Reconstructing a lattice equation: a non-autonomous approach to the ...

arXiv:1705.00298v1 [nlin.SI] 30 Apr 2017

Reconstructing a lattice equation: a non-autonomous approach to the Hietarinta equation G. Gubbiotti∗

C. Scimiterna†

Dipartimento di Matematica e Fisica, Universit`a degli Studi Roma Tre and Sezione INFN di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy

Abstract In this paper we construct a non-autonomous version of the Hietarinta equation [J. Hietarinta. A new two-dimensional lattice model that is consistent around the cube. J. Phys. A: Math. Gen., 37:L67–L73, 2004] and study its integrability properties. We show that this equation possess linear growth of the degrees of iterates, generalized symmetries depending on arbitrary functions, and consequently that is Darboux integrable. We use the first integrals to provide a general solution of this equation. Finally we show that this equation is a particular case of the non-autonomous QV equation, and we provide a non-autonomous M¨ obius transformation to another equation found in [J. Hietarinta. Searching for CAC-maps. J. Nonlinear Math. Phys., 12:223–230, 2005].

1

Introduction

Since its introduction the Consistency Around the Cube (CAC) has been a source of many results in the classification of nonlinear integrable partial difference equations on a quad graph. The importance of this criterion relies on the fact that it ensures the existence of B¨acklund transformations [5, 9, 14, 42, 43] and, as a consequence, of Lax pairs. However [58] Lax pairs and B¨acklund transforms are associated with both linearizable and integrable equations. Let us point out that to be bona fide a Lax pair has to give rise to a genuine spectral problem [10], otherwise the Lax pair is a fake Lax pair [11, 30–33]. A fake Lax pair is useless in proving (or disproving) the integrability, since it can be equally found for integrable and non-integrable equations. In the linearizable case Lax pairs are fake, even if proving it it is usually a highly nontrivial task [24]. The first attempt to classify all the multi-affine partial difference equations defined on the quad graph and possessing CAC was carried out in [1]. There the equation on the quad graph was treated as a geometric object not embedded in ∗ E-mail: † E-mail:

[email protected] [email protected]

1

any Z2 -lattice, as displayed in Figure 1. The quad-equation is an expression of the form: Q (x, x1 , x2 , x12 ; α1 , α2 ) = 0, (1) connecting some a priori independent fields x, x1 , x2 , x12 assigned to the vertices of the quad graph, see Figure 1. Q is assumed to be a multi-affine polynomial in x, x1 , x2 , x12 and, as shown in Figure 1, α1 and α2 are parameters assigned to the edges of the quad graph. x1

α1

α2

x

x12

α2

α1

x2

Figure 1: The purely geometric quad graph not embedded in any lattice. In this setting, we define the Consistency Around the Cube as follows: assume we are given six quad-equations: A (x, x1 , x2 , x12 ; α1 , α2 ) = 0, A¯ (x3 , x13 , x23 , x123 ; α1 , α2 ) = 0, B (x, x2 , x3 , x23 ; α3 , α2 ) = 0, ¯ B (x1 , x12 , x13 , x123 ; α3 , α2 ) = 0, C (x, x1 , x3 , x13 ; α1 , α3 ) = 0, C¯ (x2 , x12 , x23 , x123 ; α1 , α3 ) = 0,

(2a) (2b) (2c) (2d) (2e) (2f)

arranged on the faces of a cube as in Figure 2. Then if x123 computed from (2b), (2d) and (2f) coincide we say that the system (2) possesses the Consistency Around the Cube. In [1] the classification was carried out up to the action of a general M¨ obius transformation and up to point transformations of the edge parameters, with the additional assumptions: 1. All the faces of the cube in Figure 2 carry the same equation up to the edge parameters. 2. The quad-equation (1) possesses the D4 discrete symmetries: Q (x, x1 , x2 , x12 ; α1 , α2 ) = µQ (x, x2 , x1 , x12 ; α2 , α1 ) = µ′ Q (x1 , x, x12 , x2 ; α1 , α2 ) ,

(3)

where µ, µ′ ∈ { ±1 }. 3. The system (2) possesses the tetrahedron property, i.e. x123 is independent of x: ∂x123 = 0. (4) x123 = x123 (x, x1 , x2 , x3 ; α1 , α2 , α3 ) =⇒ ∂x 2

x23 x3

α2 α1

A¯

x

x13

α3

¯ B

B

α1 α3

C α2

α2 C¯

α3 α3

x123

α1

x2

α2

A

α1

x12

x1

Figure 2: Equations on a Cube The results were three classes of discrete autonomous equations with these properties: the H equations, Q equations and the A equations. However the A equations can be transformed in particular cases of the Q equations through non-autonomous M¨ obius transformation. Therefore the A equations are usually removed from the general classification. After the introduction of the ABS equations Hietarinta tried to weaken its hypotheses. First in [34] he made a new classification with no assumption about the symmetry and the tetrahedron property. Therein he obtained the following new equation: x1 + e2 x2 + o2 x + e2 x12 + o2 = , (5) x + e1 x12 + o1 x1 + o1 x2 + e1 where ei and oi are constants. We will refer to this equation as the Hietarinta equation. It was later proved that the Hietarinta equation (5) embedded into a Z2 -lattice with the “trivial” embedding: x → un,m ,

