Lie point symmetries of differential--difference equations

Lie p oin t symmetries of differen tial–difference equations D. Levi Dipartimen to di Ingegneria Elettronica, Univ ersit` a degli Studi Roma T re and Sezione INFN, Roma T re, Via della V asca Na v ale 84, 00146 Roma, Italy E-mail: levi@roma3.infn.it P . Win ternitz Cen tre de rec herc hes math ´ ematiques et D ´ epartemen t de math ´ ematiques et statistique, Univ ersit ´ e de Mon tr´ eal C.P . 6128, succ. Cen tre–ville, H3C 3J7, Mon tr ´ eal (Qu´ ebec), Canada E-mail: win tern@crm.umon treal.ca R.I. Y amilo v Ufa Institute of Mathematics, Russian Academ y of Sciences, 112 Chern yshevsky Street, Ufa 450008, Russian F ederation E-mail: RvlY amilo v@matem.anrb.ru No v em b er 26, 2024 Abstract W e presen t an algorithm for determining the Lie p oin t symmetries of dif- feren tial equations on fixed non transforming lattices, i.e. equations in v olving b oth contin uous and discrete independent v ariables. The symmetries of a sp ecific integrable discretization of the Krichev er-No viko v equation, the T o da lattice and T o da field theory are presen ted as examples of the general method. 1 In tro duction Tw o differen t but equiv alent infinitesimal formalisms exist for calculating Lie p oin t symmetries of differential equations [18]. One is that of ‘standard‘ vector fields ˆ X = p X i =1 ξ i ( ~ x, ~ u ) ∂ x i + q X α =1 φ α ( ~ x, ~ u ) ∂ u α (1) 1 acting on the indep enden t v ariables x i and dep enden t ones u α in the considered differen tial equation. The other is that of the evolutionary vector fields ˆ X E = q X α =1 Q α ( ~ x, ~ u, ~ u ~ x ) ∂ u α , (2) acting only on the dep endent v ariables. The equiv alence of the t w o formalisms is due to the fact that the total deriv atives D x i are themselv es ‘generalized‘ symmetry operators, so for any differential equation E ( x i , u α , u α,x i , · · · ) = 0 (3) w e hav e pr ˆ X E E | E =0 = (pr ˆ X − p X i =1 ξ i D x i ) E | E =0 = 0 . (4) Here pr ˆ X E and pr ˆ X are the appropriate differential prolongations of ˆ X E and ˆ X . Relation (4) implies that for p oint transformations we ha ve Q α = φ α − p X i =1 ξ i u α,x i . (5) F or all details we refer to e.g. P . Olver‘s textb o ok [18]. An adv antage of the standard formalism is its direct relation to the group trans- formations obtained by in tegrating the equations d ˜ x i dλ = ξ i ( ˜ ~ x, ˜ ~ u ) , d ˜ u α dλ = φ α ( ˜ ~ x, ˜ ~ u ) , (6) ˜ x i | λ =0 = x i , ˜ u α | λ =0 = u α , i = 1 , · · · , p, α = 1 , · · · , q . One adv antage of the evolutionary formalism is its direct relation to the existence of flows comm uting with the studied equation (3) d ˜ u α dλ = Q α , (7) where Q α is the characteristic of the vector field as in (5). Another adv antage is that the ev olutionary formalism can easily b e adapted to the case of higher symmetries. Let us no w consider a purely discrete equation, i.e. a difference equation. W e restrict to the case of one scalar function defined on a t w o dimensional lattice u mn . W e shall view u as a contin uous v ariable, introduce tw o further contin uous v ariables x and t and consider ( x, t, u ) as b eing ev aluated, or sampled at discrete p oints on 2 a lattice lab elled by the indices m , n . W e shall write ( x mn , t mn , u mn ) for v alues at these p oints. A difference system will consist of relations E a ( x m + k,n + l , t m + k,n + l , u m + k,n + l ) = 0 , a = 1 , · · · , A k m ≤ k ≤ k M , l m ≤ l ≤ l M , (8) b et ween the v ariables x , t , and u ev aluated at a finite num b er of p oin ts on a lattice. A Lie p oint symmetry of the system (8) will b e generated by a v ector field of the form ˆ X mn = ξ ( x mn , t mn , u mn ) ∂ x mn + τ ( x mn , t mn , u mn ) ∂ t mn + φ ( x mn , t mn , u mn ) ∂ u mn . (9) W e see that the v ector field (9) for difference equations has the same form as (1) for differential ones. Its prolongation is how ever differen t, namely pr ˆ X = X k,l ˆ X m + k,n + l , (10) where the sum is ov er all p oin ts figuring in the system (8). In the contin uous limit the system (8) go es into a partial