The Sato Grassmannian and the CH hierarchy

The Sato Grassmannian and the CH hierarc h y Gregorio F ALQUI • , Gio v anni OR TENZI ⊙ • Dipartimento di Matematica e Applicazioni, Universit` a di Milano-Bico cca Via R. Cozzi 53 – Ed. U5, I-2012 6 Milano, Italy E-mail: grego r io.falqui@unimib.it ⊙ Dipartimento di Fis ic a N ucleare e T eorica, Universit` a di P avia, Via A. Bassi, 6 27 1 00 P avia, Italy , and I.N.F.N. Sezione di Pavia E-mail:giov anni.ortenzi@ unim ib.it Abstract W e discuss how the Camassa-Holm hierarc hy can b e framed within the geometry of the Sato Grass- mannian. 1 In tro duction In this pap er w e study s ome sp ecific asp ects of the Camassa Holm hierarch y . Since it app earance in the liter a ture, it has b een recognized that the CH equation po ssesses spe c ific features, (e.g., pea k on solutions, the app earance of third order Ab elian differentials in finite gap so lutions,...) that other more ”classica l” soliton hiera r c hies (KdV, Bo us sinesq, NLS) do not exhibit. Among these , esp ecially in view of the Dubrovin–Zhang classification scheme [8], the no n- existence of a formulation via a τ function is, from o ur p oint of view, of particular in terest. The Sato theory of the τ function, basica lly views it as a section of the (dual) determinant bundle ov er the so –called Sato (or Universal) Grassmannia n (UG), and allows to a ssociate such a str ucture to any hierarch y of evolutionary PDEs that can b e represented as linear flo ws on this Grass mannian. Thus, it s e ems imp ortant to a nalyze whether (and which) flows of the CH hiera rc hy can b e r ealized as linear flows in the Sato Grassma nnian. The main aim of this pa per is to discuss this pr oblem, in the framework of a s e t up, intro duced in [11, 2], relating the (bi)–Hamiltonian s tructures of soliton hier a rc hie s of KdV type to the Sato Grass - mannian. In [12] it was shown th at the bi–Hamiltonian structure s of CH and KdV equa tions (as well of the Har ry– Dym equation) are re la ted, being geo desic motions on the Viraso ro group with resp ect to differen t metrics. Actually , the r elation with the evolution on the Sato Gra ssmannian has b een studied for KdV and the HD hierarchies, s ho wing that they ar e related to linea r flows in the big cell o f UG. In this pap er we try to complete this picture showing that the CH hiera rc hy to o is rela ted to the big cell of the Sato Grassmannia n by mea ns o f its lo cal (also called neg ativ e) flows. One of the basic differenc e s a mong this representation of the thr ee hier arc hies is given by the relation betw een the lo cal flows and the “ t ime” of the hier arc h y r elated to the conserv atio n of the linear momen- tum. This will sho w up, in the present pap er, a s the realization of the CH lo cal hierarch y in a constrained subspace of the big cell. The path leading us to this r esult is the analysis of the evolution of the No ether currents as sociated with the bi–Hamiltonian recurr e nc e rela tion of the lo cal hierar c hy . W e will a r gue as, on mor e gener al gro unds, they are asso ciated with a t wo-field (alb eit s o meho w tr ivial) bi–Hamiltonia n extension o f the CH lo cal hier arc h y . The or dinary CH bi–Hamiltonian hier arc h y is recovered – together with the non lo cal par t including the ”true” CH equation – b y Dirac restr icting this t wo-field hierarch y to a sp ecific submanifold, namely those selected b y these No ether curr en ts sa tisfying a sp ecific cons train t. Thu s the CH equation is realiz e d, in the picture herewith presented, as a n a dditio na l commuting flow of an infinite sy stem of linear flows on the Sato Grassmannian. 