Discrete exponential type systems on a quad graph, corresponding to the affine Lie algebras A (1) N−1
aa r X i v : . [ n li n . S I] M a y Discrete exponential type systems on a quad graph,corresponding to the affine Lie algebras A (1) N − . I T Habibullin , , A R Khakimova Institute of Mathematics, Ufa Federal Research Centre, Russian Academy ofSciences, 112, Chernyshevsky Street, Ufa 450008, Russian Federation Bashkir State University, 32, Validy Street, Ufa 450076 , Russian FederationE-mail: [email protected] , [email protected] Abstract.
The article deals with the problem of the integrable discretization of thewell-known Drinfeld-Sokolov hierarchies related to the Kac-Moody algebras. A class ofdiscrete exponential systems connected with the Cartan matrices has been suggestedearlier in [1] which coincide with the corresponding Drinfeld-Sokolov systems in thecontinuum limit. It was conjectured that the systems in this class are all integrableand the conjecture has been approved by numerous examples. In the present article westudy those systems from this class which are related to the algebras A (1) N − . We foundthe Lax pair for arbitrary N , briefly discussed the possibility of using the method offormal diagonalization of Lax operators for describing a series of local conservationlaws and illustrated the technique using the example of N = 3. Higher symmetriesof the system A (1) N − are presented in both characteristic directions. Found recursionoperator for N = 3. It is interesting to note that this operator is not weakly nonlocal. iscrete exponential type systems on a quad graph
1. Introduction
Exponential type systems of hyperbolic equations in partial derivatives v ix,y = exp( a i v + a i v + · · · + a iN v N ) , ≤ i ≤ N (1.1)are actively discussed in literature due to their applications in the field theory and theother areas of physics, as well as in geometry, the integrability theory etc.Exponential system in the form of the two dimensional Toda lattice v ix,y = exp( v i +1 − v i + v i − ) (1.2)has appeared many years ago within the frame of the Laplace cascade integration method(see [2]). In the context of the soliton theory this equation has been rediscovered in [3],[4]. Due to the works by Mikhailov, Olshanetsky, Perelomov, Leznov, Savel’ev, Shabat,Smirnov, Wilson, Yamilov, Drinfeld, Sokolov and many others mathematical theory ofthe exponential systems has been developed and nowadays (see [4]-[11]) it is well-knownthat in the case when the coefficient matrix A = { a i,j } coincides with the Cartan matrixof a semi-simple Lie algebra then (1.1) admits a complete set of non-trivial integralsin both characteristic directions and therefore is integrable in the sense of Darboux.Similarly, if A is the generalized Cartan matrix of an affine Lie algebra then the system(1.1) can be studied by means of the inverse scattering transform method [4]-[11].Inspired by Ward (see [12]) the problem of finding integrable discrete analogs ofequation (1.1), which are discrete in both independent variables, has been intensivelystudying since the middle of 90’s. We remark that the particular 1 + 1 dimensionalcase, obtained by setting x = y was successfully investigated in [13]. An effective way toconstruct the discrete version of (1.2) based on the discretization of the bilinear equationof the Toda lattice, was suggested by Hirota [14] and Miwa [15]. Hirota-Miwa equation isa universal soliton equation from which a large variety of integrable discrete models canbe derived by symmetry constraints (see [16]). An alternative approach to discretise thetwo dimensional Toda lattice is used by Fordy and Gibbons [17], [18], where the discreteequation is derived as the superposition formula of the Backlund transformations forthe equation (1.2). A large class of the quad systems and their applications in physicsare studied in [19].In the article [1] a class of exponential type systems of discrete equations ae − u i , + u i , + u i , − u i , − b exp i − X j =1 a i,j u j , + N X j = i +1 a i,j u j , + a i,i u i , + u i , ! (1.3)has been suggested for the particular case a = b = 1. In the present article we assumethat a and b are arbitrary nonzero constants. Here the upper index j is a natural rangingfrom 1 to N . For the shifts of the sought functions u jn,m in (1.3) we use the abbreviatednotations u jn + i,m + k = u ji,k such that u j , means u jn +1 ,m and so on.For the small values of N system (1.3) was obtained in [1] from integrable cases of(1.1) by applying the method of discretization preserving integrals and symmetries. For(1.3) with A being the generalized Cartan matrix of the algebra D (2) N the Lax pair has iscrete exponential type systems on a quad graph A N and B N are Darboux integrable. These facts partially confirmour hypothesis from [1] that the quad system inherits the integrability property of thesystem (1.1) and is integrable for the Cartan matrices of both simple and affine Liealgebras.The main goal of this paper is to present the Lax pair for the quad system (1.3)associated with the algebra A (1) N − . We outlined a method for applying the formaldiagonalization technique for finding local conservation laws and higher symmetries.In other words we approve the aforementioned hypothesis for one more series of affineLie algebras. Unfortunately, our Lax pairs for A (1) N − and D (2) N are not given in terms ofthe Cartan-Weyl basis of the algebra as it was in the case of (1.1) (see [10]), so thereare problems with generalization to other algebras. Note that alternative examples ofdiscretizations of the Toda lattices related with the algebra A (1) N − are investigated in[21]-[24].Quadrilateral lattices studied in the article are very close to those suggested in[25] as discretizations of the Gelfand-Dikii hierarchy. However these two classes ofthe discrete equations differ from each other. For instance, the first members are thediscrete d’Alembert equation and, respectively discrete potential KdV equation. Thesecond members are the couplet system (4.4) and the discrete Boussinesq equation.They are not connected by the point transformations.An interesting integrable system of partial difference equations is suggested in [26]with the arbitrarily large number of the independent variables. In a particular casewhen the number of independent variables is two this system looks very similar to (4.1)and (2.17), but the systems are not equivalent (see Proposition 3 in § § t j + Nn,m = t jn +1 ,m − of the Hirota-Miwa equation which leads to a quadrilateralsystem of the form (1.3) corresponding to the algebra A (1) N − . We show that the constraint ψ j + Nn,m = λψ jn +1 ,m − imposed on the system (2.4) associated with the Hirota-Miwaequation successfully creates a Lax pair for the resulting quad system written in theform (2.1).We also give an example of a system obtained using the purely periodic condition t j + Nn,m = t jn,m . Note that various types of the periodic conditions for the Hirota-Miwaequation are studied in [27]. For example, it is shown that the restriction t j +2 n,m = t jn,m leads to a discrete version of the sine-Gordon equation. However, to our knowledge,examples of quasi-periodic reduction of the form t j + Nn,m = t jn +1 ,m − are not consideredearlier.In § λ to a specialform allowing to determine asymptotic representation of the eigenfunctions. Findingof this special form usually causes the main problem. In section 3 we found triangulartransformations reducing the Lax equations to the required form. The case N = 3 is iscrete exponential type systems on a quad graph § N = 2 and N = 3 in the particular values of the parameters a = b = 1. For the case N = 3 the found recursion operator is not weakly nonlocal.
