On some class of homogeneous polynomials and explicit form of integrable hierarchies of differential-difference equations

On some class of homogeneou s p olynomials and explicit form of in tegrable hierarc hies of differen t ial-difference equations A K Svinin Institute for System Dynamics and Control Theory , Siberia n Branc h of Russian Academy of Sciences , Russia E-mail: svinin @icc.ru Abstract. W e int r oduce tw o classes of homo geneous polynomials and show their role in constructing of integrable hierarchies for some integrable lattices. P ACS num b ers: 02.30.Ik Keywor ds : KP hierarc h y , integrable lattices Submitted to: J. Phys. A: M at h. The or. 1. In tro duction Our main goal in this pap er is to introduce t wo classes o f homogeneous p olynomials T l s and S l s of ma ny v aria bles and to sho w its applicabilit y in the theory of integrable differen tial-difference equations (lattices). More exactly , w e construct in terms of these p olynomials explicit form of some integrable hierarc hies. W e base our studies on the relationship of inte g rable lattices with K P hierar ch y . T o this aim, w e consider bi-infinite sequence s of KP hierarc hies in the f o rm of lo cal differen tial-difference conserv a t ion la ws [10] expressed in terms of generating relations. Our approac h, in particular, go es bac k to [2 ], [4], [10]. Let us giv e a sk etc h of our approac h for in ves tig ation of some in tegrable la ttices and its hierarc hies. A presen tatio n of the sequence of K P hierarc hy in the fo r m o f tw o generating equations ∂ s h ( i, z ) = ∂ H ( s ) ( i, z ) and ∂ s a ( i, z ) = a ( i, z )  H ( s ) ( i + 1 , z ) − H ( s ) ( i, z )  , i ∈ Z , where ∂ s stands for a deriv ativ e with resp ect to the ev o lutio n parameter t s , can b e found in [4]. T aking this represen tat ion as a ba sis we hav e sho wn in [6], [7], [8] that man y in tegrable lattices can b e o bta ined as reductions of this system . Our study in these pap ers w as based on tw o theorems whic h select an infinite n umber of in v ariant On som e class of h omo gene ous p olynomials 2 submanifolds S n l − 1 . In particular the restriction of this sequence of KP hierarch ies on S n 0 giv es the totality of ev olution differen tial- difference equations on infinite num b er of fields { a k = a k ( i ) } . W e call this set of equations the n th discrete KP hierar ch y . W e denote by M the corresp onding pha se space. F urther w e consider in tersections of the form S n 0 ∩ S p l − 1 and show that this correspo nds to restriction of the n th discrete KP hierarc h y on some submanifold M n,p,l ⊂ M restricting the latter to some differen tial- difference syste m of equations on finite nu mber of fields. The restriction on M n,p,l is giv en by an infinite set of algebraic equations J ( n,p,l ) k [ a 1 , . . . , a k + l ] = 0 for k ≥ 1. In this paper w e consider only a class of one-field lattices c o rresponding to submanifolds M n,p, 1 with n ≥ 1 and p > n . The pa rticular case, when p = n + 1, is giv en by Itoh- Narita-Bogo ya vlenskii (INB) la ttice [3], [5 ], [1] or extended V olterra equation in terminology of [5]. F or general case of M n,p, 1 w e construct corr esp onding in tegrable hierarch y in explicit form. In [9 ] w e a lready ha ve sho wn explicit form of in tegrable hierarc hy of INB lattice equations in terms of homog ene o us p olynomials S l s . In this pap er w e construct more general class of suc h p olynomials and cor r esp onding in tegrable hierarc hies. The rest of the pap er is organized as follo ws. In the next tw o sections w e give explanation of the relationship of the KP hierarch y with some integrable lattices. In section 4, w e define t w o classes of homog eneous p olynomials T l s and S l s of man y v aria bles and sho w some identities whic h directly fo llow from their definition. In section 5, w e consider the restriction of n th discrete KP hierarc hy on submanifolds M n,p, 1 corresp onding to one-comp onen t lattices and pro v e there some tec hnical lemmas. In particular, w e write down an infinite n umber of linear equations on KP wa v efunction Ψ = { ψ i } whic h corr es p onds to t he sequence of in v arian t submanifolds inclusions. W e sho w explicit form of ev olution equations o f some class o f in tegrable hierarc hies. In subsections 5.5 a nd 5.6 w e pro vide the reader by examples and sho w explicit form of some integrable hierarc hies of o ne-component lattices. 