On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 4 (2008), 077, 14 pages On Miura T ransformations and V olterra-T yp e Equations Asso ciated with the Adler–Bo b enk o–Suris Equations De ci o LEVI † , Matte o PETRERA ‡† , Christian SCIMITERNA ‡† and R avil Y AMILOV § † Dip artimento di Inge gne ria Elettr onic a,Unive rsit` a de gli Studi R oma T r e and Sezi one INFN , R oma T r e, Via del la V asc a Navale 84, 00146 R oma , Italy E-mail: levi@fis.unir oma3.it ‡ Dip artimento di Fisic a E. Am aldi, Universit` a de gli Studi R oma T r e and Sezione INFN, R oma T r e, Via del la V asc a Navale 84, 00146 R oma , Italy E-mail: p etr er a@fis.unir oma3.it , scimiterna@fis.unir oma 3.it § Ufa Institute of Mathematics, 112 Chernyshevsky Str., Uf a 450077, Russia E-mail: R vlY amilov@matem.an rb.ru Received August 29, 2008, in f inal form Octob er 30 , 2008; Published online Nov em ber 08, 2008 Original article is av ailable at http ://ww w.emi s.de/journals/SIGMA/2008/077/ Abstract. W e construct Miura transformatio ns mapping the scala r sp ectral pr oblems of the int egrable lattice equations be lo nging t o the Adler– Bob enko–Suris (ABS) list into the dis- crete Schr¨ odinger sp ectral pro blem ass o ciated with V olterra-type eq uations. W e show that the ABS equa tions corres po nd to B¨ acklund tra nsformations for some particular c a ses of the discrete K richev er–Novik ov equation found by Y amilov (YdKN equation). This enables us to construct new g eneralized symmetries for the ABS eq ua tions. The sa me can b e s aid a b o ut the g e neralizations of the ABS equations introduce d by T ongas, Ts oub elis and Xenitidis. All of them gener ate B¨ acklund transfor mations for the YdKN equation. The higher o rder generalized symmetries we construct in the pres ent paper conf irm their integrabilit y . Key wor ds: Miur a tr ansformatio ns; genera lized symmetries; ABS lattice e quations 2000 Mathematics Subje ct Classific ation: 37K10; 37L20 ; 39A05 1 In tro duction The discov ery of new t w o-dimensional in teg rable partial d if ference equations (or Z 2 -lattice equa- tions) is alw a ys a v ery c hallenging problem as, by p rop er con tin uous limits, m any other results on dif ferentia l-dif ference and p artial d if ferent ial equations ma y b e obtained. M oreo v er man y physi- cal and biological applications inv olv e discrete systems, see for instance [13, 25] and references therein. The theory of n onlinear in tegrable dif ferential equations got a b o ost when Gardner, Green, Krusk al and Mi ura introd uced the Inv erse S cattering Metho d for the solutio n of the Kortew eg– de V ries equation. A s ummary of these results can b e foun d in the Encyclop edia of Mathematical Ph ysics [12]. A f ew tec hniques ha v e b een in tro duced to classify integ rable partial dif feren tial equations. Let us jus t m en tion the classif ication scheme in tro duced b y Shabat, u sing the form al symmetry appr oac h (see [21] for a review). This approac h h as b een successfully extended to the dif ferentia l-dif ference case by Y amilo v [30, 31, 20]. In th e completely discrete case the situation turns out to b e quite dif ferent . F or in stance, in the case of Z 2 -lattice equations th e formal symmetry tec hnique do es not work. In this fr amew ork, the f irst exhaustive classif ications of families of lattice equations hav e been p resen ted in [1] b y Adler and in [2, 3] by Adler, Bob enk o and Suris. 2 D. Levi, M. P etrera, C. Scimiterna and R. Y amilov In the present p ap er w e shall consider the Adler–Bob enk o–Suris (ABS) classif ication of Z 2 - lattice equatio ns def ined on the squ are lat tice [2]. W e refer to the pap ers [3, 24, 28, 17, 18, 27] for some r ecen ts r esults ab out th ese equations. Our main p urp ose is th e analysis o f their trans- formation prop erties. In fact, our aim is, on the one hand , to present new Miur a transform ations b et w een the ABS equations and V olterra-t yp e dif ference equations an d on the other hand, to sho w that the ABS equ ations corresp ond to B¨ ac klund transformations for some particular cases of the discrete Kric hev er–N o vik o v equation found b y Y amilo v (YdKN equation) [30]. Section 2 is dev oted to a short r eview of the in tegrable Z 2 -lattice equations d eriv ed in [2 ] and to present details on their matrix and scalar sp ectral problems. In S ection 3, by trans f orming the obtained scalar s p ectral problems in to the discrete Sc hr¨ odinger sp ectral problem associated with the V olterra latti ce w e w ill b e able to connect the ABS equation with V olterra-t yp e equations. In Section 4 w e pro v e that the ABS equations corresp ond to B¨ ac klund t ransformations for certain sub cases of the YdKN equation. Using this result and a master symmetry of the YdKN equ ation, w e construct new generalized symmetries for th e ABS list. Then w e discuss the in tegrabilit y of a class of non-autonomous ABS equations and of a generalizatio n of the ABS equatio ns in tro duced b y T ongas, Tsoub elis and Xe nitidis in [28]. Section 5 is devo ted to some concludin g remarks. 