2. Derivation of the Lax pair
In the article we deal with the quad system (1.3) corresponding to the algebra A (1) N − which can be written in terms of the variable t jn,m = exp (cid:8) − u jn,m (cid:9) as follows at , t , − t , t , = bt N , t , ,at j , t j , − t j , t j , = bt j − , t j +10 , , ≤ j ≤ N − , (2.1) at N , t N , − t N , t N , = bt N − , t , . Evidently system (2.1) can be obtained from the Hirota-Miwa equation at j , t j , − t j , t j , = bt j − , t j +10 , , −∞ ≤ j ≤ + ∞ (2.2)by imposing the quasi-periodicity closure constraint t j + N , = t j , − , N ≥ . (2.3)Note that for the simplest case N = 1 (2.2) implies the discrete version of the d’Alembertequation.Our aim is to derive the Lax pair for (2.1) from the overdetermined system of thelinear equations ψ j , = t j +11 , t j , t j +10 , t j , ψ j , − ψ j +10 , , ψ j , = ψ j , + b t j +10 , t j − , t j , t j , ψ j − , (2.4)associated with the equation (2.2). More precisely, when t jn,m solves equation (2.1) thenthe system (2.4) is compatible. However the converse is not true: the compatibility of thesystem does not imply (2.2). Nevertheless (2.4) can be used effectively for constructingthe true Lax pair for the quad system (2.1). Evidently (2.4) implies the hyperbolic typediscrete linear equation ψ j , − ψ j , − t j +11 , t j , t j +10 , t j , ψ j , + a t j , t j +11 , t j , t j +10 , ψ j , = 0 . (2.5)It is widely known that the Laplace invariants are important characteristics of thehyperbolic type discrete and continuous equations. Recall that for an equation of theform f , + b , f , + c , f , + d , f , = 0 (2.6) iscrete exponential type systems on a quad graph K and K are determined due to the rules (see [30, 31]) K = b , c , d , , K = b , c , d , . Further we will use theorem (see [31]) claiming that equation (2.6) and the equation˜ f , + ˜ b , ˜ f , + ˜ c , ˜ f , + ˜ d , ˜ f , = 0are related with one another by the multiplicative transformation f = λ ˜ f if and only ifthey have the same pair of the Laplace invariants, i.e. K = ˜ K and K = ˜ K . Proposition 1.
Under the quasi-periodicity condition (2.3) equation (2.5) and theequation ψ j + N , − ψ j + N , − t j + N +11 , t j + N , t j + N +10 , t j + N , ψ j + N , + a t j + N , t j + N +11 , t j + N , t j + N +10 , ψ j + N , = 0 (2.7)are related by the multiplicative transformation, or more precisely, by the equation ψ j , = A j ψ j + N , . (2.8) Proof . Let us first give the Laplace invariants in enlarged (not abbreviated!) form K ( n, m, j ) = t jn +2 ,m t jn +1 ,m +1 t jn +1 ,m t jn +2 ,m +1 a , K ( n, m, j ) = t j +1 n +1 ,m +1 t j +1 n,m +2 t j +1 n,m +1 t j +1 n +1 ,m +2 a . Now it is easily seen that constraint (2.3) implies K ( n + 1 , m, j ) = K ( n, m + 1 , j + N )and K ( n +1 , m, j ) = K ( n, m +1 , j + N ). These two relations due to the over-mentionedtheorem allow one to complete the proof. Proposition 2 . The coefficient A j in the relation (2.8) does not depend on any ofthe variables j, n, m . Proof . By applying the shift operator D m , acting according to the rule D m y m = y m +1 to equation (2.8), we evidently obtain ψ j , = A j , ψ j + N , . Next we replace ψ j , dueto the hyperbolic type equation (2.5) and find ψ j , + t j +11 , t j , t j +10 , t j , ψ j , − a t j , t j +11 , t j , t j +10 , ψ j , = A j , ψ j + N , . Then we get rid the function ψ j by virtue of the relation (2.8). As a result we arrive atthe equation ψ j + N , − A j , − A j , ψ j + N , − A j , A j , t j +12 , t j , t j +11 , t j , ψ j + N , + a A j , − A j , t j , − t j +12 , t j , − t j +11 , ψ j + N , = 0which should coincide with (2.7). Now we compare the corresponding coefficients ofthese two equations and by means of (2.8) we get three equations for A j A j , − = A j , , A j , = A j , , A j , − = A j , which approve that A j does not depend on n and m . iscrete exponential type systems on a quad graph ψ j , − t j +11 , t j , t j +10 , t j , ψ j , = A j ψ j + N , − t j +11 , t j , t j +10 , t j , ψ j + N − , ! . (2.9)The left hand side of (2.9) coincides with − ψ j +10 , . Let us replace the fraction by meansof the relation (2.3) and find that due to (2.4) the right hand side in (2.9) coincides with − A j ψ j + N +1 − , . Thus we have a relation ψ j +11 , = A j ψ j + N +10 , which together with (2.8) gives A j = A j +1 . Proposition 2 is proved.Let us consider quad system (2.1). It is easily verified that (2.1) can be quasi-periodically prolonged to the infinite interval −∞ < j < + ∞ by setting t j , = t j + N , such that the prolonged function t jn,m will solve equation (2.2). Then the Propositions1,2 provide the following gluing conditions ψ , = λ − ψ N , and ψ N +10 , = λψ , , where λ is an arbitrary parameter.In order to derive the Lax pair for the quad system (2.1) we impose the gluingconditions on the linear system (2.4) and find ψ j , = t j +11 , t j , t j +10 , t j , ψ j , − ψ j +10 , , ≤ j ≤ N − , (2.10) ψ N , = t N +11 , t N , t N +10 , t N , ψ N , − λψ , − and ψ , = ψ , + bλ − t , t , t , t , ψ N − , , (2.11) ψ j , = ψ j , + b t j +10 , t j − , t j , t j , ψ j − , , ≤ j ≤ N. Let us apply now the shift operator D m to the last equation in (2.10): ψ N , = t N +11 , t N , t N +10 , t N , ψ N , − λψ , . Due to the equation (2.5) taken at the value j = N we obtain after substitution t N +11 , = t , , t N +10 , = t , that ψ N , − a t N t , t N , t , ψ N , = − λψ , . By replacing ψ , from (2.10) we bring the equation to the suitable form ψ N , = − λ t , t , t , t , ψ , + λψ , + a t N , t , t N , t , ψ N , . iscrete exponential type systems on a quad graph ψ , = a t , t , t , t , ψ , + b λ − t N − , t , t N , t , ψ N − , + bλ − t , t N , t , t , ψ N , . Let us summarize the computations above and present the desired Lax pair: ψ j , = t j +11 , t j , t j +10 , t j , ψ j , − ψ j +10 , , ≤ j ≤ N − , (2.12) ψ N , = − λ t , t , t , t , ψ , + λψ , + a t N , t , t N , t , ψ N , and ψ , = a t , t , t , t , ψ , + b λ − t N − , t , t N , t , ψ N − , + bλ − t , t N , t , t , ψ N , , (2.13) ψ j , = ψ j , + b t j +10 , t j − , t j , t j , ψ j − , , ≤ j ≤ N. Surprisingly the Lax pair (2.12), (2.13) is obtained from (2.4) by changing only twoequations.
