On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations
Decio Levi, Matteo Petrera, Christian Scimiterna, Ravil Yamilov
aa r X i v : . [ m a t h - ph ] N ov Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2008), 077, 14 pages On Miura Transformationsand Volterra-Type Equations Associatedwith the Adler–Bobenko–Suris Equations
Decio LEVI † , Matteo PETRERA ‡† , Christian SCIMITERNA ‡† and Ravil YAMILOV §† Dipartimento di Ingegneria Elettronica,Universit`a degli Studi Roma Tre and Sezione INFN,Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
E-mail: levi@fis.uniroma3.it ‡ Dipartimento di Fisica E. Amaldi, Universit`a degli Studi Roma Tre and Sezione INFN,Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
E-mail: petrera@fis.uniroma3.it, scimiterna@fis.uniroma3.it § Ufa Institute of Mathematics, 112 Chernyshevsky Str., Ufa 450077, Russia
E-mail:
Received August 29, 2008, in final form October 30, 2008; Published online November 08, 2008Original article is available at
Abstract.
We construct Miura transformations mapping the scalar spectral problems of theintegrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the dis-crete Schr¨odinger spectral problem associated with Volterra-type equations. We show thatthe ABS equations correspond to B¨acklund transformations for some particular cases of thediscrete Krichever–Novikov equation found by Yamilov (YdKN equation). This enables usto construct new generalized symmetries for the ABS equations. The same can be said aboutthe generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis.All of them generate B¨acklund transformations for the YdKN equation. The higher ordergeneralized symmetries we construct in the present paper confirm their integrability.
Key words:
Miura transformations; generalized symmetries; ABS lattice equations
The discovery of new two-dimensional integrable partial difference equations (or Z -lattice equa-tions) is always a very challenging problem as, by proper continuous limits, many other results ondifferential-difference and partial differential equations may be obtained. Moreover many physi-cal and biological applications involve discrete systems, see for instance [13, 25] and referencestherein.The theory of nonlinear integrable differential equations got a boost when Gardner, Green,Kruskal and Miura introduced the Inverse Scattering Method for the solution of the Korteweg–de Vries equation. A summary of these results can be found in the Encyclopedia of MathematicalPhysics [12]. A few techniques have been introduced to classify integrable partial differentialequations. Let us just mention the classification scheme introduced by Shabat, using the formalsymmetry approach (see [21] for a review). This approach has been successfully extended to thedifferential-difference case by Yamilov [30, 31, 20]. In the completely discrete case the situationturns out to be quite different. For instance, in the case of Z -lattice equations the formalsymmetry technique does not work. In this framework, the first exhaustive classifications offamilies of lattice equations have been presented in [1] by Adler and in [2, 3] by Adler, Bobenkoand Suris. D. Levi, M. Petrera, C. Scimiterna and R. YamilovIn the present paper we shall consider the Adler–Bobenko–Suris (ABS) classification of Z -lattice equations defined on the square lattice [2]. We refer to the papers [3, 24, 28, 17, 18, 27]for some recents results about these equations. Our main purpose is the analysis of their trans-formation properties. In fact, our aim is, on the one hand, to present new Miura transformationsbetween the ABS equations and Volterra-type difference equations and on the other hand, toshow that the ABS equations correspond to B¨acklund transformations for some particular casesof the discrete Krichever–Novikov equation found by Yamilov (YdKN equation) [30].Section 2 is devoted to a short review of the integrable Z -lattice equations derived in [2] andto present details on their matrix and scalar spectral problems. In Section 3, by transforming theobtained scalar spectral problems into the discrete Schr¨odinger spectral problem associated withthe Volterra lattice we will be able to connect the ABS equation with Volterra-type equations. InSection 4 we prove that the ABS equations correspond to B¨acklund transformations for certainsubcases of the YdKN equation. Using this result and a master symmetry of the YdKN equation,we