High order multiscale analysis of discrete integrable equations

In this article we present the results obtained applying the multiple scale expansion up to the order $\varepsilon^6$ to a dispersive multilinear class of equations on a square lattice depending on 13 parameters. We show that the integrability conditions given by the multiple scale expansion give rise to 4 nonlinear equations, 3 of which are new, depending at most on 2 parameters and containing integrable sub cases. Moreover at least one sub case provides an example of a new integrable system.


Introduction
Discrete-or difference-equations play an important role in Mathematical Physics for their double role. First, discrete space-time seems to be basic in the description of fundamental phenomena of nature, as suggested by quantum gravity. On the other hand, discrete equations are related to differential difference and differential equations through continuous limits. A well-known classification of integrable partial difference equations was given by Adler, Bobenko and Suris [2] in the particular case of equations defined on four lattice points. They used the "consistency around the cube" condition with some symmetry constrains to be able to get definite results. Due to the constraints introduced, this classification is partial and already new equations with respect to those contained in the ABS classification have been found [1,8,10,13,17,19].
In this paper we provide necessary conditions for the integrability of a class of real, autonomous difference equations in the variable u : Z 2 → R defined on a Z 2 square-lattice Q(u n,m , u n+1,m , u n,m+1 , u n+1,m+1 ; β 1 , β 2 , ...) = 0, n, m ∈ Z, where the β i 's are real, independent parameters. Integrability conditions will be determined through a multiscale perturbative development, continuing with the theory explained in references such as [4][5][6][7]11] applicable in differential and difference equations. This approach has the distinctive advantage of providing criteria in a manner completely independent from other current approaches. Multiscale developments can be used to reinforce, enhance or augment our previous knowledge of discrete integrable systems given by other techniques.
The constraint (6) implies that one of the following two conditions must be satisfied: (1) a 00 = a 11 ≡ a 1 , a 01 = a 10 ≡ a 2 , (2) a 00 = −a 11 ≡ a 1 , a 01 = −a 10 ≡ a 2 . Then the dispersion relation (5) reduces to: ω ± (k) = arctan ± 2a 1 a 2 ± (a 2 1 + a 2 2 ) cos(k) a 2 1 − a 2 2 sin(k) We denote the family of equations (3) satisfying condition (1) with dispersion relation ω + (k) as Q + and the one with dispersion relation ω − (k) as Q − . In all the cases a 1 and a 2 cannot be zero and their ratio cannot be equal to ±1 in order to have a nontrivial dispersion relation. We will consider integrability conditions for the class of equations Q + . The study of the class Q − is left to a future work. The result of this work are a series of integrability theorems and a table of equations, invariant under a restricted Möbius transformations, passing the very stringent integrability conditions obtained with the multiple scale expansion up to ε 6 order.
In Section 2 we present the main result on the discrete multiscale integrability test, the conditions up to order ε 6 . In Section 3 we apply it to the classification of dispersive multilinear equations defined on a square lattice Q + . Section 4 is devoted to some conclusive remarks.

The discrete multiscale integrability test
Consider a dispersive discrete equation of the form Q + , i.e. a completely discrete multilinear dispersive equation defined on a lattice of four points. In such a situation the discrete multiscale integrability test may be summarized as follows.
Then (3) splits into linear and nonlinear terms: where N ∈ N is the nonlinearity order. A multilinear equation defined on a square can be at most quartic, i.e. N ≤ 4. In the formal expansion (8) each term Q i contains only homogeneous polynomials of degree i in w n,m . If the discrete equation is dispersive then the linear part Q 1 admits a solution w n,m = exp[i(κn − ωm)] = K n Ω m , where ω = ω(κ) = ω + (κ), the dispersion relation, is a real function of κ given by Eq. (7).
ii. The multiscale expansion of the basic field variable w n,m around the harmonic K n Ω m reads where u is a bounded slowly varying function of its arguments and u ℓ ,ū ℓ being the complex conjugate of u ℓ , because we look only for real solutions. Here n 1 = εn, m j = ε j m j = 1, 2, . . . are the slow-varying lattice variables.
iii. The nearest-neighbors fields are expanded according to the following formulas: The operators A i , B i , C i , are equal to 1 when i = 0, and for some lower values of i are: where δ k are the formal derivatives with respect to the index k, δ k := ∂ k and ∇ := δ m 1 + δ n 1 . The operator δ k can always be expressed in terms of powers of the difference operators by the well known identity where ∆ k is the discrete first right difference operator with respect to the variable k, i.e.
