INTEGRABILITY TEST FOR DISCRETE EQUATIONS VIA
D. LEVI AND R.I. YAMILOV
Abstract. In this article we present some integrability conditions for partial difference
equations obtained using the formal symmetries approach. We apply them to find in-
tegrable partial difference equations contained in a class of equations obtained by the
multiple scale analysis of the general multilinear dispersive difference equation defined
on the square.
DL met Marcos Moshinsky for the first time on his arrival in Mexico in February 1973.
He went there with a two years fellowship of the CONACYT (Consejo Nacional de Ciencia
y Tecnolog´ ıa), the Mexican Research Council. DL visit was part of his duties as military
service in Italy and was following a visit to Mexico in the summer of 1972 by Francesco
Calogero, with whom he graduated at the University of Rome La Sapienza in the spring of
1972 with a thesis on ”Computation of bound state energy for nuclei and nuclear matter
with One-Boson-Exchange Potentials” .
The bureaucratic process, preliminary to this visit, was long and tiring and it was suc-
cessful only for the strong concern of DL uncle, Enzo Levi, professor of Hydrulics at UNAM
 and for the help of Marcos Moshinsky who signed a contract for him which, as DL uncle
wrote, was ready to substain with his own personal funds if no other source could be found.
During DL stay in Mexico he was able, with the help of Marcos, to appreciate the plea-
sures of research and to start his career in Mathematical Physics. With Marcos Moshinsky
he published, during the two years stay at the Instituto de Fisica of UNAM, 4 articles
[6, 14, 7, 8] but he also collaborated with some of the other physicists of the Institute
[4, 11, 12]. This visit was fundamental to shape DL future life, both from the personal
point of view and from the point of view of his career.
After 1975 DL met Marcos Moshinsky frequently both abroad and in Mexico. When
DL was in Mexico, Marcos Moshinsky always invited him to his home to partecipate to his
family dinners. As a memento of Marcos, let us add a picture of his with Pavel Winternitz,
Petra Seligman and Decio Levi, taken in September 1999 in front of the Guest House of
the CIC-AC where he stayed overnight after a seminar given at the Instituto de Fisica of
UNAM, Cuernavaca branch.
Symmetries have played always an important role in physics and in the research of
Marcos Moshinsky. This presentation shows how crucial can be the notion of symmetry in
uncovering integrable structures in nonlinear partial difference equations.
The discovery of new integrable Partial Difference Equations (P∆E) is always a very
challenging problem as, by proper continuous limits, we can obtain integrable Differential
Difference Equations (D∆E) and Partial Differential Equations (PDE).
A very successful way to uncover integrable PDE has been the formal symmetry approach
due to Shabat and his school in Ufa . These results have been later extended to the case
of D∆E by Yamilov , a former student of Shabat.
Here we present some results on the application of the formal symmetry technique to
The basic theory for obtaining symmetries of differential equations has been introduced
by Sophus Lie at the end of the nineteen century and can be found together with its
extension to generalized symmetries introduced by Emma Noether, for example in the book
arXiv:1010.3400v1 [nlin.SI] 17 Oct 2010
2D. LEVI AND R.I. YAMILOV
Figure 1. Pavel Winternitz, Marcos Moshinsky, Petra Seligman and Decio Levi
in front of the Guest House of CIC-AC in Cuernavaca in September 1999.
by Olver . The extension of the classical theory to P∆E can be found in the work of
Levi and Winternitz [19, 21, 29].
Here in the following we outline the results of the geometric classification of P∆E given
by Adler, Bobenko and Suris  with some critical comments at the end. Then in Section 2
we derive the lowest integrability conditions starting from the request that the P∆E admit
generalized symmetries of sufficiently high order. In Section 3 we show how the test can be
applied. Then we apply it to a simple class of P∆E obtained by the multiple scale analysis
of a generic multilinear dispersive equation defined on the square.
