Families of Integrable Equations

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 7 (2011), 100, 14 pages F amilies of In tegr able Equations ⋆ Pavlos KASSOT AKIS † and Maciej NIESZ P ORSKI ‡ † Dep artment of M athematics and Statistics University of Cyprus, P.O. Box: 20537 , 1678 Nic osia , Cyprus E-mail: kassotakis.p avlos@ucy.ac.cy , p avlos1978@gmail.c om ‡ Kate dr a Meto d Matematy cznych Fizyki, Uniwersytet Warszawski, ul. Ho ˙ za 74, 00-682 Warszawa, Poland E-mail: maciejun@fuw.e du.pl Received May 23, 2011, in f inal for m Octo b er 20 , 201 1 ; Published online Octob er 28 , 2 0 11 ht tp://dx.doi.or g/10 .3842/ SIGMA.2011.100 Abstract. W e pr esent a metho d to obtain families of lattice equations. Sp e cif ically we fo cus on tw o of such families, which include 3-para meters and their members ar e connected through B¨ acklund transformatio ns. At least one of the members of each family is int egrable, hence the whole family inher its some integrability prop erties. Key wor ds: in tegrable la ttice equations; Y ang–Baxter ma ps ; cons istency a round the cube 2010 Mathematics Subje ct Classific ation: 82B20 ; 37K35; 39A05 1 In tro duction Discrete mathematics retur n ed on the interest of mathematicians at the b eginning of the 20 th cen tury . P oincar ´ e, Birkhof f , Ritt (1924) [1], J ulia, F a tou (1918–1 923) [2, 3] and man y others sa w the necessit y of exploring the d iscrete scene. Un fortunately , this trend was paused thr ough the tw o big wars and only after 1960, k eeping p ace w ith the rev olution caused by the disco v ery of soliton from Zabusky and Kr usk a l [4], mathematicians started to inv estig ate discrete systems in the con text of integrable systems. It w as the w ork of Hirota [5], as w ell as Ablowit z et al. [6] and s eparately Cap el and his sc ho ol [7], whic h introdu ced lattice and dif ferentia l dif ference analogues of man y integrable PDE’s. The int ro du ction of discrete v ersions of in tegrable ODE’s, surpr isingly , came late r with the QR T family of mappings by Quisp el, Rob erts and Thomson [8] and by the work of P apageorgi ou et al. [9, 10], where Liouville integ rable map s [11] w ere obtained by imp osing p erio d ic staircase initial data on integrable lattices. Another wa y to obtain integ rable mappings from an int egrable lattice equation was suggested in s eries of pap ers [12, 13, 14]. Actually with this pro cedur e one can get in v olutiv e mapp ings (comp osition of the map with itself is th e iden tit y map) whic h are set th eoretical solutions of the qu an tum Y ang–Ba xter equation the so called Y a ng–Baxter m ap s [15, 16, 12, 17 ]. As in our previous w ork [18], we fo cus here on the in v erse pr o cedure, i.e. how to obtain in tegrable lattice equations from in v olutiv e mapp ings that ma y or ma y not satisfy the Y ang–Ba xter equation. The main result of th e pap er is that th e pr o cedure we hav e in mind can lead to families of equations. It is n ecessary to mentio n that p oin ts of the lattic e ma y not b e related in a uniqu e w a y . The members of families are r elated by a B¨ ac klund transformation (see Section 6 ) and since in considered cases at least one of the members is in tegrable, the whole family inh erits ⋆ This paper is a contribution to the Proceedings of the Conference “Sy mmetries and Integrabilit y of Dif ference Equations (SIDE-9)” (June 14–18, 2010, V arna, Bulgaria). The full collection is av ailable at http://w ww.emis.de/j ournals/SIGMA/SIDE-9.html 2 P . K assotakis and M. Nieszp orski some prop erties from the distinguish ed mem b er. W e s tr ess that solutions of eac h mem b er of th e family can b e obtained fr om s olutions of the integ rable member by discrete quadratures (which can b e regarded as sort of B¨ ac klund transformation) and