R-matrix approach to integrable systems on time scales

R -matrix approac h to in tegrable s ystems on time scales Maciej B laszak † , Burcu Silin dir ‡ and B la ˙ zej M . Szab liko wski † ( † ) Departmen t of Physics, Adam Mic kiewicz Universit y Um ulto wsk a 85, 61-61 4 P ozna ´ n, Pol and e-mail’s: bla szakm@am u.edu.pl and bszabl ik@amu.e du.pl ( ‡ ) Departmen t of Mathematics, F acult y of Sciences Bilk en t Universit y , 068 00 An k ara, T u rk ey e-mail: sil indir@fe n.bilken t.edu.tr Abstract A general u nifying framew ork for integ rable soliton-lik e systems on time scales is in tro duced. Th e R -matrix formalism is applied t o the algebra of δ -differen tial op er ators in terms of w hic h one can construct in finite hierarch y of comm uting v ecto r fi elds. The theory is illustrated b y tw o infi n ite-field integ rable hierarchies on time scales whic h are d ifferen ce coun terparts of KP and mK P . The difference counte rparts of AKNS and Kaup-Bro er soliton systems are constru cted as related finite-field restrictions. 1 In tro d uction In tegrable systems are widely in v estigated in (1 + 1) dimensions, where one of t he dimensions stands for the time ev olution v ariable and the o ther one stands for the space v ariable. The space v ariable is usually considered o n contin uous in terv als, or b oth on in teger v alues and on R [1] or on K q in terv als [2, 3]. In order to em b ed the study of integrable systems in to a more general unifying framew o rk, one of the p ossible approac hes is to c onstruct the in tegrable systems on time scales. Here the space v ariable is considered on any time scale where R , ℏZ , K q are special cases. The first step in this direction w as tak en in [4], where the Gelfand- Dic k ey approach [5, 6] w as extended in order to construct in tegrable nonlinear ev olutionary equations on an y time scale. Another unifying approach is to form ulate differen t types of discrete dynamics on R . Some con tribution in this direction was made recen tly in [7]. The main goal of this work is to presen t a theory for the systematic construction of (1 + 1)- dimensional in tegrable systems on time scales in the frame of the R -matrix formalism. By an in tegrable system, we mean suc h a system whic h has an infinite-hierarch y of m utually comm uting symmetries. The R -matr ix formalism is one of the most effectiv e and systematic metho ds of constructing in tegrable systems [8, 9]. This formalism originated from the pio- neering article [5] b y G elfand and Dic k ey , who constructed the soliton systems of K dV t ype. 1 2 The crucial p oin t of the R - matrix formalism is that the construction of in tegrable sys tems pro ceeds from the Lax equations on a ppropriate Lie algebras [8, 9]. The simplest R -matrices can b e constructed by a decomp osition of a given Lie algebra into t w o Lie subalgebras. W e refer to [9, 6, 1] for abstract for malism of classic al R -matrices on Lie algebras. This pap er is organized as follo ws: In the next section, w e giv e a brief review of the time scale calculus. In the third section, w e define the δ -differentiation o p erator and form ulate the Leibniz rule for this op erator. W e in tro duce the Lie algebra as an algebra of δ -differen tial op erators equipped with t he comm utator, decomp ose it in to t w o Lie subalgebras a nd con- struct the simplest R -matrix o n this algebra. W e pr esen t the appropriate Lax op erat ors for infinite-field cases and t he admissible finite-field restrictions generating consisten t Lax hier- arc hies. In T = R case, or in the contin uo us limit of some sp ecial time scales, w e observ e that t he algebra of δ -differen tial op erators turns out to b e the algebra of pseudo-differen tial op erators. Next, w e fo rm ulate and pro v e the pro p ert y of the algebra of δ -differen tial op- erators. This prop ert y allo ws us to obta in natural constrain ts whic h are fulfilled by finite field restrictions. Therefore, the source of the constrain