1.1. Classification of linear affine discrete equations. Adler, Bobenko and Suris 
considered the following class of autonomous P∆E:
ui+1,j+1= F(ui+1,j,ui,j,ui,j+1), (1)
where i,j are arbitrary integers. Eq. (1) is a discrete analogue of the hyperbolic equations
uxy= F(ux,u,uy). (2)
which are very important in many fields of physics. Up to now the general equation (2) has
not been classified. Only the two particular cases:
uxy= F(u);ux= F(u,v) , vy= G(u,v),
which are essentially easier, have been classified by the formal symmetry approach [32, 33].
The ABS integrable lattice equations are defined as those autonomous affine linear (i.e.
polynomial of degree one in each argument, i.e. multilinear) partial difference equations of
E(u0,0,u1,0,u0,1,u1,1;α,β) = 0,
where α and β are two constant parameters , whose integrability is based on the consis-
tency around a cube (or 3D-consistency). Here and in the following, as the equations are
autonomous, and thus translational invariant, we skip the indices i and j and write the
equations around the origin.
The main idea of the consistency method is the following:
INTEGRABILITY TEST FOR DISCRETE EQUATIONS VIA GENERALIZED SYMMETRIES.3
Figure 2. A square lattice
Figure 3. Three-dimensional consistency
(1) One starts from a square lattice and defines the three variables ui,jon the vertices
(see Figure 2). By solving E = 0 one obtains a rational expression for the fourth
(2) One adjoins a third direction, say k, and imagines the map giving u1,1,1as being
the composition of maps on the various planes. There exist three different ways
to obtain u1,1,1and the consistency constraint is that they all lead to the same
(3) Two further constraints have been introduced by Adler, Bobenko and Suris:
• Tetrahedron property: u1,1,1is independent of u0,0,0.
(4) The equations are classified according to the following equivalence group:
• A M¨ obius transformation.
• Simultaneous point change of all variables.
As a result of this procedure all equations possess a symmetric (in the exchange of the
first to the second index) Lax pair, B¨ acklund transformations etc. . Thus the compatible
equations are, for all purposes, completely integrable equations.
4D. LEVI AND R.I. YAMILOV
The ABS list read:
(H1)(u0,0− u1,1)(u1,0− u0,1) − α + β = 0,
The potential discrete KdV equation [15, 23]
(u0,0− u1,1)(u1,0− u0,1) + (β − α)(u0,0+ u1,0+ u0,1+ u1,1) −
−α2+ β2= 0,
α(u0,0u1,0+ u0,1u1,1) − β(u0,0u0,1+ u1,0u1,1) + δ(α2− β2) = 0,
(Q1)α(u0,0− u0,1)(u1,0− u1,1) − β(u0,0− u1,0)(u0,1− u1,1) +
+δ2αβ(α − β) = 0,
α(u0,0− u0,1)(u1,0− u1,1) − β(u0,0− u1,0)(u0,1− u1,1) +
+αβ(α − β)(u0,0+ u1,0+ u0,1+ u1,1) − αβ(α − β)(α2− αβ + β2) = 0,
(β2− α2)(u0,0u1,1+ u1,0u0,1) + β(α2− 1)(u0,0u1,0+ u0,1u1,1) −
−α(β2− 1)(u0,0u0,1+ u1,0u1,1) −δ2(α2− β2)(α2− 1)(β2− 1)
+a1(u0,0u1,0u0,1+ u1,0u0,1u1,1+ u0,1u1,1u0,0+ u1,1u0,0u1,0) +
+a2(u0,0u1,1+ u1,0u0,1) + ¯ a2(u0,0u1,0+ u0,1u1,1) +
+˜ a2(u0,0u0,1+ u1,0u1,1) + a3(u0,0+ u1,0+ u0,1+ u1,1) + a4= 0,
The Schwarzian discrete KdV equation [16, 25]
where the seven parameters ai’s in (Q4) are related by 3 equations.