differential equation, eq. (10) go es in to the usual prolongation of a standard vector field (i.e. it also acts on functions of deriv atives). F or recent reviews of the theory of con tinuous symmetries of difference equations see Ref. [2, 11, 22]. The purp ose of this article is to consider an in termediate case, that of differential– difference equations. In Section 2 we shall tak e a ‘semicon tinuous‘ limit, i.e. leav e the v ariable x discrete but let t tend to a contin uous v ariable. This will provide us with b oth a standard and an evolutionary formalism for calculating p oint symmetries of differen tial–difference equations. In Section 3 we consider sev eral sp ecial cases and pro v e some theorems that greatly simplify the calculation of symmetries. Section 4 is devoted to examples and Section 5 to a summary of the results obtained. 2 Lie p oin t symmetries of difference systems and their semicon tin uous limit 2.1 The semicon tin uous limit A difference system is defined on a discr ete jet sp ac e , a space of indep enden t and dep enden t v ariables on a lattice. In this article we restrict to the case of tw o inde- p enden t v ariables x , t and one dep endent one u defined on a t w o dimensional lattice with p oints lab elled b y tw o indices. W e shall write { x mn , t mn , u mn ≡ u ( x mn , t mn ) } , (11) (see Fig. 1). 3 - 6 t u m + 1 , n 1 u m , n − 1 u m − 1 , n − 1 u m + 1 , n u m , n u m − 1 , n u m + 1 , n + 1 u m , n + 1 u m − 1 , n + 1 n − 1 n n + 1 m − 1 m m + 1 - x e v ie w R 7 Figure 1: Example of a tw o-dimensional lattice. The discr ete jet sp ac e will b e the set of all v ariables { x j k , t j k , u j k } on the lattice. The dep endence of x , t and u on the lab els m , n is determined from the difference system E a mn ( x j k , t j k , u j k ) = 0 , a = 1 , · · · , N , (12) and some b oundary conditions. In eq. (12) N is an integer satisfying N ≥ 3 and ( j, k ) run o v er some finite set of v alues on the lattice while ( m, n ) is a fixed reference p oin t. Eq. (12) th us determines b oth the difference equation and the lattice. Lie p oint symmetries of the system (12) are generated b y vector fields of the form ˆ X D mn = ξ ( x mn , t mn , u mn ) ∂ x mn + τ ( x mn , t mn , u mn ) ∂ t mn + φ ( x mn , t mn , u mn ) ∂ u mn (13) (the sup erscript D stands for ‘discrete‘) satisfying pr ˆ X D E a | E 1 = E 2 = ··· = E N =0 = 0 . (14) In eq. (14) pr ˆ X D is the prolongation of the v ector field ˆ X D mn to the discrete jet space pr ˆ X D = X j,k ˆ X D j k , (15) where the sum is ov er all p oin ts figuring in the difference system (12). In this approach the lattice ( x mn , t mn ) is in general determined together with u mn from the system (12) and the group transformations generated by the vector field ˆ X D mn also transform the lattice. A sp ecial case corresponds to an a priori determined non transforming lattice. In that case tw o of the equations in the system (12) hav e the form x mn = f ( m, n ) , t mn = g ( m, n ) , (16) 4 where f and g are given. Such is the case of a uniform orthogonal lattice where (16) tak es the form x mn = σ 1 n + x 0 , t mn = σ 2 m + t 0 , (17) and the scale factors ( σ 1 , σ 2 ) and the origin ( x 0 , t 0 ) are given num b ers (e.g. σ 1 = σ 2 = 1 , x 0 = t 0 = 0). W e are interested in obtaining the form of the vector field in the semicontin uous limit in whic h t mn b ecomes a contin uous v ariable t , but x mn remains discrete ( and indep enden t of t ). Th us x will dep end on one discrete lab el n only and in particular x ma y b e given as x n = f ( n ), with f ( n ) kno wn (e.g. x n = hn for a uniform lattice, or x n = λ n for an exp onen tial one). In this limit the difference system (12) will reduce to a differential–difference equation (D∆E) E n ( t, u j , ˙ u j , ¨ u j , · · · , u ( K ) j ) = 0 , n − L ≤ j ≤ n + M , (18) where dots denote t –deriv ativ es, and K , L and M are some nonnegativ e integers. T ogether with eq. (18) we ha ve another equation which determines x n = f ( n ). W e shall consider the case when f ( n ) is already giv en (a kno wn function) so that