Let’s briefly discuss the periodic closure condition (see also [27]) t j + N , = t j , (2.14)for the Hirota-Miwa equation (2.2). It is easily checked that (2.14) generates a closureconstraint for ψ : ψ j + N , = ξψ j , . As a result we get a quad system at , t , − t , t , = bt N , t , ,at j , t j , − t j , t j , = bt j − , t j +10 , , ≤ j ≤ N − , (2.15) at N , t N , − t N , t N , = bt N − , t , . In this case, the corresponding reduction of linear equations (2.4) is found immediatelyΦ , = F Φ , Φ , = G Φ , (2.16)where Φ = (cid:0) ψ , ψ , . . . , ψ N (cid:1) T and F = t , t , t , t , − . . . t , t , t , t , − . . .
00 0 t , t , t , t , . . . . . . − ξ . . . t N , t , t N , t , , iscrete exponential type systems on a quad graph G = . . . ξ − bt , t N , t , t , bt , t , t , t , . . . bt , t , t , t , . . . . . . . . . bt N − , t , t N , t N , . It can be checked by a direct computation that system (2.16) does not define aLax pair for (2.15) (In contrast to the quasi-periodic case (2.3)). More precisely, theconsistency of (2.16) does not imply (2.15). However the situation is changed if we passin (2.15) to the potential variables r j , = t j , t j , : ar , = r , + r , r N , (cid:16) ar , − r , (cid:17) ,ar j , = r j , + r j − , r j +10 , (cid:18) ar j , − r j , (cid:19) , ≤ j ≤ N − ,ar N , = r N , + r N − , r , (cid:16) ar N , − r N , (cid:17) . (2.17)Now the system (2.16) with the potentials F = r , r , − . . . r , r , − . . .
00 0 r , r , . . . . . . − ξ . . . r , r N , ,G = . . . ξ − ar , − r , r N , ar , − r , r , . . . ar , − r , r , . . . . . . . . . ar N , − r N , r N − , provides the Lax pair for (2.17).
3. Formal asymptotic solutions of the direct scattering problem and thelocal conservation laws
The method of the formal diagonalization of the Lax pairs given by differential operatorsis suggested in [10]. It is based on the ideas and technique applied earlier [32] inorder to construct asymptotic solutions to the systems of differential equations with aparameter, when the parameter goes to its singular value. Let us recall that the formal iscrete exponential type systems on a quad graph Y n +1 = f n Y n , f n = ∞ X j = − f ( j ) n λ − j , (3.1)where f ( j ) n ∈ C k × k for j ≥ − A = A A A A ! , where the blocks A , A are square matrices. Here we assume that in (3.1) thecoefficient f ( − n is of one of the forms f ( − n = A ! , det A = 0 (3.2)or f ( − n = A
00 0 ! , det A = 0 . (3.3)Now our goal is to bring (3.1) to a block-diagonal form ϕ n +1 = h n ϕ n , (3.4)where h n is a formal series h n = h ( − n λ + h (0) n + h (1) n λ − + h (2) n λ − + · · · (3.5)with the coefficients having the block structure h ( j ) n = h ( j )11 h ( j )22 ! . (3.6)To this end we use the linear transformation Y n = T n ϕ n assuming that T n is also aformal series T n = E + T (1) n λ − + T (2) n λ − + · · · . where E is the unity matrix and T ( j ) n is a matrix with vanishing block-diagonal part: T ( j ) n = T ( j )12 T ( j )21 ! . iscrete exponential type systems on a quad graph Y n = T n ϕ n in (3.1) we get T n +1 h n = ∞ X j = − f ( j ) n λ − j ! T n , (3.7)where h n = ϕ n +1 ϕ − n . Let us replace in (3.7) the factors by their formal expansions:( E + T (1) n +1 λ − + · · · )( h ( − n λ + h (0) n + · · · ) = ( f ( − n λ + f (0) n + · · · )( E + T (1) n λ − + · · · ) . By comparing coefficients at the powers of λ we derive a sequence of equations h ( − n = f ( − n , (3.8) T ( k ) n +1 h ( − n + h ( k − n − f ( − n T ( k ) n = R kn , k ≥ . (3.9)Here R kn denotes terms that have already been found in the previous steps.To find the unknown coefficients T ( j ) n , we must solve linear equations, that looklike difference equations. However due to the special form of the coefficient f ( − n theseequations are linear algebraic and therefore are solved without “integration”. In otherwords T ( j ) n and h ( j ) n are local functions of the potential since depend on a finite numbersof the shifts of the functions f ( − n , f (0) n , f (1) n , etc. Indeed, equation (3.9) obviouslyimplies D n ( T ( k )12 ) A ! + p q ! − A T ( k )21 ! = R kn . Here D n is the operator shifting the argument n : D n y n = y n +1 , and p = h ( k − , q = h ( k − . Evidently this equation is easily solved and the searched matrices T ( k ) n and h ( k − n are uniquely found for any k ≥ Y n +1 ,m = ( f ( − n,m λ + f (0) n,m + · · · ) Y n,m , Y n,m +1 = G n,m ( λ ) Y n,m , (3.10)where G n,m ( λ ) is analytic at a vicinity of λ = ∞ , is the Lax pair for the nonlinear quadsystem Q ([ u jn,m ]) = 0 , i.e. Q depends on the variable u jn,m and on its shifts with respect to the variables j, n, m .Assume that function f ( − n,m has the structure (3.2). Then due to the reasoningsabove there exists a linear transformation Y n,m ϕ n,m = T − n,m Y n,m which reduces thefirst equation in (3.10) to a block-diagonal form (3.4)-(3.6). It can be checked that thistransformation brings also the second equation of (3.2) to an equation of the same blockstructure ϕ n,m +1 = S n,m ϕ n,m , where S n,m is a formal power series S n,m = S (0) n,m + S (1) n,m λ − + S (2) n,m λ − + · · · iscrete exponential type systems on a quad graph S ( j ) n,m = S ( j )11 S ( j )22 ! . Since the compatibility property of linear systems is preserved under change of variables,we have the relation S n +1 ,m h n,m = h n,m +1 S n,m which implies due to the block-diagonal structure that( D n −
1) log det( S ii ) = ( D m −
1) log det( h ii ) , i = 1 , . By evaluating and comparing the coefficients at the powers of λ we derive the sequenceof the local conservation laws.The Lax pairs considered in the article have also the second singular point λ = 0,so we briefly discuss the Lax pair represented as Y n +1 ,m = F n,m Y n,m , Y n,m +1 = ( g ( − n,m λ − + g (0) n,m + g (1) n,m λ + · · · ) Y n,m , (3.11)where F n,m = F n,m ( λ ) is analytic at a vicinity of λ = 0. We request the here the term g ( − n,m has the block structure (3.3). In this case the block-diagonalization is performedin a way very similar to the one recalled above for the singularity λ = ∞ . A (1) N − Let us apply the scheme discussed above to the Lax pair of the quad system (2.1). Tothis end we present the Lax pair (2.12), (2.13) in a matrix formΦ , = F Φ , Φ , = G Φ , (3.12)where F and G are as follows F = t , t , t , t , − . . . t , t , t , t , − . . . t , t , t , t , . . . . . . − λ t , t , t , t , λ . . . t N , t , t N , t , ,G = a t , t , t , t , . . . b λ − t , t N − , t , t N , bλ − t , t N , t , t , bt , t , t , t , . . . bt , t , t , t , . . . . . . . . . bt N − , t , t N , t N , . iscrete exponential type systems on a quad graph HY (or Φ = ¯ HY ) , where the factor H is lower (respectively ¯ H is upper) block-diagonal matrix H = H H H ! , or ¯ H = ¯ H ¯ H H !! . Here H , H , ¯ H , ¯ H are diagonal matrices, some entries of which might depend onthe spectral parameter ξ = λ N − (or ζ = λ − N − ). Examples show that these factors areeffectively found. Below, for simplicity, we illustrate all the arguments and calculationsusing the example of N = 3. Example 1.