By a proper limiting procedure all equations of the ABS list are contained in eq. (Q4) .
The symmetries for the discrete equations of the ABS list have been constructed [28, 27] and
are given by D∆E, subcases of Yamilov’s discretization of the Krichever–Novikov equation
(YdKN) [17, 31]:
,R(u1,u0,u−1) = A0u1u−1+ B0(u1+ u−1) + C0,
0+ 2c2u0+ c3,
0+ c4u0+ c5,
0+ 2c5u0+ c6.
It is immediate to see that by defining vi= ui,jand ˜ vi= ui,j+1, the equations of the ABS
list are nothing else but B¨ acklund transformations for particular subcases of the YdKN
[13, 17]. The ABS equations do not exhaust all the possible B¨ acklund transformations for
the YdKN equation as the whole parameter space is not covered [17, 30]. Moreover, in the
list of integrable D∆E of Volterra type , there are equations different from the YdKN
which may also have B¨ acklund transformations of the form (1). So we have space for new
integrable P∆E which we will search by using the formal symmetry approach. An extension
of the 3D consistency approach has been proposed by the same authors  allowing different
equations in the different faces of the cube. However in this way ABS were able to provide
only examples of new integrable P∆E but not to present a complete classification scheme.
2. Construction of Integrability Conditions
We consider the class of autonomous P∆E
u1,1= f0,0= F(u1,0,u0,0,u0,1)(∂u1,0F, ∂u0,0F, ∂u0,1F) ?= 0. (6)
Introducing the two shifts operators, T1and T2such that T1ui,j= ui+1,j, T2ui,j= ui,j+1,
it follows that the functions ui,j are related among themselves by eq. (6) and its shifted
2f0,0= fi,j= F(ui+1,j,ui,j,ui,j+1).
INTEGRABILITY TEST FOR DISCRETE EQUATIONS VIA GENERALIZED SYMMETRIES.5
So, the functions ui,jare not all independent. However we can introduce a set of independent
functions ui,j in term of which all the others are expressed. A possible choice is given by
(ui,0, u0,j), for any arbitrary i,j integers.
A generalized symmetry, written in evolutionary form, is given by
dtu0,0= g0,0= G(un,0,un−1,0,...,un?,0,u0,k,u0,k−1,...,u0,k?),
where t is the group parameter. By shifting, we can write it in any point of the plane
n ≥ n?, k ≥ k?.(7)
2g0,0= gi,j= G(ui+n,j,...,ui+n?,j,ui,j+k,...,ui,j+k?).
In term of the functions gi,jwe can write down the symmetry invariant condition
i.e. g1,1 = (g1,0∂u1,0+ g0,0∂u0,0+ g0,1∂u0,1)f0,0. This equation involves the independent
variables (ui,0, u0,j) appearing in g0,0shifted to points laying on lines neighboring the axis,
i.e. (ui,1, u1,j). For those function we can state the following Proposition , necessary to
prove the subsequent Theorems:
Proposition 1. The functions ui,1,u1,jhave the following structure:
i > 0 : ui,1= ui,1(ui,0,ui−1,0,...,u1,0,u0,0,u0,1),
i < 0 : ui,1= ui,1(ui,0,ui+1,0,...,u−1,0,u0,0,u0,1),
j > 0 : u1,j= u1,j(u1,0,u0,0,u0,1,...,u0,j−1,u0,j),
j < 0 : u1,j= u1,j(u1,0,u0,0,u0,−1,...,u0,j+1,u0,j),
In eq. (9) and in the following, fui,j=
metry of characteristic function g0,0 depends on at least one variable of the form ui,0,
then (gun,0,gun?,0) ?= 0, and the numbers n,n?are called the orders of the symme-
try. The same can be said about the variables u0,jand the corresponding numbers k,k?if
(gu0,k,gu0,k?) ?= 0.
