From auto-Bäcklund transformations to auto-Bäcklund transformations, and torqued ABS equations
aa r X i v : . [ n li n . S I] F e b From auto-B¨acklund transformations to auto-B¨acklundtransformations, and torqued ABS equations
Dan-da Zhang , Da-jun Zhang , Peter H. van der Kamp School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China.Email: [email protected] Department of Mathematics, Shanghai University, Shanghai 200444, China.Email: djzhang@staff.shu.edu.cn Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia.Email: [email protected]
Abstract
We provide a method which takes an auto-B¨acklund transformation (auto-BT) and producesanother auto-BT for a different equation. We apply the method to the natural auto-BTs for theABS quad equations, which gives rise to torqued versions of ABS equations and explains the originof each auto-BT listed in [J. Atkinson, J. Phys. A: Math. Theor. 41 (2008) 135202]. The methodis also applied to non-natural auto-BTs for ABS equations, which yields 3D consistent cubes whichhave not been found in [R. Boll, J. Nonl. Math. Phys. 18 (2011) 337–365], and to a multi-quadraticABS* equation giving rise to a multi-quartic equation.
Consistency Around the Cube (CAC) is a central notion in the study of discrete integrable systems.Three quadrilateral equations A = B = Q = 0 , each posed on two opposite faces of a cube, as inFigure 1, is called CAC (or 3D consistent) if values can be assigned to u, v and their shifts, suchthat each equation is satisfied [1, 14, 15, 3, 7, 4]. For multi-linear equations the property is usuallycharacterised as an initial value problem, where be v is determined uniquely from u, e u, b u, v . It is possibleto define different equations on opposite faces of the cube [7, 17], however we will not consider thatsituation in this paper. u e u b u be uv e v b v be vpr q A BQ
Figure 1: Consistency: quad-equations can be posed on the six faces of a cube, we have A = 0 on thefront and back faces, B = 0 on the left and right faces, and Q = 0 on the top and bottom faces. Theblack dots indicate the initial values. Each equation depends on two of the three lattice parameters p, q, r . This is short-hand notation for the system { A = 0 , B = 0 , Q = 0 } , which is complemented by the equations on theopposite faces { ˆ A = 0 , ˜ B = 0 , ¯ Q = 0 } , where tilde and hat denote shifts in different directions, e.g. if u = u ( n, m ) and A = f ( u ), then e A ˆ = f ( u ( n + 1 , m − u = v . A = B = 0 (together with the upshifted equations b A = e B = 0) is an auto-B¨acklundtransformation (auto-BT) for the equation Q = 0 =, mapping each solution u of Q = 0 to anothersolution v of Q = 0. The auto-BT depends on a parameter, called the B¨acklund parameter, andsolutions related by auto-BTs satisfy a superposition principle (Bianchi permutability) [5].In the more special situation where the three equations have the same form, i.e. Q = Q ( p, q ), A = Q ( p, r ), B = Q ( q, r ), the equation Q = 0 is called CAC. Such equations provide their own,natural auto-BT. Multi-affine quadrilateral equations have been classified with respect to CAC in [1].Under the additional assumptions, that the equations are D4-symmetric and possess the tetrahedronproperty, i.e. a relation exists between e u , b u , v and be v , Adler, Bobenko and Suris (ABS) obtained alist of 9 equations, three equations of type H, two equations of type A, and 4 equations of type Q.For completeness, the list is included in section 3. Multi-quadratic equations, denoted ABS ∗ , withthe CAC property were given in [4]. The ABS ∗ equations define multi-valued evolution, however,due to the discriminant factorization property, they allow reformulation a single-valued system, andpossess BTs to the ABS equations. Lists of BTs between ABS equations, as well as non-naturalauto-BTs (each with a superposition principle) were given in [3]. Auto-BTs play an important role inthe construction of solutions, cf. [11, 12, 13, 16].In 2009, Adler, Bobenko and Suris classified multi-affine cube systems (with a priori differentequations on each face) without the assumptions of D4-symmetry and the tetrahedron property, butwith an additional non-degeneracy condition. Their main result [2, Theorem 4] states that each 3D-consistent system of type Q is, up to M¨obius transformations, one of the known Q-type systems (withD4-symmetry) found in [1]. This includes the equations of type A, which are related to equations Q1and Q3 by (non-autonomous) point transformations. A classification of the degenerate cases (theH-equations) has been performed in [6, 7]. To each of the 6 edges (including the diagonals) of a quadequation one associates a biquadratic, and equations with i degenerate biquadratics, i >
0, are saidto be of type H i . There are 3 quad equations of type H , 4 quad equations of type H , and a list of3D consistent systems (with the tetrahedron property) was given in [7, Theorem 3.4].Inspired by the work [16], where multi-component generalisations of CAC lattice systems wereobtained by extending the scalar variable to a diagonal matrix and applying cyclic transformations toshifted matrices, we consider the equations of the auto-BT as dependent on two 2-component vectors,and apply cyclic transformations to those. This provides a systematic method (which can be calledtorquing) which takes an auto-BT (invariant under interchanging u ↔ v ) and produces another auto-BT for a different equation. In the symmetric case, the new equation is related to the tetrahedronproperty of the original system. In this case the result is similar to [7, Theorem 3.3] in the more generalsetting where equations on opposite faces of the cube do not have to coincide. In the asymmetric case,the new equation relates to a (previously hidden) relation, which can be obtained from the auto-BT,satisfying a property we have called planarity.The paper is organized as follows. In section 2 we observe that for an ABS-equation Q ( u ) = 0the natural