\tau)=2\eta^{16}(\tau)/\eta^{8}(2\tau)\;.$

The basic formula in level $2$ is

$E_{2,2}^{4}-E_{2,4}^{2}=256\eta_{2}^{8}\;.$

This can be factored in two ways and leads to the following:

###### Definition 3.1

We define

$F_{2,j}(\tau)=E_{2,4}(\tau)-(-1)^{j}\sqrt{-256}\eta_{2}^{4}(\tau)\text{\quad
and\quad}G_{2,j}(\tau)=E_{2,2}(\tau)-4i^{j}\eta_{2}^{2}(\tau)\;,$

where $\eta_{2}$ has been defined above.

###### Proposition 3.2

We have

$\displaystyle F_{2,j}(\tau+1)=F_{2,j+1}(\tau)\;,\quad
F_{2,j}(-1/(2\tau))=-4\tau^{4}F_{2,j+1}(\tau)\;,$ $\displaystyle
F_{2,j}(\tau)F_{2,j+1}(\tau)=E_{2,2}^{4}(\tau)\;,$ $\displaystyle
G_{2,j}(\tau+1)=G_{2,j+1}(\tau)\;,\quad
G_{2,j}(-1/(2\tau))=-2\tau^{2}G_{2,j}(\tau)\;,$ $\displaystyle
G_{2,j}(\tau)G_{2,j+1}(\tau)G_{2,j+2}(\tau)G_{2,j+3}(\tau)=E_{2,4}^{2}(\tau)\;.$

Proof. Trivial.  $\sqcap$$\sqcup$

######  Theorem 3.3 (Level 2)

  1. (1)

The functions

$f_{2,j}(\tau)=4E_{1,-4}(\tau)+(-1)^{j}2\sqrt{-1}(E_{1,-4}(\tau/2)-E_{1,-4}((\tau+1)/2))$

(where we recall that $E_{1,-4}=\theta^{2}/4$) have the following properties:

$\displaystyle f_{2,j}^{4}(\tau)=F_{2,j}(\tau)\;,$ $\displaystyle
f_{2,j}(\tau)f_{2,j+1}(\tau)=E_{2,2}(\tau)\;,\quad
f_{2,j}(((-1)^{j}+\sqrt{-1})/2)=0\;,$ $\displaystyle
f_{2,j}(\tau+1)=f_{2,j+1}(\tau)\;,\quad
f_{2,j}(-1/(2\tau))=((-1)^{j}-\sqrt{-1})\tau f_{2,j+1}(\tau)\;.$

  2. (2)

The functions

$g_{2,j}(\tau)=\sum_{0\leq i\leq 3}(1-2\delta_{i,j})E_{1,-8}((\tau+i)/4)$

have the following properties:

$\displaystyle g_{2,j}^{2}(\tau)=G_{2,j}(\tau)\;,$ $\displaystyle\prod_{0\leq
j\leq 3}g_{2,j}(\tau)=E_{2,4}(\tau)\;,\quad g_{2,j}((4-j)+\sqrt{-2}/2)=0\,$
$\displaystyle g_{2,j}(\tau+1)=g_{2,j+1}(\tau)\;,\quad
g_{2,j}(-1/(2\tau))=\varepsilon_{j}\sqrt{-2}\tau g_{2,j}(\tau)\;,$

with $\varepsilon_{j}=1$ if $4\mid j$ and $\varepsilon_{j}=-1$ if $4\nmid j$.

Proof. As in level 1, one can do a non-illuminating proof by working in
finite-dimensional vector spaces of modular forms for $\Gamma(8)$.
$\sqcap$$\sqcup$

This theorem shows both that the functions
$E_{2,4}(\tau)-(-1)^{j}\sqrt{-256}\eta_{2}^{4}(\tau)$ are fourth powers of
natural modular forms, and that $E_{2,2}(\tau)=2E_{2}(2\tau)-E_{2}(\tau)$ is
the product of two conjugate Eisenstein series of weight $1$.