The δ k -operators, which in principle are formal infinite series in powers of ∆ k , when acting on slow-varying functions of finite order L reduce to polynomials in ∆ k at most of order L. We shall assume that we are dealing with functions of an infinite slow-varying order, i.e. L = ∞, so the δ k -operators may be taken as differential operators acting on the indices of the harmonics u iv. Substituting the expansions (9-12) into (8), we get an equation of the following form: i.e. we must have W The multiscale expansion of the Q + equation for functions of infinite order thus gives rise to a system of continuous partial differential equations. At the lowest order (slow-time m 2 ) one gets a Nonlinear Schrödinger equation (NLS ) for the first harmonic u (1) 1 . We will use orders beyond that to define the values of the constants appearing in Q + for which the equation is integrable. The first attempt to go beyond the NLS order in the case of partial differential equations was presented by Santini, Degasperis and Manakov in [6] and by Kodama and Mikhailov using normal forms [12]. In [6] the authors, starting from S-integrable models, through a combination of an asymptotic functional analysis and spectral methods, succeeded in removing all the secular terms from the reduced equations, order by order. Their results could be summarized in the following statements: (1) The number of slow-time variables required for the amplitudes u (2) The amplitude u (1) 1 evolves at the slow-times t σ := m σ , σ ≥ 3 according to the σ−th equation of the NLS hierarchy; (3) The amplitudes of the higher perturbations of the first harmonic u (1) j , j ≥ 2 evolve at the slow-times t σ , σ ≥ 2 according to certain linear, nonhomogeneous equations when taking into account some asymptotic boundary conditions. From these statements one can conclude that the cancellation at each stage of the perturbation process of all the secular terms is a sufficient condition to uniquely fix the evolution equations followed by every u (1) j , j ≥ 1 for each slow-time t σ . Conversely, the results in [7] imply that this expansion is secularity-free. Thus, this procedure provides necessary and sufficient conditions to get secularity-free reduced equations. Following [7] we can state the following proposition: Proposition 1. If a nonlinear dispersive partial difference equation is integrable, then under a multiscale expansion the functions u (1) l , l ≥ 1 satisfy the equations is the σ-th flow in the nonlinear Schrödinger hierarchy. All the other u (κ) j , κ ≥ 2 are expressed in terms of differential monomials of u In (14b) f σ (j) is a nonhomogeneous nonlinear forcing term depending on all the u (1) κ , 1 ≤ κ ≤ j −1, their complex conjugates and their ξ-derivatives, where ξ is a variable representing the group velocity and expressed through a linear combination of the slow-space and the first slow- the linearization of the expression K σ [u] along the direction v near the function u.
In order to characterize the flows K σ u (1) 1 and the nonlinear forcing terms f σ (j), following [5], we introduce the finite dimensional vector spaces P ℓ , ℓ ≥ 2, as being the set of all homogeneous, fully-nonlinear, differential polynomials in the functions u (1) j , j ≥ 1, their complex conjugates and their ξ-derivatives of homogeneity order ℓ in ε and 1 in the accompanying exponential e iθ = e i(κn−ωm) , where We introduce the subspaces P ℓ () of P ℓ ,  ≥ 1, ℓ ≥ 2, whose elements are homogeneous, fullynonlinear, differential polynomials in the functions u (1) k , their complex conjugates and their ξ-derivatives with 1 ≤ k ≤ . Firstly from these definitions it follows that P ℓ = P ℓ (ℓ − 2), that is  ≤ ℓ − 2. In fact the terms u 1 to give the right homogeneity degree in e iθ . For the same reasons, terms of the types ∂ κ ξ u ℓ−κ−1 , 0 ≤ κ ≤ ℓ − 2 cannot appear. So the space P ℓ () is defined as that functional space generated by the base of monomials of the following types α,β,γ,δ where the product is extended for For n ≥ 3 the subspaces P ℓ (), can be generated recursively starting from the lowest one, corresponding to ℓ = 2 by the following relation where ρ (β) ≥ 0 ∀β, σ (δ) ≥ 0 ∀δ and the product is extended for 1 ≤ β, δ ≤  ≤ ℓ − 2, so that It is then clear that in general K n u Eqs. (14) are a necessary condition for integrability and represent a hierarchy of compatible evolutions for the same function u where, as f σ (j) and f σ ′ (j) are functions of the different perturbations of the fundamental harmonic up to degree j − 1, the time derivatives ∂ tσ , ∂ t σ ′ of those harmonics appearing respectively in M σ and M σ ′ have to be eliminated using the evolution equations (14) up to the index j − 1.
