Search for integrable two-component versions of the lattice equations in the ABS-list
aa r X i v : . [ n li n . S I] D ec Search for integrable two-component versions of thelattice equations in the ABS-list
Jarmo HietarintaDepartment of Physics and AstronomyUniversity of Turku, FIN-20014 Turku, FinlandDecember 7, 2020
Abstract
We search and classify two-component versions of the quad equations in the ABSlist, under certain assumptions. The independent variables will be called y, z and inaddition to multilinearity and irreducibility the equation pair is required to have thefollowing specific properties: (1) The two equations forming the pair are related by y ↔ z exchange. (2) When z = y both equations reduce to one of the equations inthe ABS list. (3) Evolution in any corner direction is by a multilinear equation pair.One straightforward way to construct such two-component pairs is by taking someparticular equation in the ABS list (in terms of y ), using replacement y ↔ z for someparticular shifts, after which the other equation of the pair is obtained by property(1). This way we can get 8 pairs for each starting equation. One of our main resultsis that due to condition (3) this is in fact complete for H1, H3, Q1, Q3. (For H2 wehave a further case, Q2, Q4 we did not check.) As for the CAC integrability test,for each choice of the bottom equations we could in principle have 8 possible side-equations. However, we find that only equations constructed with an even numberof y ↔ z replacements are possible, and for each such equation there are two setsof “side” equation pairs that produce (the same) genuine B¨acklund transformationand Lax pair. Within the topic of integrable discrete systems [14], equations that can be definedon a single quadrilateral of the Cartesian Z × Z lattice have been studied in greatdetail. One common equation type is defined by the following: Definition 1 (Acceptable one-component quad equations) .1.1
The equation depends on all corner variables of the elementary quadrilateral.
The equation is affine linear in each corner variable. n,m u n +1 ,m pu n,m +1 q u n +1 ,m +1 Figure 1: Corner variables on an elementary quadrilateral.
The equation is irreducible.
Uniformity: Every quadrilateral in the plane carries the same equation (de-pending on corresponding corner variables)
The geometric description is in Figure 1: subscript m labels the points in thevertical direction and n in the horizontal direction. The lattice parameters p, q areassociated with horizontal and vertical directions, respectively. In practice we useshorthand notation in which a shift in the n -direction is indicated by a tilde, andin the m -direction by a hat u n,m = u, u n +1 ,m = e u, u n,m +1 = b u, u n +1 ,m +1 = be u. If the conditions in Definition 1 are satisfied one can define evolution startingfrom staircase- or corner-like initial conditions.For lattice equations a necessary property for integrability is “MultidimensionalConsistency”. It means that the equations can be consistently extended into higherdimensions, which is related to the existence of a hierarchy of integrable continuousequations ([14], Sec 3.2). For 2D quad equations it means in practice Consistency-Around-a-Cube (CAC), that is, the original quad equation can be put on an 3Dcube in a consistent way. Consider Figure 2 and assume that the original 2D latticeequation is on the bottom of the cube. Certain modifications of that equation arethen placed on the back and left sides. Typically these equations are obtained bycyclic permutation: e → b → → e p → q → r → p n → m → k → n (1)where we have also introduced a bar to denote shift in the vertical direction, wheresteps are counted by k : u n,m,k +1 = u . The equations on the opposing sides areobtained by the perpendicular shift. We then have 6 