Reductions of integrable lattices

Reduction s of in tegrable lattices A K Svinin Institute for System Dynamics and Control Theory , Siberia n Branch of Russian Academy of Sciences E-mail: svinin @icc. ru Abstract. Based on the no tion of Dar b oux-KP chain hierarch y and its inv ar iant submanifolds we construct some class of constraints compatible with integrable lattices. Some simple examples are given. R e ductions of inte gr able lattic es 2 1. In tro duction Our concern in the pap er is with differen tial-difference systems (lattices) o v er finite n um b er of fields (unkno wn functions of discrete v a r iable i ) whic h share the prop erty of ha ving infinite num b er of conserv ation la ws. In a sense one can in terpret these conserv ation law s as analog of first inte grals for finite- dimensional dynamical systems. It is suppo sed that to eac h conserv ation law corresp onds the flow on suitable infinite- dimensional phas e-space and these flo ws are pair-wise comm uting. Differen tial-difference systems of this t ype are reffered to as ‘in tegrable’. F or in tegrable latt ice under consideration corresp onding flo ws gov erned by some ev olutionary equations can b e in terpreted as generalized symmetries. One sa ys that g iven system with these prop erties admits in tegra ble hierarc h y (se e, for e xample [11]). It is kno wn from the literature a n umber of differen tial- difference systems with ab ov e-mentioned prop erties whic h hav e applications in differen t areas of natural sciences . P erhaps simplest and fro m the other hand interesting example of inte grable lattice is w ell-kno wn V olterra [23] (or Kac-v an Mo erb ek e) equation [5]. a ′ i = a i ( a i − 1 − a i +1 ) . (1) It can b e considered as single equation on function a = a ( i, x ) = a i of discre te v ariable i ∈ Z and con tin uous v ariable x ≡ t 1 ∈ R and is kno wn to be in tegrable discretization of K ortew eg-de V ries equation [5]. Due to n umerous applications of V olterra lattice (see, for example, [4], [8], [22]), it can be accepted now ada ys as classical equation of mathematical ph ysics. This equation is kno wn to b e in tegrable b y inv erse scattering transform method due to [8]. V olterra lattice hierarc h y can b e written as infinite n um b er of ev olutionary equations of the form ∂ s a i = a i ( ζ s ( i + 1) − ζ s ( i − 1)) , where ∂ s ≡ ∂ ∂ t s . (2) It turns out t hat a p olynomial discrete functions ζ k = ζ k [ a ] here are conserv ation densities for a ll the flows. O ne can write differen tial-difference conserv ation law s for (2) a s D t s ζ k ( i ) = J s,k ( i + 1) − J s,k ( i ) . The dis crete functions ζ k can be calculated making of use recursion relatio n ∆ ζ k +1 = Lζ k , with ζ 1 = − a whic h can b e deriv ed from Lax represen tation of V olterra lattice hierar ch y (see, for example, [20]). This relation is defined b y a pair of discrete op erator s: forw ard difference op erator ∆ ≡ Λ − 1 and L ≡ ( a Λ + a ) ◦ Λ − 1 − Λ ◦ ( a Λ + a ) whith Λ b eing a shift op erator acting on arbitrary function of discrete v ariable as (Λ f )( i ) = f ( i + 1). F ormally , one can construct first in tegrals of (2) as ζ k = P i ∈ Z ζ k ( i ) but, generally sp eaking, it mak es no sense b ecause of con v ergence problem. Provided p erio dicity condition a i + N = a i whic h are eviden tly c ompatible (since righ t-hand side of (2) do es R e ductions of inte gr able lattic es 3 not dep end explicitly on i ) with all flo ws (2) one is forced to consider finite-dimensional dynamical sy stem u ′ k = u k ( u k − 1 − u k +1 ) where k ∈ Z / N Z . It is kno wn to b e Liouville-in tegrable Hamiltonian system. One kno ws a class of constrain ts compatible with V olterra lattice whic h characteriz e similarit y solutions of the latt er and it s hierarch y . Simplest example is given by the first discrete Painlev ´ e equation [4 ] a i ( a i − 1 + a i + a i +1 − x ) + α i = 0 . (3) Here it is supp osed that α i ’s do not dep end on x and satisfy the relation α i +2 = α i + 1 . The restriction of V olterra latt ice (1) with the help o f (3) leads to the system o f ordinary differen tial