2. The K P hierarc h y 2.1. Th e KP hier ar chy It is kno wn tha t there exists close relationship of the KP hierarc hy with some integrable lattices and its hierarchie s. In this a nd next sections w e briefly describ e t his relationship based on our approach dev elop ed in [6], [7], [8] considering free bi-infinite c hain of KP hierarc hies and its suitable r educ tio ns. W e write ev o lutio n equations of the KP hierarc hy itself in the form of lo cal conserv ation laws [10] ∂ s h ( z ) = ∂ H ( s ) ( z ) , (1) where formal Lauren t series h ( z ) = z + P j ≥ 2 h j z − j +1 and H ( s ) ( z ) = z s + P j ≥ 1 H s j z − j are related with forma l KP wa v efunction a s h ( z ) = ∂ ψ / ψ and H ( s ) ( z ) = ∂ s ψ /ψ , resp ectiv ely . Eac h co efficien ts H s j , in fact, is calculated to b e some differential p olynomial H s k = On som e class of h omo gene ous p olynomials 3 H s k [ h 2 , . . . , h s + k ]. More exactly , the La urent series H ( s ) ( z ) is calculated a s pro jection of z s on to the linear space H + = < 1 , h, h (2) , . . . > spanned b y F a` a di Bruno iterates h ( k ) ≡ ( ∂ + h ) k (1). It can b e show n that dynamics of co efficien ts H s j in virtue of KP flo ws (1) is given b y the in v ariance relation ( ∂ s + H ( s ) ) H + ⊂ H + whic h can b e written explicitly as ∂ s H ( k ) = H ( k + s ) − H ( k ) H ( s ) + s X j =1 H k j H ( s − j ) + s X j =1 H s j H ( k − j ) . (2) Con v ersely , one can start from these equations whic h constitute a n infinite n umber of comm uting flo ws including the first flow corresp onding to evolution parameter t 1 = x on the phase space whose p oin ts are parameterized by the semi-infinite matrix ( H s j ). System (2) know n as the Cen tral System is equiv a len t in fact to the KP hierarch y and the latter can b e obtained by choosing t 1 as the spatial v aria ble [2]. 2.2. Th e system describing infinite chain of KP hier ar chi e s Let us consider bi-infinite sequence of KP hierarchie s { h ( i, z ) : i ∈ Z } . T o our aim, it is con v enien t to in t r oduce anot her Lauren t series a ( i, z ) = z + P j ≥ 1 a j ( i ) z − j +1 ≡ z ψ i +1 /ψ i , whic h as can b e chec ked to satisfy the equation ∂ s a ( i, z ) = a ( i, z )  H ( s ) ( i + 1 , z ) − H ( s ) ( i, z )  . (3) The lat t er in turn can b e rewritten in the form lo oking as differen tial- differenc e conserv ation laws ∂ s ξ ( i, z ) = H ( s ) ( i + 1 , z ) − H ( s ) ( i, z ) with ξ ( i, z ) = ln a ( i, z ) = ln z − X j ≥ 1 1 j − X k ≥ 1 a k ( i ) z − k ! j ≡ ln z + X j ≥ 1 ξ j ( i ) z − j . Th us as a starting p oin t w e consider following an infinite set of ev olution equations: ∂ s h k ( i ) = ∂ H s k − 1 ( i ) , ∂ s ξ k ( i ) = H s k ( i + 1) − H s k ( i ) (4) whic h as is sho wn b elo w to admit a bro a d class of reductions yielding an infinite n umber of differen tial- difference equations. Let us remem b er tha t the KP hierar ch y is equiv alen t to the Cen tral System (2). Therefore, w e can assume tha t the p oint of our phase space is defined b y infinite num b er o f functions { H j s ( i ) , a k ( i ) } . 3. The n t h discrete KP hierarc hy 3.1. R e ductions of e quations (4) W e ar e go ing to sho w in this and next sections how some integrable lattices can b e obtained as a result of sp ecial reductions o f equations (4). T o this aim, we need in follo wing tw o theorems. On som e class of h omo gene ous p olynomials 4 Theorem 1 [6] The submanifold S n l − 1 define d by the c ondition z l − n a [ n ] ( i, z ) ∈ H + ( i ) , ∀ i ∈ Z (5) is invariant w ith r esp e ct to flow s given by (4). Theorem 2 [7] The fol lowing chain of invarian t submanifolds inclusions S n l − 1 ⊂ S 2 n 2 l − 1 ⊂ S 3 n 3 l − 1 ⊂ · · · (6) is valid. Let us sp end some lines to clarify certain details. In the theorem 1, b y definition, a [ r ] ( i, z ) = z r ψ i + r ψ i =      Q r j =1 a ( i + j − 1 , z ) , r ≥ 1 , 1 , r = 0 , Q | r | j =1 a − 1 ( i − j, z ) , r ≤ − 1 . Th us, the co efficien t s of the Laurent series a [ r ] ( i, z ) = z r + P j ≥ 1 a [ r ] j ( i ) z − j + r are some quasi-homogeneous p olynomials a [ r ] j [ a 1 , . . . , a j ]. In what follows w e use obvious 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 ) for an y r 1 , r 2 ∈ Z , whic h yields