1 The full interpretation of the whole nonlo cal hierarch y to this Sato Grassmannia n approa c h, as well as the problem of how far this picture could b e useful to explain and understand the no n–existence o f the τ function for CH is still under consideration. 2 The geometry of the CH hierarc h y .... It is well known[1, 10] that the CH equation 1 4 v t − v xxt = 24 v x v − 4 v xx v x − 2 vv xxx is a bi–Hamiltonia n evolutionary PDE on C ∞ ( S 1 , R ) w.r .t. the Poisson p encil P λ = (4 ∂ x − ∂ 3 x ) + λ (2 m∂ x + 2 ∂ x m ) λ ∈ R where m = 4 v − v xx . The densities of the conserved laws of the hierarch y can be obtained by re c ursiv e ly solving h x + h 2 = mz 2 + 1 , z = √ λ (1) where h is the generating function of the densities of the Casimir of P λ [4, 5 , 6, 14, 17]. This Riccati equatio n admits tw o different solutions h = h − 1 z + h 0 + h 1 z + h 2 z 2 + . . . k = k 0 + k − 1 z + k − 2 z 2 + k − 3 z 3 + . . . . The tw o families of co efficien ts { h i } i ≥− 1 and { k i } i ≤ 0 give, by mea ns of the Le na rd recursion, all the f CH hiera r c hy . In particula r, the h i ’s ar e the dens ities of the neg ativ e (o r loca l) CH hier arc hy , and ca n be algebraic ally found from (1), while the k j ’s are the de ns ities of the po sitiv e (or “non– local”) C H hierarch y , whose fir st tw o member s are, resp ectiv ely x -tra nslation and the CH equation itself. The first flow of the lo c al hier arc hy is ∂ ∂ t 3 m = (4 ∂ x − ∂ 3 x ) 1 2 √ m . (2) The key ingredient used in [1 1] to r elate the Hamiltonian structure of Solito n hierarchies of KdV type to evolutions on the Sato Universal Gr assmannian manifold is given by the No ether currents. In particular, it has b een shown in [4] that the Noe ther currents as sociated with the lo cal CH hierarch y are c har acterized, in the space of formal Laurent series in the pa rameter z by th e follo wing tw o pr operties: 1. Their asymptotic b e havior is given by J ( s ) = z s + O ( z ) , s ≥ 2 (3) 2. They b elong to the span h ( ∂ x + h ) n z 2 i n ≥ 0 (4) of the F a` a di Bru n o monomials asso ciated with the generating function h , which so lv es (1 ) with asymptotic condition h ( z ) = h 1 z + h 0 + h 1 z + · · · , with co efficien ts on C ∞ ( S 1 , R ). The connection betw een the currents J ( s ) and the generating function h is given by the fa ct that, along the s -th time of the lo cal CH hie r arc hy , they evolve as ∂ s h = ∂ x J ( s ) where ∂ s = ∂ ∂ t s . (5) The asymptotic b eha vior of the lo cal No ether currents and the presence of a “ g enerator” h sugges t, in analogy with what happ ens in the KdV case, that they can b e asso ciated with linear evolutions on the Sato Gr assmannian. 1 W e ha v e herewith c hosen un usual normalizations b eca use this somewhat s implifies s o me of the formulæ we are in terested in. 2 3 .... and the Sato Grassmannian In this sectio n we shall lo ok at the pr oblem s ta rting from a slightly different p erspective. Let us consider the space J + given by the span on C ∞ ( S 1 , R ) o f the family J ( i ) = z i + J i − 1 z + J i 0 + J i 1 z − 1 + . . . i ≥ 2 in the spa ce J of Laur e nt s eries (with at most a p ole singular ity at z = ∞ ). J admits a direct splitting as J = J + ⊕ J − , where J − := h z i i i ≤ 1 . (6) Therefore the collection { J ( i ) } i ≥ 2 defines a p oint o f the big cell B of the Sa to Grassma