construct new generalized symmetries for the ABS list. Then we discuss the integrabilityof a class of non-autonomous ABS equations and of a generalization of the ABS equationsintroduced by Tongas, Tsoubelis and Xenitidis in [28]. Section 5 is devoted to some concludingremarks. A two-dimensional partial difference equation is a functional relation among the values of a func-tion u : Z → C at different points of the lattices of indices n , m . It involves the independentvariables n , m and the lattice parameters α , β E ( n, m, u n,m , u n +1 ,m , u n,m +1 , . . . ; α, β ) = 0 . For the dependent variable u we shall adopt the following notation throughout the paper u = u , = u n,m , u k,l = u n + k,m + l , k, l ∈ Z . (1)We consider here the ABS list of integrable lattice equations, namely those affine linear (i.e.polynomial of degree one in each argument) partial difference equations of the form E ( u , , u , , u , , u , ; α, β ) = 0 , (2)whose integrability is based on the consistency around a cube [2, 3]. The function E dependsexplicitly on the values of u at the vertices of an elementary quadrilateral, i.e. ∂ u i,j E 6 = 0, where i, j = 0 ,
1. The lattice parameters α , β may, in general, depend on the variables n , m , i.e. α = α n , β = β m . However, we shall discuss such non-autonomous extensions in Section 4.The complete list of the ABS equations can be found in [2]. Their integrability holds byconstruction since the consistency around a cube furnishes their Lax pairs [2, 9, 22]. The ABSequations are given by the list H(H1) ( u , − u , )( u , − u , ) − α + β = 0 , (H2) ( u , − u , )( u , − u , ) + ( β − α )( u , + u , + u , + u , ) − α + β = 0 , (H3) α ( u , u , + u , u , ) − β ( u , u , + u , u , ) + δ ( α − β ) = 0 , and the list Q(Q1) α ( u , − u , )( u , − u , ) − β ( u , − u , )( u , − u , ) + δ αβ ( α − β ) = 0 , (Q2) α ( u , − u , )( u , − u , ) − β ( u , − u , )( u , − u , )n Miura Transformations and Volterra-Type Equations 3+ αβ ( α − β )( u , + u , + u , + u , ) − αβ ( α − β )( α − αβ + β ) = 0 , (Q3) ( β − α )( u , u , + u , u , ) + β ( α − u , u , + u , u , ) − α ( β − u , u , + u , u , ) − δ ( α − β )( α − β − αβ = 0 , (Q4) a u , u , u , u , + a ( u , u , u , + u , u , u , + u , u , u , + u , u , u , )+ a ( u , u , + u , u , ) + ¯ a ( u , u , + u , u , ) + e a ( u , u , + u , u , )+ a ( u , + u , + u , + u , ) + a = 0 . The coefficients a i ’s appearing in equation (Q4) are connected to α and β by the relations a = a + b, a = − aβ − bα, a = aβ + bα , ¯ a = ab ( a + b )2( α − β ) + aβ − (cid:16) α − g (cid:17) b, e a = ab ( a + b )2( β − α ) + bα − (cid:16) β − g (cid:17) a,a = g a − g a , a = g a − g a , with a = r ( α ), b = r ( β ), r ( x ) = 4 x − g x − g .Following [2] we remark that • Equations (Q1)–(Q3) and (H1)–(H3) are all degenerate subcases of equation (Q4) [7]. • Parameter δ in equations (H3), (Q1) and (Q3) can be rescaled, so that one can assumewithout loss of generality that δ = 0 or δ = 1. • The original ABS list contains two further equations (list A)(A1) α ( u , + u , ) ( u , + u , ) − β ( u , + u , ) ( u , + u , ) − δ αβ ( α − β ) = 0 , (A2) ( β − α )( u , u , u , u , + 1) + β ( α − u , u , + u , u , ) − α ( β − u , u , + u , u , ) = 0 . Equations (A1) and (A2) can be transformed by an extended group of M¨obius transfor-mations into equations (Q1) and (Q3) respectively. Indeed, any solution u = u n,m of (A1)is transformed into a solution e u = e u n,m of (Q1) by u n,m = ( − n + m e u n,m and any solu-tion u = u n,m of (A2) is transformed into a solution e u = e u n,m of (Q3) with δ = 0 by u n,m = ( e u n,m ) ( − n + m .Some of the above equations were known before Adler, Bobenko and Suris presented theirclassification, see for instance [23, 14]. We finally recall that a more general classificationof integrable lattice equations defined on the square has been recently carried out by Adler,Bobenko and Suris in [3]. But here we shall consider only the lists H and Q contained in [2]. The algorithmic procedure described in [2, 9, 22] produces a 2 × , = L ( u , , u , ; α, λ )Ψ , , Ψ , = M ( u , , u , ; β, λ )Ψ , , (3)with Ψ = ( ψ ( λ ) , φ ( λ )) T , where the lattice parameter λ plays the role of the spectral parameter.We shall use the following notation L ( u , , u , ; α, λ ) = 1 ℓ (cid:18) L L L L (cid:19) , M ( u , , u , ; β, λ ) = 1 t (cid:18) M M M M (cid:19) , D. Levi, M. Petrera, C. Scimiterna and R. Yamilov
Table 1.