in this sense eac h memb er of the family is integrable. Ho w ev er, w e discus s here h allmarks of in tegrabilit y of the memb er s of the family suc h as consistency aroun d the cub e prop erty or τ -function f orm ulation. Notion of the family of discrete int egrable systems sh ould not b e confused w ith notion of h ierarchies of in tegrable systems. The later notion was wid ely inv est igated in th e literature w hereas for the former one w e can indicate only the articles that in v estigate the family of d iscr ete K dV equations [19] and the family of discrete Boussinesq equations [20, 21, 22]. W e discuss h ere tw o examples, the f irst one is con tin uatio n of our p revious pap er [18]. W e in tro duce a family of dif ference equations asso ciated with t yp e I I I of map s discuss ed in [12, 13] (w e introd uced families related to typ es IV and V in [18]). Example of the map of type I I I is a map C 2 ∋ ( u, v ) 7→ ( U, V ) ∈ C 2 U = v pu − q v q u − pv , V = u pu − qv q u − pv . and the three parameter family of equations (see Section 3) r eads ψ 12 = ψ + a ln pu − q v q u − pv +  p 2 − q 2   b uv q u − pv − c 1 pu − q v  , (1.1) where u and v are give n implicitly by a ln u + p  bu + c 1 u  = ψ 1 + ψ , a ln v + q  bv + c 1 v  = ψ 2 + ψ , (1.2) function ψ is dep enden t v ariable on Z 2 and we denote ψ ( m, n ) =: ψ , ψ ( m + 1 , n ) =: ψ 1 , ψ ( m, n + 1) =: ψ 2 , ψ ( m + 1 , n + 1) = : ψ 12 , p := p ( m ) and q := q ( n ) are given functions of a single v ariable and a , b and c are arbitrary constants (w e assume that one of the constants a , b or c is not equ al to zero). How ever, ough t to p ossible branching in formulas (1.2), the system (1.1), (1.2) needs sp ecifying (as it was p oin ted us by Pr ofessors F r an k Nijhof f and Y ur i Sur is). Th e s p ecif ication is ac hiev ed by demandin g that functions u and v ob ey u 2 = v pu − q v q u − pv , v 1 = u pu − q v q u − pv . (1.3) After this sp ecif ication th er e is still some fr eedom left in f indin g solutions of (1.1), (1.2) for giv en in itial conditions on ψ . T h e freedom lies in f inding the initial conditions for u and v out of initial conditions on ψ b y means of (1.2). Th e s olution need not to b e uniqu e. All th e equations within the family are consisten t around the cub e (for the consistency around the cub e prop ert y see [23, 24, 25, 26], notice we resign f rom multia f f init y assump tion of pa- p er [25]). W e f ind esp ecially interesti ng the fact that w e obtain examples of lattice equations together with transform ations whic h can b e regarded as B¨ ac klund tran s formations but not in the usual sense; we usually require B¨ ac klu n d transformation to b e linearisable (see Def inition 5) and this requirement is violated in these examples. Th er efore L ax p air could not b e easily found from this sort of B¨ ac klund transform ation and it is not clear if th e L ax pair exists in these cases. Mem b ers of the family are Hirota’s sine-Gordon equation (c hoice of parameters b = 0 = c ) referred also to as lattice p otentia l m o dif ied KdV [27, 28, 29, 30, 31, 14, 32] (see Section 2 where w e discuss v arious forms of lattice equations) p ( xx 1 + x 2 x 12 ) = q ( xx 2 + x 1 x 12 ) F amilies of In tegrable Equations 3 and lattice Sch w arzian Kd V [28] in a d isguise, see Section 2 (c hoice of p arameters a = 0 = b or a = 0 = c ) p 2 ( y 12 + y 1 )( y 2 + y ) = q 2 ( y 12 + y 2 )( y 1 + y ) . In the second example we go aw ay from the maps of pap ers [12, 13] and consider the map U = v + k  1 − v u  , V = u + k  − 1 + u v  , whic h giv es also a 3 p arameter family of equations (see Section 5) including Hirota’s Kd V lattice equation [5] x 12 − x = κ  1 x 2 − 1 x 1  and t w o fur ther bilinear equations y 1 y − y 12 y 1 = κ ( y 12 y + y 1 y 2 ) , z 