On non-multiaffine consistent-around-the-cube lattice equations

On non-m ultiaffine consisten t-around-the-cub e lattice equations P a vlos Kassotakis ∗ Maciej Nieszporski † No v ember 5, 2018 Abstract W e sho w that integrable invol u tiv e maps, due to the fact they admit three integrals in separated form, can give rise to equations, whic h are consistent around the cube and which are not in the multia ffin e form assumed in pap ers [1, 2 ]. Lattice models, which are discussed here, are related to the lattice p oten tial KdV equ ati on by nonlocal transformations (discrete q uadratures). P acs: 02.30.Ik, 0 5.50.+q Keyw ords: In tegr able lattice equations, Y ang-Bax t er maps, Consistency aro und the cub e Mathematics Sub ject Classification: 82B20, 37K 35, 39A05 1 In tro duction F or a long tim e the electroma gnetic po ten tials, Ψ and A w ere considered as convenien t purely mathematica l concepts without physical significa nce, so to say phantoms, and o nly the ele c t r omagnetic fields, E and B , were considered to b e physical. It w as thought that a change of p oten tial that did not affect the fields did no t cause measurable effects. The discov er y of the Aharonov-Bohm effect ch a nged the situation. The inv estigation of a system of equations tog ether with its potential image is now adays mor e than merely searching for a conv enient wa y of desc r iption. Here we c o ncen trate on the fact that o ne of the most impo r tan t nonlinear difference equations, the lattice po ten tial Korteweg-de V ries (lpKdV) equa tio n (6) (se e [3, 4]), arises a s a potential version of a more fundamen ta l s ystem 1 u ( m + 1 2 ,n +1) = p ( m + 1 2 ) − q ( n + 1 2 ) u ( m + 1 2 ,n ) − v ( m,n + 1 2 ) − v ( m ,n + 1 2 ) , v ( m +1 ,n + 1 2 ) = p ( m + 1 2 ) − q ( n + 1 2 ) u ( m + 1 2 ,n ) − v ( m,n + 1 2 ) − u ( m + 1 2 ,n ) , ( 1 ) where ( m, n ) ∈ Z 2 , so the function u ca n b e rega rded a s g iv en on horiz o n tal edges of Z 2 , v on vertical edges and p , q are given functions of a single v ariable (see Sectio n 5). The sy stem (1) originates fro m paper s where the interrelation b et ween discr ete in tegr able systems a nd set theoretical solutions of the Y ang-Ba xter equa tion (the so ca lled Y a ng-Baxter maps) were inv estigated ∗ Pr esent addr ess: Departmen t of Mathe m atics and Sta tistics Universit y of Cyprus, P .O Bo x: 20537, 1678 Nicosia, Cyprus; e-mails: kassotakis.pavl os@ucy.ac.cy, pavlos1 978@gmail.com † Pr esent addr ess: Katedra Meto d M atematycz nyc h Fizyki, Uniwersytet W ar sza wski, ul. Ho ˙ za 74, 00-682 W arsza wa, Po l a nd; e-mail: maciejun@ fuw.edu.pl 1 The w ord “p oten tial” in the name of the equat ion lpKdV is some r e mnant from classical terminology used in the theory of integrable discretizations of integrable systems and should not b e confused with the meaning of the scalar “poten tial” of a discrete v ector field which w e use throughout the pap er. 1 (see Section 4 for details and a list of r ef er ences). A surpr ise and the main re sult o f this pap er is that the fundamental system (1) has additiona l p otent ia ls and moreover the system can b e easily ex t ended to higher dimensions (24). T o be more sp ecific the sys tem (1) a llo ws us to in tro duce thr ee p oten tials defined on vertices of the lattice, say x , z a nd f , defined resp ectiv ely by u ( m + 1 2 , n ) = x ( m + 1 , n ) − x ( m, n ) , v ( m, n + 1 2 ) = x ( m, n + 1) − x ( m, n ) (2) u ( m + 1 2 , n ) 2 = z ( m + 1 , n ) + z ( m, n ) + p ( m + 1 2 ) , v ( m, n + 1 2 ) 2 = z ( m, n + 1) + z ( m, n ) + q ( n + 1 2 ) (3) u ( m + 1 2 ,n ) 3 − 3 p ( m + 1 2 ) u ( m + 1 2 ,n ) = f ( m + 1 ,n ) − f ( m ,n ) , v ( m,n + 1 2 ) 3 − 3 q ( n + 1 2 ) v ( m,n + 1 2 ) = f ( m,n + 1) − f ( m,n ) (4) Having in mind the philosophica l r epercussio ns that the discov er y of Aharo no v-Bohm