An unusual series of autonomous discrete integrable equations on the square lattice
aa r X i v : . [ n li n . S I] J a n An unusual series of autonomous discreteintegrable equations on the square lattice
R.N. Garifullin and R.I. Yamilov
Institute of Mathematics, Ufa Federal Research Centre,Russian Academy of Sciences,112 Chernyshevsky Street, Ufa 450008, Russian Federation
E-mails: [email protected], [email protected]
January 24, 2019
Abstract
We present an infinite series of autonomous discrete equations on the square latticepossessing hierarchies of autonomous generalized symmetries and conservation laws inboth directions. Their orders in both directions are equal to κN , where κ is an arbitrarynatural number and N is equation number in the series. Such a structure of hierarchiesis new for discrete equations in the case N > N = 2 we show that a second order generalized symmetry isclosely related to a relativistic Toda type integrable equation. As far as we know, thisproperty is very rare in the case of autonomous discrete equations. The following discrete integrable equation( u n,m +1 + 1)( u n,m −
1) = ( u n +1 ,m +1 − u n +1 ,m + 1) (1)is well-known [16, 19]. Recently a number of integrable generalizations of this equationhave been found [6, 10, 17]. All of them are non-autonomous, and here we write down thetwo most interesting. One of them reads:( u n,m +1 + χ n + m +1 )( u n,m − χ n + m ) = ( u n +1 ,m +1 − χ n + m )( u n +1 ,m + χ n + m +1 ) ,χ k = 12 (1 + ( − k ) , (2) nd this is the equation [10, (77)] up to the involution n ↔ m. The second example actually represents a series of discrete equations corresponding tosome periods of n -dependent coefficient. For any fixed N ≥
1, an equation is defined by α n ( u n,m +1 + 1)( u n,m −
1) = α n +1 ( u n +1 ,m +1 − u n +1 ,m + 1) ,α n + N = α n = 0 , for all n ∈ Z , (3)and it has been studied in [6]. In both cases these generalizations have hierarchies ofgeneralized symmetries and conservations laws in both directions as well as the L − A pairs, but all these objects are non-autonomous, i.e. explicitly depend on the discretevariables n or m .Here we are going to construct a series of autonomous integrable generalizations of(1). We show that all equations of that series have autonomous L − A pairs, generalizedsymmetries and conservations laws. In particular, that series provides us with examples ofautonomous discrete equations, such that the minimal possible orders of their autonomousgeneralized symmetries in any direction can be arbitrarily high.A series of equations we construct here is a particular case of (3), however, results ofthe present paper are not a direct consequence of the results presented in [6].In Section 2 we consider an autonomous generalization of (1) with an arbitrary con-stant coefficient, which includes the whole series under consideration, and construct for ithierarchies of generalized symmetries and of conservation laws in the m -direction. Thoseresults are necessary for the next sections.In Section 3 we construct and study a series of autonomous integrable generalizationsof (1) which is the aim of the present work. Autonomous generalized symmetries andconservation laws in the m -direction are constructed in Section 3.1, while symmetriesand conservation laws in the n -direction are discussed in Sections 3.2, 3.4. The mostinteresting case N = 2 is considered in more detail in Section 3.3, and its relationshipwith a relativistic Toda type equation is discussed. Autonomous L − A pairs for equationsof the series are constructed in Section 3.5.Based on the results of the present paper, we formulate and discuss in Conclusion animportant hypothesis about the symmetry structure of equations of the series. We alsobriefly discuss there all the new results obtained. (1) with an arbi-trary constant coefficient The most broad generalization of equation (1) we know is( u n,m +1 + a n,m +1 )( u n,m − a n,m ) = ( u n +1 ,m +1 − b n +1 ,m +1 )( u n +1 ,m + b n +1 ,m ) ,a n,m +2 = a n,m , b n,m +2 = b n,m , a n,m = b n,m . (4)This is equation [10, (40)] up to transformations n ↔ m and b n,m → − b n,m . Equations (1)and (2) are its particular cases. In the case b n,m = a n,m = 0 for all n, m , after rescaling u n,m = ˆ u n,m a n,m , we get for the function ˆ u n,m the following equation: α n ( u n,m +1 + 1)( u n,m −
1) = α n +1 ( u n +1 ,m +1 − u n +1 ,m + 1) , α n = 0 , (5) ith α n = a n,m +1 a n,m . This equation was introduced in [17, Section 3] in a little bitdifferent form.There is in (5) an obvious autonomous subcase with an arbitrary constant coefficient β :( u n,m +1 + 1)( u n,m −
1) = β ( u n +1 ,m +1 − u n +1 ,m + 1) , β = 0 . (6)It corresponds to the restriction α n +1 /α n = β for all n , i.e. up to a multiplier we get: α n = β n . (7)This equation possesses an L − A pair and hierarchies of generalized symmetries andconservation laws in the m -direction, but all these objects are non-autonomous. Equation(6) includes the whole series of equations which is the aim of this paper. The results wepresent here are necessary to the next sections.An L − A pair for equation (6) is given by:Ψ n +1 ,m = L (1) n,m Ψ n,m , Ψ n,m +1 = L (2) n,m Ψ n,m , (8)where L (1) n,m = λβ n ( u n,m + 1) − u n,m − u n,m +1 u n,m − ! , (9) L (2) n,m = (cid:18) − λβ n ( u n,m + 1)( u n,m +1 − (cid:19) , (10)Ψ n,m is the vector-function and λ is the spectral parameter. In more general form, cor-responding to (5), it was presented in [6], while it was constructed for the first timein [17]. m -direction Here we construct the generalized symmetries in the m -direction. A differential-differenceequation of the form ∂ t u n,m = h n,m ( u n,m + µ , u n,m + µ − , . . . , u n,m − µ ) , µ > , (11)is called the generalized symmetry in the m -direction of the discrete equationΦ n,m ( u n,m , u n +1 ,m , u n,m +1 , u n +1 ,m +1 ) = 0 (12)if equations (11) and (12) are compatible. The compatibility condition is obtained bydifferentiating (12) with respect to the time t in virtue of (11): X i,j ∈{ , } h n + i,m + j ∂ Φ n,m ∂u n + i,m + j = 0 , and it must be identically satisfied on the solutions of (12).We suppose that there exist numbers n , m , n , m , such that ∂h n ,m ∂u n ,m + µ = 0 , ∂h n ,m ∂u n ,m − µ = 0 . (13) he number µ is called the order of the generalized symmetry (11). The form of equation(11) is symmetric in a sense. An explanation why such a form is natural for integrabledifferential-difference equations see in [23, Section 2.4.1].First we discuss the particular case of (6) with β = 1 which is known. It is important,as generalized symmetries for the general case (3) are constructed in terms of symmetriesof this particular case.Its simplest generalized symmetry in the m -direction is ∂ t ′ u n,m = ( u n,m − u n,m +1 − u n,m − ) = f (1) n,m , (14)and this is nothing but the modified Volterra equation. The known master symmetry of(14), see [24], can be written in the form: ∂ τ ′ u n,m = ( u n,m − m + 1) u n,m +1 − ( m − u n,m − ) = g n,m . (15)The hierarchy of equation (14) can be constructed as follows: ∂ t ′ k u n,m = f ( k ) n,m ( u n,m + k , u n,m + k − , . . . , u n,m − k ) , k ≥ , (16) f ( k +1) n,m = ad g n,m f ( k ) n,m = [ g n,m , f ( k ) n,m ] = D τ ′ f ( k ) n,m − D t ′ k g n,m = k X j = − k g n,m + j ∂f ( k ) n,m ∂u n,m + j − X j = − f ( k ) n,m + j ∂g n,m ∂u n,m + j . (17)Here D τ ′ and D t ′ k are the operators of total derivatives in virtue of equations (15) and(16), respectively, with a definition shown in (17).We get in this way the standard and known symmetries of the modified Volterraequation. As [ f (1) n,m , f ( k ) n,m ] = 0 for so-constructed functions and g n,m = mf (1) n,m + ( u n,m − u n,m +1 + u n,m − ) , it is easy to prove by induction that all the functions f ( k ) n,m do not depend on m explicitly,e.g.: f (2) n,m = ( u n,m − u n,m +1 − u n,m +2 + u n,m ) − ( u n,m − − u n,m + u n,m − )] . (18)It can be shown, see an explanation below, that (16) are the generalized symmetries ofthe discrete equation (6) with β = 1 too. We also notice that both (14) and its mastersymmetry (15) are generalized symmetries of the discrete equation (6) with β = 1.In general case (6) the simplest generalized symmetry in the m -direction reads: ∂ t u n,m = β n f (1) n,m . (19)Its master symmetry can be taken in the form: ∂ τ ′′ u n,m = β n g n,m . (20)But (20) is not a generalized symmetry of (6) and, therefore, it allows one to constructgeneralized symmetries for (19), but not for (6). To solve this problem, we need tointroduce a special dependence on the master symmetry time into the discrete equation(6) and into both equations (19) and (20). uch a scheme with introducing the time of master symmetry into a discrete equationis used, probably, for the first time.Let us consider a special generalization of (6): A n ( τ )( u n,m +1 + 1)( u n,m −
1) = A n +1 ( τ )( u n +1 ,m +1 − u n +1 ,m + 1) , (21)where A n ( τ ) = ( β − n + 4 τ ) − , A ′ n ( τ ) = − A n ( τ ) , A n (0) = β n , (22)and τ is an external parameter. Here τ is the time of a master symmetry. It can bechecked that the both following equations are generalized symmetries of (21): ∂ t u n,m = F (1) n,m = A n ( τ ) f (1) n,m , (23) ∂ τ u n,m = G n,m = A n ( τ ) g n,m . (24)In particular, the important relation A ′ n = − A n of (22) is the consequence of compati-bility of (21) and (24). As (24) does not commutate with (23), it is reasonable to expectthat, for any k ≥
1, the following functions define nontrivial generalized symmetries of(21): F ( k +1) n,m = ad G n,m F ( k ) n,m = [ G n,m , F ( k ) n,m ] = D τ F ( k ) n,m − D t k G n,m = ∂F ( k ) n,m ∂τ + k X j = − k G n,m + j ∂F ( k ) n,m ∂u n,m + j − X j = − F ( k ) n,m + j ∂G n,m ∂u n,m + j . (25)We see that (24) allows one to construct a hierarchy of generalized symmetries of (21).It also generates a hierarchy of conservation laws, see the next section. For this reasonequation (24) plays the role of the master symmetry not only for (23) but also for thediscrete equation (21).Now we are going to study the structure of these generalized symmetries in order toextract later autonomous among them.We can prove by induction that the following formula takes place: F ( k ) n,m = A kn ( τ ) k − X j =0 j c k,j f ( k − j ) n,m , (26)where c k,j are some constants, e.g. c , = 1 , c , = 1 , c , = − , c , = 1 , c , = − , c , = 2 . We substitute (26) into (25) and obtain: F ( k +1) n,m = ∂A kn ( τ ) ∂τ k − X j =0 j c k,j f ( k − j ) n,m + A k +1 n ( τ ) k − X j =0 j c k,j ad g n,m f ( k − j ) n,m . (27)Taking into account (17) and (22), we get F ( k +1) n,m = − kA k +1 n ( τ ) k X j =1 j c k,j − f ( k +1 − j ) n,m + A k +1 n ( τ ) k − X j =0 j c k,j f ( k +1 − j ) n,m . (28) omparing (26) and (28) we derive the following recurrence formulae: c k +1 ,j = c k,j − kc k,j − , c k, − = c k,k = 0 , c , = 1 , ≤ j ≤ k, k ≥ . (29)We see that generalized symmetries of (21) have the form: ∂ t k u n,m = F ( k ) n,m , k ≥ , (30)where the functions F ( k ) n,m are of the form (26), while f ( k ) n,m , A n ( τ ) and c k,j are given by(17,22) and (29). The order of such a symmetry equals k . An explicit dependence on n and τ is defined by the multiplier A kn ( τ ), and there is here no explicit dependence on m .When τ = 0, equation (21) turns into (6) and the symmetries (30) turn into generalizedsymmetries of (6). Theorem 1.