ts, obtained in the Burg ers equations and KdV hierarc h y o n time scales in [4], is established. W e end up this section with the construction of the recursion op erators b y means of the metho d presen ted in [10]. In the fourth section, we illustrate tw o infinite-field inte grable hierarchies on time scales whic h a re difference coun t erparts of Kadom tsev -Pe tviash vili (K P) and mo dified Kadom tsev-P etviash vili (mKP) hierarc hies. In the last section, w e presen t finite-field restrictions whic h are difference coun terparts of Ablo witz-Kaup-New ell-Segur (AKNS) and Kaup-Bro er ( KB) hierarc hies with their recursion op erators. 2 Preliminaries In this section, we give a brief in tro duction to the concept of time scale. W e refer to [11 , 1 2] for the basic definitions a nd general theory o f time scale. What w e mean b y a time scale T , is an a rbitrary nonempty closed subset o f real n um b ers. The time scale calculus w as in tro duced b y Aulbac h and Hilg er [13, 14] in order to unify a ll p ossible interv als on the real line R , lik e con tin uous (whole) R , discrete Z , and q -discrete K q ( K q = q Z ∪ { 0 } ≡ { q k : k ∈ Z } ∪ { 0 } , where q 6 = 1 is a fixed real n um ber) inte rv als. F or the definition of the deriv ativ e in time scales, w e use forwar d and b ackwar d jump op er ators whic h are defined as follo ws. Definition 2.1 F or x ∈ T , the forwar d jump op er ator σ : T → T is define d by σ ( x ) = inf { y ∈ T : y > x } , (2.1) while the b ackwar d jump op er ator ρ : T → T is define d by ρ ( x ) = sup { y ∈ T : y < x } . (2.2) We set in addition σ (max T ) = max T if ther e ex ists a finite max T , and ρ (min T ) = min T if ther e exists a fin i te min T . The jump o p er ators σ and ρ al low the classific ation of p oints in a time sc ale in the fol lowing way: x is c al le d righ t dens e , right sc a tter e d, l e f t dense, left sc atter e d, dense and isolate d if 3 σ ( x ) = x, σ ( x ) > x, ρ ( x ) = x, ρ ( x ) < x, σ ( x ) = ρ ( x ) = x and ρ ( x ) < x < σ ( x ) , r esp e ctivel y. Mor e over, w e define the gr aininess functions µ, ν : T → [0 , ∞ ) as fol lows µ ( x ) = σ ( x ) − x, ν ( x ) = x − ρ ( x ) , for al l x ∈ T . (2.3) In literature, T κ denotes a set consisting of T except for a p ossible left-scattered maximal p oin t while T κ stands fo r a set of p oin ts of T except for a p ossible righ t-scattered minimal p oin t. Definition 2.2 L et f : T → R b e a function on a time sc ale T . F or x ∈ T κ , delta derivative of f , de note d by ∆ f , is define d as ∆ f ( x ) = lim s → x f ( σ ( x )) − f ( s ) σ ( x ) − s , s ∈ T , (2.4) while for x ∈ T κ , ∇ -de rivative of f , denote d by ∇ f , is define d as ∇ f ( x ) = lim s → x f ( s ) − f ( ρ ( x )) s − ρ ( x ) , s ∈ T , (2.5) pr ov i d e d that the lim its exis t. A function f : T → R i s said to b e ∆ -smo oth ( ∇ -smo oth) if it is in fi nitely ∆ -differ entiable ( ∇ -differ entiable). Remark 2.3 L et f : T → R b e ∆ -differ e ntiable on T κ . If x is right-sc atter e d, then the definition (2.4) turns out to b e ∆ f ( x ) = f ( σ ( x )) − f ( x ) µ ( x ) , while if x is rig h t-dense, (2.4) implies that ∆ f ( x ) = lim s → x f ( x ) − f ( s ) x − s , s ∈ T . Similarly, let f : T → R b e ∇ -differ entiable on T κ . If x is left-sc atter e d, then the definition (2.5) turns out to b e ∇ f ( x ) = f ( x ) − f ( ρ ( x )) ν ( x ) , while if x is le ft-dense, (2.5) yields as ∇ f ( x ) = lim s → x f ( x ) − f ( s ) x − s , s ∈ T . In order t o b e more precise, w e presen t ∆ and ∇ deriv ativ es for some sp ecial time scales. If T = R , then ∆- and ∇ -deriv ativ es b ecome ordinary deriv ativ es, i.e. ∆ f ( x ) = ∇ f ( x ) = d f ( x ) dx . 