$lambda$-symmetries for discrete equations

λ -symmetries for discrete equations D Levi 1 and M A Ro dr ´ ıguez 2 1 Dipartimento di Ingegneria Elettronica Univ ersit` a degli Studi Roma T re and INFN Sez io ne di Roma T re Via della V asca Nav ale 8 4, 0014 6 Ro ma, Italy 2 Departamento de F ´ ısica T e´ orica I I F acultad de F ´ ısica s, Univ ersidad Complutense 28040 -Madrid, Spain E-mail: 1 levi@r oma3.infn.it , 2 rodrig ue@fis.ucm . es P A CS num bers: 02.20 .a, 02 .3 0.Ks, 02 .3 0.Hq, 02 .70.Bf Abstract. F ollo wing the usua l definition o f λ -symmetries of differential equations, we introduce the analogous concept fo r difference e quations a nd a pply it to s o me examples. 1. In tro duction One of the most fruitful metho ds in the study of differen tial equations is the use of Lie symmetries to construct exact solutions. Once a symmetry is found, w e can reduce the order of the ordina r y differen tial equation or, in the case o f part ia l differen tial equations, construct sp ecial solutions as functions of the inv arian ts o f the symmetry group. As it is shown in Olv er [13 ] there are differen tial equations whic h can b e reduced ev en if there is no Lie p oin t symmetry . Muriel and Romero in 2001 [10] in tro duc ed the concept of λ - s ymmetries to justify the existence of these sp ecial cases of reduction for o r d inary differen tial equations. Gaeta and Morando g av e lat er a geometrical in terpretation for the λ -symmetries and extend it to partial differen tial equations ( µ - symmetries ) [6] (see also [2, 3 , 4, 9, 11, 12] and [5] f o r a review of the problem). In these w orks, these symmetries w ere show n to b e related to gauge transformat io ns . Many other approac hes to t h is problem ha v e b een prop osed b y differen t a ut ho r s . F or example, b y Catalano F erraioli [1], using p oten tia l symmetries, or b y Pucci and Saccomandi [14], using telescopic ve ctor fields. Lie symmetry approac h has b een extended with succes s to the case of difference equations [8 ]. F rom one side, we can discretize a differen tial equation with symmetries giving rise to a difference sc heme, i.e. a set of difference equations defining b oth the equation a nd the lattice and from the other we can consider a discrete equation on a predefined lattice. In the first case the symmetries e xist by construction and the purp ose is to write and solv e nume rically the difference sc heme. In the second case, λ -symmetries for discr ete e quations 2 when the equation and the lattice are a priori giv en, w e would lik e to find solutions using symmetries but we a re in a situation in whic h usually no symmetries are presen t. A wa y to get symmetries could b e to use the approach t o λ - s ymmetries. W e sho w here that effectiv ely we can construct λ -symmetries for discrete equations and w e presen t some illustrativ e examples. In Section 2 we review the Lie theory for λ -symmetries for o rdinary differen tial equations. In Section 3 we presen t o ur definition of λ -symmetries f or o rdinary difference equations together with some examples. Finally , Section 4 is dev oted to some concluding remarks and to p ossible extensions. 