The discrete equation (6) possesses generalized symmetries of the form ∂ t k u n,m = β nk k − X j =0 j c k,j f ( k − j ) n,m , k ≥ , (31) with f ( k ) n,m and c k,j defined by (17,29). These symmetries do not depend explicitly on m ,and a dependence on n is given by the multiplier β nk . In the case β = 1, we can see that not only the special linear combination of f ( k ) n,m shown in (31) defines generalized symmetries of (6), but also any of the functions f ( k ) n,m . m -direction Let us consider the relation ( T n − p n,m = ( T m − q n,m , (32)where the functions p n,m , q n,m depend on n, m, u n + i,m + j and T n , T m are the shift operatorsin the n - and m -directions: T n h n,m = h n +1 ,m , T m h n,m = h n,m +1 . This relation is calledthe conservation law of the discrete equation (12) if (32) is identically satisfied on thesolutions of (12). By using (12) we can rewrite p n,m , q n,m in terms of n, m and thefunctions u n + i,m , u n,m + j (33)only, and we will represent them in such a form. In case of the m -direction, p n,m has theform p n,m = p n,m ( u n,m + k , u n,m + k − , . . . , u n,m + k ) , k ≥ k . This function p n,m can be called the conserved density by analogy with the discrete-differential case.For k > k we shall obtain conserved densities p n,m such that ∂ p n,m ∂u n,m + k ∂u n,m + k = 0 or all n, m , then the number k − k can be called the order of this conservation law, seee.g. [23]. If k = k and p n,m is not constant, then the conservation law is not trivial, andits order is equal to 0. Conservation laws of different orders are essentially different.Conservation laws for (6) are constructed in [15] by using the L − A pair (9, 10).However, there is there only a way of construction and a few conservation laws. It isdifficult to trace on that way the structure of conservation laws and extract autonomousamong them. Here we solve the problem by using the master symmetry (24). Such a wayof construction of conservation laws is apparently new.It is known in the discrete-differential case that, differentiating a conservation law invirtue of the master-symmetry, one obtains new conservation laws, see e.g. [23]. Here weshow that the same is true for the discrete conservation laws (32). We demonstrate thisin detail by example of the discrete equation (21) and its master symmetry (24). Then,as in previous section, we pass to equation (6) by choosing τ = 0.It is easy to check that the following functions p (1) n,m = A n ( τ )( u n,m +1 − u n,m + 1) , q (1) n,m = − A n ( τ ) u n,m (34)define a conservation law of (21) in the m -direction. Using it and the master symmetry(24), we can construct a hierarchy of conservation laws for equation (21):( T n − p ( k ) n,m = ( T m − q ( k ) n,m , k ≥ , (35)all of which will not depend explicitly on m . We do this by induction, using the propertythat ( T n − D τ p ( k ) n,m = ( T m − D τ q ( k ) n,m (36)is also a conservation law, where D τ is the total derivative in virtue of the master symmetry(24). The master symmetry (24) is one of generalized symmetries of (21) in the m -direction. For this reason, the operator D τ commutes with T m automatically, while itcommutes with T n on solutions of the discrete equation (21).New conservation law (36) depends on m explicitly. To eliminate m we will use thefact that one can add to both sides of the conservation law (32) a function of the form( T n − T m − h n,m and get a new conservation law defined by:˜ p n,m = p n,m + ( T m − h n,m , ˜ q n,m = q n,m + ( T n − h n,m . (37)Besides, we use the fact that p ( k ) n,m are also conserved densities for the differential-differenceequation (23): D t p ( k ) n,m = ( T m − r ( k ) n,m . (38)This is so, as (38) is true for k = 1 with r (1) n,m = A n ( τ )( u n,m +1 − u n,m − u n,m − + 1) , and it is known from the differential-difference case that, if p ( k ) n,m is a conserved densitiesof (23), then the function D τ p ( k ) n,m is also its conserved density. et us suppose that the functions p ( k ) n,m , q ( k ) n,m , r ( k ) n,m do not explicitly depend on m forsome k ≥ . Then the total derivative D τ p ( k ) n,m has a linear dependence on m , namely: D τ p ( k ) n,m = ( m − D t p ( k ) n,m + . . . . (39)As (38) takes place, we can use transformation (37) with h n,m = − ( m − r ( k ) n,m and getas a result: p ( k +1) n,m = D τ p ( k ) n,m − ( T m − m − r ( k ) n,m ] , (40) q ( k +1) n,m = D τ q ( k ) n,m − ( T n − m − r ( k ) n,m ] . (41)The function p ( k +1) n,m is a new conserved density for the discrete equation (21) and for itssymmetry (23) and it does not explicitly depend on m .Let us explain now how to construct the function r ( k +1) n,m and why it has no explicitdependence on m . We also give a more simple construction scheme for the functions p ( k ) n,m ,which provides an important information about their structure, as well as the second morerigorous justification of why these functions are conserved densities of (23).The function v n,m = A n ( τ )( u n,m +1 − u n,m + 1) (42)satisfies the equations ∂ t v n,m = v n,m ( v n,m +1 − v n,m − ) , (43) ∂ τ v n,m = v n,m (( m + 2) v n,m +1 + v n,m − ( m − v n,m − ) . (44)This is nothing but the Volterra equation and its master-symmetry [20]. Relation (42) isa slight non-autonomous generalization of the well-known discrete Miura transformation.It transforms the problem of construction of p ( k ) n,m and r ( k ) n,m into the well-known problemfor the Volterra equation. In particular, the initial conserved density p (1) n,m takes the form p (1) n,m = v n,m and it is the common density for all generalized symmetries of the Volterraequation (43). For this reason, it can be strictly proved that for all k the functions D kτ p (1) n,m are conserved densities for (43), see [23, Theorem 20].The function r (1) n,m takes the form r (1) n,m = v n,m v n,m − . All the functions p ( k ) n,m and r ( k ) n,m can also be expressed in terms of v n,m + j , i.e. relations (38) turn into conservation laws ofthe Volterra equation (43). The structure of these conservation laws is described by thefollowing lemma: Lemma 1.