For N = 3 the quad system (2.1) reads as at , t , − t , t , = bt , t , ,at , t , − t , t , = bt , t , ,at , t , − t , t , = bt , t , . (3.13)System (3.13) corresponds to the algebra A (1)2 and relates with the Lax pairΦ , = F Φ , Φ , = G Φ , (3.14)where F = t , t , t , t , − t , t , t , t , − − t , t , t , t , λ λ at , t , t , t , , G = at , t , t , t , b t , t , λt , t , bt , t , λt , t , bt , t , t , t , bt , t , t , t , . Let us perform the formal diagonalization (really block-diagonalization) procedure tothe system (3.14) around the singular values λ = 0 and λ = ∞ of the spectral parameter.We begin with the case λ = ∞ . By changing the variablesΦ = HY, where H = t , t , t , t , ξ − t , t , t , t , , ξ = √ λ, we reduce (3.14) to the form Y , = f Y, Y , = gY. (3.15)Here f = f ( − ξ + f (0) + f (1) ξ − , g = g (0) + g (1) ξ − + g (2) ξ − + g (3) ξ − , f ( − = −
10 1 0 , f (0) = t , t , t , t , + t , t , t , t , at , t , ( t , ) at , t , t , t , , iscrete exponential type systems on a quad graph f (1) = − t , t , t , t , , g (0) = at , t , t , t , ,g (1) = − abt , t , t , ( t , ) t , − bt , t , t , t , bt , t , t , t , ,g (2) = abt , t , t , ( t , ) t , bt , t , t , t , − b t , t , t , t , − abt , t , ( t , ) ( t , ) t , t , − bt , t , t , t , , g (3) = b t , t , t , t ,
00 0 00 − b t , t , t , t , t , t , . Obviously the leading term f ( − of the potential f at ξ
7→ ∞ is of the necessaryblock-diagonal form. According to the above scheme, there is a formal series T = E + T (1) ξ − + T (2) ξ − + · · · with the coefficients having the following block structure T ( i ) = ∗ ∗∗ ∗ , i ≥ , such that replacement Y = T ϕ brings the system (3.15) to a block-diagonal form ϕ , = hϕ, ϕ , = Sϕ, (3.16)where the potentials h and S are also formal series: h = h ( − ξ + h (0) + h (1) ξ − + h (2) ξ − + · · · ,S = S (0) + S (1) ξ − + S (2) ξ − + · · · . Here we assume that the potential h has a block diagonal form h ( i ) = ∗ ∗ ∗ ∗ ∗ , i ≥ − . Then we can check that S ( j ) for all j ≥ S ( j ) = ∗ ∗ ∗ ∗ ∗ . Omitting the calculations we give several members of these series T = + − at , t , ( t , ) ξ − + iscrete exponential type systems on a quad graph −
10 0 0 − at , ( t , t , t , + t , t , t , )( t , ) t , t , ξ − ++ − at , t − , t , t , at , t , t , t , + a t , t , t , t , ( t , ) t , t , + a t , ( t , ) t , ( t , ) t , ξ − + . . . ,h = −
10 1 0 ξ + t , t , t , t , + t , t , t , t ,
00 0 at , t , t , t , + t , t , t , t , ξ − ++ at , t , ( t , ) − at , t , ( t , ) ξ − + − a t , t − , t , t , ξ − ++ − at , t , (cid:16) t , t , + at , t , t , t , t , t , + a ( t , ) t , ( t , ) t , (cid:17) h ξ − + . . . , where h = at , t , t , t , (cid:16) t , t , ( t , ) + at , t − , t , t , + at − , t , t , t , (cid:17) , S = at , t , t , t , + − bt , t , t , t , bt , t , t , t , ξ − ++ abt , t , t , ( t , ) t , − b t , t , t , t ,
00 0 − bt , t , t , t , ξ − + abt , t , t , ( t , ) t , − b t , t , t , t , t , t , ξ − ++ − abt , t , ( at , t , t , + t , t , t , )( t , ) t , t , a bt − , t , t , t , t , t ,
00 0 abt , t , ( t , ) ( t , ) t , t , ξ − + . . . . The consistency condition of the system (3.16) can be written as D n ( S ) h = D m ( h ) S .Due to the representation h = h h ! , S = S S ! the latter implies( D n −
1) log det( S ii ) = ( D m −
1) log det( h ii ) , i = 1 , . (3.17) iscrete exponential type systems on a quad graph D n − (cid:16) − bt , t , t , t , (cid:17) = ( D m − (cid:16) t , t , t , t , + at , t , t , t , + at , t , t , t , (cid:17) ,2. ( D n − (cid:16) − abt , t , t , t , ( t , ) t , t , − bt , t , t , t , t , t , − b ( t , t , ) ( t , t , ) (cid:17) =( D m − (cid:18) at , t , t , t , + at , t , t , t , ( t , ) t , t , + a t , t , t , t , + (cid:16) t , t , t , t , + at , t , t , t , + at , t , t , t , (cid:17) (cid:19) ,3. ( D n − (cid:16) b t , t , t , t , − a bt , t − , t , t , t , t , − bt , t , t , t , ( t , ) t , t , +
12 ( bt , t , ) ( t , t , ) (cid:17) =( D m − (cid:18) at , t , t , t , + a t , t − , t , t , + at , t , t , t , ( t , ) t , t , + (cid:16) t , t , t , t , + at , t , t , t , + at , t , t , t , (cid:17) (cid:19) .In a similar way we study the system (3.14) around the point λ = 0. In this casewe use the change of the variables Φ = ¯ HY ¯ H = t , t , b t , t , ζ − − t , t , bt , t , , ζ = 1 √ λ → ∞ and arrive at the system Y , = ¯ f Y, Y , = ¯ gY with potentials¯ g = ¯ g ( − ζ + ¯ g (0) + ¯ g (1) ζ − and ¯ f = ¯ f (0) + ¯ f (1) ζ − + ¯ f (2) ζ − + ¯ f (3) ζ − , where ¯ g ( − = b t , t , t , t , bt , t , t , t , , ¯ g (0) = at , t , t , t , at , t , t , b ( t , ) t , t , t , t t , t , t ,
00 0 0 , ¯ g (1) = − t , t , t , bt , t , t ,
00 0 00 bt , t , t , t , , ¯ f (0) = t , t , t , t , t , t , t , t ,
00 0 at , t , t , t , , ¯ f (1) = − − t , t , t , bt , t , t , − at , t , t , t , b ( t , ) t , t , , ¯ f (2) = t , t , t , t , b ( t , ) t , t , a ( t , ) t , t , b ( t , ) t , t , t , bt , t , − t , t , t , t , − at , t , b ( t , ) , ¯ f (3) = − t , t , b t , t ,
00 0 00 1 0 . iscrete exponential type systems on a quad graph ζ = ∞ and the coefficient ¯ g ( − at ζ is of the necessaryblock-diagonal form. Therefore we can find the formal series ¯ T and ¯ h from the equation¯ T , ¯ h = (¯ g ( − ζ + ¯ g (0) + ¯ g (1) ζ − ) ¯ T .