w e can replace the dep endence on x n b y a dep endence on the integer n (without necessarily assuming that f ( n ) is linear). Eq. (18) is th us defined on a ‘semidiscrete jet space‘ with lo cal co ordinates { t, u j , ˙ u j , ¨ u j , · · · , u ( K ) j } , (19) where j runs ov er all v alues on a one–dimensional lattice. The vector field generating symmetries of eq. (18) will ha v e the form ˆ X S D = τ n ( t, u n ) ∂ t + φ n ( t, u n ) ∂ u n , (20) ( S D = semidiscrete) and its prolongation will be defined on the semidiscrete jet space (19). Let us consider the simplest non trivial case, namely that of a difference system (12) in volving the three p oin ts ( m, n ), ( m + 1 , n ) and ( m, n + 1), i.e. relating the v ariables x j k , t j k and u j k in these 3 p oin ts: E n ( x mn , x m +1 ,n , x m,n +1 , t mn , t m +1 ,n , t m,n +1 , u mn , u m +1 ,n , u m,n +1 ) = 0 . (21) Before taking the limit we c hange notation and transform to new v ariables. W e c ho ose a reference p oin t ( m, n ) on the lattice (see Fig.1) and measure t and x from this p oint: t m + a,n + b = t +  ab , x m + a,n + b = f ( n + b ) + θ (  ab ) , (22) 5 where f ( n + b ) is a given function and θ (  ab ) is an analytic function. Instead of u m + j,n + k w e introduce a function v n ( t ) v n + b ( t +  ab ) ≡ u m + a,n + b ( x m + a,n + b , t m + a,n + b ) , (23) and assume that the dep endence on t is analytical. Th us the dep endence on x (whic h remains discrete in the limit  ab → 0) is replaced by a dep endence on the lab el n + b . F or the reference p oin t x mn , t mn w e put  00 = 0, θ (  00 ) = 0. So, in the case of the stencil ( m, n ), ( m + 1 , n ) and ( m, n + 1) w e hav e t mn ≡ t, t m +1 ,n ≡ t +  10 , t m,n +1 ≡ t +  01 , x mn ≡ f ( n ) , x m +1 ,n = f ( n ) + θ (  10 ) , x m,n +1 ≡ f ( n + 1) + θ (  01 ) , u mn ( x mn , t mn ) = v n ( t ) , u m +1 ,n ≡ v n ( t ) +  10 v n,t ( t ) , (24) u m,n +1 ( x m,n +1 , t m,n +1 ) = v n +1 ( t +  01 ) = v n +1 ( t ) + ∞ X j =1  j 01 j ! v ( j ) n +1 ( t ) , where v ( j ) n +1 ( t ) = d j v n +1 ( t ) dt j and v n,t is the ‘discrete deriv ative‘ of v n ( t ) giv en by v n ( t ) = u m +1 ,n − u m,n  10 . Since v n ( t ) is by assumption analytical, the T aylor series in (24) are con v ergent. Using eq. (24) we can also express the deriv atives { ∂ t mn , ∂ t m +1 ,n , ∂ t m,n +1 , ∂ u mn , ∂ u m +1 ,n , ∂ u m,n +1 } in terms of { ∂ t , ∂  10 , ∂  01 , ∂ v n , ∂ v n,t , ∂ v n +1 } and th us transform the prolongation of the v ector field (15). W e obtain pr ˆ X D = τ mn ∂ t + φ mn ∂ v n + ( τ m +1 ,n − τ mn ) ∂  10 + ( τ m,n +1 − τ mn ) ∂  01 (25) + h ( φ m +1 ,n − φ mn ) + ( τ mn − τ m +1 ,n ) v n,t i 1  10 ∂ v n,t + h φ m,n +1 + ( τ mn − τ m,n +1 ) ∞ X j =1 ( t m,n +1 − t mn ) j − 1 ( j − 1)! v ( j ) n +1 i ∂ v n +1 . F urther, w e put τ mn ≡ τ n ( t, v n ) , φ mn ≡ φ n ( t, v n ) , (26) and expand τ m +1 ,n and φ m +1 ,n ab out  10 = 0, τ m,n +1 and τ m,n +1 and φ m,n +1 ab out  01 = 0 and then let pr ˆ X D act on functions E n = E n ( t, v n , v n +1 , v n,t ) , (27) obtained as the limit of eq. (21). In the semicon tinuous limit w e tak e  10 → 0,  01 → 0 and we obtain lim (  10 , 01 ) → (0 , 0) pr ˆ X D = pr ˆ X S D = τ n ∂ t + φ n ∂ v n + φ [ t ] n ∂ v n,t + φ [ n +1] n ∂ v (1) n +1 , (28) φ [ t ] n = D t φ n − ( D t τ n ) ˙ v n , (29) φ [ n +1] n = φ n +1 + ( τ n − τ n +1 ) v (1) n +1 . (30) 6 The form (29) of the co efficient φ [ t ] n is the ”obvious” generalization of the first prolongation for ordinary differential equations. The presence of the second term in (30) is less obvious and follo ws from the abov e analysis of the semicontin uous limit. W e see that the prolongation of the vector field ˆ X S D to deriv atives is the same as for differen tial equations The prolongation to other p oints x n on the lattice do es ho w ever not consist of merely shifting n in φ n . W e men tion that the additional term in φ [ n +1] n w as missed in the article [9]. If we start from the set of all 9 p oints on the stencil of Fig.1 and take the semicon tin uous limit in the same w ay , w e arrive at a more general D∆E, namely E n ( t, u n , u n +1 , u n − 1 , ˙ u n , ˙ u n +1 , ˙ u n − 1 , ¨ u n , ¨ u n +1 , ¨ u n − 1 ) = 0 , (31) (with the change of notation to u n ). W e also obtain the prolongation pr ˆ X S D for suc h an equation (see b elow). 