x1 → un+1,m ,

x2 → un,m+1 ,

x12 → un+1,m+1 ,

(6)

is linearizable [44]. In a subsequent paper [35] J. Hietarinta made a new classification adding the D4 discrete symmetries (3) and he found three “new” equations, all linearizable. Releasing the hypothesis that every face of the cube carried the same equation, in [2] were presented some new equations without classification purposes. A complete classification in this extended setting was then accomplished by Boll in a series of papers culminating in his Ph.D. thesis [6–8]. In these papers the classification of all the consistent sextuples of partial difference equations on the quad graph, i.e. systems of the form (2), has been carried out. The only technical assumption used in by Boll is the tetrahedron property. The obtained equations may fall into three disjoint families depending on their bi-quadratics: hij =

∂Q ∂Q ∂2Q −Q , ∂yk ∂yl ∂yk ∂yl

Q = Q (y1 , y2 , y3 , y4 ; α1 , α2 ) ,

(7)

where we use a special notation for variables of Q, and the pair {k, l} is the complement of the pair {i, j} in {1, 2, 3, 4}. A bi-quadratic is called degenerate 3

if it contains linear factors of the form yi − c, where c is a constant, otherwise a bi-quadratic is called non-degenerate. The three families are characterized by how many bi-quadratics are degenerate: • Q-type equations: all the bi-quadratics are non-degenerate, • H 4 -type equations: four bi-quadratics are degenerate, • H 6 -type equations: all of the six bi-quadratics are degenerate. Let us notice that the Q family is the same as the one introduced in [1]. The H 4 equations are divided into two subclasses: rhombic and trapezoidal, depending on their discrete symmetries. We remark that the classification results of [6–8] hold locally in the sense that they relate to a single quadrilateral cell or a single cube displayed in Figures 1 and 2. The important problem of embedding these results into a two- or three-dimensional lattice, with preservation of the three-dimensional consistency condition, was discussed in [2, 57] by using the concept of a Black and White lattice. One way to solve this problem is to embed (1) into a Z2 -lattice with an elementary cell of size greater than one. In this case, the quad-equation (1) can be extended to a lattice, and the lattice equation becomes integrable or linearizable. To this end, following [6–8], we reflect the square with respect to the normal to its right and top sides and then complete a 2 × 2 lattice by again reflecting one of the obtained squares in the other direction. Such procedure is graphically described in Figure 3. x1

x

|Q

Q

x2

x2

x12 |Q

Q

x

x

x1

x

Figure 3: The “four stripe” lattice. It corresponds to constructing three equations obtained from (1) by flipping its arguments: Q = Q(x, x1 , x2 , x12 ; α1 , α2 ) = 0,

(8a)

|Q = Q(x1 , x, x12 , x2 ; α1 , α2 ) = 0, Q = Q(x2 , x12 , x, x1 ; α1 , α2 ) = 0,

(8b) (8c)

|Q = Q(x12 , x2 , x1 , x; α1 , α2 ) = 0.

(8d)

4

By paving the whole Z2 with such equations, we get a partial difference equation which can be in principle studied using known methods. Since a priori Q 6= |Q 6= Q 6= |Q, the obtained lattice will be a four stripe lattice, i.e. an extension of the Black and White lattice considered in [2, 37, 57]. This gives rise to lattice equations with two-periodic coefficients for an unknown function un,m , with (n, m) ∈ Z2 : (+) Fn(+) Fm Q(un,m , un+1,m , un,m+1 , un+1,m+1 ; α1 , α2 ) (+) + Fn(−) Fm |Q(un,m , un+1,m , un,m+1 , un+1,m+1 ; α1 , α2 ) (−) + Fn(+) Fm Q(un,m , un+1,m , un,m+1 , un+1,m+1 ; α1 , α2 )

(9)

(−) + Fn(−) Fm |Q(un,m , un+1,m , un,m+1 , un+1,m+1 ; α1 , α2 ) = 0,

where

1 ± (−1)k . (10) 2 This explicit formula was first presented in [24]. For more details on the construction of equations on the lattice from the single cell equations, we refer to [6–8, 57] and to the Appendix in [23]. We underline that this construction should be applied to a consistent system of quad equations (2) every time one of the equations do not possess the D4 discrete symmetries (3). Given an equation possessing the D4 discrete symmetries (3), it can be shown that it reduces to “trivial” embedding (6). A detailed study of all the lattice equations derived from the rhombic H 4 family, including the construction of their three-leg forms, Lax pairs, B¨acklund transformations and infinite hierarchies of generalized symmetries, has been presented in [57]. There are plenty of results about the Q and the rhombic H 4 equations. On the contrary, besides the CAC property little is known about the integrability features of the trapezoidal H 4 equations and of the H 6 equations. These equations where thoroughly studied in a series of papers [23–29] with some unexpected results. In [23] the Algebraic Entropy [4,36,54,55] of the trapezoidal H 4 and the H 6 equations was computed. The result of this computation showed that the rate of growth of all the trapezoidal H 4 and of all H 6 equations is linear. According to the Algebraic Entropy conjecture [15, 36] this fact implies the linearizability. Following the suggestions obtained in [24, 25] in [29] it was showed that the trapezoidal H 4 equations and all the H 6 equations are Darboux integrable [3]. Finally in [28] it was shown, applying a modification of the procedure presented in [17], that Darboux integrability provides the general solutions of these equations. Moreover in [25] it was showed that the three quad equations found in [35] were Darboux integrable. In this paper we study a new non-autonomous version of the Hietarinta equation (5) obtained using the prescriptions of [2, 6–8, 57]. We then show that this equation possesses properties analogous to those of the other three equations considered in [35]. First to construct this new equation we start from the polynomial version of the Hietarinta equation (5): (±)

Fk

=

Q = (x + e2 ) (x1 + o1 ) (x2 + e1 ) (x12 + o2 ) − (x + e1 ) (x1 + e2 ) (x2 + o2 ) (x12 + o1 ) .