equationsbottom: Q ( u, e u, b u, be u ; p, q ) = 0 . top: Q ( u, e u, b u, be u ; p, q ) = 0 , (2a)back: Q ( u, b u, u, b u ; q, r ) = 0 , front: Q ( e u, be u, e u, be u ; q, r ) = 0 , (2b)left: Q ( u, u, e u, e u ; r, p ) = 0 , right: Q ( b u, b u, be u, be u ; r, p ) = 0 . (2c)The consistency problem arrives as follows: Take u, e u, b u, u as initial values,then from bottom, back and left equations we can compute the values of be u , b u and e u , e u be u b uu e u be u b ur, kp, n q, m Figure 2: The consistency cube. respectively. After these are substituted into the top, front and right equations weget independently 3 values for be u and these values must be the same. This introducessevere conditions.Several isolated examples of integrable quad-equations were found already inthe 1980s by considering continuous equations and the permutability property oftheir B¨acklund transformations ([14], Sec. 2.4-5). A major development in this fieldwas the classification of integrable quad-equations by Adler, Bobenko and Suris[1], under the assumptions of D4 symmetry and the “tetrahedron property”. (Thetetrahedron property was essential in the classification work. It states that the triplyshifted quantity computed in three ways from (2) does not depend on u .) The resultof this classification is the so-called “ABS-list”, its main components being the Hand Q lists: H-list H : ( u − be u )( b u − e u ) = p − q (3a)H : ( u − be u )( e u − b u ) = ( p − q )( u + e u + b u + be u ) + p − q (3b)H : p ( u e u + b u be u ) − q ( u b u + e u be u ) = δ ( p − q ) (3c) Q-list :Q : p ( u − b u )( e u − be u ) − q ( u − e u )( b u − be u ) = δ pq ( q − p ) (4a)Q : p ( u − b u )( e u − be u ) − q ( u − e u )( b u − be u ) + pq ( p − q )( u + e u + b u + be u )= pq ( p − q )( p − pq + q ) (4b)Q : p (1 − q )( u b u + e u be u ) − q (1 − p )( u e u + b u be u )= ( p − q ) (cid:18) ( b u e u + u be u ) + δ (1 − p )(1 − q )4 pq (cid:19) (4c)Q from [11] : sn ( α )( u e u + b u be u ) − sn ( β )( u b u + e u be u ) − sn ( α − β )( e u b u + u be u )+ k sn ( α ) sn ( β ) sn ( α − β )(1 + u e u b u be u ) = 0 . (4d) owever, it is well known that there are other CAC-compatible equations if someconditions used by ABS are relaxed, for example be u u − e u b u = 0, which breaks thetetrahedron condition.One of the assumptions used to generate the ABS list was that equations onopposing sides are related by the corresponding shift, as can be seen in (2). Thisassumption was relaxed in the work of Boll [3, 4], while still keeping the tetrahedronproperty. On the other hand, in the classification of Hietarinta [13] the tetrahedronassumption was not made but the search was restricted to equations that werequadratic homogeneous.When we have a set of consistent equations on the sides of the cube one can usethe “side” equations to construct a Lax pair or a B¨acklund transformation, whichshould generate the “bottom” equation (see e.g., [14], Sec. 3.3). But in [13] it wasfound that many equations can pass the CAC test without being integrable, in otherwords, sometimes the Lax pair generated from the side equations is trivial. Thismeans that CAC is only a necessary test and must be verified by the existence of agenuine Lax or B¨acklund pair. Multi-component quad equations have also been studied and various types of equa-tions have been proposed. For example discrete versions of the Boussinesq equationshave been proposed, often in three-component form [12], but after eliminating onevariable one obtains in some cases a two component form still on the elementaryquadrilateral (e.g., [18], (4.8)). Several two-component equations were also pro-posed in [8]. However, none of these equations satisfy the exchange conditions in Definition 2 below.Furthermore in this paper we restrict our attention to equations