Matrix L for the ABS equations (in equation (Q4) a = r ( α ), b = r ( λ ), r ( x ) = 4 x − g x − g ). L L L L H1 u , − u , ( u , − u , ) + α − λ u , − u , H2 u , − u , + α − λ ( u , − u , ) + 2( α − λ )( u , + u , )+ 1 u , − u , − α + λ + α − λ H3 λu , − αu , λ ( u , + u , ) − αu , u , + δ ( λ − α ) α αu , − λu , Q1 λ ( u , − u , ) − λ ( u , − u , ) + δαλ ( α − λ ) − α λ ( u , − u , )Q2 λ ( u , − u , )+ − λ ( u , − u , ) + − α λ ( u , − u , ) − + αλ ( α − λ ) +2 αλ ( α − λ )( u , + u , ) − − αλ ( α − λ ) − αλ ( α − λ )( α − αλ + λ )Q3 α ( λ − u , − − λ ( α − u , u , + λ ( α −
1) ( λ − α ) u , −− ( λ − α ) u , + δ ( α − λ )( α − λ − / (4 αλ ) − α ( λ − u , Q4 − a u , u , − − ¯ a u , u , − a ( u , + u , ) − a a u , u , + ¯ a + a u , u , + a u , + − a u , − e a u , − a + a ( u , + u , ) + e a u , + a where ℓ = ℓ , = ℓ ( u , , u , ; α, λ ), t = t , = t ( u , , u , ; β, λ ), L ij = L ij ( u , , u , ; α, λ ) and M ij = M ij ( u , , u , ; β, λ ), i, j = 1 ,
2. The matrix M can be obtained from L by replacing α with β and shifting along direction 2 instead of 1. In Table 1 we give the entries of the matrix L for the ABS equations.Note that ℓ and t are computed by requiring that the compatibility condition between L and M produces the ABS equations (H1)–(H3) and (Q1)–(Q4). The factor ℓ can be written as ℓ , = f ( α, λ )[ ρ ( u , , u , ; α )] / , (4)where the functions f = f ( α, λ ) is an arbitrary normalization factor. The functions f = f ( α, λ )and ρ = ρ , = ρ ( u , , u , ; α ) for equations (H1)–(H3) and (Q1)–(Q4) are given in Table 2.A formula similar to (4) holds also for the factor t .The scalar Lax pairs for the ABS equations may be immediately computed from equation (3).Let us write the scalar equation just for the second component φ of the vector Ψ (the use of thefirst component would give similar results). For equations (H1)–(H3) and (Q1)–(Q3) it reads( ρ , ) / φ , − ( u , − u , ) φ , + ( ρ , ) / µφ , = 0 , (5)where the explicit expressions of µ = µ ( α, λ ) are given in Table 2. The corresponding scalarequation for equation (Q4) takes a different form and needs a separate analysis which will bedone in a separate work. The aim of this Section is to show the existence of a Miura transformation mapping the scalarspectral problem (5) of equations (H1)–(H3) and (Q1)–(Q3) into the discrete Schr¨odinger spec-tral problem associated with the Volterra lattice [10] φ − , + v , φ , = p ( λ ) φ , , (6)where v , is the potential of the spectral problem and the function p ( λ ) plays the role of thespectral parameter.n Miura Transformations and Volterra-Type Equations 5 Table 2.
Scalar spectral problems for the ABS equations (in equation (Q4) c = r ( λ ), r ( x ) = 4 x − g x − g ). f ( α, λ ) ρ ( u , , u , ; α ) µ ( α, λ )H1 − λ − α H2 − u , + u , + α λ − α )H3 − λ u , u , + δα α − λ αλ Q1 λ ( u , − u , ) − δ α λ − αλ Q2 λ ( u , − u , ) − α ( u , + u , ) + α λ − αλ Q3 α (1 − λ ) α ( u , + u , ) − ( α + 1) u , u , + δ ( α − α α − λ α (1 − λ )Q4 ( α − λ ) c / × ( u , u , + αu , + αu , + g / − −× » a + c + “ a + cα − λ ” − α ( a + c ) α − λ – / − ( u , + u , + α )(4 αu , u , − g ) Suppose that a function s , = s ( u , , u , , u , , . . . ) is given by the linear equation s , s , = u , − u , ( ρ , ) / . (7)By performing the transformation φ , µ n/ s , φ , , and taking into account equation (7),equation (5) is mapped into the scalar spectral problem (6) with v , = ρ , ( u , − u − , )( u , − u , ) , p ( λ ) = [ µ ( α, λ )] − / . (8)From these results there follow some remarkable consequences: (i) There exists a Miura trans-formation between all equations of the set (H1)–(H3) and (Q1)–(Q3). Some results on thisclaim can be found in [7]; (ii) The Miura transformation (8) can be inverted by solving a lineardifference equation. Therefore we can in principle use these remarks to find explicit solutionsof the ABS equations in terms of the solutions of the Volterra equation.The following statement holds. Proposition 1.