5

(11)

This equation does not possess the D4 discrete symmetries (3), then in addition to the trivial embedding (6) we can consider the non-autonomous embedding given by formula (9). Applying formula (9) to the Hietarinta equation in polynomial form (11) we obtain the following non-autonomous lattice equation:      (1) (1) (1) (1) un,m+1 + En,m un+1,m+1 + On,m+1 un,m + En,m+1 un+1,m + On,m      (2) (2) (2) (2) un+1,m+1 + On+1,m , un+1,m + En+1,m un,m+1 + On,m = un,m + En,m (12) which is similar to the original equation (11), but with the four constants ei , oi replaced by the following four two-periodic functions of the lattice variables: (−) (−) (1) (+) (+) e2 + Fn(−) Fm o1 , e1 + Fn(−) Fm o2 + Fn(+) Fm En,m = Fn(+) Fm

(13a)

(2) (+) (+) (−) (−) En,m = Fn(+) Fm e1 + Fn(−) Fm e2 + Fn(+) Fm o2 + Fn(−) Fm o1 ,

(13b)

(1) On,m

+

(−) Fn(−) Fm e1 ,

(13c)

+

(−) Fn(−) Fm e1 ,

(13d)

(2) On,m

=

(+) Fn(+) Fm o1

=

(+) Fn(+) Fm o1

+

(+) Fn(−) Fm e2

+

(+) Fn(−) Fm o2

+

(−) Fn(+) Fm o2

+

(−) Fn(+) Fm e2

We will call equation (12) the non-autonomous Hietarinta equation. For the rest of this paper we will make a completeness study of the nonautonomous Hietarinta equation (12). In Section 2 we will show that the nonautonomous Hietarinta equation (12) possess linear growth in all directions. In Section 3 we will explain the linear growth by showing that the non-autonomous Hietarinta equation (12) is Darboux integrable with first order first integrals. In Section 4 we will present the generalized symmetries of the non-autonomous Hietarinta equation (12) of every order by using the first integrals and the so called Laplace operators. In Section 5 we use the first integrals to derive the general solution of the non-autonomous Hietarinta equation (12). In Section 6 we identify the non-autonomous Hietarinta equation (12) with a sub-case of the non-autonomous QV equation [26]. Then we show that the non-autonomous Hietarinta equation (12) can be reduced to the quad equation: vn,m vn+1,m+1 + vn+1,m vn,m+1 = 0.

(14)

which was found in [35]. In Section 7 we give some conclusion and we give an outlook on presented results. The main objective of this paper is to show how the Hietarinta equation (5), which is part of a sextuple consistent on the cube without tetrahedron property, behaves under the Boll’s embedding, becoming then the non-autonomous equation (12). In this sense our procedure is similar to those adopted in [37], where it was showed that different consistent embedding can produce equations with different properties. In particular we use this example to show all the interesting properties that were found for the trapezoidal H 4 and H 6 equations. Finally what we show is the non-autonomous Hietarinta equation (12) is not an independent equation, but can be reduced to other known equations. This shows the tetrahedron property is not an intrinsic property of the equation, but it is a property of the configuration. This fact poses several threats to the classification of quad equations Consistent Around the Cube without the tetrahedron property.

6

2

Algebraic Entropy

Algebraic Entropy is as a test of integrability for discrete systems. Given a bi-rational map, which can be an ordinary difference equation, a differential difference equation or even a partial difference equation, the basic idea is to examine the growth of the degree of its iterates, and extract a canonical quantity, which is an index of complexity of the map. This canonical quantity will be the algebraic entropy (or its avatar the dynamical degree) [4, 12, 15, 46, 53]. In [52, 54] the method was developed in the case of quad equations and then used as a classifying tool [36]. A slight generalization of the method was given in [23] where it was showed that non-autonomous equations can have more that a single sequence of degrees. Given a sequence of degrees obtained iterating a rational map: 1, d0 , d1 , . . . , dk , . . . ,

(15)

we define the algebraic entropy of this map to be: η = lim

k→∞

1 log dk . k

(16)

This quantity is canonical as it is invariant with respect to bi-rational transformations. Geometrically algebraic entropy is deeply linked with the structure of the singularities of a discrete system, and in some cases it can be computed resorting to this structure [13, 47, 51, 56]. Instead of computing the whole sequence of iterates which is clearly impossible, only a finite number of iterates is computed. Then are used some tools like generating functions [39] or discrete derivatives in order to obtain the asymptotic behaviour of the series [19, 20].