12 z + z 1 z 2 = z 12 z 2 + z 12 z 1 . In this case an interesting fact is that th e pr o cedure yields τ -function representat ion of the family (see e.g. [19]) τ 112 τ − κτ 11 τ 2 = τ 12 τ 1 , τ 122 τ + κτ 22 τ 1 = τ 12 τ 2 . In S ection 2, we giv e an o v erview of p oint transform ations, B¨ ac klun d transformations and dif ference sub stitutions and touch the issue of equiv alence of lattice equations. W e pro ceed in Section 3 where w e pr esen t the metho d that leads to families of lattice equ ations. In Sec- tion 4 w e r elate our f indings to some results of the pap ers [12 , 13], follo w ed by Section 5 wh er e we deal with Hirota’s Kd V lattice equ ation. T hen we exp lain ho w to get B¨ ac klund transformation b et w een memb ers of the families (Section 6 ) and we end the pap er with some conclusions and p ersp ectiv es for fu ture work. 2 P oin t transformations, dif ference substitutions, B¨ ac klund transformations and equiv alence of lattice equations Before we start w e wo uld lik e to giv e some def initions and recall some w ell kno wn r elations [29, 30, 31, 19, 32] b et w een equations that app ear in the article (terminology used by v arious au- thors is far from b eing u nif ied). Let us consider k d ep endent v ariables of n indep endent ones: u i ( m 1 , . . . , m n ), i = 1 , . . . , k . W e denote M ≡ ( m 1 , . . . , m n ). Definition 1 (c hange of in d ep end en t v ariables) . By change of indep endent variables we u n der- stand the bijection f : Z n → Z n ˜ m i = f i ( M ) , i = 1 , . . . , n. 2D examples are ˜ m 1 = m 1 , ˜ m 2 = m 1 + m 2 , or ˜ m 1 = m 1 + 2 m 2 , ˜ m 2 = m 1 + m 2 . Definition 2 (p oin t transformations not altering ind ep endent v ariables) . By p oint tr ansfor- mation not altering indep endent variables we und erstand an inv ertible map F b etw een subsets of C k ˜ u i ( M ) = F i  u 1 ( M ) , . . . , u k ( M ); M  , i = 1 , . . . , k . 4 P . K assotakis and M. Nieszp orski Definition 3 (equiv alence of lattice equations) . Tw o lattice equations are e quivalent if and only if there exists comp osition of p oin t transformation with change of ind ep end ent v ariables whic h maps solutions of one equation to solutions of the second one. Examples of v a rious d isguises of th e same equation are • Hirota’s sine-Gordon equation q sin( ψ 12 + ψ − ψ 1 − ψ 2 ) = p sin( ψ 12 + ψ + ψ 1 + ψ 2 ) turns into ( H 3 0 ) : p ( xx 1 + x 2 x 12 ) = q ( xx 2 + x 1 x 12 ) (2.1) H 3 0 equation from ABS list [25] by means of p oint transformation x = i m + n e 2 i ( − 1) n ψ . H 3 0 in turn can b e transformed into lattice p oten tial mo dif ied Kd V p ( ww 1 − w 2 w 12 ) = q ( w w 2 − w 1 w 12 ) b y sub stitution x = i m + n w . • Sc h w arzian Kd V equation (or cross ratio equation, or equation Q 1 0 on ABS list) ( z 12 − z 1 )( z 2 − z ) ( z 12 − z 2 )( z 1 − z ) = q 2 p 2 under the p oin t transform ation z = ( − 1) m + n y turns into ( A 1 0 ) : p 2 ( y 12 + y 1 )( y 2 + y ) = q 2 ( y 12 + y 2 )( y 1 + y ) (2.2) whic h in the pap er [25 ] got its own name A 1 0 . Definition 4 (dif ference substitutions) . Let j p oin ts M i , i = 1 , . . . , j of a lattice are giv en. By differ enc e su b stitution of or der j w e under s tand a tr an s formation ˜ u i ( M ) = F i  u 1  M 1  , . . . , u k  M 1  , . . . , u 1  M j  , . . . , u k  M j  ; M  , i = 1 , . . . , k . Ev ery p oint tr an s formation is dif ference su bstitutions of order 1. Standard examples of dif ference sub stitution (of ord er 2, 3 and 4 resp ectiv ely) are • p otentia l r elation v = 1 α − β ( u 2 − u 1 ) b et w een lattice p oten tial KdV ( u 12 − u )( u 1 − u 2 ) = α 2 − β 2 and Hirota’s dif f er en ce Kd V v 12 − v = α + β α − β  1 v 1 − 1 v 2  ; F amilies of In tegrable Equations 5 • Miura-t yp e trans f ormation v = β ψ 2 − αψ 1 ( β − α ) ψ b