Differential-difference equations associated with the fractional Lax operators

Differen tial-difference equations asso ciated with the fractional Lax op erators V.E. Adler ∗ , V.V. P ostniko v † July 12, 2011 Abstract. W e study in tegrable hierarchies associated with sp ectral prob- lems of the form P ψ = λQψ where P , Q are difference operators. The cor- resp onding nonlinear differential-difference equations can b e view ed as in- homogeneous generalizations of the Bogo ya vlensky t yp e lattices. While the latter turn into the Kortew eg–de V ries equation under the con tinuous limit, the lattices under consideration provide discrete analogs of the Sa wada– Kotera and Kaup–Kup ershmidt equations. The r -matrix formulation and sev eral simplest explicit solutions are presen ted. Keyw ords: Lax pair, discretization, Bogoy avlensky lattice, Sa wada–Kotera equation, Kaup–Kup ershmidt equation MSC: 35Q53, 37K10 1 In tro duction The simplest example studied in this pap er is the lattice equation u ,t = u 2 ( u 2 u 1 − u − 1 u − 2 ) − u ( u 1 − u − 1 ) (1) where we use the shorthand notations u = u ( n, t ) , u ,t = ∂ t ( u ) , u j = u ( n + j, t ) . F or the first time, this equation w as deriv ed by Tsujimoto and Hirota [ 1 , eq. (4.12)] as the con tinuous limit of the reduced discrete BKP hierarc hy . Recall that both equations u ,t 0 = u ( u 1 − u − 1 ) and u ,t 00 = u 2 ( u 2 u 1 − u − 1 u − 2 ) (2) ∗ L.D. Landau Institute for Theoretical Physics, 1A Ak. Semeno v, Chernogolovk a 142432, Russia. E-mail: adler@itp.ac.ru † So chi Branc h of Peoples’ F riendship Universit y of Russia, 32 Kuib yshev str., 354000 So c hi, Russia. E-mail: p ostnik ofvv@mail.ru 1 are v ery well kno wn integrable mo dels: resp ectiv ely , the V olterra lattice [ 2 , 3 ] and the mo dified Narita–Itoh–Bogo ya vlensky lattice of the second or- der [ 4 , 5 , 6 ]. One can easily v erify that the flows ∂ t 0 and ∂ t 00 do not commute, that is, these equations b elong to the differen t hierarc hies. Hence, one should not exp ect a priori that their linear combination remains in tegrable. Nev- ertheless, this is the case: we will sho w that equation ( 1 ) admits the Lax represen tation L ,t = [ A, L ] with the op erator L equal to a ratio of tw o difference op erators, namely , L = ( T 2 + u ) − 1 ( uT 2 + 1) T where T denotes the shift op erator u k → u k +1 . Equation ( 1 ) can b e cast in to the Hirota’s bilinear form whic h admits a family of generalizations dep ending on a pair of integer parameters ( l , m ). These generalizations were discov ered b y Hu, Clarkson and Bullough [ 7 , eq. (4)] who searc hed for bilinear equations admitting N -soliton solutions. One of the goals of our pap er is to demonstrate that this family of equations is asso ciated with the fractional Lax op erators of the form L = ( T m + u ) − 1 ( uT m + 1) T l . (3) As usually , any such L is asso ciated with a whole commutativ e hierarc hy of equations corresp onding to the sequence of difference op erators A of in- creasing order. W e denote this hierarch y dSK ( l,m ) , since it can b e view ed as a discretization of the hierarc hy con taining the Saw ada–Kotera equation [ 8 , 9 ] U ,τ = U 5 + 5 U U 3 + 5 U 1 U 2 + 5 U 2 U 1 (SK) where we denote U = U ( x, τ ) , U ,τ = ∂ τ ( U ) , U j = ∂ j x ( U ) . F or instance, equation ( 1 ) belongs to dSK (1 , 2) . The concrete formula of the con tinuous limit in this example is the follo wing, at ε → 0: u ( n, t ) = 1 3 + ε 2 9 U  x − 4 9 εt, τ + 2 ε 5 135 t  , x = εn (4) and an analogous formula exists for any ( l , m ). It should b e noted that eac h of equations ( 2 ) apart defines a discretization of the Kortew eg–de V ries (KdV) equation U ,t = U 3 + 6 U U 1 rather than the SK one. Moreo ver, it is well kno wn that actually all Bogo ya vlensky t yp e lattices serve as discretizations of the KdV equation or its higher symmetries, so that an infinite family of discrete hierarchies corresp ond to just one con tinuous. Quite analogously , the whole family of dSK ( l,m ) hierarc hies serve as discrete analogs of the SK hierarc hy . W e hop e that this observ ation mak es clear the place of these equations in the big picture of in tegrable systems. 2 On the other hand, the differen tial and difference cases are not quite parallel. First, Lax op erator for the SK equation L = D 3 + U D = ( D − f )( D + f ) D is not fractional. Lax op erators given by the ratio of differen tial op erators w ere studied by Krichev er [ 10 ], how ev er it seems that these examples and ( 3 ) are unrelated. Second, let us consider the problem of discretization for another impor- tan t example, the Kaup–Kup ershmidt equation [ 11 , 12 , 13 ] U ,τ = U 5 + 5 U U 3 + 25 2 U 1 U 2 + 5 U 2 U 1 . (KK) Recall that it is asso ciated with the op erator L = D 3 + U D + 1 2 U ,x = ( D + f ) D ( D − f ) and b oth SK and KK equations are connected through the Miura substitu- tions obtained b y factorization of Lax op erators [ 14 , 15 ]: U SK = f ,x − f 2 , U KK = − 2 f ,x − f 2 . Despite of this close relation, it w as noted that some prop erties of the SK and KK equations are rather differen t, see e.g. [ 16 ]. It seems that distinc- tions b etw een the lattice analogs of these equations are even more deep. A discretization of the KK equation is presen