Now we can state the following Theorem, whose proof can be found in :
∂ui,j. If a generalized sym-
Theorem 1. If the P∆E u1,1 = F possesses a generalized symmetry then the following
relations must take place:
n > 0,(Tn
1− 1)logfu1,0= (1 − T2)T1loggun,0,
1 − 1)logfu0,0
= (1 − T2)loggun?,0,(11)
k > 0,(Tk
2− 1)logfu0,1= (1 − T1)T2loggu0,k,
= (1 − T1)loggu0,k?.(13)
− 1 = (Tl− 1)(1 + Tl+ ··· + Tm−1
− 1 = (1 − Tl)(T−1
m > 0,
m < 0,
+ ··· + Tm
l = 1,2,
it follows from Theorem 1 that we can write the equations (10, 11, 12, 13) as standard
conservation laws. Thus, the assumption that a generalized symmetry exist implies the
existence of some conservation laws.
If we assume that a second generalized symmetry exists, i.e. we can find a nontrival
function˜G such that
u0,0,˜ t= ˜ g0,0=˜G(u˜ n,0,u˜ n−1,0,...,u˜ n?,0,u0,˜k,u0,˜k−1,...,u0,˜k?),
where ˜ n, ˜ n?,˜k,˜k?are its orders, then we can state the following Theorem:
6D. LEVI AND R.I. YAMILOV
Theorem 2. Let the P∆E u1,1= F possess two generalized symmetries of orders (n,n?,k,k?)
and (˜ n, ˜ n?,˜k,˜k?), u00,t= g00and u00,˜ t= ˜ g00, and let their orders satisfy one of the following
Case 1 : n > 0, ˜ n = n + 1
Case 3 : k > 0,˜k = k + 1
Case 2 : n?< 0, ˜ n?= n?− 1
Case 4 : k?< 0,˜k?= k?− 1
Then in correspondence with each of the previous cases the P∆E u1,1= F admits a con-
0,0= (T2− 1)q(m)
0,0,m = 1,2,3,4, (15)
So the assumption that the P∆E u1,1= F have two generalized symmetries implies that
we must have four necessary conditions of integrability, i.e. there must exist some functions
of finite range q(1)
Summarizing the results up to now obtained we can say that a nonlinear partial difference
equation will be considered to be integrable if it has a generalized symmetry of finite order,
i.e. depending on a finite number of fields. This provide some conditions which imply the
existence of functions p(m)
0,0of finite range whose existence is proved by solving a
For a D∆E, when all shifted variables are independent the proof that a total difference
has a solution depending on a finite number of fields, i.e. is a finite range function, is carried
out by applying the discrete analogue of the variational derivative, i.e. a function qnis (up
to a constant) a total difference of a function of finite range iff
see, e.g. . For P∆E this is no more valid as the shifted variables are not independent
as they are related by the nonlinear P∆E, in our case u1,1= F(u1,0,u0,0,u0,1). This turns
out to be the main problem for the application of the formal symmetry approach to P∆E.
To get a definite result we limit our considerations to five points generalized symmetries,
i.e. when :
0,0satisfying the conservation laws (15) with p(1)
0,0may depend only on the variables ui,0, and p(3)
defined by eq. (16).
˙ u0,0= g0,0= G(u1,0,u−1,0,u0,0,u0,1,u0,−1),
The existence of a 5 points generalized symmetry will be taken by us as an integrability
criterion. This may be a severe restriction as there might be integrable equations with
symmetries depending on more lattice points. However just in this case we are able to
get sufficiently easily a definite result and, as will be shown in the next Section, we can
even solve a classification problem. In this case we can state the following Theorem, which
specifies the results obtained so far to the case of five point symmetries:
Theorem 3. If the P∆E u1,1 = F possesses a 5 points generalized symmetry, then the
m = 1,2,
m = 3,4,
must satisfy the conditions (15, 16).