2 A short review of the ABS equations A tw o-dimens ional partial d if ference equation is a fu nctional relation among the v alues of a fun c- tion u : Z 2 → C at dif ferent p oin ts of the lattic es of indices n , m . It inv olve s the indep endent v ariables n , m and the lattice parameters α , β E ( n, m, u n,m , u n +1 ,m , u n,m +1 , . . . ; α, β ) = 0 . F or the d ep endent v ariable u we shall adopt the follo wing notation throughout the pap er u = u 0 , 0 = u n,m , u k ,l = u n + k ,m + l , k , l ∈ Z . (1) W e consider here the A BS list of in te grable lattice equati ons, namely those af f ine linear (i.e. p olynomial of degree one in eac h argumen t) partial dif ference e quations of the form E ( u 0 , 0 , u 1 , 0 , u 0 , 1 , u 1 , 1 ; α, β ) = 0 , (2) whose integ rabilit y is based on the c onsistency ar ound a cub e [2, 3]. The function E d ep ends explicitly on the v al ues of u at the vertic es of an elementa ry quadrilateral , i.e. ∂ u i,j E 6 = 0, where i, j = 0 , 1. The latt ice parameters α , β ma y , in general, d ep end on the v ariables n , m , i.e. α = α n , β = β m . Ho w ev er, w e shall discuss such non-autonomous extensions in Section 4. The complete list of the ABS equations can b e found in [2]. Their integrabilit y holds b y construction since the consistency around a cu b e furnishes their Lax pairs [2, 9, 22]. Th e ABS equations are giv en b y th e list H (H1) ( u 0 , 0 − u 1 , 1 )( u 1 , 0 − u 0 , 1 ) − α + β = 0 , (H2) ( u 0 , 0 − u 1 , 1 )( u 1 , 0 − u 0 , 1 ) + ( β − α )( u 0 , 0 + u 1 , 0 + u 0 , 1 + u 1 , 1 ) − α 2 + β 2 = 0 , (H3) α ( u 0 , 0 u 1 , 0 + u 0 , 1 u 1 , 1 ) − β ( u 0 , 0 u 0 , 1 + u 1 , 0 u 1 , 1 ) + δ ( α 2 − β 2 ) = 0 , and the list Q (Q1) α ( u 0 , 0 − u 0 , 1 )( u 1 , 0 − u 1 , 1 ) − β ( u 0 , 0 − u 1 , 0 )( u 0 , 1 − u 1 , 1 ) + δ 2 αβ ( α − β ) = 0 , (Q2) α ( u 0 , 0 − u 0 , 1 )( u 1 , 0 − u 1 , 1 ) − β ( u 0 , 0 − u 1 , 0 )( u 0 , 1 − u 1 , 1 ) On Miura T rans formations and V olterra-T yp e Equations 3 + αβ ( α − β )( u 0 , 0 + u 1 , 0 + u 0 , 1 + u 1 , 1 ) − αβ ( α − β )( α 2 − αβ + β 2 ) = 0 , (Q3) ( β 2 − α 2 )( u 0 , 0 u 1 , 1 + u 1 , 0 u 0 , 1 ) + β ( α 2 − 1)( u 0 , 0 u 1 , 0 + u 0 , 1 u 1 , 1 ) − α ( β 2 − 1)( u 0 , 0 u 0 , 1 + u 1 , 0 u 1 , 1 ) − δ 2 ( α 2 − β 2 )( α 2 − 1)( β 2 − 1) 4 αβ = 0 , (Q4) a 0 u 0 , 0 u 1 , 0 u 0 , 1 u 1 , 1 + a 1 ( u 0 , 0 u 1 , 0 u 0 , 1 + u 1 , 0 u 0 , 1 u 1 , 1 + u 0 , 1 u 1 , 1 u 0 , 0 + u 1 , 1 u 0 , 0 u 1 , 0 ) + a 2 ( u 0 , 0 u 1 , 1 + u 1 , 0 u 0 , 1 ) + ¯ a 2 ( u 0 , 0 u 1 , 0 + u 0 , 1 u 1 , 1 ) + e a 2 ( u 0 , 0 u 0 , 1 + u 1 , 0 u 1 , 1 ) + a 3 ( u 0 , 0 + u 1 , 0 + u 0 , 1 + u 1 , 1 ) + a 4 = 0 . The coef f icien ts a i ’s app earing in equation (Q4) are connected t o α and β by the r elations a 0 = a + b, a 1 = − aβ − bα, a 2 = aβ 2 + bα 2 , ¯ a 2 = ab ( a + b ) 2( α − β ) + aβ 2 −  2 α 2 − g 2 4  b, e a 2 = ab ( a + b ) 2( β − α ) + bα 2 −  2 β 2 − g 2 4  a, a 3 = g 3 2 a 0 − g 2 4 a 1 , a 4 = g 2 2 16 a 0 − g 3 a 1 , with a 2 = r ( α ), b 2 = r ( β ), r ( x ) = 4 x 3 − g 2 x − g 3 . F ollo w ing [2] w e remark that • Equations (Q1)–(Q3) and (H1 )–(H3) are all degenerate sub cases of equation (Q4) [7]. • P arame ter δ in equations (H3), (Q 1) and (Q3) can b e rescaled, so that one can assume without loss of generalit y that δ = 0 o r δ = 1. • The original ABS l ist con tains tw o further equations ( list A) (A1) α ( u 0 , 0 + u 0 , 1 ) ( u 1 , 1 + u 1 , 0 ) − β ( u 0 , 0 + u 1 , 0 ) ( u 1 , 1 + u 0 , 1 ) − δ 2 αβ ( α − β ) = 0 , (A2) ( β 2 − α 2 )( u 0 , 0 u 1 , 0 u 0 , 1 u 1 , 1 + 1) + β ( α 2 − 1)( u 0 , 0 u 0 , 1 + u 1 , 0 u 1 , 1 ) − α ( β 2 − 1)( u 0 , 0 u 1 , 0 + u 0 , 1 u 1 , 1 ) = 0 . Equations (A1) and (A2) can b e transformed b y an extend ed group of M¨ obiu s transfor- mations int o equations (Q1) and (Q3) resp ectiv ely . Indeed, an y solution u = u n,m of (A1) is tr ansformed into a solution e u = e u n,m of (Q1) by u n,m = ( − 1) n + m e u n,m and any solu- tion u = u n,m of (A2) is transformed in to a solution e u = e u n,m of (Q3) with δ = 0 b y u n,m = ( e u n,m ) ( − 1) n + m . Some of th e ab ov e equ ations w ere kn o wn b efore Adler, Bob enk o and Suris p resen ted their classif ication, see for instance [23, 14]. W e f inally r ecall that a more general cla ssif ication of integrable lattice equations def ined on the square has b een recent ly carried out by Adler, Bob enk o and Su ris in [3]. But here w e shall consider only the lists H and Q conta ined in [2]. 2.1 Sp ectral problems of the ABS equations The algorithmic pro cedure describ ed in [2, 9, 22] pro duces a 2 × 2 matrix Lax pair for the ABS equations, th u s e nsurin g their in tegrabilit y . It ma y b e w ritten as Ψ 1 , 0 = L ( u 0 , 0 , u 1 , 0 ; α, λ )Ψ 0 , 0 , Ψ 0 , 1 = M ( u 0 , 0 , u 0 , 1 ; β , λ )Ψ 0 , 0 , (3) with Ψ = ( ψ ( λ ) , φ ( λ )) T , where the lattice paramet er λ pla ys t he role of the sp ectral p arameter. W e shall use the fol lo wing notatio n L ( u 0 , 0 , u 1 , 0 ; α, λ ) = 1 ℓ  L 11 L 12 L 21 L 22  , M ( u 0 , 0 , u 0 , 1 ; β , λ ) = 1 t  M 11 M 12 M 21 M 22  , 4 D. Levi, M. P etrera, C. Scimiterna and R. Y amilov T able 1. Matrix L for the ABS eq uations (in equation (Q4) a 2 = r ( α ), b 2 = r ( λ ), r ( x ) = 4 x 3 − g 2 x − g 3 ). L 11 L 12 L 21 L 22 H1 u 0 , 0 − u 1 , 