an infinite set of iden t ities 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 ) . (7) It is w orth remarking that theorem 1 w as pro ven in [4] in the case n = 1. It is useful t o define another set of quasi-homo g eneous p olynomials { q ( n,r ) j = q ( n,r ) j [ a 1 , . . . , a j ] } with the help of the generating relation ‡ z r = a [ r ] + X j ≥ 1 q ( n,r ) j z j ( n − 1) a [ r − j n ] . (8) Clearly , in terms of the w av efunction Ψ = { ψ i } this relatio n t ak es the form z r ψ i = z r ψ i + r + X j ≥ 1 z r − j q ( n,r ) j ( i ) ψ i + r − j n . (9) Relation (8) generate triangular infinite system a [ r ] k + k − 1 X j =1 a [ r − j n ] k − j q ( n,r ) j + q ( n,r ) k = 0 , k ≥ 1 . (10) One can c hec k that a more general relation than (10), namely a [ m ] k ( i ) = a [ r ] k ( i ) + k − 1 X j =1 a [ r − j n ] k − j ( i ) q ( n,r − m ) j ( i + m ) + q ( n,r − m ) k ( i + m ) (11) ‡ F or simplicity we sometimes do no t indica te dep endence on discrete v aria ble i ∈ Z in for m ulae which contain no shifts with resp ect to this v ar iable. On som e class of h omo gene ous p olynomials 5 with an y r, m ∈ Z , is v alid [8]. Resolving the latter in fav or of q ( n,r − p ) k ( i + p ) yields q ( n,r − m ) k ( i + m ) = a [ m ] k ( i ) + k − 1 X j =1 q ( n,r − ( k − j ) n ) j ( i ) a [ m ] k − j ( i ) + q ( n,r ) k ( i ) . It should b e noted that p olynomials q ( n,r ) j satisfy follow ing iden tities: 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 ) (12) for an y r 1 , r 2 ∈ Z . These iden tities can b e obtained from (9) if w e rewrite it in op erator form: z r Ψ = Q ( n,r ) (Ψ). Then w e deriv e the set of iden tities (12 ) from the relation z r 1 + r 2 Ψ = Q ( n,r 1 ) · Q ( n,r 2 ) (Ψ) = Q ( n,r 2 ) · Q ( n,r 1 ) (Ψ) . Let us remark t ha t the condition (5) is equiv alent to the relation z l − n a [ n ] = H ( l ) + l X k =1 a [ n ] k H ( l − k ) . 3.2. Th e n th discr ete KP hier a r chy Let us consider now the restriction of equations (4) on S n 0 defined b y simple relation z 1 − n a [ n ] = h + a [ n ] 1 from which w e get h k = a [ n ] k first of all. Therefore, we know that on S n 0 one has H 1 k = h k +1 = a [ n ] k +1 . T o obtain explicit express ions for all H s k as quasi- homogeneous p olynomials of a k w e need in theorem 2. As a result w e ha v e [8] H s k = F ( n,s ) k [ a 1 , . . . , a k + s ] ≡ a [ sn ] k + s + s − 1 X j =1 q ( n,sn ) j a [( s − j ) n ] k + s − j . This tota lit y of relations can b e written dow n as a whole with t he help of one generating relation H ( s ) = F ( n,s ) = z s − sn a [ sn ] + s X j =1 z j ( n − 1) q ( n,sn ) j a [( s − j ) n ] ! , (13) where F ( n,s ) ≡ z s + P j ≥ 1 F ( n,s ) j z − j . Th us, the restriction of dynamics giv en b y (4) on S n 0 leads to ev olution equations in the fo rm of differen tial- difference conserv ation laws ∂ s ξ k ( i ) = F ( n,s ) k ( i + 1) − F ( n,s ) k ( i ) (14) on infinite num b er of fields { a k = a k ( i ) } . The cor r esp onding phase space we denote b y M . The hierarc hy of the flows on M giv en by ev olution equations (1 4 ) we call n th discrete KP hierarc h y . These equations also admit a ric h family of r eductions. On som e class of h omo gene ous p olynomials 6 F ollowing remark is in order. A reform ulation of the generating relatio n (13) in terms of the w av efunction Ψ = { ψ i } is ∂ s ψ i = z s ψ i + sn + s X j =1 z s − j q ( n,sn ) j ( i ) ψ i +( s − j ) n . (15) Chec king compatibility of (1 5) with (9) w e o btain ∂ 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 ) . (16) 3.3. ...a nd its r e ductions Ob viously , tha t restriction of dynamics giv en by (4) o n in tersection S n 0 ∩S p l − 1 is equiv alen t to restriction of n th discrete KP hierarch y on some submanifold M n,p,l ⊂ M . W e can easily to write do wn the equations defining M n,p,l . They eviden tly fo llo w f r o m generating relation z l − p a [ p ] = F ( n,l ) + l X j =1 a [ p ] j F ( n,l − j ) . F rom here w e o btain that M n,p,l is defined b y infinite num b er o f equations J ( n,p,l ) k [ a 1 , . . . , a k + l ] = 0 , ∀ k ≥ 1 (17) with J ( n,p,l ) k = a [ p ] k + l − F ( n,l ) k − l − 1 X j =1 a [ p ] j F ( n,l − j ) 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 ) . As a consequence of theorem 3, we hav e the follo wing. Theorem 3 The fol lowing c hain of i n variant submanifolds inclusions: M n,p,l ⊂ M n, 2 p, 2 l ⊂ M n, 3 p, 3 l ⊂ · · · (18) is valid. 