et w een H 3 0 (Hirota’s sine-Gordon or lattice mo d if ied p otent ial Kd V) α ( ψ 2 ψ 12 − ψ ψ 1 ) = β ( ψ 1 ψ 12 − ψ ψ 2 ) and Hirota’s dif f er en ce Kd V; • and f inally th e in tro duction of τ f u nction v = τ 12 τ τ 1 τ 2 , whic h transform ev ery solution of the compatible system τ 112 τ − κτ 11 τ 2 = τ 12 τ 1 , τ 122 τ + κτ 22 τ 1 = τ 12 τ 2 to solution of Hirota’s dif ference KdV. T o the end we prop ose dr aft def inition of B¨ ac klund transform ation which is con v enien t for our purp oses. How ever we are a w are that the d ef inition is not exhaustiv e (some transformation that deserv e this name can b e not co v ered by the def inition). Definition 5 (B¨ ac klun d transformations (in narrow sense)) . By B¨ acklund tr ansformation w e un d erstand h ere a transf ormation b et w een tw o equations F ( u 12 , u 1 , u 2 , u ) = 0 and ˜ F ( ˜ u 12 , ˜ u 1 , ˜ u 2 , ˜ u ) = 0 ˜ u 1 = f ( ˜ u, u, u 1 ) , ˜ u 2 = g ( ˜ u , u, u 2 ) , whic h is inv ertible to u 1 = ˜ f ( u, ˜ u, ˜ u 1 ) , u 2 = ˜ g ( u, ˜ u, ˜ u 2 ) , where functions f and g are fractional linear in ˜ u and functions ˜ f , ˜ g are function fractional linear in u . A classical example of B¨ ac klund trans formation b et w een p ( xx 1 + x 2 x 12 ) − q ( xx 2 + x 1 x 12 ) = 0 and p 2 ( y 12 + y 1 )( y 2 + y ) = q 2 ( y 12 + y 2 )( y 1 + y ) is the transformation y 1 + y = px 1 x, y 2 + y = q x 2 x. (2.3) 3 Outline of the metho d W e consider the Z 2 lattice together with its horizon tal edges (whic h can b e view ed as set of ordered pair of p oints of Z 2 , i.e. E h =  (( m, n ) , ( m + 1 , n )) | ( m, n ) ∈ Z 2  ) and the vertica l ones ( E v =  (( m, n ) , ( m, n + 1)) | ( m, n ) ∈ Z 2  ). W e tak e in to accoun t a fu nction u wh ic h is give n on h orizon tal edges u : E h → C and a function v giv en on vertical ones v : E v → C . S hift op erators T 1 and T 2 act on horizon tal edges in standard wa y T 1 (( m, n ) , ( m + 1 , n )) := (( m + 1 , n ) , ( m + 2 , n )), T 2 (( m, n ) , ( m + 1 , n )) := (( m, n + 1) , ( m + 1 , n + 1)) (and similarly f or v ertical edges). W e u se con v en tio n to den ote sh ift action on a fun ction by sub scripts T 1 u := u 1 . No w, the outline of the metho d w e dev elop ed in [18] can b e presente d as follo ws. 6 P . K assotakis and M. Nieszp orski 3.1 F rom equations to in v olutiv e maps. I dea system T ake a function x giv en on vertic es of the lattice an d whic h ob eys H 3 0 equation p ( xx 1 + x 2 x 12 ) = q ( xx 2 + x 1 x 12 ) . (3.1) In tro duce f ie lds u and v giv en on horizon tal and v ertical edges resp ectiv ely (f ields u and v are actually the in v ariant s of a sym m etry group of the lattice equation (3.1) as it w as shown in [13]) u = xx 1 , v = xx 2 . W e get u 2 u = v 1 v , p ( u 2 + u ) = q ( v 1 + v ) and w e arrive at the system of equations u 2 = v pu − q v q u − pv , v 1 = u pu − q v q u − pv . (3.2) The main idea is to inv estigate system (3.2) rather than equation (3.1) itself. W e dare to refer to the system (3.2) as to 2D Id ea system I I I . The p oin t is that the system (3.2) ad m its, as we sh all see, three parameter family of p oten tials ψ giv en on vertic es of the lattice . Every “p oten tial image” of (3.2) we refer to as id olon (adopting Plato terminology of I deas and idolons). First w e apply the standard pro cedu r e f or reinterpretatio n of equ ations on a lattice as a map. The rein terpretation is based on identif ication (see Fig. 1) u ( m, n ) = u, v ( m, n ) = v , u ( m, n + 1) = U, v ( m + 1 , n ) = V , (3.3) whic h turns system (3.2) into C 2 → C 2 map U = v pu − q v q u − pv , V = u pu − qv q u − pv . (3.4) W e arriv e at an inv olutive Y ang–Baxter m ap that b elongs to family of maps denoted by F II I (see [12]). Figure 1. V ariables on edg e s of a Z 2 lattice (left picture) and ar guments and v alues of a C 2 7→ C 2 map (right picture). 