For any k ≥ , the function p ( k ) n,m is an autonomous and homogeneous poly-nomial of the degree k and it is of the form: p ( k ) n,m = P ( k ) ( v n,m , v n,m +1 , . . . , v n,m + k − ) , ∂ P ( k ) ∂v n,m ∂v n,m + k − = 0 . (45) The function r ( k ) n,m is also an autonomous and homogeneous polynomial of degree k + 1 andit is of the form: r ( k ) n,m = R ( k ) ( v n,m − , v n,m , v n,m +1 , . . . , v n,m + k − ) . (46)Let us recall that, for any k ≥
1, such two functions define a conservation law of order k − p ( k ) n,m and r ( k ) n,m are autonomous in the sense that they do not explicitly depend neither on n nor on m . ketch of proof. Lemma 1 is true for k = 1. Let us suppose that it is true for a number k ≥ k + 1. We will use the same formula (40) for the construction of p ( k +1) n,m . In this case, one easily can check that this function satisfies the assertions ofthe lemma. The function p ( k +1) n,m is the next conserved density of (43). So there existsa function r ( k +1) n,m satisfying the relation (38) and depending on the functions v n,m + j . Iteasily can be constructed directly from (38), see e.g. [23]. Moreover, the left hand side of(38) is an autonomous homogeneous polynomial of v n,m + j of the degree k + 2. If we lookfor r ( k +1) n,m as a homogeneous polynomial, then it exists and is unique and autonomous.The resulting function satisfies Lemma 1.If in both functions p ( k ) n,m and r ( k ) n,m we replace the functions v n,m + j by u n,m + j , using(42), we get a conservation law for the symmetry (23), and its order will be k , see [23,Theorem 18]. It is clear that the so-constructed functions p ( k ) n,m and r ( k ) n,m do not explicitlydepend on m . As the function p ( k ) n,m in (45) is the homogeneous polynomial of degree k ,then its structure in terms of u n,m + j is: p ( k ) n,m = A kn ( τ ) ˆ P ( k ) ( u n,m , u n,m +1 , . . . , u n,m + k ) , (47)where ˆ P ( k ) is an autonomous polynomial. The dependence on n and τ is defined here bythe multiplier A kn ( τ ) only.Now we can show that the function q ( k +1) n,m has no explicit dependence on m , and thestructure of q ( k ) n,m is similar to (47). Lemma 2.
For any k ≥ , the function q ( k ) n,m is of the form q ( k ) n,m = A kn ( τ ) ˆ Q ( k ) ( u n,m , u n,m +1 , . . . , u n,m + k − ) , (48) where ˆ Q ( k ) is an autonomous polynomial. The function q ( k +1) n,m can be constructed by thefollowing recurrence formula: q ( k +1) n,m = ∂q ( k ) n,m ∂τ + A n ( τ ) k − X j =0 ( u n,m + j − j + 2) u n,m + j +1 − ju n,m + j − ) ∂q ( k ) n,m ∂u n,m + j . (49) Proof.
It follows from relations (39) and (41) that the function q ( k +1) n,m has a lineardependence on m : q ( k +1) n,m = ( m − W ( k ) n,m + Z ( k ) n,m , W ( k ) n,m = D t q ( k ) n,m − ( T n − r ( k ) n,m . Relation (35) with k replaced by k + 1 and the fact that p ( k +1) n,m does not depend on m imply that ( T m − W ( k ) n,m = 0 on the solutions of (21).The function W ( k ) n,m can be expressed in terms of n, τ and u n,m + j only. This is obviousfor the function D t q ( k ) n,m and it is true for r ( k ) n,m in virtue of (42) and (46). Definition (42)of v n,m and the discrete equation (21) imply v n +1 ,m = A n ( τ )( u n,m +1 + 1)( u n,m − , (50) herefore, the function T n r ( k ) n,m = R ( k ) ( v n +1 ,m − , v n +1 ,m , . . . , v n +1 ,m + k − )also can be expressed so. It is important that the dependence on u n,m + j in W ( k ) n,m ispolynomial.Such a function W ( k ) n,m satisfies ( T m − W ( k ) n,m = 0 if and only if it does not depend on u n,m + j , i.e. W ( k ) n,m = η ( k ) n ( τ ). This function equals zero if u n,m + j = 1 for all j , therefore, W ( k ) n,m ≡ . Now we get for q ( k +1) n,m the formula q ( k +1) n,m = ( D τ − ( m − D t ) q ( k ) n,m which can berewritten as (49). The structure (48) for q ( k +1) n,m follows from (22,34) and the recurrenceformula (49).In this way we get the following explicit formulae: p (1) n,m = v n,m , q (1) n,m = − A n ( τ ) u n,m , r (1) n,m = v n,m v n,m − , (51) p (2) n,m = v n,m (2 v n,m +1 + v n,m ) ,q (2) n,m = − A n ( τ )( u n,m +1 u n,m − u n,m +1 − u n,m ) ,r (2) n,m = 2 v n,m v n,m − ( v n,m +1 + v n,m ) , (52) p (3) n,m = 2 v n,m (3 v n,m +2 v n,m +1 + 3 v n,m +1 v n,m + 3 v n,m +1 + v n,m ) ,q (3) n,m = − A n ( τ )[3( u n,m − u n,m +2 u n,m +1 + u n,m +1 u n,m − u n,m +2 − u n,m +1 − u n,m ) + 16 u n,m ] ,r (3) n,m = 6 v n,m v n,m − ( v n,m +2 v n,m +1 + 2 v n,m +1 v n,m + v n,m +1 + v n,m ) , (53)with v n,m given by (42), illustrating the construction scheme described above.When τ = 0, the discrete equation (21) turns into (6) and its conservation laws turninto conservation laws of (6). As A n (0) = β n , then we get the following result for theconservation laws of (6): Theorem 2.