ted in section 4 , ho w ever, w e w ere able to find just one op erator L in this case comparing to infinite family ( 3 ) in the SK case, and no discrete analog of Miura type substitution b et ween dSK and dKK is known. The con tents of the pap er is the following. Section 2 con tains some necessary information on the lattices of Bogoy a vlensky type, see also b o oks [ 17 , 18 ]. Section 3 dev oted to discretization of the SK equation contains the main results of the pap er. A general construction of the Lax pairs with op erator ( 3 ) is giv en in section 3.1 . In section 3.3 , the r -matrix approac h in the difference setting [ 19 , 20 , 18 ] is used to obtain explicit form ulas for the op erator A and to prov e the comm utativity of the dSK ( l,m ) hierarc hy . The con tinuous limit, the bilinear represen tation, the simplest breather t ype solutions are presented in sections 3.4 , 3.5 . Section 4 is dev oted to discretiza- tion of the KK equation and section 5 contains sev eral examples of coupled lattice equations associated with more general fractional Lax op erators. 3 2 Preliminaries 2.1 Definitions and notations W e consider differential-difference (lattice) equations of the ev olutionary form u ,t = f ( u m , . . . , u − m ) , u = u ( n, t ) , u ,t = ∂ t ( u ) , u j = u ( n + j, t ) . (5) Suc h equations can be viewed as discrete analogs of con tinuous ev olutionary equations like KdV or SK U ,τ = F ( U k , . . . , U ) , U = U ( x, τ ) , U ,τ = ∂ τ ( U ) , U j = ∂ j x ( U ) (the orders m and k may not coincide under the con tinuous limit). The shift op erator T : u j 7→ u j +1 pla ys the same role for equations ( 5 ) as the total x -deriv ativ e D : U j 7→ U j +1 pla ys in the contin uous case. Differential op erators are polynomials with respect to D , with the m ultiplication defined b y the Leibniz rule D A = D ( A ) + AD and the conjugation defined by the rule D † = − D . In con trast, difference op erators are in general Laurent p olynomials, that is con tain p ow ers of b oth T and T − 1 , and the rules for the multiplication and the conjugation are T A = T ( A ) T and T † = T − 1 . F or short, w e will use subscripts also for denoting action of T on op erators, A j = T j ( A ). A lattice equation u ,t 0 = g ( u k , . . . , u − k ) is called symmetry of ( 5 ) if the compatibility condition D ,t ( g ) = D ,t 0 ( f ) is fulfilled, that is [ f , g ] ∗ := m X s = − m ∂ u s ( f ) T s ( g ) − k X s = − k ∂ u s ( g ) T s ( f ) = 0 . (6) The lattice is called in tegrable if it admits an infinite sequence of symmetries with the order k greater than any fixed n umber. The linear space of all symmetries is called hierarc hy . A conserv ation law is a relation of the form D ,t ( ρ ( u k , . . . , u )) = ( T − 1)( σ ( u k + m − 1 , . . . , u − m )) whic h holds true in virtue of equation ( 5 ). The discussion of these notions and applications to the problem of classification of in tegrable lattice equa- tions can be found in the review article by Y amilo v [ 21 ]. 2.2 Bogo y avlensky lattices Understanding the structure of dSK ( l,m ) hierarc hy is not p ossible without understanding the homogeneous hierarchies of Bogoy avlensky t yp e. A gen- eral pattern of (lo cal) equations from dSK ( l,m ) is given b y the form ula u ,t k = F ( L + K M ) + · · · + F ( L + M ) + F ( L ) 4 where F ( s ) denotes a homogeneous polynomial of degree s with resp ect to the v ariables u j and K, L, M are related somehow with the parameters l , m and the order k of the flow. Moreov er, the first and the last terms in the sum alwa ys corresp ond to some (mo dified) lattices of Bogo ya vlensky type b elonging to the different hierarc hies. This structure is explained by the following arguments, starting from the Lax representation with the op erator L ( 3 ). Let us consider the scaling u → δ − m u , T → δ T , then it is easy to see that the limit δ → ∞ sends L to the op erator L 0 = u − m T l + T l − m and the limit δ → 0 leads to L 00 = T m + l + u − 1 T l . Eac h of these operators corresp onds to its o wn hierarch y of homogeneous lattice equations. The total inhomogeneous equation con tains b oth of them together with the intermediate terms whic h are necessary for preserving commutativit y of the flo ws. Let us consider the concrete example. One can c heck that the lattice u ,t 0 = u  w 1 ( w 3 + w 2 + w 1 + w ) − w − 1 ( w + w − 1 + w − 2 + w − 3 ) − u 1 ( w 3 + w − 1 ) + u − 1 ( w 1 + w − 3 )  , w := u (1 − u 1 u − 1 ) (7) is a higher symmetry of equation ( 1 ). Collecting the homogeneous terms yields u ,t = F (4) + F (2) , u ,t 0 = G (7) + G (5) + G (3) and the consistency condition of the flo ws splits to relations [ F (4) , G (7) ] ∗ = 0 , [ F (4) , G (5) ] ∗ + [ F (2) , G (7) ] ∗ = 0 , [ F (4) , G (3) ] ∗ + [ F (2) , G (5) ] ∗ = 0 , [ F (2) , G (3) ] ∗ = 0 where commutator [ , ] ∗ is defined b y equation ( 6 ). As it w as already said in In tro duction, p olynomials F (4) and F (2) corresp ond to the modified Bo- go ya vlensky and V olterra lattices. Polynomials G (7) and G (3) corresp ond to their symmetries and the intermediate polynomial G (5) comp ensates incon- sistency of the hierarc hies. The Bogo ya vlensky hierarc hy B ( m ) is asso ciated with the op erator L = T + uT − m and w e recall here several