equations u ′ 1 = − u 1 ( u 1 + 2 u 2 − x ) − a, u ′ 2 = u 2 (2 u 1 + u 2 − x ) + b, (4) where u 1 = a i , u 2 = a i +1 and a = α i , b = α i +1 with some i = i 0 . This system, as is kno wn, is Hamiltonian and equiv alent to f ourth P ainlev ´ e equation in tegrable via isomono dromy metho d [9 ]. Compatibilit y of (3) with V o lt erra lattice means that shifting i → i + 1 yields discrete symmetry transformation ¯ u 1 = u 2 , ¯ u 2 = − u 1 − u 2 − b u 2 + x, ¯ a = b, ¯ b = a + 1 for (4 ). It t urns out that there exists an infinite num b er of conditions compatible with V olterra lattice hierarch y whic h can b e written as ordinary auto nomous N th- o rder difference equation a i + N = R ( a i , ..., a i + N − 1 ) (5) with righ t-hand s ide R b eing s ome rational function of its arg uments. As a result w e are led to the system of ordinary differen t ia l eq uations u ′ 1 = u 1 ( S ( u 1 , ..., u N ) − u 2 ) , u ′ k = u k ( u k − 1 − u k +1 ) , k = 2 , ..., N − 1 (6) u ′ N = u N ( u N − 1 − R ( u 1 , ..., u N )) on u 1 = a i , ..., u N = a i + N − 1 with some initial v alue i = i 0 . The function S here is defined through in v ersion form ula a i − 1 = S ( a i , ..., a i + N − 1 ). Shif t ing i → i + 1 yields discrete symmetry transfomation ¯ u k = u k +1 , k = 1 , ..., N − 1 , ¯ u N = R ( u 1 , ..., u N ) for (6) . Some examples of reductions generated b y a kind of relations (5) w ere presen ted in [17 ]. R e ductions of inte gr able lattic es 4 It is our main goa l in this pap er is to presen t unified geometric appro ac h for constructing some class of restrictions compatible with integrable lattices and its hierarc hies. W e consider a comm unit y of in tegrable lattices whic h a r e supp osed to b e derive d as a result of suitable reduction of the so-called Darb oux-K P (DKP) chain hierarc h y whic h, in fact, is a ch ain of KP hierarc h y solutions related with eac h other b y Darb oux map [7 ]. F rom geometric p oin t of view it is conv enien t to prese n t DKP c hain in the fo rm of differen tial and differential-difference conserv ation la ws. One writes its equations in terms of infinite colle ctions { h k , a k } of functions of ev olution parameters t s and discrete v ariable i . It is relev an t in this approa ch to consider these unknown functions as co efficien ts of some formal Lauren t series. DKP chain hierarc hy admits an infinite n um b er of inv ariant submanifolds S n l − 1 and V olterra eq uation and man y other integrable latt ices arise as a result of restriction o f DKP chain on submanifold constructed as an interse ction S n 0 ∩ S p l − 1 with some n, p and l . In particular S 1 0 ∩ S 2 0 corresp onds to V olterra lattice. It is imp ortant that all in tegrable lattices under consideration in this pap er ma y b e written in terms of univ ersal co ordinates { a k = a k ( i ) : k ≥ 1 } parametrizing the p o in ts of some infinite-dimensional phase-space M . Restricting DKP chain only on S n 0 yields a hierarc hy w hic h is more con v enien t to write in terms of infinite n um b er of functions { a k } . This hierarc h y w e call n th discrete KP , since in t he case n = 1 one has equations of ordinary discrete K P hierarc h y . In tegrable lattices ov er finite n um b er of fields can b e considered as a result of restriction of M to some submanifold M n,p,l with the help of algebraic equations I k [ a 1 , a 2 , ... ] = 0 , with k ≥ 1 where I k ’ are some s uitable p olynomial discrete functions on M . T o mak e the matter more clear, let us illustraite our g eneral result stated b elo w in Theorem 3 b y discus sing simple example. Consider one-field lattice a ′ i + a ′ i +1 = ( a i + a i +1 ) ( a i − 1 − a i +2 ) whic h corresp o nds t o S 1 0 ∩ S 3 0 and require that a is also solution to V olterra equation. It is equiv alen t to conditio n a i a i +2 = a i +1 a i +3 (7) whic h, as can b e chec ke d b y direct calculations, is compatible with V oltera lattice. A t the first sigh t it seems that V olterra equation supplemen ted b y constrain t (7) corresp onds to triple interse ction S 1 0 ∩ S 2 0 ∩ S 3 0 , but one can c hec k that it give s only