The commutativity conditions (15) turn out to be an integrability test.
We finally define the degree of integrability of a given equation: If the relations (15) are satisfied up to the index j, j ≥ 2, we say that our equation is asymptotically integrable of degree j or A j integrable.
Conjecturing that an A ∞ degree of asymptotic integrability actually implies integrability, we have that under this assumption the relations (14,15) are a sufficient condition for the S-integrability or that integrability is a necessary condition to have a multiscale expansion where all the Eqs. (14) are satisfied. So the multiscale integrability test tell us that Q + will be integrable if its multiscale expansion will follow all the infinite relations (14,15). The higher the degree of asymptotic integrability, the nearer the equation will be to an integrable one. However, as we can test the conditions (14,15) only up to a finite order (currently A 4 ), from them we can only derive necessary conditions for integrability, so we will not be able to state with certainty that the discrete equation is integrable. The results obtained at a finite but sufficiently high order will have a good probability to correspond to an integrable equation, but we need to use other techniques to prove it with certainty.
Let us present for completeness the lowest order conditions for asymptotic S-integrability of order k or A k -integrability conditions. To simplify the notation, we will use for u (1) j the concise form u(j), j ≥ 1. Moreover, for the convenience of the reader, we list the fluxes K σ [u] of the NLS hierarchy for u up to σ = 5: and the corresponding K ′ σ [u]v up to σ = 4: where A, ρ 1 , ρ 2 , B, C and D are all non null and, if ρ 2 = 0, real arbitrary constants.
The A 1 -integrability condition is given by the reality of the coefficient ρ 2 of the nonlinear term in the NLS. It is obtained commuting the NLS flux K 2 [u] with the flux B u ξξξ + τ |u| 2 u ξ + µu 2ū ξ with τ and µ constants. This commutativity condition gives, if ρ 2 = 0, We remark that, when ρ 2 = 0, by the same method it is possible to determine all the coefficients of all the higher NLS -symmetries (16) together with the reality conditions of the coefficients A, C and D.

The A 2 -integrability conditions.
The A 2 -integrability conditions are obtained choosing j = 2 in the compatibility conditions (15) with σ = 2 and σ ′ = 3 or alternatively σ ′ = 4, respectively In this case f 2 (2), f 3 (2) and f 4 (2) will be identified by respectively two, (a, b), five, (α, β, γ, δ, ǫ), and eight, (θ 1 , · · · , θ 8 ), complex constants As ρ 2 = 0, eliminating from Eq. (19a) the derivatives of u(1) with respect to the slow-times t 2 and t 3 , using the evolutions (14a) with σ = 2 and σ ′ = 3 and equating term by term, we obtain the following two A 2 -integrability conditions So we have two conditions obtained requiring the reality of the coefficients a and b. The expressions of α, β, α, δ in terms of a and b are: The same integrability conditions (21) can be derived using Eq. (19b). As in our analysis we will need them, here follows the explicit expressions of the coefficients of the forcing term f 4 (2) The A 3 -integrability conditions are derived in a similar way setting j = 3 in the compatibility conditions (15) In this case f 2 (3) and f 3 (3) will be respectively identified by 12 and 26 complex constants Eliminate from Eq. (19a) with j = 3 the derivatives of u(1) with respect to the slow-times t 2 and t 3 using the evolutions (14a) respectively with σ = 2 and σ ′ = 3 and the derivatives of u(2) using the evolutions (14b) with σ = 2 and σ ′ = 3. Equating the remaining terms term by term, with ρ 2 = 0 and, indicating with R i and I i the real and imaginary parts of τ i , i = 1, . . . , 12, we obtain the following 15 A 3 -integrability conditions , I 7 = 0, I 9 = 2I 8 , I 10 = I 12 , I 11 = I 6 + I 12 .