4 If T = ℏZ , then ∆ f ( x ) = f ( x + ℏ ) − f ( x ) ℏ and ∇ f ( x ) = f ( x ) − f ( x − ℏ ) ℏ . If T = K q , then ∆ f ( x ) = f ( q x ) − f ( x ) ( q − 1) x and ∇ f ( x ) = f ( x ) − f ( q − 1 x ) (1 − q − 1 ) x , for all x 6 = 0, and ∆ f (0) = ∇ f (0) = lim s → 0 f ( s ) − f (0) s , s ∈ K q , pro vided that this limit exis ts. As an imp ortan t prop ert y of ∆-differen tiation on T , w e giv e the pro duct rule. If f , g : T → R are ∆-differen tiable functions at x ∈ T κ , then their pro duct is also ∆-differen tiable and t he follo wing Lebniz-lik e rule hold ∆( f g )( x ) = g ( x )∆ f ( x ) + f ( σ ( x ))∆ g ( x ) = f ( x )∆ g ( x ) + g ( σ ( x ))∆ f ( x ) . (2.6) Besides, if f is ∆-smo oth function, then f ( σ ( x )) = f ( x ) + µ ( x )∆ f ( x ) . (2.7) If x ∈ T is righ t-dense, then µ ( x ) = 0 and the r elation (2.7) is trivial. Definition 2.4 A time sc ale T is r e gular if b oth of the fol lowing two c onditions ar e satisfie d: (i) σ ( ρ ( x )) = x for al l x ∈ T , (ii) ρ ( σ ( x )) = x for al l x ∈ T . Set x ∗ = min T if there exists a finite min T , and set x ∗ = −∞ otherwise. Also s et x ∗ = max T if there exis ts a finite max T , and set x ∗ = ∞ otherwise. Prop osition 2.5 [4] A time sc ale is r e gular if and only if the fol lowin g two c onditions hold: (i) the p oint x ∗ = min T is right dense and the p oint x ∗ = max T is left-de n se; (ii) e ach p oint of T \ { x ∗ , x ∗ } is either two-side d dense or two-side d sc atter e d. In particular R , ℏZ ( ℏ 6 = 0) and K q are regular time scales , as a re [0 , 1] a nd [ − 1 , 0] ∪ { 1 /k : k ∈ N } ∪ { k / ( k + 1) : k ∈ N } ∪ [1 , 2]. 5 Throughout this w o rk, let T be a regular time scale. By ∆, we denote the delta-differen tiation op erator whic h assigns eac h ∆-differentiable function f : T → R to its delta- deriv ative ∆( f ) , defined b y [∆( f )]( x ) = ∆ f ( x ) , for x ∈ T κ . (2.8) The shift op er ator E is defined b y the form ula ( E f )( x ) = f ( σ ( x )) , x ∈ T . (2.9) The in v erse E − 1 is defined by ( E − 1 f )( x ) = f ( σ − 1 ( x )) = f ( ρ ( x )) , (2.10) for all x ∈ T . Note that E − 1 exists only in the case of regular time scales and tha t in general E and E − 1 do not commute with ∆ and ∇ op erators. Prop osition 2.6 [15] L et T b e a r e gular time sc ale. (i) If f : T → R is a ∆ -smo oth function on T κ , then f is ∇ - s m o o th and for al l x ∈ T κ , ∇ f ( x ) = E − 1 ∆ f ( x ) . (2.11) (ii) If f : T → R is a ∇ -sm o oth function on T κ , then f is ∆ -smo oth and for al l x ∈ T κ , ∆ f ( x ) = E ∇ f ( x ) . (2.12) Th us the prop erties of ∆- and ∇ - smo othness for functions on regular t ime scales are equiv a- len t. In some sp ecial cases, b y prop erly in tro ducing t he deformation parameter, it is p ossible to consider a con tin uous limit of a time scale. F or instance, the con tin uous limit of ℏZ is the whole real line R , i.e. T = ℏZ ℏ → 0 − − − → T = R ; (2.13) and the con tin uous limit of K q is the closed half line R + ∪ 0, thus T = K q q → 1 − − − → T = R + ∪ 0 . (2.14) F or more ab out the calculus o n time scales w e refer the readers t o [11, 12]. 3 Algebra of δ -diffe r e n tial op erator s 3.1 Basic notions In this section, w e deal with the algebra o f δ -differential op erators defined on a regular time scale T . W e denote the delta differen tiation o p erator by δ instead of ∆, for con v enience in the 6 op erational relations. The op erator δ f whic h is a comp o sition of δ a nd f , where f : T → R , is in tro duced as follo ws δ f := ∆ f + E ( f ) δ, ∀ f . (3.1) Note that, the definition (3.1) is consisten t with the Lebniz-lik e rule on time scales (2.6). Theorem 3.1 The L eibniz rule on time sc ales for the op er ator δ is given as fol lows. (i) F or n > 0 : δ n f = n X k =0 X i 1 + i 2 + ... + i k +1 = n − k (∆ i k +1 E ∆ i k E ... ∆ i 2 E ∆ i 1 ) f δ k , (3.2) wher e i γ > 0 for al l γ = 1 , 2 , .., k + 1 . Her e the formula incl ude s al l p ossible strings c on tain ing n − k times ∆ and k times E . (ii) F or n < 0 : δ n f = ∞ X k = − n X i 1 + i 2 + ... + i k + n +1 = k ( − 1) k + n ( E − i k + n +1 ∆ E − i k + n ∆ ...E − i 2 ∆ E − i 1 ) f δ − k , (3.3) wher e i γ > 0 for al l γ = 1 , 2 , .., k + n + 1 > 0 . Her e the formula includes al l p ossible strings c ontaining k + n + 1 times E and k + n times ∆ . The ab o v e theorem is a straightforw ard consequence of definition (3.1). Note that δ − 1 f has the form of the formal series δ − 1 f = ∞ X k =0 ( − 1) k (( E − 1 ∆) k E − 1 ) f δ − k − 1 , (3.4) whic h w as previously giv en in [4], in terms o f ∇ . Th us (3.3) is the appropriate generalization of (3.4). 