Difference Krichever-Novikov operators
aa r X i v : . [ n li n . S I] A ug Difference Krichever–Novikov operators
Gulnara S. Mauleshova and Andrey E. Mironov
Abstract
In this paper we study commuting difference operators of rank two. We in-troduce an equation on potentials V ( n ) , W ( n ) of the difference operator L =( T + V ( n ) T − ) + W ( n ) and some additional data. With the help of this equa-tion we find the first examples of commuting difference operators of rank twocorresponding to spectral curves of higher genus. I.M. Krichever and S.P. Novikov [1], [2] discovered a remarkable class of solutions ofsoliton equations — algebro–geometric solutions of rank l >
1. This class is determinedby the following condition: common eigenfunctions of auxiliary commuting ordinarydifferential or difference operators form a vector bundle of rank l over the spectralcurve. Rank two solutions of the Kadomtsev–Petviashvili (KP) equation and 2D-Todachain corresponding to the spectral curves of genus g = 1 were found in [1], [2]. Tofind higher rank solutions one has to find higher rank commuting operators and theirappropriate deformations. The problem of classification of commuting differential anddifference operators was solved in [2]–[4]; however, finding the operators themselves hasremained an open problem. Moreover, no examples of commuting differential operatorsof rank l > g > L k , L s theoperators of orders k = N − + N + and s = M − + M + L k = N + X j = N − u j ( n ) T j , L s = M + X j = M − v j ( n ) T j , n ∈ Z , where T is the shift operator. The condition of their commutativity is equivalent to acomplicated system of nonlinear difference equations on the coefficients. These equa-tions have been studied since the beginning of the 20th century (see [7]). An analogueof the Burchnall–Chaundy lemma [8] holds. Namely, if L k L s = L s L k , then there existsa nonzero polynomial F ( z, w ) such that F ( L k , L s ) = 0 [9]. The polynomial F definesthe spectral curve Γ = { ( z, w ) ∈ C | F ( z, w ) = 0 } . L k ψ = zψ, L s ψ = wψ, then ( z, w ) ∈ Γ . The dimension of the space of common eigenfunctions with the fixedeigenvalues is called the rank of the pair L k , L s l = dim { ψ : L k ψ = zψ, L s ψ = wψ } , where the point ( z, w ) ∈ Γ is in general position. Thus the spectral curve and the rankare defined exactly the same way as in the case of the differential operators.Before discussing difference operators we briefly discuss differential operators. Thefirst important results relating to commuting differential operators of rank l > l withperiodic coefficients were studied in [12]. Rank two operators at g = 1 were found byI.M. Krichever and S.P. Novikov [1]. These operators were studied in [13]–[20] (see also[21]–[24] for g = 2–4 , l = 2 , g = 1 , l = 3 operators were found in [25]. Methodsof [5] allow to construct and study higher rank operators at g > L k and L s is iso-morphic to the ring of meromorphic functions on an algebraic spectral curve Γ withpoles q , . . . , q m ∈ Γ (see [2]). Such operators are called m -points operators . We notethat any ring of commuting differential operators is isomorphic to a ring of meromorphicfunctions on a spectral curve with a unique pole. The commuting difference operatorsof rank one were found by I.M. Krichever [9] and D. Mumford [32]. Eigenfunctions(Baker–Akhiezer functions) and coefficients of such operators can be found explicitlywith the help of theta-functions of the Jacobi varieties of spectral curves. In the caseof l > g = 1 were found in [2], operators withpolynomial coefficients among them were obtained in [33].In this paper we consider one-point operators of rank two L , L g +2 correspondingto the hyperelliptic spectral curve Γ w = F g ( z ) = z g +1 + c g z g + c g − z g − + ... + c , (1)herewith L = X i = − u i ( n ) T i , L g +2 = g +1 X i = − (2 g +1) v i ( n ) T i , u = v g +1 = 1 , (2) L ψ = zψ, L g +2 ψ = wψ, ψ = ψ ( n, P ) , P = ( z, w ) ∈ Γ . (3)Common eigenfunctions of L and L g +2 satisfy the equation ψ ( n + 1 , P ) = χ ( n, P ) ψ ( n − , P ) + χ ( n, P ) ψ ( n, P ) , (4)where χ ( n, P ) and χ ( n, P ) are rational functions on Γ having 2 g simple poles, depend-ing on n (see [2]). The function χ ( n, P ) additionally has a simple pole at q = ∞ . To2nd L and L g +2 it is sufficient to find χ and χ . Let σ be the holomorphic involutionon Γ , σ ( z, w ) = σ ( z, − w ) . The main results of this paper are Theorems 1–4.
