Darboux transformation with Dihedral reduction group

Transcription

1 Darbou transformation with Dihedral reduction group Aleander V. Mikhailov, Georgios Papamikos and Jing Ping Wang School of Mathematics, Statistics & Actuarial Science, University of Kent, UK Applied Mathematics Department, University of Leeds, UK Abstract We construct the Darbou transformation with Dihedral reduction group for the 2 dimensional generalisation of the periodic Volterra lattice. The resulting Bäcklund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the La representation for this symmetry. Using formal diagonalisation of the Darbou matri we obtain local conservation laws of the system. Introduction The connection among integrable partial differential equations, Darbou transformations of the corresponding La operators and partial difference equations and the corresponding Bäcklund transformations is well established see, for instance, [, 2, 3, 4]. Namely, the Darbou transformations serve as La representations for integrable difference equations, while the Bäcklund transformations are symmetries of these difference equations and are integrable differential-difference equations in their own right. The motivation of this research is to etend the reduction groups [5, 6, 7] of La representations of integrable partial differential equations to integrable difference equations via Darbou transformations, and ultimately to describe all elementary Darbou transformations corresponding to affine Lie algebras and automorphic Lie algebras with finite reduction groups. Recently, the authors of [8, 9] have completed a comprehensive study for the La operators of the nonlinear Schrödinger equation type and they derived some new discrete equations and new maps. Here we go beyond sl 2 La representations and take outer automorphisms into account. The concept of reduction groups for La representations of partial differential equations was first introduced in [5, 6, 7]. Later, a new class of infinite dimensional quasi-graded Lie algebra called automorphic Lie algebras based on a reduction group was studied in [0]. The reduction groups have been used to classify La representations and their corresponding integrable equations [, 2]. The study of automorphic Lie algebras has become a research topic in its own right [3, 4]. In this paper, we construct the Darbou transformation for the 2 dimensional generalisation of the Volterra lattice [5]: { φ i t = θ i + θ i φ i e 2φi + e 2φi+, + φ i = 0, θ i+ θ i + φ i+ φ i+n = φ i, θ i+n = θ i, n φ i = i= n θ i = 0. This system can be viewed as a discretisation of the Kadomtsev-Petviashvili equation. Indeed, in the limit n, φ i, t = h 2 uξ, η, τ, h = n, system goes to τ = h 3 t, ξ = ih + 4ht, η = h 2, u τ = 2 3 u ξξξ + 8uu ξ 2D ξ u ηη + Oh 2. i=

2 The eact solutions of system have much in common with 2 + -dimensional integrable equations, which have recently been investigated by Bury and Mikhailov [2, 5]. For fied period n, it is a bi- Hamiltonian system. When n = 3, its recursion operator and bi-hamiltonian structure are eplicitly constructed from its La representation in [6]. The La representation of is invariant under the the dihedral reduction group D n [5, 7, 0,, 2] with both inner and outer Lie algebra automorphisms. To the best of our knowledge, this is the first eample in which symmetries of a Darbou transformation are also generated by outer automorphisms. We rigorously prove that a Bäcklund transformation for the 2 dimensional generalisation of the Volterra lattice is S φ k = n 2 Ω 2 a k b k p k epφ k, 2 where S is a shift operator and Ω is an n periodic shift operator acting on the upper inde, and a k, b k and p k are defined in terms of φ i and φ i = Sφ i, i, k =,, n by p i = ep{ k=i k= k n φk φ k }; a i2 = pi2 2n i i l= pl2 2n ; b i = ai l=i p l2 p n2. l= pl2 n 2n i p i Moreover, for the nonevolutionary integrable differential difference equation 2, we find its generalise symmetry φ k n τ = < b, a > Ω b k a k, where < b, a >= n i= bi a i and conserved densities ρ 0 = log < b, a >; ρ = < b, a >< b, a > n Γ k q b k ak b i a i, Γ l = l mod n n. 2 k,i= The arrangement of this paper is as follows: In Section 2, we give basic definitions related to the La Darbou scheme such as Darbou transformation, Darbou map and Bäcklund transformation. Meanwhile, we also fi notations. In Section 3, we parametrise the rank matri with simple poles invariant under the dihedral group D n. Then in Section 4 we provide a rigorous proof for the rank Darbou transformation of the 2 dimensional generalisation of Volterra lattice and give the corresponding Bäcklund transformation, which can be viewed as an integrable nonevolutionary differential difference equation. In Section 5, we construct a local generalised symmetry for this nonevolutionary equation using the Darbou matri. We also provide the La representation for this symmetry flow viewed as an evolutionary integrable equation. In Section 6, we transform the La pair for this symmetry flow into a formal block-diagonal form to get its conservation laws. The nonevolutionary integrable equation and its symmetry share the same conserved densities. Finally, we give a brief conclusion on what we have done in this paper and a short discussion about open problems. To improve the readability, we put the technical results required in the Appendi. 2 Generalised Volterra lattice and related structures In this section, we discuss the known La representation for the 2 dimensional generalisation of the Volterra lattice and its reduction group. Meanwhile we fi notations and introduce the definitions related to the La Darbou scheme such as a Darbou transformation, Darbou map and Bäcklund transformation. 2