The field u for equations (H1)–(H3) and (Q1)–(Q3) can be expressed in termsof the potential v of the spectral problem (6) through the solution of the following linear differenceequations H1 : u , − ( v , + v − , ) v , u , + v − , v , u − , = 0 , (9)H2 : u , − v , + v − , v , u , + v − , v , u − , − v , = 0 , (10)H3 : u , − v , + v − , v , u , + v − , v , u − , = 0 , (11)Q1 : u , − v , u , + 2 − v , − v − , v , u , − v , u − , + v − , v , u − , = 0 , (12)Q2 : u , − v , u , + 2 − v , − v − , v , u , − v , u − , + v − , v , u − , + 2 α v , = 0 , (13)Q3 : u , − αv , u , + α + 1 − v , − v − , v , u , − αv , u − , + v − , v , u − , = 0 . (14) D. Levi, M. Petrera, C. Scimiterna and R. Yamilov Proof .
From equation (8) we get v , ( u , − u , ) = ρ , u , − u − , , v − , ( u , − u − , ) = ρ − , u , − u − , . Subtracting these relations and taking into account that (see equation (A.11) in [28]) ∂ u , ρ , + ∂ u − , ρ − , = 2 ρ , − ρ − , u , − u − , , one arrives at v , ( u , − u , ) − v − , ( u , − u − , ) = 12 (cid:0) ∂ u , ρ , + ∂ u − , ρ − , (cid:1) . (15)Writing equation (15) explicitly for equations (H1)–(H3) and (Q1)–(Q3) we obtain equa-tions (9)–(14). (cid:4) Lie symmetries of equation (2) are given by those continuous transformations which leave theequation invariant. We refer to [19, 31] for a review on symmetries of discrete equations.From the infinitesimal point of view, Lie symmetries are obtained by requiring the infinite-simal invariant condition (cid:0) pr b X , (cid:1) E (cid:12)(cid:12) E =0 = 0 , (16)where b X , = F , ( u , , u , , u , , . . . ) ∂ u , . (17)By pr b X , we mean the prolongation of the infinitesimal generator b X , to all points appearingin E = 0.If F , = F , ( u , ) then we get point symmetries and the procedure to construct themfrom equation (16) is purely algorithmic [19]. If F , = F , ( u , , u , , u , , . . . ) the obtainedsymmetries are called generalized symmetries . In the case of nonlinear discrete equations, theLie point symmetries are not very common, but, if the equation is integrable, it is possible toconstruct an infinite family of generalized symmetries.In correspondence with the infinitesimal generator (17) we can in principle construct a grouptransformation by integrating the initial boundary problem du , ( ε ) dε = F , ( u , ( ε ) , u , ( ε ) , u , ( ε ) , . . . ) , (18)with u , ( ε = 0) = v , , where ε ∈ R is the continuous Lie group parameter and v , is a solutionof equation (2). This can be done effectively only in the case of point symmetries as in thegeneralized case we have a nonlinear differential-difference equation for which we cannot findthe general solution , but, at most, we can construct particular solutions.Equation (16) is equivalent to the request that the ε -derivative of the equation E = 0, writtenfor u , ( ε ), is identically satisfied on its solutions when the ε -evolution of u , ( ε ) is given byequation (18). This is also equivalent to say that the flows (in the group parameter space) givenby equation (18) are compatible or commute with E = 0.In the papers [24, 28] the three and five-point generalized symmetries have been found forall equations of the ABS list. We shall use these results to show that the ABS equations maybe interpreted as B¨acklund transformations for the differential-difference YdKN equation [30].This observation will allow us to provide an infinite class of generalized symmetries for thelattice equations belonging to the ABS list. We shall also discuss the non-autonomous case andthe generalizations of the ABS equations considered in [28].n Miura Transformations and Volterra-Type Equations 7 In the following we show that the ABS equations may be seen as B¨acklund transformations ofthe YdKN equation. Moreover we prove that the symmetries of the ABS equations [24, 28] aresubcases of the YdKN equation. For the sake of clarity we consider in a more detailed wayjust the case of equation (H3). Similar results can be obtained for the whole ABS list (seeProposition 2).According to [24, 28] equation (H3) admits the compatible three-point generalized symmetries du , dε = u , ( u , + u − , ) + 2 αδu , − u − , , (19) du , dε = u , ( u , + u , − ) + 2 βδu , − u , − . (20)Notice that under the discrete map n ↔ m , α ↔ β , equation (19) goes into equation (20), whileequation (H3) is left invariant.The compatibility between equation (H3) and equation (19) generates a B¨acklund transfor-mation (see an explanation below) of any solution u , of equation (19) into its new solution e u , = u , , e u , = u , . (21)Thus equation (H3) can be rewritten as a B¨acklund transformation for the differential-differenceequation (19) α ( u , u , + e u , e u , ) − β ( u , e u , + u , e u , ) + δ ( α − β ) = 0 . (22)Moreover, the discrete symmetry n ↔ m , α ↔ β implies the existence of the B¨acklund trans-formation for equation (20) b u , = u , , b u , = u , . This interpretation of lattice equations as B¨acklund transformations has been discussed forthe first time in the differential-difference case in [16]. Examples of B¨acklund transformationssimilar to equation (22) for Volterra-type equations can be found in [29, 11].In [24, 28] generalized symmetries have been obtained for autonomous ABS equations, i.e.such that α , β are constants. We present here some results on the non-autonomous case when α and β depend on n and m . Similar results can be found in [24].Let the lattice parameters in equation (2) be such that α is a constant and β = β = β m .Let us consider the following two forms of equation (2) u , = ξ ( u , , u , , u , ; α, β ) , u , = ζ ( u , , u , , u , ; α, β ) , (23)and a symmetry du , dε = f , = f ( u , , u , , u − , ; α ) , (24)given by equation (19). We suppose that u k,l depends on ε in all equations and write down thecompatibility condition between equation (23) and equation (24) f , = f , ∂ u , ξ + f , ∂ u , ξ + f , ∂ u , ξ. (25)As a consequence of equations (23), (24) the functions f , , f , and f , may be expressed interms of the fields u k, , u ,l . Therefore, equation (25) depends explicitly only on the variab-les u k, , u ,l , which can be considered here as independent variables for any fixed n , m . For all D. Levi, M. Petrera, C. Scimiterna and R. Yamilovautonomous ABS equations, the compatibility condition (25) is satisfied identically for all valuesof these variables and of the constant parameter β . In the non-autonomous case, equation (25)depends only on β and α . Therefore the compatibility condition is satisfied also for any m .So, equation (19) is compatible with equation (H3) also in the case when α is constant, but β = β m . In a similar way, one can prove that equation (20) is the generalized symmetry ofequation (H3) if β is constant, but α = α n .Let us now discuss the interpretation of the ABS equations as B¨acklund transformations.Let u , be a solution of equation (24), and the function e u , = e u n,m ( ε ) given by equation (21)be a solution of equation (23), which is compatible with equation (24). equation (23) can berewritten as the ordinary difference equation e u , = ξ ( u , , u , , e u , ; α, β ) , (26)where α is constant, β = β m , m is fixed, n ∈ Z . Differentiating equation (26) with respect to ε and using equation (24) together with the compatibility condition (25), one gets d e u , dε − d e u , dε ∂ e u , ξ = f , ∂ u , ξ + f , ∂ u , ξ = e f , − e f , ∂ e u , ξ, where e f k, = f ( e u k +1 , , e u k, , e u k − , ; α ) = f k, , e u k, = u k, . The resulting equation is expressed in the formΞ , = Ξ , ∂ e u , ξ, Ξ k, = d e u k, dε − e f k, . (27)There is for the ABS equations a formal condition ∂ e u , ξ = ∂ u , ξ = 0. We suppose here that,for the functions u , , e u , under consideration, ∂ e u , ξ = 0 for all n ∈ Z . The function e u , isdefined by equation (26) up to an integration function µ = µ m ( ε ). We require that µ satisfiesthe first order ordinary differential equation given by Ξ , | n =0 = 0. Then equation (27) impliesthat Ξ , = 0 for all n , i.e. e u , is a solution of equation (24).So, we start with a solution of a generalized symmetry of the form (24), define a function e u , by the difference equation (26) which is a form of corresponding ABS equation, then we specifythe integration function µ by the ordinary differential equation Ξ , | n =0 = 0, and thus obtaina new solution of equation (24). This solution depends on an integration constant ν = ν m and the parameter β . We can construct in this way the solutions u , , u , , . . . , u ,N , and thelast of them will depend on 2 N arbitrary constants ν , β , ν , β , . . . , ν N − , β N − . Using suchB¨acklund transformation and starting with a simple initial solution, one can obtain, in principle,a multi-soliton solution. See [6, 8] for the construction of some examples of solutions.The symmetries (19), (20) are Volterra-type equations, namely du dε = f ( u , u , u − ) , (28)where we have dropped one of the independent indexes n or m , since it does not vary. TheVolterra equation corresponds to f ( u , u , u − ) = u ( u − u − ). An exhaustive list of differential-difference integrable equations of the form (28) has been obtained in [30] (details can be foundin [31]). All three-point generalized symmetries of the ABS equations, with no explicit depen-dence on n , m , have the same structure as equation (19) (see details