3.2 Classical R -matrix formalism In order to construct integrable hierarc hies of m utually comm uting vec tor fields on time scales, w e deal with a systematic method, so-called the clas s ic al R -matrix formalism [9, 6, 1], presen ted in the follo wing sc heme. Let G b e an alg ebra, with some asso ciative m ultiplication op eration, ov er a commutativ e field K of complex or r eal num bers, based on an additional bilinear pro duct g iv en b y a Lie brac k et [ · , · ] : G → G , whic h is sk ew-sym metric and satisfies the Jacobi identit y . Definition 3.2 A li n e ar map R : G → G such that the br acket [ a, b ] R := [ Ra, b ] + [ a, R b ] , (3.5) is a se c ond Lie br acket on G , is c al le d the classi c al R -m atrix. 7 Sk ew-symme try of (3.5) is o b vious. When one c hec ks the Jacobi iden tity of (3.5), it can be clearly deduced that a sufficien t condition for R to be a classical R -matr ix is [ Ra, R b ] − R [ a, b ] R + α [ a, b ] = 0 , (3.6) where α ∈ K , called the Y ang-Ba x ter e quation YB( α ) . There are o nly tw o relev an t cases o f YB( α ), namely α 6 = 0 and α = 0 , as Y ang-Baxter equations for α 6 = 0 are equiv alen t and can b e reparametrized. Additionally , assume that the Lie brack et is a deriv ation of multiplic ation in G , i.e. the relation [ a, bc ] = b [ a, c ] + [ a, b ] c a, b, c ∈ G (3.7) holds. If the Lie brac k et is g iv en b y the comm utator, i.e. [ a, b ] = ab − bc , the condition (3.7) is satisfied a utomatically , since G is associative. Prop osition 3.3 L et G b e a Lie algebr a fulfil ling al l the ab ove assumptions and R b e the classic al R -matrix satisfying the Y ang- Baxter e quation, YB ( α ) . The n the p ower functions L n on G , L ∈ G and n ∈ Z + , gen er ate the so- c al le d L ax hie r ar ch y dL dt n = [ R ( L n ) , L ] , (3.8) of p airwise c ommuting ve ctor fields on G . Her e, t n ’s ar e r elate d evolution p ar ameters. We additional ly a ssume that R c ommutes with derivatives with r esp e c t to these evolution p ar am- eters. Pro of. It is clear that the p ow er functions on G are w ell defined. Then ( L t m ) t n − ( L t n ) t m = [ RL m , L ] t n − [ RL n , L ] t m = [( R L m ) t n − ( RL n ) t m , L ] + [ RL m , [ RL n , L ]] − [ RL n , [ RL m , L ]] = [( R L m ) t n − ( RL n ) t m + [ RL m , RL n ] , L ] . Hence, the v ector fields (3.8) m utually commute if the so-called zer o-curvatur e (or Z akharo v- Shabat) e q uations ( RL m ) t n − ( RL n ) t m + [ RL m , RL n ] = 0 , are satisfied. F rom (3.8) and b y the Leibniz rule (3.7) w e ha v e that ( L m ) t n = [ R L n , L m ]. Using Y ang-Baxter equation for R and the fact that R comm utes with ∂ t n , w e deduce R ( L m ) t n − R ( L n ) t m + [ RL m , RL n ] = = R [ RL n , L m ] − R [ RL m , L n ] + [ R L m , RL n ] = [ RL m , RL n ] − R [ L m , L n ] R = − α [ L m , L n ] = 0 . Hence, the vec tor fields pairwise comm ute.  In practice the p o w ers of L ax op erators in (3.8) are fractional. Notice that, t he Y ang-Baxter equation is a sufficien t condition for mutual comm utation of v ector fields (3 .8), but not nec- essary . Th us c ho osing an algebra G prop erly , the Lax hierarc h y yields abstract in tegrable systems . In practice, the elemen t L of G m ust be appropriately chose n, in suc h a w a y that the ev olution systems (3.8) a re consisten t on the subs pace of G . 