auto-BT is not only an auto-BT for Q ( u ), but also for an equation R ( u, v ) = 0, whichdepends on both u, v . We define the notion of planarity for such equations. We prove a usefulresult (Lemma 1), which states that one can torque an auto-BT in two different ways (symmetricand non-symmetric). Full details are provided for H1, where an additive transformation of the latticeparameters is required. A pictorial representation of the consistent cubes on which symmetric andasymmetric torqued auto-BTs, as well as their superposition principles, is also provided. In section3, we provide the results of torquing the natural auto-BTs for all ABS equations. For 6 of them (H1,H2, A1 δ , Q1 δ , Q2, Q4) the same additive parameter transformation applies. For the remaining 3 ABSequations (H3 δ , A2, Q3 δ ) a multiplicative transformation of the lattice parameters is required. Detailsare provided for H3 δ . We obtain, amongst other cube systems, all auto-BTs listed in [3, Table 2],together with their superposition principles. In section 5, we apply Lemma 1 to non-natural auto-BTsfor H3 δ , A2 and Q3 δ , which were found in [18]. These auto-BTs, and their torqued versions seemto missing in the classification of [7, Theorem 3.3]. In section 6, we show that the multi-quadraticmodel H3 ∗ [4, Equation (22)] can also be torqued, giving rise to a multi-quartic equation. In the finalsection, we summarise our findings and mention some related results.2 Torqued equations
In this paper, we will write the dependence on the field values, and the parameters, as Q ( u ) = Q ([ u, b u ] , [ e u, be u ]; p, q ) = 0 , (1)instead of the more standard Q ( u, e u, b u, be u ; p, q ) = 0. We assume equations on opposite faces of thecube are identical, and that the auto-BT A = Q ([ u, v ] , [ e u, e v ]; p, r ) = 0 , B = Q ([ u, v ] , [ b u, b v ]; q, r ) = 0 , (2)which we denote shortly by A = B = 0, is symmetric under interchanging u ↔ v .In this section, we lay out and illustrate our method using the equation H1:H1([ u, b u ] , [ e u, be u ]; p, q ) := ( u − be u )( e u − b u ) − p + q = 0 . Given an auto-BT, one is able to determine the equation it is an auto-BT for. As we shall see, insection 2.1, it may actually be an auto-BT for more than one equation.
We start from a consistent cube equipped with A = 0 on the front face, b A = 0 on the back face, and B = 0 and e B = 0 on the side faces, where the two equations Q ( u ) = 0, Q ( v ) = 0 on the bottomand top faces are omitted. Taking u, e u, b u and v as initial values, as in Figure 1, the values of e v, b v are uniquely determined by the equations A = B = 0 (2). We then solve the system of equations b A = e B = 0 for be u, be v . For multi-affine equations, this gives rise to two solutions, one of which will satisfythe decoupled set of equations Q ( u ) = Q ( v ) = 0 . (3)In the above procedure, the equation Q ( u ) = 0 is obtained directly. However, to obtain Q ( v ) = 0 oneneeds to substitute the solution of A = B = 0 with respect to e u, b u . In doing so the dependence on u disappears.The second solution corresponds to a coupled set of equations, which we denote, assuming thatthe auto-BT is u ↔ v symmetric, by R ( u, v ) = R ( v, u ) = 0 . (4)Here, the equation R ( u, v ) = 0 is the equation for be v , where we have substituted the solution of A = B = 0 with respect to e u, b v , and R ( v, u ) = 0 is the equation for be u , having used the solution of A = B = 0 with respect to b u, e v . If R ( u, v ) = 0 depends on u, b u, e v, be v but not on v , and R ( v, u ) = 0depends on v, b v, e u, be u but not on u , then the set of equations (4) is called planar , cf. Figure 2. u e u b u be uv e v b v be v u e u b u be uv e v b v be v Figure 2: The stencils for a set of planar equations R ( u, v ) = 0 (left) and R ( v, u ) = 0 (right) areindicated by the black dots. .Importantly, the system of equations A = B = R = b A = e B = R = 0, where R ( u, v ) = R ( v, u ), isCAC as well (irrespective of planarity). Indeed, from initial values u, e u, b u, v , there are three ways of3alculating be v , which yield the same value. Moreover, we consider the system A = B = b A = e B = 0 tobe an auto-BT for the equation R = 0. Additive example (asymmetric case).
The natural auto-BT for H1 is given by A = ( u − e v )( e u − v ) − p + r = 0 , B = ( u − b v )( b u − v ) − q + r = 0 . We solve for e v, b v , b v = u + q − rv − b u , e v = u + p − rv − e u , (5)and substitute the result into b A = ( b u − be v )( be u − b v ) − p + r = 0 , e B = ( e u − be v )( be u − e v ) − q + r = 0 . The obtained system for be u, be v has two solutions: • The first solution is be u = u + p − q b u − e u , be v = e u + ( q − r )( v − e u )( b u − e u ) p ( b u − v ) + q ( v − e u ) + r ( e u − b u ) , (6)which, together with (5), satisfies H1([ u, b u ] , [ e u, be u ]; p, q ) = H1([ v, b v ] , [ e v, be v ]; p, q ) = 0. • The second solution is be u = u + p − rv − e u + q − rv − b u , be v = b u + e u − v, (7)which does not decouple. Substituting e u = v + ( p − r ) / ( u − e v ) into the second equation yields R ( u, v ) = ( u − e v )( b u − be v ) + p − r = 0 , (8)and substituting b u = u + ( q − r ) / ( u − b v ) into the first equation gives R ( v, u ) = 0. We introduce an involution σ which switches u ↔ v , that is σ [ u, v ] = [ v, u ], and which com-mutes with shifts. We define an action of σ on functions of 2-vectors, such as A ([ u, v ] , [ e u, e v ]) by σA ([ u, v ] , [ e u, e v ]) = A ( σ [ u, v ] , σ [ e u, e v ]). For functions of u, v and their shifts, such as (3) or (4), we write Q ( u, v ) = Q ([ u, v ] , [ e u, e v ] , [ b u, b v ] , [ be u, be v ]) and then we simply have σQ ( u, v ) = σQ ([ u, v ] , [ e u, e v ] , [ b u, b v ] , [ be u, be v ])= Q ( σ [ u, v ] , σ [ e u, e v ] , σ [ b u, b v ] , σ [ be u, be v ])= Q ([ v, u ] , [ e v, e u ] , [ b v, b u ] , [ be v, be u ])= Q ( v, u ) . We now formulate a useful lemma, which enables us to derive auto-BTs from auto-BTs.