trivial solution a = 0 of it. The p oint is that there exist inv arian t conditions w eak er than those defining M 1 , 3 , 1 . It turns out that this conditions are written in the f o rm of p erio dicit y relations on some discrete p olynomial functions. The submanifold defined by this ‘w eak’ conditions w e denote as N 1 , 3 , 1 and V olterra equation supplemen ted by (7) app ear as a result of restriction of discrete KP hierarc h y on M 1 , 2 , 1 ∩ N 1 , 3 , 1 This pap er is organized as follo ws: in Section 2, w e prov ide the reader b y some basic facts ab out DKP chain hierarch y follow ing alo ng the lines suggested in [7]. In Section 3, w e f orm ulate theorem whic h provide us b y an infinite num b er of in v ariant submanifolds R e ductions of inte gr able lattic es 5 for D KP chain hierarc h y . This result is the basis to establish relationship of man y in tegrable lattices with the KP hierarc hy . W e pro vide the r eader b y some examples of in tegrable differen tial- difference systems ov er finite num b er of fields. In Section 4 w e fo rm ulate our main result whic h allo ws to construct a broad class o f constrain ts compatible with integrable lattices and show, in Section 5, how this result can b e a pplied on example of ex tended V olterra equation [1 0 ] a ′ i = a i   n X j =1 a i − j − n X s =1 a i + j   (8) kno wn also as Bogo y a vlenskii lattices [3]. 2. DKP c hain hierarc h y Let us first to giv e some basics on D KP c hain [7]. This can b e defined through t w o relations ∂ s h ( i ) = ∂ H ( s ) ( i ) , (9) ∂ s a ( i ) = a ( i )  H ( s ) ( i + 1) − H ( s ) ( i )  (10) first of which yields ev olution equations of KP hierarch y in the form of lo cal conserv ation la ws with h = z + P k ≥ 2 h k z − k +1 b eing a generating f unction for conserv ed densities of KP hierar c h y . The for ma l Lauren t series H ( s ) = z s + X k ≥ 1 H s k z − k , attac hed to an y integer s ≥ 2, is the generating function for suitable fluxes and can b e uniquely define d as pro jection of z s on the space H + = < 1 , h, h (2) , ... > spanned b y F a` a di Bruno iterates h ( k ) ≡ ( ∂ + h ) k (1). F or instance, one has H (1) = h, H (2) = h (2) − 2 h 2 , H (3) = h (3) − 3 h 2 h − 3 h 3 − 3 h ′ 2 , ... (11) Th us the co efficien ts H s k are defined as differen tial p olynomials of h 2 , h 3 , ... Linear relations (11) a re in v ertible and this means that an y elemen t o f H + can b e written as suitable linear combination o v er H ( k ) . Represen tation of the KP hierarc h y in the form (9) is equiv alen t to Sa t o ’s form ulation of that on the lev el of Lax eq uation ∂ s Q = [( Q s ) + , Q ] , on forma l pseudo differen tial op erator Q = ∂ + P k ≥ 2 u k ∂ − k +1 . One has the follo wing: ∂ = Q + X k ≥ 2 h k Q − k +1 and ( Q s ) + = Q s + X k ≥ 2 H s k Q − k +1 . R e ductions of inte gr able lattic es 6 One can write, f o r example, the fo llowing: h 2 = − u 2 , h 3 = − u 3 , h 4 = − u 4 − u 2 2 , ... and H 2 1 = 2 h 3 + h ′ 2 = − 2 u 3 − u ′ 2 , H 2 2 = 2 h 4 + h ′ 3 + h 2 2 = − 2 u 4 − u ′ 3 − u 2 2 , ... The relatio n (10) o n forma l Lauren t series a ( i ) = z + P k ≥ 1 a k ( i ) z − k +1 relate tw o neigh b ours h ( i ) and h ( i + 1) by Darb oux map h ( i ) → h ( i + 1) = h ( i ) + a x ( i ) /a ( i ) and garan tee compat ibility o f the latter w ith KP flo ws. It is ob vious that this equation can b e rewritten in the form of differen tial-difference conserv ation la ws ∂ s ξ ( i ) = H ( s ) ( i + 1) − H ( s ) ( i ) with ξ = ln a = ln z + X k ≥ 1 a k z − k − 1 2   X k ≥ 1 a k z − k   2 + 1 3   X k ≥ 1 a k z − k   3 − · · · ≡ ln z + X k ≥ 1 ξ k z − k Th us, w e consider follo wing equ ations of DKP c hain hierarc h y ∂ s h k ( i ) = ∂ H s k − 1 ( i ) , ∂ s ξ k ( i ) = H s k ( i + 1) − H s k ( i ) . (12) 3. In tegrable lattices 3.1. Invarian t submanifolds of DKP chain As was men tioned in In tro duction one knows f r o m the literature man y examples of differen tial-difference systems o v er finite n um b er of fields whic h share the prop ert y of having infinite n um b er of conserv ation laws and corresponding o ne-pa rametric generalized symmetry groups