3 Consider the following linear differential operators with n n matri coefficients [5, 7] L = D t V, V = λu λ u, L 2 = D t2 V 2, V 2 = λ 2 u u + λa λ a λ 2 u u, 3 where t =, t 2 = t, u = diagepφ i, a = diagθ i epφ i are diagonal matrices and ij = δ i,j whose indees are count modulo n, i.e., = This convention to count indees modulo n will be used throughout the whole paper, thus φ i+n = φ i and θ i+n = θ i. We introduce an n periodic shift operator as follows: Ωφ i = φ i+ for i =,, n and Ωφ n = φ. We set n i= φi = n i= θi = 0 without losing generality. The condition of commutativity of these operators [L, L 2 ] = D t V D V 2 + [V, V 2 ] = 0 5 leads to the 2 dimensional generalisation of the Volterra lattice. This is often called a zero curvature representation or La representation of equation. These two operators, L and L 2, are conventionally called the La pair. The commutativity of operators can be seen as a compatibility condition for the linear problems D Ψ = V u; λψ, D t Ψ = V 2 u; λψ, 6 i.e. the condition for the eistence of a common fundamental solution Ψ det Ψ 0. The operators L i are invariant under the group of automorphisms generated by following two transformations: s : L i λ QL i λωq and r : L i λ L it λ, Q = diagω i, ω = ep 2πi n. 7 Here T is the transpose of a matri. These two transformations satisfy s n = r 2 = id, rsr = s and therefore generate the dihedral group D n [5, 7, 0,, 2]. Note that the transformation r is an outer automorphism of the Lie algebra sln over the Laurent polynomial ring C[λ, λ ]. A Darbou transformation is a linear map acting on a fundamental solution Ψ Ψ = M Ψ, det M 0 8 such that the matri function Ψ is a fundamental solution of the linear problems D Ψ = V u; λψ, D t Ψ = V 2 u; λψ 9 with new potentials u. The matri M is often called the Darbou matri. From the compatibility of 8 and 9 it follows that D ti M = V i u; λm M V i u; λ, i =, 2. 0 Equations 0 are differential equations which relate two solutions u and ū of. In the literature they are also often called Bäcklund transformations. A Darbou transformation maps one compatible system 6 into another one 9. It defines a Darbou map S : u u. The map 8 is invertible det M 0 and it can be iterated Ψ S Ψ S Ψ S Ψ S. 3

4 It suggests notations... Ψ = Ψ, Ψ 0 = Ψ, Ψ = Ψ, Ψ 2 = Ψ,...,... u = u, u 0 = u, u = u, u 2 = u,.... With a verte k of the one dimensional lattice Z we associate variables Ψ k and u k ; with the edges joining vertices k and k + we associate the matri S k M. In these notations the Darbou maps S and S increase and decrease the subscript inde by one, and therefore we shall call it a S shift, or shift operator S. In what follows we often shall omit zero in the subscript inde and write u instead of u 0. The map S is an automorphism of the La structure 6 and a discrete symmetry of system. In these notations the resulting Bäcklund transformations from 0 are integrable differential difference equations and 0 are their La-Darbou representations. A Darbou transformation with matri M with a parameter results in the S shift. If we also consider a Darbou transformation M ν with a different choice of the parameter, then the corresponding shift automorphism of the La structure 6 we denote T. The compatibility condition similar to 0 reads D ti M ν = T V i M ν M ν V i, i =, 2.. These shifts act on Z 2 lattice and with the verte n, m we associate variables u n,m, so that Su n,m = u n+,m, T u n,m = u n,m+. Commutativity of the shifts S and T leads to a system of partial-difference equations T M M ν SM ν M = 0. Differential difference equations 0, are the symmetries of this partial-difference equation. Indeed D ti T M M ν SM ν M = T SV i T M M ν SM ν M T M M ν SM ν M V i = 0. The aim of this paper is to construct a Darbou matri M for the operators L and the corresponding integrable Bäcklund transformation chain. We also show how to construct local symmetries for this chain and the corresponding La operators. 3 Parametrisation of D n invariant matrices M λ with simple poles In this section, we show that a rational in λ matri which is invariant under the dihedral group D n and has simple poles at generic orbit {ω i, i = 0,,, n } can be completely parametrised by n variables. We assume that the Darbou matri M inherits the same reduction group 7, although the action of the automorphsm r is different since M is an element of a Lie group rather than Lie algebra. The matri M satisfies M λ = M T λ, QM λωq = M λ 2 and has simple poles at ω i, k =, 2,, n ω i 0, ±. We thus take it of the following form: Q i AQ i M λ = C + λω i, 3 where C and A are n n matrices. We now study the constraints on the Darbou matri M following from the condition 2. It follows from the second identity in 2 that matri C satisfies QCQ = C and thus C is diagonal. We write C = diagp i. 4

5 Proposition. Let I n denote the identity n n matri. The matri M λ given by 3 is invariant under group D n if and only if the matri A and diagonal matri C satisfy the relations: C 2 Q i A T CQ i = I n ; CA + Q j A T Q j A ω j 2 = 0. 4 Proof. To prove the statement, we need to check the identities in 2. It is obvious that the given M λ satisfies QM λωq = M λ. We now compute M T λ Q M λ = C j A T Q j + Q i AQ i ω j λ C + λω i which is the identity matri according to 2. Taking the limit λ we obtain the first equation in 4. Requesting the vanishing of the residue at λ = ν we obtain the second equation in 4. The residues at all other points of the reduction group orbit will vanish due to the manifest invariants of the epression with respect to the reduction group. Using the first two lemmas in the Appendi, it follows from this proposition that the Darbou matri M λ is trivial if the matri A is non-singular det A 0. Corollary. If det A 0, then C 2 = 2n I n and M λ = λn n λ n n C. Proof. If det A 0, it follows from the second identity in 4 that C + Q j A T Q j ω j 2 = 0. It follows from Lemma 2 in Appendi that matri A is diagonal since C is diagonal, and thus C = Q j A T Q j ω j 2 = ω j 2 A = n2 2n A, where we used Lemma 2 in Appendi for the last equality. Substituting this into the first identity in 4, we have C 2 n AC = 2n C2 = I n, that is, C 2 = 2n I n. Moreover, we have M λ = C + A λω i = C 2n λn 2 λ ω i = C 2n λ λ 2 λ n. Simplifying the above epression, we obtain the formula for M λ in the statement. We define the rank of the Darbou transformation as the rank of matri A. In this paper we restrict ourself with Darbou matrices M of rank. We represent the matri A by a bi-vector A = a >< b, where a >= a,..., a n T and < b = b,..., b n are column and row vectors respectively and thus A i,j = a i b j. Theorem. For a rank matri A = a >< b, and a diagonal matri C = diagp i, the matri M λ given by 3 is invariant under group D n if where i =, 2,, n and n l= pl2 = 2n. l= pl2 a i2 = pi2 i 2i p n 2 ; 5 b i = ai l=i p l2 p n2 n 2n i p i, 6 5

6 Proof. To prove this statement we need to check the conditions 4 in Proposition for the given matri A. It follows from Lemma that matri Q i A T CQ i is diagonal. The diagonal entries of the first identity of 4 are p i2 n ai b i p i =, i =, 2,, n. 7 The matri entries of the second identity of 4 can be represented as 0 = p k a k b l + r= n ω jr k b k a r2 b l ω j 2, k, l =, 2,, n. Notice that all terms have a factor b l. For a nonzero vector b, it is equivalent to 0 = p k a k + r= n ω jr k b k a r2 ω j 2, k =, 2,, n. 8 Notice that there are only n independent relations, from which we can determine all b k as follows: b k = p k a k n r= γ r k 2 a, γ r k 2 = r2 ω jr k which is the direct result from Lemma 2 in Appendi. This leads to b k = 2n a k p k 2 ω j = n2{r k mod n} 2n, n. 9 n r= ar2 2{r k mod n} These formulas for b k have appeared in [2], where the author used them when computing soliton solutions for the Volterra system. Combining 7 and 9, we obtain the following system of linear homogeneous equations for a i2 : 2 p i2 n k=,k i a k2 2{k i mod n} + p i2 2n a i2 = 0. This system has a nontrivial solution if n p l2 = 2n. 20 l= Setting a n2 = we obtain a unique solution of this linear system a j2 = pj2 2n j j l= pl2 2n l= pl2, j =, 2,, n, which can be brought to the form 5 in the statement using 20. Here we use the convention that 0 l= pl2 =. Substituting 5 into 7, we get b i = pi2 = ai 2n na i p i n 2n i p i i l= pl2 = ai l= pl2 where we used 2n = n l= pl2, and we complete the proof. l=i p l2 p n2, n 2n i p i Following from this theorem, the matri M λ is completely parametrised by p i, i =, 2,, n. 6