Then knowing ¯ T we find the series ¯ S from¯ T , ¯ S = ¯ f ¯ T .
As a result we obtain¯ T = + − at , ( t , ) b ( t , ) t , ζ − ++ at , t , ( t , ) b t , ( t , ) t , + at , t , t , b t , t , t , at , − t , bt , t , − ζ − ++ − at , ( t , ) b t , t , t , − a ( t , ) ( t , ) b ( t , ) t , t , − a t , ( t , ) t , b ( t , ) t , t , − at , t , − ( bt , ) ζ − + . . . , ¯ h = b t , t , t , t , bt , t , t , t , ζ + at , t , t , t , t , t , t , t , t , t ,
00 0 0 ++ − t , t , t , bt , t , t ,
00 0 00 0 0 ζ − + a t , − t , t , b t , t , t , − at , t , t , b ( t , ) t , − at , t , b t , t , ζ − ++ − a t , t , − t , b t , t , t ,
00 0 00 0 0 ζ − + . . . , ¯ S = t , t , t , t , t , t , t , t ,
00 0 at , t , t , t , + − − t , t , t , bt , t , t , ζ − ++ t , t , t , t , b ( t , ) t , t , t , t , bt , t ,
00 0 − at , t , ( bt , ) ζ − + iscrete exponential type systems on a quad graph − t , t , b t , t , − at , t , t , ( at , − t , − bt , t , ) b ( t , ) t , t , t , ζ − + . . . . Passing to the blocks¯ h = ¯ h
00 ¯ h ! , ¯ S = ¯ S
00 ¯ S ! we can write the relations( D n −
1) log det( ¯ S ii ) = ( D m −
1) log det(¯ h ii ) , i = 1 , , from which we can derive the local conservation laws:1. ( D n − (cid:16) t , t , bt , t , + at , t , bt , t , + at , t , bt , t , (cid:17) = ( D m − (cid:16) − t , t , t , t , (cid:17) ,2. ( D n − (cid:16)
12 ( t , t , ) ( t , t , ) − at , t , ( t , ) t , t , ( t , ) − a t , t , t , t , t , ( t , ) t , − t , t , t , t , t , t , t , t , −
12 ( at , t , ) ( t , t , ) − a t , t , t , t , t , t , ( t , ) −
12 ( at , t , ) ( t , t , ) − at , t , t , t , − at , t , t , t , − at , t , t , t , (cid:17) = ( D m − (cid:16) bt , t , t , t , t , t , + abt , t , t , t , ( t , ) t , t , +
12 ( bt , t , ) ( t , t , ) (cid:17) ,3. ( D n − h at , − t , − bt , t , t , t , t , (cid:16) at , t , t , + at , t , t , t , t , (cid:17) + a t , t , − t , t , +
12 ( t , t , ) ( t , t , ) + at , t , t , t , t , ( t , ) t , + at , t , t , t , + a ( t , t , t , + t , t , t , ) ( t , t , t , ) i =( D m − (cid:16) ab t , − t , t , t , t , t , − a bt , − t , t , t , t , t , − b t , t , t , t , − abt , − t , t , t , t , t , t , t , + b t , t , t , t , t , ( t , ) t , +
12 ( bt , t , ) ( t , t , ) (cid:17) .
4. Higher symmetries
Quad system (2.1) possess higher symmetries. However, presented in the variables t jn,m the symmetries have non-localities. They become local in new variables introduced bypotentiation or, more precisely, by setting r j , = t j , t j , . Actually in terms of these variablesquad system (2.1) turns into ar , = r , + r , r N , (cid:16) ar , − r , (cid:17) ,ar j , = r j , + r j − , r j +10 , (cid:18) ar j , − r j , (cid:19) , ≤ j ≤ N − ,ar N , = r N , + r , r N − , (cid:16) ar N , − r N , (cid:17) . (4.1)More precisely in what follows we present higher symmetries to this quad system. Wenote that obtained system (4.1) does not already contain the parameter b. Everywherebelow we write r j instead of r j , .The Lax pair of the system (4.1) is written asΦ , = F Φ , (4.2)Φ , = G Φ , (4.3) iscrete exponential type systems on a quad graph F and G are matrices F = r r − . . . r r − . . . r r . . . . . . − λ r r λ . . . ar , r N ,G = ar , r . . . λ − ar , − r )( ar N , − r N ) r r N − λ − ar , − r r ar , − r r . . . ar , − r r . . . . . . . . . ar N , − r N r N − . Example 2.
Consider the system (4.1) for N = 2. For the sake of simplicity herewe use notations u := r , v := r au , = u , + v , (cid:16) au − u , (cid:17) ,av , = v , + u , (cid:16) av − v , (cid:17) . (4.4)The simplest higher symmetry of (4.4) in the direction of n is given by u t = v + au v − , ,v t = au , + v u . (4.5)The symmetry admits Lax pair Φ , = F Φ, Φ t = A Φ where F = vu − − λ vu λ + au , v ! , A = auv − , + λ vu λ vu ! . For the particular case a = 1, quad system (4.4) and also its symmetry (4.5) havealready been found in [1]. We also indicate the symmetry in the direction of m [33] u τ = u av , − v ,v τ = uu , − au − u , − (4.6)with the Lax pair Φ , = G Φ, Φ τ = B Φ where G = au , u + ( au , − u )( av , − v ) λu au , − uλuav , − vu ! , B = uav , − v uλ ( av , − v ) u , − au − u , − − λ ! . For the special choice a = 1 of the parameter the quad system (4.4) and itssymmetries take the most simple form. Namely the coupled lattice (4.5) in that case isconverted to a scalar lattice R n,m,t = R n +1 ,m + R n,m R n − ,m iscrete exponential type systems on a quad graph R n,m = u n,m and R n +1 ,m = v n,m . Hence the desire to transform the discretesystem itself to a scalar equation. However, the symmetry (4.6) in the other directionis reduced to a scalar form P n,m,τ = 1 P n,m − − P n,m +1 under the different transformation P n, m = u n,m , P n, m − = v n,m . Therefore the systemof equations (4.4) is not reduced to a scalar autonomous equation [34]. If a = 1 thennone of the symmetries is reduced to an autonomous scalar lattice. Example 3.