defined b y respectiv e ev olutionary equations. There are differen t metho ds for constructing in tegrable lattices a nd its explicit solutions, suc h tha t Lax pairs, recursion op erators etc. F or consulting, see, for example , [1], [2], [3], [5], [6], [10], [1 1 ], [13 ], [18]. In Refs. [15] and [1 6] w e hav e prov ed that DKP c hain is a conv enient and simple notion to sho w relationship of in tegrable lattices with KP hierarch y . This relationship is v ery useful due to remark able Sato theory whic h giv es description o f analitic solutions of KP hierar c h y in terms of infinite G r a ssmanian manifo ld and τ -function [12 ]. F o llowing t w o theorems giv e a f r amew ork for constructing integrable lattices whose hierar chies directly related with KP hierarc h y . Theorem 1. [15] The submanifold S n l − 1 defined b y condition z l − n a [ n ] ( i ) ∈ H + ( i ) , ∀ i ∈ Z (13) is ta ngen t with resp ect to DKP c hain flows defined b y ( 1 2). Theorem 2. [16] The c hain of inclusions of inv arian t submanifolds S n l − 1 ⊂ S 2 n 2 l − 1 ⊂ S 3 n 3 l − 1 ⊂ · · · ⊂ S k n k l − 1 ⊂ · · · R e ductions of inte gr able lattic es 7 is v alid. Here, by definition a [ s ] ( i ) =        a ( i ) × · · · × a ( i + s − 1 ) , s ≥ 1 1 , s = 0 a − 1 ( i − 1) × · · · × a − 1 ( i − | s | ) , s ≤ − 1 are discrete F a ` a di Bruno iterates o f Lauren t series a ( i ). In what follo ws, we use simple ob vious iden tit y a [ r 1 + r 2 ] ( i ) = a [ r 1 ] ( i ) a [ r 2 ] ( i + r 1 ) = a [ r 2 ] ( i ) a [ r 1 ] ( i + r 2 ) , ∀ r 1 , r 2 ∈ Z . (14) Using the co efficien ts a [ r ] k defined t hr o ugh the relation a [ k ] = z k + P s ≥ 1 a [ k ] s z k − s one reco v ers from (14 ) a [ r 1 + r 2 ] k ( i ) = a [ r 1 ] k ( i ) + k − 1 X j =1 a [ r 1 ] j ( i ) a [ r 2 ] k − j ( i + r 1 ) + a [ r 2 ] k ( i + r 1 ) = a [ r 2 ] k ( i ) + k − 1 X j =1 a [ r 2 ] j ( i ) a [ r 1 ] k − j ( i + r 2 ) + a [ r 1 ] k ( i + r 2 ) ∀ r 1 , r 2 ∈ Z . (15) W e observ e that the condition (13) can b e written in the f orm of the following generating relation: z l − n a [ n ] = H ( l ) + l X k =1 a [ n ] k H ( l − k ) . After a lo ok at negativ e p o w ers of z in the latt er form ula, one gets explicit form of (13) as f ollo ws: G ( n,l ) k ≡ a [ n ] k + l − H l k − l − 1 X j =1 a [ n ] j H l − j k = 0 , ∀ k ≥ 1 . 3.2. R estriction of DKP chain on S n 0 Let us consider the case l = 1 corresp onding to the in v ar ia n t submanifold S n 0 . It is defined b y conditions G ( n, 1) k ≡ a [ n ] k +1 − H 1 k = 0 , ∀ k ≥ 1 . So, on S n 0 one has H 1 k = h k +1 = a [ n ] k +1 . Theorem 2 say s that S n 0 ⊂ S 2 n 1 ⊂ S 3 n 2 ⊂ · · · ⊂ S k n k − 1 ⊂ · · · and, hen ce, G ( n, 1) k = 0 ⇒ G (2 n, 2) k = 0 ⇒ G (3 n, 3) k = 0 ⇒ · · · Solving s uccessiv ely these relations in fa v or of H s k giv es H s k = F ( n,s ) k [ a 1 , a 2 , ... ] ≡ a [ sn ] k + s + s − 1 X j =1 q ( n,sn ) j a [( s − j ) n ] k + s − j , (16) R e ductions of inte gr able lattic es 8 where q ( n,r ) k = q ( n,r ) k [ a 1 , a 2 , ... ] are p olynomial discrete functions defined through the relation z r = a [ r ] + X k ≥ 1 q ( n,r ) k z k ( n − 1) a [ r − k n ] (17) or, more exactly , a [ r ] k + k − 1 X j =1 a [ r − j n ] q ( n,r ) j + q ( n,r ) k = 0 , ∀ k ≥ 1 . (18) Belo w the first few q ( n,r ) k are written q ( n,r ) 1 = − a [ r ] 1 , q ( n,r ) 2 = − a [ r ] 2 + a [ r ] 1 a [ r − n ] 1 , q ( n,r ) 3 = − a [ r ] 3 + a [ r ] 1 a [ r − n ] 2 + a [ r − 2 n ] 1 a [ r ] 2 − a [ r ] 1 a [ r − n ] 1 a [ r − 2 n ] 1 . It can b e shown that the func tions q ( n,r ) k are related with eac h other b y the relation q ( n,r 1 + r 2 ) 1 ( i ) = q ( n,r 1 ) k ( i ) + k − 1 X j =1 q ( n,r 1 ) j ( i ) q ( n,r 2 ) k − j ( i + r 1 − j n ) + q ( n,r 2 ) k ( i + r 1 ) = q ( n,r 2 ) k ( i ) + k − 1 X j =1 q ( n,r 2 ) j ( i ) q ( n,r 1 ) k − j ( i + r 2 − j n ) + q ( n,r 1 ) k ( i + r 2 ) . (19) F urther, there is a need in more general t ha n (18) relation a [ r ] k ( i ) + k − 1 X j =1 a [ r − j n ] k − j ( i ) q ( n,r − p ) j ( i + p ) + q ( n,r − p ) k ( i + p ) = a [ p ] k ( i ) (20) with an y integers r and p . The latter can b e easily obtained as follo ws. T aking