7 4 Darbou transformation for the generalised Volterra lattice We construct the rank Darbou matri M, invariant under the D n reduction group, for the La operator L. From the compatibility condition 0, we know M satisfy D M = SV M M V 2 First we write out the right hand of the above identity. It equals to λu λ u C + Q i AQ i λω i C + = λu C Cu + λ C u u C + + ω i + ω i λω i u Q i AQ i Q i AQ i u λ ω i λω i u Q i AQ i Q i AQ i u. Q i AQ i λω i λu λ u Notice that ω i Q i Q i =. Then we compare the residues at different poles λ = +, 0, and constant terms on both sides of 2. The zero curvature condition 2 is equivalent to the following four identities: u C Cu = 0; 22 C u u C + u Q i AQ i Q i AQ i u = 0; 23 A = u A Au u A A u; 24 C = It follows from 22 that ω i u Q i AQ i Q i AQ i u. 25 p i+ = p i epφ i φ i = p ep Substituting 26 into the identity n i= pi2 = 2n from Theorem, we have p 2n which leads to n i=2 i ep2 k= Using the relation 26, we obtain φ k φ k i φ k φ k. 26 k= = p2n ep2 n kφ k φ k = 2n, k= p = ep k n φk φ k. p i = ep{ It follows from Lemma in Appendi that k=i k= k= ω i Qi AQ i = n k n φk φ k }. 27 n a j b j e j,j. j= 7

8 From 25 we get p k = n a k+ b k epφ k ak b k epφ k, k =, 2,, n, 28 where the upper indees are again counted modulo n. So far, we only deal with two of four equivalent identities of the zero curvature condition 2. We used 22 and 25 to obtain 27 and 28, respectively. To claim that we have obtain a Darbou transformation and further the Bäcklund transformation, we need to show that the other two identities are satisfied due to the reduction group invariance of the matri M. Indeed, it can be easily checked that the identity 23 holds from 22 as follows: We know that Qi AQ i = n diag A following from Lemma in Appendi. So the non-zero entries of the left side of 23 are = p i+ epφ i p i epφ i + n ai b i epφ i n ai+ b i+ epφ i p i+ epφi epφi p i = 0, where we used formulas 7 and 26. Using 28, 5 and 6, we are able to check identity 24 is valid as shown in Proposition 3 in Appendi. Thus we have the following result: Theorem 2. The Bäcklund transformation for system is S φ k = n 2 Ω 2 a k b k p k epφ k, 29 where p i, a i and b i can be epressed via 27, 5 and 6 in terms of φ k, and Ω is the n periodic shift operator. Proof. We have shown that the zero curvature condition 2 are equivalent to 27 and 28. It leads from 26 that the dependent variables φ k, k =, 2,, n are determined by p i as follows: S φ k = ln p k ln p k+ = Ω ln p k. 30 Differentiating both sides of 30 with respect to and using 28, we get S φ k a k+ b k epφ k = nω ak b k epφ k p k p k a k+ b k epφ k = nω ak b k epφ k p k+ p k = nω 2 a k b k epφ k p k It follows from 6 that 2 a k b k = a k b k p k p k. Substituting it into the above formula, we get the Bäcklund transformation as stated. The matri M satisfies the relation in Theorem, together with the relations 27 and 28 it is the Darbou transformation for the operator L given by 3. It can be verified that the right-hand side of 29 is not in the image of S of any function of a finite number of variables φ m l, m =,..., n. Thus the differential-difference equation for φ i is not evolutionary. However, equation 29 possesses a proper zero curvature representation and is integrable in this sense. A direct search for its symmetries is a rather hard task for arbitrary n. In the net section, we show that there eists another La operator with matri part U such that the zero curvature condition between M and U defined by 34 naturally leads to the symmetry flow for the nonevolutionary equation

9 5 An evolutionary flow of the Bäcklund transformation The Bäcklund transformation 29 given in the previous section can be viewed as an integrable differential difference equation. For arbitrary n, it is not easy to directly compute the generalised symmetries for this nonevolutionary system according to the following definition. In this section, we compute one of its higher symmetries using the Darbou matri M. Meanwhile, we also provide its La representation. We first give the general definition of symmetry. More details on it can be found in [7]. Definition. Given a k-component system Q[u] = 0, we say an evolutionary vector P[u] is its symmetry if and only if the system is invariant along the flow u τ = P, that is, D τ Q = 0. In what follows, we construct a symmetry using the Darbou matri M and check the resulting flow is indeed a symmetry of 29 according to the above definition. Notice that 2 is equivalent to M D M = M SV M V. 3 Its left hand side has simple poles at points of the generic orbit, that is, ω i, ω i i = 0,,, n, and matri V has simple poles at points of a degenerated orbit λ = 0,. We replace matri V by a matri U and require that U has poles consistent to the left hand side of 3. In other words, we have SΨ = Ψ = M Ψ and D τ Ψ = U Ψ. 32 It follows from 32 that matri U inherits the same reduction group 7, that is, it satisfies U λ = U T λ, QU λωq = U λ. 33 Thus it is of the following form: Q i BQ i U λ = λω i Q i B T Q i ω i λ, 34 where B is an n n matri. Unlike what we have seen for the Darbou matri M, such U λ satisfying 33 is automatically invariant under the D n group. We now derive the differential-difference equation for p i, i =, 2,, n using the zero curvature condition or it can be written eplicitly in the form C τ + + C + D τ M = SU M M U, 35 Q i A τ Q i λω i Q i AQ i λω i Q i B Q i λω i Q i B T Q i ω i λ Q i BQ i λω i Q i B T Q i ω i λ C + = 0. Q i AQ i λω i The vanishing conditions for the second order pole at λ =, the value at λ =, and the residues at 9