The expressions of the γ j , j = 1, . . . , 26 as functions of the τ i , i = 1, . . . , 12 are: The A 4 -integrability conditions are derived similarly from (15) . Now f 2 (4) and f 3 (4) are respectively defined by 34 and 77 complex constants If we indicate with S j and T j respectively the real and imaginary parts of η j , j = 1, . . . , 34, when ρ 2 = 0, the A 4 -integrability conditions are represented by 48 real relations whose expressions we leave for a specific Appendix.
Other integrability conditions corresponding to in the subspaces with u (2n) = 0, n ≥ 1 for purely imaginary coefficients can be found in [16]. They are respectively given by 1 and 14 real relations, the first of which can be deduced from (25) and corresponds to The results presented in this Section will be used in the following Sections to classify integrable nonlinear equation on the square lattice.
To perform a classification of the equations Q + , we need to find the set of transformations that leave it invariant, i.e. the equivalence transformation. As mentioned before, a generic multilinear  (1) is invariant under a Möbius transformation (2). The constant term f 0 and the differences a 00 − a 11 , a 01 − a 10 transform according to +2BD 3 (a 00 + a 11 + a 01 + a 10 ) , These formulas allow to determine when a given linear-affine equation (1) can be transformed into one belonging to class Q + . For this to happen all three terms must be null, so setting the l.h.s. of (29) to zero we get three polynomial equations over B/D or D/B. If simultaneously solvable (over the reals), we have an equation of the class Q + . One could try to write the conditions over the coefficients of a general linear-affine equation (1) by using resultant calculations on the three polynomial conditions, but they turn out to be too complicated to merit further attention. Thus (29) tells that the class Q + is invariant under restricted simultaneous Möbius transformations R of the form u n,m → u ′ n,m = u n,m /(Cu n,m + D), which will be our equivalence transformation. Under (30) the coefficients of Eq. (28) undergo the following transformations: We will indicate by N the number of free parameters (although not all of them essential under R) appearing in each subcase of (28). Its maximum number is N = 13, the number of free coefficients in (28).
Proof: Following the procedure described in Section 2 we expand the fields appearing in equation Q + according to formulas (9)(10)(11)(12). The lowest order necessary conditions for the Sintegrability of Q + are obtained by considering the equation W 3 (see Eq. (13)), namely the order ε 3 of the multiscale expansion. At this order we get the m 2 -evolution equation for the harmonic u where the coefficients ρ 1 and ρ 2 will depend on the parameters of the equation Q + and on the wave parameters κ and ω = ω + , with ω + expressed in terms of κ through the dispersion relation (7). According to our multiscale test the lowest order necessary condition for Q + to be an S-integrable lattice equation is that Eq. (38) be integrable itself, namely ρ 1 and ρ 2 have to be real coefficients. Let us outline the construction of Eq. (38). At O(ε) we get: • for α = 1 a linear equation which is identically satisfied by the dispersion relation (7).
• for α = 0 a linear equation whose solution is u (0) 1 = 0. At O(ε 2 ), taking into account the dispersion relation (7), we get: • for α = 2 an algebraic relation between u 1 , whose solution is given by u Notice that from the O(ε 2 ) we find that the dependence of all the harmonics on the slow-variables n 1 and m 1 is given by ξ.