For any k ≥ , the discrete equation (6) possesses a conservation law (35) of the order k defined by functions of the form: p ( k ) n,m = β nk ˆ P ( k ) ( u n,m , u n,m +1 , . . . , u n,m + k ) , (54) q ( k ) n,m = β nk ˆ Q ( k ) ( u n,m , u n,m +1 , . . . , u n,m + k − ) , (55) where the polynomials ˆ P ( k ) and ˆ Q ( k ) do not explicitly depend neither on n nor on m . In Section 2 we have considered the autonomous discrete equation (6) possessing an L − A pair and hierarchies of generalized symmetries and conservation laws in the m -direction.All those objects are, however, essentially non-autonomous. Symmetries, conservation aws and L − A pairs of autonomous discrete equations we consider here will be au-tonomous, and those equations will have hierarchies of generalized symmetries and con-servation laws in both directions n and m .It has been shown in [6] that the discrete equation (3), which has the periodic coeffi-cient α n , should have hierarchies of generalized symmetries and conservation laws in bothdirections n and m . In case of conservation laws, this was shown by using an L − A pair.In case of symmetries, we studied some particular cases.As we are interested in the autonomous equations, we are going to consider the inter-section of equations (3) and (6). It follows from (7) that β N = 1 . So, we will consider thefollowing equations( u n,m +1 + 1)( u n,m −
1) = β N ( u n +1 ,m +1 − u n +1 ,m + 1) , (56)where β NN = 1 , N ≥ N , we consider here the primitiveroots of unit. It is clear that β = 1, and this case is well-known, see (1). If N >
1, then β NN = 1 , β jN = 1 for all 1 ≤ j < N. (57)In particular, β = 1 , β = − , β = − ± i √ , β = ± i, (58)i.e. in the last two cases one has two primitive roots corresponding to the signs + and − .For any N >
4, at least two primitive roots exist, which are given by β N = exp( ± iπ/N ).So, we consider below the series of equations (56), such that β N are the primitive rootsof unit.Currently we know only one similar series of integrable discrete equations [9]. Thoseequations are Darboux integrable and of the Burgers type, and the minimal order of theirfirst integrals may be arbitrarily high. Equations of the series (56) are integrable by theinverse scattering method.For equation (56,57)with N = 2 we have β = −
1, i.e. it reads:( u n,m +1 + 1)( u n,m −
1) = − ( u n +1 ,m +1 − u n +1 ,m + 1) . (59)This is the most interesting example in the series, as it has real coefficients. It was foundin [4], where the authors searched discrete equations on the square lattice, using as ageneralized symmetry five-point differential-difference equations obtained in the recentsymmetry classification [12, 13]. m -direction Here we construct autonomous generalized symmetries and conservation laws in the m -direction for equations (56,57), using the results of Section 2.In Theorem 1 we constructed symmetries (31), where an explicit dependence on n was given by the multiplier β nk . It follows from this theorem that equations (56,57) haveinfinitely many autonomous generalized symmetries in the m -direction, which are givenby (31) with k = N, N, N, . . . . orollary 1. For any N ≥ , the discrete equation (56,57) has autonomous generalizedsymmetries in the m -direction given by (31,17,29) with k = κN, κ ∈ N . For equation (59) the simplest autonomous generalized symmetry in the m -directionis given by ∂ t u n,m = c , f (2) n,m + 4 c , f (1) n,m . (60)We find from (29) that c , = 1 , c , = − f (1) n,m , f (2) n,m ,we get the following explicit formula: ∂ t u n,m = ( u n,m − (cid:2) ( u n,m +1 − u n,m +2 + u n,m ) − ( u n,m − − u n,m + u n,m − ) − u n,m +1 − u n,m − ) (cid:3) . (61)This symmetry was first found in [4].In Theorem 2 we constructed conservation laws for equation (6) given by (54,55),where an explicit dependence on n was given by the multiplier β nk . It follows from thistheorem that equations (56,57) have infinitely many autonomous conservation laws in the m -direction, and they are given by (54,55) with k = N, N, N, . . . .
Corollary 2.
For any N ≥ , the discrete equation (56,57) has infinitely many au-tonomous conservation laws, and their orders are multiples of the number N . In the case of equation (59), the simplest autonomous conservation law, taken from(52), has the order 2 and is given by: p (2) n,m = v n,m (2 v n,m +1 + v n,m ) , v n,m = ( u n,m +1 − u n,m + 1) ,q (2) n,m = − u n,m +1 u n,m − u n,m +1 − u n,m ) . (62) n -direction Let us consider generalized symmetries in the n -direction. The discrete equation (12) hasa generalized symmetry in the n -direction: ∂ θ u n,m = ζ n,m ( u n + ν,m , u n + ν − ,m , . . . , u n − ν,m ) , ν > , (63)if (12) and (63) are compatible, i.e. the equation D θ Φ n,m = X i,j ∈{ , } ζ n + i,m + j ∂ Φ n,m ∂u n + i,m + j = 0 (64)is identically satisfied on the solutions of (12). It is natural to suppose that there existnumbers n , m , n , m , such that ∂ζ n ,m ∂u n + ν,m = 0 , ∂ζ n ,m ∂u n − ν,m = 0 . (65)The number ν is called the order of symmetry (63). The form of equation (63) is symmetricas in Section 2.1 for the same reason.In [6, Section 4] two theorems for equation (5) and its ”nondegenerate” symmetries oforders 1 and 2 have been proved. Here we prove analogues theorems for equation (6) andits symmetries of orders 1,2 and 3 without the use of any non-degeneracy conditions. heorem 3. The following two statements take place:1. If equation (6) has a generalized symmetry (63) in the n -direction of order N , suchthat ≤ N ≤ , then β N = 1 , i.e. equation (6) has the form (56) ;2. Equation (56) with ≤ N ≤ and β N being a primitive root of unit has a generalizedsymmetry of the order N and does not have generalized symmetries of lower orders. Sketch of Proof.
For the construction of generalized symmetries for the discreteequations, we use a method developed in [5, 17, 18], see [5] for its most advanced version.The compatibility condition (64) is a functional equation for the unknown function ζ n,m .The method allows one to get consequences in the form of partial differential equationsfor ζ n,m , using so-called annihilation operators introduced in [14].1. If equation (6) has a generalized symmetry (63) of the order N , with 1 ≤ N ≤
3, thenthe simplest differential consequences of (64) have the form:( β N − ∂ζ n,m ∂u n + N,m = 0 , ( β N − ∂ζ n,m ∂u n − N,m = 0 , (66)and these relations must be satisfied for all n, m . Conditions (65,66) imply β N = 1.2. For equation (56) with 1 ≤ N ≤ β N being a primitive root of unit, we look forsymmetries of the form (63) with ν = N and we use no restriction like (65). We find thefollowing generalized symmetries.In case N = 1 it has the form: ∂ θ u n,m = ( u n,m − (cid:18) a n +1 u n +1 ,m + u n,m − a n u n,m + u n − ,m (cid:19) . (67)Here a n = b + cn , where b, c are arbitrary constants.In case N = 2 it is of the form: ∂ θ u n,m =( u n,m − T n − (cid:18) a n +1 ( u n +1 ,m + u n,m ) U n,m + a n ( u n − ,m + u n − ,m ) U n − ,m (cid:19) ,U n,m =( u n +1 ,m + u n,m )( u n,m + u n − ,m ) − u n,m − . (68)The function a n is given by a n = b n + cn , where c is a constant and b n +2 ≡ b n is anarbitrary two-periodic function on n . It can be represented as b n = b (1) + b (2) ( − n withtwo arbitrary constants b (1) , b (2) .In case N = 3 the generalized symmetry has the form: ∂ θ u n,m =( u n,m − T n − (cid:18) a n +2 V n,m U n,m + a n W n,m U n − ,m + ( T n + 1) a n +1 Z n,m U n − ,m (cid:19) ,V n,m = β ( u n +1 ,m −
1) + u n +1 ,m ( u n +2 ,m − u n − ,m ) − u n +2 ,m u n − ,m + 1 ,W n,m = β ( u n − ,m −
1) + u n − ,m ( u n − ,m + u n − ,m ) + u n − ,m u n − ,m + 1 ,Z n,m =( u n +1 ,m + u n,m )( u n − ,m + u n − ,m ) ,U n,m = β ( u n +1 ,m − u n,m + u n − ,m ) + β ( u n,m − u n +2 ,m + u n +1 ,m )+( u n +1 ,m u n,m + 1)( u n +2 ,m + u n − ,m ) + ( u n +1 ,m + u n,m )( u n +2 ,m u n − ,m + 1) . (69) ere β is any of two primitive roots shown in (58). The function a n is given by a n = b n + cn , where c is a constant and b n +3 ≡ b n is an arbitrary three-periodic function. It canbe represented as b n = b (1) + b (2) β n + b (3) β n , where b (1) , b (2) , b (3) are arbitrary constants.We see that such an equation (56) has a generalized symmetry of the order N in allthree cases. We also can see that generalized symmetries of lower orders do not exist incases N = 2 , ∂ζ n,m ∂u n + N,m ≡ ∂ζ n,m ∂u n − N,m ≡ ζ n,m ≡ . In case N = 2 we have the only primitive root β = −
1, and formulae for b n incases N = 2 and N = 3 are analogous. We have the following important consequence ofTheorem 3 for autonomous equations (56): Corollary 3.
Any of equations (56) with ≤ N ≤ and β N being a primitive root of unithas an autonomous generalized symmetry of the order N , given by (67-69) with a n ≡ ,and does not have autonomous generalized symmetries of lower orders. These autonomous symmetries exemplify integrable differential-difference equationswith one continuous variable θ N and one discrete variable n , while the parameter m is notessential. The symmetry (67) with a n ≡ a n ≡ N = 2, the subcases a n ≡ a n ≡ ( − n of (68) are compatible, i.e. we have heretwo commuting generalized symmetries of the second order. When N = 3, the subcases a n ≡ a n ≡ β n and a n ≡ β n of (69) are compatible, i.e. we have three commutinggeneralized symmetries of the third order.Equation (67) with a n ≡ n is a known master symmetry for the differential-differenceequation (67) with a n ≡
1, see [3]. It is important for us here that in all the three cases N = 1 , , a n ≡ n plays the role of the master symmetryfor discrete equation (56) with β N being a primitive root of unit. Compared with Section2.1, these master symmetries are more convenient to use, as they do not depend explicitlyon the time of the master symmetry.Let us denote by ∂ ˆ θ N u n,m = Ξ ( N ) n,m (70)the generalized symmetry of discrete equation (56) with N = 2 or N = 3 correspondingto a n ≡ n in (68) or (69), which plays the role of the master symmetry. We show how toconstruct generalized symmetries of higher orders: ∂ ˜ θ k,N u n,m = Υ ( k,N ) n,m , k ∈ N , (71)starting from symmetries (68) or (69) with periodic coefficient a n ≡ b n , which correspondto (71) with k = 1. The order of such a symmetry (71) will be equal to kN . The righthand sides of these symmetries are constructed by using the recurrence formula:Υ ( k +1 ,N ) n,m = ad Ξ ( N ) n,m Υ ( k,N ) n,m = D ˆ θ N Υ ( k,N ) n,m − D ˜ θ k,N Ξ ( N ) n,m = kN X j = − kN Ξ ( N ) n + j,m ∂ Υ ( k,N ) n,m ∂u n + j,m − N X j = − N Υ ( k,N ) n + j,m ∂ Ξ ( N ) n,m ∂u n + j,m . (72)Here D ˆ θ N and D ˜ θ k,N are the total derivatives in virtue of (70) and (71). .3 Comparison of the case N = 2 with a known example.Relation with relativistic Toda type equations Let us consider in more detail the discrete equation (59). We know the only autonomousexample analogous to (59) from the viewpoint of generalized symmetry structure. It wasfound in [8] and then studied in [7].This example reads: u n +1 ,m +1 ( u n,m − u n,m +1 ) − u n +1 ,m ( u n,m + u n,m +1 ) + 2 = 0 . (73)Its generalized symmetries of the first and second order in the m -direction are ∂ t u n,m = ( − n u n,m +1 u n,m − + u n,m u n,m +1 + u n,m − , (74) ∂ t u n,m = ( u n,m +2 − u n,m − )( u n,m +1 − u n,m )( u n,m − u n,m − )( u n,m + u n,m − )( u n,m +1 + u n,m − ) ( u n,m +2 + u n,m ) . (75)The simplest symmetry in the n -direction has the second order: ∂ ˜ θ u n,m = ( u n +1 ,m u n,m − u n,m u n − ,m − b n +1 u n +2 ,m − b n u n − ,m ) , (76)where b n +2 ≡ b n is an arbitrary two-periodic function, i.e. b n = b (1) + b (2) ( − n witharbitrary constant coefficients b (1) , b (2) . In case of the discrete equation (59), a symmetryanalogous to (74) reads: ∂ t u n,m = ( − n ( u n,m − u n,m +1 − u n,m − ) . (77)Generalized symmetries of equation (59) similar to (75) and (76) are (61) and (68) with a n ≡ b n .These autonomous discrete equations (59) and (73) have hierarchies of autonomousgeneralized symmetries in both directions. The orders of those autonomous symmetries areeven and, as we see from above examples, the simplest autonomous generalized symmetriesin both directions have the order 2.In [7] we showed that the differential-difference equation (76) was equivalent to asystem of Tsuchida [21], see details below. However, in the class of five-point differential-difference equations, this is an interesting integrable example as itself. Equation (68) with a n ≡ b n seems a new integrable example of the five-point differential-difference equation.In [7] we outlined that equation (76) was similar to relativistic Toda type equations,see the review articles [2, Sections 4.2,4.3] and [23, Secion 3.3], according to its generalizedsymmetry properties. In [6] we demonstrated such a relation with relativistic Toda typeequations in a more explicit way for a non-autonomous equation. Here, following [6], wewill demonstrate such an explicit relation for the equations (76) and (68) with a n ≡ b n .Let us first consider the generalized symmetry (76). For any fixed m we introduce v k = u k,m , w k = u k − ,m , ς = b k , η = b k − , and rewrite (76) in the form of a system: ∂ ˜ θ v k = ( ηv k +1 − ςv k − )( w k +1 v k − v k w k − ,∂ ˜ θ w k = ( ςw k +1 − ηw k − )( v k w k − w k v k − − . (78) his is nothing but the Tsuchida system [21, (3.13)]. In any of two cases ς = 1 , η = 0 or ς = 0 , η = 1 , we introduce U k = log v k or U k = − log w k and get in any of these four cases the following relativistic Toda type equation:¨ U k = ˙ U k (cid:16) ˙ U k +1 − ˙ U k − − e U k +1 − U k + e U k − U k − (cid:17) , where we denote ˙ U k = ∂ ˜ θ U k . This is a known equation presented in [23, Section 3.3.4,(Ld3) with µ = 0 , ν = 1].Now we consider the symmetry (68) with a n ≡ b n . For any fixed m we introduce ˜ u n : u n,m = ˜ u n + ˜ u n +1 ˜ u n − ˜ u n +1 . This transformation is not invertible, but it is linearizable, i.e. not of the Miura type,in the terminology of the theoretical paper [11]. As a result we obtain the followingintegrable modification of (68) with a n ≡ b n : ∂ ˜ θ ˜ u n = (˜ u n +2 − ˜ u n )(˜ u n +1 − ˜ u n )(˜ u n − ˜ u n − )2(˜ u n +2 ˜ u n + ˜ u n +1 ˜ u n − ) − (˜ u n +2 + ˜ u n )(˜ u n +1 + ˜ u n − ) b n +1 + (˜ u n +1 − ˜ u n )(˜ u n − ˜ u n − )(˜ u n − ˜ u n − )2(˜ u n +1 ˜ u n − + ˜ u n ˜ u n − ) − (˜ u n +1 + ˜ u n − )(˜ u n + ˜ u n − ) b n . (79)Now we pass to the notations v k = ˜ u k , w k = ˜ u k − , ς = b k , η = b k − and are led to the following system: ∂ ˜ θ v k = ( v k +1 − v k )( w k +1 − v k )( v k − w k )2( v k +1 v k + w k +1 w k ) − ( v k +1 + v k )( w k +1 + w k ) η + ( w k +1 − v k )( v k − w k )( v k − v k − )2( w k +1 w k + v k v k − ) − ( w k +1 + w k )( v k + v k − ) ς,∂ ˜ θ w k = ( w k +1 − w k )( v k − w k )( w k − v k − )2( w k +1 w k + v k v k − ) − ( w k +1 + w k )( v k + v k − ) ς + ( v k − w k )( w k − v k − )( w k − w k − )2( v k v k − + w k w k − ) − ( v k + v k − )( w k + w k − ) η. (80)In any of two subcases ς = 2 , η = 0 or ς = 0 , η = 2 , we introduce U k = v k or U k = w k and get in any of these four cases the relativistic Toda type equation:¨ U k = ˙ U k ˙ U k − ( U k − U k − ) − ˙ U k +1 ( U k − U k +1 ) + 1 U k − U k − + 1 U k − U k +1 ! , where we denote ˙ U k = ∂ ˜ θ U k . This is a known equation presented in [23, Section 3.3.3,(L2) with r ( x, y ) = ( x − y ) / a n ≡ b n and (76) have appeared in [1]. .4 Conservation laws in the n -direction As equation (56) with N = 1 is well-known, we consider here the cases N = 2 , n -direction, which will be autonomous.The relation ( T n − p n,m = ( T m − q n,m , (81)with ˇ p n,m , ˇ q n,m depending on n, m, u n + i,m + j , is called the conservation law of the discreteequation (56,57) if it is identically satisfied on the solutions of (56,57). By using equation(56), we can rewrite ˇ p n,m , ˇ q n,m in terms of n, m and of the functions (33) only, and theywill be presented in such a form. In case of the n -direction, ˇ q n,m will be of the form:ˇ q n,m = ˇ q n,m ( u n + k ,m , u n + k − ,m , . . . , u n + k ,m ) , k ≥ k . The function ˇ q n,m can be called the conserved density by analogy with the discrete-differential case.For k > k we will obtain the densities ˇ q n,m , such that ∂ ˇ q n,m ∂u n + k ,m ∂u n + k ,m = 0 for all n, m, then the number k − k will be called the order of such a conservation law. If k = k andthe function ˇ q n,m is not constant, then (81) will be the nontrivial zero-order conservationlaw. Conservation laws of different orders are essentially different.For the construction of conservation laws in the cases N = 2 ,
3, we use the mastersymmetries (70). They are simpler than in Section 2.1 in the sense that do not dependexplicitly on the time ˆ θ N of master symmetry. However, such a way of construction ofconservation laws is new even in ˆ θ N -independent case.We construct a hierarchy of n -independent conservation laws for equations (56,57)with N = 2 ,
3: ( T n − p ( k ) n,m = ( T m − q ( k ) n,m , k ≥ . (82)Similarly to Section 2.2, we can construct the conservation laws by induction, using aproperty that ( T n − D ˆ θ N ˇ p ( k ) n,m = ( T m − D ˆ θ N ˇ q ( k ) n,m (83)is also the conservation law. Here D ˆ θ N is the total derivative in virtue of (70).In both cases N = 2 , ∂ ˜ θ N u n,m = Ω ( N ) n,m (84)the autonomous symmetries (68) and (69) with a n ≡ u n +1 ,m +1 = ϕ ( N ) ( u n,m , u n +1 ,m , u n,m +1 ) . Then starting conservation laws read:(1 − T Nn ) T − n log ∂ϕ ( N ) ∂u n +1 ,m = ( T m −
1) log ∂ Ω ( N ) n,m ∂u n + N,m . (85) uch a conservation law is autonomous with the conserved density ˇ q (1) n,m = log ∂ Ω ( N ) n,m ∂u n + N,m , however, the conservation law (83) explicitly depends on the variable n . In Section 2.2it is explained how to remove this dependence on n from such a conservation law. Wecan do it because the master symmetry (70) has the linear dependence on n , the functionˇ q (1) n,m is also a conserved density of equation (84), and we can add a function of the form(37) to both parts of conservation law (83).If N = 2 we can rewrite the conservation law (85) in the form:( T m − q ( k ) n,m = ( T n − p ( k ) n,m , (86)where k = 1 andˇ q (1) n,m = log ( u n +1 ,m − u n,m − U n +1 ,m , ˘ p (1) n,m = log u n,m + 1 u n,m +1 − , (87)while U n,m is defined in (68). The form of this conservation law is specific, but it is aparticular case of (82) with ˇ p ( k ) n,m = ( T n + 1)˘ p ( k ) n,m . By using the master symmetry (70), weget the next conservation law which can be made autonomous and rewritten in the samespecific form (86). It is given byˇ q (2) n,m = − ( u n +4 ,m + u n +3 ,m )( u n +2 ,m − u n +1 ,m + u n,m ) U n +3 ,m U n +1 ,m + u n +1 ,m − U n +1 ,m , ˘ p (2) n,m = ( u n +2 ,m + u n +1 ,m )( u n,m − U n +1 ,m . The orders of these conservation laws are equal to 2 and 4. These conservation laws wereconstructed in [6] in a little bit different form by using an L − A pair.If N = 3 we can rewrite the conservation law (85) in the form:( T m − q ( k ) n,m = ( T n − p ( k ) n,m , (88)where k = 1, ˘ p (1) n,m is defined by (87) as before, andˇ q (1) n,m = log ( u n +2 ,m − u n +1 ,m − u n,m − U n +1 ,m , with U n,m defined in (69). The specific form (88) is also a particular case of (82). Thisconservation law is autonomous and has the order 3. By using the master symmetry (70),we can get the next