10 λ = and λ = have the form: B A AB = 0; 36 C τ = B T C CB T Q i B T C CB T Q i ; 37 = A τ = B C CB + Q i AQ i B T B T Q i AQ i j= ω i 2 ; 38 B Q j A AQ j BQ j + Q j B Q j A AQ j B ω j AQ j B T Q j Q j B T Q j A ω j Due to the reduction group symmetry all singularities vanishes and thus the above conditions are equivalent to 35. It is easy to check that B = a >< b < b, a > 40 is a solution of 36 when A = a >< b. It follows from 37 and Lemma see Appendi that p i τ = n b i ai p i < b, a > bi a i pi = n b i a i < b, a > pi S < b, a >, 4 where a and b satisfy 5 and 6. The consistent condition 38 can be obtained from 40. To see this, we are going to show that B T C = B T Q i AQ i 2 ω i and CB T = Q i AQ i B T 2 ω i. Indeed, we write out the entries for the matrices by substituting 40 and A = a >< b into them. They both are equivalent to the identity 8. Finally, according to Proposition 4 in Appendi, the identity 39 follows from 4. Thus we obtain the following result: Theorem 3. Let A = a >< b and B be defined by 40. The evolutionary differential-difference equation 4 possesses a La representation 35 with Q i AQ i M = C + λω i, U = Qi BQ i λω i Q i B T Q i ω i λ, where C = diagp i, Q = diagω i, ω = ep 2πi n, and a and b satisfy 5 and 6 respectively. Using 30, we obtain the evolutionary differential-difference equations for φ k from 4, that is, φ i p i τ τ = Ω = n b i p i Ω a i < b, a > 42 Notice that both equations 29 and 42 are obtained from the same Darbou matri M. This implies that both of them share the same generalised symmetries and conserved densities derived from the zero curvature conditions [8, 9, 20]. Equation 42 is evolutionary, which can be viewed as a symmetry of the nonevolutionary equation 29. We are going to show it by direct calculation in the following theorem. Theorem 4. Equation 42 is a symmetry of the nonevolutionary equation 29. 0

11 Proof. To prove this statement we need to show that equation 29 is invariant along the flow 42, that is, to check S D φ i τ = n Ω 2 a i b i p i epφ i. 2 τ Therefore, we only need to show b i a i S D = Ω a i b i p i epφ i. 43 < b, a > τ Using the epressions for a i This leads to and b j b i a i + bi a i, = Ω +a i bi S in the proof of Proposition 3 in Appendi, we get ai b i a pn epφn b i a i < b, a > = bi a i + bi a i, < b, a > = Ω < b, a > ai p an + a i bi epφ i b epφ p n 2 n b bi a i < b, a, > + < b, a > < b, a > 2 b i To prove the identity 43, we now only need to show that S < b, a > ai b i + 2 a i bi epφ i = τ which is the identity 70 proved in Appendi. + a i bi epφ i.. 44 a i b i p i epφ i, 6 Formal diagonalisation of the La-Darbou scheme and Conservation laws In this section, we study the conservation laws for integrable equation 29. Given the La representation of an equation, its conservation laws, both conserved densities and conserved flues, can be computed by preforming formal diagonalisation of its La pair [8]. This idea has been adapted to differential difference and partial difference equations. See, for eample, [9, 20]. Instead of directly working on the La pair of 29, we work on the La pair of its symmetry. Then we prove that the obtained conserved densities are the conserved densities of 29. Consider the gauge transformation Ψ = W λ Ψ. It follows from 32 that the La operators transform into M λ M λ = SW λm λw λ; U λ U λ = W λu λw λ W λd τ W λ. In this section we shall show that both operator U λ and the Darbou matri M λ defined in Theorem 3 can be simultaneously brought to a formal block-diagonal form by a suitable transformation W λ. In other words, we show that there eists W λ such that U λ = π λ + U 0 + λ U + λ 2 U 2 + ; M λ = < b, a > π λ + M 0 + λ M + λ 2 M 2 +,