that can beconsidered as multi-component generalizations of the equations in the ABS-list.One such equation was given in [5] (table 5, with name change x → y, y → z ) ( ( y − be y )( e z − b z ) − p + q = 0 , ( z − be z )( e y − b y ) − p + q = 0 . (5)Clearly the limit z → y (for any shift: none, tilde, hat, tilde-hat) takes both equa-tions to H1. Furthermore the equations are related by y ↔ z exchange for all shifts.Also note that in the first equation the once shifted variables have been changed by y → z .Is this equation integrable? At least it should have the CAC property. With (5)as the bottom equation we have to choose the side equations. It would be natural o try the cyclic rule (1), which producesbottom: ( ( y − be y )( e z − b z ) + q − p = 0 , ( z − be z )( e y − b y ) + q − p = 0 , (6a)back: (cid:26) ( y − b y )( b z − z ) + r − q = 0 , ( z − b z )( b y − y ) + r − q = 0 , (6b)left: (cid:26) ( y − e y )( z − e z ) + p − r = 0 , ( z − e z )( y − e y ) + p − r = 0 . (6c)This indeed passes the CAC test with the triply shifted variables being be y = p e y ( y − b y ) + q b y ( − y + e y ) + ry ( b y − e y ) p ( y − b y ) + q ( − y + e y ) + r ( b y − e y ) , (7a) be z = p e z ( z − b z ) + q b z ( − z + e z ) + rz ( b z − e z ) p ( z − b z ) + q ( − z + e z ) + r ( b z − e z ) . (7b)Note that this has the tetrahedron property (no unshifted y, z ) and that the y and z variables are separated in the final formulae.The above construct in which variables with an odd number of shifts are ex-changed, was described already in [2] as Toeplitz extension. It was generalized toall the equations in the ABS list in [9] by the same rule: exchanging the singlyshifted variables, see also [16]. This approach was developed further by includingother replacements by symmetry arguments [19]. Another approach in derivingmulti-component versions of the ABS list was given in [17] where such equationswere derived by from the star-triangle relations.One result in [19] was the following two-component version of H1 (Eqs. (2.25),(2.28),(2.29)) consisting ofbottom: ( ( z − be y )( e z − b y ) + q − p = 0 , ( y − be z )( e y − b z ) + q − p = 0 , (8a)back: (cid:26) ( z − b y )( b y − z ) + r − q = 0 , ( y − b z )( b z − y ) + r − q = 0 , (8b)left: (cid:26) ( y − e y )( y − e y ) + p − r = 0 , ( z − e z )( z − e z ) + p − r = 0 , (8c)Here the bottom and back equations are related by cyclic permutation, but the leftequation is of entirely different type, in fact separating the y and z variables. Thispeculiar combination passes the CAC test, and the triply shifted variables are be y = p e z ( b y − z ) + q b y ( z − e z ) + rz ( − b y + e z ) p ( b y − z ) + q ( z − e z ) + r ( − b y + e z ) , (9a) be z = p e y ( y − b z ) + q b z ( − y + e y ) + ry ( − e y + b z ) p ( y − b z ) + q ( − y + e y ) + r ( − e y + b z ) , (9b)and they have the tetrahedron property. Classification of two-component generaliza-tions of the ABS list
The puzzling triplet (8) suggests that there may be interesting phenomena specificfor two-component equations. The purpose of this paper is to search and classifysuch equations.
Since the fully generic case of the problem is too hard to tackle we restrict ourattention to equation pairs with the following properties:
Definition 2 (Acceptable two-component quad equations) .2.1
Both equations of the pair are affine multilinear and irreducible.
Exchange rule: The two equations that form the pair are related by the ex-change rule y ↔ z, e y ↔ e z, b y ↔ b z, be y ↔ be z . Evolution: From the pair of equations one can solve for any of the cornervariable pairs { y, z } , { e y, e z } , { b y, b z } , { be y, be z } Strong multilinearity: When any resolved variable pair is written as a pair ofpolynomial equations, the polynomials are again multilinear and irreducible.