effect tr ig gered, we b eliev e that this nontrivial a rbitrariness in intro ducing the scala r p oten tial will b e int er esting in itself. How ever, we focus here on relations that the p otent ia ls satisfy and also we aim to contribute to the mo dern area o f discrete integrable sy stems related to the no tio n of the consistency around the cube prop ert y [1, 5, 6 ]. Namely , in the case of the p oten tial f by confining ours elv es to real- v alued functions a nd everywhere negative para m eter s p and q , we get a n equation on the lattice f 12 = f + ( p − q )  v − u + f 1 − f 2 ( u − v ) 2 + ( p − q ) 2 ( u − v ) 3  (5) u := 3 r f 1 − f 2 + q ( f 1 − f ) 2 4 − p 3 + 3 r f 1 − f 2 − q ( f 1 − f ) 2 4 − p 3 , v := 3 r f 2 − f 2 + q ( f 2 − f ) 2 4 − q 3 + 3 r f 2 − f 2 − q ( f 2 − f ) 2 4 − q 3 , where by subsc r ipt we denote the for w ard shift op erator and at the same time we o mit the indep enden t v ariables in formulas e.g. u ( m + 1 2 , n + 1 ) := u 2 , v ( m + 1 , n + 1 2 ) = v 1 , f ( m + 1 , n + 1) = f 12 (see Figure 3). T o the b est of our knowledge equation (5) is the first ex ample o f a 2 D non-multiaffine consis tent-around- the-cub e lattice equatio n. W e discus s also here a more complicated case of complex-v alued functions. W e sta rt the paper with the basic r elations that the p otentials satisfy (Section 2). Next we r ecall some basic prop erties of the lpKdV equation including its cons istency-around-the-cub e prop ert y (Section 3), follow ed by recalling s o me Y ang- B axter maps related to lpKdV (Section 4). W e stress, anticipating facts , that the Y ang-Ba xter prop ert y is not essential her e. The o rigin o f the p oten tials, namely the existence of sp ecific integrals of the maps, is explained in Section 4 as w ell. The main results of the pap er are in Section 5 and our int ention, while wr itin g this letter, was to keep Sectio n 5 self- c on tained. 2 Lattice mo dels asso ciated with scalar p oten tials Here we determine the rela t io ns that the p oten tials x , z and f satisfy a nd we discuss in what s e ns e these relations are in teg rable. Specific a lly from (1), (2), (3) and (4) w e infer that the p oten tials ob ey ( x 12 − x )( x 1 − x 2 ) = p − q , (6) z 12 = z + ( p − q ) z 2 − z 1 ( u − v ) 2 , (7) f 12 = f + ( p − q )  v − u + f 1 − f 2 ( u − v ) 2 + ( p − q ) 2 ( u − v ) 3  , (8) 2 Equation (6) is referred to as the la ttice po ten tial KdV equa tion [3, 4] and is deno t ed in [1] by H 1. W e can consider the system co nsisting of equations (3), (7) and the o ne that consists of equatio ns (4) and (8) as relations imp osed on functions z and f respectively . Equations (3) and (4) viewed a s the definition of the functions u and v in terms of z a nd f respectively are no t single -v alued in the case of consideratio n of complex-v alued functions (the single-v alued r eal case o f (4 ) and (8) leads to equation (5)). It means that re lations (3), (7) a nd (4), (8) are “ill- defined” and should b e regarded rather as corres pondences than equations (we a re g rateful to the r eferees for this c o mmen t). Instead of that we s uggest a differen t p oin t of view. W e cons ider b o th systems (3), (7) and (4), (8) together with equation (1). This turns corre s pondences (3), (7) and (4), (8) into well-defined lattice mo dels (1), (3), (7) (we shall denote the mo del by G1, the mo del can be fo und in disguise in [7]) and (1), (4), (8) (we shall r efer to this mo del as F1) defined on bo t h vertices and edges of a 2D sq ua re la tt ice (nD lattice in the multid imensio nal extension). Similar mo dels have also b een introduce d in the r ecen t work of Hietarinta and Viallet [8]. W e po stpone to Section 5 a remark on prop er initial v alue problem of mo dels F1 and G1. The ques tion arises: in what sense are these sy s tems in teg