2.2 The Ev olutionary F ormalism and Comm uting Flows for Differen tial–Difference Equations An alternativ e metho d of calculating symmetries of D∆E on a fixed lattice is to construct commuting flo ws in t wo v ariables. Let us again consider eq. (27), this time solv ed for the first deriv ative, and c hange the notation from ˙ v n to u n,t , which no w denotes an ordinary time deriv ative; ˙ u n ≡ u n,t = F n ( t, u n , u n +1 ) . (32) W e introduce an additional v ariable λ , the group parameter and consider the flo w on u n ( t, λ ) in this v ariable u n,λ = Q n ( t, u n , ˙ u n ) . (33) Let us no w require that the flo ws (32) and (33) b e compatible, i.e. comm ute. Th us we imp ose u n,tλ = u n,λt . (34) W e replace u n,λ using eq. (33), ˙ u n and ¨ u n using (32) and its differen tial consequences and obtain Q n,t + Q n,u n F n + Q n, ˙ u n  F n,t + F n,u n F n + F n,u n +1 F n +1  = (35) F n,u n Q n + F n,u n +1 Q n +1 . This deriv ation of (35)is completely equiv alent to the following pro cedure. W e first introduce an ev olutionary vector field ˆ X E = Q n ( t, u n , ˙ u n , · · · ) ∂ u n , (36) 7 and its prolongation pr ˆ X E = Q n ∂ u n + Q n +1 ∂ u n +1 + ( D t Q n ) ∂ ˙ u n + · · · . (37) W e then apply this prolonged field to eq. (32), require pr ˆ X E  ˙ u n − F n ( t, u n , u n +1 )     ( ˙ u n = F n , ¨ u n = D t F n ) = 0 , (38) and reobtain eq. (35). Let us no w sp ecialize to the case of p oint symmetries. The quantit y Q n in (33) and (36) is the char acteristic of the v ector field ˆ X E . F or p oin t symmetries it has the form Q n ( t, u n , ˙ u n , · · · ) = φ n ( t, u n ) − τ n ( t, u n ) ˙ u n . (39) The total deriv ative D t is itself a (generalized) symmetry of the D∆E (18) and in particular (32). This provides us with a relation b etw een ordinary and ev olutionary v ector fields and their prolongations, namely pr ˆ X = pr ˆ X E + τ n ( t, u n ) D t . (40) Putting (39) and (37) into (40) w e reobtain eqs. (28 – 30). W e see that the ”obvious” prolongation (37)of the evolutionary vector field (36) pro vides, via eq. (40) the correct prolongation (28) of the ordinary vector field (20). 2.3 General Algorithm for Calculating Lie P oin t Symme- tries of a Differen tial Difference Equation Let us consider a D∆E in v olving L + M + 1 p oin ts and t deriv ativ es up to order K as in eq. (18). The Lie p oint symmetries of eq. (18) can b e obtained using the ev olutionary formalism b y imp osing pr ˆ X E E n   E n =0 , D k t E n =0 = 0 , k = 1 , · · · , K. (41) Th us the expression pr ˆ X E E n is anihilated on the solution set of the equation (18) and of all differen tial consequences of the equation. The v ector field ˆ X E has the form (36) with Q n as in (39). The prolongation of ˆ X E is pr ˆ X E = X j Q j ∂ u j + K X k =1 X j ‘ ( D k t Q j ) ∂ u ( k ) j , (42) where the j summation is o ver all p oin ts figuring in eq. (18) and u ( k ) j denotes the k –th t –deriv ative of u j . 8 The standard vector field ˆ X generating Lie p oin t symmetries and its prolongation are given b y the formula (40). Explicitly the prolongation formula is pr ˆ X = φ n ∂ u n + τ n ∂ t + X j 6 = n φ j ∂ u j (43) + X j K X k =1 φ [ k ] j ∂ u [ k ] j + X j K X k =1 ( τ n − τ j )( D k +1 t u j ) ∂ u [ k ] j , φ [ k ] j = D t φ [ k − 1] j − ( D t τ j ) u [ k ] j , D k t u j ≡ u [ k ] j . (44) Notice that φ [ k ] j is the same as for a differen tial equation [18] but the last term in (43) has no analog in the con tin uous case. The co efficien ts φ n and τ n in the vector field ˆ X itself are a priori functions of n , t and u n (see eq. (20)). In the following section we will examine some cases when τ n ( t, u n ) simplifies. Eq. (43) is also obtained as the semicon tin uous limit of the discrete prolongation (15) 3 Theorems Simplifying the Calculation of Sym- metries of D ∆ E. 3.1 General commen ts Lie