in to accoun t (14), w e can write z r − p = a [ r − p ] ( i + p ) + X k ≥ 1 q ( n,r − p ) k ( i + p ) z k ( n − 1) a [ r − p − k n ] ( i + p ) = a [ − p ] ( i + p )   a [ r ] ( i ) + X k ≥ 1 q ( n,r − p ) k ( i + p ) z k ( n − 1) a [ r − k n ] ( i )   and z r − p a [ p ] ( i ) = a [ r ] ( i ) + X k ≥ 1 q ( n,r − p ) k ( i + p ) z k ( n − 1) a [ r − k n ] ( i ) . Then writting explicitly the latter relation w e get (20). Solving (20) in fav or of q ( n,r − p ) k ( i + p ) yields a [ p ] k ( i ) + k − 1 X j =1 q ( n,r − ( k − j ) n ) j ( i ) a [ p ] k − j ( i ) + q ( n,r ) k ( i ) = q ( n,r − p ) k ( i + p ) . (21) Observ e that relations (16) are co ded in H ( s ) = z s (1 − n ) a [ sn ] + s X j =1 z ( s − j )(1 − n ) q ( n,sn ) j a [( s − j ) n ] . This me ans that when restricting on S n 0 , one has H + = < 1 , z 1 − n a [ n ] , z 2(1 − n ) a [2 n ] , ... > . (22) R e ductions of inte gr able lattic es 9 W e see that on S n 0 DKP chain equations can b e written in the form of differen tial- difference conserv ation la ws ∂ s ξ k ( i ) = F ( n,s ) k ( i + 1) − F ( n,s ) k ( i ) . (23) More exactly the equations (23) app ear as a result o f pro j ection of restricted DKP c hain flo ws on the space M whose points are defined b y infinite n um b er of functions { a k = a k ( i ) } . As was sho wn in [7], with n = 1, ev olution equations (23) are equiv alen t to the discrete KP hierarc h y (dKP) [19]. W e can sho w that giv en an y solution of DKP c hain hierarc h y restricted to S n 0 , noninv ertible map g n : a ( i ) → z 1 − n a [ n ] ( ni ) give s solution of the dKP one. More g enerally , it is kno wn that g k ( S k n l − 1 ) ⊂ S n l − 1 [16]. W e refer, for simplicit y , to equations (23) , with some fixed as n th dK P hierarc h y . Since all n th dKP hierarc h y flo ws ‘liv e’ on the same phase-space M it is natural to call (23) (w ith an y n ) as ex tended dKP hierarch y [14]. It is useful for our a ims to write eq uations D t s q ( n,r ) k ( i ) = q ( n,r ) k + s ( i + sn ) + s X j =1 q ( n,sn ) j ( i ) q ( n,r ) k + s − j ( i + ( s − j ) n ) − q ( n,r ) k + s ( i ) − s X j =1 q ( n,sn ) j ( i + r − ( k + s − j ) n ) q ( n,r ) k + s − j ( i ) (24) whic h are fulfilled in virtue of (23). These equations can be found b y making of use suitable differen tia l-difference Lax equation (see, for example [16]). 3.3. R estriction of DKP chain on S n 0 ∩ S p l − 1 Let us no w consider nontrivial inte rsections S n 0 ∩ S p l − 1 . The w ord ‘non trivial’ means that ln − p 6 = 0 to b e supp o sed. According to the Theorem 1, the restriction of D KP c hain on S n 0 ∩ S p l − 1 is defined b y the generating relation z l − p a [ p ] − F ( n,l ) − l X j =1 a [ p ] j F ( n,l − j ) = X k ≥ 1 J l k z − k = 0 with La uren t se ries F ( n,s ) ≡ z s + X k ≥ 1 F ( n,s ) k z − k . Th us, restriction of DKP c hain flo ws on S n 0 ∩ S p l − 1 is giv en b y the follow ing p olynomial conditions: J l k ≡ a [ p ] k + l − F ( n,l ) k − l − 1 X j =1 a [ p ] j F ( n,l − j ) k = 0 , ∀ k ≥ 1 (25) whic h, in f act, define some submanifold M n,p,l ⊂ M inv arian t with respect to flo ws of n th dis crete KP hierarch y (2 3 ). Making of use relation (21) w e get the follo wing F ( n,l ) k ( i ) + l − 1 X j =1 a [ p ] j ( i ) F ( n,l − j ) k ( i ) = a [ ln ] k + l ( i ) + l − 1 X j =1 q ( n,ln − p ) j ( i + p ) a [( l − j ) n ] k + l − j ( i ) R e ductions of inte gr able lattic es 10 and, hen ce, J l k = a [ p ] k + l ( i ) − a [ ln ] k + l ( i ) − l − 1 X j =1 q ( n,ln − p ) j ( i + p ) a [( l − j ) n ] k + l − j ( i ) . Replacing in (20) k → k + l a nd setting r = l n w e get J l k = Q l k + k − 1 X j =1 a [ − ( k − j ) n ] j Q l k − j , k ≥ 1 , (26) with Q l k ( i ) ≡ q ( n,ln − p ) k + l ( i + p ). Solving thes e relations in fa v or of Q l k yields Q l k = J l k + k − 1 X j =1 q ( n, − ( k − j ) n ) j J l k − j , k ≥ 1 . (27) F ro m (24) w e ha v e D t s Q l k ( i ) = Q l s + k ( i + sn ) + s X j =1 q ( n,sn ) j ( i + p ) Q l s + k − j ( i + ( s − j ) n ) − Q l s + k ( i ) − s X j =1 q ( n,sn ) j ( i − ( s + k − j ) n ) Q l s + k − j ( i ) . It is w orth while to notice t ha t the co efficien ts of this equation do not dep end on l . The same is true for co efficien ts of transformatio n (27). 