Theorem 1 If χ ( n, P ) = χ ( n, σ ( P )) , χ ( n, P ) = − χ ( n, σ ( P )) , (5) then L has the form L = ( T + V n T − ) + W n , (6) where χ = − V n Q n +1 Q n , χ = wQ n , Q n ( z ) = z g + α g − ( n ) z g − + . . . + α ( n ) . (7) Functions V n , W n , Q n satisfy F g ( z ) = Q n − Q n +1 V n + Q n Q n +2 V n +1 + Q n Q n +1 ( z − V n − V n +1 − W n ) . (8)In Theorem 1 and further we use the notations V n = V ( n ) , W n = W ( n ). It is aremarkable fact that (8) can be linearized. Namely, if we replace n → n + 1 and takethe difference with (8), then the result can be divided by Q n +1 ( z ) . Finally we obtainthe linear equation on Q n ( z ). Corollary 1
Functions Q n ( z ) , V n , W n satisfy Q n − V n + Q n ( z − V n − V n +1 − W n ) − Q n +2 ( z − V n +1 − V n +2 − W n +1 ) − Q n +3 V n +2 = 0 . (9)At g = 1, the equation (8) allows us to express V n , W n via a functional parameter γ n . Corollary 2
The operator L = ( T + V n T − ) + W n , where V n = F ( γ n )( γ n − γ n − )( γ n − γ n +1 ) , W n = − c − γ n − γ n +1 , (10) commutes with L = T + ( V n + V n +1 + V n +2 + W n − γ n +2 ) T ++ V n ( V n − + V n + V n +1 + W n − γ n − ) T − + V n − V n − V n T − . The spectral curve of L , L is w = F ( z ) . In the theory of commuting ordinary differential operators there are equations whichare similar to (8), (9). Let us compare (8), (9) with their smooth analogues. First, weconsider the one-dimensional finite-gap Schr¨odinger operator L = − ∂ x + V ( x ) commut-ing with a differential operator L g +1 of order 2 g + 1. The theory of such operators is3losely related to the theory of periodic and quasiperiodic solutions of the Korteweg–deVries equation (see [34]–[36]). Denote by ψ a common eigenfunction( − ∂ x + V ( x )) ψ = zψ, L g +1 ψ = wψ. The point P = ( z, w ) belongs to the spectral curve (1). Function ψ ( x, P ) satisfies ψ ′ ( x, P ) = iχ ( x, P ) ψ ( x, P ) , where χ = Q x i Q + w Q , Q = z g + α g − ( x ) z g − + . . . + α ( x ) . Polynomial Q satisfies the equation4 F g ( z ) = 4( z − V ) Q − ( Q x ) + 2 QQ xx , which is linearized as well as (8) (see [37], [38]) Q xxx − Q x ( V − z ) − V x Q = 0 . Equations (8), (9) are analogues of the last two.Let us consider one more example. We denote by L , L g +2 rank two commutingdifferential operators with the spectral curve (1). The common eigenfunctions of L and L g +2 satisfy ψ ′′ = χ ( x, P ) ψ ′ + χ ( x, P ) ψ. In [5] it was proved that L is self-adjoint if and only if χ ( x, P ) = χ ( x, σ ( P )) , herewith L = ( ∂ x + V ( x )) + W ( x ) ,χ = − Q xx Q + w Q − V , χ = Q x Q , Q = z g + α g − ( x ) z g − + . . . + α ( x ) , polynomial Q satisfies4 F g ( z ) = 4( z − W ) Q − V ( Q x ) + Q xx − Q x Q xxx + 2 Q (2 V x Q x + 4 VQ xx + Q xxxx ) , (11)and also satisfies ∂ x Q + 4 VQ xxx + 2 Q x (2 z − W − V xx ) + 6 V x Q xx − QW x = 0 . (12)Equations (8), (9) are discrete analogues of (11), (12).Theorem 1 allows us to construct the examples. Theorem 2
