# Integrability Test for Discrete Equations via Generalized Symmetries

**ABSTRACT** In this article we present some integrability conditions for partial difference equations obtained using the formal symmetries approach. We apply them to find integrable 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. Comment: Proceedings of the Symposium in Memoriam Marcos Moshinsky

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**ABSTRACT:**We examine two lattice equations, obtained through the application of multiscale perturbative analysis, from the point of view of integrability. We show that both equations are integrable and related to the discrete sine-Gordon. We compute the limit of both systems whereby they become linearizable, obtaining the discrete Liouville equation and a linearizable lattice system recently proposed by Hydon and Viallet. We present the explicit solution of the latter.Journal of Physics A Mathematical and Theoretical 12/2010; 44(3):032002. · 1.77 Impact Factor

Page 1

INTEGRABILITY TEST FOR DISCRETE EQUATIONS VIA

GENERALIZED SYMMETRIES.

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.

1. Introduction

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” [5].

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

[9] 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 [22]. These results have been later extended to the case

of D∆E by Yamilov [31], a former student of Shabat.

Here we present some results on the application of the formal symmetry technique to

P∆E.

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

1

arXiv:1010.3400v1 [nlin.SI] 17 Oct 2010

Page 2

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 [26]. 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 [2] 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 [2]

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

the form

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:

(3)

Page 3

INTEGRABILITY TEST FOR DISCRETE EQUATIONS VIA GENERALIZED SYMMETRIES.3

u0,0

β

α

u1,0

u1,1

u0,1

Figure 2. A square lattice

u

u

u

u0,0,1

u

e

e

e

e

"""""

"

"""""

u1,1,0

"

"""""

u1,0,0

"

"

"

"

"

"

"

u0,0,0

u0,1,0

u1,0,1

u0,1,1

u1,1,1

α

β

γ

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

one.

(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

result.

(3) Two further constraints have been introduced by Adler, Bobenko and Suris:

• D4-symmetry:

E(u0,0,u1,0,u0,1,u1,1;α,β)=

±E(u0,0,u0,1,u1,0,u1,1;β,α)

±E(u1,0,u0,0,u1,1,u0,1;α,β).=

• 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.

Page 4

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,

(4)

(H2)

(H3)

(5)

(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)

a0u0,0u1,0u0,1u1,1+

+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]

(Q2)

(Q3)

4αβ

= 0,

(Q4)

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) [24].

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]:

du0

d?

=R(u1,u0,u−1)

u1− u−1

,R(u1,u0,u−1) = A0u1u−1+ B0(u1+ u−1) + C0,

where

A0= c1u2

B0= c2u2

C0= c3u2

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 [31], 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 [3] 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

values

ui+1,j+1= Ti

1Tj

2f0,0= fi,j= F(ui+1,j,ui,j,ui,j+1).

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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

d

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)

d

dtui,j= Ti

1Tj

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 [20], necessary to

prove the subsequent Theorems:

g1,1−df0,0

dt

?????

u1,1=f0,0

= 0.(8)

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),

∂ui,0ui,1= Ti−1

∂ui,0ui,1= −Ti

∂u0,ju1,j= Tj−1

∂u0,ju1,j= −Tj

1

fu1,0;

fu0,0

fu0,1;

fu0,1;

fu0,0

fu1,0.

1

2

2

(9)

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 [20]:

∂f0,0

∂ui,jand gui,j=

∂g0,0

∂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

fu0,1

(10)

n?< 0,(Tn?

= (1 − T2)loggun?,0,(11)

k > 0,(Tk

2− 1)logfu0,1= (1 − T1)T2loggu0,k,

(Tk?

fu1,0

(12)

k?< 0,

2− 1)logfu0,0

= (1 − T1)loggu0,k?.(13)

As

Tm

l

Tm

l

− 1 = (Tl− 1)(1 + Tl+ ··· + Tm−1

− 1 = (1 − Tl)(T−1

l

),

l),

m > 0,

m < 0,

l

+ T−2

l

+ ··· + 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:

(14)

Page 6

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

conditions:

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-

servation law

(T1− 1)p(m)

where

0,0= logfu0,0

fu0,1

0,0= (T2− 1)q(m)

0,0,m = 1,2,3,4, (15)

p(1)

0,0= logfu1,0,p(2)

,q(3)

0,0= logfu0,1,q(4)

0,0= logfu0,0

fu1,0

. (16)

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)

q(1)

on u0,j.