Remarks: • We use • to indicate when the exchange rule has been applied, That is,if B is obtained from A by the exchange rule we write B = • A . Obviously( • A ) • = A . • Multilinearity does not imply unique evolution. Consider the pair z be y + b y b z + 2 b y e z + e z e y = 0 ,y be z + b y b z + 2 b z e y + e z e y = 0 . As given it is resolved for { be y, be z } and for { y, z } . However, if one tries solve for { e y, e z } or { b y, b z } there will be square roots and therefore evolution in the NWor SE direction is not uniquely determined. Note also that the one-componentreduction of this pair is not multilinear. • Multilinearity does not imply strong multilinearity. Consider the pair of equa-tions (2 y − z − be y )(2 b z − b y − e z + e y ) = p − q , (2 z − y − be z )(2 b y − b z − e y + e z ) = p − q , which is resolved for be y, be z . It is obviously multilinear and reduces to H1. Whenthis pair is solved for { e y, e z } one obtains the pair3( b y − e y )(2 y − be y − z )( y − z + be z ) = ( p − q )(2 be y + be z − y ) , b z − e z )(2 z − be z − y )( z − y + be y ) = ( p − q )(2 be z + be y − z ) , hich is resolved for both { e y, e z } and { b y, b z } . However, this is not multilinearbecause y and z appear quadratically. (As a consequence, if we attempt toresolve for y, z from this pair there is a superfluous solution.) Proposition 3.1.
1. For any given equation in the ABS list one can get a two com-ponent version satisfying the conditions in Definition 2 by applying to the originalequation any one of the following eight replacements none , (10a)1 : y → z, (10b)2 : e y → e z, (10c)3 : b y → b z, (10d)4 : y → z, e y → e z, (10e)5 : y → z, b y → b z, (10f)6 : e y → e z, b y → b z, (10g)7 : y → z, e y → e z, b y → b z, (10h) after which the other member of the pair is obtained by the exchange rule
2. For H1, H3, Q1 and Q3 this result is complete.
For H2 we have a counterexample on completeness, given below, while for Q2and Q4 uniqueness is open.
Proof.
1. It is easy to verify that from an equation in the ABS list, any of thesubstitutions (10) results in a pair satisfying all properties of Definition 2.2. It is a bit more laborious to show that there are no others. For this purpose wegenerate multilinear equations (with arbitrary coefficients) for all four resolutions,i.e. equation pairs of the type ( be y L ( y, z, e y, e z, b y, b z ) + P ( y, z, e y, e z, b y, b z ) + C = 0 , be z L ( z, y, e z, e y, b z, b y ) + P ( z, y, e z, e y, b z, b y ) + C = 0 , (11a) ( e y L ( y, z, b y, b z, be y, be z ) + P ( y, z, b y, b z, be y, be z ) + C = 0 , e z L ( z, y, b z, b y, be z, be y ) + P ( z, y, b z, b y, be z, be y ) + C = 0 , (11b) ( b y L ( y, z, e y, e z, be y, be z ) + P ( y, z, e y, e z, be y, be z ) + C = 0 , b z L ( z, y, e z, e y, be z, be y ) + P ( z, y, e z, e y, be z, be y ) + C = 0 , (11c) ( y L ( e y, e z, b y, b z, be y, be z ) + P ( e y, e z, b y, b z, be y, be z ) + C = 0 ,z L ( e z, e y, b z, b y, be z, be y ) + P ( e z, e y, b z, b y, be z, be y ) + C = 0 , (11d)where L j are linear and P j quadratic multilinear polynomials in the indicated vari-ables. Next some coefficients in L j , P j are fixed by the condition that the z y reduction leads to one of the equations H1, H3, Q1, Q3. This still leaves 3 free co-efficients in each L j and 9 in P j . The pairs in (11) describe the same evolution and herefore if we solve { be y, be z } from (11a), say, and substitute to the other equationsthey should all vanish. This leads to 384 smallish equations, which can be solvedby starting with the simplest ones and proceeding step by step. This is not diffi-cult, only tedious. For each of the equations H1, H3, Q1, Q3 the solution processeventually splits into eight branches as listed in (10). For H2 we found an equation that does not fit into the result of Proposition 3.1:( be y − ( y + z ) / b y − e y + b z − e z )+ ν ( y + e y + z + e z + ǫ p b y − b z )+ ν ( y + b y + z + b z + ǫ q e y − e z )+ ν ( b y − e y + b z − e z − ǫ ( p − q )2)( y − z ) − ǫ ( p − q )(2 be y + y + z + b y + e y + b z + e z ) − ǫ ( p − q ) = 0 , (12)together with its z ↔ y reflection. This pair satisfies the strong multilinearitycondition if all parameters ν i are nonzero. It reduces to H2 when all z = y , andclearly the parameters ν j disappear in this reduction. We do not know whether (12)is integrable or linearizable. The example (6) shows that if one uses the replacement rule 6 and its cyclic variantsfor the bottom back and left equations (which we denote as (6 , , , , possibili-ties have the CAC property. The result is as follows: Proposition 4.1.