12 where π is a matri a projector with at the, position and 0 elsewhere, and also ad π U k = ad π M k = 0, k = 0,, 2,..., i.e. the coefficient n n matrices U k and M k have a block-diagonal form We shall also show that the entries of M k and U k are local, which means that they can be epressed in terms of variables φ i and their S-shifts. Obviously, it follows from 35 that the transformed operators also satisfy the zero curvature condition The projection to the element, leads to D τ M λ = SU λm λ M λu λ. 45 D τ mz = mzs uz. Here we introduced a new parameter z = λ and denoted mz = M z +, and uz = U z +,. Thus log mz and uz are generating functions of local conservation laws and corresponding flues D τ ρ k = S σ k, log mz = logz + z k ρ k, uz = z + z k σ k. 46 We represent the transformation W z + in the form k=0 W = W W, W = I + zw + z 2 W 2 + z 3 W 3 +, where W is a z-independent invertible matri and W is a formal series in z with off block-diagonal coefficients W k in the image of ad π. The entries of W and W k are local. The gauge transformation W brings simultaneously the residues to the diagonal form W res λ= U λ = a >< b < b, a >, res λ=m λ = a >< b 47 a >< b < b, a > W = π, SW a >< bw =< b, a > π, while the formal series W takes care of the regular in z parts. To construct the gauge transformation W we note that vector < b is non-zero and therefore for some k it has a non-zero component b k 0. Let α k > denotes a vector-column with at the k-th position and zeros elsewhere and < α k is the transpose of α k >, i.e., < α k = α k > T. Then matri W can be written in the form W = b k I α k >< b + a >< α k k where is a matri defined in 4. One can check that det W = b k n 2 < b, a > 0 The gauge transformation W brings the singular parts of U z + and M z + to a diagonal form Ûz = W U z + W W D τ W = z π + Û0 + zû + z 2 Û 2 + ˆMz = SW M z + W = z < p q > π + ˆM 0 + z ˆM + z 2 ˆM2 +. k=0 2

13 The coefficients of the regular part can be easily found, but they are not yet in the block-diagonal form. For eample, we obtain Û 0 = < b, a > W Q i a >< bq i n Q i b >< a Q i ω i ω i 2 W W D τ W, i= i= ˆM 0 = SW Q i a >< bq i C + ω i W. In particular, we have i= Û0, = < b, a.τ > < b, a > < b, a > 2 ˆM 0, = < b, Ca > < b, a > < b, a > n k,q= n k,q= Γ k q b k a k bq a q where the functions Γ k, γ k were introduced in Lemma 2 in Appendi. 2 γ q k 2 b k a q 2,48 Γ k q b k ak b q a q, 49 Now we can construct a formal series W which transforms simultaneously Ûz and ˆMz into a blockdiagonal form. Actually, it follows from equation 45 that if we formally block diagonalise the Darbou matri, then the corresponding La operator must also be block-diagonal and vice-versa. Proposition 2. There eists a unique formal series W = I + zw + z 2 W 2 + z 3 W 3 + with W k ad π Mat n n F such that all coefficients M k of are block-diagonal. Mz = SW ˆMzW = z < b, a > π + M 0 + zm + z 2 M Proof. All matrices in the equation 50 we split into four blocks associated with the projectorπ: m < 0 ˆm < f < q Mz =, ˆMz =, W =, 0 > N g > ˆN r > I where 0 >, g >, r > and < 0, < f, < q are n dimensional column and row vectors respectively, N, ˆN, I are square n n matrices. We rewrite equation 50 in the form ˆMzW = SWMz 5 and split it into four blocks as above. It leads to one scalar, two vector and one matri equations Using 52 and 55 we eliminate m and N from 53 and 54 Substitution of the formal epansions ˆm+ < f, r >= m, 52 ˆN r > +g >= Sr >m, 53 ˆm < q+ < f = S< qn, 54 g >< f + ˆN = N. 55 ˆN r > +g >= Sr > ˆm + Sr > < f, r > 56 ˆm < q+ < f = S< qg >< f + S< q ˆN 57 ˆm = z < b, a > + ˆm 0 + z ˆm + z 2 ˆm 2 + z 3 ˆm 3, ˆN = ˆN 0 + z ˆN + z 2 ˆN2 + z 3 ˆN3, r >= zr > +z 2 r 2 > +z 3 r 3 >, < q = z < q + z 2 < q 2 + z 3 < q 3 3