(−, −) (+, −)

(−, +)

(+, +)

Figure 4: Principal growth directions. The classification of lattice equations based on the algebraic entropy test is:

7

Linear growth: The equation is linearizable. Polynomial growth: The equation is integrable. Exponential growth: The equation is chaotic. We have performed the Algebraic Entropy analysis in the principal growth directions [54], shown in Figure 4, of the non-autonomous Hietarinta equation (12) to identify its behaviour. To this end we used the SymPy [50] module ae2d.py [21, 22]. We found that the non-autonomous Hietarinta equation (12) is isotropic, i.e. its sequence of degrees is the same in every direction, and despite being non-autonomous and two-periodic it possess a single sequence of degrees given by: 1, 3, 5, 7, 9, 11, 13, . . . . (17) The sequence (17) is asymptotically (in this case exactly) fitted by: dn = 2n + 1,

∀n ∈ N,

(18)

and the algebraic entropy from (16) is clearly zero. Since the growth is linear we expect the non-autonomous Hietarinta equation (12) to be linearizable. In the next Section we will show why the linearization arises.

3

Darboux integrability

A quad-equation, maybe non-autonomous: Qn,m (un,m , un+1,m , un,m+1 , un+1,m+1 ) = 0,

(19)

is Darboux integrable if two independent first integrals, one containing only shifts in the first direction and the other containing only shifts in the second direction exist. This means that there exist two functions: W1 = W1,n,m (un+l1 ,m , un+l1 +1,m , . . . , un+k1 ,m ), W2 = W2,n,m (un,m+l2 , un,m+l2 +1 , . . . , un,m+k2 ),

(20a) (20b)

where l1 < k1 and l2 < k2 are integers, such that the relations where Tn hn,m = hn+1,m , Tm hn,m = hn,m+1 , and Id hn,m = hn,m : (Tn − Id)W2 = 0,

(21a)

(Tm − Id)W1 = 0

(21b)

hold true identically on the solutions of (19). The number ki −li , where i = 1, 2, is called the order of the first integral Wi . Any Darboux integrable equation is linearizable [3]. Indeed equation (21) implies that the following two transformations: un,m → u˜n,m = W1,n,m ,

(22a)

un,m → uˆn,m = W2,n,m ,

(22b)

bringing the quad-equation (19) into two trivial linear equations: u˜n,m+1 − u˜n,m = 0, 8

(23a)

uˆn+1,m − uˆn,m = 0.

(23b)

It follows that any Darboux integrable equation is linearizable in two different ways, i.e. using transformation (22a) bringing to (23a) or using the transformation (22b) bringing to (23b). Methods for calculating first integrals of non-autonomous quad equations (19) with two-periodic coefficients were given in [18, 29]. In particular in [29] was presented a new algorithm that relies on the fact that in the case of nonautonomous quad equations (19) with two-periodic coefficients we can, in general, represent the first integrals in the form: (−,+)

(+,+)

(+) + Fn(−) Fm Wi

(+,−)

(−) + Fn(−) Fm Wi

(+) Wi = Fn(+) Fm Wi

(−,−)

(−) + Fn(+) Fm Wi (±)

(24) ,

(±,±)

are functions. where Fk are given by (10) and the Wi Applying the algorithm presented in [29] we find that the non-autonomous Hietarinta equation (12) possesses two first order first integrals in both directions: (α1 )

W1 = α1 W1 W2 =

(α ) α2 W2 2

(β1 )

+ β1 W1 +

,

(25a)

(β ) β2 W2 2 ,

(25b)

where: (α1 )

W1

(β1 )

W1

(α2 )

W2

(β2 )

W2

4

(un,m + e2 ) (un+1,m + o1 ) (un,m + e1 ) (un+1,m + e2 ) (u n,m + o2 ) (un+1,m + o1 ) (−) + Fn(+) Fm (un,m + e1 ) (un+1,m + o2 ) (+) (un,m + o1 ) (un+1,m + e2 ) = Fn(−) Fm (un,m + e2 ) (un+1,m + e1 ) (u n,m + o1 ) (o2 + un+1,m ) (−) + Fn(−) Fm , (un,m + o2 ) (un+1,m + e1 ) (+) (un,m + e2 ) (un,m+1 + e1 ) = Fn(+) Fm (un,m + e1 ) (un,m+1 + o2 ) (+) (un,m + e2 ) (o1 + un,m+1 ) + Fn(−) Fm (un,m + o1 ) (un,m+1 + o2 ) (−) (un,m + o2 ) (un,m+1 + e1 ) = Fn(+) Fm (un,m + e1 ) (un,m+1 + e2 ) (−) (un,m + o2 ) (un,m+1 + o1 ) + Fn(−) Fm . (un,m + o1 ) (un,m+1 + e2 ) (+) = Fn(+) Fm

(26a)

(26b)

(26c)

(26d)

Generalized symmetries

A vector field of the form:
ˆ = gn,m uD ∂un,m , X n,m

uD n,m = {un+i,m+j }i=l1 ,...,k1 ,j=l2 ,...,k2 , 9

(27)

where l1 < k1 and l2 < k2 is said to be a generalized symmetry for the quad equation (19) if its discrete prolongation ˆ = gn,m ∂un,m + Tn gn,m ∂un+1,m prD X + Tm gn,m ∂un,m+1 + Tn Tm gn,m ∂un+1,m+1 ,