For each equation in the ABS list the following eight replacementrules have the CAC property: (0,0,0), (0,4,5), (4,5,0), (4,6,5), (5,0,4), (5,4,6),(6,5,4), (6,6,6), where the numbers in the triplet are the replacement rules usedfor bottom, back, and left equations on the cube. The rules given in (10) must bemodified cyclically to fit the corresponding side. The top, front and right equationsare obtained by a perpendicular shift.Proof.
By direct computation. Since there are no free parameters the computationsfor the 512 cases are easy to automatize.Remarks: • The only replacements appearing in the list are 0,4,5,6, which correspond toreplacements of even number of variables. The cases (0,4,5), (4,5,0), (5,0,4), are related by rotation around the ( y, z ) − ( be y, be z ) axis, the same holds for (4,6,5), (6,5,4), (5,4,6). There are therefore onlyfour essentially different triplets. • The fact that there are two kinds of side equation pairs for each bottom equa-tion pair follows from the y ↔ z symmetry. For if we do this exchange only onthe variables with a bar-shift, the top pair does not change (and neither doesthe bottom pair), but the side equations will change. • Our end result agrees with the result of [19], which was derived by an entirelydifferent approach.
It is easy to see that in the (0,0,0) case the equations are decoupled, since in eachpair one equation depends only on the y variables and the other only on the z variables. We will now look whether the other sets can also be decoupled somehow.Since the CAC analysis is completely algebraic the variable names do not matter:instead of y, z, e y, e z, b y, . . . we could have used a, b, c, d, . . . and the algebra would havebeen the same.In the case of (4,5,0) given in (8) we see that (8c) is already decoupled and we canseparate variables into two set, S a = { y, e y, y, e y } and S b = { z, e z, z, e z } . Insisting thatany particular equation depends only on variables from one set, we can augmentthese sets using (8) and its shifted version to(4 , ,
0) : S a = { y, e y, y, e y, b z, b z, be z, be z } , S b = { z, e z, z, e z, b y, b y, be y, be y } . (13a)Thus there are 6 equations depending on variables from S a and they satisfy CAC allbe themselves, similarly for S b . It seems that all equations that pass the CAC testdo decouple in the described manner. For example while (6,5,4) passes CAC anddecouples, (6,4,5) does not decouple nor pass CAC. The sets for the other integrablecombinations are as follows:(0 , ,
0) : S a = { y, e y, y, e y, b y, b y, be y, be y } , S b = { z, e z, z, e z, b z, b z, be z, be z } , (13b)(6 , ,
4) : S a = { y, e z, y, e z, b z, b z, be y, be y } , S b = { z, e y, z, e y, b y, b y, be z, be z } , (13c)(6 , ,
6) : S a = { y, e z, z, e y, b z, b y, be y, be z } , S b = { z, e y, y, e z, b y, b z, be z, be y } . (13d)These can be described as follows: (0,0,0) is decoupled as it stands; (4,5,0) is de-coupled if for odd number of hat-shits we exchange z ↔ y ; (6,5,4) can be decoupledif for odd total number of tilde and hat-shifts we exchange, while bar shift has noeffect; for (6,6,6) exchange is needed when the total number of all kinds of shifts isodd.If the exchanges needed for decoupling are transferred to a property of the latticeitself, then for (0,0,0) the lattice is uniform; for (4,5,0) we should change the latticeon alternate planes in the hat direction; for (6,5,4) we have checkerboard lattice intilde and hat direction without change in bar direction; for (6,6,6) we need a latticethat is alternating in every direction. he possibility of decoupling follows in part from our assumption that the twoequations are related by y ↔ z replacement. But there are other two-componentequation pairs for which decoupling is not possible, for example the discrete Boussi-nesq equation given