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

total difference.

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. [31]. 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 :

0,0,q(2)

0,0,p(3)

0,0and q(2)

0,0,p(4)

0,0satisfying the conservation laws (15) with p(1)

0,0may depend only on the variables ui,0, and p(3)

0,0,p(2)

0,0,q(3)

0,0and p(4)

0,0,q(4)

0,0

defined by eq. (16).

0,0

0,0or q(m)

δqn

δun

=

j

T−j∂qn

∂un+j

= 0,(17)

˙ 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:

gu1,0gu−1,0gu0,1gu0,−1?= 0.(18)

Theorem 3. If the P∆E u1,1 = F possesses a 5 points generalized symmetry, then the

functions

q(m)

p(m)

0,0= Q(m)(u2,0,u1,0,u0,0),

0,0= P(m)(u0,2,u0,1,u0,0),

m = 1,2,

m = 3,4,

(19)

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

symmetry G:

q(1)

0,0= −T1logG,u1,0,

p(3)

0,0= −T2logG,u0,1,

q(2)

0,0= T1logG,u−1,0,

p(4)

0,0= T2logG,u0,−1.

(20)

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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

0,0and p(m)

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

q(m)

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

functions q(1)

0,0.

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, q(2)

0,0, p(3)

0,0and p(4)

p(1)

0,0= log(fu1,0), q(1)

0,0= Q(1)(u2,0,u1,0,u0,0), T2q(1)

0,0= Q(1)(u2,1,u1,1,u0,1).(23)

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

p(m)

0,0− p(m)

p(m)

−1,0

= 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)

(24)

0,−1− p(m)

−1,−1

(25)

where, to simplify the notation, we introduce in the following the functions

u1,1= f(1,1)(u1,0,u0,0,u0,1),

u1,−1= f(1,−1)(u1,0,u0,0,u0,−1),

to indicate f0,0 and its analogues. Moreover, we introduce the following two differential

operators

u−1,1= f(−1,1)(u−1,0,u0,0,u0,1),

u−1,−1= f(−1,−1)(u−1,0,u0,0,u0,−1),

A

=∂u0,0−f(1,1)

u0,0

f(1,1)

u1,0

∂u1,0−f(−1,1)

u0,0

f(−1,1)

u−1,0

∂u−1,0,(26)

B

=∂u0,0−f(1,−1)

u0,0

f(1,−1)

u1,0

∂u1,0−f(−1,−1)

f(−1,−1)

u0,0

u−1,0

∂u−1,0.

in such a way that the functional equations (24, 25) reduce to differential monomials [1]:

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).

(27)

(28)

Page 8

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)

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.

(29)

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 [10]

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) +

ζu0,0u1,1u1,0u0,1= 0.

+

+

Eq. (30) is a dispersive multi–linear partial difference equation which passes the A3multiple

scales integrability test [18]. 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

two cases:

(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:

u0,0;t

=

(u2

0,0− 2)(2u2

0,0− 1)

u0,0

1

A

?

1

1

u1,0u0,0+ 1−

??

1

u−1,0u0,0+ 1

?

+(33)

+B

u0,1u0,0+ 1−

u0,−1u0,0+ 1

.

(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

u0,0;t= A(u2

0,0− 1)u1,0− u−1,0

u−1,0u1,0− 1+ B(u2

0,0− 1)u0,1− u0,−1

u0,−1u0,1− 1

(35)

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 [31]. This shows that the eqs.

(32, 34) do not belong to the ABS classification. Moreover this calculation shows that the

Page 9

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.

Acknowledgments.

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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E-mail address: levi@roma3.infn.it

Dipartimento di Ingegneria Elettronica, Universit` a di Roma Tre, and INFN, Sezione di Roma

Tre, via della Vasca Navale, 84, Roma, Italy

Ufa Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky Street,

450008 Ufa, Russian Federation