(28)

∂Qn,m ∂Qn,m + Tn gn,m ∂un,m ∂un+1,m ∂Qn,m ∂Qn,m + Tm gn,m + Tn Tm gn,m = 0, ∂un,m+1 ∂un+1,m+1

(29)

is such that: gn,m

identically on the solutions of (19) [16]. Symmetries of this kind with ki = −li = 1, i.e. three-point generalized symmetries were first considered in [45]. Moreover three-point generalized symmetries were threated in [40, 41], and the general case was discussed in [16]. In [16] it was also proved that quad equations (19) do not posses “mixed” symmetries, i.e. the function gn,m in (27), known as the characteristic of the generalized symmetry, is the sum of two simpler functions depending only on variables shifted only in one direction:
(1) (2) gn,m uD n,m = gn,m (un+li ,m , . . . , un+k1 ,m )+gn,m (un,m+l2 , . . . , un,m+k2 ) . (30)

The relationship between generalized symmetries and Darboux integrability is well known in the continuous case [59] and in the discrete case have been investigated in [3, 48, 49]. As conjectured in [25] and proved in [49], a quad equation is Darboux integrable if and only if it possesses generalized symmetries depending on arbitrary function in the following form: (1) = R(1) (Fn (Tnp1 W1 , . . . , Tnq1 W1 )) , gn,m

(31a)

(2) gn,m

(31b)

=R

(2)

q2 p2 W2 )) , W2 , . . . , Tm (Gm (Tm

where pi < qi and R(i) are operators of the form: R

(1)

=

h1 X

r=j1

R(2) =

h2 X

r=j2


λr un+l′i ,m , . . . , un+k1′ ,m Tnr ,

(32a)


r µr un,m+l′2 , . . . , un,m+k2′ Tm ,

(32b)

with ji < hi and li′ < ki′ . The operators R(i) (32) are called Laplace operators. It is then easy to show that the non-autonomous Hietarinta equation possesses the following two-point generalized symmetries, i.e. such that ki = 1, li = 0 in (27):     (α ) (+) (−) (un,m + e1 ) (un,m + o2 ) (1) (+) (+) (un,m + e1 ) (un,m + e2 ) Fn(1) W1 1 + Fn Fm gn,m = Fn Fm e1 − e2 e1 − o2     (u + o2 ) (o1 + un,m ) (u + e ) (u + o ) n,m n,m 2 n,m 1 (β ) (−) (−) (−) (+) G(1) W1 1 , − Fn Fm + Fn Fm n e2 − o1 o1 − o2 (33a)

10

(2) gn,m

"

# (+) (+)   Fn Fm (un,m + e1 ) (un,m + e2 ) (α ) (2) (−) (+) (un,m + e2 ) (un,m + o1 ) = Fm W2 1 − Fn Fm e1 − e2 e2 − o1     (α1 ) (−) (un,m + o2 ) (un,m + o1 ) (−) (un,m + e1 ) (un,m + o2 ) , W G(2) + Fn(−) Fm + Fn(+) Fm m 2 (e1 − o2 ) o1 − o2 (33b) (i)

(i)

where Fk and Gk are arbitrary functions of their arguments. This implies that we have two multipliticative Laplace operators R(i) (32) in each direction: (+) R(1,α1 ) = Fn(+) Fm

(+) R(1,β1 ) = Fn(−) Fm

(+)

R(2,α2 ) =

(un,m + e1 ) (un,m + e2 ) (−) (un,m + e1 ) (un,m + o2 ) + Fn(+) Fm e1 − e2 e1 − o2 (34a) (un,m + e2 ) (un,m + o1 ) (−) (un,m + o2 ) (o1 + un,m ) − Fn(−) Fm e2 − o1 o1 − o2 (34b)

(+)

Fn Fm (un,m + e1 ) (un,m + e2 ) (+) (un,m + e2 ) (un,m + o1 ) − Fn(−) Fm e1 − e2 e2 − o1 (34c)

(−) R(2,β2 ) = Fn(+) Fm

(un,m + e1 ) (un,m + o2 ) (−) (un,m + o2 ) (un,m + o1 ) , + Fn(−) Fm (e1 − o2 ) o1 − o2 (34d)

and then that the generalized symmetry of arbitrary order Ni = qi − pi are given by (31):   (α ) (α ) (1) (35a) gn,m = R(1,α1 ) Fn(1) Tnp1 W1 1 , . . . , Tnq1 W1 1   (β ) (β ) Tnp1 W1 1 , . . . , Tnq1 W1 1 + R(1,β1 ) G(1) n   (α ) (α ) (2) (35b) gn,m = R(2,α2 ) Fn(2) Tnp2 W2 2 , . . . , Tnq2 W2 2   (β ) (β ) Tnp2 W2 2 , . . . , Tnq2 W2 2 + R(2,β2 ) G(2) n (i)

(i)

where Fk (x1 , . . . , xNi ) and Gk (x1 , . . . , xNi ) are arbitrary functions of their arguments.