in Section 4.4 below) andare particular cases of the YdKN equation du dε = R ( u , u , u − ) u − u − , R ( u , u , u − ) = R = A u u − + B ( u + u − ) + C , (29)n Miura Transformations and Volterra-Type Equations 9where A = c u + 2 c u + c , B = c u + c u + c , C = c u + 2 c u + c , and the c i ’s are constants. equation (29) has been found by Yamilov in [30], discussed in [21, 4],and in most detailed form in [31]. Its continuous limit goes into the Krichever–Novikov equa-tion [15]. This is the only integrable example of the form (28) which cannot be reduced, ingeneral, to the Toda or Volterra equations by Miura-type transformations. Moreover, equa-tion (29) is also related to the Landau–Lifshitz equation [26]. A generalization of equation (29)with nine arbitrary constant coefficients has been considered in [20].By a straightforward computation we get the following result: all three-point generalizedsymmetries in the n -direction with no explicit dependence on n , m for the ABS equations areparticular cases of the YdKN equation. For the various equations of the ABS classification thecoefficients c i , 1 ≤ i ≤
6, readH1 : c = 0 , c = 0 , c = 0 , c = 0 , c = 0 , c = 1 , H2 : c = 0 , c = 0 , c = 0 , c = 0 , c = 1 , c = 2 α, H3 : c = 0 , c = 0 , c = 0 , c = 1 , c = 0 , c = 2 αδ, Q1 : c = 0 , c = 0 , c = − , c = 1 , c = 0 , c = α δ , Q2 : c = 0 , c = 0 , c = 1 , c = − , c = − α , c = α , Q3 : c = 0 , c = 0 , c = − α , c = 2 α ( α + 1) , c = 0 , c = − ( α − δ , Q4 : c = 1 , c = − α, c = α , c = g − α , c = αg + g , c = g + αg . Proposition 2.
The ABS equations (H1)–(H3) and (Q1)–(Q4) correspond to B¨acklund trans-formations of the particular cases of the YdKN equation (29) listed above. The same holds forthe non-autonomous ABS equations, such that α is constant and β = β m or α = α n and β is constant. Equation (29) and the replacement u i → u i, provide the three-point generalizedsymmetries in the n -direction of the ABS equations with a constant α and β = β m , while equa-tion (29) and the replacement u i → u ,i , α → β provide symmetries in the m -direction for thecase α = α n and a constant β . The non-autonomous case is briefly discussed in [24] where they state that if α is not constant,then the ABS equations have no local three-point symmetries in the n -direction. We shall presentthree-, five- and many-point generalized symmetries in the m -direction for such equations inSubsection 4.3.A relation between the ABS equations and differential-difference equations is discussedin [2, 5]. In [2] most of the ABS equations are interpreted as nonlinear superposition principlesfor differential-difference equations of the form( ∂ x u n +1 ) ( ∂ x u n ) = h ( u n +1 , u n ; α ) , (30)where h is a polynomial of u n +1 , u n . Equations of the form (30) define B¨acklund transformationsfor subcases of the Krichever–Novikov equation ∂ t u = ∂ xxx u −
32 ( ∂ xx u ) − P ( u ) ∂ x u , (31)where P is a fourth degree polynomial with arbitrary constant coefficients. In the case ofequations (H1) and (H3) with δ = 0, the corresponding differential-difference equations havea different form, and the resulting KdV-type equations differ from equation (31).In [5] it is shown that the continuous limit of equation (Q4) goes into a subcase of theYdKN equation. It is stated that equation (Q4) defines a B¨acklund transformation for the samesubcase. The same scheme holds for equations (Q1)–(Q3), but it is not clear if the resultingVolterra-type equations are of the form (29).0 D. Levi, M. Petrera, C. Scimiterna and R. Yamilov It is possible to revise the Miura transformations constructed in Section 3 from the point of viewof the generalized symmetries.Let us introduce the following function r = r ( u , u − ) = A u − + 2 B u − + C = R ( u − , u , u − ) . It can be checked that r ( u , u − ) = r ( u − , u ) and, in terms of r the right hand side of equa-tion (29) reads R u − u − = r u − u − + 12 ∂ u − r = r u − u − − ∂ u r . (32)All the ABS equations, up to equation (Q4), are such that c = c = 0, so that the polyno-mial R is of second degree. In this case equation (29) may be transformed [31] into equation (28)with f ( u , u , u − ) = u ( u − u − ) (Volterra equation) by the Miura transformation e u = − r ( u − u )( u − u − ) . The above map brings any solution u of equation (29) with c = c = 0 into a solution e u of the Volterra equation. This is exactly the same Miura transformation we have already pre-sented in Section 3. So, also at the level of the generalized symmetries, we may see that there isa deep relation between equations (H1)–(H3) and (Q1)–(Q3) and the Volterra equation. If equa-tion (29) cannot be transformed