8 3.3 Classical R -matrix on time-scales W e introduce the algebra G as an algebra of formal Lauren t series of (pseudo-) δ - differen tial op erators equipp ed with the comm utator, and define its decomp osition suc h as: G = G > k ⊕ G k u i ( x ) δ i } ⊕ { X i k , G k − P k and P k and G k , L i = − h  L n N  k ⊕ G k u i ( x ) ∂ i } ⊕ { X i 0 . (3.22) Then the fol lo w ing r elation r X i =0 ( − µ ) i C i = F (3.23) is va lid. Pro of. W e make use of induction. Assume that (3.23) holds fo r r . Then δ r +1 F = δ r ( E F ) δ + δ r ∆ F = r X i =0 A i δ r − i +1 + r X i =0 B i δ r − i = r +1 X i =0 C i δ r +1 − i . (3.24) By the assumption we hav e P r i =0 ( − µ ) i A i = E F and P r i =0 ( − µ ) i B i = ∆ F . Hence r +1 X i =0 ( − µ ) i C i = r X i =0 ( − µ ) ( i +1) B i + r X i =0 ( − µ ) i A i = − µ ∆ F + E F = F . (3.25)  Let us explain the source of Lemma 3.4. Consider the equalit y A = X i > 0 a i δ i = 0 , (3.26) where the sum is finite, and A is purely δ -differential op erator. W e expand A with resp ect to the shift o p erator E : E u = E ( u ) E . F rom the relatio n (2 .7) w e ha v e E = 1 + µδ. (3.27) The equality from Lemma 3.4 is trivially satisfied for dense x ∈ T , since in this case µ = 0. Th us, it is enough to consider r emaining p oin ts in a time scale so assume that µ 6 = 0. Hence, from (3.27), we ha v e the fo rm ula δ = µ − 1 E − µ − 1 . (3.28) Th us, using (3.28) the relation (3.26) can b e rewritten as A = X i a ′ i E i = 0 . (3.29) Ob viously , it m ust hold fo r terms of all orders. The equalit y for the zero-order terms, i.e. a ′ 0 = 0, can b e simply obta ined b y replacing δ with − µ − 1 in (3.26). The same substitution in (3.22) allo ws us to find ( − µ ) − r F = r X i =0 C i ( − µ ) − r + i , (3.30) 11 whic h is equiv alent to (3.23). The ab o v e pro cedure can b e extended also to op erators A t hat are not purely δ - differen tial and con tain finitely man y terms with δ − 1 , δ − 2 , . . . . As an illustrat ion consider the equality [ Aδ r , ψ δ − 1 ϕ ] = r − 1 X i =0 C i δ r − 1 − i + ˆ C r δ − 1 ϕ + ψ δ − 1 C r . (3.31) The ab ov e equalit y is w ell-form ulated since it follows immediately from the definition a nd the prop erty of the δ op erator. Replacing δ with − µ − 1 , the comm ut ator v anishes, and w e ha v e 0 = r − 1 X i =0 C i ( − µ ) − r +1+ i + ˆ C r ( − µ ) ϕ + ψ ( − µ ) C r ⇐ ⇒ (3.32) r − 1 X i =0 ( − µ ) i C i + ( − µ ) r ( ˆ C r ϕ + ψC r ) = 0 . (3.33) Straigh tforward consequence of s uc h a behavior of δ -differen tial op erators is the next theorem. Theorem 3.5 (i) The c a se k = 0 . The c onstr aint b etwe en dynam ic al fields of (3.15) , g e ner a ting L ax hier a r chy ( 3.12) , has the form ( − µ ) N − 1 d ˜ u N − 1 dt n + N − 2 X i =0 ( − µ ) i du i dt n − µ X s d ( ψ s ϕ s ) dt n = 0 = ⇒ ( − µ ) N − 1 ˜ u N − 1 + N − 2 X i =0 ( − µ ) i u i − µ X s ψ s ϕ s = a n , (3.34) wher e n ∈ Z + and a n is a time-indep endent function. (ii) The c ase k = 1 . The c onstr aint b etwe en dynamic al fields of (3.17) , gener ating (3.12) , has the form ( − µ ) N d ˜ u N dt n + N − 1 X i = − 1 ( − µ ) i du i dt n − µ X s d ( ψ s ϕ s ) dt n = 0 = ⇒ ( − µ ) N ˜ u N + N − 1 X i = − 1 ( − µ ) i u i − µ X s ψ s ϕ s = a n , (3.35) wher e n ∈ Z + and a n is a time-indep endent function. Pro of. W e already kno w tha t Lax operato rs (3.15) a nd (3.17) generate consisten t Lax hi- erarc hies (3.12). Thus , the righ t-hand side of (3.12) can be represe n ted in the form of L t n . Replacing δ with − µ − 1 in (3.12), w e hav e L t n | δ = − µ − 1 = [( L n ) > k , L ] | δ = − µ − 1 = 0 . (3.36) 12 Hence, the constraints (3.34) and (3 .35) fo llo w.  The a b ov e theorem can b e generalized to further restrictions. As a conseq uence, the con- strain ts (3.3 4) or ( 3.35) with fixed common v alue of all a n , are v alid for the whole L ax hierarc h y ( 3.12). 