3.4. Examples of in te gr able lattic es Let us giv e b elo w some examples of differen tial- difference equations whic h app ear a s a result of restricting of n th dKP hierarc h y on M n,p,l . (i) One-field lattic es . The s ubmanifold M n,p, 1 is defined by infinite set of conditions J 1 k = a [ p ] k − a [ n ] k = 0 , ∀ k ≥ 2. Without loss of g enerality o ne can suppose that p ≥ 1 a nd n < p and n 6 = 0. F rom (15) one has a [ p ] 2 ( i ) = a [ n ] 2 ( i ) + a [ n ] 1 ( i ) a [ p − n ] 1 ( i + n ) + a [ p − n ] 2 ( i + n ) = a [ n ] 2 ( i + p − n ) + a [ n ] 1 ( i + p − n ) a [ p − n ] 1 ( i ) + a [ p − n ] 2 ( i ) and a [ p − n ] 2 ( i ) = − a [ n ] 1 ( i − n ) a [ p − n ] 1 ( i ) and a [ n ] 2 ( i + p − n ) − a [ n ] 2 ( i ) = a [ p − n ] 1 ( i )  a [ n ] 1 ( i − n ) − a [ n ] 1 ( i + p − n )  . F ro m the latter one has follo wing diffe rential-difference equation ‡ ∂ a [ p − n ] 1 ( i ) = a ′ i + · · · + a ′ i + p − n − 1 = a [ n ] 2 ( i + p − n ) − a [ n ] 2 ( i ) = a [ p − n ] 1 ( i )  a [ n ] 1 ( i − n ) − a [ n ] 1 ( i + p − n )  . (28) ‡ Here and in what follows a i ≡ a 1 ( i ) R e ductions of inte gr able lattic es 11 One needs to conside r tw o cases. Let n ≥ 1 and p ≥ n + 1 . Then (28) is sp ecified as p − n X s =1 a ′ i + s − 1 = p − n X s =1 a i + s − 1   n X s =1 a i − s − p − 1 X s = p − n a i + s   . (29) Imp ortant case to consider is n ≥ 1 and p = n + 1 whic h corr espo nds to Bogo y avle nskii lattice (8). Let n ≤ − 1 and p ≥ 1. In this case (28) b ecomes p + | n | X s =1 a ′ i + s − 1 = p + | n | X s =1 a i + s − 1   p + | n | X s = p +1 a i + s − 1 − | n | X s =1 a i + s − 1   . (30) It should b e noted tha t tw o pairs of in tegers ( n, p ) and ( − p, | n | ) corresp onds to the same eq uation of the form (30 ). (ii) T o da lattic e . When res tricting dKP hierarch y on M 1 , 1 , 2 , one requires J 2 k = a k +2 ( i ) − a [2] k +2 ( i ) − q (1 , 1) 1 ( i + 1) a k +1 ( i ) = 0 . The latter is solved by a k +2 ( i ) = − k X j =1 a j ( i − 1) a k − j +2 ( i ) . Equations of t he firs t flo w of dKP hierarc hy in this case are reduced to a pair of ev olution equations a ′ 1 ( i ) = a 2 ( i + 1) − a 2 ( i ) , a ′ 2 ( i ) = a 2 ( i )( a 1 ( i − 1) − a 2 ( i )) whic h are equiv alent to T o da lattice in its ex p onential form [18] (iii) Belov-Ch altikian lattic e . Restricting first flo w of dKP hie rarch y on M 1 , 3 , 2 yields t w o-field system a ′ 1 ( i ) = a 2 ( i + 1) − a 2 ( i ) , a ′ 2 ( i ) = a 1 ( i ) a 2 ( i − 1) − a 1 ( i − 1) a 2 ( i + 1) + a 2 ( i ) { a 1 ( i − 2) + a 1 ( i − 1) − a 1 ( i ) − a 1 ( i + 1) } + a 1 ( i − 1) a 1 ( i )( a 1 ( i − 2) − a 1 ( i + 1)) w cic h, in turn, via inv ertible ansatz L i = − a 1 ( i ) , W i = a 2 ( i + 1) + a 1 ( i ) a 1 ( i + 1) can be transformed into Belo v-Chaltikian lattice [1] L ′ i = W i − 1 − W i + L i ( L i +1 − L i − 1 ) , W ′ i = W i ( L i +2 − L i − 1 ) . (iv) Shab at dr essing lattic e . Consider M 1 , 0 , 2 whic h is defined b y conditio ns J 2 k = − a [2] k +2 − q (1 , 2) 1 a k +1 = 0 . (31) One can c hec k that in virtue of (31) with k = 1, D x q (1 , 2) 2 ( i ) = D x ( − a 2 ( i ) − a 2 ( i + 1) + a 2 i ) = 0 . So, w e can write a 2 ( i ) + a 2 ( i + 1) = a 2 i − µ i , where µ i do not dep end on x and a ′ i + a ′ i +1 = a 2 ( i + 2) − a 2 ( i ) = a 2 i +1 − a 2 i − µ i +1 + µ i . The latter is not hing but Shabat dressing lattice [13]. R e ductions of inte gr able lattic es 12 4. Reductions of n th dKP hierarc h y The main go al of this section is to sho w s ome class of restrictions compatible with n th dKP hierar ch y (2 3). Corr esp o nding constraints are suppo sed to be written in the form of p erio dicit y conditions I l k ( i + n ) = I l k ( i ) , ∀ k ≥ 1 (32) with suitable infinite collection of p olynomial discrete functions { I l k = I l k : k ≥ 1 } . W e are looking for these functions through in v ertible relations § Q k = I k + k − 1 X j =1 ζ k − 1 ,j I k − j , k ≥ 1 (33) with some unkno wn co efficien ts ζ k ,j . W e hav e D t s I 1 ( i ) = D t s Q 1 ( i ) = Q s +1 ( i + sn ) + s