2. λ -symmetries for differen tial equations Let us briefly review the main ideas of the con tin uous λ - s ymmetry . If ˆ X is a v ector field with v ariables x and u , ˆ X = ξ ( x, u ) ∂ x + φ ( x, u ) ∂ u (1) the co efficie n ts of ∂ u k , where u k = d k u ( x ) dx k , in the standard prolonga tion ˆ X ( m ) = ξ ( x, u ) ∂ x + φ ( x, u ) ∂ u + m X k =1 φ ( k ) ∂ u k (2) are defined as [13]: φ ( k +1) = D x φ ( k ) − u k +1 D x ξ , φ (0) = φ, (3) where b y D x one means the total deriv ative with r e sp ect to x . The Lie symmetries of a n m -th order ODE u m = f ( x, u, u 1 , . . . , u m − 1 ) (4) are obtained applying the m -prolo nged infinitesimal generator (2,3) to the equation, i.e., requiring that the inv ariance condition ˆ X ( m ) ( u m − f ) | u m = f = 0 (5) b e satisfied. The λ -symm etries are defined as those symmetries for which the infinitesimal prolongation is mo dified with resp ect to the standard one (3) and is giv en by [10]: φ ( k +1 , λ ) = ( D x + λ ( x, u, u 1 )) φ ( k, λ ) − u k +1 ( D x + λ ( x, u, u 1 )) ξ , (6) where λ ( x, u , u 1 ) is a smo oth function to b e determined at the same time as the co effic ien ts of the infinitesimal generators, ξ and φ . T he λ - s ymmetry for an m - order differen tial eq uation is then obtained b y applying the follo wing m -prolong ed infinitesimal symmetry generator onto the differential equation: ˆ X ( m,λ ) = ξ ∂ x + φ ∂ u + m X k =1 φ ( k, λ ) ∂ u k . (7) λ -symmetries for discr ete e quations 3 If the ODE (4) is in v ariant under the symmetry generator ˆ X ( m,λ ) , t h en we sa y that it has a λ -symmetry . The λ -symmetries can b e used to reduce the original equation and to find symmetry inv arian t solutions. As an example of OD E whic h has no Lie p oin t symmetry but p osses ses λ - symmetries , let us consider the following OD E ([13], p. 182) u 2 = [( x + x 2 )e u ] x . (8) This is an equation whic h can b e integrated b y quadrature but whic h has no symmetries. Equation (8) is written in the form of a conserv ation la w, u 2 = D x F ( x, u ) (9) whic h ha s the o b vious reduction u 1 = F ( x, u ) + C . (10) Equation (9) ma y hav e no symmetries but w e can prov e that t here alw a ys exists a λ - symmetry of infinitesimal generator giv en b y ξ = 0 and φ = 1 with λ = F u ( x, u ) whic h is at the origin of this reduction. The λ -symmetry generator is ˆ X (2 ,λ ) = ∂ u + F u ∂ u 1 + [( F u ) 2 + u 1 F uu + F xu ] ∂ u 2 , (11) whic h has tw o obvious in v arian ts z = x and w = u 1 − F and the differential in v arian t w z = D x w /D x z = u 2 − F x = 0. So the general solution of our equation is given b y the solution of the equation (10). F or the particular c hoice of the function F give n b y equation (8), the reduced equation is inte grable and thus equation (8) is integrable b y quadrature [13]. This trivial example is not the only one w e can find in the literature. F or example, the follow ing equation has b een studied in [7] u 2 − ( u − 1 ( u 1 ) 2 + g ( x ) pu p u 1 + g ′ ( x ) u p +1 ) = 0 . (12) This equation is in tegrable b y quadrature. It has Lie p oin t symmetries only when the function g ( x ) is giv en b y o ne of the tw o follo wing expressions , g ( x ) = k 1 e k 2 x ( k 3 + k 4 x ) k 5 or g ( x ) = k 6 + e k 7 x 2 , (13) where k j , j = 1 , . . . , 7 are arbitrary constant parameters. How ev er, for a generic g ( x ), there exists a λ -symmetry given by ˆ X = ∂ u and λ = u − 1 ( u 1 + g ( x ) pu p +1 ). Ho w can w e obta in the prolo ng ation (6) from the know n theory? In what sense the λ -symmetries are a generalization of Lie p oin t symmetries? These are some of the questions w e need to answ er to b e a b le to construct λ -symmetries for equations on the lattice. A constructiv e wa y to get λ -symmetries is con tained in the w ork o f Catalano F erraio li [1]. There he in tro duces λ -symmetries exploiting nonlo cal symmetries. This approac h is easily extendable to ordinary difference equations. T he case of higher dimensional lattices will b e discussed elsewhere. λ -symmetries for discr ete e quations 4 The construction presen ted in [1] consists fundamen tally in adding to the OD E at study (4) an additional differential equation w 1 = λ ( x, u, u 1 ) , (14) for a new dep enden t function w ( x ). This means tha t instead of considering an ODE we are solving a system o f OD Es written in triangular form. One can stat e the following prop osition ( see [1] for the details): Prop osition: An OD E a dmits a λ -symmetry gener a t or ˆ X ( m,λ ) if and only if the symmetry gener ator ˆ Y ( m ) = ξ ( x, u, w ) ∂ x + φ ( x, u, w ) ∂ u + η ( x, u, u 1 , w ) ∂ w + m X i =1 φ ( i ) ∂ u i + η (1) ∂ w 1 (15) with φ ( i ) = ˜ Dφ ( i − 1) − u i ˜ D ξ , η (1) = ˜ Dη − w 1 ˜ D ξ , φ (0) = φ, u 0 = u , w 0 = w , (16) le aves the e q uations (4 , 14) invariant a nd i s such that [ ∂ w , ˆ Y ( m ) ] = ˆ Y ( m ) . (17) In equation (16) ˜ D is the tota l deriv ativ e op erator, ˜ D = ∂ x + X i ≥ 0 u i +1 ∂ u i + X i ≥ 0 w i +1 ∂ w i . (18) Equation (17) implies ξ ( x, u, w ) = e w ˜ ξ ( x, u ) , φ ≡ φ (0) ( x, u, w ) = e w ˜ φ ( x, u ) , φ ( i ) = e w ˜ φ ( i ) , η (0) ≡ η ( x, u, u 1 , . . . , u m − 1 , w ) = e w ˜ η ( x, u , u 1 , . . . , u m − 1 ) , η (1) = e w ˜ η (1) , (19) and consequen tly ˆ Y ( m ) = e w  ˜ ξ ∂ x + m X i =0 ˜ φ ( i ) ∂ u i + 1 X i =0 ˜ η ( i ) ∂ w i  . (20) When we a pply the generator (2 0) o n to the equation (14) ˆ Y (1) ( w 1 − λ ( x, u, u 1 )) = e w ( ˜ ξ ∂ x + ˜ φ∂ u + ˜ η (1) ∂ w 1 )( w 1 − λ ( x, u, u 1 )) (21) w e get the determining equation for ˜ η ( x, u , u 1 , · · · , u m − 1 ): ˜ η (1) − ˜ ξ λ x − ˜ φ λ u − ˜ φ (1) λ u 1 = 0 (22) where, from (16, 19) ˜ η (1) = e − w ( ˜ D x η − w 1 ˜ D x ξ ) = ˜ η x + w 1 ˜ η + m X k =1 u k ˜ η u k − 1 − ( ˜ ξ x + w 1 ˜ ξ + u 1 ˜ ξ u ) w 1 ˜ φ (1) = e − w ( ˜ D x φ − u 1 ˜ D x ξ ) = ˜ φ x + w 1 ˜ φ − ( ˜ ξ x + w 1 ˜ ξ − ˜ φ u + u 1 ˜ ξ u ) u 1 (23) Applying the generator ˆ Y ( m ) (20) on to ( 4 ), w e can factorize t he w - dep endence and the infinitesimal generator ˆ Y ( m ) reduces to the λ -symmetry generator (7). When equation λ -symmetries for discr ete e quations 5 (14) is satisfied, equation (22) is a w -indep enden t partial differen tial equation fo r ˜ η in terms of ˜ φ and λ : ˜ η x + m − 1 X k =1 u k ˜ η u k − 1 + f ˜ η u m − 1 = ˜ ξ λ x + ˜ φλ u +( ˜ φ x + λ ˜ φ − ( ˜ ξ x + λ ˜ ξ − ˜ φ u + u 1 ˜ ξ u ) u 1 ) λ u 1 + ( ˜ ξ x + λ ˜ ξ + u 1 ˜ ξ u − ˜ η ) λ (24) In this w a y , λ symmetries are just classical symmetries for the system formed b y the ODE ( 4 ) and the equation (14) . Ho w eve r, t hey may no t corresp ond t o just Lie p oint symmetries as equation (24) ma y not hav e a solutio n when ˜ η dep ends just on x and u . See [1] for more details on the equiv alence of this metho d to that o f Muriel and Romero [10]. 3. λ -symmetries in the discrete case W e will construct λ -symmetries in the discrete case, closely following the approac h w e ha v e discussed in the last section. Let us consider a difference sche me in a one- dimensional lattice: f i ( x n − a , . . . , x n + b , u n − a , . . . , u n + b ) = 0 , a, b ∈ N , i = 1 , 2 (25) Equations (25) corresp ond to a discrete sc heme where the t w o equations define a t the same time the solution and the lat t ice. In the contin uous limit when the distance b et w ee n the p oints go es to zero, one of the equations is iden tically satisfied while the other go es ov er to a differential equation [8]. In order to find λ - symmetries for this equation, w e introduce a first order difference equation for a new dep enden t v ariable w n : w n +1 − w n − ( x n +1 − x n ) λ n ( x n , u n ) = 0 (26) Here, w e sho w explicitly the λ n dep endence on the v a lues of u n in t he p oint of the lattice of index n but we m ust think that λ n ma y dep end on more lattice p oints, i.e., λ n ( x n , { u j } j = β j = − α ), α, β ∈ N . F ro m equation (26) w e can express t he function w n in an y p oint o f the latt ice in terms of the function u n and the initial data. In fact, by solving it w e g et: w n + k = w n + k − 1 X i =0 ( x n + i +1 − x n + i ) λ n + i ( x n + i , u n + i ) , k ∈ N w n − k = w n − k − 1 X i =0 ( x n − i − x n − i − 1 ) λ n − i − 1 ( x n − i − 1 , u n − i − 1 ) , k ∈ N . (27) The symmetry generator fo r equations (25, 26) is: ˆ Y = ξ n ( x n , u n , w n ) ∂ x n + φ n ( x n , u n , w n ) ∂ u n + η n ( x n , u n , w n ) ∂ w n (28) where, as in the case of the function λ n , η n ma y dep end on mo r e lattice p oints. The prolongation of ˆ Y is g iv en b y ˆ Y ( a,b ) = b X i = − a ξ n + i ( x n + i , u n + i , w n + i ) ∂ x n + i + b X i = − a φ n + i ( x n + i , u n + i , w n + i ) ∂ u n + i + λ -symmetries for discr ete e quations 6 + 1 X i =0 η n + i ( x n + i , u n + i , w n + i ) ∂ w n + i . (29) If w e apply ˆ Y (0 , 1) on to equation (26) w e get, − ( ξ n +1 − ξ n ) λ n + ( x n +1 − x n )( ξ n ∂ x n λ n + φ n ∂ u n λ n ) + η n +1 − η n = 0 . (30) F or simplicit y w e consider here that λ n = λ n ( x n , u n ) only , but the result, with appropriated c hanges, is v a lid in general. Imp osing the condition ( 