Now we represent the higher symmetries to the system (4.1) for N = 3. Here we use notations u := r , v := r and w := r au , = u , + v , w , (cid:16) au − u , (cid:17) ,av , = v , + u , w , (cid:16) av − v , (cid:17) ,aw , = w , + u , v , (cid:16) aw − w , (cid:17) . (4.7)The Lax pair to the system takes the formΦ , = F Φ , Φ , = G Φ , (4.8)where F = vu − wv − − vu λ λ au , w , G = au , u ( au , − u )( aw , − w ) λuv au , − uλuav , − vu aw , − wv . The simplest higher symmetry of the system (4.7) is given by u t = w + auvw − , + au v − , ,v t = au , + vwu + av w − , ,w t = av , + au , wv + w u . (4.9)The linear system of the formΦ , = F Φ , Φ t = A Φis a Lax pair for (4.9), where F is given in (4.8) and A is as A = λ + auv − , auw − , vu λ avw − , vuwu λ wu . The quad system (4.7) admits also the classical symmetries u t = u, v t = v, w t = w and u t = ( − n u, v t = ( − n +1 v, w t = ( − n w. iscrete exponential type systems on a quad graph m the system has a symmetry of the form u τ = u v ( av , − v )( aw , − w ) ,v τ = uvu , − ( aw , − w )( au − u , − ) ,w τ = uu , − v , − ( au − u , − )( av − v , − ) . The symmetry is related to the Lax pair Φ , = G Φ, Φ τ = B Φ with G and B definedby (4.8) and, respectively, by B = uv ( av , − v )( aw , − w ) uv ( av , − v )( aw , − w ) λ − vu , − ( aw , − w )( au − u , − ) − vaw , − w λ − − u , − au − u , − auu , − ( au − u , − )( av − v , − ) λ − . Example 4.
The next example concerns the algebra A (1)3 . In this case the quadsystem (4.1) reads as ar , = r , + r , r , (cid:16) ar − r , (cid:17) ,ar , = r , + r , r , (cid:16) ar − r , (cid:17) ,ar , = r , + r , r , (cid:16) ar − r , (cid:17) ,ar , = r , + r , r , (cid:16) ar − r , (cid:17) . (4.10)The Lax pair for (4.10) is given by a system of the formΦ , = F Φ , Φ , = G Φ , where the potentials are F = r r − r r − r r − − λ r r λ ar , r ,G = ar , r ( ar , − r )( ar , − r ) r r λ − ar , − r r λ − ar , − r r ar , − r r ar , − r r . The following multifield lattices are symmetries for the quad system (4.10) r t = r + a ( r ) r − , + ar r r − , + ar r r − , ,r t = ar , + a ( r ) r − , + ar r r − , + r r r ,r t = ar , + a ( r ) r − , + r r r + ar r , r ,r t = ar , + ( r ) r + ar r , r + ar r , r iscrete exponential type systems on a quad graph r τ = ( r ) r r ( ar , − r )( ar , − r )( ar , − r ) ,r τ = r r r r , − ( ar − r , − )( ar , − r )( ar , − r ) ,r τ = r r r , − r , − ( ar − r , − )( ar − r , − )( ar , − r ) ,r τ = r r , − r , − r , − ( ar − r , − )( ar − r , − )( ar − r , − ) . The Lax pairs for these lattices Φ , = F Φ, Φ t = A Φ and Φ , = G Φ, Φ τ = B Φ aregiven by A = ar r − , + λ ar r − , ar r − , r r λ ar r − , ar r − , r r r r λ ar r − , r r r r λ r r ,B = − r r r P (3) ( r ,r ,r ) − r r r λ − P (3) ( r ,r ,r ) − r , − r r P (3) ( r , − ,r ,r ) r r λ − P (2) ( r ,r ) r , − r P (2) ( r , − ,r ) − ar r , − r P (3) ( r , − ,r , − ,r ) − r λ − ar , − r − r , − ar − r , − ar r , − P (2) ( r , − ,r , − ) − ar r , − r , − P (3) ( r , − ,r , − ,r , − ) λ − , where P (3) ( h, k, l ) = ( ah , − h )( ak , − k )( al , − l ) , P (2) ( h, k ) = ( ah , − h )( ak , − k ) . Example 5.
Finally, we concentrate on the general case of the system (4.1).Symmetries of the system are given as r t = r N + a ( r ) r − , + ar r r − , + ar r r − , + . . . + ar r N − r N − , ,r t = ar , + a ( r ) r − , + ar r r − , + ar r r − , + . . . + ar r N − r N − , + r r N r ,r t = ar , + a ( r ) r − , + ar r r − , + . . . + ar r N − r N − , + r r N r + ar r , r ,. . . ,r N − t = ar N − , + a ( r N − ) r N − , + r N − r N r + ar N − r , r + . . . + ar N − r N − , r N − ,r Nt = ar N − , + ( r N ) r + ar N r , r + ar N r , r + . . . + ar N r N − , r N − and r τ = ( r ) r r ...r N − ( ar , − r )( ar , − r )( ar , − r ) ... ( ar N , − r N ) ,r τ = r r r ...r N − r , − ( ar − r , − )( ar , − r )( ar , − r ) ... ( ar N , − r N ) ,r τ = r r ...r N − r , − r , − ( ar − r , − )( ar − r , − )( ar , − r ) ... ( ar N , − r N ) ,. . . ,r N − τ = r r N − r , − r , − r , − ...r N − , − ( ar − r , − )( ar − r , − ) ... ( ar N − − r N − , − )( ar N , − r N ) ,r Nτ = r r , − r , − r , − ...r N − , − ( ar − r , − )( ar − r , − )( ar − r , − ) ... ( ar N − − r N − , − ) . iscrete exponential type systems on a quad graph p jn,m +1 = p jn +1 ,m + p jn,m +1 p jn +1 ,m p jn +1 ,m +1 p j +1 n,m (cid:0) p j +1 n,m +1 − p j +1 n +1 ,m (cid:1) , j = 1 , . . . , N − ,p Nn,m +1 = p Nn +1 ,m + p Nn,m +1 p Nn +1 ,m p Nn +1 ,m +1 p n,m (cid:0) p n,m +1 − p n +1 ,m (cid:1) . (4.11)Evidently (4.11) is a periodic reduction p j + Nn,m = p jn,m (4.12)of the Hirota-Miwa equation written as p jn,m +1 = p jn +1 ,m + p jn,m +1 p jn +1 ,m p jn +1 ,m +1 p j +1 n,m (cid:0) p j +1 n,m +1 − p j +1 n +1 ,m (cid:1) , −∞ < j < ∞ . (4.13)In turn, system (4.1) can be obtained from another form of the Hirota-Miwa equation ar jn +1 ,m +1 = r jn +1 ,m + r j − n +1 ,m r j +1 n,m +1 r jn,m r jn,m +1 (cid:0) ar jn,m +1 − r jn,m (cid:1) , −∞ < j < ∞ (4.14)by imposing a restriction r j + Nn,m = r jn +1 ,m − . (4.15)It can easily be checked that in the case a = 1 equations (4.13) and (4.14) are relatedby p jn,m = r n + m − j,m . (4.16)For a = 1 the equations are essentially different. Direct computation convinces thattriple of the equations (4.12), (4.15), (4.16) is not consistent, therefore Proposition 3.
The systems (4.1) and (4.11) are not related by a pointtransformation.