Note that ρ 1 is a real coefficient depending only on the parameters of the linear part of Q + , while ρ 2 is a complex one. Hence the integrability of the NLS equation (38) is equivalent to the request R 2 = 0 ∀ κ, that is R (41) Eq. (41) is a nonlinear algebraic system of six equations in twelve unknowns. By solving it one gets the six solutions contained in Proposition 1. These solutions are computed taking into account that a 1 , a 2 ∈ R \ {0} with |a 1 | = |a 2 |. One can solve two of the six equations (41) for ξ 1 and ξ 3 , thus expressing them in terms of the remaining ten coefficients. The resulting system of four equations turns out to be ξ 2 and ξ 4 -independent and linear in the four variables α 1 , β 1 , γ 1 and γ 2 . Therefore we may write the remaining four equations as a matrix equation with coefficients nonlinearly depending on α 2 , β 2 , a 1 and a 2 . The rank of the matrix is three. The six solutions are obtained by requiring that the matrix be of rank 3, 2, 1 and 0, and correspond to six classes of equations (28) that pass integrability conditions up to order O(ε 3 ). A direct calculation proves the invariance of the six classes with respect to the restricted Möbius transformation R. Corollary 1. If the coefficients a 1 , a 2 , α 1 , α 2 , β 1 , β 2 , γ 1 , γ 2 , ξ 1 , ..., ξ 4 of equation Q + do not satisfy one of the conditions given in Eqs. (32-37) then Q + is not integrable.
Quadratic difference equations are a subclass of Q + which have attracted a deal of attention. These equations are not Möbius invariant, but we can spot those that belong to the class Q + and pass our integrability conditions, just by inspection of (32-35).

Classification at order ε 4 .
For what concerns the A 2 -asymptotically integrable cases satisfying the integrability conditions (21), the following statement holds Proposition 3. At order ε 4 , the necessary conditions for the S-integrability of equations Q + read: • Case 1 (N = 9): • Case 4 (N = 8): The corresponding two subclasses of equations are non overlapping and invariant under the restricted Möbius transformation (30).
Notice that of the six A 1 -asymptotically integrable cases listed in Proposition 2, Case 1 and Case 4 automatically satisfy the A 2 -integrability conditions (21), while the remaining four cases 2, 3, 5 and 6 specify to some subcases of theirs. Notice that only two out the previous four quadratic cases in Remark 1 survive, the Cases Q1 and Q4: the first one is a subcase of Case 1, while the second is a subcase of Case 4.

Classification at order ε 5 .
It is possible to find all the cases satisfying the A 3 -integrability conditions (25). They are given by the following proposition Case (c): (N = 5) Case (e): (N = 4) Notes: In all of the cases a 2 /a 1 = (0, ±1); the values a 2 /a 1 = (2, 1 2 ) are excluded in Case (e) because we would obtain respectively a subcase of Case (a) or of Case (b). All of the Cases (a)-(e) are subcases of Case 1. So nothing survives out of Case 4 at order ε 5 . Cases Q α -Q δ are subcases both of the Case Q1 and Case (a); the Cases Q η -Q λ are subcases both of the Case Q1 and Case (b).

3.3.1.
Canonical forms for ε 5 S-asymptotically integrable cases. Comparison with the ABS list. We will use now the Möbius transformation to reduce the equation to normal form, i.e. to eliminate the maximum number of free parameters appearing in the nonlinear difference equation and reduce the coefficients of the linear part in v n,m and v n+1,m+1 to 1.