The operator L ♯ = ( T + ( r n + r n + r n + r ) T − ) + g ( g + 1) r n, r = 0 commutes with a difference operator L ♯ g +2 . heorem 3 The operator L X = ( T + ( r a n + r ) T − ) + r ( a g +1 − a g +1 − a g + 1) a n − g , r , a = 0 , where a g +1 − a g +1 − a g + 1 = 0 , commutes with a difference operator L X g +2 . Theorem 4
The operator L ♮ = ( T + ( r cos( n ) + r ) T − ) − r sin( g g + 12 ) cos( n + 12 ) , r = 0 commutes with a difference operator L ♮ g +2 . In Section 2 we recall the Krichever–Novikov equations on Tyurin parameters.In Section 3 we prove Theorems 1–4 and consider examples.In Appendix we consider the differential–difference system on V n ( t ) , W n ( t )˙ V n = V n ( W n − − W n + V n − − V n +1 ) , (13)˙ W n = ( W n − W n − ) V n + ( W n +1 − W n ) V n +1 . (14)From (13), (14) it follows that ϕ n ( t ), where e ϕ n ( t ) = V n ( t ), satisfies the generalized Todachain ¨ ϕ n = e ϕ n − + ϕ n − − e ϕ n − + ϕ n + e ϕ n +1 + ϕ n +2 − e ϕ n +1 + ϕ n . From (13), (14) it follows also that[ L , ∂ t − V n − ( t ) V n ( t ) T − ] = 0 , where L = ( T + V n ( t ) T − ) + W n ( t ) . Following [1], [2] we call the solution V n ( t ) , W n ( t )of (13), (14) the solution of rank two , if additionally [ L , L g +2 ] = 0 for some differenceoperator L g +2 . In the case of rank two solutions an evolution equation on Q n ( t ) isobtained in Theorem 5. At g = 1 this equation is reduced to a discrete analogue of theKrichever–Novikov equation, which appeared in the theory of rank two solutions of KP. As mentioned above, in the case of rank one operators the eigenfunctions can be foundexplicitly in terms of theta-functions of the Jacobi varieties of spectral curves. Let usconsider the simplest example. Let Γ be an elliptic curve Γ = C / { Z + τ Z } , τ ∈ C , Im τ > , and θ ( z ) the theta-function θ ( z ) = P n ∈ Z exp( πin τ + 2 πinz ) . The Baker–Akhiezerfunction has the form ψ ( n, z ) = θ ( z + c + nh ) θ ( z ) (cid:18) θ ( z − h ) θ ( z ) (cid:19) n , c, h / ∈ { Z + τ Z } . (15)5or the meromorphic function λ = θ ( z − a ) ... ( z − a k ) θ k ( z ) , a + . . . + a k = 0 there is a uniqueoperator L ( λ ) = v k ( n ) T k + . . . + v ( n ) such that L ( λ ) ψ = λψ. Coefficients of L ( λ )can be found from the last identity (see [39]). Operators L ( λ ) for different λ form acommutative ring of difference operators. This example can be generalized from theelliptic spectral curves to the principle polarized abelian spectral varieties. It allowsto construct commuting difference operators in several discrete variables with matrixcoefficients (see [39]).At l > l have the form L = Nr + X i = − Nr − u i ( n ) T i , A = Mr + X i = − Mr − v i ( n ) T i , where l = r − + r + , ( N, M ) = 1. Consider the space H ( z ) of solutions of the equation Ly = zy. We have dim H ( z ) = N ( r − + r + ) . The operator A defines the linear operator A ( z ) on H ( z ) . Let us choose the basis ϕ i ( n ) in H ( z ), satisfying the normalizationconditions ϕ i ( n ) = δ in , − N r − ≤ i, n < N r + . The components of A ( z ) in the basis ϕ i ( n ) are polynomials in z . The characteristic polynomial of A ( z ) has the form det( w − A ( z )) = R l ( w, z ) . Polynomial R defines the spectral curve Γ, i.e. Lψ = zψ, Aψ = wψ, R ( z, w ) = 0 . Common eigenfunctions of L and A form a vector bundle of rank l over the affine partof Γ. Let us choose the basis in the space of common eigenfunctions such that ψ in ( P ) = δ i,n , − r − ≤ i, n < r + , P = ( z, w ) ∈ Γ . Functions ψ in ( P ) have the pole divisor γ = γ + . . . + γ lg of degree lg . We have thefollowing identities α js Res γ s ψ in ( P ) = α is Res γ s ψ jn ( P ) . The pair ( γ, α ) is called the
Tyurin parameters , where α is the set of vectors α , . . . , α lg , α s = ( α − r − s , . . . , α r + − s ) . The Tyurin parameters define a stable holomorphic vector bundle on Γ of degree lg with holomorphic sections ζ − r − , . . . , ζ r + − , where γ is the divisor of their linear depen-dence P r + − j = − r − α js ζ j ( γ s ) = 0 . Let Ψ( n, P ) be the Wronski matrix with the componentsΨ ij ( n, P ) = ψ in + j ( P ) , − r − ≤ i, j < r + . Function detΨ( n, P ) is holomorphic in theneighbourhood of q = ∞ . The pole divisor of detΨ( n, P ) is γ , the zero divisor ofdetΨ( n, P ) is γ ( n ) = γ ( n ) + · · · + γ lg ( n ) , herewith γ (0) = γ . Consider the matrixfunction χ ( n, P ) = Ψ( n + 1 , P )Ψ − ( n, P ) ,χ ( n, P ) = . . .
00 0 1 . . . · · · · · · · · · · · · · · · . . . χ − r − ( n, P ) χ − r − +1 ( n, P ) χ − r − +2 ( n, P ) . . . χ r + − ( n, P ) .
6n the neighbourhood of q we have χ i ( n, k ) = k − δ i, − f i ( n, k ) , where k is a localparameter near q , f i ( n, k ) is an analytical function in the neighbourhood of q . Theorem (I.M. Krichever, S.P. Novikov)
The matrix function χ ( n, P ) has simple poles in γ s ( n ) , and α js ( n )Res γ s ( n ) χ i ( n, P ) = α is ( n )Res γ s ( n ) χ j ( n, P ) . (16) Points γ s ( n + 1) are zeros of det χ ( n, P )det χ ( n, γ s ( n + 1)) = 0 . (17) Vectors α j ( n + 1) = ( α − r − s ( n + 1) , . . . , α r + − s ( n + 1)) satisfy α s ( n + 1) χ ( n, γ s ( n + 1)) = 0 . (18)Equations (16)–(18) define the discrete dynamics of the Tyurin parameters. In [2]solutions of (16)–(18) are found at g = 1 , l = 2. The corresponding operators in thesimplest case have the form L = L − ℘ ( γ n ) − ℘ ( γ n − ) ,L is the difference Schr¨odinger operator L = T + v n + c n T − with the coefficients c n = 14 ( s n − − F ( γ n , γ n − ) F ( γ n − , γ n − ) , v n = 12 ( s n − F ( γ n , γ n − ) − s n F ( γ n − , γ n )) , where F ( u, v ) = ζ ( u + v ) − ζ ( u − v ) − ζ ( v ) . Here, ℘ ( u ) , ζ ( u ) are the