14 leads to recurrence relations for determining elements r k > and < q k. From 56 it follows that k i+j k r k+ >= < b, a 2 > S g k > + ˆN i r k i > Sr k i > ˆm i Sr i > < f k i j, r j >. Then the coefficients m k in the epansion are determined by 52 i,j= m = z < b, a > +m 0 + zm + z 2 m 2 + z 3 m 3 m k = ˆm k + 58 k < f k i, r i >. 59 i= Coefficients N k in the epansion N = N 0 + zn + z 2 N 2 + and < q k can be uniquely found from 54, 55 in a similar way. It follows from 59 and 49 that m = z < b, a > < b, a > n k,q= Γ k q b k ak b q a q + Oz and thus the conserved densities densities 46 for equation 4 are of the form ρ 0 = log < b, a >, 60 n ρ = Γ k q b k < b, a >< b, a > b q a q. 6 We have formally diagonalised the Darbou matri M in Proposition 2. It follows from 2 that the corresponding La operator V must also be block-diagonal. Therefore, we have the following result: Corollary 2. The above epressions 60 and 6 are conserved densities for the integrable nonevolutionary equation 29 given in Theorem 2, where a and b are defined in Theorem. In fact, we can directly check D ρ k ImS for k = 0, for equation 29. Here we only present it for k = 0. Using 44 we have D ρ 0 = < b, a > = S n i= a pn epφn k,q= b i a i + bi a i, p an b epφ p n 2 n b. 7 Discussion In this paper, we construct a Darbou transformation with Dihedral reduction group for the 2-dimensional generalisation of the Volterra lattice with period n. The reduction group enables us to parametrise the Darbou matri by n dependent variables. The Dihedral reduction group is generated by both inner automorphisms and outer automorphisms. To the best of our knowledge, this is the first eample to deal with arbitrary n and the outer automorphisms. The Bäcklund transformation resulting from the Darbou transformation can be viewed as a nonevolutionary multi-component integrable differential difference equation. Assuming the La operator having 4

15 the same simple poles as the logarithmic derivative of Darbou matri, we obtain its local higher symmetry. In the similar way, local symmetries can be found for the ABS equations [2, 22]. In this paper, we only investigated the conservation laws of this nonevolutionary multi-component integrable differential difference equation. To obtain them we transform the Darbou matri and the La operator into block-diagonal form. The corresponding recursion operator and bi-hamiltonian structure are not studied yet. Knowing the La representation, in principle, we are able to construct the recursion operator for a fied period n as it was done for equation in [6] applying the method given in [23]. It would be interesting to see whether that can be done for arbitrary n as in the case for the Narita-Itoh-Bogoyavlensky lattice [24]. Following from the Bianchi commutativity, we can construct a new integrable discrete equation with Dihedral reduction group using the Darbou matri, which we have not included in this paper since we could not write it down in a neat way. The integrable nonevolutionary system and the symmetry flow are its nonlocal and local symmetry, respectively. It would be interesting to find continuous limits of the systems obtained. Appendi: Technical results used in the proofs We first give two simple lemmas, which will be used to simplify epression throughout the paper. We then give the detailed computation to check the consistency required for the zero curvature conditions. Lemma. Let P be an n n matri, Q = diagω, ω 2,..., ω,, and ω = ep 2πi n. Then n n ω km Q m P Q m = m= n P i,i+k e i,i+k, 62 where all indees are counted modulo n, matri e i,j has a unit entry at the position i, j and zero elsewhere. In the case k = 0 it is a projection to the diagonal part of the matri P, i.e., n i= n Q m P Q m = diag P. Proof. Let us compute the i, j entry of n m= ωkm Q m P Q m, which equals to n { ω km+im jm 0 k + i j 0 mod n P i,j = np i,j k + i j 0 mod n. m= m= This is eactly the right hand of the identity in the statement. Lemma 2. Let ω = ep 2πi n. Then γ l = ω lj ω j = nl mod n n and Γ l = j= ω lj ω n = l mod n. j 2 Proof. We first prove the identity for γ 0. Since ω is a primitive n th root of unity, we have n = ω j 63 We now take logarithm of both sides of 63 and get ln n = ln ωj. We then differentiate it with respect to. This leads to the value for l = 0, that is, γ 0 = ω j = n n = n mod n n. 5

16 Notice that ω lj l = ω j l r=0 ωrj l r for 0 < l < n. Thus we have γ l = ω lj ω j = l = l γ 0 r=0 l ω j + ω lj l ω j ω rj l r = nn+l n nl = nl We know that ω r = ω l if l r 0 mod n. Thus γ l = γ r if r l mod n. n = nl mod n n. We now prove the second part of the statement based on the formula for γ l. We have Γ l = j= ω lj ω j = lim and hence we complete the proof. j= ω lj n l mod n ω j = lim n n l mod n = lim n l=0 l = l mod n n 2 In Theorem, we give the relation between the entries of matri A and those of matri C. In the following lemma we give the formulas on their derivatives. Lemma 3. Consider p j, j =, 2,, n, are the smooth function of variable τ. Then a i τ = a i p i p i i τ p i2 + p r τ p p n τ ; 64 r r= p n p n2 b i τ = b i a i τ a + 2p r τ pi τ i p r p + 2pn p n τ. 65 i p n2 r=i Proof. Taking the logarithmic derivatives in τ of 5 and 6 we obtain 64 and 65 respectively. Proposition 3. For the matri M satisfying Theorem, identity 24 holds given formula 28. Proof. To compute the left-hand side of 24, we first compute a and b. Using the above lemma, Theorem and 28, we have a i = a i p i p i i p i2 + p r p p n r r= p n p n2 pi a i+ b i epφ i i ai b i epφ i = na p i2 = n p i2 ai b i p i i +na i na i r= a r+ b r epφ r ar b r epφ r p r a b n epφ n an b epφ p n p n2 = a i+ epφ i ai b i epφ i + nai2 b i epφ i b i p i epφr p r = epφr p r+ p n2 = n an b n p n nai a b n epφ n p ai a epφ n p a n p n + ai b epφ p n2 b n = a i+ epφ i ai b i epφ i ai a p n epφ n + ai b epφ b i p i2 p a n p n2 b n 6