r able? It turns out that bo t h G1 and F1 are related to the lpKdV (H1) equation though the dep endence is no t lo cal. Namely , if x ob eys the lpKdV then the function z g iv en by disc r ete quadratures z ( m + 1 , n ) + z ( m, n ) = [ x ( m + 1 , n ) − x ( m, n )] 2 − p ( m ) , z ( m, n + 1) + z ( m, n ) = [ x ( m, n + 1 ) − x ( m, n ) ] 2 − q ( n ) (9) satisfies G1, whereas the function f given by f ( m + 1 , n ) − f ( m, n ) = ( x ( m + 1 , n ) − x ( m, n )) 3 − 3 p ( m + 1 2 )( x ( m + 1 , n ) − x ( m, n )) , f ( m, n + 1) − f ( m, n ) = ( x ( m, n + 1) − x ( m, n )) 3 − 3 q ( n + 1 2 )( x ( m, n + 1 ) − x ( m, n )) (10) is a solution o f F1. So we understand the in tegr abilit y of G1, F1 as the possibility of finding their solutio ns from kno wn solutions of an integrable system (lpKdV in this c a se) by solv ing line ar eq uations (9), (10) resp ectiv ely . W e call the equations (9) and (10) non-a ut o -B¨ acklund tr a nsformations betw een (6) a nd G1, F1 resp ectiv ely . How ever, there is a tendency to re fer to suc h transformations as to Miura-type transformations (compare [9, 10]). Note that non-auto-B¨ acklund transformations betw een equations in the Adler-Bob enk o- Suris (ABS) list [1] are presented in [11, 12, 9]. Moreov er , as we alr eady mentioned, eq ua tion H1 and the mo dels G1 and F1 which are g iv en on a tw o- dimensional lattice, can be extended to a co mpatible system on an n-dimensio nal lattice, see Section 5. This prop ert y (of “c o mpatible extendibility” to higher dimensio ns) as it has b een pointed out in [13, 1 4 , 5, 6, 1] is a hallmar k o f integrabilit y . In other w o rds, following the terminolo g y introduced in [1], equation H1 and mo dels G1 and F1 p ossess the pro perty of consistency a r ound the cub e. T o the end of this section w e mention tha t the three c ases above ca n be combined giving rise to a family of p oten tials ψ c 0 + c 1 u + c 2 ( − 1) m + n ( u 2 − p ) + c 3 ( u 3 + pu ) = ψ 1 − ψ , c 0 + c 1 v + c 2 ( − 1) m + n ( v 2 − q ) + c 3 ( v 3 + q v ) = ψ 2 − ψ , (11) where c i ∈ C , i = 0 , . . . 3 . 3 Consistency around the c ub e of the lattice p oten tial KdV equa- tion Here w e recall some essential facts concerning the co ns istency around the cub e of the lattice p oten tial KdV equation and ex plain how the mo dels F1 and G1 differ from equations in the ABS list. All the 3 formulas c a n be found in the clas sical pap er by W ahlquist and Es t a brook [15]. How ever, one sho uld b e aw are of an ingenious obse r v ation that B¨ acklund tr ansformations ca n b e reinterpreted as increments in additional discrete v aria bles [16] and the difference-differ en tial or fully discre t e equations obtained in this wa y are integrable. With this observ ation the nonlinear s up erp osition principle for the potential KdV w t = 6( w y ) 2 − w y yy presented in [1 5 ] can b e reinterpreted as a difference equation, namely the lpK dV (6). Going further in reinterpretation of the pap er by W ahlquis t and Es ta brook one can consider Z 3 lattice with three copies of equation (6) given on it w ( m + 1 , n + 1 , s ) = w ( m, n, s ) + p − q w ( m + 1 , n, s ) − w ( m, n + 1 , s ) w ( m + 1 , n , s + 1) = w ( m, n, s ) + p − λ w ( m + 1 , n, s ) − w ( m, n, s + 1) w ( m, n + 1 , s + 1) = w ( m, n, s ) + q − λ w ( m, n + 1 , s ) − w ( m, n, s + 1) (12) There are three different wa ys in which we can o btain w 123 (according to our no t a tion w 123 ≡ w ( m + 1 , n + 1 , s + 1 )) in terms of w , w 1 , w 2 , w 3 . Due to the suitable choice of for m of the n umera to rs in formulas (12), no -matter which path we follow, the r esulting v alue w 123 is the sa me (see figure 1 ). This prop ert y is referred to as 3D-consistency or consistency around the cub e [1, 5, 6]. W e have [15]: w 123 = pw 1 ( w 2 − w 3 ) + q w 2 ( w 3 − w 1 ) + λw 