Lemma 1.
Suppose the system A ([ u, v ] , [ e u, e v ]) = 0 , B ([ u, v ] , [ b u, b v ]) = 0 , (9) is invariant under σ and that it composes, together with b A = e B = 0 , an auto-BT for Q ( u, v ) = Q ([ u, v ] , [ e u, e v ] , [ b u, b v ] , [ be u, be v ]) = 0 , (10) i.e. solutions of (10) are mapped to solutions of Q ( v, u ) . Then A ([ u, v ] , σ a [ e u, e v ]) = 0 , B ([ u, v ] , σ b [ b u, b v ]) = 0 , (11) is an auto-BT for Q ([ u, v ] , σ a [ e u, e v ] , σ b [ b u, b v ] , σ a + b [ be u, be v ]) = 0 . (12) where a, b ∈ { , } . roof. For the BT given by (11) the equations b A = e B = 0 are equivalent to A ( σ b [ b u, b v ] , σ a + b [ be u, be v ]) = 0 , B ( σ a [ e u, e v ] , σ a + b [ be u, be v ]) = 0because they are invariant under σ . The consistency of the corresponding cube system then followsby relabeling of variables, cf. the proof of Lemma 1 in [17].Lemma 1 boils down to two cases. The anti-symmetric torqued auto-BT, (11) with a = 1, b = 0(or a = 0, b = 1), A ([ u, v ] , [ e v, e u ]) = 0 , B ([ u, v ] , [ b u, b v ]) = 0 (13)yields Q ([ u, v ] , [ e v, e u ] , [ b u, b v ] , [ be v, be u ]) = 0 , (14)and the symmetric torqued auto-BT, (11) with a = b = 1, A ([ u, v ] , [ e v, e u ]) = 0 , B ([ u, v ] , [ b v, b u ]) = 0 , (15)yields Q ([ u, v ] , [ e v, e u ] , [ b v, b u ] , [ be u, be v ]) = 0 , (16)where the function Q is either solution, (3) or (4), to the natural auto-BT. Remark 2.
Note that although in Lemma 1, and in equations (14), (16), the function Q depends onboth fields u, v , in applications we are interested in the case where the procedure yields an equation Q ( u, v ) = 0 which depends only on the field u . This happens due to either planarity, in the asymmetriccase, or the tetrahedron property, in the symmetric case. If the second solution to a BT, equation (4), is planar, then by relabeling e u ↔ e v and be u ↔ be v , theequations for u, v decouple. The planar equation R ( u, v ) = 0 will depend on u only, and the planarequation R ( v, u ) = 0 will depend on v only, cf. Figure 3. u e u b u be uv e v b v be v u e v b u be vv e u b v be u Figure 3: By relabeling of variables the diagonal planar stencil for R ( u, v ) (left) becomes a quadrilateralfor a decoupled equation R ( u ) = 0 (right). Remark 3.
The cube on the right in Figures 3 and 4 is useful to define equations. However, oneshould be aware that the fields u, v are defined in the usual way, e.g. ˜ u ( n, m ) = u ( n + 1 , m ) , cf. [17,Remark 2.4]. Additive example, asymmetric case (continued).
For the H1 equation, the asymmetric auto-BT(13) reads A = ( u − e u )( e v − v ) − p + r = 0 , B = ( u − b v )( b u − v ) − q + r = 0 (17)Switching e u ↔ e v and be u ↔ be v in (4) with (8), gives( u − e u )( b u − be u ) + p − r = 0 , ( v − e v )( b v − be v ) + p − r = 0 , u, b u ] , σ [ e u, be u ]; p, r ) = 0 , H1([ v, b v ] , σ [ e v, be v ]; p, r ) = 0 . (18)These are torqued versions of H1, similar to the A -part of the auto-BT, except that the dependenceon the parameters is different. The B -part of the auto-BT is the standard H1 equation.In order for (17) to be a proper auto-BT, the equations (18) should not depend on the B¨acklundparameter r . We achieve this by introducing a new parameter p ′ = p − r. (19)We can write the A-part of the auto-BT asH1([ u, v ] , [ e v, e u ]; p ′ + r, r ) = 0 , in terms of the new parameter. We note that for H1 this may seem a bit odd as the equation doesnot depends on r . Notation.
We will adopt the notation Q a = 0 with Q a ( p, q )[ u ] = Q a ([ u, b u ] , [ e u, be u ]; p, q ) = Q ([ u, b u ] , σ [ e u, be u ]; p + q, q ) (20)for the torqued version of a lattice equation Q = 0, with an a dditive transformation of the latticeparameters. Additive example, asymmetric case, recap.