0 ( u 0 , 0 − u 1 , 0 ) 2 + α − λ 1 u 0 , 0 − u 1 , 0 H2 u 0 , 0 − u 1 , 0 + α − λ ( u 0 , 0 − u 1 , 0 ) 2 + 2( α − λ )( u 0 , 0 + u 1 , 0 )+ 1 u 0 , 0 − u 1 , 0 − α + λ + α 2 − λ 2 H3 λu 0 , 0 − αu 1 , 0 λ ( u 2 0 , 0 + u 2 1 , 0 ) − 2 αu 0 , 0 u 1 , 0 + δ ( λ 2 − α 2 ) α αu 0 , 0 − λu 1 , 0 Q1 λ ( u 1 , 0 − u 0 , 0 ) − λ ( u 1 , 0 − u 0 , 0 ) 2 + δ αλ ( α − λ ) − α λ ( u 1 , 0 − u 0 , 0 ) Q2 λ ( u 1 , 0 − u 0 , 0 )+ − λ ( u 1 , 0 − u 0 , 0 ) 2 + − α λ ( u 1 , 0 − u 0 , 0 ) − + αλ ( α − λ ) +2 αλ ( α − λ )( u 1 , 0 + u 0 , 0 ) − − αλ ( α − λ ) − αλ ( α − λ )( α 2 − αλ + λ 2 ) Q3 α ( λ 2 − 1) u 0 , 0 − − λ ( α 2 − 1) u 0 , 0 u 1 , 0 + λ ( α 2 − 1) ( λ 2 − α 2 ) u 0 , 0 − − ( λ 2 − α 2 ) u 1 , 0 + δ ( α 2 − λ 2 )( α 2 − 1)( λ 2 − 1) / (4 αλ ) − α ( λ 2 − 1) u 1 , 0 Q4 − a 1 u 0 , 0 u 1 , 0 − − ¯ a 2 u 0 , 0 u 1 , 0 − a 3 ( u 0 , 0 + u 1 , 0 ) − a 4 a 0 u 0 , 0 u 1 , 0 + ¯ a 2 + a 1 u 0 , 0 u 1 , 0 + a 2 u 0 , 0 + − a 2 u 1 , 0 − e a 2 u 0 , 0 − a 3 + a 1 ( u 0 , 0 + u 1 , 0 ) + e a 2 u 1 , 0 + a 3 where ℓ = ℓ 0 , 0 = ℓ ( u 0 , 0 , u 1 , 0 ; α, λ ), t = t 0 , 0 = t ( u 0 , 0 , u 0 , 1 ; β , λ ), L ij = L ij ( u 0 , 0 , u 1 , 0 ; α, λ ) and M ij = M ij ( u 0 , 0 , u 0 , 1 ; β , λ ), i, j = 1 , 2. The matrix M can b e obtained from L by replacing α with β and shifting along direction 2 instead of 1. In T able 1 we giv e the en tries of the m atrix L for t he ABS equations. Note that ℓ and t are computed by requiring that the compatibilit y condition b etw een L and M pr o duces the ABS e quations (H1)– (H3) and (Q1)–( Q4). Th e factor ℓ ca n b e wr itten as ℓ 0 , 0 = f ( α, λ )[ ρ ( u 0 , 0 , u 1 , 0 ; α )] 1 / 2 , (4) where the functions f = f ( α, λ ) is an arbitrary normalization f actor. Th e fun ctions f = f ( α, λ ) and ρ = ρ 0 , 0 = ρ ( u 0 , 0 , u 1 , 0 ; α ) f or equ ations (H1)– (H3) and (Q1)–(Q4) are given in T able 2. A form ula similar to (4) holds also for the factor t . The scala r Lax pairs for the ABS equ ations ma y b e immediately computed from equation (3). Let u s wr ite the scalar equation just for the second comp onen t φ of the vec tor Ψ (the use of the f irst co mp onent would give similar results). F or equations (H1 )–(H3) and (Q1)–(Q3) it rea ds ( ρ 1 , 0 ) 1 / 2 φ 2 , 0 − ( u 2 , 0 − u 0 , 0 ) φ 1 , 0 + ( ρ 0 , 0 ) 1 / 2 µφ 0 , 0 = 0 , (5) where the explicit expressions of µ = µ ( α, λ ) are giv en in T able 2. The corresp onding scalar equation for equation (Q 4) tak es a dif ferent form and needs a separate analysis wh ic h will b e done i n a separate w ork. 3 Miura transformat ions for equations (H1)–(H3) and (Q1) –(Q3) The aim of this S ection is to sh o w the exi stence of a Miura transformation mapping t he scalar sp ectral pr oblem (5) of equati ons (H1 )–(H3) an d (Q1) –(Q3) into the discr ete Sc hr¨ odinger s p ec- tral problem asso ciated with the V olterra lattic e [10] φ − 1 , 0 + v 0 , 0 φ 1 , 0 = p ( λ ) φ 0 , 0 , (6) where v 0 , 0 is th e p oten tial of the sp ectral problem and the function p ( λ ) pla ys the role of the sp ectral parameter. On Miura T rans formations and V olterra-T yp e Equations 5 T able 2. Scalar spectr a l problems for the ABS equations (in eq uation (Q4 ) c 2 = r ( λ ), r ( x ) = 4 x 3 − g 2 x − g 3 ). f ( α, λ ) ρ ( u 0 , 0 , u 1 , 0 ; α ) µ ( α, λ ) H1 − 1 1 λ − α H2 − 1 u 0 , 0 + u 1 , 0 + α 2( λ − α ) H3 − λ u 0 , 0 u 1 , 0 + δ α α 2 − λ 2 αλ 2 Q1 λ ( u 1 , 0 − u 0 , 0 ) 2 − δ 2 α 2 λ − α λ Q2 λ ( u 1 , 0 − u 0 , 0 ) 2 − 2 α 2 ( u 1 , 0 + u 0 , 0 ) + α 4 λ − α λ Q3 α (1 − λ 2 ) α ( u 2 0 , 0 + u 2 1 , 0 ) − ( α 2 + 1) u 0 , 0 u 1 , 0 + δ 2 ( α 2 − 1) 2 4 α α 2 − λ 2 α 2 (1 − λ 2 ) Q4 ( α − λ ) c 1 / 2 × ( u 0 , 0 u 1 , 0 + αu 0 , 0 + αu 1 , 0 + g 2 / 4) 2 − − × » 2 a + c + 1 4 “ a + c α − λ ” 3 − 3 α ( a + c ) α − λ – 1 / 2 − ( u 0 , 0 + u 1 , 0 + α )(4 αu 0 , 0 u 1 , 0 − g 3 ) Supp ose that a f u nction s 0 , 0 = s ( u 0 , 0 , u 1 , 0 , u 0 , 1 , . . . ) is give n by the linear equation s 0 , 0 s 1 , 0 = u 2 , 0 − u 0 , 0 ( ρ 0 , 0 ) 1 / 2 . (7) By p erf orming the tr ansformation φ 0 , 0 7→ µ n/ 2 s 0 , 0 φ 0 , 0 , and taking in to accoun t equ ation (7), equation (5) is mapp ed into the scalar spectral problem ( 6) with v 0 , 0 = ρ 0 , 0 ( u 1 , 0 − u − 1 , 0 )( u 2 , 0 − u 0 , 0 ) , p ( λ ) = [ µ ( α, λ )] − 1 / 2 . (8) F rom these results there follo w some remark able consequences: (i) There exists a Miura trans- formation b et w een all equ ations of the set (H1)–(H 3) and (Q1)–(Q3). Some results on this claim can b e foun d in [7 ]; (ii) The Miura transformation (8) c an be i n v erted by solving a linear dif ference equation. Therefore we can in principle use these r emarks to f in d explicit solutions of the ABS equations in terms o f the solutions of the V olterra equation. The foll o wing statemen t holds . Prop osition 1. The field u for e quations (H1)–(H3) and (Q1)–(Q3) c an b e expr esse d in terms of the p otential v of the sp e ctr al pr oblem (6) thr ough the solution of the fol lowing line ar differ enc e e quations H1 : u 2 , 0 − ( v 0 , 0 + v − 1 , 0 ) v 0 , 0 u 0 , 0 + v − 1 , 0 v 0 , 0 u − 2 , 0 = 0 , (9) H2 : u 2 , 0 − v 0 , 0 + v − 1 , 0 v 0 , 0 u 0 , 0 + v − 1 , 0 v 0 , 0 u − 2 , 0 − 1 v 0 , 0 = 0 , (10) H3 : u 2 , 0 − 1 + v 