3.4. Line ar e quations o n KP wavefunction c orr esp onding to subman ifold M n,p,l Some remarks ab out linear equations o n KP formal w av e function { ψ i } which fo llo ws as a result of restriction on M n,p,l are in order. Let J ( n,p,l ) ( z ) ≡ P j ≥ 1 J ( n,p,l ) j z − j . W e observ e that an infinite n umber of equations (17) can b e presen ted b y single generating relation J ( n,p,l ) ( i, z ) = z l − p a [ p ] ( i, z ) − z l − ln a [ ln ] ( i, z ) + l X j =1 z j ( n − 1) q ( n,ln − p ) j ( i + p ) a [( l − j ) n ] ( i, z ) ! = 0 . On som e class of h omo gene ous p olynomials 7 Clearly , in terms of KP w av efunction, we can rewrite the latter relation as J ( n,p,l ) ( i, z ) = z l ψ i + p − z l ψ i + ln − P l j =1 z l − j q ( n,ln − p ) j ( i + p ) ψ i +( l − j ) n ψ i = 0 . Th us in terms o f KP w av e function the r estriction of n th discrete KP hierarc hy on M n,p,l is giv en by the linear equation z l ψ i + ln + l X j =1 z l − j q ( n,ln − p ) j ( i + p ) ψ i +( l − j ) n = z l ψ i + p . (19) Second linear equation whic h we should hav e in mind is (15). When considering restriction of n th discrete KP hierarch y o n M n,p,l one can find man y examples of in tegrable lattices known fro m the literature (some examples can b e found in the pap er [8 ]) and construct “new” ones. Classical examples are V olterra and T o da lattices. In what follo ws, w e restrict ourselv es by consideration only a class of one-field lattices corresp onding to M n,p, 1 . 4. The p olynomials S l s and T l s In the next section w e consider restriction of n th discrete KP hierarc hy on the submanifold M n,p, 1 , with n ≥ 1 and p > n . F o r further use, let us prepare in this section t wo classes of p olynomials through § S l s ( y 1 | y sp − ( s − l + 1) n +1 ) = X 1 ≤ λ l ≤···≤ λ 1 ≤ s +1 l Y j =1 y ( p − n ) λ j + j n − p + 1 ! (20) and T l s ( y 1 | y sp − ( s − l + 1) n +1 ) = X 1 ≤ λ 1 < ··· <λ l ≤ s +1 l Y j =1 y ( p − n ) λ j + j n − p + 1 ! (21) for s ≥ 0 . k Let us show some identities fo r t hese p olynomials. Firstly consider S l s . A partition of the set B l,s = { λ j : 1 ≤ λ l ≤ · · · ≤ λ 1 ≤ s + 1 } in to a pair of disjoint subsets B l,s = B l 1 ∪ B l,s − 1 with B l 1 = { λ j : λ 1 = s + 1 , 1 ≤ λ l ≤ · · · ≤ λ 2 ≤ s + 1 } and B l,s − 1 = { λ j : 1 ≤ λ l ≤ · · · ≤ λ 1 ≤ s } , as can b e c hec ked, leads to the iden tity S l s ( y 1 | y sp − ( s − l + 1) n +1 ) = S l s − 1 ( y 1 | y ( s − 1) p − ( s − l ) n +1 ) + y ( p − n ) s +1 S l − 1 s ( y n +1 | y sp − ( s − l + 1) n +1 ) . (2 2) § W e us e per haps unusual but quite co nvenien t no tation wr iting “fir st” a nd “las t” arg ument through the vertical bar. k It is conv enie nt to think tha t T l s ≡ 0 for s = 0 , . . . , l − 2. On som e class of h omo gene ous p olynomials 8 Let B l k = { λ j : λ 1 = s − k + 2 , 1 ≤ λ l ≤ · · · ≤ λ 2 ≤ s − k + 2 } and B l,s − k +1 = { λ j : 1 ≤ λ l ≤ · · · ≤ λ 1 ≤ s − k + 2 } for k = 1 , . . . , s + 1 . Clearly , B l,s − k +1 = B l k ∪ B l,s − k and B l, 0 = B l s +1 . Th us, we hav e the follo wing: B l,s = s +1 [ j =1 B l j . This decomp osition of B l,s in turn yields the iden tity S l s ( y 1 | y sp − ( s − l + 1) n +1 ) = s +1 X j =1 y ( s − j +1)( p − n )+1 S l − 1 s − j +1 ( y n +1 | y ( s − j +1) p − ( s − l − j +2) n +1 ) . (23) On the other ha nd a partitio n of B l,s in to ¯ B l 1 = { λ j : λ l = 1 , 1 ≤ λ l − 1 ≤ · · · ≤ λ 1 ≤ s + 1 } and ¯ B l,s − 1 = { λ j : 2 ≤ λ l ≤ · · · ≤ λ 1 ≤ s + 1 } giv es the following identit y: S l s ( y 1 | y sp − ( s − l + 1) n +1 ) = S l s − 1 ( y p − n +1 | y sp − ( s − l + 1) n +1 ) + y ( l − 1) n +1 S l − 1 s ( y 1 | y sp − ( s − l + 2) n +1 ) . (24) Making use of the partition ¯ B l,s − k +1 = ¯ B l k ∪ ¯ B l,s − k and ¯ B l, 0 = ¯ B l s +1 with ¯ B l k = { λ j : λ l = k , k ≤ λ l − 1 ≤ · · · ≤ λ 1 ≤ s + 1 } and ¯ B l,s − k +1 = { λ j : k ≤ λ l ≤ · · · ≤ λ 1 ≤ s + 1 } w e are led to the partition B l,s = s +1 [ j =1 ¯ B l j and corresp onding identit y S l s ( y 1 | y sp − ( s − l + 1) n +1 ) = s +1 X j =1 y ( j − 1) p − ( j − l ) n +1 S l − 1 s − j +1 ( y ( j − 1)( p − n )+1 | y sp − ( s − l + 2) n +1 ) . (25) It is worth to remark that identities (23) a nd (25) b eing in nature recurrence relations b oth uniquely define p olynomials S l s starting f rom S 0 s ≡ 