##  4 Results in Level $3$

For simplicity, we set $\eta_{3}(\tau)=\eta(\tau)\eta(3\tau)$, of weight $1$
(not to be confused with $\eta(3\tau)$ itself).

In this level, first note the identity $E_{3,2}=(6E_{1,-3})^{2}$. In addition,
note that the function $E_{3,4}^{2}/E_{3,2}$ is a modular form (i.e., is
holomorphic) since one easily shows that
$E_{3,4}^{2}/E_{3,2}=E_{3,2}^{3}-108\eta_{3}^{6}$. Even better, its square
root $E_{3,4}/E_{3,2}^{1/2}\in M_{3}(\Gamma_{0}(3),\chi_{-3})$, since
$E_{3,4}/E_{3,2}^{1/2}=-9(E_{3,-3,1}+3E_{3,1,-3})$, where

$E_{3,-3,1}=-1/9+\sum_{n\geq 1}\bigl{(}\sum_{d\mid
n}d^{2}\mbox{$\left(\frac{-3}{d}\right)$}\bigr{)}q^{n}\text{\quad
and\quad}E_{3,1,-3}=\sum_{n\geq 1}\bigl{(}\sum_{d\mid
n}d^{2}\mbox{$\left(\frac{-3}{n/d}\right)$}\bigr{)}q^{n}$

are the two normalized Eisenstein series in $M_{3}(\Gamma_{0}(3),\chi_{-3})$.

Thus, we have two basic formulas in level $3$:

$E_{3,2}^{3}-E_{3,4}^{2}/E_{3,2}=108\eta_{3}^{6}\text{\quad
and\quad}E_{3,2}^{4}-E_{3,4}^{2}=3888\eta_{3}^{6}E_{1,-3}^{2}\;,$

each of which can be factored in two ways and leads to the following:

###### Definition 4.1

We define

$\displaystyle F_{3,j}(\tau)$
$\displaystyle=(E_{3,4}/E_{3,2}^{1/2})(\tau)-(-1)^{j}6\sqrt{-3}\eta_{3}^{3}(\tau)\;,$
$\displaystyle G_{3,j}(\tau)$ $\displaystyle=E_{3,2}(\tau)-3\root 3
\of{4}\rho^{j}\eta_{3}^{2}(\tau)\;,$ $\displaystyle F^{\prime}_{3,j}(\tau)$
$\displaystyle=E_{3,4}(\tau)-(-1)^{j}36\sqrt{-3}(\eta_{3}^{3}E_{1,-3})(\tau)\;,$
$\displaystyle G^{\prime}_{3,j}(\tau)$
$\displaystyle=E^{2}_{3,2}(\tau)-(-1)^{j}36\sqrt{3}(\eta_{3}^{3}E_{1,-3})(\tau)\;.$

We have evidently $F^{\prime}_{3,j}=6E_{1,3}F_{3,j}$.

###### Proposition 4.2

We have

$\displaystyle F_{3,j}(\tau+1)=F_{3,j+1}(\tau)\;,\quad
F_{3,j}(-1/(3\tau))=-3\sqrt{-3}\tau^{3}F_{3,j+1}(\tau)\;,$ $\displaystyle
F_{3,j}(\tau)F_{3,j+1}(\tau)=E_{3,2}^{3}(\tau)\;,$ $\displaystyle
G_{3,j}(\tau+1)=G_{3,j+1}(\tau)\;,\quad
G_{3,j}(-1/(3\tau))=-3\tau^{2}G_{3,j}(\tau)\;,$ $\displaystyle
G_{3,j}(\tau)G_{3,j+1}(\tau)G_{3,j+2}(\tau)=(E_{3,4}^{2}/E_{3,2})(\tau)\;,$
$\displaystyle F^{\prime}_{3,j}(\tau+1)=F^{\prime}_{3,j+1}(\tau)\;,\quad
F^{\prime}_{3,j}(-1/(3\tau))=-9\tau^{4}F^{\prime}_{3,j+1}(\tau)\;,$
$\displaystyle
F^{\prime}_{3,j}(\tau)F^{\prime}_{3,j+1}(\tau)=E_{3,2}^{4}(\tau)\;,$
$\displaystyle G^{\prime}_{3,j}(\tau+1)=G^{\prime}_{3,j+1}(\tau)\;,\quad
G^{\prime}_{3,j}(-1/(3\tau))=9\tau^{4}G^{\prime}_{3,j}(\tau)\;,$
$\displaystyle
G^{\prime}_{3,j}(\tau)G^{\prime}_{3,j+1}(\tau)=E_{3,4}^{2}(\tau)\;.$