Weierstrass functions, s n , γ n are the functional parameters. Let Γ be the hyperelliptic spectral curve (1), L , L g +2 are operators of the form (2)with the properties (3). Matrix χ ( n, P ) = Ψ( n + 1 , P )Ψ − ( n, P ) has the form χ ( n, P ) = (cid:18) χ ( n, P ) χ ( n, P ) (cid:19) . The functions χ , χ have the following expansions in the neighbourhood of q = ∞ : χ ( n ) = b ( n ) + b ( n ) k + . . . , χ ( n ) = 1 /k + e ( n ) + e ( n ) k + . . . , (19)where k = √ z . emma 1 The operator L = T + u ( n ) T + u ( n ) + u − ( n ) T − + u − ( n ) T − has the coefficients: u ( n ) = − e ( n ) − e ( n + 1) , u ( n ) = e ( n ) + e ( n ) − e ( n + 1) − b ( n ) − b ( n + 1) ,u − ( n ) = b ( n ) (cid:18) e ( n ) + e ( n − − b ( n − b ( n − (cid:19) − b ( n ) , u − ( n ) = b ( n ) b ( n − . If b ( n ) = 0 , e ( n ) = 0 , then L can be written in the form (6), where V n = − b ( n ) , W n = − e ( n ) − e ( n + 1) . Proof
Using (4) let us express ψ n +2 ( P ) and ψ n − ( P ) via ψ n − ( P ) , ψ n ( P ) , χ ( n, P ) ,χ ( n, P ) ψ n +2 = ψ n − χ ( n ) χ ( n +1)+ ψ n ( χ ( n +1)+ χ ( n ) χ ( n +1)) , ψ n − = ψ n − ψ n − χ ( n − χ ( n − , and substitute it in L ψ n = zψ n . We get P ( n, P ) ψ n ( P ) + P ( n, P ) ψ n − ( P ) = zψ n ( P ) , where P ( n ) = χ ( n + 1) + χ ( n + 1) χ ( n ) + u ( n ) χ ( n ) + u ( n ) + u − ( n ) χ ( n − ,P ( n ) = χ ( n + 1) χ ( n ) + u ( n ) χ ( n ) + u − ( n ) − u − ( n ) χ ( n − χ ( n − . Consequently we have P = z = 1 k , P = 0 . (20)From (19), (20) it follows that P − k = e ( n ) + e ( n + 1) + u ( n ) k + ( b ( n + 1) + e ( n ) e ( n + 1) + e ( n )+ e ( n + 1) + u ( n ) + e ( n ) u ( n ) + u − ( n ) b ( n − (cid:19) + O ( k ) = 0 ,P = b ( n ) − u − ( n ) b ( n − k + ( b ( n ) + b ( n ) e ( n + 1) + b ( n ) u ( n )+ u − ( n ) + b ( n − u − ( n ) b ( n − − e ( n − u − ( n ) b ( n − (cid:19) + O ( k ) = 0 . This yields the formulas for the coefficients of L . By direct calculations one can checkthat if b ( n ) = e ( n ) = 0, then L has the form (6). Lemma 1 is proved.8hus if χ , χ satisfy (5), then b ( n ) = e ( n ) = 0 , and hence L has the form (6).Operators L − z and L g +2 − w have the common right divisor T − χ ( n ) − χ ( n ) T − , i.e. L − z = l ( T − χ ( n ) − χ ( n ) T − ) , L g +2 − z = l ( T − χ ( n ) − χ ( n ) T − ) , where l and l are operators of orders 2 and 4 g . Let us assume that (5) holds. Then( T + V n T − ) + W n − z = ( T + χ ( n + 1) − V n − V n χ ( n − T − )( T − χ ( n ) − χ ( n ) T − ) , where χ , χ satisfy the equations V n − V n + χ ( n − V n + V n +1 − z + W n + χ ( n + 1) + χ ( n ) χ ( n + 1)) = 0 , (21) − V n − V n χ ( n −