17) to the symmetry generator (28) w e get ξ n + i = e w n + i ˜ ξ n + i ( x n + i , u n + i ) , φ n + i = e w n + i ˜ φ n + i ( x n + i , u n + i ) , η n + i = e w n + i ˜ η n + i ( x n + i , u n + i ) . (31) If w e substitute (31) in (30), w e obtain e w n +1 − w n ( ˜ η n +1 − ˜ ξ n +1 λ n ) − ( ˜ η n − ˜ ξ n λ n ) + ( x n +1 − x n )( ˜ ξ n ∂ x n λ n + ˜ φ n ∂ u n λ n ) = 0 , (32) and using (26), we find tha t ˜ η n m ust satisfy the follo wing w n -indep enden t equation e ( x n +1 − x n ) λ n ( ˜ η n +1 − ˜ ξ n +1 λ n ) − ( ˜ η n − ˜ ξ n λ n ) + ( x n +1 − x n )( ˜ ξ n ∂ x n λ n + ˜ φ n ∂ u n λ n ) = 0 (33) It is w orth while to notice t ha t equation (33) implies that the function ˜ η n will dep end in general on sev eral p oin ts of the lattice; its solutio n when ˜ η n dep ends just o n u n ma ybe trivial as equation (33 ) imp oses in this case strict constrain ts on λ n , ˜ ξ n and ˜ φ n . The prolongatio n o f the infinitesimal generator of the λ - symmetry is giv en b y ˆ X ( a,b ; λ ) = ˜ ξ n ∂ x n + k + ˜ φ n ∂ u n + b X k =1 e w n + k − w n ˜ φ n + k ∂ u n + k + a X k =1 e w n − k − w n ˜ φ n − k ∂ u n − k , (34) and, when w e apply (34) onto equations (25), w e get b X k = − a e w n + k ( ˜ ξ n + k ∂ x n + k f i + ˜ φ n + k ∂ u n + k f i ) | [ f i =0 , w n +1 = w n + λ n ( x n +1 − x n )] = 0 , i = 1 , 2 , (35) whic h, t a king into accoun t equations (27), provide a facto r ized common factor e w n in fron t of a n equation dep ending only on u n . So, as the dep endence on w n factorizes, the effectiv e infinitesimal generator (34) when applied to the O∆E (25) will dep end just on ξ n , φ n and λ n . In this w a y w e get the extra freedom necessary to p ossibly get nontrivial symmetries . T o c hec k the corresp ondence of this approach with the contin uous case in [10], let us consider the tw o-p oin t prolongation ˆ X (0 , 1; λ ) = ˜ ξ n ∂ x n + ˜ φ n ∂ u n + e ( x n +1 − x n ) λ n ( ˜ ξ n +1 ∂ x n +1 + ˜ φ n +1 ∂ u n +1 ) . (36) In this case w e can construct the approx imation to the first deriv ative and consequen tly w e can get the contin uo us limit formula corresp onding to the first prolongation (equation (7) with m = 1). Let us consider the f ollo wing change of v a riables, f r o m ( x n , x n +1 , u n , u n +1 ) to ( ¯ x n , h n +1 , ¯ u n , u x,n ): ¯ x n = x n , ¯ u n = u n , h n +1 = x n +1 − x n , u x,n +1 = u n +1 − u n x n +1 − x n (37) λ -symmetries for discr ete e quations 7 whic h corr esp o nds to the infinitesimal transformation: ∂ x n = ∂ ¯ x n + u x,n +1 h n +1 ∂ u x,n +1 − ∂ h n +1 , ∂ x n +1 = − u x,n +1 h n +1 ∂ u x,n +1 + ∂ h n +1 ∂ u n = ∂ ¯ u n − 1 h n +1 ∂ u x,n +1 , ∂ u n +1 = 1 h n +1 ∂ u x,n +1 . (38) Then the v ector field (36) is rewritten a s: ˆ X (0 , 1; λ ) = ˜ ξ n ∂ ¯ x n + ˜ φ n ∂ ¯ u n + (e h n +1 λ n ˜ ξ n +1 − ˜ ξ n ) ∂ h n +1 +  e h n +1 λ n ˜ φ n +1 − ˜ φ n h n +1 − e h n +1 λ n ˜ ξ n +1 − ˜ ξ n h n +1 u x,n +1  ∂ u x,n +1 . (39) In the con tin uous limit, h n and h n +1 go t o zero, so that w e ha v e tak e e h n +1 λ n ∼ 1 + h n +1 λ n , and u x,n +1 go es into u 1 . Then ˆ X (0 , 1; λ ) ∼ lim h → 0  ˜ ξ n ∂ ¯ x n + ˜ φ n ∂ ¯ u n + ( ˜ ξ n +1 − ˜ ξ n + h n +1 λ n ˜ ξ n +1 ) ∂ h n +1 +  ˜ φ n +1 − ˜ φ n h n +1 − ˜ ξ n +1 − ˜ ξ n h n +1 u x,n +1 + λ n ( ˜ φ n +1 − ˜ ξ n +1 u x,n +1 )  ∂ u x,n +1  = ˜ ξ ∂ x + ˜ φ ∂ u + [( D x + λ ) ˜ φ − u 1 ( D x + λ ) ˜ ξ ] ∂ u 1 (40) i.e., the result of Muriel and Romero [10]. 