0 , 0 + v − 1 , 0 v 0 , 0 u 0 , 0 + v − 1 , 0 v 0 , 0 u − 2 , 0 = 0 , (11) Q1 : u 2 , 0 − 1 v 0 , 0 u 1 , 0 + 2 − v 0 , 0 − v − 1 , 0 v 0 , 0 u 0 , 0 − 1 v 0 , 0 u − 1 , 0 + v − 1 , 0 v 0 , 0 u − 2 , 0 = 0 , (12) Q2 : u 2 , 0 − 1 v 0 , 0 u 1 , 0 + 2 − v 0 , 0 − v − 1 , 0 v 0 , 0 u 0 , 0 − 1 v 0 , 0 u − 1 , 0 + v − 1 , 0 v 0 , 0 u − 2 , 0 + 2 α 2 v 0 , 0 = 0 , (13) Q3 : u 2 , 0 − α v 0 , 0 u 1 , 0 + α 2 + 1 − v 0 , 0 − v − 1 , 0 v 0 , 0 u 0 , 0 − α v 0 , 0 u − 1 , 0 + v − 1 , 0 v 0 , 0 u − 2 , 0 = 0 . (14) 6 D. Levi, M. P etrera, C. Scimiterna and R. Y amilov Pro of . F rom equation (8) we get v 0 , 0 ( u 2 , 0 − u 0 , 0 ) = ρ 0 , 0 u 1 , 0 − u − 1 , 0 , v − 1 , 0 ( u 0 , 0 − u − 2 , 0 ) = ρ − 1 , 0 u 1 , 0 − u − 1 , 0 . Subtracting these relations and taking into accoun t that (see equation (A.11) in [28]) ∂ u 1 , 0 ρ 0 , 0 + ∂ u − 1 , 0 ρ − 1 , 0 = 2 ρ 0 , 0 − ρ − 1 , 0 u 1 , 0 − u − 1 , 0 , one arriv es at v 0 , 0 ( u 2 , 0 − u 0 , 0 ) − v − 1 , 0 ( u 0 , 0 − u − 2 , 0 ) = 1 2  ∂ u 1 , 0 ρ 0 , 0 + ∂ u − 1 , 0 ρ − 1 , 0  . (15) W riting equation (15) explicitly for equations (H1)–(H3) and (Q1)–(Q3) w e obtain equa- tions (9)–( 14).  4 Generalized symmetrie s of the ABS equations Lie symmetries of equation (2) are giv en b y those cont in uous transformations wh ic h lea v e the equation in v arian t. W e refer to [19, 31] for a review on s y m metries of discrete equatio ns. F rom the inf in itesimal p oin t of view, Lie symmetries are obtained b y requiring the inf inite- simal in v arian t co ndition  pr b X 0 , 0  E   E =0 = 0 , (16) where b X 0 , 0 = F 0 , 0 ( u 0 , 0 , u 1 , 0 , u 0 , 1 , . . . ) ∂ u 0 , 0 . (17) By pr b X 0 , 0 w e mean the pr olongatio n of th e inf initesimal ge nerator b X 0 , 0 to all p oin ts app earing in E = 0. If F 0 , 0 = F 0 , 0 ( u 0 , 0 ) then we get p oint symmetries and the pro cedu re to construct them from equation (16) is purely algorithmic [19]. If F 0 , 0 = F 0 , 0 ( u 0 , 0 , u 1 , 0 , u 0 , 1 , . . . ) the obtained symmetries are called gener alize d symmetries . In the case of nonlinear d iscr ete equations, the Lie p oint symmetries are not v ery common, but, if the equation is in tegrable, it is p ossible to construct an inf in ite family of generalized symmetries. In corresp ondence with the inf initesimal generator (17) we can in principle construct a group transformation b y inte grating the i nitial b oundary p roblem du 0 , 0 ( ε ) dε = F 0 , 0 ( u 0 , 0 ( ε ) , u 1 , 0 ( ε ) , u 0 , 1 ( ε ) , . . . ) , (18) with u 0 , 0 ( ε = 0) = v 0 , 0 , wher e ε ∈ R is the con tin uous Lie group parameter and v 0 , 0 is a solution of equation (2). This can b e done ef fectiv ely only in the case of p oin t symmetries as in th e generalized case we ha v e a nonlinear dif ferential -dif ference equation for whic h we cannot f ind the ge neral solution , but, at most, w e can construct particular solutions. Equation (16) is equiv alen t to the request that the ε -deriv ativ e of the equation E = 0, wr itten for u 0 , 0 ( ε ), is identica lly satisf ied on its solutions when the ε -ev olution of u 0 , 0 ( ε ) is giv en by equation (18). This is also equ iv alen t to sa y that the f lo ws (in the group p arameter space) give n b y equ ation (18) are compatible o r co mm ute with E = 0. In the pap ers [24, 28] the thr ee and f iv e-p oin t generalized sym m etries ha v e b een found for all equ ations of th e ABS list. W e shall use these results to sho w that the ABS equations may b e in terpreted as B¨ ac klund transf ormations for the dif feren tial-dif ference YdK N equatio n [30]. This obser v ation will allo w us to provide an inf inite class of generalized symmetries for the lattice equations b elonging to the ABS list . W e sh all also discus s the n on-autonomous case and the ge neralizations of the ABS equat ions considered in [28]. On Miura T rans formations and V olterra-T yp e Equations 7 4.1 The A BS equations as B¨ ac klund transformations of the YdKN equation In the follo wing we sho w that the ABS equations ma y b e seen as B¨ ac klund transformations of the YdKN equation. Moreo v er w e p r o v e that the sym metries of the A BS equati ons [24, 28] are sub cases of the YdKN equation. F or the sake of clarit y w e consider in a more detailed wa y just the case of equation (H3). Similar r esults can b e obtained for th e whole ABS list (see Prop osition 2). According to [24 , 28] equation (H3) admits the compatible three-p oin t generalized symmetries du 0 , 0 dε = u 0 , 0 ( u 1 , 0 + u − 1 , 0 ) + 2 αδ u 1 , 0 − u − 1 , 0 , (19) du 0 , 0 dε = u 0 , 0 ( u 0 , 1 + u 0 , − 1 ) + 2 β δ u 0 , 1 − u 0 , − 1 . (20) Notice that u nder the discrete map n ↔ m , α ↔ β , equation (19) go es in to equation (20), while equation (H 3) is left in v arian t. The compatibilit y b et w een equation (H3) and equation (19) generates a B¨ ac klund tr ansfor- mation (se e an explanation b elo w) of any solution u 0 , 0 of equatio n (19) into its new sol ution e u 0 , 0 = u 0 , 1 , e u 1 , 0 = u 1 , 1 . (21) Th us equation (H3) can b e r ewritten as a B¨ ac klund transformation for the dif feren tial-dif feren ce equation (1 9) α ( u 0 , 0 u 1 , 0 + e u 0 , 0 e u 1 , 0 ) − β ( u 0 , 0 e u 0 , 0 + u 1 , 0 e u 1 , 0 ) + δ ( α 2 − β 2 ) = 0 . (22) Moreo v er, the discrete symmetry n ↔ m , α ↔ β implies the existence of the B¨ ac klund trans - formation for equation (20) b u 0 , 0 = u 1 , 0 , b u 0 , 1 = u 1 , 1 . This interpretatio n of lattice equations as B¨ ac klund transformations h as b een discussed for the f irst time in the dif ferential -dif ference case in [16]. Examples of B¨ ac klund transformations similar to equation (22) for V olterra-t yp e equations can be found in [29, 11]. In [24, 28] generalized symmetries hav e b een obtained f or autonomous ABS equ ations, i.e. suc h that α , β are constants. W e present here some results on the n on-autonomous case w hen α and β dep end on n a nd m . S imilar results can b e fou n d in [24]. Let the lattice parameters in equation (2) b e suc h that α is a constan t and β = β 0 = β m . Let us consider the f ollo wing tw o forms of equation (2) u 1 , 1 = ξ ( u 0 , 0 , u 1 , 0 , u 0 , 1 ; α, β 0 ) , u 0 , 1 = ζ ( u 0 , 0 , u 1 , 0 , u 1 , 1 ; α, β 0 ) , (23) and a symmetry du 0 , 0 dε = f 0 , 0 = f ( u 1 , 0 , u 0 , 0 , u − 1 , 0 ; α ) , (24) giv en by equation (19). W e su pp ose that u k ,l dep end s on ε in all e quations and write do wn the compatibilit y condition b et w een equation (23) a nd equation (24) f 1 , 1 = f 0 , 0 ∂ u 0 , 0 ξ + f 1 , 0 ∂ u 1 , 0 ξ + f 0 , 1 ∂ u 0 , 1 ξ . (25) As a consequence of equations (23), (24) the fu nctions f 1 , 1 , f 1 , 0 and f 0 , 1 ma y b e expressed in terms of the f ields u k , 0 , u 0 ,l . Therefore, equation (25) dep ends explicitly only on th e v ariab- les u k , 0 , u 0 ,l , whic h can b e co nsidered here a s indep endent v ariables for any f ixed n , m . F or all 8 D. Levi, M. P etrera, C. Scimiterna and R. Y amilov autonomous ABS equations, the compatibilit y condition (25) is satisf ied iden tical ly for all v alues of these v ariables and of the constant paramete r β . In the n on-autonomous case, equation (25) dep end s only on β 0 and α . Therefore the compatibilit y condition i s sat isf ied also for an y m . So, equation (19) is compatible with equation (H3) al so in the ca se when α is constant, but β = β m . In a similar wa y , one can pro v e that equation (20 ) is the generalized symm etry of equation (H 3) if β is constan t, but α = α n . Let us n o w d iscu ss the inte rpretation of the ABS equations as B¨ ac klund transformations. Let u 0 , 0 b e a solution of equat ion (24), and the function e u 0 , 0 = e u n,m ( ε ) giv en b y equatio n (21) b e a solution of equation (23), whic h is compatible with equation (24). equation (23) can b e rewritten as the ordinary dif ference equatio n e u 1 , 0 = ξ ( u 0 , 0 , u 1 , 0 , e u 0 , 0 ; α, β 0 ) , (26) where α is constant, β 0 = β m , m is f ixed, n ∈ Z . Dif f eren tiating equation (26) with resp ect to ε and using equation (24) together with the compatibilit y condition (2 5), one gets d e u 1 , 0 dε − d e u 0 , 0 dε ∂ e u 0 , 0 ξ = f 0 , 0 ∂ u 0 , 0 ξ + f 1 , 0 ∂ u 1 , 0 ξ = e f 1 , 0 − e f 0 , 0 ∂ e u 0 , 0 ξ , where e f k , 0 = f ( e u k +1 , 0 , e u k , 0 , e u k − 1 , 0 ; α ) = f k , 1 , e u k , 0 = u k , 1 . The resulting equation is expr essed in the form Ξ 1 , 0 = Ξ 0 , 0 ∂ e u 0 , 0 ξ , Ξ k , 0 = d e u k , 0 dε − e f k , 0 . (27) There is for the ABS equations a formal cond ition ∂ e u 0 , 0 ξ = ∂ u 0 , 1 ξ 6 = 0. W e sup p ose here that, for the fun ctions u 0 , 0 , e u 0 , 0 under consideration, ∂ e u 0 , 0 ξ 6 = 0 for all n ∈ Z . The function e u 0 , 0 is def ined b y equation (26) up to an integrat ion function µ 0 = µ m ( ε ). W e require that µ 0 satisf ies the f irst o rder ord inary d if ferent ial e quation giv en b y Ξ 0 , 0 | n =0 = 0. Then equation (27) implies that Ξ 0 , 0 = 0 fo r all n , i.e. e u 0 , 0 is a solution of equation (24). So, we start with a solution of a generalized sy m metry of the form (24), def ine a fu nction e u 0 , 0 b y the dif ference equation (26) wh ic h is a form of corresp onding ABS equation, then we sp ecify the in tegratio n function µ 0 b y the ordinary dif ferential equatio n Ξ 0 , 0 | n =0 = 0, and th us obtain a new solution of equation (24). Th is solution d ep ends on an int egration constan t ν 0 = ν m and the parameter β 0 . W e can constr u ct in this w a y th e solutions u 0 , 2 , u 0 , 3 , . . . , u 0 ,N , and the last of th em will dep end on 2 N arbitrary constants ν 0 , β 0 , ν 1 , β 1 , . . . , ν N − 1 , β N − 1 . Using such B¨ ac klund transformation and starting with a simple in itial solution, one can obtain, in p rinciple, a m u lti-soliton sol ution. See [6, 8 ] f or the construction of so me e xamples of solutions. The symmetries (19), (20) a re V o lterra-t yp e equations, namely du 0 dε = f ( u 1 , u 0 , u − 1 ) , (28) where we ha v e d ropp ed one of the indep endent indexes n or m , since it do es not v ary . The V olterra equation corresp onds to f ( u 1 , u 0 , u − 1 ) = u 0 ( u 1 − u − 1 ). An exhaustiv e list of d if ferent ial- dif ference in tegrable equations of the form (28) has b een o btained in [30] (details c an b e f ound in [31]). All three-p oint generalized symmetries of the ABS equations, with no explicit dep en- dence on n , m , ha ve the sa me