5

General solutions

In this Section we construct the general solution of the non-autonomous Hietarinta equation (12) using the method presented in [28, 29], a modification of the procedure presented in [17]. By general solution we mean a representation of the solution of a quad equation (19) in terms of the right number of arbitrary functions of one lattice variable n or m. Since quad equations are the discrete analogue of second order hyperbolic partial differential equations, the general solution must contain an arbitrary function in the n direction and another one in the m direction. To obtain the solution we will need only the W1 integrals we derived in Section 3 and the fact that the relation (21b) implies W1 = ξn with ξn an 11

arbitrary function of n. The equation W1 = ξn is an ordinary difference equation in the n direction depending parametrically on m. Then from every W1 integral we can derive two different ordinary difference equations, one corresponding to m even and one corresponding to m odd. In both the resulting equation we can get rid of the two-periodic terms by considering the cases n even and n odd and using the definitions: u2k,2l = vk,l , u2k+1,2l = wk,l , u2k,2l+1 = yk,l , u2k+1,2l+1 = zk,l .

(36a) (36b)

This transformation brings both equations into a system of coupled difference equations. Apply the even/odd splitting (36) of the lattice variables to describe a general solution we will need two arbitrary functions in both directions, i.e. we will need a total of four arbitrary functions. We assume without loss of generality α1 = β1 = 1 in (25a): (un,m + e2 ) (un+1,m + o1 ) (un,m + e1 ) (un+1,m + e2 ) (−) (un,m + o2 ) (un+1,m + o1 ) + Fn(+) Fm (un,m + e1 ) (un+1,m + o2 ) (u n,m + o1 ) (un+1,m + e2 ) (+) + Fn(−) Fm (un,m + e2 ) (un+1,m + e1 ) (−) (un,m + o1 ) (o2 + un+1,m ) = ξn . + Fn(−) Fm (un,m + o2 ) (un+1,m + e1 ) (+) Fn(+) Fm

(37)

Then we treat the cases m even and odd separately. Case m = 2l: In this case equation (37) becomes: (un,2l + e2 ) (un+1,2l + o1 ) (un,2l + e1 ) (un+1,2l + e2 ) (un,2l + o1 ) (un+1,2l + e2 ) + Fn(−) = ξn . (un,2l + e2 ) (un+1,2l + e1 ) Fn(+)

(38)

We can then separate the even and odd part in n of equation (38) and apply the transformation (36a). We obtain the following system: (vk,l + e2 ) (wk,l + o1 ) = ξ2k , (vk,l + e1 ) (wk,l + e2 ) (wk,l + o1 ) (vk+1,l + e2 ) = ξ2k+1 . (wk,l + e2 ) (vk+1,l + e1 )

(39a) (39b)

This system is linear and equation (39a) is not a difference equation, but defines wk,l in terms of vk,l : wk,l = −

(−o1 + ξ2k e2 ) vk,l − e2 o1 + ξ2k e1 e2 . (−1 + ξ2k ) vk,l − e2 + ξ2k e1

(40)

Inserting wk,l from (40) into equation (39b) we obtain that vk,l solves the following discrete Riccati equation: vk+1,l =

(−ξ2k e2 + ξ2k+1 e1 ) vk,l − ξ2k e1 e2 + ξ2k+1 e1 e2 . (ξ2k − ξ2k+1 ) vk,l − ξ2k+1 e2 + ξ2k e1 12

(41)

Equation (41) is linearized through the M¨ obius transformation: vk,l = −e2 −

e2 , Vk,l

(42)

and yields the following linear equation for Vk,l : Vk+1,l =

ξ2k ξ2k+1

Vk,l −

e2 (ξ2k − ξ2k+1 ) . ξ2k+1 (e1 − e2 )

(43)

Exploiting the arbitrariness of ξ2k we introduce a new arbitrary function ak by defining: ak ξ2k = ξ2k+1 . (44) ak+1 Then equation (43) becomes: ak+1 Vk+1,l = ak Vk,l −

e2 (ak − ak+1 ) . e1 − e2

(45)

Equation (45) is a total difference hence its solution is: Vk,l =

e2 αl + . ak e1 − e2

(46)

Then inserting (46) into (42) and (40) we obtain the solution of the system (39): e2 (αl (e1 − e2 ) + ak e1 ) , αl (e1 − e2 ) + ak e2 e2 (ξ2k+1 αl e1 + o1 ak+1 − ξ2k+1 e2 αl ) =− . (ak+1 − ξ2k+1 αl ) e2 + ξ2k+1 αl e1

vk,l = −

(47a)

wk,l

(47b)

Case m = 2l + 1: In this case equation (37) becomes: (un,2l+1 + o2 ) (un+1,2l+1 + o1 ) (un,2l+1 + e1 ) (un+1,2l+1 + o2 ) (un,2l+1 + o1 ) (o2 + un+1,2l+1 ) + Fn(−) = ξn . (un,2l+1 + o2 ) (un+1,2l+1 + e1 ) Fn(+)

(48)

We can then separate the even and odd part in n of equation (48) and apply the transformation (36b). We obtain the following system: ak (yk,l + o2 ) (zk,l + o1 ) ξ2k+1 , = (yk,l + e1 ) (zk,l + o2 ) ak+1 (zk,l + o1 ) (yk+1,l + o2 ) = ξ2k+1 , (zk,l + o2 ) (yk+1,l + e1 )