3 ( w 1 − w 2 ) w 1 ( w 2 − w 3 ) + w 2 ( w 3 − w 1 ) + w 3 ( w 1 − w 2 ) . (13) A t this moment the indep e nd enc e o f w 123 on w in for mula (13) s hould be underlined. This pr operty is referred as the tetrahedro n pr operty since equation (13) relates four p oin ts of the 3D-lattice only . Finally , one can interpret the tw o last eq ua tions o f (12) as a B¨ acklund transformation for the first one. It is necessary to mention that b oth the la st equatio ns of (12) a re first o rder fractio nal linear differ e nc e equations (discr ete Riccati e q uations), when reg a rded a s recurrence relatio ns in the v aria ble s , hence their solving can be reduced to solving linear equations only . Extensions of the ABS list, interesting structure s c losely related to the consistency-ar ound-the-cube prop ert y and s olutions of the equa t io ns from the list, have b een studied since the publication o f pap er [1 ] (see [17, 18, 6 , 19, 20, 21, 22, 23, 2 4 , 25, 26, 27, 28, 2 9, 3 0, 3 1, 1 1, 32, 10]). w 23 w 2 w 123 w 3 w 1 w 12 w w 13 k 2 k 1 k 3 Figure 1: 3-dimensional cons is tency Recapitulating, for the system (12) we hav e 4 1. Each o f the equations include tw o pa r ameters 2. Due to sp e c ific de p endence on the pa rameters the system is c o mpatible (consis ten t around the cub e) 3. The tw o last equations of (12) are fractional line a r recurr ence relations in the v ariable s 4. F orm ula (13) relates w 123 with w 1 w 2 and w 3 only , the so called tetrahedro n pr operty . Equation (5) and mo dels G1 and F1 poss ess prop erties 1. and 2. only . It means for instance that equation (5) is not in the form assumed in [1, 2], namely it is not in the for m Q ( x 12 , x 1 , x 2 , x ) = 0 where Q is a m ultiaffine p olynomial i.e. polynomia l of degree o ne in each argument. 4 In v olutiv e maps There are several pro cedures for obtaining integrable mapping s [33] from in tegr able lattice equations [34, 35, 36, 3 7 , 38, 39, 40]. F ollowing the pro cedure of [3 9 , 40] one can get inv olutive maps whic h are set-theor etical solutions o f the Y ang- B axter equatio n the so-called Y ang -Baxter maps [4 1 , 42, 3 9 , 43]. W e complement this pro cedure with a systematic wa y of finding lattices from inv olutive mappings. W e confine o urselv es to a family of inv olutive maps C 2 ∋ ( u, v ) 7→ ( U, V ) ∈ C 2 (see fig ure 2) whic h are related to the Y ang- Baxter map denoted by F V in the clas sification list of qua drirational Y ang-Baxter maps given in [39]. ( F V ) : U = v + p − q u − v , V = u + p − q u − v (14) Apart from F V , we num b er a mong the family , the Y ang-B axter map denoted by H V in article [43] (c.f. u v U V Figure 2: A map on C 2 . [42]) ( H V ) : U = v + p − q u + v , V = u − p − q u + v (15) and its companion (i.e. the map ( u, v ) → ( U, V ) that ar is es from the switching of v and V in (15)) ( cH V ) : U = − v + p − q u − v , V = − u + p − q u − v . (16) Note that (16) is not a Y ang-B axter map [43], and in order to illustra te that the Y a ng -Baxter pr o perty is not crucial from the po in t of view of the pro cedures describ ed in this pap er, we deal here with the map (16) mainly . 5 The sta ndard pro cedure fo r r ein terpretation of a map as equations on a lattice is based on the ide ntifi- cation (see Figure 3.) u = u ( m + 1 2 , n ) , v = v ( m, n + 1 2 ) , U = u ( m + 1 2 , n + 1) , V = v ( m + 1 , n + 1 2 ) . (17) F or the map (16) we get ex a ctly the system (1). v 1 u u 1 u 22 u 122 v 2 v v 112 v 11 u 2 u 12 v 12 Figure 3: The map as a lattice. It is imp ortant for us to find functions F and G such that F ( U ) + G ( V ) = f ( u ) + g ( v ) . (18) In the case of the map (16) it leads to an equation in F and G (w e differen tiate (18) with r espect to u and v ) F ′′ ( U )[( U − V ) 2 − ( p − q )] − 2 F ′ ( U )( U − V ) − G ′′ ( V )[( U − V ) 2 + ( p − q )] − 2 G ′ ( V )( U − V ) = 0 . (19) T o find a s o lution o f equatio n (19) that is v alid fo r all v alues of U and V w e observe tha t the necessary conditions that the functions F and G should satisfy are F ′′ ( U )[ U 2 − ( p − q )] − 2 U F ′ ( U ) = c 1 U 2 + c 2 U + c 3 G ′′ ( V )[ V 2 + ( p − q )] − 2 V G ′ ( V ) = c 4 V 2 + c 5 V + c 6 (20) where the c i are s o me constants. Solving the ODEs for F(U) and G(V) and plugg ing the r esults into equation (19) we find the following general solution of (19) F ( U ) + G ( V ) = c 1 ( U − V ) + c 2 ( U 2 − V 2 ) + c 3 ( U 3 + pU − V 3 − q V ) + c 4 . (21) As a result we arrive at the following in tegr als with additively separ ated v ar iables H 1 ( u, v ) = u − v , H 3 ( u, v ) = u 3 + pu − v 3 − q v, (22) and a 2-integral with separa ted v ar iables as well H 2 ( u, v ) = u 2 − v 2 . (23) After the reinterpretation (17) of the map as a system of eq uations on a lattice, in teg rals (22) and (23) will give rise to equations that guara n tee the existence of p oten tials. 6 The q ue s tion arises whether o ne can extend the sys tem (1) to a c ompatible multid imensio nal system on the Z n lattice. T he answer is p ositive and we will take up this is sue in the next section. W e end the sectio n with a pro position of a n eq uiv alence rela t io n in the set of 2D maps. According to the prop osition all three maps (14), (15) and (16) are equiv alent. Prop osition 1 Two 2D maps ar e e quivalent if their systems arising fr om identific ation (17), let say on function ˜ u i and u i , c an b e r elate d by an invertible p oint t r ansformation: ˜ u i = F i ( u 1 , u 2 , m, n ) , i = 1 , 2 , ˜ m = m, ˜ n = n. 5 Idea I V and its idolons W e co nsider the Z n = { ( m 1 , . . . , m n ) : m i ∈ Z , i = 1 , . . . , n } lattice tog e ther with its edges i.e. segments connecting tw o consec ut ive p o in ts. The edges can b e viewed as (and desc r ibed by) a pair of vertices. W e refer to the elements of the set of ordered pairs of p oin ts { (( m 1 , . . . , m i , . . . , m n ) , ( m 1 , . . . , m i + 1 , . . . , m n )) : m i ∈ Z , i = 1 , . . . , n } , as edges in the i-th direction. W e denote by subscript the forward shift in the i-th direction T i i.e. T i f ( m 1 , . . . , m i , . . . , m n ) ≡ f i ( m 1 , . . . , m i , . . . , m n ) = f ( m 1 , . . . , m i + 1 , . . . , m n ). W e omit independent v ariables to make the formulas shorter (e.g. T i f ≡ f i ). By ∆ i we denote the fo rw ard difference op erator ∆ i = T i − 1. W e take into consider ation n functions u i , i = 1 , . . . , n (mind that we enumerate functions by a s uper- script!). The i-th function u i is g iv en on e dg es in the i-th directio n o nly . A function u = ( u 1 , . . . , u n ) from the set of edges to R n (or C n ) can b e viewed as a vector field on a lattice. W e a re searching fo r functions that ob ey the following system of difference equa t io ns ( I V ) : u i j + u j = p i − p j u i − u j i, j = 1 , . . . , n i 6 = j (24) where the p i are g iv en functions of the i-th a r gumen t o nly ( p i ( m i + 1 2 )). T o emphasise the meaning of the system (1 ) and its mu ltidimensio nal extension (24) and the fact that the system ca n hav e v arious po ten tial “r eflections” we sug g est to adopt P lato’s terminology a nd refer to the sys t em as the Ide a system (or the representative of an Ide a , see Pr o position 2) a s sociated with the map (16) and to underline the connection with (16) we deno te it by I V . Whereas the mo dels H1, G1 and F1 (and their higher dimensional extensions, see b e lo w) we refer to as ido lons of the Idea system (keeping the original ancie nt Greek form of the word idol and having in mind rather idols from Baco n’s works than any contempor ary meaning of the word). Note that in this termino logy we do no t exclude the case where a given idolon participates in several Ideas. The following facts hold • system (24) is compatible i.e. u i j k = u i kj i, j, k = 1 , . . . , n i 6 = j 6 = k 6 = i (25) Prescr ibin g the v alue of each function u i on the i -th initial condition line ( m j = 0 for j 6 = i ) a nd using (24) one can find recur siv ely u i in the whole domain (singularities ca n o ccur because of the v a nishing of the deno mina tor on the right hand side of (24)). 