Omitting the dependence on the fields, the equations(18) can both be written as H1 a ( p ′ , q ) = 0 , (21)and the auto-BT, (17), is written as A = H1 a ( p ′ , r ) = 0 , B = H1( q, r ) = 0 . (22)One may wonder why q appears in (21). In fact, the equation does not depend on q , cf. H1 a inthe list provided in section 6. Note that the other additive torqued ABS (t-ABS) equations do have q -dependence. Here we exploit the tetrahedron property. Consider again the equation Q ( v ) in (3), but now let ussubstitute the solution of A = B = 0 with respect to e v, b v . This gives rise to an equation for be v whichdepends on v, e u, b u , and u . If this equation does not depend on u , the equation is said to have the tetrahedron property. By relabeling e u ↔ e v and b u ↔ b v one obtains a decoupled equation, as inFigure 4. u e u b u be uv e v b v be v u e v b v be uv e u b u be v Figure 4: An equation with the tetraheron property (left), the black dots indicate the stencil on whichthe equation is defined. After relabeling variables the equation depends on v only (right). Here Q a ( p, r ) and Q ( p, r ) mean Q a ( p, r )[ u ] = Q a ([ u, u ] , [ e u, e u ]; p, r ) , Q ( p, r )[ u ] = Q ([ u, u ] , [ e u, e u ]; p, r ) , respectively, where u = v . If replacing ( e u, p ) by ( b u, q ), one gets Q a ( q, r ) and Q ( q, r ). dditive example, symmetric case I. We consider the symmetric auto-BT (16), which consists of A = ( u − e u )( e v − v ) − p + r = 0 , B = ( u − b u )( b v − v ) − q + r = 0 . (23)We now switch e u ↔ e v and b u ↔ b v in the first solution, (6). This gives be u = u + p − q b v − e v , be v = e v + ( q − r )( v − e v )( b v − e v ) p ( b v − v ) + q ( v − e v ) + r ( e v − b v ) . The latter equation is decoupled, and the first one takes the same form after elimination of e v, b v makinguse of (23). The equation can be identified as Q1 ( p − r, q − r ) = 0, which is one of the ABS equationsprovided in the next section, again with different dependence on the parameters. Using (19) anddefining also q ′ = q − r, (24)the equation Q1 ( p ′ , q ′ ) = 0 admits the auto-BTH1 a ( p ′ , r ) = H1 a ( q ′ , r ) = 0 . (25) Additive example, symmetric case II.
We can also switch e u ↔ e v and b u ↔ b v in the secondsolution, (7). This gives be u = u + p − rv − e v + q − rv − b v , be v = b v + e v − v, of which the first equation gives rise to the linear (difference) equation D : ( be u + u ) − ( e u + b u ) = 0 , (26)due to (23). Thus, the auto-BT (25) is an auto-BT for both Q1 ( p ′ , q ′ ) = 0 and D = 0. This is quitespecial; for only two ABS equations (H1 and H3 δ ) we find decoupling in the second solution, applyingthe symmetric switch.Note that in the asymmetric case both the equation (21) and the auto-BT (22) only depend on p ′ ,not on p . Similarly, for the symmetric case we have dependence on p ′ , q ′ , and not on p, q . Therefore,in the sequel we will omit the prime. We will represent Q a ( p, q )[ u ] = 0, equation (20), pictorially as in Figure 5. The edge [ e u, be u ] beingtorqued does not mean we switch e u and be u on the lattice, but only in the equation. u e u b u be u Q a Figure 5: Representation of a torqued equation.The results for the H1 equation obtained in the previous subsection can then be represented bythe CAC systems in Figure 6. 7 e u b u be uv e v b v be v H1 a H1H1 a (a) u e u b u be uv e v b v be v H1 a H1 a Q1 (b) u e u b u be uv e v b v be v H1 a H1 a D (c)Figure 6: Consistent cubes, with torqued H1-equations.The consistent cubes in Figure 6 represent auto-BTs. Each auto-BT maps a solution, u , to theequation on the bottom face, to a new solution (with parameter r ), v , to the equation on the topface. We now apply the same auto-BT but with parameter s to both solutions u, v to create twonew solutions, called z, w respectively. A relation of the form S ( u, v, z, w, r, s ) = 0 between the foursolutions u, v, z and w is called a superposition principle . z e z b z be zw e w b w be wu e u b u be uv e v b v be v H1 a H1H1 a H1 (a) z e z b z be zw e w b w be wu e u b u be uv e v b v be v H1 a H1 a Q1 H1 (b)Figure 7: 4D cubes representing auto-BTs with their superposition principles.It can be seen from Figure 7 that the superposition principles for both the asymmetric auto-BTand the symmetric auto-BT is equal to the equation H1. Each 4D-cube consists of 7 3D-cubes (center,top, bottom, front, back, left and right), excluding the outer 3D-cube. We have placed the 3D cubefrom Figure 6(a) in the center of Figure 7(a) and, as we apply the same auto-BT to both u and v , weplace Figure 6(a) also in the top and bottom cubes of Figure 7(a). As parallel faces carry the sameequation, in the front and back cubes the same configuration as in Figure 6(a) appears, whereas inthe left and right cubes we find no torqued equations. All four cubes left, right, front and back areCAC if we impose the equation H1 = 0 on the inner faces, i.e. the face z − u − v − w and their shifts.Similarly, placing the cube from Figure 6(b) in the center, top and bottom cube of a 4D-cube yieldsthe cube from Figure 6(a) on the remaining 4 cubes, as depicted in Figure 7(b), where the equationQ1 may be replaced by D (26). 8 Torqued auto-BTs for ABS equations and torqued ABS equations