to the case with c = c = 0, using a M¨obius transformation,then it cannot be mapped into the Volterra equation by e u = G ( u , u , u − , u , u − , . . . ) [31].Equation (Q4) is of this kind and thus is the only equation of the ABS list which cannot berelated to the Volterra equation. Generalized symmetries of equation (29) will also be compatible with the ABS equations, whichare, according to Proposition 2, their B¨acklund transformations. Such symmetries can be con-structed, using the master symmetry of equation (29) presented in [4].Let us rewrite equation (29) by using the equivalent n -dependent notation (see equation (1)),namely du n dε = f (0) n = R ( u n +1 , u n , u n − ) u n +1 − u n − , (33)where ε is the continuous symmetry parameter (previously denoted with ε ). We shall denotewith ε i , i ≥
1, the parameters corresponding to higher generalized symmetries du n dε i = f ( i ) n , such that df ( j ) n dε i − df ( i ) n dε j = 0 , i, j ≥ . Let us introduce the master symmetry du n dτ = g n , such that df ( i ) n dτ − dg n dε i = f ( i +1) n , i ≥ . (34)Once we know the master symmetry (34) we can construct explicitly the infinite hierarchy ofgeneralized symmetries.n Miura Transformations and Volterra-Type Equations 11The master symmetry of equation (33) is given by g n = nf (0) n . (35)According to a general procedure described in [31] we need to introduce an explicit dependenceon the parameter τ into the master symmetry (35) and into equation (33) itself. Let thecoefficients c i , appearing in the polynomials A n , B n , C n , be functions of τ . This τ -dependenceimplies that r n satisfies the following partial differential equation2 ∂ τ r n = r n ∂ u n ∂ u n − r n − ( ∂ u n r n ) (cid:0) ∂ u n − r n (cid:1) . (36)On the left hand side of the above equation, we differentiate only the coefficients of r n withrespect to τ . The right hand side has the same form as r n , but with different coefficients.Collecting the coefficients of the terms u in u jn − for various powers i and j , we obtain a systemof six ordinary differential equations for the six coefficients c i ( τ ), whose initial conditions are c i (0) = c i . Generalized symmetries constructed by using equation (34) explicitly depend on τ .They remain generalized symmetries for any value of τ , as τ is just a parameter for them andfor equation (33). So, going over to the initial conditions, we get generalized symmetries ofequation (33) and of the corresponding ABS equations.Let us derive, as an illustrative example, a formula for the symmetry f (1) n from equation (34).From equations (33)–(35) it follows that f (1) n = ∂ τ f (0) n + f (0) n +1 ∂ u n +1 f (0) n − f (0) n − ∂ u n − f (0) n . (37)Using equations (32) and (36) one obtains ∂ u n +1 f (0) n = − r n ( u n +1 − u n − ) , ∂ u n − f (0) n = r n +1 ( u n +1 − u n − ) , and ∂ τ R n = R n = R ( u n +1 , u n , u n − ) = A n u n +1 u n − + B n u n +1 + u n − ) + C n , with A n = B n ∂ u n A n − A n ∂ u n B n , B n = C n ∂ u n A n − A n ∂ u n C n , C n = C n ∂ u n B n − B n ∂ u n C n . From equation (37) we get the first generalized symmetry du n dε = f (1) n = R n u n +1 − u n − − r n f (0) n +1 + r n +1 f (0) n − ( u n +1 − u n − ) . (38)Up to our knowledge this formula is new. It provides five-point generalized symmetries inboth n - and m -directions for the ABS equations. Examples of such five-point symmetries forequations (H1) and (Q1) with δ = 0 can be found in [24, 27].Let us clarify the construction of the symmetry f (1) n for equations (H1)–(H3). In these casesthe function r n takes the form r n = 2 c ( τ ) u n u n − + 2 c ( τ )( u n + u n − ) + c ( τ ) , and equation (36) is equivalent to the system ∂ τ c ( τ ) = 0 , ∂ τ c ( τ ) = 0 , ∂ τ c ( τ ) = c ( τ ) c ( τ ) − c ( τ ) . (39)2 D. Levi, M. Petrera, C. Scimiterna and R. YamilovThe initial conditions of system (39) are (see the list above Proposition 2)H1 : c (0) = 0 , c (0) = 0 , c (0) = 1 , H2 : c (0) = 0 , c (0) = 1 , c (0) = 2 α, H3 : c (0) = 1 , c (0) = 0 , c (0) = 2 αδ, and its solutions are given byH1 : c ( τ ) = 0 , c ( τ ) = 0 , c ( τ ) = 1 , H2 : c ( τ ) = 0 , c ( τ ) = 1 , c ( τ ) = 2( α − τ ) , H3 : c ( τ ) = 1 , c ( τ ) = 0 , c ( τ ) = 2 αδe τ . Note that the master symmetry with the above c i ( τ ) generates τ -dependent symmetries fora τ -dependent equation, but by fixing τ we obtain τ -independent symmetries for a τ -independentequation. Let us remark that the τ -dependence is independent of the order of the symmetryand it may be used for the construction of all higher symmetries.So, according to formula (38), we may construct the generalized symmetry f (1) n , in the caseof the list H, from