7 • The following equality holds ∆ j u i = ∆ i u j i, j = 1 , . . . , n i 6 = j (26) which implies the existence of a p otent ia l x (given on vertices of the lattice) such that u i = ∆ i x i = 1 , . . . , n (27) In terms of the po ten tial x the sys t em (2 4 ) rea ds ( x ij − x )( x i − x j ) = p i − p j i, j = 1 , . . . , n i 6 = j (28) so we g e t the system of lpKdV equations. Since the e xistence of x is gua r an teed for any initial data for x (ex c luding the ones that leads to u i − u j = 0) on initial lines, system (28) is nD-consistent. • F rom sys tem (24) one infers that ( T j + 1)  ( u i ) 2 − p i  = ( T i + 1)  ( u j ) 2 − p j  i, j = 1 , . . . , n i 6 = j (29) Equations (29) imply the ex istence of a p oten tial z (given o n v er tices of the la tt ice) s uch that ( u i ) 2 − p i = ( T i + 1) z i = 1 , . . . , n (30) In terms of po ten tial z the system (24) can b e rewr itt en as z ij − z = ( p i − p j ) z j − z i ( u i − u j ) 2 i, j = 1 , . . . , n i 6 = j. (31) W e g et the sy stem of equations (24), (30) and (31), which we refer to as sys t em of G1 mo dels a nd which is nD-consistent. An explicit ma nif esta tion of 3D-consistency is the formula z ij k + z = [ p i u i ( u j − u k )+ p j u j ( u k − u i )+ p k u k ( u i − u j )] 2 [ p i ( u j − u k )+ p j ( u k − u i )+ p k ( u i − u j )] 2 + [ u i ( p j − p k )( p j + p k − p i )+ u j ( p k − p i )( p i + p k − p j )+ u k ( p i − p j )( p i + p j − p k )] p i ( u j − u k )+ p j ( u k − u i )+ p k ( u i − u j ) . Finally , we infer that z must sa tisfy ( z ij − z ) 2 ( z i − z j + p i − p j ) 2 + ( p i − p j ) 2 ( z i − z j ) 2 + 2( z ij − z )( p i − p j )( z i − z j )( z i + z j + 2 z + p i + p j ) = 0 (32) compare with [7 ]. • The third identit y giving rise to a po t ential is ∆ j  ( u i ) 3 + a i u i  = ∆ i  ( u j ) 3 + a j u j  i, j = 1 , . . . , n i 6 = j (33) where a i are functions of the i-th argument only ( a i ( m i + 1 2 )) such that a i − a j = 3( p j − p i ) i, j = 1 , . . . , n i 6 = j (34) In fact by the introduction of the a i functions we tacitly smuggled in the p ossibilit y of combination of integrals (22). Equations (33) imply the existence of a po ten tial f (g iv en o n vertices o f the la tt ice again) such that ( u i ) 3 + a i u i = ∆ i f i = 1 , . . . , n (35) 8 In terms of the po ten tial f the s ystem (24) can b e rewritten as f ij − f = ( p i − p j )  ( p i − p j ) 2 ( u i − u j ) 3 + f i − f j ( u i − u j ) 2 − u i + u j  i, j = 1 , . . . , n i 6 = j. (36) Just like in the previo us cas e s we g et the sys tem o f equations (24), (35) and (36), which we re fer to as s ystem o f F1 mo dels, system which is nD-consistent. The 3D-c o nsistency is manifested in the formula f ij k − f = ( p i u i u jk + p j u j u ki + p k u k u ij ) 3 ( p i u jk + p j u ki + p k u ij ) 3 − 3 p ij p jk p ki p i u jk + p j u ki + p k u ij + 3 { p i u i p ij p ki [( u j ) 2 +( u k ) 2 ]+ p j u j p ij p jk [( u k ) 2 +( u i ) 2 ]+ p k u k p jk p ki [( u i ) 2 +( u j ) 2 ]+2 u i u j u k [( p ij − p k )( p jk − p i )( p ki − p j )+ p i p j p k ] } [ p i u jk + p j u ki + p k u ij ] 2 where for the sake of brevity we hav e intro du ce d nota t io n u ij := u i − u j and p ij := p i − p j . The real ca se with p ositiv e pa rameters a i provides us with an example o f a system of non-multiaffine consistent-around-the-cub e lattice equa t io ns (5). • The prop er initial