3.2 Finding functions such that F ( U ) + G ( V ) = f ( u ) + g ( v ) The next step is to f ind suc h functions F and G su c h that for the map (3.4) F ( U ) + G ( V ) = f ( u ) + g ( v ) . (3.5) F amilies of In tegrable Equations 7 holds. An ticipating facts, the fu n ctions will allo w u s to introdu ce a family of p otenti als in th e next sub s ection. Dif ferentia tion of (3.5) with resp ect to u and v yields − F ′′ ( U ) qU 2  pU 2 − 2 qU V + pV 2  + F ′ ( U )2( q U − pV ) q U V + G ′′ ( V ) pV 2  q U 2 − 2 pU V + q V 2  + G ′ ( V )2( q U − pV ) pU V = 0 . The equation ab o v e shou ld hold for ev ery v alue of U and V resp ectiv ely . T he equation has the form − pq U 4 F ′′ ( U ) + 2 q 2 U 2 ( U F ′′ ( U ) + F ′ ( U )) V − pq U ( U F ′′ ( U ) + F ′ ( U )) V 2 + α ( V ) U 2 + β ( V ) U + γ ( V ) = 0 , so F ( U ) must satisfy (n ecessary but not suf f icient condition) the O DE − pq U 4 F ′′ ( U ) + c 2 U 2 + c 1 U + c 0 = 0 . with some constan ts c 1 , c 2 and c 0 . Similarly we get pq V 4 G ′′ ( V ) + d 2 V 2 + d 1 V + d 0 = 0 . Chec king obtained by this wa y solutions we obtain F ( U ) + G ( V ) = a ln( U /V ) + b ( pU − q V ) + c  p U − q V  + d and w e f ind that for the map (3.4) the follo wing equalit y holds a ln( U /V ) + b ( pU − q V ) + c  p U − q V  = − h a ln( u/v ) + b ( pu − q v ) + c  p u − q v i . (3.6) 3.3 P oten tials of t he Idea systems. Idolons Returning to equations on the lattice (b y means of (3.3)) one can r ewrite (3.6) as ( T 2 + 1)  a ln u + bpu + c p u + d  = ( T 1 + 1)  a ln v + bq v + c q v + d  . It means there exists fun ction ψ such that a ln u + p  bu + c 1 u  + d = ψ 1 + ψ , a ln v + q  bv + c 1 v  + d = ψ 2 + ψ (3.7) where a , b , c and d are arbitrary constan ts (w e assum e that one of the constan ts a , b , c is not equal zero). The constant d can b e alw a ys r emo v ed by r edef inition ψ → ψ + 1 2 d and we neglect it a ln u + p  bu + c 1 u  = ψ 1 + ψ , a ln v + q  bv + c 1 v  = ψ 2 + ψ . (3.8) System (3.7) and Idea sys tem (3.2 ) give rise to ψ 12 = ψ + a ln pu − q v q u − pv +  p 2 − q 2   b uv q u − pv − c 1 pu − q v  , (3.9) so w e get thr ee parameter family of equations. Note that in general, (3.2) d o es not follo ws from (3.7) and (3.9) and therefore we will treat (3.2) as an additional cond ition that m ust b e satisf ied. As w e h a v e said in the in tro duction, c hoice of p arameters b = 0 = c leads to equation 8 P . K assotakis and M. Nieszp orski H 3 0 (2.1) whereas choice of parameters either a = 0 = b or a = 0 = c leads to equation A 1 0 (2.2). Ev ery such p oten tial repr esen tation of the Id ea system we refer to as idolon of th e Idea system. T o the end let us write another idolon. Namely , a = 0 yields the equation ψ 2 − ψ 1 ψ 12 − ψ = p 2 + q 2 p 2 − q 2 − pq p 2 − q 2  u v + v u  , where u and v are solutions of the follo wing quadr atic equ ations p  bu 2 + c  = ( ψ 1 + ψ ) u, q  bv 2 + c  = ( ψ 2 + ψ ) v and w e still assume that (3.2 ) h olds. 3.4 Extension to m ultidimension, m ultidimensional consistency of idolons of I II I The system (3.2) can b e extended to multidimension. W e denote b y s i (mind su p erscrip t!) function giv en on edges in i -th direction of the Z n lattice , by su bscript w e denote forwa rd shift in indicated direction. The extension reads ( I II I ) : s i j = s j p i s i − p j s j p j s i − p i s j , i, j = 1 , . . . , n, i 6 = j, (3.10) where p i is giv en fun ction and can dep end only on i -th indep endent v ariable. The crucial fact is the system is compatible s i j k = s i k j . (3.11) Moreo v er, w e ha v e ( T j + 1)  a ln s i + p i  bs i + c 1 s i  = ( T i + 1)  a ln s j + p i  bs j + c 1 s i  . ( 3.12) It means that there exists scalar function ψ suc h that a ln s i + p i  bs i + c 