the following expressionsH1 : f (0) n = 1 u n +1 − u n − , r n = 1 , R n = 0 , H2 : f (0) n = u n +1 + u n − + 2( u n + α ) u n +1 − u n − , r n = 2( u n + u n − + α ) , R n = − , H3 : f (0) n = u n ( u n +1 + u n − ) + 2 αδu n +1 − u n − , r n = 2( u n u n − + αδ ) , R n = 2 αδ. It is possible to verify that the symmetries (38) with f (0) n , r n , R n given above are compatiblewith both equations (33) and (H1)–(H3).By using the master symmetry constructed above we can construct infinite hierarchies ofmany-point generalized symmetries of the ABS equations in both directions. In the non-autonomous cases (see Proposition 2) we provide one hierarchy in the n - or m -direction. Themaster symmetry and formula (38) will also be useful in the case of the generalizations of theABS equations presented in the next Subsection. It should be remarked that in [24] the authorsconstructed master symmetries for all autonomous and non-autonomous ABS equations, whichare of a different kind with respect to the ones presented here. Here we discuss the generalization of the ABS equations introduced by Tongas, Tsoubelis andXenitidis (TTX) in [28]. The TTX equations are autonomous lattice equations of the form (2)which possess only two of the four main properties of the ABS equations: they are affine linearand possess the symmetries of the square.In terms of the polynomial E , see equation (2), one generates the following function hh ( u , , u , ; α, β ) = E ∂ u , ∂ u , E − (cid:0) ∂ u , E (cid:1) (cid:0) ∂ u , E (cid:1) , which is a biquadratic and symmetric polynomial in its first two arguments. It has been provedin [28] that the TTX equations admit three-point generalized symmetries in the n -direction ofthe form du , dε = hu , − u − , − ∂ u , h. (40)n Miura Transformations and Volterra-Type Equations 13Of course, there is a similar symmetry in the m -direction. Comparing equations (29), (32)and (40), we see that the symmetry (40) is nothing but the YdKN equation in its general form.This shows that all TTX equations can also be considered as B¨acklund transformations forthe YdKN equation. However, they probably describe the general picture for B¨acklund trans-formations of the YdKN equation, which have the form (2). The general formula (38) and themaster symmetry discussed in the previous Subsection, provide five- and many-point generalizedsymmetries of the TTX equations in both directions, thus confirming their integrability. In this paper we have considered some further properties of the ABS equations. In particularwe have shown that equations (H1)–(H3) and (Q1)–(Q3) can be transformed into equations as-sociated with the spectral problem of the Volterra equation. Therefore all known results for thesolution of the Volterra equation can be used to construct solutions of the ABS equations. More-over, all equations of the ABS list, except equation (Q4), can be transformed among themselvesby Miura transformations.The situation of equation (Q4) is somehow different. It is shown that this equation canbe thought as a B¨acklund transformation for a subcase of the Yamilov discretization of theKrichever–Novikov equation. But it cannot be related by a Miura transformation to a Volterra-type equation and this explains the complicate form of its scalar spectral problem. The mastersymmetry constructed for the YdKN equation can, however, be used also in this case to constructgeneralized symmetries.It turns out that a generalizations of the ABS equations introduced by Tongas, Tsoubelisand Xenitidis are B¨acklund transformations for the YdKN equation.Further generalizations of the TTX and ABS equations can be probably obtained by a properexplicit dependence on the point of the lattice not only in the lattice parameters α and β , butalso in the Z -lattice equation itself. The existence of an n -dependent generalization of theYdKN equation, introduced in [20], could help in solving this problem. Such a generalization isintegrable in the sense that it has a master symmetry [4] similar to the one presented here. Acknowledgments
DL, MP and CS have been partially supported by PRIN Project
Metodi geometrici nella teoriadelle onde non lineari ed applicazioni-2006 of the Italian Minister for Education and ScientificResearch. RY has been partially supported by the Russian Foundation for Basic Research(Grant numbers 07-01-00081-a and 06-01-92051-KE-a) and he thanks the University of RomaTre for hospitality. This work has been done in the framework of the Project
Classification ofintegrable discrete and continuous models financed by a joint grant from EINSTEIN consortiumand RFBR.
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