v alue problem for systems of mo dels H1, G1 a nd F1 is to pre scribe a v alue of a po ten tial at the intersection o f initial lines and v alues of u i on the edges that b elong to the initial lines. Then we have a unique solution to the models. Howev er, w e find it interesting to discuss the case when the v a lues of po ten tials are given on the initial lines. In the case of complex- v alued functions, we ca n find from equations (27), (30 ) o r (35), r espectively at mos t k = 1 , 2 or 3 v alues of u i on an e dg e b elonging to the initial line. Therefore for an elementary n-cube having prescrib ed v alues of po ten tial at a given vertex a nd its clo sest neighbo urs one can get at mo st k n v alues of the po ten tial a t the farthest (from the given one) vertex. • One can combine all of the cases ab o ve by int r oducing the family of p oten tials ψ c 0 + c 1 u i + c 2 ( − 1) m 1 + ... + m n [( u i ) 2 − p i ] + c 3 [( u i ) 3 + p i u i ] = ψ i − ψ i = 1 , . . . , n (37) W e p ostpo ne the inv estigation of the lattice eq uation on ψ , which includes four par ameters c 0 , c 1 , c 2 and c 3 , to a for t hco ming pap er [44]. • Eliminating u i from (30) a nd from (35) b y mea ns of (27 ) we get ( T i + 1) z = (∆ i x ) 2 − p i i = 1 , . . . , n (38) ∆ i f = (∆ i x ) 3 + a i ∆ i x i = 1 , . . . , n (39) from whic h we infer – If x ob eys ∆ i ∆ j x = 0 then f giv en by (39) ob eys ∆ i ∆ j f = 0 as w ell. – If x ob eys (2 8 ) then f giv en by (39) ob eys (36). – If x ob eys (2 8 ) then z given by (29) ob eys (31). • If we replace ( p i , u i ) with ( − p i , ( − 1 ) m 1 + ... + m n u i ) we get u i j − u j = p i − p j u i − u j (40) In the tw o- dimens io nal ca se, after making the identification (17), one obtains the F V map (14 ). The whole pro cedure we hav e describ ed so far can b e r e peated using the F V map instead of the cH V map. Since “entities must not be m ultiplied be yond necessity” w e arr iv e at the pro position. 9 Prop osition 2 Two ide a systems , let’s say on funct i ons ˜ u i and u i , ar e e quivalent iff they ar e r elate d by an invertible p oint tr ansformation ˜ u i = F i ( u 1 , . . . , u n , m 1 , . . . , m n ) , ˜ m i = m i , i = 1 , . . . , n (41) An e quivalenc e class of ide a systems we re fer to as Ide a. Idea systems (24) and (40) ar e representativ es of the same Idea, the Idea as sociated with the family of maps discussed in previous section. W e denote the Idea by I V just like its repre s en tative (24 ). • If we replace u 2 with − u 2 we get equations u i 2 = u 2 + p i − p 2 u i + u 2 , u 2 i = u i − p i − p 2 u i + u 2 i 6 = 2 which in the t wo-dimensional case, after making the iden tification (17), changes to the H V map (15). How ever, it is not p ossible to turn a ll equations int o the form u j i = u i − p i − p j u i + u j . In this case u j ik 6 = u j ki . 6 Commen ts W e fo cused in this pa p er o n tw o mo dels G1 a nd F1 which a re multidimensionally consistent. The mo dels are related to the sys tem o f lattice p oten tial K dV equatio ns (28) by non-auto-B ¨ acklund transformatio ns (38) and (39 ). Equation (5) and its multidimensional extension is a particular (real) ca se of F1. Because of the la c k o f the tetrahedr on prop erty , for n > 2, equation (5) c a nnot b e r elated by a p oin t transfor mation to a n y sys tem of equatio ns from Adler-Bob enk o- Suris list [1]. An op en question is whether o ne can find difference substitution (i.e. a gener alization of po in t tr ansformations (4 1 ) where functions F i can dep end on functions u i given in several p oin ts) that reduce solving system (31) or system (36) to solving equations from the Adler-Bob enko-Suris list or to solving linear equations. W e shall pick up this issue in the near future. There is no doubt that the notion of co nsistency (compatibility) is an