(49a) (49b)

where we used the definition of ξ2k (44). This system is linear and equation (49a) is not a difference equation, but defines zk,l in terms of yk,l : zk,l =

−ak o2 (yk,l + e1 ) ξ2k+1 + o1 ak+1 (yk,l + o2 ) . ak (yk,l + e1 ) ξ2k+1 − ak+1 (yk,l + o2 ) 13

(50)

Inserting zk,l from (50) into equation (49b) we obtain that yk,l solves the following discrete Riccati equation: yk+1,l =

e1 (yk,l + o2 ) ak+1 − ak o2 (yk,l + e1 ) . − (yk,l + o2 ) ak+1 + ak (yk,l + e1 )

(51)

Equation (51) is linearized through the M¨ obius transformation: yk,l = −o2 −

o2 , Yk,l

(52)

which yields the following equation for Yk,l : ak+1 Yk+1,l = ak Yk,l −

o2 (ak − ak+1 ) − o2 . e1

(53)

Equation (53) is already a total difference, so its solution is: Yk,l =

o2 βl + . ak e1 − o2

(54)

Then inserting (54) into (52) and (50) we obtain the solution of the system (49): o2 (βl (e1 − o2 ) + ak e1 ) , βl (e1 − o2 ) + ak o2 o2 (ξ2k+1 βl e1 − βl ξ2k+1 o2 + o1 ak+1 ) =− . (ak+1 − βl ξ2k+1 ) o2 + ξ2k+1 βl e1

yk,l = −

(55a)

zk,l

(55b)

The solution of the non-autonomous Hietarinta equation (12) is then given by formulæ (47) and (47). The four arbitary functions are ξ2k+1 , ak , αl and βl . It can be checked that this is the general solution by inserting it into (12)

14

6

Relation to other quad equations

In [26] it was proved that there exists a non-autonomous, two-periodic generalization of the QV equation [55] given by: a1 un,m un+1,m un,m+1 un+1,m+1 i h n m n+m + a2,0 − (−1) a2,1 − (−1) a2,2 + (−1) a2,3 un,m un,m+1 un+1,m+1 h i n m n+m + a2,0 + (−1) a2,1 − (−1) a2,2 − (−1) a2,3 un+1,m un,m+1 un+1,m+1 i h + a2,0 + (−1)n a2,1 + (−1)m a2,2 + (−1)n+m a2,3 un,m un+1,m un+1,m+1 i h n m n+m + a2,0 − (−1) a2,1 + (−1) a2,2 − (−1) a2,3 un,m un+1,m un,m+1 m

+ [a3,0 − (−1) a3,2 ] un,m un+1,m + [a3,0 + (−1)m a3,2 ] un,m+1 un+1,m+1 i h n+m + a4,0 − (−1) a4,3 un,m un+1,m+1 h i n+m + a4,0 + (−1) a4,3 un+1,m un,m+1 n

+ [a5,0 − (−1) a5,1 ] un+1,m un+1,m+1 n + [a5,0 + (−1) a5,1 ] un,m un,m+1 h i n m n+m + a6,0 + (−1) a6,1 − (−1) a6,2 − (−1) a6,3 un,m i h + a6,0 − (−1)n a6,1 − (−1)m a6,2 + (−1)n+m a6,3 un+1,m i h n m n+m + a6,0 + (−1) a6,1 + (−1) a6,2 + (−1) a6,3 un,m+1 h i n m n+m + a6,0 − (−1) a6,1 + (−1) a6,2 − (−1) a6,3 un+1,m+1 +a7 = 0

(56) This equation is integrable according to the Algebraic Entropy test, possessing quadratic growth of the degrees of the iterates and contains as particular cases all the equations coming from the Boll’s classifications, i.e. the rhombic H 4 equations, the trapezoidal H 4 equations, the H 6 equations and the Q equations. Upon the substitution a2,1 = a2,2 = a2,3 = a3,2 = a4,3 = a5,1 = a6,1 = a6,2 = a6,3 = 0 the non-autonomous QV equation reduces to the original QV equation presented in [55]. For this reasons equation (56) has been called the non-autonomous, or two-periodic, QV equation. It is just a matter of computation to show that the original autonomous Hietarinta equation (5) is not a sub-case of the autonomous QV equation, but that the non-autonomous Hietarinta equation is a sub-case of the non-autonomous QV equation (56) with the following values of the coefficients: a2,0 = 0, a2,1 = e2 − o2 , a2,2

a1 = 0, = e1 − o1 , a2,3 = e1 + o1 − o2 − e2 ,

a3,0 = −(e2 − o2 )(e1 − o1 ), a3,2 = −(e2 + o2 )(e1 − o1 ), a5,0

a4,0 = 0, a4,3 = 2(e2 o2 − o1 e1 ), = (e2 − o2 )(e1 − o1 ), a5,1 = −(e2 − o2 )(o1 + e1 ), 15

(57a) (57b) (57c) (57d) (57e)

a6,0 = 0, a6,1 = −o1 e1 (e2 − o2 ), a6,2 = −(e1 − o1 )e2 o2 , a6,3 = o2 e1 o1 + e2 o1 e1 − e2 o2 e1 − e2 o2 o1 , a7 = 0.