e.g. as the pair (3.3) of [15] has e w, e z, b w, b z in both equations ofthe pair.The above decoupling approach explains the algebra quite well, but in practicethe variables are not featureless symbols but the tilde-hat-bar decorations have adynamic meaning. That is, if we have a solution y = y ( n, m, k ) , z = z ( n, m, k ) forthe set of equations then b y, b z are obtained by changing m → m + 1 in the concreteformula for y and z . If the y and z solutions are different (for example solitonstraveling in different directions) it may be better to keep the dependent variablesuncoupled and the equations coupled than vice versa. It has been observed that CAC is necessary but not always sufficient for integra-bility [13]. However, the existence of a genuine B¨acklund Transformation (BT) (orequivalently, a nontrivial Lax pair) is a proof of integrability.
Proposition 4.2.
For each equation in the ABS list the following eight replacementrules (0,0,0), (0,4,5), (4,5,0), (5,0,4), (4,6,5), (6,5,4), (5,4,6), (6,6,6), generatea genuine BT. That is, if one assigns any of the listed triplet replacements onthe bottom, back, and left pairs of equations (or their cyclic permutations), thenone can freely choose two pairs of “side” equations for BT and together with theirperpendicular shift equations they generate by variable elimination the third pair.
For example if the bottom pair is generated by replacement 5, back pair by 4then they together generate left pair of type 6, after three of the four variables onthe right pair are eliminated. The same left pair is also generated if bottom andback equations are both generated by replacement 6.
One of the more important proofs of integrability is by construction of a Lax pair.But here one must note that there are “fake” Lax pairs [7, 13] and therefore onemust verify that the Lax pair is genuine and able to generate the equation(s) inquestion. The general formula for constructing Lax pairs from a CAC consistentsystem is given e.g., in [14] Section 3.3.1, and furthermore there are now evencomputer programs that can do that [5, 6].It is perhaps sufficient to consider an example, for which we choose type 5 Q1.As noted before type 5 bottom equation goes together with side equations 4,6 aswell as 0,4. .4.1 Example: Q1, sides 4,6 generate bottom 5 We construct the Lax pair for Q1;5 from a back equation pair of type 4 q ( z − y )( b z − b y ) − r ( z − b z )( y − b y ) = δ qr ( r − q ) , (14a) q ( y − z )( b y − b z ) − r ( y − b y )( z − b z ) = δ qr ( r − q ) , (14b)and a left equation of type 6 r ( y − e z )( z − e y ) − p ( y − z )( e z − e y ) = δ rp ( p − r ) , (14c) r ( z − e y )( y − e z ) − p ( z − y )( e y − e z ) = δ rp ( p − r ) , (14d)Together they should generate a bottom equation of type 5, i.e., p ( z − b z )( e y − be y ) − q ( z − e y )( b z − be y ) = δ pq ( q − p ) , (14e) p ( y − b y )( e z − be z ) − q ( y − e z )( b y − be z ) = δ pq ( q − p ) . (14f)In order to construct the Lax pair we solve the first 4 equations of (14) for thedouble shifted quantities and then replace the barred quantities as follows: y = fk , z = gl , e y = e f e k , e z = e g e l , b y = b f b k , b z = b g b l . For the left equations this leads to e f e k = g [ p e z + r ( y − e z )] + pl [ δ r ( r − p ) − y e z ] pg + l [ − py + r ( y − e z )] , (15a) e g e g = f [ p e y + r ( z − e y )] + pk [ δ r ( r − p ) − z e y ] pf + k [ − pz + r ( z − e y )] . (15b)These can be written in matrix form e ψ = L Q ψ where ψ = ( f, k, g, l ) T and L Q = • L 0 ! with (16a) L = λ (cid:18) p e z + r ( y − e z ) p [ δ r ( r − p ) − y e z ] p − py + r ( y − e z ) (cid:19) , (16b)where the parameter λ is the splitting factor (and may depend on y, e z and thereforeit is possible that λ = • λ ).Similarly, from the back equation we get b ψ = M Q ψ where M Q = M 00 • M ! with (17a) M = κ (cid:18) q b z + r ( z − b z )] q [ δ r ( r − q ) − z b z ] q − qz + r ( z − b z )] (cid:19) . (17b)The commutativity condition b LM = f ML now implies b L • M = f M L , or equivalently • b L M = • f M • L . n order to get the equations from this we have to fix the separations constants λ, κ .One elegant way to do that is to require det L = det M = 1. Here it leads to λ =1 / [( δ p − ( y − e z ) )( p − r ) r ] , • λ = 1 / [( δ p − ( z − e y ) )( p − r ) r ] , (18a) κ =1 / [( δ q − ( y − b y ) )( q − r ) r ] , • κ = 1 / [( δ p − ( z − b z ) )( q − r ) r ] . (18b)From these one can relatively easily derive ( e κ • κ q ) = ( λ b λp ) , but in practice we need e κ • κ q = λ b λp, (19)and its exchanged version, in order to derive the bottom Q
1; 5-equation. (Theapparent asymmetry in (19) is due to the block structure of the Lax matrices).
A back equation pair of type 0 is given by q ( y − y )( b y − b y ) − r ( y − b y )( y − b y ) = δ qr ( r − q ) , (20a) q ( z − z )( b z − b z ) − r ( z − b z )( z − b z ) = δ qr ( r − q ) , (20b)and a left equation of type 4 r ( z − e y )( z − e y ) − p ( z − z )( e y − e y ) = δ rp ( p − r ) , (20c) r ( y − e z )( y − e z ) − p ( y − y )( e z − e z ) = δ rp ( p − r ) . (20d)Now comparing equations (14) and (20) we find that the sets are the same if weexchange all barred quantities and only them: y ↔ z, e y ↔ e z, b y ↔ b z . From the Laxmatrix point of view this means permuting the blocks, i.e., L Q = • LL 0 ! , M Q = • M 00 M ! , while keeping the previous definitions (16b) and (17b). Thus we end up with thesame conditions. In this paper we have searched for two-component versions of the equations in theABS list. In addition to the standard assumptions placed on quad equations weassumed the following (the dependent variables are named y, z )1. The two equations forming the pair are related by y ↔ z exchange (for allshifts).2. When z = y both equations reduce to one of the equations in the ABS list.3. Evolution in any corner direction is by a pair of multilinear equations. ondition 3 in more detail: one must be able to solve for any corner variable pair(e.g., e y = A/B, e z = C/D ) and when this is written as equations (e.g., B e y − A =0 , D e z − C = 0) they must be multilinear in all of the dependent variables. We callthis strong multilinearity .The above conditions turn out to be quite strong and as a result we found thatthe only possibility is that the pair of equations is obtained from the original onecomponent equation by a simple replacement (Proposition 3.1), the only caveats areH2, for which we have a counterexample, and Q2 and Q4 which we did not check.Since one can get several candidate pairs for each original equation there rises aquestion related to multidimensional consistency, namely how we should populatethe sides of the consistency cube. We found eight combinations that satisfy theCAC-condition, as described in Proposition 4.1. Our end result is essentially thesame as the one obtained in [19] by symmetry arguments. Note also that if the CACcondition is satisfied for some bottom equation, it is satisfied with two different setsof side equations. Furthermore both pairs of side equations work as a B¨acklundtransformations generating the same bottom equation, and from both pairs onegets the same Lax pair. We gave the details for Q1 of type 5.With the above equations one can ask some natural questions which include:what are their semi-continuous and fully continuous limits, and what are their soli-ton solutions, in particular how do the different components of a soliton solutioninteract. Acknowledgement
I would like to thank D-j Zhang for comments. I would also like to thank the refereesfor additional references and for bringing the decoupling issue to my attention. Allcomputations were done with REDUCE [10].