basic form ulas regarding this case. A detailed theory can b e found in the bo oks [ 6 , 18 ]. More general op erators of the form L = T l + uT − m w ere considered recen tly in the pap er [ 22 ]. The simplest equation from the B ( m ) hierarc hy reads u t = u ( u m + · · · + u 1 − u − 1 − · · · − u − m ) . (8) This equations and its higher symmetries are asso ciated with the difference sp ectral problem ψ 1 + uψ − m = λψ and admit the Lax representations L ,t k = [ A ( k ) , L ] , L = T + uT − m , A ( k ) = π +  L ( m +1) k  (9) 5 where π + denotes the pro jection of an y formal series A = P j < ∞ a ( j ) T j on to the linear space of p olynomials with resp ect to T : π + ( A ) = X 0 ≤ j < ∞ a ( j ) T j , π − ( A ) = X j < 0 a ( j ) T j . In particular, A (1) = T m +1 + v , v := u m + · · · + u and equation ( 9 ) at k = 1 is equiv alent to lattice ( 8 ). The chec k is easy: L ,t − [ A (1) , L ] = u ,t T − m − [ T m +1 + v , T + uT − m ] = u ,t T − m − ( u m +1 − u + v − v 1 ) T − u ( v − v − m ) T − m , (10) the terms with T cancel and the rest yields the equation. In order to pro v e that equation ( 9 ) correctly defines the lattice for any k , w e hav e to chec k that all p o wers of T except for T − m v anish in the comm utator [ A ( k ) , L ]. Since L m +1 is a Laurent p olynomial with resp ect to T m +1 , hence A ( k ) is a p olynomial with resp ect to T m +1 . Therefore the comm utator con tains only pow ers of the form T ( m +1) j +1 , j ≥ − 1. On the other hand, [ A ( k ) , L ] = − [ π −  L ( m +1) k  , L ] , so that the commutator do es not con tain positive p o wers of T and only one p ossible p o w er T − m remains. It can b e prov en that equations ( 9 ) define a sp ecial reduction in the Lax pair with a generic op erator L = T + u (0) + u (1) T − 1 + · · · + u ( m ) T − m . In this case one can c ho ose op erators A in the form A = π + ( L k ) with arbitrary k . F or instance, the T oda lattice hierarc hy app ears at m = 1. This t yp e of m ulti-field systems w as studied, for instance, in pap ers [ 20 , 22 ]. 3 Discretizations of the Saw ada–Kotera equation 3.1 Lax representation Let us consider the difference sp ectral problem uψ m + l + ψ l = λ ( ψ m + uψ ) (11) where m, l are in tegers. W e assume that m, l are p ositiv e and coprime, with- out loss of generalit y , since the general case can b e obtained b y refinemen t of the mesh and/or c hange of its directions. It is less ob vious that the num b ers m and l can be exc hanged: sp ectral problem ( 11 ) is equiv alen t to uϕ m + l + ϕ m = µ ( ϕ l + uϕ ) 6 under the c hange ψ ( n ) = κ n ϕ ( n ) , λ = − κ l , µ = − κ − m . (12) In the operator form, equation ( 11 ) reads P ψ = λQψ , P = ( uT m + 1) T l , Q = T m + u. (13) The isosp ectral deformations are defined b y equation ψ ,t = Aψ with some difference op erator A . The corresp onding Lax equation L ,t = [ A, L ] , L = Q − 1 P (14) can b e rewritten as the system P ,t = B P − P A, Q ,t = B Q − QA (15) where one of equations can be considered just as a definition of B . Let P, Q b e as in ( 13 ), then this system is equiv alen t to equations u ,t = B ( T m + u ) − ( T m + u ) A, B ( T 2 m − 1) = A m T 2 m − A l + uAT m − uA m + l T m . (16) In order to resolv e the latter w e mak e the assumption that operator A is of the form A = F ( T m − T − m ) (17) then B is found as the difference operator B = F m T m − F 1 T − m + u ( F − F m +1 ) (18) while first equation ( 16 ) turns into u ,t = T m F u + uF T − m − uT m F l − F l T − m u + F m − F l + u ( F − F m + l ) u. (19) It is clear that the same ev olution of the v ariable u is defined by the conju- gated op erator F † and, moreov er, all terms T j , j - m can be thrown a wa y . This means that we can find F as a self-adjoin t op erator F = F † whic h is a Lauren t polynomial with resp ect to the pow ers T m : F = f ( k ) T km + · · · + f (1) T m + f (0) + T − m f (1) + · · · + T − km f ( k ) , k ≥ 0 . (20) Certainly , the co efficien ts dep end on k , l , m , so that it w ould b e more rig- orous to write f ( j,k,l ,m ) instead of f ( j ) , but we will consider these num b ers fixed at the momen t. Collecting the coefficients at T j m , j > 0, yields the relations u j m f ( j − 1) m − uf ( j − 1) m + l = f ( j ) l − f ( j ) m + uu j m ( f ( j ) m + l − f ( j ) ) + u j m f ( j +1) l − uf ( j +1) , j = 1 , . . . , k + 1 , (21) 7 where it is assumed for con venience that f ( j ) = 0 at j > k . The co efficien t at T 0 giv es an ev olutionary equation for u : u ,t = 2 u ( f (1) − f (1) l ) + u 2 ( f (0) − f (0) m + l ) + f (0) m − f (0) l . (22) System of equations ( 21 ), ( 22 ) defines the k -th flow in the hierarch y dSK ( l,m ) . If we are interested in the lo cal evolution only then w e require that all f ( j ) can b e recurren tly found as functions of an finite set of v ariables u i . In this case a certain restriction on the v alues of k app ears and a part of the flo ws is rejected. Indeed, consider equation ( 21 ) at j = k + 1, u ( k +1) m f ( k ) m = uf ( k ) m + l , (23) or ( T l − 1)(log f ( k ) m ) = ( T ( k +1) m − 1)(log u ) . It can b e pro ven that it is solv able with resp ect to f ( k ) if and only if ( k + 1) m is divisible b y l and the solution is, up to a constant factor, f ( k ) = u − m u l − m · · · u ( s − 1) l − m , ( k + 1) m = sl . (24) Since l and m are coprimes, hence the lo cal flows may app ear only if k = pl − 1 and s = mp . The fact that the rest equations ( 21 ) for such k are solv able indeed will b e v erified later in section 3.3 . The case l = 1 is the only one when there are no restrictions on k and the simplest c hoice k = 0 brings in this case to the follo wing family of lattices. Theorem 1. F or any m > 0 , the simplest e quation in the hier ar chy dSK (1 ,m ) u ,t = u 2 ( u m · · · u 1 − u − 1 · · · u − m ) − u ( u m − 1 · · · u 1 − u − 1 · · · u 1 − m ) (25) p ossesses L ax r epr esentation ( 14 ) with the op er ators P = uT m +1 + T , Q = T m + u, A = f ( T − m − T m ) , B = f 1 T − m − f m T m + u ( f m +1 − f ) wher e f = u − 1 · · · u − m . Pr o of. A direct computation (cf with ( 10 )) pro ves that both equations ( 15 ) with given P, Q, A, B are equiv alen t to relations u m f m = uf m +1 , u ,t = u 2 ( f m +1 − f ) − f m + f 1 . The former defines the v ariable f (up to a constan t factor) and the latter is equiv alen t to lattice ( 25 ). 8 In particular, equation ( 25 ) at m = 2 coincide with ( 1 ) and at m = 1 it is just the mo dified V olterra lattice u ,t = u 2 ( u 1 − u − 1 ) . It should b e remark ed that gauge equiv alence ( 12 ) b et ween the sp ectral problems can b e extended on the level of nonlinear equations and the same flo w ( 25 ) appears also as a member of dSK ( m, 1) hierarc hy . Ho w ev er, op- erator ( 20 ) is muc h more complicated in this case: it contains all p o wers T m − 1 , T m − 2 , . . . , T 1 − m comparing with just F = f (0) in dSK (1 ,m ) case. Computing of higher symmetries quickly b ecomes inv olved, b ecause find- ing of F requires (discrete) in tegration of rather bulky expressions. F or in- stance, the second flo w in the hierarch y dSK (1 ,m ) is, according to ( 22 ), of the form u ,t 0 = 2 u ( f (1) − f (1) 1 ) + u 2 ( f (0) − f (0) m +1 ) + f (0) m − f (0) 1 where functions f (1) , f (0) are defined b y relations u 2 m f (1) m = uf (1) m +1 , u m f (0) m − uf (0) m +1 = f (1) 1 − f (1) m − uu m ( f (1) − f (1) m +1 ) . This yields, up to integration constan ts, f (1) = u m − 1 · · · u − m , f (0) = ( w + · · · + w − 2 m +1 ) u − 1 · · · u − m , w := (1 − u m − 1 u − 1 ) u m − 2 · · · u 0 (at m = 2 equation ( 7 ) app ears). One can chec k straightforw ardly that the obtained flo w commutes with ( 25 ) indeed. A general pro of and a wa y to b ypass the in tegration are giv en below in section 3.3 . Adopting nonlo cal v ariables leads to some extension of the hierarc h y . In this case w e consider equation ( 23 ) as a constrain t which defines the v ariable f ( k ) for an y k . Then w e arriv e to the following system which generalizes ( 25 ) for an y l , making the picture more uniform. W e will return to this system in section 3.5 . Theorem 2. F or any c oprime m, l , the simplest system in the extende d dSK ( l,m ) hier ar chy u m f m = uf m + l , u ,t = u 2 ( f − f m + l ) + f m − f l (26) p ossesses L ax r epr esentation ( 14 ) with op er ators P = uT m + l + T l , Q = T m + u, A = f ( T − m − T m ) , B = f l T − m − f m T m + u ( f m + l − f ) . 9 m = 2 : u ,t = u 2 ( u 2 u 1 − u − 1 u − 2 ) − u ( u 1 − u − 1 ) u = v 1 v v ,t = v 1 v 3 v − 1 ( v 2 v 1 − v − 1 v − 2 ) − v 2 ( v 1 − v − 1 ) m = 3 : u ,t = u 2 ( u 3 u 2 u 1 − u − 1 u − 2 u − 3 ) − u ( u 2 u 1 − u − 1 u − 2 ) v = u 1 u v ,t = v ( v 3 v 1 + v 2 v − v v − 2 − v − 1 v − 3 ) − v ( v 2 + v 1 − v − 1 − v − 2 ) u = v 2 v 1 v v ,t = v 2 v 2 1 v 4 v 2 − 1 v − 2 ( v 3 v 2 v 1 − v − 1 v − 2 v − 3 ) − v 1 v 3 v − 1 ( v 2 v 1 − v − 1 v − 2 ) m = 4 : u ,t = u 2 ( u 4 u 3 u 2 u 1 − u − 1 u − 2 u − 3 u − 4 ) − u ( u 3 u 2 u 1 − u − 1 u − 2 u − 3 ) u = v 2 v v ,t = v 2 v 1 v 3 v − 1 v − 2 ( v 4 v 3 v 2 v 1 − v − 1 v − 2 v − 3 v − 4 ) − v 1 v 2 v − 1 ( v 3 v 2 v 1 − v − 1 v − 2 v − 3 ) T able 1. Examples of lattices ( 25 ) from dSK (1 ,m ) and their modifications 3.2 Mo dified lattices Equations under consideration can b e rewritten in several w ays by use of difference substitutions. The simplest kind of substitution is introducing a p oten tial. Let A b e a constan t op erator, then substitution u = A ( v ) maps solutions of equation v ,t = f [ A ( v )] into solutions of equation u ,t = A ( f [ u ]). T able 1 contains several instances of such kind, up to the c hange u → e u , v → e v . Another kind of substitutions are Miura t yp e transformations. Let ϕ b e a particular solution of spectral problem ( 11 ) corresponding to a v alue λ = α of the sp ectral parameter. Then one readily finds that the ratio h = ϕ 1 /ϕ is related with the p oten tial u b y formula M − : u = αh m − 1 · · · h − h l − 1 · · · h h m + l − 1 · · · h − α . This defines a difference substitution, according to the follo wing statement. Theorem 3. L et u satisfies an e quation ( 22 ) fr om dSK ( l,m ) , then h = ϕ 1 /ϕ also satisfies a lattic e e quation which c an b e written as a c onservation law (log h ) ,t = ( T − 1) S [ h ] . (27) Pr o of. Since ϕ is gov erned by equation ϕ ,t = Aϕ = F ( ϕ m − ϕ − m ), hence (log h ) ,t = ( T − 1)(log ϕ ) ,t = ( T − 1)  1 ϕ F ( ϕ m − ϕ − m )  . 