3.1. Exam ples Let us consider as examples, second order difference equations on a fixed un transformable lattice of spacing x n +1 − x n = h whic h has λ -symmetries: u n +1 − 2 u n + u n − 1 h 2 = F n ( u n , u n − 1 ) (41) T aking into accoun t the results presen ted ab o v e, the determining equation for the λ - symmetries of equation ( 4 1) is obtained by applying the v ec tor field ˆ X (1 , 1; λ ) = ˜ φ n ∂ u n + e hλ n ( u n ) ˜ φ n +1 ∂ u n +1 + e − hλ n − 1 ( u n − 1 ) ˜ φ n − 1 ∂ u n − 1 (42) on to the equation (41). If w e define χ n = e hλ n ( u n ) , the v ector field reduces to ˆ X (1 , 1; λ ) = ˜ φ n ∂ u n + χ n ˜ φ n +1 ∂ u n +1 + ˜ φ n − 1 χ n − 1 ∂ u n − 1 (43) and the determining equation is χ n ˜ φ n +1 + ˜ φ n − 1 χ n − 1 − 2 ˜ φ n = h 2  ˜ φ n F n,u n + ˜ φ n − 1 χ n − 1 F n,u n − 1  . (44) Example 1. Let us c ho ose a f unction F n whic h is the discrete deriv ativ e o f a function f n ( u n ), i.e., u n +1 − 2 u n + u n − 1 h 2 = F n ( u n , u n − 1 ) = 1 h [ f n ( u n ) − f n − 1 ( u n − 1 )] . (45) This equation has no p oin t symmetry for a g eneric f unction f n ( u n ). It is ob vious that this equation, as in the example of Olv er ( 9 ), reduces t o a first order difference equation u n +1 − u n h = f n ( u n ) + C . (46) λ -symmetries for discr ete e quations 8 W e will show that it has a non trivial λ -symmetry whic h will b e t he origin of the reduction of the order of the equation. If we tak e ˜ φ n = 1 the determining equation (44) reduces to χ n + 1 χ n − 1 − 2 = h  ∂ u n f n − 1 χ n − 1 ∂ u n − 1 f n − 1  (47) and is satisfied by χ n = 1 + h∂ u n f n . The v ector field (43) is in this case ˆ X (1 , 1; λ ) = ∂ u n + (1 + h∂ u n f n ) ∂ u n +1 + 1 1 + h∂ u n − 1 f n − 1 ∂ u n − 1 (48) and its inv a r ian t is v n = u n +1 − u n − hf n ( u n ) , (49) Substituting into equation (4 5) w e get v n = C , i.e., u n satisfies (46). Example 2. Let us consider the difference equation u n +1 − 2 u n + u n − 1 h 2 = u n − 1  1 + h 2 u n − 1  u n − u n − 1 h − h 8 u 4 n − 1 , (50) whose con tin uous limit giv es the ODE u 2 = uu 1 + 1 2  u 2 u 1 − uu 2 − 2 u 2 1 − 1 4 u 4  h + O ( h 2 ) . (51) Equation (50), b eing on an uniform lattice, could only hav e dila tion symmetries. It is easy to see that it has no Lie symmetry . T aking in to accoun t the results presen ted ab ov e, the determining equation for the λ - symmetries o f equation (50) is g iv en b y χ n ˜ φ n +1 − 2 ˜ φ n + 1 χ n − 1 ˜ φ n − 1 =  1 + 1 2 hu n − 1  hu n − 1 ˜ φ n + h  u n − 2 u n − 1 + hu n − 1  u n − 3 2 u n − 1  − 1 2 h 2 u 3 n − 1  1 χ n − 1 ˜ φ n − 1 (52) when equation (50) is satisfied, i.e. when u n +1 is expressed in terms of u n and u n − 1 using this equation. T o get a