The ABS equations areH1 : ( u − be u )( e u − b u ) − p + q = 0 , H2 : ( u − be u )( e u − b u ) − ( p − q )( u + e u + b u + be u + p + q ) = 0 , H3 δ : p ( u e u + b u be u ) − q ( u b u + e u be u ) + δ ( p − q ) = 0 , A1 δ : p ( u + b u )( e u + be u ) − q ( u + e u )( b u + be u ) − δ pq ( p − q ) = 0 , A2 : p (1 − q )( u e u + b u be u ) − q (1 − p )( u b u + e u be u ) − ( p − q )(1 + u e u b u be u ) = 0 , Q1 δ : p ( u − b u )( e u − be u ) − q ( u − e u )( b u − be u ) + δ pq ( p − q ) = 0 , Q2 : 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 − p + pq − q ) = 0 , Q3 δ : p (1 − q )( u b u + e u be u ) − q (1 − p )( u e u + b u be u ) − ( p − q ) (cid:18)e u b u + u be u + δ (1 − p )(1 − q )4 pq (cid:19) = 0 , Q4 : sn( p )( u e u + b u be u ) − sn( q )( u b u + e u be u ) + sn( p − q ) (cid:16) k sn( p )sn( q )( u e u b u be u + 1) − e u b u − u be u (cid:17) = 0 , where we have taken the Hietarinta form of Q4, found in [9], in which k is the elliptic modulus of theJacobi sine function sn.For some of the ABS equations the parameter transformation in their torqued counterpart isdifferent than the additive one we have seen for H1. We provide details for H3 , where the parametertransformation is multiplicative. Multiplicative example, H3 . The natural auto-BT for H3 is given by A = p ( u e u + v e v ) − r ( uv + e u e v ) = 0 , B = q ( u b u + v b v ) − r ( uv + b u b v ) = 0 . (27)Solving the equations A = ˆ A = B = ˜ B = 0 yields two solutions, which gives rise to: • a decoupled system, p ( u e u + b u be u ) − q ( u b u + e u be u ) = 0 , p ( v e v + b v be v ) − q ( v b v + e v be v ) = 0 , (28)which is just H3 [ u ] = H3 [ v ] = 0, • and a coupled system, e u b u − v be v = 0 , e v b v − u be u = 0 , (29)which is written in the form (4) as( u be v + b u e v ) p − ( u b u + e v be v ) r = 0 , ( v be u + b v e u ) p − ( v b v + e u be u ) r = 0 . (30) Asymmetric auto-BT.
The asymmetric auto-BT is obtained by switching e u ↔ e v in (27), A = p ( u e v + v e u ) − r ( uv + e u e v ) = 0 , B = q ( u b u + v b v ) − r ( uv + b u b v ) = 0 . (31)Switching e u ↔ e v and be u ↔ be v in the coupled system (30) leads to the decoupled system p ( u be u + e u b u ) − r ( u b u + e u be u ) = 0 , p ( v be v + e v b v ) − r ( v b v + e v be v ) = 0 . (32)These equations should not depend on r . So we introduce the variable p ∗ = p/r and the followingnotation. Notation.
We will adopt the notation Q m = 0 with Q m ( p, q )[ u ] = Q m ([ u, b u ] , [ e u, be u ]; p, q ) = Q ([ u, b u ] , σ [ e u, be u ]; pq, q ) (33)for the multiplicative torqued version of a lattice equation Q = 0.9y doing so, the equations (32) are captured byH3 m ( p ∗ , q )[ u ] = H3 m ( p ∗ , q )[ v ] = 0 , and the A -part of the auto-BT (31) becomes A = H3 m ([ u, v ] , [ e u, e v ]; p ∗ , r ) = 0 . Symmetric auto-BT.
The symmetric auto-BT is obtained by switching e u ↔ e v and b u ↔ b v in (27), A = p ( u e v + v e u ) − r ( uv + e u e v ) = 0 , B = q ( u b v + v b u ) − r ( uv + b u b v ) = 0 , (34)which is conveniently written as H3 m ( p ∗ , r ) = H3 m ( q ∗ , r ) = 0, where q ∗ = q/r . Switching e u ↔ e v and b u ↔ b v in the decoupled system (28), and eliminating the dependence on either e u, b u or e v, b v , using (34)leads to the decoupled system (up to an irrelevant factor r )Q3 ( p ∗ , q ∗ )[ u ] = Q3 ( p ∗ , q ∗ )[ v ] = 0 . (35)As for the H1 equation, switching e u ↔ e v and b u ↔ b v in the coupled system (29), and eliminatingthe dependence on either e u, b u or e v, b v , using (34) leads to another decoupled system, of which the u -equation is the quotient equation K : u be u − e u b u = 0 . (36)For equations H3 δ , A2 and Q3 δ the dependence on the parameters is the same as in the H3 case.As we no longer have dependence on p, q , we will omit the asterix in the sequel (remember, we werealso going to omit the prime we used for the additive parameters).For each ABS equation the natural auto-BT is an auto-BT for an equation which depends on both u, v and which is planar. As we also know each ABS equation has the tetrahedron property, this leadsto the following theorem, whose proof is obtained by direct calculation. Theorem 4.
For each ABS equation both system (13) and system (15) provide an auto-BT for aquad-equation. For equations H2 , A1 , Q1 , Q2 and Q4 the dependence on the parameters is additive.For equations H3 , A2 and Q3 the dependence on the parameters is multiplicative. The asymmetric andsymmetric torqued auto-BTs (t-auto-BTs) and the equations they give rise to are provided in Table 1and Table 2.