(57f) (57g)

Since the non-autonomous QV equation (56) contains all the Boll’s class, which are equations consistent on the cube possessing the tetrahedron property, this result suggests that the non-autonomous Hietarinta equation can be mapped into a quad equation which is consistent on the cube and possesses the tetrahedron property. Indeed it can be shown that the non-autonomous, two-periodic M¨ obius transformation: (+)

un,m = −

(−)

(+)

(−)

(+)

(−)

(Fm e2 + Fm o2 )vn,m + Fn e1 + Fn (Fm − Fm )o1 (+)

(+)

vn,m + Fm + (Fn

(−)

(−)

− Fn )Fm

maps the non-autonomous Hietarinta equation (12) into equation (14). Equation (14) is a particular case of the 1 D4 equation from [6–8, 23]:   δ1 Fn(−) un,m un,m+1 + Fn(+) un+1,m un+1,m+1   (−) (+) un,m un+1,m + Fm un,m+1 un+1,m+1 + δ2 Fm

(58)

(59)

+ un,m un+1,m+1 + un+1,m un,m+1 + δ3 = 0,

with δ1 = δ2 = δ3 = 0. So we have that the non-autonomous Hietarinta equation (12) is part of the Boll’s classification! This result is clearly unexpected, since in its original derivation the Hietarinta equation (5) arises from a sextuple of equations (2) with no tetrahedron property, which is the crucial hypothesis in the derivation of Boll’s classification. This is of course a consequence of the fact that the tetrahedron property is a property of the sextuple of equation (2) rather then a property of the equation itself. Moreover this result suggests that to the same side equation on the single cell many different systems (2) with different properties can be attached. Since the non-autonomous Hietarinta equation (12) is linearizable its Lax pairs obtained both from sextuple in [35] and from the sextuple in [6–8] (obtained reversing the transformation (58)) should be both fake. This cannot give much information, but it is possible at least that also integrable equations can have Lax pairs coming from sextuple without tetrahedron property. Given a quad equation with such properties it will be an interesting issue to investigate the relationship between these Lax pairs as it was done for the continuous Painlev´e I and II equations [38].

7

Conclusions

In this paper we presented a new approach to the Hietarinta equation (5). From the consideration that the Hietarinta equation possesses the property of the Consistency Around the Cube [35], but do not possesses the discrete symmetries of the square (3) we applied to it the Boll’s construction developed in [2, 6–8, 37, 57]. The result was a seemingly new non-autonomous, two-periodic quad equation which we called the non-autonomous Hietarinta equation (12). In this sense we “reconstructed” the Hietarinta equation (5), since from the same singlecell equation instead of using the trivial embedding (6) we adopted a different one resulting in a different equation on the Z2 lattice. 16

We devoted the rest of the paper to the study of the integrability properties of this equation. The results of Sections 2–5 can be seen as a nice example of the properties that were found for the two classes of linearizable equations belonging to Boll’s classifications, namely the trapezoidal H 4 equations and the H 6 equations [23–29]. Indeed we proved in Section 6 that the non-autonomous Hietarinta equation can be mapped into a particular case of the 1 D4 equation, an equation belonging to the H 6 class. We underline that the identification of the non-autonomous Hietarinta equation (12) with equation (14) is possible only in the extended framework of the Consistency Around the Cube given in [2, 6–8], since in order to transform the non-autonomous Hietarinta equation (12) into (14) the non-autonomous M¨ obius transformation (58) must be used. In the framework of [1] this is not possible since transformations of this kind are not allowed. Therefore the construction carried out in this paper is “necessary” to obtain this result. In this work we prove that the problem of the embedding of single cell quad equations possessing the Consistency Around the Cube is crucial. As already discussed in [37] there might exist multiple embeddings preserving on the whole lattice the Consistency Around the Cube. To understand how many and what they are is then a relevant task. We conjecture that in the procedure of extension might be essential to form bi-quadratic patterns [6–8]. To form a bi-quadratic pattern means that, given a quad equation on the single cell (1), to construct an equation on the Z2 lattice in a way such that the bi-quadratics (7) pertaining to the sides of neighboring cells are of the same type. However preserving the bi-quadratic patterns might be ineffectual in the case of the H 6 due to the lack of a well-defined “Black/White” assignment to the vertexes of the elementary cell. Moreover as a byproduct of the existence of the transformation (58) we have that an equation coming from a sextuple possessing the tetrahedron property can be linked to an equation coming from a sextuple without tetrahedron property. As noted in Section 6 this shows that, in principle, also other known quad equations possessing the Consistency Around the Cube can arise as well from sextuples without tetrahedron property. What kind of information these non-tetrahedron sextuples can give is for the moment unknown. Finally we point out that whereas it was very easy to prove the Darboux integrability for the non-autonomous Hietarinta equation (12), if an analogous result holds for the Hietarinta equation (5) with the trivial embedding (6) it is not known. It was conjectured in [21] that the the Hietarinta equation (5) with the trivial embedding (6) is Darboux integrable with first integrals of order greater than two, but technical difficulties prevent a full proof of this statement.

Acknowledgements We would like to thank prof. Decio Levi for the many interesting and fruitful discussion during the preparation of this paper. GG is supported by INFN IS-CSN4 Mathematical Methods of Nonlinear Physics.

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