1 s i  = ψ i + ψ , i = 1 , . . . , n . (3.13) F rom (3.10) and (3.13) we infer that ψ ij = ψ + a ln p i s i − p j s j p j s i − p i s j + [( p i ) 2 − ( p j ) 2 ]  b s i s j p j s i − p i s j − c 1 p i s i − p j s j  , (3.14) i, j = 1 , . . . , n, i 6 = j, where s i and s j are giv en implicitly by means of (3.13). Due to (3.11) the system (3.14) is m ultidimensionaly consisten t (compatible) and we clarify what w e mean by that in the follo wing theorem (by i-th initial line we understand in wh at follo w s the s et l i = { ( m 1 , . . . , m n ) ∈ Z n | ∀ k 6 = i : m k = 0 } and b y set of initial lines we m ean l = l 1 ∪ · · · ∪ l n ) Theorem 1. F or arbitr ary initial c onditio n on initial lines ψ ( l ) ther e e xi sts solution ( we do not exclude singularities ) ψ of the multidimensional system (3.13) , (3.14) that ob e ys (3.10) . F amilies of In tegrable Equations 9 Pro of . Indeed, take arbitrary in itial condition on initial lines ψ ( l ). Th en choose a solution s i ( l i ) (in general the v alue of s i is given on the edge b et w een vertic es that ψ and ψ i are give n on) of the equation a ln s i ( l i ) + p i  bs i ( l i ) + c 1 s i ( l i )  = ψ i ( l i ) + ψ ( l i ) , i = 1 , . . . , n (3.15) (this is a p lace wh en non-un iqueness ma y enter). W e treat s i ( l ) as initial conditions for the system (3.10) . Due to (3.11 ) the solution s i of (3.10) with in itial conditions s i ( l ) exists (w e admit singularities that come from zero es of p j s i − p i s j ). No w, du e to ident it y (3.12) there exists fu nction ψ su c h that (3.13 ) h olds and the v alue of ψ at the int ersection of in itial lines is equal to initial condition at the in tersection of in itial lines. Since s i ob eys (3.10) ψ satisf ies (3.14) as w ell. Finally ψ satisf ies the assu med arbitrary initial cond ition since form ulas (3.13) at initial lines coincides with (3.15).  W e refer to the system (3.10) as to n-d imensional Idea s ystem I I I and th at is why w e hav e denoted it by I II I . 4 Maps As we ha v e already mentioned our insp iration wa s a survey on Y ang–Baxter maps. Our goal no w is to relate our f indings to some results of the pap ers [12, 13] and ju stify wh y it makes sense to talk ab ou t the Id ea s ystems s i j = s j p i s i − p j s j p j s i − p i s j , i = 1 , . . . , n (4.1) asso ciated with maps of type I I I rather than single Id ea system. The Idea systems are r elated b y p oint transform ation. Indeed, f irst we p erform a cosmetic p oint transformation s i = p i v i , p i 2 → p i and w e get v i j = v j p i p i v i − p j v j v i − v j , whic h in t w o-dimensional case after iden tif ication analogous to th e one sho w ed on the Fig. 1 yields F II I map of pap er [12] ( F II I ) : U = v p pu − q v u − v , V = u q pu − qv u − v . (4.2) In fact by F II I w e u nderstand equiv alence class of Y ang–Ba xter maps (cf. [13]) the equations (4.1) and (4.2) b elongs to. No w after the p oin t transform ation v i = u i ( − 1) m 1 + ··· + m n w e get u i j = − u j p i p i u i − p j u j u i − u j asso ciated 2D map of w hic h is ( cH A II I ) : U = − v p pu − q v u − v , V = − u q pu − q v u − v . (4.3) After another p oint trans formation u i = w i ( − 1) m 1 + ··· + m n p i 1 2 [( − 1) m 1 + ··· + m n − 1] w e obtain w i j = − 1 w j w i − w j p i w i − p j w j 10 P . K assotakis and M. Nieszp orski and its asso ciated map ( cH B II I ) : U = − 1 v u − v pu − q v , V = − 1 u u − v pu − q v . (4.4) Maps (4.3) and (4.4) are n ot Y ang–Baxter maps but they are companions (if f : ( u, v ) 7→ ( U, V ) is in v olutiv e map then th e map ( u, V ) 7→ ( U, v ) w e refer to as companion of map f , cf. [12]) of Y ang– Baxter m aps H A II I , H B II I of p ap er [17]. The maps H A II I , H B II I can b e obtained in tw o-dimensional case by the p oint transf