Proof. Trivial.  $\sqcap$$\sqcup$

######  Theorem 4.3 (Level 3)

The functions

$f_{3,j}(\tau)=6E_{1,-3}(2\tau)+(-1)^{j}\sqrt{-3}(E_{1,-3}(\tau/2)-E_{1,-3}((\tau+1)/2))$

have the following properties:

$\displaystyle f_{3,j}^{3}(\tau)=F_{3,j}(\tau)\;,$ $\displaystyle
f_{3,j}(\tau)f_{3,j+1}(\tau)=E_{3,2}(\tau)\;,\quad
f_{3,j}(((-1)^{j}3+\sqrt{-3})/6)=0\;,$ $\displaystyle
f_{3,j}(\tau+1)=f_{3,j+1}(\tau)\;,\quad
f_{3,j}(-1/(3\tau))=(((-1)^{j}3-\sqrt{-3})/2)\tau f_{3,j+1}(\tau)\;.$

Proof. As in previous levels, work over $\Gamma(6)$.  $\sqcap$$\sqcup$

This shows both that $F_{3,j}$ is the cube of a modular form, and that
$(3E_{2}(3\tau)-E_{2}(\tau))/2$ is the product of two conjugate weight $1$
Eisenstein series.

Contrary to levels $1$, $2$, and $4$, I have not been able to find
corresponding results for the functions $G_{3,j}$, $F^{\prime}_{3,j}$, and
$G^{\prime}_{3,j}$.

##  5 Results in Level $4$

For simplicity, we set $\eta_{4}(\tau)=\eta(\tau)\eta(4\tau)/\eta(2\tau)$, of
weight $1/2$ (not to be confused with $\eta(4\tau)$ itself). The space of
Eisenstein series of weight $2$ is now two-dimensional (it was one-dimensional
in levels $2$ and $3$), so in addition to $E_{4,2}$ we need to introduce an
additional Eisenstein series, and we choose

$E^{\prime}_{4,2}(\tau)=4E_{2}(4\tau)-4E_{2}(2\tau)+E_{2}(\tau)$

(we will not use derivatives in this paper so this notation will not lead to
any confusion).

This level has many more eta quotient identities:

$\displaystyle E_{4,2}(\tau)=\theta^{4}(\tau)$
$\displaystyle=\eta(2\tau)^{20}/(\eta(\tau)\eta(4\tau))^{8}\;,$ $\displaystyle
E_{4,2}(\tau)-E^{\prime}_{4,2}(\tau)$
$\displaystyle=32\eta(4\tau)^{8}/\eta(2\tau)^{4}\;,$ $\displaystyle
E_{4,2}(\tau)+E^{\prime}_{4,2}(\tau)$
$\displaystyle=2\eta(\tau)^{8}/\eta(2\tau)^{4}\;,$ $\displaystyle
E_{4,2}^{2}(\tau)-{E^{\prime}_{4,2}}^{2}(\tau)$
$\displaystyle=64(\eta(\tau)\eta(4\tau)/\eta(2\tau))^{8}=64\eta_{4}^{8}(\tau)\;,$
$\displaystyle\eta(2\tau)^{24}$
$\displaystyle=(\eta(\tau)\eta(4\tau))^{8}(\eta(\tau)^{8}+16\eta(4\tau)^{8})\;.$

The basic identity $E^{2}_{4,2}-{E^{\prime}_{4,2}}^{2}=64\eta_{4}^{8}(\tau)$
can thus be factored in two additional ways, and leads to the following:

###### Definition 5.1

We define

$F_{4,j}(\tau)=E^{\prime}_{4,2}(\tau)-(-1)^{j}\sqrt{-64}\eta_{4}^{4}(\tau)\text{\quad
and\quad}G_{4,j}(\tau)=E_{4,2}(\tau)-(-1)^{j}8\eta_{4}^{4}(\tau)\;.$

###### Proposition 5.2

We have

$\displaystyle F_{4,j}(\tau+1)=F_{4,j+1}(\tau)\;,\quad
F_{4,j}(-1/(4\tau))=4\tau^{2}F_{4,j+1}(\tau)\;,$ $\displaystyle
F_{4,j}(\tau)F_{4,j+1}(\tau)=E_{4,2}^{2}(\tau)\;,$ $\displaystyle
G_{4,j}(\tau+1)=G_{4,j+1}(\tau)\;,\quad
G_{4,j}(-1/(4\tau))=-4\tau^{2}G_{4,j}(\tau)\;,$ $\displaystyle
G_{4,j}(\tau)G_{4,j+1}(\tau)={E^{\prime}_{4,2}}^{2}(\tau)\;.$

Proof. Trivial.  $\sqcap$$\sqcup$

######  Theorem 5.3 (Level 4)

  1. (1)

The functions $f_{4,j}(\tau)=\theta(\tau/2-(-1)^{j}/4)$ have the following
properties:

$\displaystyle f_{4,j}^{4}(\tau)=F_{4,j}(\tau)\;,$ $\displaystyle
f_{4,j}(\tau)f_{4,j+1}(\tau)=E_{4,2}^{1/2}(\tau)=\theta^{2}(\tau)\;,$
$\displaystyle f_{4,j}(\tau+1)=f_{4,j+1}(\tau)\;,\quad
f_{4,j}(-1/(4\tau))=(1+(-1)^{j}\sqrt{-1})(\tau/i)^{1/2}f_{4,j+1}(\tau)\;.$

  2. (2)

The functions $g_{4,j}(\tau)=2\theta^{2}(2\tau)-\theta^{2}((\tau+j)/2)$ have
the following properties:

$\displaystyle g_{4,j}^{2}(\tau)=G_{4,j}(\tau)\;,$ $\displaystyle
g_{4,j}(\tau)g_{4,j+1}(\tau)=E^{\prime}_{4,2}(\tau)\;,$ $\displaystyle
g_{4,j}(\tau+1)=g_{4,j+1}(\tau)\;,\quad
g_{4,j}(-1/(4\tau))=(-1)^{j}2\sqrt{-1}\tau g_{4,j}(\tau)\;.$

Proof. As usual, work over $\Gamma(8)$.  $\sqcap$$\sqcup$

This shows both that $F_{4,j}$ is the fourth power of a natural function, and
that $((4E_{2}(4\tau)-E_{2}(\tau))/3)^{1/2}$ is a product of two conjugate
unary theta series of weight $1/2$. Perhaps even more surprising is the
equality $f_{4,j}f_{4,j+1}=\theta^{2}$, which shows that this function is a
product of two forms of weight $1/2$ in two different ways. Note that this
identity also follows trivially from the product formula
$\theta(\tau)=\prod_{n\geq 1}(1-q^{2n})(1+q^{2n-1})^{2}$ by rearranging terms.

##  6 Conclusion

The existence of the above identities leads to several questions. First, it is
quite plausible that similar identities exist in higher levels, for instance
in levels $N$ such that $(N-1)\mid 24$, or more generally such that
$\Gamma_{0}^{*}(N)$ has genus $0$.

Second, as suggested by D. Zagier, are these factorizations unique, more
generally to what extent do holomorphic modular forms have unique
factorizations (up to scalars) into irreducible ones, is there a description
of the _complete_ set of holomorphic modular forms dividing a given one, for
instance $E_{4}$ ?

## References

  * [1] [Coh] H. Cohen, Algebraic Triangle Groups, Hauptmoduln, Complex Multiplication, and Hypergeometric Functions, preprint.