1) + χ ( n − χ ( n ) χ ( n + 1) = 0 . (22)We have det χ ( n, P ) = − χ ( n, P ) = detΨ( n + 1 , P )(detΨ( n, P )) − . The degree of thezero divisor γ ( n ) of detΨ( n, P ) is 2 g. Since χ is invariant under the involution σ , thedivisor γ ( n ) has the form γ ( n ) = γ ( n ) + σγ ( n ) + . . . + γ g ( n ) + σγ g ( n ) . Let γ i ( n ) have the coordinates ( µ i ( n ) , w i ( n )) . We introduce the polynomial in zQ n = ( z − µ ( n )) . . . ( z − µ g ( n )) . From (17) we have χ ( n, P ) = b ( n ) Q n +1 Q n , where b ( n ) is some function. In the neigh-bourhood of q we have χ = b ( n ) + b ( n ) k + O ( k ) . By Lemma 1 V n = − b ( n ) , so we get χ ( n, P ) = − V n Q n +1 Q n . Since the pole divisor of χ ( n, P ) is γ ( n ) and in the neighborhood of q we have (19),then χ ( n, P ) = wQ n . If χ ( n, P ) = − V n Q n +1 Q n and χ ( n, P ) = wQ n , then (22) holds identically, and (21) isreduced to (8). Theorem 1 is proved.To prove Theorems 2–4 it is sufficient to prove that for potentials V n , W n fromTheorems 2–4 there are polynomials Q n ( z ) of degree g in z which satisfy (9) (and hencesatisfy (8)). Let V n = r n + r n + r n + r , W n = g ( g + 1) r n, then (9) takes the form Q n − ( n r + n r + nr + r )+ Q n ( z − n r − n (2 r +3 r ) − n (2 r +2 r +3 r + g ( g +1) r ) − (2 r + r + r + r )) − Q n +2 (cid:0) z − n r − n (2 r + 9 r ) − (2 r + 6 r + 15 r + g (1 + g ) r ) − (2 r + 3 r + 5 r + 9 r + g ( g + 1) r )) − Q n +3 (cid:0) n r + n ( r + 6 r ) + n ( r + 4 r + 12 r ) + r + 2 r + 4 r + 8 r (cid:1) = 0 . (23)Let us take the following ansatz for Q n ( z ) Q n = δ g n g + . . . + δ n + δ , δ i = δ i ( z ) , then (23) can be rewritten in the form β g +3 ( z ) n g +3 + β g +2 ( z ) n g +2 + . . . + β ( z ) = 0for some β s ( z ). Potentials V n , W n have the following remarkable properties: it turnsout that β g = β g +1 = β g +2 = β g +3 = 0automatically (this can be checked by direct calculations). From (23) we find β s β s = r (2 s + 1)( g ( g + 1) − s ( s + 1)) δ s + g X m =1 (cid:0) ( − m (cid:0) C ms + m r − C m +1 s + m r +C m +2 s + m r − C m +3 s + m r (cid:1) + 2 m (cid:0) C ms + m (2 r + 3 r + 5 r + 9 r + g ( g + 1) r − z )+2C m +1 s + m (2 r + 6 r + 15 r + g ( g + 1) r ) + 4C m +2 s + m (2 r + 9 r ) + 16C m +3 s + m r (cid:1) − m (cid:0) C ms + m ( r + 2 r + 4 r + 8 r ) + 3C m +1 s + m ( r + 4 r + 12 r )+3 m (cid:0) C ms + m ( r + 2 r + 4 r + 8 r ) + 3C m +1 s + m ( r + 4 r + 12 r )+9C m +2 s + m ( r + 6 r ) + 27C m +3 s + m r (cid:1)(cid:1) δ s + m , (24)where 0 ≤ s < g − , C km = m ! k !( m − k )! at m ≥ k, C km = 0 at m < k, δ g is a constant and δ s = 0 , if s > g. From β s = 0 we express δ s via δ s +1 , . . . , δ g . In particular, δ g − = δ g (2 g r + g ( g + 1) r + 2 z )2(2 g − r . For a suitable δ g we have Q n = z g + α g − ( n ) z g − + . . . + α ( n ) . So we proved that thereexists Q n satisfying (9). Theorem 2 is proved.In [5] it was proved that L ♯ = ( ∂ x + r x + r x + r x + r ) + g ( g + 1) r x commutes with a differential operator L ♯ g +2 of order 4 g + 2. The operator L ♯ is adiscrete analogue of L ♯ . At g = 1 the operators L ♯ , L ♯ were found by Dixmier [10]. Theoperators L ♯ , L ♯ g +2 define a commutative subalgebra in the first Weyl algebra.Let us consider the algebra W generated by two elements p and q with the relation[ p, q ] = p. Since [