10 Co efficien ts of the operator F are functions on the v ariables h j , b eing func- tions on u j ’s. The ratios of the form ϕ k /ϕ can be expressed through h j as w ell and therefore an equation of the form ( 27 ) holds. It is worth noticing that an infinite sequence of conserv ation la ws for the original lattice ( 22 ) can b e obtained from ( 27 ) b y use of the classical trick with the in version of Miura map u = M − ( h, α ) as a formal pow er series with resp ect to α [ 23 ]. Second Miura map is obtained b y replacing h → 1 /h , α → 1 /α which results in the mapping M + : u = αh m + l − 1 · · · h l − h m + l − 1 · · · h m h m + l − 1 · · · h − α . This substitution relates the same equations as M − , due to inv ariance of the sp ectral problem with resp ect to the change n → − n , λ → 1 /λ . There- fore, the composition M − ( M + ) − 1 defines a B¨ ac klund transformation whic h relates tw o copies of the dSK ( l,m ) hierarc hy . Recall that B¨ ac klund transfor- mation for the con tinuous SK equation was deriv ed in [ 24 ]. A particular e xample at l = 2 , m = 1 is giv en b y substitutions M − : u = ( α − h 1 ) h h 2 h 1 h − α , M + : u = h 2 ( α − h 1 ) h 2 h 1 h − α whic h map solutions of the mo dified equation h ,t = h ( α − h ) h 1 hh − 1 − α  h ( α − h 1 )( α − h − 1 )( h 2 h 1 − h − 1 h − 2 ) ( h 2 h 1 h − α )( hh − 1 h − 2 − α ) − h 1 + h − 1  in to solutions of ( 1 ). 3.3 r -matrix form ulation In this section w e pro v e that: (i) if the constraint ( 23 ) is resolved b y formula ( 24 ) then the further recurren t relations ( 21 ) are solved in the lo cal form as well, so that the (lo cal) hierarc hy dSK ( l,m ) is correctly defined; (ii) the flo ws corresp onding to the differen t k comm ute. In ac hieving this goal the r -matrix approac h is an indisp ensable tool, see e.g. [ 19 , 20 , 18 ]. Let us consider the Lie algebra of the formal Laurent series with resp ect to the p o w ers T m of the shift op erator: g ( m ) = n X j < ∞ g ( j ) T j m o with the comm utator [ A, B ] = AB − B A . It is easy to see that an y element G = g ( k +1) T ( k +1) m + g ( k ) T km + g ( k − 1) T ( k − 1) m + . . . 11 of this Lie algebra admits an unique decomp osition of the form G = F ( T m − T − m ) + H (28) where F = F † is a self-conjugated difference op erator and H is a formal series whic h contains only nonp ositiv e p o wers of T m . Each of the linear spaces g ( m ) + =  F ( T m − T − m ) | F = F †  , g ( m ) − = n X j ≤ 0 h ( j ) T j m o constitutes a Lie algebra: for g ( m ) − this is ob vious and for g ( m ) + w e ha v e [ F ( T m − T − m ) , F 0 ( T m − T − m )] = ( P + P † )( T m − T − m ) where P = F ( T m − T − m ) F 0 . Th us, form ula ( 28 ) is the decomp osition (in the v ector space sense) g ( m ) = g ( m ) + ⊕ g ( m ) − of the Lie algebra in to the direct sum of tw o Lie subalgebras. This decom- p osition defines the pro jections π ± on the g ( m ) ± comp onen t and the r -matrix r = 1 2 ( π + − π − ). Now w e can form ulate the following theorem ab out Lax equations ( 13 ), ( 14 ) with fractional L op erator. Theorem 4. L et l , m b e c oprime, P = ( uT m + 1) T l , Q = T m + u and let L = Q − 1 P b e exp ande d as a formal L aur ent series. Then the flows L ,t p = [ π + ( L pm ) , L ] (29) ar e c orr e ctly define d for al l p = 1 , 2 , . . . , c oincide with the dSK ( l,m ) flows intr o duc e d by e quations ( 21 ), ( 22 ) and c ommute with e ach other. Pr o of. After expanding, L takes the form L = (1 − u − m T − m + ( u − m T − m ) 2 − . . . )( u − m + T − m ) T l = u − m T l + (1 − u − m u − 2 m ) T l − m + . . . . Differen tiating this series turns ( 29 ) in to an infinite system of equations for a single v ariable u , and the correctness means that all these equations m ust coincide. T o prov e this, we compare represen tation ( 29 ) with Lax equation ( 14 ) in fractional form. Notice that L itself do es not belong to the Lie algebra g ( m ) , but its pow er G = L pm do es, so that the pro jection A = π + ( G ) = F ( T m − T − m ) makes sense. W e denote the order of op erator F as k = pl − 1, in agreemen t with 12 ( 20 ) and ( 24 ). The co efficien ts of F are uniquely computed from co efficien ts of G accordingly to the recurrent relations f ( k +2) = f ( k +1) = 0 , f ( j ) = g ( j +1) + f ( j +2) , j = k , k − 1 , . . . , 0 so that all co efficien ts are lo cal functions of u j (in particular, f ( k ) is given b y ( 24 )). Moreov er, the order of ( 29 ) righ t hand side is equal to l , b ecause [ π + ( G ) , L ] = − [ π − ( G ) , L ]. This prov es that F pro vides a solution of the recurren t relations ( 21 ) as w ell (whic h is unique up to integration constants). Indeed, these relations w ere derived from the condition that terms with T ( k +1) m ,. . . , T m in equation ( 19 ) cancel whic h is equiv alen t to cancellation of the p o w ers T ( k +1) m + l ,. . . , T m + l in the original Lax equation ( 14 ). Th us, flo w ( 29 ) coincides with a flo w from dSK ( l,m ) whic h is, therefore, lo cal. On the other hand, this prov es correctness of ( 29 ), since the whole infinite set of equations turns out to b e equiv alen t to the single equation ( 22 ). The pro of of the commutativit y is standard. Let G 0 = L p 0 m and A 0 = π + ( G 0 ) then ( L ,t p ) ,t p 0 − ( L ,t p 0 ) ,t p = [ A t p 0 − A 0 t p + [ A, A 0 ] , L ] , so it is sufficien t to pro ve that A t p 0 − A 0 t p + [ A, A 0 ] = 0 . Since A t p 0 = π + ([ A 0 , G ]) and [ G, G 0 ] = 0, this is equiv alen t to π +  [ A 0 , G ] − [ A, G 0 ] + [ A, A 0 ]  = π +  [ G 0 − π − ( G 0 ) , G ] − [ G − π − ( G ) , G 0 ] + [ G − π − ( G ) , G 0 − π − ( G 0 )]  = π +  [ π − ( G ) , π − ( G 0 )]  = 0 as required. 