λ -symmetry w e c ho ose ˜ φ n = 1 and equation (52) reduces to just a nonlinear first difference equation for the function χ n ( u n ) χ n − 2(1 + 1 2 hu n − 1 ) 2 + (1 + hu n − 1 )[(1 + hu n − 1 ) 2 + 1 − 2 hu n ] 2 χ n − 1 = 0 . (53) If w e differen tiate equation (53) with respect t o u n t wice we get ¨ χ n = 0, i.e., χ n = α + β u n where α and β are tw o arbitrary constants . Substituting this necessary result in to equation (53) we get t ha t α = 1, β = h and taking in to a ccount the definition of χ n w e get as the only p ossible solution λ n = 1 h log(1 + hu n ) (54) W e can no w use t he λ -symmetry to reduce the difference equation. The infinitesimal generator of the λ - symmetry is ˆ X (1 , 1; λ ) = 1 χ n − 1 ∂ u n − 1 + ∂ u n + χ n ∂ u n +1 (55) λ -symmetries for discr ete e quations 9 The in v ariants are obtained by inte grating the f o llo wing shift related equations χ n − 1 d u n − 1 = d u n 1 = d u n +1 χ n . (56) whic h, a fter substituting (5 4), yields (1 + hu n − 1 )d u n − 1 = d u n = d u n +1 1 + hu n . (57) The compatible solution of equations (57) is giv en b y u n +1 − u n = κ n + h 2 u 2 n (58) where κ n app ear a s an integration constant and thus dep ends just on n , the in v arian t index. If we in tr o duce equation (58) in to the nonlinear equation (50) w e get as the reduced equation, a log istic–t ype difference equation κ n +1 = κ n  1 − h 2 κ n  . (59) This is a first order recurrence relatio n. O ne could lo ok again for a λ -symmetry . In this case o ne can alw a ys find one but w e can not use it to get a solution. 4. Conclusions In this pap er w e hav e considered the λ -symmetries for difference equations. They are determined b y extending t o the discrete case the p ot en tial symmetries in the form giv en b y Catalano F erraio li. As o ne can see from equation (34) the discrete λ - prolongation is quite different form the con tin uous case (6, 7) and it w ould b e difficult to guess it without go ing through the p otential symmetries. The examples we presen ted, sho w that a lso in the discrete case the λ -symmetries can b e use d to reduce the equations to simpler ones as in the con tin uous case. It is w orth w hile to notice that discrete equations hav e usually m uc h less symmetries that the con tin uo us one, so λ -symmetries can b e in this case mo r e useful. W ork is in pro gress to extend this result to the case of µ -symmetries, i.e., to the case o f partial difference equations. Ac kno wledgm en ts DL has b een partly supp orted b y t he Italian Ministry of Education and R esearch , PRIN “Nonlinear w a v es: in tegrable fine dimensional reductions a nd discretizations” f r om 2007 to 2009 and PRIN “Con tin uous and discrete nonlinear in tegrable ev olutions: from w ater w a ves to symplectic maps” from 2010. The w ork o f MAR has b een partly supp o rted b y MCI (Spain), grant FIS2008-00209, and UCM-Banco Santander, g ran t GR58/08- 910556. W e are greatly indebted to G. Gaeta for suggesting us this pro blem and to C. 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