Equation Asymmetric t-auto-BT Symmetric t-auto-BT Q ( p, q ) = 0 Q a ( p, r ) = Q ( q, r ) = 0 Q a ( p, r ) = Q a ( q, r ) = 0H1 H1 a Q1 or D H2 H2 a Q1 A1 δ A1 aδ Q1 δ Q1 δ Q1 aδ Q1 δ Q2 Q2 a Q2Q4 Q4 a Q4Table 1: Torqued auto-B¨acklund transformations, additive cases.Equation Asymmetric t-auto-BT Symmetric t-auto-BT Q ( p, q ) = 0 Q m ( p, r ) = Q ( q, r ) = 0 Q m ( p, r ) = Q m ( q, r ) = 0H3 H3 m K H3 δ H3 mδ Q3 A2 A2 m Q3 Q3 δ Q3 mδ Q3 δ Table 2: Torqued auto-B¨acklund transformations, multiplicative cases.10 or each auto-BT mentioned in Tables 1 and 2, the superposition principle is given by the originalequation, i.e. it takes the form Q ([ u, z ] , [ v, w ]; r, s ) = 0 , (37) cf. Figure 7. Each additive torqued equation of type Q satisfies Q a ( p, q ) = Q ( − p, q ) (for Q4 this is due tothe anti-symmetry of the Jacobi sine function), whereas for the multiplicative equation Q3 mδ ( p, q ) = qp Q3(1 /p, q ), and for the A -equations we have A1 a ([ u, b u ] , [ e u, be u ]; p, q ) = A1([ u, b u ] , [ − e u, − be u ]; − p, q ),and A2 a ([ u, b u ] , [ e u, be u ]; p, q ) = qp e u be u A2([ u, b u ] , [1 / e u, / be u ]; 1 /p, q ). It is not difficult to verify that we canadjust the signs on the whole cube, cf. [2, Proof of Theorem 4], to show that, for these equations, thetorqued cubes are equivalent to the natural ones.The consistent cubes with H-type equations are special cases of consistent cubes listed in [6, 7].This connection is made precise in section 6. Whereas for ǫ = 0 the rhombic version of H ǫi correspondsto ABS equation H i , the trapezoidal version of H i corresponds to a t-ABS equation.The B¨acklund transformations given in [3, Table 2] consist of torqued ABS equations. Our resultexplains how the corresponding consistent cubes relate to the CAC property of ABS-equations. δ , A2 and Q3 One does not have to take the natural auto-BT as the starting point. For example, consider theequations A = u e u + v e v + δp = 0 , B = u b u + v b v + δq = 0 , (38)which provide an auto-BT for H3 δ [18], and note that it does not depend on a B¨acklund parameter.Performing the symmetric switch we obtain u e v + v e u + δp = 0 , u b v + v b u + δq = 0 , (39)which provides an auto-BT for H3 . Here, the parameter δ now acts as a B¨acklund parameter. Theasymmetric switch does not lead to a decoupled system.To find a superposition principle for the auto-BT (38), we pose the following equations on thefront cube of a 4D cube (see Figure 7 for the labeling of the vertices): u e u + v e v + δp = 0 (back) z e z + w e w + δp = 0 (front) u e u + z e z + δp = 0 (bottom) v e v + w e w + δp = 0 (top) . (40)These equations are coupled and we can neither eliminate all variables u, v, w, z nor their upshifts e u, e v, e w, e z . We can however derive the equations u e u − w e w = 0 , v e v − z e z = 0 , which show that a superposition principle is given by the (reducible) equation( u − w )( v − z ) = 0 . (41)The equations (40) decouple after application of the symmetric switch, here we can derive both equa-tions u = w and v = z , as well as their upshifted versions. Consequently, (41) is also a superpositionprinciple for (39).The non-natural auto-BT (38) can be conveniently written as Q ( p,
0) = Q ( q,
0) = 0 , (42)where Q = H3 δ . Both equations A2 and Q3 give rise to non-natural auto-BTs of the same form (42),cf. [18], and to torqued versions thereof, as per the below theorem.11 otation. We will adopt the notation Q t = 0 with Q t ( p, q )[ u ] = Q t ([ u, b u ] , [ e u, be u ]; p, q ) = Q ([ u, b u ] , σ [ e u, be u ]; p, q ) (43)for the (plain) torqued version of a lattice equation Q = 0. Theorem 5.