X j =1 q ( n,sn ) j ( i + p ) Q s − j +1 ( i + ( s − j ) n ) − Q s +1 ( i ) − s X k =1 q ( n,sn ) j ( i − ( s − j + 1 ) n ) Q s − j +1 ( i ) . (34) Substituting ( 33) into righ t-ha nd side of (34) w e require that it is iden tically zero pro vided that the conditions (3 2) are imposed. This leads to r elat io ns ζ s,m ( i + sn ) + m − 1 X j =1 q ( n,sn ) j ( i + p ) ζ s − j,m − j ( i + ( s − j ) n ) + q ( n,sn ) m ( i + p ) = ζ s,m ( i ) + m − 1 X j =1 q ( n,sn ) j ( i − ( s − j + 1) n ) ζ s − j,m − j ( i ) + q ( n,sn ) m ( i − ( s − m + 1 ) n ) with s ≥ 1 and m = 1 , ..., s . T aking in to accoun t (19), one can easily c hec k that the solution of these eq uations is giv en b y ζ s,m ( i ) = q ( n, − p − ( s − m +1) n ) m ( i + p ) . (35) Th us, w e hav e the follo wing: if conditions (32) for I k defined through the r elat io ns Q k ( i ) = I k ( i ) + k − 1 X j =1 q ( n, − p − ( k − j ) n ) j ( i + p ) I k − j ( i ) , k ≥ 1 (36) are v alid then D t s I 1 = 0 , ∀ s ≥ 1. Solving (36) in fa v or of I k yields I k ( i ) = Q k ( i ) + k − 1 X j =1 a [ − p − ( k − j ) n )] j ( i + p ) Q k − j ( i ) , k ≥ 1 . (37) W e can prov e, b y induction, that prov ided (3 2 ) the quan tities I k do not depend on ev olution parameters t s for a ll k ≥ 1. W e ha v e D t s I k +1 ( i ) + k X j =1 D t s ζ k ,j ( i ) I k − j +1 ( i ) + k X j =1 ζ k ,j ( i ) D t s I k − j +1 ( i ) = D t s Q k +1 ( i ) § In what follows we use simplified no tations I k ≡ I l k , Q k ≡ Q l k etc R e ductions of inte gr able lattic es 13 = Q s + k +1 ( i + sn ) + s X j =1 q ( n,sn ) j ( i + p ) Q s − j + k +1 ( i + ( s − j ) n ) − Q s + k +1 ( i ) − s X j =1 q ( n,sn ) j ( i − ( s − j + k + 1) n ) Q s − j + k +1 ( i ) . Let us supp o se that w e already pro v ed that D t s I j for j = 1 , ..., k in virtue of (3 2) with I k giv en b y (36) . Then D t s I k +1 = 0 under (32) if the relations ζ s + k ,m ( i + sn ) + m − 1 X j =1 q ( n,sn ) j ( i + p ) ζ s + k − j,m − j ( i + ( s − j ) n ) + q ( n,sn ) m ( i + p ) = ζ s + k ,m ( i ) + m − 1 X j =1 q ( n,sn ) j ( i − ( s + k − j + 1) n ) ζ s + k − j,m − j ( i ) + q ( n,sn ) m ( i − ( s + k − m + 1) n ) with m = 1 , ..., s and ζ s + k ,s + m ( i + sn ) + s X j =1 q ( n,sn ) j ( i + p ) ζ s + k − j,s + m − j ( i + ( s − j ) n ) = ζ s + k ,s + m ( i ) + s X j =1 q ( n,sn ) j ( i + ( s + k − j + 1) n )) ζ s + k − j,s + m − j ( i ) + D t s ζ k ,m ( i ) with m = 1 , ..., k are v alid. Again w e can c hec k that these relatio ns are solv ed b y (35). As an ob vious cons equence of the ab ov e calculations w e o bta in following theorem: Theorem 3. P erio dicit y conditions (32) with I l k giv en b y (37) are compatible with n th dis crete KP hierarch y . Let us denote the submanifold of M defined by conditions of p erio dicit y (32) as N n,p,l . Infinite set o f constraints I l k = 0 defining M n,p,l giv e particular solutio n of ( 3 2) and, hen ce, w e hav e M n,p,l ⊂ N n,p,l . It is useful to es tablish relationship b et w een J l k and I l k . Making of use (20), (26) and (36), w e get the relation J k = I k + k − 1 X j =1 a [ p ] j I k − j and its in v erse I k ( i ) = J k ( i ) + k − 1 X j =1 a [ − p ] j ( i + p ) J k − j ( i ) . 5. Reductions of Bogo y a vlenskii latt ice The goal o f this Section is to sho w how Theorem 3 can b e applied for constructing of some class of constraints compatible with Bogoy avlens kii lattice (8 ). More ex actly , we w ould lik e to sho w a c lass of restrictions whic h correspo nd to in t ersection M n,n +1 , 1 ∩ N n,p, 1 whic h is eviden tly equiv a lent t o M n +1 ,n, 1 ∩ N n +1 ,p, 1 . The submanifold M n,n +1 , 1 is defined by equations a [ n ] k = a [ n +1] k with k ≥ 2 whic h uniquely solv ed as a k = P [ n ] k [ a ] R e ductions of inte gr able lattic es 14 with s ome discrete p olynomials P [ n ] k . Next w e require that the perio dicit y conditions I 1 k ( i + n ) = I 1 k ( i ) and I 1 k ( i + n + 1) = I 1 k ( i ) m ust b e v alid simutaneous ly . This is equiv alen t to I 1 k ( i + 1) = I 1 k ( i ). It is natural t o consider three different cases: (i) Let p ≥ n + 2 . Keeping in mind the condition a [ n ] 2 = a [ n +1] 2 , one calculate to obtain I 1 1 ( i ) = J 1 1 ( i ) = a [ p ] 2 ( i ) − a [ n ] 2 ( i ) = a i ( a i + n +1 + · · · + a i + p − 1 ) + a i +1 ( a i + n +2 + · · · + a i + p − 1 ) + · · · + a i + p − n − 2 a i + p − 1 . The condition I 1 1 ( i + 1) = I 1 1 ( i ) can b e written as a i + p = a i Q i + n Q i , with Q i ≡ p − n − 1 X j =1 a i + j . (38) It is natural to supp ose tha t this difference equation has a n um b er of in tegrals enough fo r its in tegrability [21]. If so, then this collection of integrals is pr ovided by { I 1 k } but one mus t to calculate its explicit form I 1 k = I 1 k [ a ] in eac h case. Nev erthless , w e can write explic it form of t w o in tegrals for (38): K i = Q p s =1 a i + s − 1 Q n j =1 Q i + j − 1 and P i = Q n +1 j =1 ( a i + j − 1 + Q i + j − 1 ) Q n j =1 Q i + j − 1 . whic h directly follows from the form of this equation. Second integral here is deriv ed as f ollo ws. As a consequence of (38) we can write a i + p ( a i + Q i ) = a i ( a i + n +1 + Q i + n +1 ) and Q i + n ( a i + Q i ) = Q i ( a i + n +1 + Q i + n +1 ) . It is ob vious that the latter relatio n can b e written a s P i = P i +1 . (ii) Let p ≤ − 1. W e calculate to write down I 1 1 ( i ) = a i − 1  a i −| p | + · · · + a i + n − 1  + a i − 2  a i −| p | + · · · + a i + n − 2  + · · · + a i −| p |− n a i −| p | . The corres p o nding cons traint can be written as a i + | p | + n Q i + n = a i + n Q i − 1 , with Q i ≡ | p | + n X j =1 a i + j . Solving this relation in fav o r of a i + | p | +2 n yields a i +2 n + | p | = a i + n a i + | p | + n Q i − 1 − 2 n + | p |− 1 X j = n +1 a i + j . (39) In t his case w e also a ble to write tw o integrals for (39) in its explicit form K i = | p | Y j =1 a i + j − 1 n +1 Y j =1 Q i − j and P i = Q n j =1 ( a i − j + Q i − j ) Q n +1 j =1 Q i − j . R e ductions of inte gr able lattic es 15 (iii) Let p = 1 , ..., n − 1. Then I 1 1 ( i ) = a i − 1 ( a i + p + · · · + a i + n − 1 ) + a i − 2 ( a i + p + · · · + a i + n − 2 ) + · · · + a i + p − n a i + p . W e can write do wn corres p o nding cons traint in the form a i + n − p Q i + n = a i + n Q i − 1 , with Q i ≡ n − p X j =1 a i + j and a i +2 n − p = a i + n a i + n − p Q i − 1 − 2 n − p − 1 X j = n +1 a i + j . Tw o in tegrals K and P are: K i = Q p j =1 a i − j Q n +1 j =1 Q i − j and P i = Q n j =1 ( a i − j + Q i − j ) Q n +1 j =1 Q i − s . One sees that in eac h of three cases corresp onding constrain ts ar e written in the form of ordinar y difference equation a i + N = R ( a i , ..., a i + N − 1 ) with N = p , N = 2 n + | p | and N = 2 n − p , resp ective ly . Finally , let us sho w simple example corr esp o nding to p = 4 and n = 1, that is, when V olterra lattice (1) is cons trained by condition giv en b y fourth-order difference equation a i +4 = a i a i +2 + a i +3 a i +1 + a i +2 . with three in tegrals K i = a i a i +1 a i +2 a i +3 a i +1 + a i +2 , P i = ( a i + a i +1 + a i +2 )( a i +1 + a i +2 + a i +3 ) a i +1 + a i +2 . J i = a i ( a i +2 + a i +3 ) + a i +1 a i +3 , Observ e that I 1 1 = J a nd I 1 2 = − K − J P . Iden tify no w a i = u 1 , a i +1 = u 2 , a i +2 = u 3 , a i +3 = u 4 . A ttac hed system of ordinary differential equations lo oks as follows: u ′ 1 = u 1  u 4 u 1 + u 2 u 2 + u 3 − u 2  , u ′ 2 = u 2 ( u 1 − u 3 ) , u ′ 3 = u 3 ( u 2 − u 4 ) , u ′ 4 = u 4  u 3 − u 1 u 3 + u 4 u 2 + u 3  , This s ystem has three first in tegr a ls K = u 1 u 2 u 3 u 4 u 2 + u 3 , P = ( u 1 + u 2 + u 3 )( u 2 + u 3 + u 4 ) u 2 + u 3 . J = u 1 ( u 3 + u 4 ) + u 2 u 4 . Shifting i → i + 1 giv es discrete symmetry transformation ¯ u k = u k +1 , k = 1 , 2 , 3 , ¯ u 4 = u 1 u 3 + u 4 u 2 + u 3 (40) By direct calculations one can c hec k that in tegrals K , P and J a r e inv ariant under (40). R e ductions of inte gr able lattic es 16 6. Conclusion W e hav e presen ted a sc heme for constructing a broad class o f constrain ts compatible with in tegrable lat t ices whic h can b e deriv ed as reductions of DKP chain. F or particular case of the Bogo y a vlenskii lattice w e show ed some simple examples in its explicit form. 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