T, n ] = T , the algebra is isomorphic to the algebra of differenceoperators with polynomial coefficients. The algebra W has the following automorphisms H : W → W, H ( p ) = p, H ( q ) = q + G ( p ) , where G is an arbitrary polynomial. Operators L ♯ , L ♯ g +2 define the commutative sub-algebra in W . Consequently, if we replace n → n + G ( T ) in L ♯ , L ♯ g +2 , then we obtainthe new commuting difference operators with polynomial coefficients.10 .2 Theorem 3 Let V n = r a n + r , W n = ( a g +1 − a g +1 − a g + 1) r a n − g , then (9) takes the form Q n − ( r + a n r ) + Q n (cid:0) z − r − a n − g r − a n + g +1 r (cid:1) + Q n +2 (cid:0) r + a n +1 − g r + a n + g +2 r − z (cid:1) − Q n +3 (cid:0) r + a n +2 r (cid:1) = 0 . (25)Let Q n = B g a gn + B g − a ( g − n + . . . + B a n + B , B i = B i ( z ) . We introduce the notation y = a n , then Q n = B g y g + · · · + B , and (25) takes the form g X s =0 B s ( a − g − s ( a g − a s )( a g + s +1 − a s +1 − r y s +1 − a − s ( a s − a s − r + a s z ) y s ) = g X s =1 y s ( B s a − s (1 − a s )(( a s − r + a s z )+ B s − a − g − s ( a g − a s − )( a g + s − a s − − r ) = 0 . Hence we obtain B s − = B s a − s ( a s − a s − r + a s z ) a − g − s ( a g − a s − )( a g + s − a s − − r , s = 1 , . . . , g. Thus we found the polynomial Q n , satisfying (9). Theorem 3 is proved.The operator L X is a discrete analogue of L X = ( ∂ x + r a x + r ) + g ( g + 1) r a x from [26], which commutes with a differential operator of order 4 g + 2. Let V n = r cos( n ) + r , W n = − r sin( g ) sin( g +12 ) cos( n + ) . Equation (9) takes theform Q n − ( r + r cos( n )) + Q n (cid:0) z − r − r cos( g + ) cos( n + ) (cid:1) − Q n +2 (cid:0) z − r − r cos( g + ) cos( n + ) (cid:1) − Q n +3 ( r + r cos( n + 2)) = 0 . (26)Let us take the following ansatz Q n = A g cos( gn ) + A g − cos(( g − n ) + . . . + A cos( n ) + A , A i = A i ( z ) . We substitute Q n in (26) and after some simplifications we obtain A = A ( z − r + 2 r cos(1)) sin(1)2 r (cos( g + ) − cos( )) sin( ) ,A s − = A s ( z − r + 2 r cos( s )) sin( s ) + A s +1 r (cos( s − ) − cos( g + )) sin( s − ) r (cos( g + ) − cos( s − )) sin( s − ) , ≤ s ≤ g , A g +1 = 0, A g is a suitable constant. We found Q n satisfying (9).Theorem 4 is proved.The operator L ♮ is a discrete analogue of L ♮ = ( ∂ x + r cos( x ) + r ) + g ( g + 1) r cos( x )from [29], which commutes with a differential operator of order 4 g + 2.Let us consider several examples. Example 1