conservation law which can be made autonomous and rewritten inthe same specific form (88) with k = 2. It will be of the order 6. It is, however, toocumbersome to show it here.It should be remarked that, using the non-autonomous generalized symmetries (68)and (69) with a n ≡ b n and the same formula (85) for starting conservation laws, we cantry to get non-autonomous conservation laws. However, nothing new arises because theoperator T m − n . .5 Autonomous L − A pairs Here we construct autonomous L − A pairs for the discrete equations of series (56,57),using the non-autonomous L − A pair (8-10) for equation (6). In the case N = 1, one has β = 1 and this L − A pair is obviously autonomous.Applying the operator T N − n to the first of equations (8), we get a consequence:Ψ n + N,m = L (1 ,N ) n,m Ψ n,m , Ψ n,m +1 = L (2) n,m Ψ n,m , (89)where N ≥ L (1 ,N ) n,m = L (1) n + N − ,m L (1) n + N − ,m . . . L (1) n +1 ,m L (1) n,m , (90)and β is replaced by β N in matrices L (1) n,m , L (2) n,m . The compatibility condition for (89) is L (1 ,N ) n,m +1 L (2) n,m = L (2) n + N,m L (1 ,N ) n,m . (91)As β NN = 1, we see that the multiplier β nN is not changed not only in the matrix L (1 ,N ) n,m +1 but also in L (2) n + N,m . That is why it plays no role in relation (91), and we can replace β nN by a constant. It can be removed by scaling the spectral parameter λ , and we get thefollowing relation Λ (1 ,N ) n,m +1 Λ (2) n,m = Λ (2) n + N,m Λ (1 ,N ) n,m (92)defined by the matrices: Λ (1 ,N ) n,m = Θ ( N − n,m Θ ( N − n,m . . . Θ (1) n,m Θ (0) n,m , (93)Θ ( k ) n,m = λβ kN ( u n + k,m + 1) − u n + k,m − u n + k,m +1 u n + k,m − ! , (94)Λ (2) n,m = (cid:18) − λ ( u n,m + 1)( u n,m +1 − (cid:19) . (95)For any N ≥ N = 2 , , N ≥
2, and we get forequation (56-57) the following autonomous L − A pair:Ψ n + N,m = Λ (1 ,N ) n,m Ψ n,m , Ψ n,m +1 = Λ (2) n,m Ψ n,m . (96) We have constructed a series of autonomous integrable discrete equations (56) with β N being a primitive root of unit. Equation (56) with N = 1 is well-known. Equations(56,57) have hierarchies of autonomous generalized symmetries and conservation laws inboth directions as well as autonomous L − A pairs.Symmetries and conservation laws of (56,57) were constructed by using the mastersymmetries. Those master symmetries arise as generalized symmetries of the discreteequations (56,57) and linearly depend on one of two discrete variables, see Sections 2.1,3.2.One of them has also an explicit dependence on its time. In the case of conservation laws, uch a construction scheme seems to be new. Entering the time of master symmetry intothe corresponding discrete equation also seems to be a new point in the method.The following hypothesis on the generalized symmetry structure should be true: Hypothesis.
Any of the autonomous equations (56,57) has an infinite hierarchy of au-tonomous generalized symmetries in both directions of the orders κN, κ ≥ . The minimalpossible order of an autonomous generalized symmetry in any direction is equal to N . We do not know any examples of this kind in case of the hyperbolic partial differentialequations which are analogous to discrete equations of the form (12). We justified thishypothesis by results presented in Sections 3.1,3.2. Corollary 1 states the existence ofautonomous generalized symmetries of orders κN in the m -direction. In Theorem 3we have proved, in particular, that equations (56,57) with N = 2 , N in the n -direction and do not have autonomoussymmetries of lower orders. We can prove a similar result for the m -direction: Theorem 4.
Equations (56,57) with N = 2 , have autonomous generalized symmetriesof the order N in the m -direction and do not have autonomous symmetries of lower orders. The proof is similar to one of Theorem 3, but it is too cumbersome to demonstratehere.This hypothesis is important from the viewpoint of the generalized symmetry methodfor discrete equations when we classify discrete equations, using the existence of general-ized symmetries of a fixed order [8,18]. Since the minimal order of autonomous generalizedsymmetry may be arbitrarily high, we cannot classify in this way all the integrable discreteequations (12) in the autonomous case.As our results show, the hierarchies of autonomous conservation laws should havea similar structure. Any of the autonomous equations (56,57) should have an infinitehierarchy of autonomous conservation laws of the orders κN, κ ≥ , in both directions.This is true in the m -direction, see Corollary 2.The case N = 2 is most interesting, as the discrete equation (59) has no complexcoefficients. We consider it in more detail in Section 3.3. For this equation (59) andits known analogue (73), we show that their second order generalized symmetries in the n -direction have a close relation to integrable differential-difference equations of the rela-tivistic Toda type. We do not know any autonomous discrete example, except for (59,73),with generalized symmetries of this kind.Finally we remark that, despite the fact that the autonomous equations (56,57) aremore or less obvious particular cases of non-autonomous equations (5), many resultsrelated to these autonomous equations are new. Acknowledgments.
RIY gratefully acknowledges financial support from a RussianScience Foundation grant (project 15-11-20007).
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