structure as equati on (19) (see details in S ection 4.4 b elo w) and are particular cases of the YdKN equat ion du 0 dε = R ( u 1 , u 0 , u − 1 ) u 1 − u − 1 , R ( u 1 , u 0 , u − 1 ) = R 0 = A 0 u 1 u − 1 + B 0 ( u 1 + u − 1 ) + C 0 , (29) On Miura T rans formations and V olterra-T yp e Equations 9 where A 0 = c 1 u 2 0 + 2 c 2 u 0 + c 3 , B 0 = c 2 u 2 0 + c 4 u 0 + c 5 , C 0 = c 3 u 2 0 + 2 c 5 u 0 + c 6 , and th e c i ’s are constan ts. equation (29) h as b een found b y Y amilo v in [30], d iscussed in [21, 4], and in m ost detailed form in [31 ]. Its contin u ous limit go es into the Krichev er–Novi k o v equ a- tion [15]. Th is is the only integ rable example of the form (28) whic h cannot b e reduced, in general, to the T o d a or V olterra equations b y Miura-t yp e transformations. Moreo v er, equa- tion (29) is also relate d to th e Land au–Lifshitz equ ation [26]. A generaliza tion of equation (2 9) with nine arbitrary constan t co ef f icien ts has b een c onsidered in [20]. By a straigh tforw ard computation w e get the follo wing result: all thr ee-p oin t generalized symmetries in the n -direction with no explicit dep endence on n , m for the ABS equations are particular case s of the YdKN equation. F or the v arious equations of the AB S classif ication the co ef f icien ts c i , 1 ≤ i ≤ 6, read H1 : c 1 = 0 , c 2 = 0 , c 3 = 0 , c 4 = 0 , c 5 = 0 , c 6 = 1 , H2 : c 1 = 0 , c 2 = 0 , c 3 = 0 , c 4 = 0 , c 5 = 1 , c 6 = 2 α, H3 : c 1 = 0 , c 2 = 0 , c 3 = 0 , c 4 = 1 , c 5 = 0 , c 6 = 2 αδ , Q1 : c 1 = 0 , c 2 = 0 , c 3 = − 1 , c 4 = 1 , c 5 = 0 , c 6 = α 2 δ 2 , Q2 : c 1 = 0 , c 2 = 0 , c 3 = 1 , c 4 = − 1 , c 5 = − α 2 , c 6 = α 4 , Q3 : c 1 = 0 , c 2 = 0 , c 3 = − 4 α 2 , c 4 = 2 α ( α 2 + 1) , c 5 = 0 , c 6 = − ( α 2 − 1) 2 δ 2 , Q4 : c 1 = 1 , c 2 = − α, c 3 = α 2 , c 4 = g 2 4 − α 2 , c 5 = αg 2 4 + g 3 2 , c 6 = g 2 2 16 + αg 3 . Prop osition 2. The ABS e quations (H1)–(H3) and (Q 1)–(Q4) c orr esp ond to B¨ acklund tr a ns- formations of the p articular c ases of the YdKN e quation (29) liste d ab ove. The same holds for the non-autonomous ABS e quations, such that α is c onstant and β = β m or α = α n and β is c onstant. Equation (29) and the r eplac ement u i → u i, 0 pr ovide the thr e e-p oint gener alize d symmetries in the n -dir e ction of the ABS e q u ations with a c onsta nt α and β = β m , while e qua- tion (29 ) and the r epla c ement u i → u 0 ,i , α → β pr ovide symmetries in the m -dir e ction for the c ase α = α n and a c onstant β . The non-autonomous case is brief ly discussed in [24 ] where they state that if α is not constan t, then th e ABS equations h a v e no lo cal three-p oin t symmetries in the n -direction. W e shall present three-, f iv e- and man y-p oint generalized symmetries in the m -direction f or such equ ations in Subsection 4. 3. A r elation b et w een the ABS equations and d if feren tial-dif fer en ce equations is discus sed in [2, 5]. In [2] most of the ABS equations are inte rpreted as n onlinear su p erp osition principles for dif ferentia l-dif ference equations of the form ( ∂ x u n +1 ) ( ∂ x u n ) = h ( u n +1 , u n ; α ) , (30) where h is a p olynomial of u n +1 , u n . Equations of the form (30) d ef ine B¨ ac klund transformations for sub cases of the K r ic hev er–No viko v equation ∂ t u = ∂ xxx u − 3 2 ( ∂ xx u ) 2 − P ( u ) ∂ x u , (31) where P is a fourth degree p olynomial with arbitrary constant coef f icien ts. In the ca se of equations (H1) and (H3) with δ = 0, the corresp on d ing dif feren tial-dif feren ce equations h a v e a dif ferent form, and the resulting KdV-t yp e equations d if fer f rom e quation (31). In [5] it is sh o wn that the con tin uous limit of equ ation (Q4) goes in to a sub case of the YdKN equation. I t is stated that equation (Q4) def ines a B¨ ac klund transformation for the same sub case. The same sc heme holds for equations (Q1)–(Q3), but it is not clear if the r esulting V olterra-t y p e equations are of the form (29). 10 D. Levi, M. P etrera, C. Scimiterna and R. Y amilov 4.2 Miura transformations revised It is p ossible to revise the Miura trans f ormations constructed in Section 3 from the p oint of view of the generalized symmetries. Let us introdu ce the follo wing fu nction r 0 = r ( u 0 , u − 1 ) = A 0 u 2 − 1 + 2 B 0 u − 1 + C 0 = R ( u − 1 , u 0 , u − 1 ) . It can b e c hec ked that r ( u 0 , u − 1 ) = r ( u − 1 , u 0 ) and, in term s of r 0 the right hand side of equa- tion (2 9) reads R 0 u 1 − u − 1 = r 0 u 1 − u − 1 + 1 2 ∂ u − 1 r 0 = r 1 u 1 − u − 1 − 1 2 ∂ u 1 r 1 . (32) All the ABS equations, up to equation (Q4), are such that c 1 = c 2 = 0, so that the p olyno- mial R 0 is of s econd degree. In th is case equation (29 ) ma y b e transformed [31] in to equation (28) with f ( u 1 , u 0 , u − 1 ) = u 0 ( u 1 − u − 1 ) (V ol terra equation) b y the Miura trans f ormation e u 0 = − r 1 ( u 2 − u 0 )( u 1 − u − 1 ) . The ab o v e map br ings an y solution u 0 of equation (29) with c 1 = c 2 = 0 in to a solution e u 0 of the V olterra equation. Th is is exactly the same Miura trans f ormation we ha v e already pre- sen ted in Section 3. So, also at the lev el of the generalized symmetries, we ma y s ee that th ere is a deep r elation b et w een equations (H1)–(H3) and (Q1)–(Q3) and th e V olte rra equation. If equa- tion (29) cann ot b e transform ed to the case with c 1 = c 2 = 0 , u sing a M¨ obius tr an s formation, then it cann ot b e mapp ed into the V olterra equation b y e u 0 = G ( u 0 , u 1 , u − 1 , u 2 , u − 2 , . . . ) [31]. Equation (Q 4) is of this kin d and thus is the only equ ation of th e ABS list w hic h cannot b e related t o the V olterra equation. 