Then, using the relations (10–13) with n = k = 1 and n?= k?= −1, we get the fol-
lowing relations between the solutions of the total difference conditions and the generalized
INTEGRABILITY TEST FOR DISCRETE EQUATIONS VIA GENERALIZED SYMMETRIES.7
So, to prove the integrability, which for us means find a generalized 5 point symmetry, for
a nonlinear P∆E u11= F, we have to check the integrability conditions (15, 16). If they
are satisfied, i.e. there exist some finite range functions q(m)
the partial derivatives of G. The compatibility of these partial derivatives of G, given by
eqs. (20), provides the additional integrability condition
0,0, we can construct
G,u1,0,u−1,0= G,u−1,0,u1,0,G,u0,1,u0,−1= G,u0,−1,u0,1.(21)
If these additional integrability conditions are satisfied, we find g0,0 up to an arbitrary
unknown function of the form ν(u0,0), which may correspond to a Lie point symmetry.
This function can be specified, using the determining equations (8).
The 5 point generalized symmetry g0,0, so obtained, will be of the form:
g0,0= Φ(u1,0,u0,0,u−1,0) + Ψ(u0,1,u0,0,u0,−1) + ν(u0,0). (22)
3. Application of the test: an example
To check the integrability conditions (15, 16) we need to find the finite range functions
0,0(m = 3,4). This is not an easy task even if they are linear first
order difference equations. A solution always exists but nothing ensure us a priory that
the solution is a finite range function. So let us present a scheme for solving explicitly
the integrability conditions we found for the equations on the square i.e. for finding the
As an example of this procedure let us consider the solution of eq. (15) for m = 1, where
0,0(m = 1,2) and p(m)
0,0= log(fu1,0), q(1)
0,0= Q(1)(u2,0,u1,0,u0,0), T2q(1)
In eq. (23) we have the dependent variables u2,1and u1,1where u2,1= F(u2,0,u1,0,u1,1)
while u1,1= F(u1,0,u0,0,u0,1). So eq. (15) for m = 1 will contain the unknow function
F which characterize the class of equations we are considering twice, one time to calculate
u1,1in terms of independent variables and then to calculate u2,1in term of u1,1and of the
independent variables. This double dependence makes the calculations extremely difficult.
To overcome this difficulty we take into account that we are considering autonomous equa-
tions which are shift invariant. So we can substitute eq. (15) for m = 1 with the following
equivalent independent equations
= Q(m)(u1,1,u0,1,u−1,1) − Q(m)(u1,0,u0,0,u−1,0)
= Q(m)(u1,0,u0,0,u−1,0) − Q(m)(u1,−1,u0,−1,u−1,−1)
where, to simplify the notation, we introduce in the following the functions
to indicate f0,0 and its analogues. Moreover, we introduce the following two differential
in such a way that the functional equations (24, 25) reduce to differential monomials :
AQ(m)(u1,1,u0,1,u−1,1) = 0,
AQ(m)(u1,0,u0,0,u−1,0) = r(m,1),
Eqs. (27) are, by construction, identically satisfied while eqs. (28) provide a set of equa-
tions for the derivatives of Q(m)(u1,0,u0,0,u−1,0) with respect to its three arguments. By
BQ(m)(u1,−1,u0,−1,u−1,−1) = 0,
BQ(m)(u1,0,u0,0,u−1,0) = r(m,2).
8D. LEVI AND R.I. YAMILOV
commuting the two operators (26) we can obtain a third equation for the derivatives of
Q(m)(u1,0,u0,0,u−1,0) with respect to its three arguments:
[A,B]Q(m)(u1,0,u0,0,u−1,0) = r(m,3).
Eqs. (28, 29), if independent, define uniquely the derivatives of the function Q(m)(u1,0,u0,0,u−1,0)
and, if their consistency is satisfied, from them we get the functions themselves.