References [1] Adler V., Bobenko A., and Suris Yu., Classification of integrable equations onquad-graphs. The consistency approach.
Comm. Math. Phys. , 233(3):513–543,2003.[2] Atkinson J.,
Integrable lattice equations: Connection to the M¨obius group,B¨acklund transformations and solutions . PhD thesis, University of Leeds, July2008. http://etheses.whiterose.ac.uk/9081/ [3] Boll R., Classification of 3D consistent quad-equations.
J. Nonlinear Math.Phys. , 18(3):337–365, 2011.[4] Boll R., Corrigendum: “Classification of 3D consistent quad-equations.”
J.Nonlinear Math. Phys. , 19(4):1292001, 3.pp, 2012.[5] Bridgman T., Hereman W., Quispel G.R.W. and van der Kamp P.H., SymbolicComputation of Lax Pairs of Partial Difference Equations using ConsistencyAround the Cube.
Found. Comput. Math.
6] Bridgman T.J.,
LaxPairPartialDifferenceEquations.m : A Mathematicapackage for the symbolic computation of Lax pairs of nonlinear partial differ-ence equations defined on quadrilaterals. http://inside.mines.edu/ ∼ whereman/software/LaxPairPartialDifferenceEquations/V2 , 2017.[7] Butler S. and Hay M., Two definitions of fake Lax pairs. AIP Conference Pro-ceedings 1648, 180006 (2015).[8] Fordy A.P., and Xenitidis P., Z N graded discrete Lax pairs and integrabledifference equations. J. Phys. A: Math. Theor. Chin. Phys. Lett. (9) 090202, 2014.[10] Hearn A.C. and Sch¨opf R., REDUCE User’s Manual https://reduce-algebra.sourceforge.io/ , 2019.[11] Hietarinta J., Searching for CAC-maps, J. Nonlinear Math. Phys. , 12(suppl.2):223–230, 2005.[12] Hietarinta J., Boussinesq-like multi-component lattice equations and multi-dimensional consistency,
J. Phys. A: Math. Theor. , 44 (2011) No.165204(22pp).[13] Hietarinta J., Search for CAC-integrable homogeneous quadratic triplets ofquad equations and their classification by BT and Lax.
J. Nonlinear Math.Phys. (3):358–389, 2019.[14] Hietarinta J., Joshi N., Nijhoff F., Discrete Systems and Integrability, Cam-bridge University Press, Cambridge, 2016.[15] Hietarinta J. and Zhang D.-J., Discrete Boussinesq-type equations, arXiv:2012.00495 (2020).[16] Kassotakis P., Nieszporski M., Papageorgiou V., Tongas A., Integrable two-component systems of difference equations, Proc. R. Soc. A. (2020),20190668.[17] Kels A.,Extended Z-Invariance for Integrable Vector and Face Models andMulti-component Integrable Quad Equations, J. Stat. Phys. , 1375–1408(2019).[18] Nijhoff F.W., Discrete Painlev´e Equations and Symmetry Reductions on theLattice, in:
Discrete Integrable Geometry and Physics , eds. A.I. Bobenko andR. Seiler, 209–234 (1999).[19] Zhang D.-D., van der Kamp P.H., Zhang D.-J., Multi-component generalisationof CAC systems.
SIGMA
060 (30pp.) 2020.060 (30pp.) 2020.