4.3 Master symmetries Generalized symmetries of equation (29) will also b e compatible w ith the ABS equations, whic h are, according to Prop osition 2, their B¨ ac klund transformations. Suc h symmetries can b e con- structed, using the master symmetry of equat ion (29) presen ted in [4]. Let us rewrite equation (29) by using the equiv alen t n -dep enden t notatio n (see equation (1)), namely du n dε 0 = f (0) n = R ( u n +1 , u n , u n − 1 ) u n +1 − u n − 1 , (33) where ε 0 is th e con tin uous symmetry parameter (pr eviously denoted w ith ε ). W e shall d enote with ε i , i ≥ 1, the parameters corresp ond ing to higher g eneralized symmetries du n dε i = f ( i ) n , suc h that d f ( j ) n dε i − d f ( i ) n dε j = 0 , i, j ≥ 0 . Let us introdu ce the master sym metry du n dτ = g n , suc h that d f ( i ) n dτ − dg n dε i = f ( i +1) n , i ≥ 0 . (34) Once w e kno w the master symmetry (34) we can construct explicitly the inf inite h ierarch y of generalized symmetries. On Miura T rans formations and V olterra-T yp e Equations 11 The maste r symmetry o f e quation (33) is giv en b y g n = nf (0) n . (35) According to a general pro cedur e describ ed in [31] w e need to introdu ce an explicit depen d ence on th e parameter τ in to the master s ymmetry (35) and in to equ ation (33) itself. Let the co ef f icien ts c i , app earing in the polynomials A n , B n , C n , b e functions of τ . Th is τ -depend ence implies that r n satisf ies the follo wing partial dif ferential equation 2 ∂ τ r n = r n ∂ u n ∂ u n − 1 r n − ( ∂ u n r n )  ∂ u n − 1 r n  . (36) On th e left h and s ide of the ab o v e equation, w e d if ferenti ate only th e co ef f icien ts of r n with resp ect to τ . The right hand side has th e same form as r n , but with d if feren t co ef f icien ts. Collecting the co ef f icients of the terms u i n u j n − 1 for v arious p o w ers i and j , we obta in a system of six ord inary dif ferent ial equations f or the six co ef f icien ts c i ( τ ), wh ose initial conditions are c i (0) = c i . Generalized sy m metries constructed by using equation (34) explicitly dep end on τ . They remain generalized symmetries for an y v alue of τ , as τ is just a parameter f or them and for equation (33). So, going o ver to the initial conditions, w e get generalized symmetries of equation (3 3) and of the c orresp ond ing ABS equations. Let us deriv e, as an illus trativ e example, a formula for th e symmetry f (1) n from equation (34). F rom equations (33)–(35) it follo ws that f (1) n = ∂ τ f (0) n + f (0) n +1 ∂ u n +1 f (0) n − f (0) n − 1 ∂ u n − 1 f (0) n . (37) Using equatio ns (32) a nd (36 ) o ne obtains ∂ u n +1 f (0) n = − r n ( u n +1 − u n − 1 ) 2 , ∂ u n − 1 f (0) n = r n +1 ( u n +1 − u n − 1 ) 2 , and ∂ τ R n = R n = R ( u n +1 , u n , u n − 1 ) = A n u n +1 u n − 1 + B n 2 ( u n +1 + u n − 1 ) + C n , with A n = B n ∂ u n A n − A n ∂ u n B n , B n = C n ∂ u n A n − A n ∂ u n C n , C n = C n ∂ u n B n − B n ∂ u n C n . F rom equation (37 ) we get the f irst generaliz ed symmetry du n dε 1 = f (1) n = R n u n +1 − u n − 1 − r n f (0) n +1 + r n +1 f (0) n − 1 ( u n +1 − u n − 1 ) 2 . (38) Up to our kn o wledge this formula is n ew. It provides f ive-point generalized symmetries in b oth n - and m -directions f or th e ABS equations. E x amp les of such f ive-point symmetries for equations (H1 ) and ( Q1) with δ = 0 can b e f ound i n [24, 27]. Let us clarify the construction of the symmetry f (1) n for equations (H1)–( H3). I n these c ases the function r n tak es the form r n = 2 c 4 ( τ ) u n u n − 1 + 2 c 5 ( τ )( u n + u n − 1 ) + c 6 ( τ ) , and equati on (36) is equiv alen t to the system ∂ τ c 4 ( τ ) = 0 , ∂ τ c 5 ( τ ) = 0 , ∂ τ c 6 ( τ ) = c 4 ( τ ) c 6 ( τ ) − 2 c 2 5 ( τ ) . (39) 12 D. Levi, M. P etrera, C. Scimiterna and R. Y amilov The initia l conditions of system (39) are (see t he list ab ov e Prop osition 2) H1 : c 4 (0) = 0 , c 5 (0) = 0 , c 6 (0) = 1 , H2 : c 4 (0) = 0 , c 5 (0) = 1 , c 6 (0) = 2 α, H3 : c 4 (0) = 1 , c 5 (0) = 0 , c 6 (0) = 2 αδ , and its solutions are giv en b y H1 : c 4 ( τ ) = 0 , c 5 ( τ ) = 0 , c 6 ( τ ) = 1 , H2 : c 4 ( τ ) = 0 , c 5 ( τ ) = 1 , c 6 ( τ ) = 2( α − τ ) , H3 : c 4 ( τ ) = 1 , c 5 ( τ ) = 0 , c 6 ( τ ) = 2 αδ e τ . Note that the master symmetry with the abov e c i ( τ ) generates τ - dep end en t symmetries for a τ -dep endent equation, but by f ixing τ w e obtain τ -ind ep end ent symmetries f or a τ -indep endent equation. Let us remark that the τ -dep endence is ind ep endent of the order of the symmetry and it ma y b e used for th e co nstruction of all higher symmetries. So, according to formula (38), w e ma y construct the generalized symmetry f (1) n , in the case of the list