p oint symmetries of D∆E of the form (18) are generated b y vector fields of the form (20). W e shall no w in v estigate 3 imp ortant cases when the coefficient τ n ( t, u n ) actually dep ends on t alone. The 3 classes of D∆E are ˙ u n = f n ( t, u n − 1 , u n , u n +1 ) , (45) ¨ u n = f n ( t, ˙ u n , u n − 1 , u n , u n +1 ) , (46) u n,xy = f n ( x, y , u n,x , u n,y , u n − 1 , u n , u n +1 ) . (47) Eq. (45) contains integrable V olterra, mo dified V olterra and discrete Burgers type equations [23]. A list of in tegrable T o da t yp e equations of the form (46) can b e found in the reference [24]. The class (47) in volv es 2 contin uous v ariables and contains the tw o dimensional T o da mo del [4, 14]. A list of integrable cases exists [19] and Lie p oin t symmetries of this class ha v e b een studied. F or equations (45) and (46) Lie point symmetries correspond to comm uting flo ws of the form (33) with Q n giv en by eq. (39) while for eq. (47) the form is u n,λ = ψ n ( x, y , u n , u n,x , u n,y ) , (48a) ψ n ( x, y , u n , u n,x , u n,y ) = φ n ( x, y , u n ) − ξ n ( x, y , u n ) u n,x − η n ( x, y , u n ) u n,y . (48b) 9 F or all equations (45, 46, 47), we assume ev erywhere b elo w that at least one of the following t wo conditions is satisfied: ∂ f n ∂ u n +1 6 = 0 , for all n, or ∂ f n ∂ u n − 1 6 = 0 , for all n. (49) 3.2 V olterra type equations and their generalizations. Let us consider eq. (45). Theorem 3.1. If (45) satisfies at le ast one of the c onditions (49) and (33) r epr e- sents a p oint symmetry of (45) then we have τ n ( t, u n ) = τ ( t ) . (50) Pro of. The compatibilit y condition (34) of eqs. (45) and (33, 39) implies 1 X l = − 1 f n,u n + l [ φ n + l − τ n + l f n + l ] + ( τ n,t + τ n,u n f n ) f n + (51) + τ n [ f n,t + 1 X l = − 1 f n,u n + l f n + l ] − φ n,t − φ n,u n f n = 0 , where indices t , u n + l denote partial deriv ativ es. T aking the deriv ativ e of eq. (51) with resp ect to u n +2 and separately with resp ect to u n − 2 , we obtain t w o relations: f n +1 ,u n +2 f n,u n +1 ( τ n − τ n +1 ) = 0 , (52) f n − 1 ,u n − 2 f n,u n − 1 ( τ n − τ n − 1 ) = 0 . In view of the conditions (49), eqs. (52) imply τ n +1 ( t, u n +1 ) = τ n ( t, u n ) or τ n − 1 ( t, u n − 1 ) = τ n ( t, u n ) . (53) Eac h of these conditions must be satisfied for any n and they are equiv alen t. Since u 0 , u 1 , u − 1 , . . . are indep endent, we find that τ n ( t, u n ) dep ends on t alone and this pro v es Theorem 3.1. A somewhat weak er theorem can b e pro ved for a more general differential– difference equation, namely , ˙ u n = f n ( t, u n + k , u n + k +1 , · · · , u n + m ) , k ≤ m. (54) Theorem 3.2. L et Eqs. (33, 39) r epr esent a symmetry of e q. (54). If the function f n in (54) satisfies m > 0 , ∂ f n ∂ u n + m 6 = 0 for al l n, (55) 10 then the function τ n ( t, u n ) is such that τ n ( t, u n ) = τ n ( t ) , τ n + m ( t ) = τ n ( t ) . (56) If the function f n satisfies k < 0 , ∂ f n ∂ u n + k 6 = 0 for al l n, (57) then we have τ n ( t, u n ) = τ n ( t ) , τ n + k ( t ) = τ n ( t ) . (58) Pro of. The compatibilit y condition for Eqs.(33, 39) and (54) will b e the same as Eq. (51) but all sums will b e from l = k to l = m . If Eq. (55) is satisfied w e can differen tiate eq. (51) with resp ect to u n +2 m and obtain τ n ( t, u n ) = τ n + m ( t, u n + m ) whic h implies (56). If (57) is satisfied we differen tiate (51) with resp ect to u n +2 k and obtain τ n ( t, u n ) = τ n + k ( t, u n + k ) which implies (58). This result is v alid, in particular, for Burgers type equations for whic h k = 0, m > 0 or k < 0, m = 0 F or all equations in the class (54), under the assumptions of this theorem, the function τ n is independent of u n and is perio dic in n . In particular, if k = − 2, m = 2 it is tw o-p erio dic and we can write τ n ( t ) = 1 + ( − 1) n 2 τ 0 ( t ) + 1 − ( − 1) n 2 τ 1 ( t ) . (59) 3.3 T o da t yp e equations The compatibility condition for eq. (46) and Eqs. (33, 39) is u n,ttλ = u n,λtt and implies f n, ˙ u n [ φ n,t + ( φ n,u n − τ n,t ) ˙ u n − τ n,u n ( ˙ u n ) 2 ] (60) + 1 X k = − 1 f n,u n + k [ φ n + k + ( τ n − τ n + k ) ˙ u n + k ] − φ n,tt + ( − 2 φ n,tu n + τ n,tt ) ˙ u n + ( − φ n,u n u n + 2 τ n,tu n )( ˙ u n ) 2 + τ n,u n u n ( ˙ u n ) 3 + τ n f n,t + (2 τ n,t − φ n,u n + 3 τ n,u n ˙ u n ) f n = 0 . W e use Eq. (60) to prov e the follo wing theorem. Theorem 3.3. L et Eqs. (33, 39) r epr esent a p oint symmetry of Eq. (46) and let the function f n in Eq. (46) satisfy at le ast one of the c onditions (49) for al l n . Then the function τ n ( t, u n ) in e q. (39) satisfies Eq. (50), i.e. τ n ( t, u n ) dep ends on t alone. 