1. Consider no w p olynomials T l s . A partition of the set D l,s = { λ j : 1 ≤ λ 1 < · · · < λ l ≤ s + 1 } , in to tw o subsets D l 1 = { λ j : λ 1 = 1 , 2 ≤ λ 2 < · · · < λ l ≤ s + 1 } On som e class of h omo gene ous p olynomials 9 and D l,s − 1 = { λ j : 2 ≤ λ 1 < · · · < λ l ≤ s + 1 } leads t o the iden tity T l s ( y 1 | y sp − ( s − l + 1) n +1 ) = T l s − 1 ( y p − n +1 | y sp − ( s − l + 1) n +1 ) + y 1 T l − 1 s − 1 ( y p +1 | y sp − ( s − l + 1) n +1 ) . (26) Let D l k = { λ j : λ 1 = k , k + 1 ≤ λ 2 < · · · < λ l ≤ s + 1 } and D l,s − k +1 = { λ j : k ≤ λ 1 < · · · < λ l ≤ s + 1 } for k = 1 , . . . , s − l + 2. T aking in to account that D l,s − k +1 = D l k ∪ D l,s − k and D l,l − 1 = D l s − l +2 w e are led to the partition D l,s = s − l +2 [ j =1 D l j and corresp onding identit y T l s ( y 1 | y sp − ( s − l + 1) n +1 ) = s − l +2 X j =1 y ( j − 1)( p − n )+1 T l − 1 s − j ( y j p − ( j − 1) n +1 | y sp − ( s − l + 1) n +1 ) (27) Finally , consider the partit ion D l,s = ¯ D l 1 ∪ ¯ D l,s − 1 with ¯ D l 1 = { λ j : λ l = s + 1 , 1 ≤ λ 1 < · · · < λ l − 1 ≤ s } and ¯ D l,s − 1 = { λ j : 1 ≤ λ 1 < · · · < λ l ≤ s } and corresp onding identit y T l s ( y 1 | y sp − ( s − l + 1) n +1 ) = T l s − 1 ( y 1 | y ( s − 1) p − ( s − l ) n +1 )+ y sp − ( s − l + 1) n +1 T l − 1 s − 1 ( y 1 | y ( s − 1) p − ( s − l +1) n +1 ) . (28) Let ¯ D l k = { λ j : λ l = s − k + 2 , 1 ≤ λ 1 < · · · < λ l − 1 ≤ s − k + 1 } and ¯ D l,s − k +1 = { λ j : 1 ≤ λ 1 < · · · < λ l ≤ s − k + 1 } . W e hav e ¯ D l,s − k +1 = ¯ D l k ∪ ¯ D l,s − k and ¯ D l,l − 1 = ¯ D l s − l +2 . The partition D l,s = s − l +2 [ j =1 ¯ D l j leads t o the iden tity T l s ( y 1 | y sp − ( s − l + 1) n +1 ) = s − l +2 X j =1 y ( s − j +1) p − ( s − l − j +2) n +1 T l − 1 s − j ( y 1 | y ( s − j ) p − ( s − l − j +2) n +1 ) . (29) On som e class of h omo gene ous p olynomials 10 Let us iden tify y k = r i + k − 1 for k = 1 , . . . , sp − ( s − l + 1) n + 1, where r = r ( i ) ≡ r i is some unknow n function of discrete v ariable i ∈ Z . W e define discrete p olynomial functions S l s [ r ] and T l s [ r ] by S l s ( i ) ≡ S l s ( r i | r i + sp − ( s − l +1) n ) a nd T l s ( i ) ≡ T l s ( r i | r i + sp − ( s − l +1) n ) , resp ectiv ely . Let us write do wn b elo w all iden tit ies for S l s [ r ] and T l s [ r ] corresp onding to relations (22)-(29) in their order. W e hav e the follow ing: S l s ( i ) = S l s − 1 ( i ) + r i + s ( p − n ) S l − 1 s ( i + n ) = s +1 X j =1 r i +( s − j +1)( p − n ) S l − 1 s − j +1 ( i + n ) = S l s − 1 ( i + p − n ) + r i +( l − 1) n S l − 1 s ( i ) = s +1 X j =1 r i +( j − 1) p − ( j − l ) n S l − 1 s − j +1 ( i + ( j − 1)( p − n )) and T l s ( i ) = T l s − 1 ( i + p − n ) + r i T l − 1 s − 1 ( i + p ) = s − l +2 X j =1 r i +( j − 1)( p − n ) T l − 1 s − j ( i + j p − ( j − 1) n ) = T l s − 1 ( i ) + r i + sp − ( s − l +1) n T l − 1 s − 1 ( i ) = s − l +2 X j =1 r i +( s − j +1) p − ( s − l − j +2) n T l − 1 s − j ( i ) . Since T l s ≡ 0 for s = 0 , . . . , l − 2, then we also can write T l s ( i ) = s X j =1 r i +( j − 1)( p − n ) T l − 1 s − j ( i + j p − ( j − 1) n ) = s X j =1 r i +( s − j +1) p − ( s − l − j +2) n T l − 1 s − j ( i ) . (30) 5. The manifold M n,p, 1 5.1. T e chnic al lem m as Our aim is to show ho w p olynomials S l s [ r ] and T l s [ r ] constructed in previous section app ear when constructing in tegra ble hierarchie s for some differen tia l-difference equations in its explicit form. Let us consider the submanifold M n,p, 1 ⊂ M with n ≥ 1 and p > n defined b y equations J ( n,p, 1) k = a [ p ] k +1 − a [ n ] k +1 ≡ 0 for k ≥ 1 whic h, using (7) can b e rewritten equiv alen tly as k X j =1 a [ n ] k − j +1 ( i ) a [ p − n ] j ( i + n ) + a [ p − n ] k +1 ( i + n ) = 0 . On som e class of h omo gene ous p olynomials 11 Making use again of the identit y (7) we see that solution of this equation is giv en by a [ p − n ] k ( i ) = a [ − n ] k − 1 ( i ) r i with r i ≡ a [ p − n ] 1 ( i ) (31) for all k ≥ 2. Lemma 1 On M n,p, 1 a [( p − n ) s ] k ( i ) = s X j =1 a [ − j n +( j − 1) p ] k − 1 ( i ) r i +( j − 1)( p − n ) , (32) and a [( n − p ) s ] k ( i ) = − s X j =1 a [ − j p + ( j − 1) n ] k − 1 ( i ) r i + j ( n − p ) (33) for al l inte gers s ≥ 1 . Pro of . In virtue of (7) and (31), a [( p − n ) s ] k ( i ) = a [( p − n )( s − 1)] k ( i ) + k − 1 X j =1 a [( p − n )( s − 1)] k − j ( i ) a [ p − n ] j ( i + ( p − n )( s − 1) ) + a [ p − n ] k ( i + ( p − n )( s − 1) ) = a [( p − n )( s − 1)] k ( i ) + n a [( p − n )( s − 1)] k − 1 ( i ) + k − 2 X j =1 a [( p − n )( s − 1)] k − j − 1 ( i ) a [ − n ] j ( i + ( p − n )( s − 1) ) + a [ − n ] k − 1 ( i + ( p − n )( s − 1)) ) r i +( p − n )( s − 1) = a [( p − n )( s − 1)] k ( i ) + a [( s − 1) p − sn ] k − 1 ( i ) r i +( p − n )( s − 1) . Making use of this r ecurrence relation w e immediately obtain (32) and (33 ) . Lemma 2 On M n,p, 1 q ( n, ( p − n ) s ) k ( i ) = ( − 1) k S k s − 1 ( i − ( k − 1) n ) , (34) q ( n, ( n − p ) s ) k ( i ) = T k s − 1 ( i − ( k − 1) n + ( n − p ) s ) (35) for s ≥ 1 . Let us remark that this lemma, in particular, sa ys that on M n,p, 1 w e ha ve q ( n, ( n − p ) s ) k ≡ 0 for s = 1 , . . . , k − 1 . Pro of of lemma 2 . In par ticular case r = 1 iden t it y (11) is sp ecified as a [ m ] k ( i ) = k − 1 X j =1 a [ − j n ] k − j ( i ) q ( n, − m ) j ( i + m ) + q ( n, − m ) k ( i + m ) . Let m = ( p − n ) s . W e observ e tha t in virtue of the latter iden tit y , (34) is equiv alen t to the relation a [( p − n ) s ] k ( i ) = k − 1 X j =1 a [ − j n ] k − j ( i ) T k − j s − 1 ( i − ( j − 1) n ) + T k s − 1 ( i − ( k − 1) n ) . (36) On som e class of h omo gene ous p olynomials 12 So, let us pro ve ( 3 6) instead of (35). F or k = 1 , the latter b ecomes a [( p − n ) s ] 1 ( i ) = T 1 s − 1 ( i ) what is eviden t, b ecause ¶ a [( p − n ) s ] 1 ( i ) = ( p − n ) s X j =1 a i + j − 1 = s X j =1 r i +( j − 1)( p − n ) ≡ T 1 s − 1 ( i ) (37) simply b y definition and without reference to an y in v arian t submanifold. F urther, w e pro ceed by induction. Supp ose w e ha v e prov ed (36) for k = 1 , . . . , k 0 , where k 0 ≥ 1. Then it is easy to see, making use of (7), that for these v a lues of k the relations of the form a [ m +( p − n ) s ] k ( i ) − a [ m ] k ( i ) = T k s − 1 ( i + m − ( k − 1) n ) + k − 1 X j =1 a [ m − ( k − j ) n ] j ( i ) T k − j s − 1 ( i + m − ( k − j − 1) n ) hold for an y m ∈ Z . In par t icular, let m = − n ; then a [ − n +( p − n ) s ] k ( i ) − a [ − n ] k ( i ) = T k s − 1 ( i − k n ) + k − 1 X j =1 a [ − ( k − j +1) n ] j ( i ) T k − j s − 1 ( i − ( k − j ) n ) . (38) With the help o f (30), (32), (37) and (38) w e ha ve the following: a [( p − n ) s ] k +1 ( i ) − a [ − n ] k ( i ) T 1 s − 1 ( i ) = s − 1 X j =1 n a [ − ( j +1) n + j p ] k ( i ) − a [ − n ] k ( i ) o r i + j ( p − n ) = s − 1 X j =1 ( T k j − 1 ( i − k n ) + k − 1 X j 1 =1 a [ − ( k − j 1 +1) n ] j 1 ( i ) T k − j 1 j − 1 ( i − ( k − j 1 ) n ) ) r i + j ( p − n ) = s − 1 X j =1 T k j − 1 ( i − k n ) r i + j ( p − n ) + k − 1 X j 1 =1 a [ − ( k − j 1 +1) n ] j 1 ( i ) s − 1 X j =1 T k − j 1 j − 1 ( i − ( k − j 1 ) n ) r i + j ( p − n ) ! = T k +1 s − 1 ( i − k n ) + k − 1 X j 1 =1 a [ − ( k − j 1 +1) n ] j 1 ( i ) T k − j 1 +1 s − 1 ( i − ( k − j 1 ) n ) . Th us, we ha ve prov ed that if (3 6) is v alid for k = 1 , . . . , k 0 then it is v alid for k = k 0 + 1 and therefore we can use no w induction with resp ect to k . The relation (34) is prov ed b y using similar reasonings. 5.2. A dditional identities for S l s [ r ] and T l s [ r ] Making use of lemma 2 and iden tities (12) with r 1 = ( p − n )( s 1 +1) and r 2 = ( p − n )( s 2 +1) w e are able to obtain following iden tit ies: S l s 1 ( i ) + l − 1 X j =1 ( − 1) j S l − j s 1 ( i ) T j s 2 ( i + ( l − j ) n ) + ( − 1) l T l s 2 ( i ) ¶ Her e a i ≡ a 1 ( i ). On som e class of h omo gene ous p olynomials 13 = S l s 1 ( i + ( s 2 + 1)( p − n )) + l − 1 X j =1 ( − 1) j S l − j s 1 ( i + ( s 2 + 1)( p − n ) + j n ) T j s 2 ( i + ( s 1 + 1)( p − n )) + ( − 1) l T l s 2 ( i + ( s 1 + 1)( p − n )) . (39) In particular, let s 1 = s 2 = s . Then w e ha ve S l s ( i ) + l − 1 X j =1 ( − 1) j S l − j s ( i ) T j s ( i + ( l − j ) n ) + ( − 1) l T l s ( i ) = 0 and S l s ( i ) + l − 1 X j =1 ( − 1) j S l − j s ( i + j n ) T j s ( i ) + ( − 1) l T l s ( i ) = 0 . 