H, fr om the follo wing expressions H1 : f (0) n = 1 u n +1 − u n − 1 , r n = 1 , R n = 0 , H2 : f (0) n = u n +1 + u n − 1 + 2( u n + α ) u n +1 − u n − 1 , r n = 2( u n + u n − 1 + α ) , R n = − 2 , H3 : f (0) n = u n ( u n +1 + u n − 1 ) + 2 αδ u n +1 − u n − 1 , r n = 2( u n u n − 1 + αδ ) , R n = 2 αδ . It is p ossible to v erify that th e symmetries (38) with f (0) n , r n , R n giv en ab o v e are compatible with both equations (33) and (H1) –(H3). By using th e master symm etry constructed ab ov e we can construct inf inite hierarc hies of man y-p oint generalized symmetries of the ABS equations in b oth directions. In th e non- autonomous cases (see Prop osition 2) w e pro vide one hierarch y in the n - or m -direction. Th e master sym metry an d formula (38) will also b e us eful in the case of the generalizat ions of the ABS equations presented in the next Sub s ection. It s h ould b e remarked that in [24] the au th ors constructed ma ster symmetries for all autonomous and non-autonomous ABS equations, whic h are of a dif fer ent kin d with resp ect to the ones presen te d here. 4.4 Generalizations of the AB S equations Here we discuss the generalization of the ABS equations introd u ced by T ongas, Tsoub elis and Xenitidis (TTX) in [28]. The TTX equ ations are autonomous lattice equations of the form (2) whic h p ossess only t w o of the four main prop erties of the ABS equatio ns: they are af f ine linear and possess the sym m etries of the square. In te rms of th e p olynomial E , see equation (2), o ne g enerates the follo wing function h h ( u 0 , 0 , u 1 , 0 ; α, β ) = E ∂ u 0 , 1 ∂ u 1 , 1 E −  ∂ u 0 , 1 E   ∂ u 1 , 1 E  , whic h is a biqu ad r atic and symmetric p olynomial in its f irst t w o argum en ts. It has b een prov ed in [28] that the TTX equations admit thr ee-p oin t generalized s ymmetries in the n -direction of the form du 0 , 0 dε = h u 1 , 0 − u − 1 , 0 − 1 2 ∂ u 1 , 0 h. (40) On Miura T rans formations and V olterra-T yp e Equations 13 Of course, there is a similar symmetry in the m -direction. Comparing equations (29), (32) and (40), we see that the sym m etry (40) is nothing but the YdKN equation in its general form. This sho ws that all TTX equations can also b e considered as B¨ ac klund transformations for the Yd K N equation. Ho wev er, they probably describ e the general picture for B¨ acklund trans- formations of the YdKN equation, wh ic h ha ve the form (2). The general form ula (38) and the master symm etry discussed in the previous Subsection, pr o vide f ive- an d man y-p oin t generalized symmetries of the TTX equ ations in b oth directio ns, th us c onf irming their i n tegrabilit y . 5 Concluding remarks In this p ap er we h a v e considered some fu rther p rop erties of the ABS equations. I n particular w e hav e sho wn that equations (H1) –(H3) a nd (Q1 )–(Q3) can be transformed into equations a s- so ciated with the sp ectral prob lem of the V olterra equation. Th erefore all kno wn r esults for th e solution of the V olte rra equation can b e us ed to construct solutions of the ABS equations. More- o v er, all equ ations of the ABS list, except equation (Q4), can b e transformed among themselve s b y Miu r a transformations. The situation of equation (Q4) is somehow dif ferent. It is s ho wn that this equation can b e th ou ght as a B¨ ac klund transformation for a sub case of the Y amilo v discretiza tion of the Kric hev er–No viko v equation. But it cannot b e r elated by a Miura transformation to a V olterra- t yp e e quation and this explains the complicate form of its scalar sp ectral pr oblem. The mast er symmetry constructed f or the Yd KN equ ation can, ho wev er, b e used also in this case to construct generalized symmetries. It turns out th at a generalizations of the ABS equ ations introdu ced b y T ongas, Tsoub elis and Xe nitidis are B ¨ ac klund transformations for the YdKN equation. F urther generaliza tions of the TTX and ABS equations can b e p robably obtained by a prop er explicit dep end ence o n the p oint o f the lattice not only in the lattice parameters α and β , but also in the Z 2 -lattice equation itself. The existence of an n -dep endent generalization of the YdKN equation, in trodu ced in [20], co uld h elp in solving this problem. Suc h a generalization is in tegrable in the sense that it has a master sym metry [4 ] similar to the one present ed here. Ac kno wledgmen ts DL, MP a nd CS ha ve b een partially supp orted b y PRIN Pro ject Meto di ge ometrici nel la te oria del le onde non line ari e d applic azioni-2006 of the Italian Minister for Edu cation and Scien tif ic Researc h. R Y h as b een partially su pp orted by the Ru ssian F oundation for Basic Research (Gran t num b ers 07-01 -00081 -a and 06-01-9205 1-KE-a) and he thanks the Unive rsit y of Roma T re for hospitalit y . This w ork has b een done in the fr amew ork of the Pro ject Classific atio n of inte gr able discr ete and c ontinuous mo dels f in anced by a joint gran t from EINSTEIN consortium and RFB R. References [1] Adler V.E., O n th e structure of the B¨ acklund transformations for the relativistic lattices, J. 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