For the ABS equations H3 δ , A2 and Q3 , the system (42) provides a non-natural auto-BT. In each case, the system (15) provides a torqued auto-BT for H3 , and for A2 and Q3 thesystem (13) provides an auto-BT for a quad-equation which depends on only lattice parameter. Thenon-natural auto-BTs and their torqued counterparts are provided in Table 3. Equation Auto-BT Symmetric t-auto-BT Asymmetric t-auto-BT Q ( p, q ) = 0 Q ( p,
0) = Q ( q,
0) = 0 Q t ( p,
0) = Q t ( q,
0) = 0 Q t ( p,
0) = Q ( q,
0) = 0H3 δ H3 δ H3 − A2 A2 or P : u e u b u be u = 1 H3 A2(0 , q − )Q3 Q3 or K H3 Q3 (0 , q )Table 3: Degenerate auto-B¨acklund transformations. All the auto-BTs in Table 3 admit (41) as their superposition principle. The symmetric torquedauto-BTs obtained from A2 and Q3 δ furthermore admit the superposition principle ( uw − vz −
1) = 0 . (44)The quad equation H3 δ ( p,
0) is of type H , a special case of D , with δ = δ = 0, cf. [6]. Theequations A2 δ ( p,
0) and Q3 ( p,
0) are of type H , equivalent to a special case of H3. By suitableM¨obius transformations, equation D (26) relates to the equation of type H named D , and equations K (36) and P relate to equation D , with δ = δ = δ = 0. The equations (41) and (44) are of typeH , and relate to H1, with p = q = 0. The consistent cubes in Table 3 which contain P and K do nothave the tetrahedron property. All other consistent cubes in Table 3 do have that property. However,we have not been able to identify any of them in the classification given in [6, 7]. ∗ equation A list of multi-quadratic quad equations: H2 ∗ , H3 ∗ δ , A1 ∗ , A2 ∗ , Q1 ∗ , Q2 ∗ , Q3 ∗ δ , Q4 ∗ c , where δ ∈ { , } and c ∈ { , ± , ± i } , was given in [4, Section 4]. Due to the special factorisation of their discriminantsthese models can be reformulated as single-valued systems. The models can be consistently posedon the cube, however sign choices lead to four solutions. Here, we show that the torqued version ofH3 ∗ forms a symmetric auto-BT for a multi-quartic quad equation, and that it admits an asymmetricauto-BT itself. Whether other multi-quadratic quad equations admit torqued versions remains to beseen.For the H3 ∗ δ equationH3 ∗ δ ([ u, b u ] , [ e u, be u ]; p, q ) =( p − q ) (cid:16) p ( u b u − e u be u ) − q ( u e u − b u be u ) (cid:17) + pq ( u − be u ) ( e u − b u ) − δ ( p − q )( u − be u )( e u − b u ) = 0 , (45)we find that be v satisfies u (cid:16) p ( q − r ) ( b uv − be v e u ) + q ( p − r ) ( e uv − be v b u ) + r ( p − q ) ( e u b u − be vv ) (cid:17) + 2( σ σ − δ )( r − p )( r − q )( be v − v ) + 2( σ σ − δ )( q − p )( q − r )( be v − b u )+ 2( σ σ − δ )( p − q )( p − r )( be v − e u ) = 0 (46)with σ = pu e u + δ , σ = qu b u + δ , σ = ruv + δ . (47)12lthough equation (46) involves five points, when δ = 0 the variable u can be eliminated (as σ i σ j /u does not depend on u ) and therefore the equation (45) with δ = 0 possesses the tetrahedron property.The multiplicative torqued version, cf. notation (33), of H3 ∗ isH3 ∗ m ( p, q )[ u ] = ( p − p ( u b u − e u be u ) − ( u be u − b u e u ) ) + p ( u − e u ) ( be u − b u ) = 0 (48)with vanishing parameter q . Theorem 6.
The symmetric systemH3 ∗ m ( p, r ) = ( p − p ( uv − e u e v ) − ( u e v − v e u ) ) + p ( u − e u ) ( v − e v ) = 0 , H3 ∗ m ( q, r ) = ( q − q ( uv − b u b v ) − ( u b v − v b u ) ) + q ( u − b u ) ( v − b v ) = 0 (49) acts as an auto-BT for p ( q − ( b uu − be u e u ) + q ( p − ( e uu − be u b u ) + ( p − q ) ( e u b u − be uu )+ 2 σ ( p − q )( p − be u − e u ) + 2 σ ( q − p )( q − be u − b u ) + 2 σ σ (1 − p )(1 − q )( be u/u −
1) = 0 (50) which is equivalent to a multi-quartic quad equation, and the asymmetric systemH3 ∗ m ( p, r ) = H3 ∗ ( q, r ) = 0 provides an auto-BT for H3 ∗ m .Proof. The equations (48) and (50) are obtained by replacing p → pq and switching e u ↔ be u inrespectively (45) and (46) taking δ = 0. The symmetric claim follows from the tetrahedron propertyof H3 ∗ . Next, consider the cube systemH3 ∗ m ( p, q ) = 0 , H3 ∗ m ( p, r ) = 0 , H3 ∗ ( q, r ) = 0 , H3 ∗ m ( p, q ) = 0 , c H3 ∗ m ( p, r ) = 0 , f H3 ∗ ( q, r ) = 0 . (51)Solving H3 ∗ m ( p, q ) = 0 , H3 ∗ ( q, r ) = 0 leads toH3 ∗ m ( u, e u, b u, be u ; p, σ ) = p ( u b u − e u be u ) − ( u be u − e u b u ) + 2 σ ( b u − be u ) = 0 , (52)H3 ∗ m ( b u, be u, u, e u ; p, b σ ) = p ( u b u − e u be u ) + ( u be u − e u b u ) + 2 b σ ( u − e u ) = 0 , (53)H3 ∗ ( u, b u, u, b u ; q, r, σ σ ) = u ( q ( uu − b u b u ) + r ( u b u − u b u )) − u − b u ) σ σ = 0 . (54)From equation (52) and (54) we derive be u = pu + e u + 2 σ p e u + u + 2 σ b u, e u = pu + e u + 2 σ p e u + u + 2 σ u, b u = quu + ru b u − σ σ qu b u + ruu − σ σ u. (55)Making use of the form (53), we have b σ = − p ( u b u − e u be u ) + u be u − e u b u u − e u ) , σ = − p ( uu − e u e u ) + u e u − e uu u − e u ) . (56)Substituting (55) and (56) in H3 ∗ m ( p, q ) = 0 , c H3 ∗ m ( p, r ) = 0, we can determine be u = quu + ru b u − σ σ qu b u + ruu − σ σ e u. (57)Based on (55) and the relation p σ σ ] σ σ = qrσ b σ σ , solving f H3 ∗ ( q, r ) = 0, we derive the equivalentvalue be u = 2 qru b uu − σ σ ( qu + r b u )2 qru b uu − σ σ ( q b u + ru ) e u. This shows that the cube system (51) is consistent.13
Concluding remarks