In the Case (a) of Proposition 4, performing the Möbius transformation u n,m = αv n,m + β γv n,m + δ , we obtain the canonical form: where (τ 1 , τ 2 ) := α 1 − 3γ 1 2 , β 1 − 3γ 1 2 = (0, 0). Performing a further rescaling on (44), we can fix, in all generality, the coefficients to either τ 1 = 0 and τ 2 = 1 or τ 1 = 1 and we obtain the following two canonical forms respectively representing the two non overlapping subclasses of Case (a) defined respectively by the additional conditions α 1 = 3γ 1 2 and α 1 = 3γ 1 2 . As under a restricted Möbious transformation τ 2 is invariant, we see that two canonical forms (45b), specified by two invariants τ 2a and τ 2b , form two disconnected components of the same conjugacy subclass unless τ 2a = τ 2b ; In the Case (b) of Proposition 4, performing the Möbius transformation u n,m = αv n,m + β γv n,m + δ , we obtain the canonical form: where (τ 1 , τ 2 ) := α 1 − 3γ 2 2 , β 1 − 3γ 2 2 = (0, 0). Performing a further rescaling on (46) we can fix, in all generality, the parameters either to τ 1 = 0 and τ 2 = 1 or to τ 1 = 1 and we obtain respectively the two canonical forms representing the two non overlapping subclasses of Case (b) defined respectively by the additional conditions α 1 = 3γ 2 2 and α 1 = 3γ 2 2 . As τ 2 is invariant under a restricted Möbious transformation, we see that two canonical forms (47b), specified by two invariants τ 2a and τ 2b , form two disconnected components of the same conjugacy subclass unless τ 2a = τ 2b ; In the Cases (c) and (d) of Proposition 4, performing the Möbius transformation u n,m = αv n,m + β γv n,m + δ , we obtain the canonical forms: where ǫ := a 2 /a 1 = 0, ±1 and ζ ′ := 8s π 2 and s := ±1. As under a restricted Möbius transformation ρ → ρ (α/δ) 2 and π → π (α/δ) 3 , we see that the absolute value of ζ ′ and sgn (ρ) are invariant under such a transformation. With another rescaling we can always fix ζ ′ ≥ 0 and the two canonical forms, specified by the two set of invariants (ǫ a , sgn (ρ a ) , ζ ′ a ) and (ǫ b , sgn (ρ b ) , ζ ′ b ), form two disconnected components of the conjugacy class unless the two sets are the same; In the Case (e) of Proposition 4, performing the Möbius transformation u n,m = αv n,m + β γv n,m + δ , we obtain the canonical form: where ǫ := a 2 /a 1 = 0, ±1, 2, 1/2. As ǫ is invariant under a restricted Möbius transformation, we see that two canonical forms, specified by the two invariants ǫ a and ǫ b , form two disconnected components of the conjugacy class unless ǫ a = ǫ b ; As our allowed transformations are subcases of the full Möbius transformations allowed in the ABS approach [2], any conjugacy class of ours is either completely contained into one of the ABS classification or is totally disjointed from them. Considering that no one out of the (left hand members of the) canonical forms (a ′ )-(e ′ ) possesses the invariance (up to an overall sign) under v n,m ↔ v n+1,m , v n,m+1 ↔ v n+1,m+1 , we can conclude that no intersection can exist between our classes and those generated by the ABS list. Even more, no equation in our list is of Klein-type or, that is the same [14], a subcase of the Q V equation.
We can enlarge our class of transformations by including also an exchange n ↔ m between the two independent variables. The subclass (45a) can be discarded because under this exchange we would get it from subclass (45b) with τ 2 = 0; similarly the subclass (47a) can be discarded because under this exchange we would get it from subclass (47b) with τ 2 = 0; finally the subclasses (48-50) are invariant under this transformation.