ormation u 1 = x , u 2 = − y and w 1 = x , w 2 = − 1 q y resp ectiv ely x 2 = y p px + q y x + y , y 1 = x q px + q y x + y and x 2 = y q xy + 1 pxy + 1 , y 1 = x pxy + 1 q xy + 1 and then by ment ioned identif ication (see Fig. 1) ( H A II I ) : U = v p pu − q v u − v , V = u q pu − q v u − v , ( H B II I ) : U = v q uv + 1 puv + 1 , V = u q uv + 1 puv + 1 . Idea systems ( H A II I ) and ( H B II I ) cannot b e extended to multidimension (in the sense of [25]). Finally , we list in the T able 1 basic ident ities of th e maps that leads to existence of p otenti als of the Idea systems to illustrate ho w the basis c hanges when one change s a map. T able 1. Basic identities of the maps that leads to existence of p otentials of the Idea system. Type of the ma p Example of the ma p Identit ies U = v p pu − q v u − v U V = q v pu F II I pU − q V = − ( pu − q v ) V = u q pu − q v u − v 1 U − 1 V = −  1 u − 1 v  U = − v p pu − q v u − v U V = q v pu cH A II I pU − q V = pu − q v V = − u q pu − q v u − v 1 U − 1 V = 1 u − 1 v U = 1 v u − v q v − pu U V = u v cH B II I pU + 1 U − q V − 1 V = pu + 1 u − q v − 1 v V = 1 u u − v q v − pu pU − 1 U − q V + 1 V = −  pu − 1 u − q v + 1 v  U = v ( pu + q v ) p ( u + v ) U V = q v pu H A II I pU + q V = pu + q v V = u ( pu + q v ) q ( u + v ) 1 U + 1 V = 1 u + 1 v U = v q uv + 1 puv + 1 U V = u v H B II I pU + q V + 1 U + 1 V = pu + q v + 1 u + 1 v V = u puv + 1 q uv + 1 pU − q V − 1 U + 1 V = −  pu − q v − 1 u + 1 v  F amilies of In tegrable Equations 11 5 Hirota’s KdV lattice equation As the second example we consider Hirota’s K dV lattice equation [5] x 12 − x = κ  1 x 2 − 1 x 1  . By the substitution u = x 1 x , v = x 2 x , we get u 2 = v + κ  1 − v u  , v 1 = u + κ  − 1 + u v  . (5.1) On applying identif ication (3.3) u = u ( m, n ) , v = v ( m, n ) , U = u ( m, n + 1) , V = v ( m + 1 , n ) (5.2) w e obtain an inv olutive mapping asso ciated to system (5.1) U = v + κ  1 − v u  , V = u + κ  − 1 + u v  . (5.3) Mapping (5.3) satisf ies (this is the outcome of searc hing for suc h fun ctions F and G that F ( U ) + G ( V ) = f ( u ) + g ( v ) as describ ed in th e pr evious section): U V = v u , ( U − κ )( V + κ ) = ( u − κ )( v + κ ) , V ( U − κ ) U ( V + κ ) = v ( u − κ ) u ( v + κ ) , hence (coming bac k to lattice v ariables (5.2 )) we can in tro duce the p oten tials x , y and z u = x 1 x, v = x 2 x, u − κ = y 1 /y , v + κ = y /y 2 , u − κ u = z 1 /z , v + κ v = z 2 /z . (5.4) Eliminating u and v from (5.1) w e arrive at the follo wing lattice equations x 12 − x = κ (1 /x 2 − 1 / x 1 ) , y 1 y − y 12 y 1 = κ ( y 12 y + y 1 y 2 ) , z 12 z + z 1 z 2 = z 12 z 2 + z 12 z 1 . (5.5) One can treat the equations as r epresen tativ es of a three-parameter family of equations on φ φ 12 φ φ 1 φ 2 =  ( u − κ )( v + κ ) + κ 2  a ( − 1) m + n +1 − b u b − c v b + c , φ 1 φ = u a ( − 1) m + n − b ( u − κ ) b + c , φ 2 φ = v a ( − 1) m + n − b ( v + κ ) b − c , (5.6) corresp ondin g to th e choic e of parameters b = 0 = c , a = 0 = b and a = 0 = c r esp ectiv ely . What more imp ortan t is th at from (5.4) w e infer z 1 z = y 1 x 1 xy , z 2 z = y y 2 x 2 x . Compatibilit y condition that guarantees existence of function z reads  x 2 x 1  2 =  y 12 y y 1 y 2  2 , from where we get x = τ 12 τ τ 1 τ 2 , y = τ 2 τ 1 , z = τ τ 12 . Eliminating x , y and z from (5.4) we arrive at a compatible p air of bilinear forms of Hirota’s KdV (cf. [19]) τ 112 τ − κτ 11 τ 2 = τ 12 τ 1 , τ 122 τ + κτ 22 τ 1 = τ 12 τ 2 . 