17 and b j = b j aj a + 2p r j p r r=j p + 2pn p n j p n2 pj = aj+ b j epφ j bj epφ j bj a p n epφ n a j p j2 p a n + bj b epφ + 2nbj a n b epφ nbj a j b j epφ j p n2 b n p n p j nbj2 a j+ epφ j p j + 2bj a epφ n 2bj b epφ a n b n = aj+ b j epφ j a j p j2 b j epφ j + bj a p n epφ n p a n bj b epφ p n2 b n. Notice that 2 a i+ b i = a i b i+ p i p i+ following from 6. Therefore, we have a i b j = a i b j + a i b j = epφ i ai+ b j a i b j epφ j epφ i a i b j a i b j+ epφ j, which is the i, j entry of the right-hand side of 24 and thus we prove the statement. Proposition 4. For the matri M satisfying Theorem, the identity 39 holds if p i satisfy 4. Proof. We write out the l, k entry of the right-hand side of 39, simplify it using Lemma 2 and 8 and obtain a l b k pk < b, a > pl a l bk n < b, a > a l b k ω r kj + ω l rj b r ar ω j < b r= j=, a > br a r < b, a > n a l ω k rj b r2 a k + ω j 2 < b, a > bl ωr lj a r2 b k ω j 2 < b, a > r= = al b k pk < b, a > pl a l bk < b, a > al b k + pl b k b l al b l < b, a > al p k b k a k a k < b, a > n b r a r r k n + l r n n + S < b, a > r= To compute the left-hand side of 39 for the corresponding entry, we first compute a τ and b τ using 4. Substituting it into 64 and 65 in Lemma 3, we have i a i τ = nai b r a r S < b, a > S bn n a + pi b i a i S p n2 < b, a > b i < b, a > 67 and b i τ r= i = nbi + nbi = nbi b r a r S < b, a r= > + S bn n a p n2 < b, a > n 2b r a r S < b, a r=i > S b i a i < b, a > i b r a r S < b, a > + S bn n a p n2 < b, a > r= + pi a i a i b S i < b, a > + pi a i a i 66 b S i < b, a > 68. 7

18 Thus, a l τ b k + a l b k τ = nal b k l r= k b r a r S < b, a > r= + pl b k b l al b l < b, a > pl b k a l < b, a > + al p k b k < b, a > al p k b k a k a k < b, a >, 69 which equals to 66 and we complete the proof. The following identity will be used in the proof of Theorem 4. Proposition 5. Given the relations 5, 6, 26 and 4, the following identity holds: S < b, a > ai bi+ + 2 a i+ b i epφ i = τ ai b i+ p i epφ i. 70 Proof. We prove the identity by direct calculation. The left-hand side equals = = a i b i+ + 2 a i+ b i epφ i < b, a > ai bi+ + 2 a i+ b i epφ i < b, a > a i b i+ + 2 a i+ b i p i ai bi+ + 2 a i+ b i epφ i < b, a > p i+ < b, a > a i p i a i+ b i+ a i+ p i+ < b, a > + bi+ p i2 a i b i epφ i b i < b, a > b i+ b i+ b i a i bi < b, a > + ai p i p i+ a i+ a i+ < b, a > epφi p i = epφi p i+ epφ i 2 a i+ b i = a i b i+ p i p i+. We now compute the right-hand side of 70. First using formula 69, we have a i τ b i+ + a i b i+ τ = bi+ b i p So the right-hand side of 70 is equal to a i τ = bi+ i a i b S i < b, a > + a i b i+ p i + a i b i+ τ p i + a i b i+ p τ i epφ i b i a i S b i < b, a > + ai p i epφ i b i ai < b, a > bi+ a i+, < b, a > + nai b i+ p i a i+ p S bi+ a i+ i+ < b, a >. + na i b i+ p i φ i τ epφ i a i+ p i+ epφi S bi+ a i+ < b, a > which is the same as the left-hand side after we use 7 and substitute n ai p i = pi2 b i containing b and n bi+ = pi+2 a i+ p i+ for the term containing a. for the term Acknowledgements The paper is supported by AVM s EPSRC grant EP/I038675/ and JPW s EPSRC grant EP/I038659/. All authors gratefully acknowledge the financial support. 8

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