11 Pro of. None of the functions f n , φ n , τ n figuring in eq. (60) dep ends on ˙ u n +1 or ˙ u n − 1 . These t w o expressions do how ever figure explicitly in (60). Their co efficien ts m ust hence v anish and we obtain f n,u n +1 ( τ n − τ n +1 ) = 0 , (61) f n,u n − 1 ( τ n − τ n − 1 ) = 0 . In view of the conditions (49), w e can use one of eqs. (61), and b oth of them pro vide the same: τ n ( t, u n ) = τ n +1 ( t, u n +1 ) (62) for any n . Hence w e again obtain the result (50), as stated in Theorem 3.3. 3.4 T o da field theory t yp e equations Let us now consider the equation (47) and assume that it has a Lie p oint symmetry represen ted by (48). Theorem 3.4. L et (48) r epr esent a Lie p oint symmetry of the field e quation (47) and let the function f n ( x, y , u n,x , u n,y , u n − 1 , u n , u n +1 ) satisfy at le ast one of the c on- ditions ∂ f n ∂ u n − 1 6 = 0 , or ∂ f n ∂ u n +1 6 = 0 . (63) The functions ξ n and η n in the symmetry (48) then ar e given by ξ n ( x, y , u n ) = ξ ( x, y ) , η n ( x, y , u n ) = η ( x, y ) . (64) Pro of. The compatibilit y condition u n,xy λ = u n,λxy in this case can b e written as 1 X k = − 1 ∂ f n ∂ u n + k ψ n + k + ∂ f n ∂ u n,x D x ψ n + ∂ f n ∂ u n,y D y ψ n − D x D y ψ n = 0 (65) with ψ n as in Eq. (48b); D x and D y are the total deriv ative op erators. The terms u n ± 1 ,x , u n ± 1 ,y only figure in ψ n ± 1 and in D x D y ψ n where we ha ve D x D y ψ n = − ξ n ( D x f n ) − η n ( D y f n ) + · · · with D x f n = ∂ f n ∂ u n − 1 u n − 1 ,x + ∂ f n ∂ u n +1 u n +1 ,x + · · · D y f n = ∂ f n ∂ u n − 1 u n − 1 ,y + ∂ f n ∂ u n +1 u n +1 ,y + · · · . 12 Substituting in to (65) and setting the coefficients of u n ± 1 ,x and u n ± 1 ,y equal to zero separately , w e obtain ( ξ n − 1 − ξ n ) ∂ f n ∂ u n − 1 = 0 , ( η n − 1 − η n ) ∂ f n ∂ u n − 1 = 0 , (66) ( ξ n +1 − ξ n ) ∂ f n ∂ u n +1 = 0 , ( η n +1 − η n ) ∂ f n ∂ u n +1 = 0 . Th us, under the assumption (63) w e obtain (64) and this completes the pro of. 4 Examples Let us now consider examples of each of the classes of differential–difference equa- tions discussed in Section 3. 4.1 The YdKN equation The Krichev er–Novik ov equation [5] ˙ u = 1 4 u xxx − 3 8 u 2 xx u x + 3 2 P ( u ) u x , (67) where P ( u ) is an arbitrary fourth degree p olynomial with constan t co efficien ts, is an integrable PDE with many in teresting prop erties [1, 3, 5 – 8, 15 – 17, 20, 21]. Y amilov and collab orators ha v e prop osed in tegrable discretizations of eq. (67) [13, 23, 25]. The original form of the YdKN equation [23, 25] is u n,t = P n u n +1 u n − 1 + Q n ( u n +1 + u n − 1 ) + R n u n +1 − u n − 1 , (68) P n = αu 2 n + 2 β u n + γ , (69) Q n = β u 2 n + λu n + δ, R n = γ u 2 n + 2 δ u n + ω , where α, · · · , ω are pure constants. A complete symmetry analysis of this equation and its generalizations is in prepa- ration [12]. Here w e will just consider one sp ecial case as an example of a V olterra t yp e equation. Let us set α = 1, β = · · · = ω = 0 in (69). The YdKN equation reduces to u n,t = u 2 n u n +1 u n − 1 u n +1 − u n − 1 . (70) According to Theorem 3.1 a compatible flow corresp onding to a point symmetry will hav e the form u n,λ = Φ n ( t, u n ) − τ ( t ) u n,t . (71) 13 W e replace u n,t in (71) using (70) and then imp ose the compatibility condition u n,tλ = u n,λt . First of all, from terms con taining u n +2 and u n − 2 w e find that Φ n and τ m ust satisfy τ = τ 0 + τ 1 t, Φ n = a n + b n u n + c n u 2 n , (72) a n = a + ˆ a ( − 1) n , b n = b + ˆ b ( − 1) n , c n = c + ˆ c ( − 1) n , where τ 0 , τ 1 , a, ˆ a, b, ˆ b, and c, ˆ c are pure constan ts. This is actually the case for the general YdKN equation (68). Substituting (72) into the compatibility condition w e obtain an equation that is p olynomial in u n + k . Setting co efficien ts of u a n − 1 u b n u c n +1 equal to zero for each indep endent term we obtain the following basis of the Lie p oin t symmetry algebra of eq. (70) X 1 = ∂ t , X 2 = u 2 n ∂ u n , X 3 = ( − 1) n u 2 n ∂ u n , (73) X 4 = t∂ t − 1 2 u n ∂ u n , X 5 = ( − 1) n u n ∂ u n . This is a solv able Lie algebra with { X 1 , X 2 , X 3 } its Ab elian niilradical. The t wo nonnilp oten t elements satisfy [ X 4 , X 5 ] = 0 and their action on the nilradical is given b y   [ X 4 , X 1 ] [ X 4 , X 2 ] [ X 4 , X 3 ]   =   − 1 0 0 0 − 1 2 0 0 0 − 1 2     X 1 X 2 X 3   , (74)   [ X 5 , X 1 ] [ X 5 , X 2 ] [ X 5 , X 3 ]   =   0 0 0 0 0 1 0 1 0     X 1 X 2 X 3   . 4.2 The T o da lattice The T o da lattice itself u n,tt = exp( u n − 1 − u n ) − exp( u n − u n +1 ) , (75) is the b est kno wn example of an equation of the type (46). According to Theo- rem 3.3 the flo w corresp onding to its p oin t symmetries will satisfy (71). F rom the compatibilit y condition u n,ttλ = u n,λtt w e obtain the Lie p oint symmetry algebra X 1 = ∂ t , X 2 = t∂ u n , X 3 = ∂ u n , X 4 = t∂ t + 2 n∂ u n . (76) This Lie algebra is solv able, its nilradical { X 1 , X 2 , X 3 } is isomorphic to the Heisen- b erg algebra. W e note that the Ansatz made in [9] w as not correct and lead to X 3 = q ( n ) ∂ u n in (76) with q ( n ) arbitrary . It w as how ever noted there that a closed Lie algebra is obtained only for q ( n ) = const. 14 4.3 The t wo–dimens ional T o da lattice equation. The equation to b e considered [4, 14] is u n,xy = exp( u n − 1 − u n ) − exp( u n − u n +1 ) . (77) According to T eorem 3.4 the flow corresp onding to p oin t symmetries will take the form u n,λ = φ n ( x, y , u n ) − ξ ( x, y ) u n,x − η ( x, y ) u n,y . (78) The Lie p oint symmetry algebra obtained from the compatibility condition u n,xy λ = u n,λxy is infinite–dimensional and dep ends on 4 arbitrary function of one v ariable eac h X ( f ) = f ( x ) ∂ x + f 0 ( x ) n∂ u n , U ( k ) = k ( x ) ∂ u n , (79) X ( g ) = g ( y ) ∂ y + g 0 ( y ) n∂ u n , W ( ` ) = ` ( y ) ∂ u n . This algebra happens to coincide with the one found in [10] through the prolongation form ula used there w as incorrect. This is a Kac–Mo o dy–Virasoro algebra as is t ypical for integrable equations with more than 2 indep enden t v ariables (in this case x, y and n ). 5 Conclusions The main results of the present article are: 1. The prolongation form ulas (41) and (42), (43) for evolutionary and ordinary v ector fields generating commuting flo ws and Lie p oint symmetry transforma- tion for differen tial–difference equations. These are view ed here as differen tial equations on fixed non–transforming lattices. 2. The prolongation form ulas and the corresponding algorithm for calculating Lie p oint symmetries of differential–difference equations are greatly simplified for 3 rather general classes of equations ( including the T o da lattice, the tw o– dimensional T o da lattice and the V olterra equations). The results are summed up in Theorems 3.1–3.4. 3. W e hav e presented an example of each class of equations cov ered by the ab ov e Theorems and identified a class of equations dep ending on 6 parameters (with generalizations depending on 9 parameters). These are the Y amilov discretiza- tions of the Kric hev er–Novik ov equation (68, 69). A complete analysis of the symmetries of the YdKN equation and its general- izations will b e published separately . 15 Ac kno wledgmen ts. RIY has b een partially supp orted by the Russian F oun- dation for Basic Research (gran t n umbers 08-01-00440-a and 09-01-92431-KE-a). The researc h of P .W. w as partially supp orted by a researc h gran t from NSERC of Canada. LD has been partly supported b y the Italian Ministry of Education and Researc h, PRIN ”Nonlinear wa v es: integrable fine dimensional reductions and discretizations” from 2007 to 2009 and PRIN ”Contin uous and discrete nonlinear in tegrable evolutions: from water w a ves to symplectic maps” from 2010. References [1] V.E. Adler, A.B. Shabat and R.I. Y amilov, Symmetry approac h to the integrabilit y problem, T e or et. Mat. Fiz. 125 (2000), no. 3, 355–424 (in Russian); English transla- tion in The or. Math. Phys. 125 (2000), no. 3, 1603–1661. [2] V.A. Dorodnitsyn The Gr oup Pr op erties of Differ enc e Equations (Moscow, Fizmatlit, 2001) (in Russian). [3] B. Dubro vin, I. Kric hev er, and S. Novik ov, ”Integrable systems. I,” in: Itogi Nauki i T ekhn. Ser.: Sovr. Probl. Matem. [in Russian] (R. V. Gamkrelidze, ed.), V ol. 4, Dynamic Systems-4, VINITI, Mosco w (1985), pp. 179-285. [4] A.P . F ordy and J. Gibb ons Inte gr able nonline ar Klein–Gor don e quations and T o da lattic es Comm. Math. Ph ys. 71 (1980) 21–30. [5] 1. I. M. Kric hever and S. P . Novik ov,Holomorphic bundles and nonlinear equations. Finite-gap solutions of rank 2, Sov. Math. Dokl. 20 (1979) 650-654. [6] I.M. Krichev er and S.P . Novik ov, Holomorphic bundles o ver algebraic curves and non-linear equations, Usp ekhi Mat. Nauk 35 (1980), 47–68 (in Russian); English translation in Russ. Math. Surv. 35 (1980), 53–80. [7] G. Latham and E. Previato,Darb oux transformations for higher-rank Kadom tsev- P etviashvili and Krichev er-Novik ov equations. A cta Appl. Math. , 39 (1995) 405-433. [8] D. Levi, M. P etrera, C. Scimiterna and R. Y amilov, On Miura transformations and V olterra-type equations asso ciated with the Adler-Bob enk o-Suris equations, SIGMA 4 (2008), 077, 14 pages. [9] D. Levi and P . Winternitz, Contin uous symmetries of discrete equations. Phys. L ett. A 152 (1991) 335–338. [10] D. Levi and P . Winternitz, Symmetries and conditional symmetries of differen tial- difference equations. J. Math. Phys. 34 (1993) 3713–3730. [11] D. Levi and P . Winternitz, Contin uous symmetries of difference equations, J. Phys. A: Math. Gen. 39 (2006) R1-R63. 16 [12] D. Levi, P . Win ternitz and R. Y amilov, Point Symmetries of the Y amilov Discr etiza- tion of the Krichever-Novikov Equation and its Gener alizations in preparation. [13] D. Levi and R. Y amilov, Conditions for the existenc e of higher symmetries of evolu- tionary e quations on the lattic e , J. Math. Ph ys. 38 (1997), 6648–6674. [14] A.V. Mikhailov, Inte gr ability of two dimensional T o da chain So v. Ph ys. JETP Lett. 30 (1979) 414–418. [15] O. I. Mokhov, Canonical Hamiltonian representation of the Krichev er-Novik ov equa- tion, Math. Notes , 50, 939-945 (1991). [16] D. P . No viko v, A lgebr aic–ge ometric solutions of the Krichever–Novikov e quation , Theoretical and Mathematical Ph ysics, 121 , 1999 1567–1573. [17] S. P . No viko v, S. V. Manak o v, L. P . Pitaevsky , and V. E. Zakharo v, Theory of Solitons: The In v erse Scattering Metho d [in Russian], Nauk a, Mosco w (1980); English transl., Plenum, New Y ork (1984). [18] P .J. Olv er Applic ations of Lie Gr oups to Differ enc e Equations (New Y ork, Springer 2000). [19] A.B. Shabat and R.I. Y amilov, T o a tr ansformation the ory of two-dimensional inte- gr able systems , Ph ys. Lett. A 227 (1997) 15–23. [20] V. V. Sokolo v, Hamiltonian prop erty of the Krichev er– No viko v equation Sov. Math. Dokl. , 30 (1984) 44-46. [21] S. I. Svinolup ov, V. V. Sok olov, and R. I. Y amilov, B¨ acklund transformations for in tegrable ev olution equations. Sov. Math. Dokl. , 28 (1983) 165-168. [22] P . Win ternitz, Symmetries of Discrete Systems, Discr ete Inte gr able Systems edited b y B. Grammaticos, Y. Kosmann–Sch warzbac h and T. T amizhmani (Berlin, Springer, 2004) p. 185–243. [23] R.I. Y amilov, Classification of discrete evolution equations, Usp ekhi Mat. Nauk 38 (1983), no. 6, 155–156 (in Russian). [24] R.I. Y amilov, Classification of T o da type scalar lattices, In: Pro ceedings of In t. W orkshop NEEDS’92 (eds: V. Makhanko v, I. Puzynin, O. P ashaev), W orld Scien tific Publishing, 1993, 423–431. [25] R. Y amilo v, Symmetries as integrabilit y criteria for differential difference equations, J. Phys. A: Math. Gen. 39 (2006) R541–R623. 17