We have provided a method which takes an auto-BT and produces another auto-BT for a differentequation. In other words, the method provides relationship between equations and their auto-BTs.The crucial property which lies at the heart of the symmetric torqued auto-BTs is the tetrahedronproperty, see Figure 4. The corresponding key property for the asymmetric torqued auto-BTs has beenthe planar property, illustrated in Figure 2. By torquing the natural ABS auto-BTs we obtained auto-BTs for an ABS equation of type Q , in the symmetric case, or a t-ABS equation, in the asymmetriccase. Thus, each ABS equation has a torqued variant (additive or multiplicative):H1 a : ( u − e u )( be u − b u ) − p = 0 , H2 a : ( u − e u )( be u − b u ) − p ( u + e u + b u + be u + p + 2 q ) = 0 , H3 mδ : p ( u be u + b u e u ) − ( u b u + e u be u ) + δ ( p − q = 0 , A1 aδ : ( p + q )( u + b u )( e u + be u ) − q ( u + be u )( b u + e u ) − δ pq ( p + q ) = 0 , A2 m : p (1 − q )( u be u + b u e u ) − (1 − p q )( u b u + e u be u ) − ( p − q (1 + u e u b u be u ) = 0 , Q1 aδ : ( p + q )( u − b u )( be u − e u ) − q ( u − be u )( b u − e u ) + δ pq ( p + q ) = 0 , Q2 a : ( p + q )( u − b u )( be u − e u ) − q ( u − be u )( b u − e u ) + pq ( p + q )( u + e u + b u + be u − p + pq − q ) = 0 , Q3 mδ : p (1 − q )( u b u + e u be u ) − (1 − p q )( u be u + b u e u ) − ( p − q (cid:18) u e u + b u be u + δ (1 − p q )(1 − q )4 pq (cid:19) = 0 , Q4 a : sn( p + q )( u be u + b u e u ) − sn( q )( u b u + e u be u ) + sn( p ) (cid:16) k sn( p + q )sn( q )( u e u b u be u + 1) − be u b u − u e u (cid:17) = 0 . Our results imply that each t-ABS equation is integrable in the sense that it is CAC, and hence it hasan auto-BT, and admits a Lax-representation. The consistent cubes in which they appear are givenin Tables 1 and Table 2. For example, the second item in Table 1 yields the following two consistentcubes, cf. Figure 1: A = H2 aδ ( p, r ) = 0 , B = H2 δ ( q, r ) = 0 , Q = H2 aδ ( p, q ) = 0 , where the auto-BT A = B = 0 is asymmetric, and A = H2 aδ ( p, r ) = 0 , B = H2 aδ ( q, r ) = 0 , Q = Q1 ( p, q ) = 0 , where the auto-BT A = B = 0 is symmetric. The torqued equations which appear in these cubes canbe represented as in Figure 6, and they can be extended to 4D cubes, cf. Figure 7, representing thesuperposition principles, which take the form H2([ u, z ] , [ v, w ]; r, s ) = 0 for each of the above cases.We have rediscovered all symmetric auto-BTs given in [3, Table 2]. They are given by the secondand third item in Table 1 and the first two items in Table 2 (right column). Thus we have shown howsuch auto-BTs arise from the natural auto-BTs which are implied by the CAC property of certainABS equations. The torqued Q -equations were found before in [2], where it was shown that theirconsistent cubes are equivalent to the natural one. The torqued H-equations are special cases oftrapezoidal versions of H equations classified in [6, 7]. In Table 4 we relate our H -type consistentcubes to consistent cubes given in [6]Table(row,column) 1(1,1) 1(1,2) Q1 a .The consistent cube in Table 1 with D , as well as the one in Table 2 with K , do not possess thetetrahedron property and hence fall outside the scope of [6, 7]. The non-natural auto-BTs for H3 δ , A214nd Q3 were found in [18]. We have not been able to identify their 3D consistent cubes, nor thoseof their torqued variants, as presented in Table 3, in the classification [6, 7]. As these systems, withthe exception of those carrying P and K , do admit the tetrahedron property this indicates that thatclassification is not complete. To the best of our knowledge, the torqued multi-quadratic equation(48) and the multi-quartic quad equation which can be derived from equation (50) are new.By cyclic rotation or interchange of shifts and parameters, cf. [10, Section 2.1] or [16, Lemma2.1], all equations which are part of an auto-BT admit an auto-BT themselves. This implies thateach symmetric torqued auto-BT obtained in the current paper can be used to create an (asymmetric)auto-BT for the corresponding t-ABS equation which is different than the asymmetric torqued auto-BT provided here. In [16] such an auto-BT is explicitly given and subsequently used to find a seedand a 1-soliton solution for H2 a . Similarly, auto-BTs for each equation in the auto-BTs in Table 3 areobtained.We would like to mention another mechanism which yields alternative auto-BTs, using parametertransformations. Let A ( p, q ) = B ( p, q ) = 0 be an auto-BT for Q ( p, q ) = 0. If a transformation( p, q ) ( p † , q † ) leaves Q invariant, then A ( p † , r ) = B ( q † , r ) = 0 provides an alternative auto-BT for Q ( p, q ) = 0. The ABS and t-ABS equations admit the parameter scaling symmetries presented inTable 5. ( p † , q † ) ABS and torqued ABS equations( p + c, q + c ) H1( cp, cq ) H3 , Q1 , Q1 a ( − p, − q ) A1 δ , A1 aδ , Q1 δ , Q1 aδ , Q2, Q2 a , Q4, Q4 a ( p − , q − ) A2, A2 m , Q3 δ , Q3 mδ Table 5: Parameter scaling symmetries.Finally, we note that multi-component t-ABS equations with corresponding auto-BTs exist, andcan be constructed using the technique explained in [17].
Acknowledgements
Financial support was provided by a La Trobe University China studies seed-funding research grant,by the department of Mathematics and Statistics of La Trobe University, and by the NSF of China[Grants 11631007, 11875040, 11801289].