impo rtan t ingredient o f the theo ry of in tegr able systems. How ever under standing consistency itself as in tegr abilit y ca n lead to confusion. One of the main features of integrable systems is that their so lv ability can be r educed to solv ing linear equations. Leaving the assumption of a multiaffine form o f the q ua d-graph equa tion we face terr a inc o gnita . One cannot exclude the situation when the consisten t equation cannot be linked to any integrable equatio n b y a linearizable transformatio n. It ca n pro duce equations with essentially nonlinear B¨ a c klund transfo rmations and that is why the co nception of consistency aro und the cube is interesting in itself. W e ta k e the stand that we should not mix it with integrability at this stage. The cla ss o f in teg rable s ystems may b e iden tical with the class of consistent-around-the-cub e equations o r may b e not. F or now we are going to confine ourselves to systems that are linked to the ABS list by a lineariza ble B¨ acklund or Miur a transfor mation (see e.g . [1 1, 9]) i.e. by a fractiona l linea r difference equation. So we are going to extend table 3 of pap er [11] (see also [9]) to non- multiaffine ca ses. In a for t hco ming paper we deal with the simplest case when the link is given by dis crete quadratur es (so to say “ s implest” fractional linear transforma t io n). It will also cov er multip lica tiv e c ases, for instance the substitution u = x 1 x, v = x 2 x (42) 10 and after making the identifi ca tion (17) changes the lpKdV equation (6) in to the Y ang-Bax t er map F I V from the list given in [39] V = u  1 + p − q u − v  , U = v  1 + p − q u − v  . (43) The most general function with prop ert y (18) for this map is a linear combination of functions log U V = log v u , U − V = − ( u − v + p − q ) , U 2 − V 2 + 2 pU − 2 qV = − ( u 2 − v 2 + 2 pu − 2 q v + p 2 − q 2 ) (44) and a constant function. I t lea ds to the following po ten tials x , y and z u = x 1 x, v = x 2 x 2 u + p = y 1 + y , 2 v + q = y 2 + y u 2 + 2 pu + 1 2 p 2 = z 1 + z , v 2 + 2 q v + 1 2 q 2 = z 2 + z (45) The first p oten tial satisfies the lpKdV equation (6), the second p oten tial satisfies e quation H 2 from the ABS list [1] ( y 12 − y )( y 1 − y 2 ) − ( p − q )( y 12 + y 1 + y 2 + y ) + p 2 − q 2 = 0 and the third one giv es rise a nother non-multiaffine mo del we denote tent a tiv ely denote by G 2 for eq uiv alence and classification in the clas s of non-multiaffine equatio ns is yet to b e defined z 12 − z = p − q u − v h 2 uv + 1 2 ( p + q )( u + v ) + uv p − q u − v i . (46) The Idea asso ciated with F I V is (we denote it by I I V ) ( I I V ) : u i j = u j  1 + p i − p j u i − u j  i, j = 1 , . . . , n i 6 = j (47) W e see that idolo n H 1 participates b oth in the I I V and I V Ideas which we illustra t e in the final figure, Figure 4. I V I I V F 1 G 1 H 1 H 2 G 2 Figure 4: Ideas I I V and I V and their idolons Ac kno wle dgemen ts W e are gra tef ul to the o rganizers of the Isaac Newton Institute for Ma t hema t ica l Sciences, prog ramme of Discrete Integrable Systems, for giving us the o pportunity to take part in this even t, extra ordinary in every way , shap e and form. W e are also grateful for the supp ort of the or ganizers of the 9th SIDE meeting in V a r na, whe r e the pap er gained its tentativ e form. MN would like to express gratitude to T o masz Steifer for fruitful discussio ns on Plato ’s ideas and idolo ns and referring us to the Liddell & Sco tt Gr eek lex icon a nd B acon’s works. MN also thanks Paolo Maria Santini, Decio Le vi and F rank Nijhoff not only for literature guidelines but most of all fo r encour agemen t when our ardo ur b egan to co ol. P .K. thanks T asos T ongas for his useful comments and for referr ing us to ar ticle [7]. 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