If in (51a), when τ = 0, we apply the (not allowed) transformation v n,m := √ 3w n,m − 1, we obtain w n,m w n+1,m + w n+1,m w n,m+1 + w n,m+1 w n+1,m+1 − 1 = 0, which in the direction n satisfies two first order necessary integrability conditions given in [14] but doesn't admit the corresponding three-point generalized symmetry either autonomous or not, while in the direction m the first order integrability conditions are not satisfied. Following [8] we were able to prove the integrability of (52) constructing two five-point symmetries, one in the n direction depending on the points (n + 2, m), (n + 1, m), (n, m), (n − 1, m) and (n − 2, m) and the other one in the m direction. In [18] its integrability was finally proven providing a 3 × 3 Lax pair. Moreover this equation has the singularity confinement property, can be bilinearized, possesses multisoliton solutions and has a continuous limit into the mKdV equation, [9]. If in (51a), when τ = 1, we apply the (not allowed) transformation v n,m := − 2 1/3 w n,m + 1 , we obtain w n+1,m w n,m+1 (w n,m + w n+1,m+1 ) + 1 = 0, an integrable system introduced in [15], where it was proved to satisfy the second order, but not the first order, integrability conditions, to posses a 3× 3 Lax pair and to be a degeneration of the discrete integrable Tzitzeica equation proposed by Adler in [1]. Moreover this equation has the singularity confinement property, can be trilinearized and possesses multisoliton solutions, [9]. Finally, if in (51a), when τ = 0, 1 we apply the (not allowed) transformation v n,m := 1−τ τ w n,m − 1, we obtain w n,m w n+1,m + w n,m+1 w n+1,m+1 + w n+1,m w n,m+1 (1 + w n,m + w n+1,m+1 ) + χ = 0, (54) where χ := (τ −3)τ 2 (1−τ ) 3 , which doesn't satisfy the first order integrability conditions for three-point generalized symmetries either autonomous or not, either in direction n or m. In [18] we showed the integrability of the subcase χ = 0 constructing two five-point generalized symmetries, one in the n direction and the other one in the m direction, and a 3 × 3 Lax pair. An indication of the integrability of the general case (54) for arbitrary χ was provided showing its algebraic entropy vanishes. Other strong indications of integrability for arbitrary χ, such as the singularity confinement property, bilinear form, multisoliton solutions and a continuous limit into the mKdV equation when χ = −1, were established in [9]. In the case χ = −1 we can provide the following five-point symmetry in the n direction depending on the points (n + 2, m), (n + 1, m), (n, m), respectively given by w n,m,t = δw 2 n,m − ǫ δǫw 2 n,m − 1 (w n+1,m − w n−1,m ) (1 + δw n,m w n+1,m ) (1 + δw n,m w n−1,m ) , where t andt are two group parameters, and, in the m direction, by similar expressions obtained changing w n+1,m → w n,m+1 and w n−1,m → w n,m−1 . The second integrable system shows a two parameters non autonomous point symmetry tail too. We note that both the integrable systems are invariant under w n,m := −v n,m ; the first integrable system is covariant under the inversion w n,m := 1/v n,m as ǫ is changed into 1/ǫ, while the second one is invariant; under the non autonomous transformation w n,m := (−1) n+m v n,m both the integrable systems are covariant as in the first case ǫ is changed into −ǫ and δ into −δ, while in the second one ǫ is changed into −ǫ. This implies that in those systems we can limit ourselves to the range 0 < ǫ < 1. Moreover the second integrable system under the non autonomous transformationw n,m := (v n,m ) (−1) n+m is invariant when δ = 1 and covariant when δ = −1 as ǫ is changed into δǫ. Finally both the integrable systems are covariant under the transformation w n,m := iv n,m as δ is changed into −δ. This implies that in those systems we can always take δ = 1 but in general, if we allow such a transformation, the solution will be no more a real field but a complex one. Let's also note that the non autonomous transformation w n,m := (−1) n v n,m or w n,m := (−1) m v n,m brings both the integrable systems from class Q + into class Q − . An indication that the general cases (58a, 58b) are not integrable when τ = 0 can be obtained showing their algebraic entropy [3] doesn't vanish.

Concluding remarks
In this paper we have considered the application of a multiple scale expansion to a class of dispersive multilinear partial difference equation on the square lattice, Q + . The integrability conditions we obtain when we require that the multiple scale expansion of the discrete class of equations is equivalent to the equations of the NLS hierarchy reduce the N = 13 initial parameters defining the Q + class to a maximum of N = 2 free (continuous) parameters defining four equations. A great effort has been directed to extend the expansion up to order ε 6 , the related integrability conditions appearing in this paper for the first time. As a result of our efforts we have been able to compare the A 3 integrable equations to the A 4 integrable equations. They turn out to be the same, so that one could presume we might be already in the asymptotic regime and that the obtained equations might be integrable. However a non vanishing algebraic entropy is an indication that the general cases (58a, 58b) are not integrable when τ = 0.
An open problem seems of major importance now: the consideration of the second class of dispersive multilinear partial difference equations on the square lattice, Q − is a major task which will provide by sure new classes of integrable equations. However in this case the lowest order integrability conditions appear already at order ε 2 and will not be an equation of the NLS type but more likely a coupled wave equations. Work is in progress on it.