We introduce the notation f ( n ) = r n + r n + r n + r . The operator L ♯ = ( T + f ( n ) T − ) + 2 r n commutes with L ♯ = T + 3( f ( n ) + f ′ ( n ) + f ′′ ( n ) + 4 r ) T + 3( f ( n ) + 3 r n + r ) T − +( f ( n − f ′ ( n ) + 2 f ′′ ( n ) − r )( f ( n ) − f ′ ( n ) + 3 r n − r + r ) f ( n ) T − . The spectral curve is w = z + (2 r + 3 r ) z + ( r r + ( r + r )( r + 3 r )) z + r (( r + r )( r + r + r ) − r r ) . Example 2
The operator L X = ( T + ( r a n + r ) T − ) + r ( a − a − a + 1) a n − commutes with L X = T + (cid:0) r ( a + 1 + a − ) + r a n − ( a + a + 1) (cid:1) T + ( a + 1 + a − )( r a n + r ) × ( r a n +1 − r a n + r a n − + r ) T − + ( r a n + r )( r a n − + r )( r a n − + r ) T − . The spectral curve is w = z + r ( a − a z + r ( a − a z. Example 3
The operator L ♮ = ( T + ( r cos( n ) + r ) T − ) − r sin( 12 ) sin(1) cos( n + 12 )commutes with L ♮ = T + (2 cos(1) + 1)( r (2 cos(1) −
1) cos( n + 1) + r ) T +(2 cos(1) + 1)( r cos( n ) + r )( r (2 cos(1) −
1) cos( n ) + r ) T − +( r cos( n −
1) + r )( r cos( n −
2) + r )( r cos( n ) + r ) T − . The spectral curve is w = z − r sin ( ) z − r (cos(1) + 1) − r ) sin ( ) z. Remark
We see that the pairs of commuting operators L , L g +2 and L , L g +2 havesimilar properties, and there are similar examples of such operators. It would be inter-esting to explain this duality. So far, our attempts to do so by some discretization werenot successful. Acknowledgements
The authors are supported by a Grant of the Russian Federationfor the State Support of Researches (Agreement No 14.B25.31.0029).12 ppendix
Consider the differential–difference system (13), (14).
Theorem 5
Let us assume that the potentials V n ( t ) , W n ( t ) of L = ( T + V n ( t ) T − ) + W n ( t ) satisfy (13), (14). We additionally assume that [ L , L g +2 ] = 0 for some differ-ence operator L g +2 . Then Q n ( t ) associated with L satisfies the evolution equation ˙ Q n = V n ( Q n +1 − Q n − ) . (27)Equation (27) defines the symmetry of (8). At g = 1 functions V n ( t ) , W n ( t ) can beexpressed via γ n ( t ) using (9). In this case the system (13), (14) and the equation (27)are reduced to one equation ˙ γ n = F ( γ n )( γ n − − γ n +1 )( γ n − − γ n )( γ n − γ n +1 ) . (28)This equation is a discrete analogue of the Krichever–Novikov equation, which appearedin the theory of rank two solutions of KP [1]. Equations similar to (13), (14) and (28)were considered in [40], [41]. Proof
Using ( ∂ t − V n − ( t ) V n ( t ) T − ) ψ n = 0 , and (4) let us express ˙ ψ n − , ˙ ψ n , ˙ ψ n +1 , ψ n − ,ψ n − in terms of ψ n − , ψ n , χ ( n ) , χ ( n )˙ ψ n − = V n − V n − ψ n − , ˙ ψ n = V n − V n ψ n − , ˙ ψ n +1 = V n V n +1 ψ n − ,ψ n − = ψ n − ψ n − χ ( n − χ ( n − ,ψ n − = ψ n − ( χ ( n −
1) + χ ( n − χ ( n − − ψ n χ ( n − χ ( n − χ ( n − . From (4) it follows that˙ ψ n +1 − χ ( n ) ˙ ψ n − − ˙ χ ( n ) ψ n − − χ ( n ) ˙ ψ n − ˙ χ ( n ) ψ n = A n ψ n + B n ψ n − = 0 , where A n = V n − V n − χ ( n ) χ ( n − − χ ( n − V n − V n χ ( n ) + χ ( n −
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