5. Evaluation of the recursion operators
In this section we discuss the recursion operators for the quad systems associated with A (1) N − . Recursion operators are closely related with higher symmetries, local conservationlaws and multi-hamiltonian structures. There are several methods for constructing therecursion operators, based on the Lax representation [35], [36], Hamiltonian operators[37], [38] and generalized invariant manifolds [39]. In our opinion the most convenientof them is one using the Lax pair, moreover in some cases it is reasonable to derive thebihamiltonian structure from the recursion operator.Since the quad system admits two hierarchies of symmetries, it admits two recursionoperators as well, corresponding to the variables n and m . Below we concentrate onthat related to n . To study the problem we use the ideas of [35], [36].It can be shown that the procedure of the formal diagonalization allows us toconstruct a formal series (see [28]) A = A (0) + A (1) λ − + A (2) λ − + . . . iscrete exponential type systems on a quad graph L = D − n F such that[ L, A ] := LA − AL = 0 , where F = F (0) + F (1) λ is the potential of the Lax equation (4.2) and the expression D − n F means a composition of the backward shift operator D − n and the operation ofmultiplication by F .The series A provides an effective tool for constructing the higher symmetries ofthe quad system (4.1). To explain the method we consider the formal series B obtainedfrom A by multiplying by λ k , where k is a positive integer: B := λ k A = B ( k ) λ k + B ( k − λ k − + . . . . (5.1)Then we take a polynomial part of the series (5.1) B + = B ( k ) λ k + B ( k − λ k − + . . . + B (1) λ + B (0)+ . Here the last summand is chosen in a nontrivial way. Its upper diagonal part coincideswith that of the matrix B (0) . The other entries vanish except one located at the leftupper corner, denote it by x . The crucial point is that the consistency condition of thelinear equation ddτ y = B + y (5.2)with (4.2) gives exactly N + 1 equations which allow us to determine the unknown x and to derive a symmetry of the quad system (4.1) with time τ .In addition to (5.1) we take one more series C := λ k +1 A , such that C = C ( k +1) λ k +1 + C ( k ) λ k + C ( k − λ k − + . . . . (5.3)By applying the algorithm used above to (5.3) we get the polynomial C + = C ( k +1) λ k +1 + C ( k ) λ k + . . . + C (1) λ + C (0)+ which provides the time part of the Lax pair ddτ y = C + y. (5.4)Our goal now is to find the relation between two symmetries defined by the polynomials B + , C + . Evidently we have C − λB = 0 . (5.5)We replace in (5.5) the summands B and C with their representations (5.1) and (5.3).As a result we get C ( k +1) λ k +1 + . . . + C (1) λ + C (0)+ + C (0) − + C ( − λ − + . . . −− λ (cid:16) B ( k ) λ k + . . . + B (1) λ + B (0)+ + B (0) − + B ( − λ − + . . . (cid:17) = 0 , (5.6)where B (0) − := B (0) − B (0)+ , C (0) − := C (0) − C (0)+ . iscrete exponential type systems on a quad graph R N := C ( k +1) λ k +1 + . . . + C (1) λ + C (0)+ − λ (cid:16) B ( k ) λ k + . . . + B (1) λ + B (0)+ (cid:17) = − C (0) − − C ( − λ − − . . . + λ (cid:16) B (0) − + B ( − λ − + . . . (cid:17) . It is easily checked that R N is a linear function of λ : R N = B (0) − λ + C (0)+ := rλ + s. (5.7)The consistency conditions of the equations (4.2), (5.2) and (4.2), (5.4) can be writtenin the form L τ = [ L, C + ] , L τ = [ L, B + ] . Therefore we obtain L τ − λL τ = [ L, C + − λB + ] = [ L, R N ] . Since L = ( α + βλ ) D − n , where α = D − n (cid:0) F (0) (cid:1) and β = D − n (cid:0) F (1) (cid:1) (recall that F = F (0) + F (1) λ ) then due to (5.7) we have an equation (cid:0) α τ + ( β τ − α τ ) λ − β τ λ (cid:1) D − n = (cid:2) ( α + βλ ) D − n , rλ + s (cid:3) . which produces a system of the equations α τ = αs − − sα,β τ − α τ = αr − + βs − − rα − sβ, (5.8) β τ = rβ − βr − for determining unknown coefficients r and s and a relation between the symmetries { r jτ } Nj =1 and (cid:8) r jτ (cid:9) Nj =1 , which should allow us to derive the recursion operator. Let us construct the recursion operator in the direction of n for a particular case of thesystem (4.1) for N = 2. In an explicit form the system is presented in (4.4). In thiscase, functions α and β used above have the form α = u − , v − , − vu − , ! , β = − u − , v − , ! . (5.9)We look for the coefficients r and s in the form r = r (11) r (21) ! , s = s (11) s (12) s (22) ! . (5.10)Let’s rewrite system (5.8) taking into account (5.9) and (5.10):1) (cid:16) u − , v − , (cid:17) τ − u − , v − , (cid:16) s (11) − , − s (11) (cid:17) = 0,2) s (22) − , − s (11) + vu − , s (12) − u − , v − , s (12) − , = 0,3) (cid:16) vu − , (cid:17) τ − vu − , (cid:16) s (22) − , − s (22) (cid:17) = 0, iscrete exponential type systems on a quad graph (cid:16) u − , v − , (cid:17) τ − r (21) − , + u − , v − , r (11) − , − u − , v − , r (11) + u − , v − , s (12) = 0,5) s (12) − r (11) = 0,6) (cid:16) u − , v − , (cid:17) τ − u − , v − , s (11) − , + u − , v − , s (22) + vu − , r (21) − , − u − , v − , r (21) = 0,7) (cid:16) vu − , (cid:17) τ − s (22) + s (22) − , + r (21) − u − , v − , s (12) − , = 0,8) (cid:16) u − , v − , (cid:17) τ − r (21) − , + u − , v − , r (11) − , = 0.Thus, to find the necessary coefficients r (11) , r (21) , s (11) , s (12) and s (22) , we obtainedan overdetermined system of equations. Using equation 5), we exclude the function s (12) by taking s (12) = r (11) . Combining equations 1), 2), 4), 6) and 7) we find the function r (11) r (11) = v τ v . (5.11)From equation 4), by virtue of (5.11), we find r (21) = u τ v . We find the remaining two coefficients s (11) and s (22) . To do this, use the equations 1),2) and 6). From which we obtain s (22) = u τ v + v τ u − , +( D n − − (cid:20)(cid:18) v − v , u (cid:19) u τ + (cid:18) u − , − uv (cid:19) v τ (cid:21) , (5.12) s (11) = v τ u − , + v ( u − , ) τ u − , +( D n − − (cid:20)(cid:18) v − v , u (cid:19) u τ + (cid:18) u − , − uv (cid:19) v τ (cid:21) . (5.13)Let’s write two more equations connecting u τ and u τ , v τ and v τ . Using equations 1)and 3) we obtain u τ = − u (cid:16) s (11)1 , + s (22) (cid:17) , v τ = − v (cid:0) s (11) + s (22) (cid:1) . (5.14)We substitute (5.12) and (5.13) in (5.14) and get uv ! τ = R uv ! τ , where R = ! D n + uv uu − , − u v vu − , ! + v u − , ! D − n − uv ! ( D n − − (cid:18) v , u − v , uv − u − , (cid:19) . (5.15)Applying the operator (5.15) to the classical symmetry u τ = u , v τ = v of the system(4.4) we obtain the higher symmetry (4.5). Operator (5.15) has been found in our article[40]. iscrete exponential type systems on a quad graph N = 3 (see (4.7)). Forthis system, we construct the recursion operator using the method presented above.Omitting the calculations, we give only the final answer R = R (0) + S (cid:2) ( D n − − R (1) + ( D n + 1) − R (2) (5.16)+ ( D n + 1) − A (1) ( D n − − R (3) + ( D n + 1) − A (2) ( D n + 1) − R (4) (cid:3) , where R (0) = uv D n + uw + w , v uu − , − uw , v uv − , − u w + w , u − , + uvu − , wvw vu − , + uw vv − , − uvw wu − , wv − , + vu − , + v u − , vwu − , w v − , D − n ,S = u v