3.4 Recursion op erators One of the c haracteristic features of in tegrable systems p ossessing infinite-hierarc h y o f m u- tually comm uting symmetries is the existe nce of a recursion op erato r [1 8, 1]. A recursion op erator of a g iv en system, is an op erator of suc h prop erty that when it acts on one sym- metry o f the system considered, it pro duces ano ther symmetry . G ¨ urses e t al. [10] presen ted a general and ve ry efficien t metho d of constructing recursion o p erators for Lax hierarchies . Among others, the authors illustrat ed the metho d b y applying it to finite-field reductions of the KP hierarch y . In [19] the metho d w as a pplied to the reductions of mo dified KP hierarc h y as w ell as to the lattice systems. Our further conside rations are based on the sc heme from [10] and [19]. The recursion o p erator Φ has the following prop ert y: Φ( L t n ) = L t n + N , n ∈ Z + , and hence it allo ws reconstruction of the whole hierarc h y (3.12) when applied to the first ( N − 1) symmetries. Lemma 3.6 (i) The c ase k = 0 . L et the L ax op er ator b e given in the gener al form (3.15) . Then, the r e cursion op er ator of the r elate d L ax hier ar ch y c a n b e c onstructe d solving L t n + N = L t n L + [ R , L ] (3.37) with the r emainder in the form R = a N − 1 δ N − 1 + · · · + a 0 + X s a − 1 ,s δ − 1 ϕ s , (3.38) wher e N is the highest or der of L . (ii) The c a s e k = 1 . Similarly for the L ax op er ator (3.1 7) , the r e cursion op er ator c a n b e c on s tructe d fr om (3 .37) with R = a N δ N + · · · + a 0 + X s a − 1 ,s δ − 1 ϕ s . (3.39) 13 Pro of. Consider the case k = 0. Then for (3.15) w e ha v e ( L n + N N ) > 0 = (( L n N ) > 0 L ) > 0 + (( L n N ) < 0 L ) > 0 = ( L n N ) > 0 L − X s [( L n N ) > 0 ψ s ] 0 δ − 1 ϕ s + (( L n N ) < 0 L ) > 0 = ( L n N ) > 0 L + R, where [ P i aδ i ] 0 = a 0 and R is giv en by (3.38). Similarly for k = 1 , we hav e ( L n + N N ) > 1 = (( L n N ) > 1 L ) > 1 + (( L n N ) < 1 L ) > 1 = ( L n N ) > 1 L − [( L n N ) > 1 L ] 0 − X s [( L n N ) > 0 ψ s ] 0 δ − 1 ϕ s + (( L n N ) < 1 L ) > 1 = ( L n N ) > 1 L + R, where R has the for m (3.39). Thus, in b oth cases (3.37) fo llo ws from (3 .12). Hence w e can extract the recursion op erator from (3.37).  Note that in general, recursion op erators on time scales are non-lo cal. This means that they con tain non-lo cal terms with ∆ − 1 b eing formal inv erse of ∆ op erator. Ho w ev er, suc h recurs ion op erators acting on an appropria te domain pro duce only lo cal hierarc hies. 4 Infinite- fi eld in tegrable systems on time scales 4.1 Difference KP , k = 0 : Consider the follo wing infinite field Lax op era tor L = δ + ˜ u 0 + X i > 1 u i δ − i , (4.1) whic h generates the Lax hierarc h y (3.1 2) as the difference counterpart of the Kadomtse v- P etviash vili (KP) hierarc h y . F or ( L ) > 0 = δ + ˜ u 0 , the first flo w is giv en b y d ˜ u 0 dt 1 = µ ∆ u 1 du i dt 1 = i − 1 X k =0 ( − 1) k +1 u i − k X j 1 + j 2 + ... + j k +1 = i ( E − j k +1 ∆ E − j k ∆ . . . E − j 2 ∆ E − j 1 ) ˜ u 0 + µ ∆ u i +1 + ∆ u i + u i ˜ u 0 ∀ i > 0 , (4.2) where j γ > 0 for all γ > 1. F or ( L 2 ) > 0 = δ 2 + ξ δ + η , where ξ := E ˜ u 0 + ˜ u 0 η := ∆ ˜ u 0 + ˜ u 2 0 + u 1 + E u 1 , (4.3) 14 one calculates the second flow du 0 dt 2 = µ ∆( E + 1) u 2 + µ ∆(∆ u 1 + u 1 ˜ u 0 + u 1 E − 1 ˜ u 0 ) du i dt 2 = i − 1 X k = − 1 ( − 1) k +2 u i − k X j 1 + j 2 + ... + j k +2 = i +1 ( E − j k +2 ∆ E − j k +1 ∆ . . . E − j 2 ∆ E − j 1 ) ξ + i − 1 X k =0 ( − 1) k +1 u i − k X j 1 + j 2 + ... + j k +1 = i ( E − j k +1 ∆ E − j k ∆ . . . E − j 2 ∆ E − j 1 ) η + ∆ 2 u i + ( E ∆ + ∆ E ) u i +1 + µ ∆( E + 1) u i +2 + ξ (∆ u i + E u i +1 ) + η u i , (4.4) where j γ > 0 for all γ > 1. The simplest case in (2 + 1) dimensions: W e rewrite the first t w o members of the first flow b y setting ˜ u 0 = w and t 1 = y and the first mem b er of the second flow b y setting t 2 = t . Since E and ∆ do not comm ute, note that in the calculations up to the last step, w e use E − 1 instead of µ ∆, to av oid confusion. w y = ( E − 1) u 1 , (4.5) u 1 ,y = ( E − 1) u 2 + ∆ u 1 + u 1 (1 − E − 1 )( w ) , (4.6 ) w t = ( E 2 − 1) u 2 + ( E − 1)(∆ u 1 + u 1 w + u 1 E − 1 ( w )) (4.7) Applying E + 1 t o (4.6) from the left yields: ( E 2 − 1) u 2 = ( E + 1) u 1 ,y − ( E + 1)∆ u 1 − ( E − 1) u 1 (1 − E − 1 ) w . (4.8) Applying ( E − 1) to (4.7) from the left and subs tituting (4.5) and (4.8) into the new deriv ed equation w e finally o btain the (2 + 1)- dimensional one-field system of the form µ ∆ w t = ( E + 1) w y y − 2∆ w y + 2 µ ∆( w w y ) . (4.9) whic h do es not ha v e a con tin uous counterpart. F or the case of T = h Z , one can sho w that (4.9) is equiv alen t to the ( 2 + 1)-dimensional T o da lattice sys tem. The difference analo gue of one-field con tin uous K P equation is to o complicated to write it do wn explicitly . Remark 4.1 Her e we want to il lustr ate the b eha v ior of ˜ u 0 in al l symmetries of the differ enc e KP hier ar chy. L et ( L n ) < 0 = X i > 1 v ( n ) i δ − i , then by the right-han d of the L ax e quation (3.12) , we obtain the first m emb ers of al l flows d ˜ u 0 dt n = µ ∆ v ( n ) 1 . (4.10) Thus ˜ u 0 is time- i ndep endent for dense x ∈ T sinc e µ = 0 . Henc e when T = R , ˜ u 0 app e ars to b e a c on stant. 15 In T = R case, or in the contin uous limit of some special time scales, with ˜ u 0 = 0, the Lax op erator (4.1) turns out t o be a Lauren t series of pseu do-differential op erators L = ∂ + X i > 1 u i ∂ − i . (4.11) Moreo v er, the first flow (4.2) turns out to b e exactly the first flo w o f the KP system du i dt 1 = u i,x , ∀ i > 1 (4.12) while the sec ond flow (4.4) b ecomes exactly the second flow of the KP system du i dt 2 = ( u i ) 2 x + 2( u i +1 ) x + 2 i − 1 X k =1 ( − 1) k +1  i − 1 k  u i − k ( u 1 ) k x ∀ i > 1 . (4.13) 4.2 Difference mKP , k = 1 : Consider the Lax op erator of the for m L = ˜ u − 1 δ + X i > 0 u i δ − i (4.14) whic h generates the difference coun terpart of the modified Kadomstsev-P etviash vili ( mKP) hierarc h y . Then, ( L ) > 1 = ˜ u − 1 δ implies the first flow d ˜ u − 1 dt 1 = µ ˜ u − 1 ∆ u 0 du i dt 1 = i − 1 X k = − 1 ( − 1) k +2 u i − k X j 1 + j 2 + ··· + j k +2 = i +1 ( E − j k +2 ∆ E − j k +1 ∆ . . . E − j 2 ∆ E − j 1 ) ˜ u − 1 + ˜ u − 1 E u i +1 + ˜ u − 1 ∆ u i ∀ i > 0 , (4.15) where j γ > 0, γ = 1 , 2 , . . . , k + 2. Next, for ( L 2 ) > 1 = ξ δ 2 + η δ , where ξ := ˜ u − 1 E ˜ u − 1 , η := ˜ u − 1 ∆ ˜ u − 1 + ˜ u − 1 E u 0 + u 0 ˜ u − 1 , (4.16) w e ha v e the second flo w as follows d ˜ u − 1 dt 2 = ξ ( E ∆ u 0 + E 2 ( u 1 )) + µ ˜ u − 1 ∆ u 2 0 − u 1 E − 1 ξ − ˜ u 2 − 1 ∆ u 0 du i dt 2 = i − 1 X k = − 2 ( − 1) k +3 u i − k X j 1 + j 2 + ... + j k +3 = i +2 ( E − j k +3 ∆ E − j k +2 ∆ . . . ∆ E − j 1 ) ξ + i − 1 X k = − 1 ( − 1) k +2 u i − k X j 1 + j 2 + ... + j k +2 = i +1 ( E − j k +2 ∆ E − j k +1 ∆ . . . ∆ E − j 1 ) η + ξ 2 (∆ 2 u i + ( E ∆ + ∆ E ) u i +1 + E 2 u i +2 ) + η (∆ u i + E u i +1 ) , (4.17) where i > 0 and j γ > 0 for all γ > 1. 16 Remark 4.2 Similarly in or der to il lustr ate the b ehavior of ˜ u − 1 in al l symmetries o f the dif- fer en c e mKP hier ar chy let us c onsider ( L n ) < 1 = X i > 0 v ( n ) i δ − i . Then we obtain the first memb e rs of al l flows d ˜ u − 1 dt n = µ ˜ u − 1 ∆ v ( n ) 0 , (4.1 8) Thus ˜ u − 1 is time-indep endent for den se x ∈ T . Henc e when T = R , ˜ u − 1 app e ars to b e a c on s tant. In T = R case, or in the con tinu ous limit of some sp ecial time scales, with ˜ u − 1 = 1, the Lax op erator (4.14) turns out to b e the pseudo-differen tial op erato r L = ∂ + X i > 0 u i ∂ − i , (4.19) F urthermore, the system of equations (4.15) is exactly the first flo w o f the mKP system du i dt 1 = u i,x , ∀ i > 0 , (4.20) while the sec ond flow (4.17) t urns out to b e the second flo w of the mKP system du i dt 2 = ( u i ) 2 x + 2( u i +1 ) x + 2 u 0 ( u i ) x + 2 u 0 u i +1 + 2 i X k =0 ( − 1) k +1  i k  u i +1 − k ( u 0 ) k x ∀ i > 0 . (4.21) 5 Finite-fiel d in tegrab l e systems on time scales 5.1 Difference AKNS, k = 0 : Let the Lax op erator (3.15) fo r N = 1 and c 1 = 1 is of the form L = δ + ˜ u + ψ δ − 1 ϕ. (5.1) The constrain t (3.34) betw een fields, with a n = 0, b ecomes ˜ u = µψ ϕ. (5.2) F or ( L ) > 0 = δ + ˜ u , one finds the first flo w d ˜ u dt 1 = µ ∆( ψ E − 1 ϕ ) , dψ dt 1 = ˜ uψ + ∆ ψ , dϕ dt 1 = − ˜ uϕ + ∆ E − 1 ϕ. (5.3) 17 Eliminating field ˜ u b y (5.2) w e ha v e dψ dt 1 = µψ 2 ϕ + ∆ ψ, dϕ dt 1 = − µϕ 2 ψ + ∆ E − 1 ϕ. (5.4) Next w e calculate ( L 2 ) > 0 = δ 2 + ξ δ + η whe re ξ := ( E + 1) ˜ u, η := ∆ ˜ u + ˜ u 2 + ϕE ( ψ ) + ψE − 1 ( ϕ ) . (5.5) Th us, the sec ond flow take s the form d ˜ u dt 2 = µ ∆  ∆( ψ E − 1 ( ϕ )) + ψ E − 1 ( ˜ uϕ ) + ˜ u ψ E − 1 ϕ  − µ ∆( E + 1) ψ E − 1 ∆ E − 1 ( ϕ ) dψ dt 2 = ψ η + ξ ∆ ψ + ∆ 2 ψ dϕ dt 2 = − ϕη + ∆ E − 1 ( ξ ϕ ) − (∆ E − 1 ) 2 ϕ. (5.6) By the use of the constrain t (5.2), the second flow can b e written as dψ dt 2 = ψ (∆ µψ ϕ + ( µψ ϕ ) 2 + ϕE ( ψ ) + ψE − 1 ( ϕ )) + ( E + 1) µψ ϕ ∆ ψ + ∆ 2 ψ , dϕ dt 2 = − ϕ (∆ µψ ϕ + ( µψ ϕ ) 2 + ϕE ( ψ ) + ψE − 1 ( ϕ )) + ∆ E − 1 ( ϕ ( E + 1) µψ ϕ ) − ( ∆ E − 1 ) 2 ϕ. (5.7) In order to obtain the r ecursion op erator one finds that fo r the Lax op erator (5.1) the appro- priate reminder (3.38) has the form R = ∆ − 1  µ − 1 ˜ u t n  − ψ t n δ − 1 ϕ. (5.8) Then, (3.37) implies the following recursion fo rm ula a s   ˜ u ψ ϕ   t n +1 =   ˜ u − µ − 1 φE ψ E − 1 ψ + ψ ∆ − 1 µ − 1 ∆ + ˜ u + ψ ∆ − 1 ϕ ψ ∆ − 1 ψ − ϕ ∆ − 1 µ − 1 − ϕE ∆ − 1 ϕ ˜ u − ∆ E − 1 − ϕE ∆ − 1 ψ     ˜ u ψ ϕ   t n (5.9) v alid for isolated p oin ts x ∈ T , i.e. when µ 6 = 0. F or dense p oints one m ust use its reduction b y constrain t (5.2):  ψ ϕ  t n +1 =  ∆ + ˜ u + µψ ϕ + 2 ψ ∆ − 1 ϕ µψ 2 + 2 ψ ∆ − 1 ψ − ϕ ( E + 1)∆ − 1 ϕ ˜ u − ∆ E − 1 − ϕ ( E + 1)∆ − 1 ψ   ψ ϕ  t n , (5.10) where ˜ u is giv en b y (5.2). 18 In T = R case, or in the con tin uous limit of some sp ecial time scales, with the c hoice ˜ u = 0, the Lax op erator (5.1) tak es the form L = ∂ + ψ ∂ − 1 ϕ . Then, the contin uo us limits of (5.3) and (5.6) respectiv ely , imply that the first flo w is the translational symme try dψ dt 1 = ψ x dϕ dt 1 = ϕ x (5.11) and the first non- trivial equation from the hierarch y is the AKNS equation dψ dt 2 = ψ xx + 2 ψ 2 ϕ, dϕ dt 2 = − ϕ xx − 2 ϕ 2 ψ . (5.12) F or that special case t he recursion form ula (5.10) is of the follo wing form:  ψ ϕ  t n +1 =  ∂ x + 2 ψ ∂ − 1 x ϕ 2 ψ ∂ − 1 x ψ − 2 ϕ∂ − 1 x ϕ − ∂ x − 2 ϕ∂ − 1 x ψ   ψ ϕ  t n . (5.13) 5.2 Difference Kaup-Bro er, k = 1 : ¿F rom the a dmissible finite field restrictions ( 3.17), we consider the follow ing simplest Lax op erator L = ˜ uδ + v + δ − 1 w . (5.14) The constrain t (3.35), with a n = 1, implies ˜ u = 1 + µv − µ 2 w . (5.15 ) Then, for ( L ) > 1 = ˜ uδ , the first flo w is give n as d ˜ u dt 1 = µ ˜ u ∆ v , dv dt 1 = ˜ u ∆ v + µ ∆ E − 1 ( ˜ uw ) , dw dt 1 = ∆ E − 1 ( ˜ uw ) . (5.16) By the constraint (5.15) one can rewrite the first flo w as dv dt 1 = ( µv − µ 2 w )∆ v + µ ∆ E − 1 ( w ( µv − µ 2 w )) , dw dt 1 = ∆ E − 1  µv w − µ 2 w 2  . (5.17) Next, w e calculate ( L 2 ) > 1 = ξ δ 2 + η δ , where ξ := ˜ uE ˜ u, η := ˜ u ∆ ˜ u + ˜ uE v + v ˜ u, (5.18) 19 that yields the second flow d ˜ u dt 2 = µ ˜ u ∆( E − 1 + 1) ˜ uw + µ ˜ u ∆ v 2 + µ ˜ u ∆( ˜ u ∆ v ) , dv dt 2 = ξ (∆ 2 v + ∆ w ) + µ ∆ E − 1 ( w η ) + E − 1 ∆ E − 1 ( w ξ ) + η ∆ v , dw dt 2 = − ∆ E − 1 ∆ E − 1 ( w ξ ) + ∆ E − 1 ( w η ) . (5.19) One can rewrite the ab o v e system reducing it b y the constrain t, but the fina l equation has complicated form. F or the Lax op erator (5.14) the appropriate reminder (3.39) is giv en b y R = ˜ u ∆ − 1 ( µ ˜ u ) − 1 ˜ u t n δ − v t n − ∆ − 1 w t n . (5.20) Hence, from (3 .37) w e ha v e the fo llo wing, v alid when µ 6 = 0, recursion form ula   ˜ u v w   t n +1 =   R ˜ u ˜ u ˜ uE µ ˜ u R v ˜ u v + ˜ u ∆ (1 + E − 1 ) ˜ u R w ˜ u w − ∆ E − 1 ˜ u + v − µw     ˜ u v w   t n , (5.21) where R ˜ u ˜ u = E ( v ) − µ − 1 ˜ u + µ ˜ u ∆( v )∆ − 1 ( µ ˜ u ) − 1 R v ˜ u = ∆( v ) + w + ˜ u ∆( v )∆ − 1 ( µ ˜ u ) − 1 + (1 − E − 1 ) ˜ uw ∆ − 1 ( µ ˜ u ) − 1 R w ˜ u = ∆ E − 1 ˜ uw ∆ − 1 ( µ ˜ u ) − 1 . (5.22) Its reduction b y the constraint ( 5.15) is  v w  t n +1 =  v + ˜ u ∆ + R v ˜ u µ (1 + E − 1 ) ˜ u − R v ˜ u µ 2 w + R w ˜ u µ − ∆ E − 1 ˜ u + v − µw − R w ˜ u µ 2   v w  t n , (5.23) with ˜ u giv en b y (5.15). In the case of T = R , or in the contin uous limit of some sp ecial time scales, with the choice ˜ u = 1, t he Lax op erator (5.14) t ak es the fo rm L = ∂ + v + ∂ − 1 w . 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