In a similar manner from (T1− 1)p(m)
This procedure works if the function F is known, i.e. if we check a given equation for
its integrability. It also works if F is known up to some unknown arbitrary constants to be
specified. In such case we solve a classification problem with unknown constants. However,
the problem is much more difficult if F depends on unknown arbitrary functions of one,
two or three variables. In such a case the coefficients of the operators (26) and functions
r(m,k)will depend on unknown functions, and r(m,k)may even depend on the composition
of unknown functions. In this case a more complicated procedure might be necessary.
0,0= (T2− 1)q(m)
0,0with m = 3,4 we get the function
0,0= P(m)(u0,2,u0,1,u0,0) and consequently the symmetry (22).
3.1. A concrete example. Let us consider the following P∆E 
2(u0,0+ u1,1)+u1,0+ u0,1+ γ[4u0,0u1,1+ 2u1,0u0,1+ 3(u0,0+ u1,1)(u1,0+ u0,1)] + (30)
(ξ2+ ξ4)u0,0u1,1(u1,0+ u0,1) + (ξ2− ξ4)u1,0u0,1(u0,0+ u1,1) +
Eq. (30) is a dispersive multi–linear partial difference equation which passes the A3multiple
scales integrability test . Applying the M¨ obious transformation ui,j= 1/(ˆ ui,j− γ) we
can rewrite it in a simplified form as
(u0,0u1,1+ α)(u1,0+ u0,1) + (2u1,0u0,1+ β)(u0,0+ u1,1) + δ = 0,(31)
where α, β and δ are well defined functions of γ, ξ2, ξ4 and ζ. We now apply to eq.
(31) the procedure outlined at the beginning of this section. Eq. (31) depends on three
free parameters and we look for conditions on the three parameters, if any, such that the
equation admits generalized symmetries. We get that the conditions are satisfied only in
(1) α = 2β ?= 0, δ = 0 and, as β ?= 0 we can always set β = 1. This choice of the
parameters α and β corresponds to ξ2= 3ξ4+ 3γ2and ζ = 12γξ4in eq. (30). The
corresponding integrable P∆E reads:
(u0,0u1,1+ 2)(u1,0+ u0,1) + (2u1,0u0,1+ 1)(u0,0+ u1,1) = 0. (32)
In correspondence with the eq. (32) we get the generalized symmetry
(2) β = 2α ?= 0, δ = 0 and, as α ?= 0 we can always set α = 1. This choice of the
parameters α and β corresponds to ξ2= 6γ2−3ξ4and ζ = 12γ(γ2−ξ4) in eq. (30).
The corresponding integrable P∆E reads:
(1 + u0,0u1,1)(u1,0+ u0,1) + 2(1 + u0,1u1,0)(u0,0+ u1,1) = 0. (34)
In correspondence with the eq. (34) we get the generalized symmetry
0,0− 1)u1,0− u−1,0
u−1,0u1,0− 1+ B(u2
0,0− 1)u0,1− u0,−1
Here A,B are constant coefficients, and in both cases, (A = 0,B ?= 0) and (A ?= 0,B = 0),
the nonlinear D∆E (33, 35) are, up to a point transformation, equations belonging to the
classification of Volterra type equations done by Yamilov . This shows that the eqs.
(32, 34) do not belong to the ABS classification. Moreover this calculation shows that the
INTEGRABILITY TEST FOR DISCRETE EQUATIONS VIA GENERALIZED SYMMETRIES.9
A3integrability in the multiple scale integrability test is not sufficient to select integrable
P∆E on the square having five points generalized symmetries.
R.I.Y. has been partially supported by the Russian Foundation for Basic Research (Grant
number 10-01-00088-a and 08-01-00440-a). LD has been partly supported by the Italian
Ministry of Education and Research, PRIN “Nonlinear waves: integrable finite dimensional
reductions and discretizations” from 2007 to 2009 and PRIN “Continuous and discrete
nonlinear integrable evolutions: from water waves to symplectic maps” from 2010.
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