12 P . K assotakis and M. Nieszp orski 6 B¨ ac klund transformations b et w een idolons In b oth p resen ted examples one can f i nd B¨ ac klund transf ormation b etw een idolons. F or instance eliminating u and v from f irst t w o lines of (5.4 ) one gets B¨ ac klun d transform ation b et w een f irst t w o equations of (5.5) y 1 y = x 1 x − k, y y 2 = x 2 x + k. Similarly in the case of I II I one can obtain B¨ ac k lu nd transformation (2.3). Finally , w e pr esen t the B¨ ac klund transform ation b et w een A 1 0 (2.2) and the idolon (3.8), (3.9). Namely , if y satisf ies A 1 0 then • function ψ giv en by a ln p y 1 + y + bp 2 y 1 + y + c ( y 1 + y ) = ψ 1 + ψ , a ln q y 2 + y + bq 2 y 2 + y + c ( y 2 + y ) = ψ 2 + ψ exists (compatibilit y conditions are satisf ied du e to the fact that y satisf ies A 1 0 ); • functions u and v giv en by u = p y 1 + y , v = q y 2 + y ob ey Id ea system (3.2); • function ψ ob eys (3.8), (3.9). 7 Conclusions In this pap er w e fo cused on t w o 3- parameter families of lattice equations. The f irst one (1.1), (1.2) and (1.3) is related to mappings of type I I I whic h w ere in tro duced in [12, 13]. Two members (idolons) of the later are, the Hirota’s sine-Gordon equation and the lattice Sch warzian KdV [28] in a disguise. Generally , all id olons are connected thr ough B¨ ac klun d transf orm ations and they are 3D-consisten t in th e sense describ ed in the p ap er. In the not-to o-distan t fu ture we are going to in v estigat e families of equations related to give n inte grable systems n ot only by discrete quadratures bu t also by B¨ ac klund transf ormation fr om the Def inition 5. The second family describ ed by (5.6) and (5.1) is not 3D-consisten t. Nev ertheless, all of its idolons are conn ected thr ough B¨ ac klund transformations, and since an idolon of this family is the Hirota’s KdV equation, the whole family inh er its some in tegrabilit y pr op erties e.g. τ -function form ulation. W e would like to emphasize once more that th e main ob j ect und er consideration are Id ea systems (3.2) (or its n -dimensional ve rsion (3.10)) an d (5.1). The m ain obser v ation is that the Idea systems admit thr ee-dimensional vecto r sp ace of scalar p otentia ls (formulas (3.8) in case of t w o-dimensional I d ea I II I and (3.13) in the n-d imensional case, see also second and third formulas of (5.6)). In a forthcoming pap er we will d iscuss all Idea sys tems that arise from equations of Adler–Bob enk o–Suris list. In other words, we plan to inv esti gate all mappin gs in [12, 13], determine their asso ciate Idea systems and put m ore light into int egrabilit y prop erties of th e asso ciated family of lattice equations. Also, it will b e in teresting to in v estigat e the mapp ings that arise wh en one imp oses p erio dic staircase initial d ata on these families of lattice equations. Another ob jectiv e is to der ive the discrete P ainlev ´ e equations asso ciated with these families. Finally , we will d iscuss the case of r eal-v alued fun ctions, wh ic h can lead to standard 3D- consisten t lattice equations. F or instance for the id olon w e prop osed in [18] f 12 = f + ( p − q )  v − u + f 1 − f 2 ( u − v ) 2 + ( p − q ) 2 ( u − v ) 3  , (7.1) F amilies of In tegrable Equations 13 u 3 + au = f 1 − f , v 3 + bv = f 2 − f , a − b = 3( q − p ) . assuming f : Z 2 → R and a, b > 0 the only real solutions of the cubic equations are u = 3 s f 1 − f 2 + r ( f 1 − f ) 2 4 + a 3 27 + 3 s f 1 − f 2 − r ( f 1 − f ) 2 4 + a 3 27 , v = 3 s f 2 − f 2 + r ( f 2 − f ) 2 4 + b 3 27 + 3 s f 2 − f 2 − r ( f 2 − f ) 2 4 + b 3 27 . (7.2) Then the real lattice equation (7.1 ), with u and v giv en by (7.2), is 3D-consisten t. F rom another p ersp ectiv e, instead of dealing with the family of lattice equations, it seems more fund amental to d ef in e a mo del that consists of the Idea system and the asso ciate p oten tial equation (e.g. equ ations (3.2 ), (3.7 ) or (3.10), (3.13) for the m ultidimensional extension). Then the f amily of 3D-consisten t (see Theorem 1) lattice equations follo ws natur ally . 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