3.4 Con tin uous limit Here we compute the contin uous limit for the basic flow of the extended hierarc hy dSK ( l,m ) defined by equation ( 26 ). There is a certain tec hnical difficult y in the prolongation of the contin uous limit on the v ariable f which is not lo cal at l 6 = 1. In order to solve the constrain t, this v ariable should b e considered as a series with resp ect to the small parameter. Up to this complication the con tin uous limit is very similar to example ( 4 ) from Intro- duction. W e p ostulate that, at ε → 0, the v ariables u, f are of the form u ( n, t ) = a + abε 2 U ( x + cεt, τ + dε 2 t ) , f ( n, t ) = 1 + ∞ X s =2 ε s Y s ( x + cεt, τ + dε 2 t ) , x = εn (30) 13 with undetermined co efficien ts a, b, c, d . F unctions Y s are expressed through the function U and its partial deriv atives with respect to x after substituting in to first equation ( 26 ) and taking the T a ylor expansion ab out ε = 0 (clearly , one can neglect the dependence on t here). W e find, omitting the unessen tial in tegration constan ts: Y 2 = mb l U, Y 3 = − m ( m + l ) b 2 l U 1 , Y 4 = m ( m + l )(2 m + l ) b 12 l U 2 + m ( m − l ) b 2 2 l 2 U 2 , Y 5 = − m 2 ( m + l ) 2 b 24 l U 3 − m ( m 2 − l 2 ) b 2 2 l 2 U U 1 , Y 6 = m ( m + l )(2 m + l )(3 m 2 + 3 ml − l 2 ) b 720 l U 4 + m ( m 2 − l 2 )(3 m + 2 l ) b 2 24 l 2 U 2 1 + m ( m 2 − l 2 )(2 m + l ) b 2 12 l 2 U U 2 + m ( m − l )( m − 2 l ) b 3 6 l 3 U 3 . This is enough, since we need only terms up to ε 7 when substituting into second equation ( 26 ). The coefficients a, c are found from the requirement that the lo w order terms v anish while the co efficien ts b, d are resp onsible for the scaling of U and t and can b e chosen arbitrarily . Finally , we come to the following statemen t. Theorem 5. Continuous limit ( 30 ) with the values of p ar ameters a = m − l m + l , b = ml 6 , c = 2 m, d = m 3 ( l 2 − m 2 ) 180 sends systems ( 26 ) into the Sawada–Koter a e quation U ,τ = U 5 + 5 U U 3 + 5 U 1 U 2 + 5 U 2 U 1 . The higher flows of the SK hierarch y can b e deriv ed analogously from suitable linear combinations of the dSK ( l,m ) flo ws. Ho wev er, the general form ulas b ecome rather complicated and w e restrict ourselves by the fol- lo wing concrete example corresponding to the lo cal hierarch y dSK (1 , 2) . Let u ,t = 88 u ,t 1 + 27 u ,t 2 where the flo ws ∂ t 1 and ∂ t 2 are defined b y equations ( 1 ) and ( 7 ) respectively , then the formula u ( n, t ) = 1 3 + ε 2 9 U  x − 200 9 εt, τ − 16 ε 7 189 t  , x = εn defines the con tin uous limit to the 7-th order symmetry of SK equation U ,τ = U 7 + 7 U U 5 + 14 U 1 U 4 + 21 U 2 U 3 + 14 U 2 U 3 + 42 U U 1 U 2 + 7 U 3 1 + 28 3 U 3 U 1 . 14 It is well known that there are gaps in the sequence of orders k of equations from the SK hierarch y , namely , the restrictions k - 2 , 3 are fulfilled, so that the next higher symmetry is of 11-th order. The natural question appears, ho w this agrees with relations ( 20 )–( 22 ) or ( 29 ) which sho w that in the discrete case there are no gaps m ultiple 3. It turns out that their app earance is an artefact of the con tinuous limit. A straigh tforw ard computation sho ws that if w e consider a linear com bination with the next dSK (1 , 2) flo w u ,t = u ,t 1 + αu ,t 2 + β u ,t 3 and set u ( n, t ) = a + bε 2 U ( x + cεt, τ + dε 9 t ) , x = εn then all paramete rs are uniquely determined b y the condition of v anishing the terms up to ε 10 , ho wev er then the coefficients at ε 11 cancel automatically and only the trivial flow U ,τ = 0 app ears. 3.5 Bilinear equations The constrain t ( 23 ) can b e solved b y introducing additional v ariables and this leads to a con venien t representation of the basic system ( 26 ) of the extended dSK ( l,m ) hierarc hy . Let u = v l v , f = v v − m then first equation ( 26 ) is s atisfied identically and the second one is equiv a- len t to ( T l − 1) v ,t v = ( T m − 1)  v v l − m − v l v − m  . F urther substitutions v = w m w ⇒ u = w m + l w w m w l , f = w m w − m w 2 bring to the bilinear equation w l,t w − w l w ,t = w m w l − m − w − m w l + m . (31) F or the first time, it app eared in paper [ 7 ], in a sligh tly more general form w l,t w − w l w ,t = w m w l − m − αw − m w l + m + β w w l whic h is reduced to ( 31 ) by the p oin t change ˜ w ( n, t ) = e β t α n 2 w ( n, t ). In particular, it w as pro ven in [ 7 ] that this equation admits N -soliton solutions. Here, we consider in more details a sp ecification of 2-soliton formula which leads to the breather solution. The substitution of the 2-soliton Ansatz w ( n, t ) = 1 + e 1 + e 2 + A 12 e 1 e 2 , e i = q n i exp( − ω i t + δ i ) (32) 15 - 1 .5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 l = 1, m = 2 - 1.5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 - 1 .5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 l = 1, m = 3 - 1.5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 - 1 .5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 l = 1, m = 4 - 1.5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 - 1 .5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 l = 1, m = 5 - 1.5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 - 1 .5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 l = 1, m = 6 - 1.5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 - 1 .5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 l = 1, m = 7 - 1.5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 - 1 .5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 l = 2, m = 3 - 1.5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 - 1 .5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 l = 2, m = 5 - 1.5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 - 1 .5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 l = 2, m = 7 - 1.5 - 1 - 0.5 0.5 1 - 1.5 - 1 - 0.5 0.5 1 1.5 Figure 1. The v alues of q = ρe i ϕ inside the b ounded domains in C corresp ond to the regular p oten tials u ( n, t ). The v alues along the dashed lines correspond to the p oten tials perio dic in t . in to ( 31 ) giv es us the disp ersion relation and the phase shift: ω i = q m i − q − m i , A ij = ( q l i − q l j )( q m i − q m j ) (1 − q l i q l j )(1 − q m i q m j ) . (33) The direct c hec k pro v es that then the 3-soliton Ansatz w = 1 + e 1 + e 2 + e 3 + A 12 e 1 e 2 + A 13 e 1 e 3 + A 23 e 2 e 3 + A 12 A 13 A 23 e 1 e 2 e 3 satisfies ( 31 ) automatically . It is in teresting to compare these formulas with their counterparts for the contin uous SK equation [ 8 , 9 , 25 , 26 ] e i = exp( κ i x − ω i t + δ i ) , ω i = κ 5 i , A ij = ( κ i − κ j ) 2 ( κ 2 i − κ i κ j + κ 2 j ) ( κ i + κ j ) 2 ( κ 2 i + κ i κ j + κ 2 j ) . 16 - 5 0 5 n - 20 - 10 0 10 t 0.5 1 1.5 2 2.5 u n H t L - 5 0 n - 5 0 5 n - 5 - 2.5 0 2.5 5 t 0 2.5 5 7.5 10 u n H t L - 5 0 n Figure 2. A moving and a stable breathers. The v alues of parameters: ρ = 1 . 2, ϕ = 2 π/ 3 (left); ρ = 1 . 6, ϕ = 3 π / 4 (righ t); in both cases l = 1, m = 2, α = β = 0. F orm ula ( 32 ) allows us to obtain the breather-t yp e solutions as well, if w e c ho ose q 1 = ρe i ϕ , q 2 = ρe − i ϕ , δ 1 = α + i β , δ 2 = α − i β . The regularity of the potential u ( n, t ) is ac hieved under certain restrictions on the v alue of q . In order to show this, rewrite relations ( 33 ) as follo ws: ω = µ + i ν , µ = ( ρ m − ρ − m ) cos mϕ, ν = ( ρ m + ρ − m ) sin mϕ, A 12 = − 4 ρ m + l sin lϕ sin mϕ (1 − ρ 2 l )(1 − ρ 2 m ) , then a simple algebra brings ( 32 ) to the form w = 1 + 2 z cos y + A 12 z 2 , y = ϕn − ν t + β , z = ρ n e α − µt . In particular, if ϕ = 2 k +1 2 m π then µ = 0 and solution u is p eriodic in t . The necessary and sufficien t condition for u to b e regular is that the function w do es not v anish at an y n, t . In the generic case the v ariables y , z are indep enden t and then this is equiv alen t to the condition that the trinomial 1 + 2 z + A 12 z 2 do es not v anish at real z , that is ( ρ l − ρ − l )( ρ m − ρ − m ) + 4 sin lϕ sin mϕ < 0 . Th us, w e see that already tw o-phase solutions in these mo dels exhibit a non trivial zone structure of the sp ectrum. The corresp onding domains in the plane q = ρe i ϕ are shown on fig. 1 , and the examples of solutions u ( n, t ) are shown on fig. 2 . 17 4 A discrete analog of the Kaup–Kup ershmidt equa- tion The Kaup–Kup ershmidt equation U ,τ = U 5 + 5 U U 3 + 25 2 U 1 U 2 + 5 U 2 U 1 is asso ciated with the sp ectral problem Lψ = λψ where L is the sk ew- symmetric ordinary differen tial op erator of third order L = D 3 + U D + 1 2 U ,x = ( D − f ) D ( D + f ) , U = 2 f ,x − f 2 . When we find a discrete analog, a difficult y is that a symmetric or sk ew- symmetric difference op erator can b e of ev en order only . A w a y to ov ercome this is to consider a 6th order difference problem, but on the o dd no des of the lattice only , so that effectiv ely it is of 3rd order with resp ect to the double shift T 2 (ho wev er, the co efficien ts ma y dep end on the v ariables asso ciated with the ev en no des as well). Let us consider the spe ctral problem u − 3 ψ − 3 + ψ − 1 = λ ( ψ 1 + uψ 3 ) (34) or, in the op erator form, denoting K = uT 3 + T : K † ψ = λK ψ . The Lax equation for the operator L = K − 1 K † can be written in the form of system ( 15 ). It admits the reduction B = − A † whic h yields the equation K ,t + A † K + K A = 0 . (35) The op erator A is found as a Laurent polynomial with resp ect to the ev en p o w ers of T , A = a ( k ) T 2 k + · · · + a ( − k ) T − 2 k , and a direct analysis of equation ( 35 ) at k = 1 , 2 prov es the follo wing state- men t. Theorem 6. Equation ( 35 ) with K = uT 3 + T is e quivalent to the nonlo c al lattic e e quation u ,t 1 = u ( f 2 u 2 − f 1 u 1 + f − 1 u − 1 − f − 2 u − 2 ) + f 1 − f − 1 , f 3 u = f − 1 u 2 under the choic e A = − f T 2 + f − 2 u − 2 − f − 1 u − 1 + f − 3 T − 2 ; and it is e quivalent to the lo c al lattic e e quation u ,t 2 = u ( v 3 − v 2 + v 1 − v − 1 + v − 2 − v − 3 − u 2 + u − 2 ) , v := u 1 uu − 1 (36) 18 under the choic e A = u 1 T 4 − u − 4 T − 4 + (1 − u − 1 u − 2 )( T 2 − T − 2 ) + u − 1 − u − 2 − v + v − 1 − v − 2 + v − 3 . It is w orth noticing that, alternativ ely , one can use the following pair of op erators (cf with the gauge equiv alence ( 12 )): ˜ K = uT 3 + T − 1 , ˜ A = − u 1 u − 1 T 4 + u − 2 u − 4 T − 4 − v + v − 1 − v − 2 + v − 3 . (37) The con tinuous limit to the KK equation is of the same general form as b efore, namely , for the flo w ( 36 ) it reads u ( n, t 2 ) = 1 3 + 4 9 ε 2 U  x − 8 9 εt 2 , τ + 64 ε 5 135 t 2  , x = εn. 5 Examples related to generic op erators Recall that, according to [ 20 ], the Bogo y av lensky t yp e lattices can b e view ed as reductions of more general m ulti-field mo dels asso ciated with the sp ec- tral problems Lψ = λψ for generic difference op erators L = u ( m ) T m + u ( m − 1) T m − 1 + · · · + u (1 − l ) T 1 − l + u ( − l ) T − l . Here m, l are any p ositiv e in- tegers, and one can adopt the normalization u ( m ) = 1 or u ( l ) = 1 without loss of generalit y . A part of the flows from the corresp onding hierarc hy is consisten t with the constraints u ( m − 1) = · · · = u (1 − l ) = 0 and this reduc- tion brings to the Bogo ya vlensky lattices. A detailed study of some other reductions can be found in [ 22 ]. The lattices introduced in the previous sections are related with the sp ectral problems P ψ = λQψ where operators P , Q are binomial. It is natural to expect that these lattices also define reductions for some m ulti- field equations related with more general op erators P, Q . The study of suc h mo dels is b eyond the scop e of the presen t pap er and w e restrict ourselves b y three t ypical examples. Example 1 . First, let us consider the Lax equations P ,t = B P − P A , Q ,t = B Q − QA for the binomial op erators P , Q with differen t p oten tials: P = uT 3 + T , Q = T 2 + v . If v = u then op erators A, B are given b y formulas ( 18 ), ( 20 ) with a self- adjoin t op erator F whic h con tains only ev en p o w ers of T 2 . In the general case tw o sets of op erators A, B app ear, containing p ositiv e or negativ e p ow- 19 ers of T 2 . The simplest operators and corresponding flows are the following: A − = v − 2 v − 1 T − 2 + f − 3 + f − 2 , B − = v − 1 v T − 2 + f − 1 + f , u ,t − = u ( f − 1 − f 1 ) , v ,t − = v ( f + f − 1 − f − 2 − f − 3 − v 1 + v − 1 ) , f := uv 1 v 2 ; A + = u − 2 u − 1 T 2 + g − 1 + g , B + = uu 1 T 2 + g + g 1 , u ,t + = u ( g + g 1 − g 2 − g 3 − u − 1 + u 1 ) , v ,t + = v ( g 1 − g − 1 ) , g := u − 2 u − 1 v . The flows ∂ t − and ∂ t + comm ute, and the flow ∂ ,t = ∂ t − − ∂ t + admits the reduction v = u which brings to the dSK equation ( 1 ). It should b e noted that the same flows can be obtained starting from the gauge equiv alent op erators P = uT 3 + T 2 , Q = T + v . Example 2 . Now let us consider trinomial op erators P = uT 3 + pT 2 + T , Q = T 2 + q T + v . In this case op erators A, B con tain the odd p o w ers of T as well. The simplest op erators and the corresp onding flows are of the form A − = v − 1 T − 1 + v − 1 p − 2 , B − = v T − 1 + v 1 p, u ,x − = u ( u − 1 q − u 1 q 2 − p + p 1 ) , p ,x − = p ( u − 1 q − uq 1 ) + u − u − 1 , v ,x − = v ( u − 1 q − u − 2 q − 1 ) , q ,x − = uv 1 − u − 2 v ; A + = u − 2 T + u − 2 q − 1 , B + = uT + u − 1 q , u ,x + = u ( v 1 p − v 2 p 1 ) , p ,x + = u − 1 v − uv 2 , v ,x + = v ( v 1 p − v − 1 p − 2 + q − 1 − q ) , q ,x + = q ( v 1 p − v p − 1 ) + v − v 1 . Example 3 . Let us consider the follo wing generalization of the sp ectral prob- lem ( 37 ): K † ψ = λK ψ , K = uT 3 + v − 1 T + T − 1 . The isosp ectral deformations are defined b y the op erators A = a ( k ) T 2 k + a ( k − 1) T 2 k − 2 + · · · + a ( − k ) T − 2 k . The simplest case k = 1 results in A = u − 1 T 2 − u − 2 T − 2 + u − 1 v − 1 − u − 2 v − 2 20 and equation K ,t + A † K + K A = 0 is equiv alen t to the lattice u ,t = − u ( u 2 v 2 − u 1 v 1 + u − 1 v − 1 − u − 2 v − 2 − v 1 + v − 1 ) , v ,t = − v ( u 1 v 1 − u − 1 v − 1 ) + u 2 u 1 − u − 1 u − 2 + u 1 − u − 1 . The higher symmetry corresp onding to k = 2 is to o bulky and w e do not write it do wn, how ev er one can chec k that it admits the reduction v = 0 to the dKK equation ( 36 ). In contrast, the flow ∂ t itself does not admit this reduction. 6 Conclusion In this article w e in tro duced a family of integrable lattice hierarchies asso ci- ated with fractional Lax op erators. In particular, these hierarc hies con tain equations found earlier in [ 1 , 7 ] by use of the Hirota bilinear formalism. W e prov ed that these equations serv e as semi-discrete analogs of SK and KK equations. An imp ortan t question which remains op en is ab out the Hamiltonian structure of the presen ted equations. As usually , the existence of Lax represen tation allows to obtain a set of conserved quantities which presumably are Hamiltonians, and moreo ver, the applicability of r -matrix approac h suggests that some more or less standard P oisson brack et should exist. How ev er, no explicit answ er is found yet. Another in triguing question is about possible relations with the models in tro duced in [ 27 , 28 ] within the theory of the lattice W algebras. Ac kno wledgemen ts W e are grateful to Y a.P . Pugay , Y u.B. Suris and A.K. Svinin for man y stim ulating discussions. The researc h of V.A. was supp orted by grant NSh– 6501.2010.2. References [1] S. Tsujimoto, R. Hirota. Pfaffian representation of solutions to the dis- crete BKP hierarc hy in bilinear form. J. Phys. So c. Jap an 65 (1996) 2797–2806 . [2] V.E. Zakharo v, S.L. Musher, A.M. Rub enc hik. 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