5.3. Line ar e quations o n KP wave function a n d its c om p atibility Let us discuss linear equations on KP w av e function Ψ = { ψ i } . On M n,p, 1 w e ha ve the linear equation z ψ i + n + T 1 0 ( i + n ) ψ i = z ψ i + p (40) with T 1 0 ( i ) = r i whic h is a sp ecification of (19). On the other hand, from theorem 3 w e ha ve M n,p, 1 ⊂ M n, 2 p, 2 ⊂ M n, 3 p, 3 ⊂ · · · and corresp onding infinite set of linear equations z k ψ i + k n + k X j =1 T j k − 1 ( i + ( k − j + 1) n ) z k − j ψ i +( k − j ) n = z k ψ i + k p (41) for k ≥ 2 whic h can b e obtained also as a consequences of linear equation (40). Remark that w e o btain co efficien ts of (41) making use of lemma (2). Let us remem b er that T l s [ r ]’s in (4 1) are discrete p olynomials defined f o r some fixed n ≥ 1 a nd p > n via (21). Let us consider linear ev olution equation ψ ′ i = z ψ i + n − s i ψ i with s i ≡ a [ n ] 1 ( i ) for t he first flo w of n th discrete KP hierarc hy . By straigh tfor ward calculations w e c hec k that the compatibility of the latter equation with (41) is guarantee d when t he relation ∂ T l k ( i ) = T l +1 k ( i ) − T l +1 k ( i − n ) + T l k ( i )  s i − n − s i +( k + 1) p − ( k − l + 2) n  holds. In particular, ∂ T 0 1 ( i ) = ∂ r i = r i ( s i − n − s i + p − n ) or p − n X j =1 a ′ i + j − 1 = p − n X j =1 a i + j − 1 n X j =1 a i − j − p − 1 X j = p − n a i + j ! . On som e class of h omo gene ous p olynomials 14 5.4. I nte gr able hier ar chies of diffe r ential-diff er enc e e quations asso ciate d w ith M n,p, 1 In this subsection w e are going to show a class of integrable hierar c hies corresp onding to M n,p, 1 . Making use of lemma 2 w e obtain ∂ ∗ s ψ i ≡ ∂ ( p − n ) s ψ i = z ( p − n ) s ψ i +( p − n ) sn + ( p − n ) s X j =1 ( − 1) j z ( p − n ) s − j S j sn − 1 ( i − ( j − 1) n ) ψ i +( p − n ) sn − j n . (42) The condition of compatibilit y of ( 4 2) with (40) yields ∂ ∗ s T 1 0 ( i ) = ∂ ∗ s r i = ( − 1) ( p − n ) s r i n S ( p − n ) s sn − 1 ( i − ( p − n ) sn + p ) − S ( p − n ) s sn − 1 ( i − ( p − n ) sn ) o (43) while when chec king compatibility of (42) with (41) w e get ∂ ∗ s T l k ( i ) = T l +( p − n ) s k ( i ) + ( p − n ) s X j =1 ( − 1) j S j sn − 1 ( i + ( p − n )( k + 1) + ( l − j ) n ) T l +( p − n ) s − j k ( i ) − T l +( p − n ) s k ( i − ( p − n ) sn ) − ( p − n ) s X j =1 ( − 1) j S j sn − 1 ( i − ( p − n ) sn ) T l +( p − n ) s − j k ( i − ( p − n ) sn + j n ) . (44) Remark that all other relations obtained as a result o f chec king compatibilit y of (42) with (41) are just identities of the f o rm (39). Remark a lso that equations (44) could b e obtained in an easier wa y by using (16) with lemma 2. F ollowing along this line, in addition to (44), w e get the following ∂ ∗ s S l k ( i ) = ( − 1) ( p − n ) s    S l +( p − n ) s k ( i ) + ( p − n ) s X j =1 S j sn − 1 ( i + ( k − j ) n ) S l +( p − n ) s − j k ( i ) − S l +( p − n ) s k ( i − ( p − n ) sn ) − ( p − n ) s X j =1 S j sn − 1 ( i + ( p − n )( k − sn + 1)) S l +( p − n ) s − j k ( i − ( p − n ) sn + j n )    . (45) 5.5. Th e c ase M n,n +1 , 1 Let us consider the submanifold M n,n +1 , 1 . Since in this case p − n = 1 then r i = a i . Linear equation (40) in this case b ecomes z ψ i + n + a i + n ψ i = z ψ i + n +1 (46) while ev olution equation (42) tak es the form ∂ s ψ i = z s ψ i + sn + s X j =1 z s − j ( − 1) j S j sn − 1 ( i − ( j − 1) n ) ψ i +( s − j ) n On som e class of h omo gene ous p olynomials 15 and (43) is sp ecifie d as ∂ s a i = ( − 1) s a i  S s sn − 1 ( i − ( s − 1) n + 1) − S s sn − 1 ( i − sn )  . (47) The first flo w in this hierarc hy is g iv en b y differen tial-difference equation a ′ i = − a i  S 1 n − 1 ( i + 1) − S 1 n − 1 ( i − n )  = a i n X j =1 a i − j − n X j =1 a i + j ! . (48) This equation with quadratic nonlinearity is nothing but INB lattice men tioned in the introduction. It is kno wn by [1] that INB la ttice, being in a sense in tegrable generalization of the V olterra lattice a ′ i = a i ( a i − 1 − a i +1 ), for any n ≥ 1, giv es in t egr a ble discretization of the Kortew eg-de V ries (KdV) equation. Therefore as a particular case w e constructed in tegra ble hierarc hy fo r INB equation (48) in its explicit form [9]. Remark that p olynomials S l s and T l s in this case are sp ecified as S l s ( y 1 | y s +( l − 1) n +1 ) = X 1 ≤ λ l ≤···≤ λ 1 ≤ s +1 l Y j =1 y λ j + n ( j − 1) ! and T l s ( y 1 | y s +( l − 1) n +1 ) = X 1 ≤ λ 1 < ··· <λ l ≤ s +1 l Y j =1 y λ j + n ( j − 1) ! . These p olynomials w ere introduced in [9]. As consequences of linear equation (46) corresp onding to t he chain of inclusions M n,n +1 , 1 ⊂ M n, 2 n +2 , 2 ⊂ M n, 3 n +3 , 3 ⊂ · · · w e hav e an infinite n umber o f linear equations z k ψ i + k n + k X j =1 T j k − 1 ( i + ( k − j + 1) n ) z k − j ψ i +( k − j ) n = z k ψ i + k n + k for k ≥ 2. Finally , (44) and (45) in this case b ecomes ∂ s T l k ( i ) = T l + s k ( i ) + s X j =1 ( − 1) j S j sn − 1 ( i + ( l − j ) n + k + 1) T l + s − j k ( i ) − T l + s k ( i − sn ) − s X j =1 ( − 1) j S j sn − 1 ( i − sn ) T l + s − j k ( i − ( s − j ) n ) and ∂ s S l k ( i ) = ( − 1) s ( S l + s k ( i ) + s X j =1 S j sn − 1 ( i + ( k − j ) n ) S l + s − j k ( i ) − S l + s k ( i − sn ) − s X j =1 S j sn − 1 ( i + k − sn + 1) S l + s − j k ( i − ( s − j ) n ) ) , resp ectiv ely . In the pa per [9] we hav e pro ve d the follow ing. On som e class of h omo gene ous p olynomials 16 Theorem 4 Each one of the c onstr aints T l +1 s ( i + 1) = T l +1 s ( i ) , s ≥ l and S l +1 s ( i + 1) = S l +1 s ( i ) , s ≥ 0 is c o nsistent with the INB lattic e hier ar c hy (47). This theorem gives an infinite n um b er of constrain ts for INB lattice hierarc hy including p erio dicit y and stationary conditions. 5.6. Th e c ase M 1 ,g +1 , 1 As a second example, let us consider the submanifold M 1 ,g +1 , 1 . In this case p − n = g and consequen tly r i = a [ g ] 1 ( i ). Linear equation (40) and its consequences (41) are sp ecified as z ψ i +1 + r i +1 ψ i = z ψ i + g +1 and z k ψ i + k + k X j =1 T j k − 1 ( i + k − j + 1) z k − j ψ i + k − j = z k ψ i + k g + k , resp ectiv ely . Linear ev olution equation (42) b ecomes ∂ ∗ s ψ i ≡ ∂ g s ψ i = z g s ψ i + g s + g s X j =1 ( − 1) j z g s − j S j s − 1 ( i − j + 1 ) ψ i + g s − j . Corresp onding hierarc hy of differen tial- differenc e equations is given by ∂ ∗ s r i = ( − 1) g s r i  S g s s − 1 ( i − g s + g + 1) − S g s s − 1 ( i − g s )  . In particular, the first flo w of this hierarc hy is managed b y ∂ ∗ 1 r i = ( − 1) g r i { S g 0 ( i + 1) − S g 0 ( i − g ) } = ( − 1) g r i g Y j =1 r i + j − g Y j =1 r i − j ! . This equation is know n by [1]. It is, as w ell as the INB equation, represen ts, for any g ≥ 1, integrable discretization of t he K dV equation. It is a simple exercise to c hec k that in virtue of the latter (cf. [1]) ∂ ∗ 1 S g 0 ( i ) = ( − 1) g S g 0 ( i ) g X j =1 S g 0 ( i + j ) − g X j =1 S g 0 ( i − j ) ! , On som e class of h omo gene ous p olynomials 17 i.e., S g 0 satisfies INB equation. Finally , let us write do wn b elo w (44) and (45) in this case. W e hav e ∂ ∗ s T l k ( i ) = T l + g s k ( i ) + g s X j =1 ( − 1) j S j s − 1 ( i + g ( k + 1) + l − j ) T l + g s − j k ( i ) − T l + g s k ( i − g s ) − g s X j =1 ( − 1) j S j s − 1 ( i − g s ) T l + g s − j k ( i − g s + j ) and ∂ ∗ s S l k ( i ) = ( − 1) g s ( S l + g s k ( i ) + g s X j =1 S j s − 1 ( i + k − j ) S l + g s − j k ( i ) − S l + g s k ( i − g s ) − g s X j =1 S j s − 1 ( i + g ( k − s + 1 )) S l + g s − j k ( i − g s + j ) ) . 6. Conclusion Our main goal in the pap er w as to construct explicit f o rm for inte g rable hierarc hies for some class of differen tial-difference equations. P erhaps a most imp ortant case in this class o f equations is giv en b y V olterra lattice. What one learned fr om this presen tation is tha t ev olution equations of in tegrable hierarc hies from t his class are essen tially form ulated with the he lp of sp ecial homogeneous p olynomials whic h w e presen t b y explicit f o rm ulas (20) and ( 2 1). In forthcoming pap ers, w e will show ho w this information can b e exploited for in ves tig a tion of some problems concerning equations from this class. Remark that in the presen t pa per w e consider only one-comp onen t lattices. It migh t b e interes ting to extend these results to m ulti-comp onen t ones. Ac knowledgm ents This w o r k has b een supp orted b y Russian F oundatio n for Basic R esearch grant No. 09-01- 0 0192-a. References [1] Bog o yavlenskii O I 1988 Some co ns tructions of integrable dynamica l sy s tems Math. U SSR-Izv. 31 47-76 [2] Casa ti P , F alqui G., Ma gri F and Pedroni M 1 996 The KP theo ry revis ited I, I I, I II, IV SISSA pr eprints, r ef. SIS SA/2-5/9 6/ [3] Itoh Y 197 5 An H -theorem for a s ystem of comp eting sp ecies Pr o c. Jap an A c ad. 51 3 74-9 [4] Mag r i F, Pedroni M and Zub elli J P 1997 O n the geometry of Darb oux transformatio ns for the KP hierarch y and its co nnection with the discre te KP hier arc hy Commun . Math. 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