00 0 w ,R (1) = − v , u − w w , v − u − ,
00 0 uw − v − , ,R (2) = v , u − w − w , αu αw u , u − , v , + uw − vαw − βu − , βv w , v − uβv − u − , βu − , + uw + v − , γw − u − , − w , v u w − v − , − vγw ,R (3) = w , u − v uv − w
00 0 vw − u − , ,R (4) = w , u + v − uv − w
00 0 vw + u − , ,A (1) = − w , u , + v , w , + uv w , u + vw + u , v ,
00 0 wu − , + v , w , + uv ,A (2) = 12 w , u , + v , w , + uv − w , u − vw − u , v ,
00 0 wu − , + v , w , + uv , where α = w , u , + v , w , + uv , β = w , u + vw + u , v , , γ = wu − , + v , w , + uv .Usually, recursion operators for integrable systems are pseudodifferential (orpseudo-difference for the case of lattices) operators with so-called weak nonlocalities iscrete exponential type systems on a quad graph Conclusions
We conjecture that the discrete exponential type system on a quad graph (1.3) is anintegrable discretization of the Drinfeld-Sokolov hierarchy [10]. Evidently (1.3) goes to(1.1) in the continuum limit for appropriate choice of the parameters a and b . As forthe proof of integrability of (1.3), this is much more complicated problem. So far it wasproved in [20] that (1.3) is integrable in the sense of Darboux for the simple Lie algebras A N , B N . For the system (1.3) corresponding to the affine algebras D (2) N and A (1)1 theLax pairs were found in [1]. For the cases A (1)1 and A (2)2 the higher symmetries havebeen constructed [1]. For the cases C , G , D the Darboux integrability was proved[1]. In the present article we studied the quad system corresponding to A (1) N − . Wederived for it the Lax representation, allowing to describe the local conservation laws.We constructed the higher symmetries by using this Lax pair. We derived the recursionoperator for N = 3, which turned out to be rather complicated. Acknowledgements
The authors gratefully acknowledge financial support from a Russian Science Foundationgrant (project 15-11-20007).
References [1] Garifullin R N, Habibullin I T and Yangubaeva M V 2012 Affine and finite Lie algebras andintegrable Toda field equations on discrete space-time
SIGMA Paris: Gauthier-Villars
513 p., 579 p., 512 p., 547 p.[3] Toda M 1967 Vibration of a Chain with Nonlinear Interaction
J. Phys. Soc. Japan JETPLett. Theor. Math. Phys. Commun. Math. Phys. Preprint,OFM BFAN SSSR, Ufa [8] Leznov A N, Smirnov V G and Shabat A B 1982 The group of internal symmetries and theconditions of integrability of two-dimensional dynamical systems
Theor. Math. Phys. Ergod. Theory Dyn. Syst. iscrete exponential type systems on a quad graph [10] Drinfeld V G and Sokolov V V 1985 Lie algebras and equation of KdV type J. Sov. Math. Inv. math. Phys. Lett. A
Leningrad Math. J. J. Phys. Soc. Jpn. Proc. Japan Acad.
J. Math. Sci. Univ.Tokyo Commun. Math. Phys. Proc.R. Ir. Acad. A J. Phys.A: Math. Theor. Theor. andMath. Phys. :2, 189–210[21] Fordy A P and Xenitidis P 2017 Zn graded discrete Lax pairs and integrable difference equations
J. Phys. A: Math. Theor. J. Phys. A:Math. Theor. :33 334001[23] Atkinson J, Lobb S B and Nijhoff F W 2012 An integrable multicomponent quad-equation and itsLagrangian formulation Theoret. and Math. Phys. :3 1644–1653[24] Habibullin I T and Poptsova M N 2015 Asymptotic diagonalization of the discrete Lax pair aroundsingularities and conservation laws for dynamical systems
J. Phys. A: Math. Theor. :11 115203[25] Nijhoff F W, Papageorgiou V G, Capel H W and Quispel G R W 1992 The lattice Gelfand-Dikiihierarchy Inverse Problems J. Phys. A: Math. Theor. :20, 205202[27] Zabrodin A V 1997 Hirota’s difference equations Theor. and Math. Phys. :2 1347–1392[28] Habibullin I T and Yangubaeva M V 2013 Formal diagonalization of a discrete Lax operator andconservation laws and symmetries of dynamical systems
Theor. and Math. Phys. :3 1655–1679[29] Mikhailov A V 2015 Formal diagonalisation of Lax-Darboux schemes
Model. Anal. Inform. Sist. :6, 795–817[30] Novikov S P and Dynnikov I A 1997 Discrete spectral symmetries of low-dimensional differentialoperators and difference operators on regular lattices and two-dimensional manifolds RussianMath. Surveys :5 1057–1116[31] Adler V E and Startsev S Ya 1999 Discrete analogues of the Liouville equation Theor. and Math.Phys. :2 1484–1495[32] Wasow W 1987 Asymptotic Expansions for Ordinary Differential Equations
Dover Books onAdvanced Mathematics (Dover: Dover Pubns)
374 pp.[33] Pavlova E V, Habibullin I T and Khakimova A R 2017 On one integrable discrete system [inRussian]
Differential Equations: Mathematical Physics (Itogi Nauki Tekh. Ser. Sovrem. Mat.Prilozh. Temat. Obz.), VINITI, Moscow
Journalof Nonlinear Mathematical Physics :3, 333–357[35] G¨ u rses M, Karasu A and Sokolov V V 1999 On construction of recursion operators from Laxrepresentation Journal of Mathematical Physics :12 6473–6490 iscrete exponential type systems on a quad graph [36] Khanizadeh F, Mikhailov A V and Wang J P 2013 Darboux transformations and recursionoperators for differential-difference equations Theoret. and Math. Phys. :3 1606–1654[37] Zhang H, Tu G Z, Oevel W and Fuchssteiner B 1991 Symmetries, conserved quantities, andhierarchies for some lattice systems with soliton structure
Journal of mathematical physics :71908–1918[38] Maltsev A Y and Novikov S P 2001 On the local Hamiltonian systems in the weakly non-localPoisson brackets Physica D
J.Phys. A: Math. Theor. :42 22 pp.[40] Habibullin I T, Khakimova A R 2017 Invariant manifolds and Lax pairs for integrable nonlinearchains Theoret. and Math. Phys.191