nnian transla ted by z 2 w.r.t. the standar d Sato repres en tation [1 8 ]. On this space we ca n define an infinite family of flows setting ( ∂ s + J ( s ) ) J + ⊂ J + s ≥ 2 (7) that, more explicitly , c a n b e wr itt en as ( ∂ s + J ( s ) ) J ( r ) = J ( s + r ) + r − 2 X i = − 1 J s i J r − i + s − 2 X i = − 1 J r i J s − i + J r − 1 J s − 1 J (2) . (8) Prop osition 1 The flows (7) c ommu te. Pro of W e have to show tha t [ ∂ s , ∂ r ] J + = 0, i.e. [ ∂ s , ∂ r ] J ( n ) = 0 , ∀ s, r, n ≥ 2 . (9) Thanks to (7) it holds the symmetr y ∂ s J ( r ) = ∂ r J ( s ) and then the equation (9 ) can b e wr itten as [ ∂ s , ∂ r ] J ( n ) = [ ∂ s + J ( s ) , ∂ r + J ( r ) ] J ( n ) . (10) F rom the explic it form of the cur ren ts it holds [ ∂ s , ∂ r ] J ( n ) ∈ J − , but fro m (7) [ ∂ s + J ( s ) , ∂ r + J ( r ) ] J ( n ) ∈ J + .  Prop osition 2 The lo c al curr ents of CH satisfy (7). Pro of The curr e n ts (3) are elements of J + . Moreov er fr om the prop erty (4) follows that every element of J + can be written as J ( i ) lC H = P k c i k ( ∂ x + h ) k z 2 . Using this ex pa nsion (5) we see that ( ∂ s + J ( s ) ) r X k =0 c r k ( ∂ x + h ) k z 2 = r X k =0 ( ∂ s c r k )( ∂ x + h ) k z 2 + r X k =0 c r k ( ∂ s + J ( s ) )( ∂ x + h ) k z 2 = r X k =0 ( ∂ s c r k )( ∂ x + h ) k z 2 + r X k =0 ( ∂ x + h ) k z 2 J ( s ) ⊂ J + ⊕ z 2 J + . In [4] it is shown that, for the lo c al cur ren ts o f CH, z 2 J + ⊂ J + and then they satisfy (7 ).  3 Therefore, taking in to a ccoun t the res ults of [11] we can conclude that the lo cal (negative) flows of CH hierarch y are given, by means o f the construction outlined ab o ve, linear flows on the big cell B of the Grassmannia n. Remark . The ba sic issue to r e c o ver a hiera rc hy of 1 +1 dimensional P DEs from a dy na mical system o f the fo r m (7) is to specify (or define) the “ ph ysica l” space v ariable x . F or instance, in the ordinary K P -KdV case, x can b e, a s it is well known, identified with the firs t ”time” of the hiera rc hy . A s it was shown in [3], fractiona l KdV hier arc hies ca n be obtained identifying x with a different time t s of a system similar to (7). Actually , in our case, x is not con tained in the dynamical system, a nd thus s hould b e added b y means of the introduction of another cur r en t h = h − 1 z + h 0 + h 1 z + . . . . In tur n, this additiona l cur ren t has to b e related with the action of x -translation o n the currents J ( s ) of the Gr assmannian. The most natura l way to a dd this new curre nt is to c o nsider the enlargement of the s ystem (7) to ( ∂ s + J ( s ) ) J + ⊂ J + , ( ∂ s + J ( s ) ) h ∈ J + ( s ≥ 2) , ( ∂ x + h ) J + ⊂ J + , (11) which explicitly is given, in addition to E qn.s (8), by ( ∂ x + h ) J ( s ) = s − 2 X i = − 1 h i J ( s − i ) + h − 1 J s − 1 J (2) s ≥ 2 ( ∂ s + J ( s ) ) h = s − 2 X i = − 1 h i J ( s − i ) + h − 1 J s − 1 J (2) s ≥ 2 . (12) How ever, these flows are not in genera l commuting, s o that further conditions hav e to b e imp o sed. It is outside the s ize of this paper to discuss this problem in full g eneralit y; we simply remark the restriction to the subs pa ce of the translated big cell defined by J (2) = z 2 and z 2 J + ⊂ J + . (13) is a cons is ten t o ne 2 . The following Lemma helps clarifying the meaning of the co nstrain t(13): Lemma 3 F or any choic e of J (2) , the curr ents J ( i ) satisfying (12) ar e elements of F = sp h ( ∂ x + h ) n J (2) i n ≥ 0 . Pro of Expanding the relation (12) it follows that J ( s +1) = 1 h − 1 ( ∂ x + h ) J ( s ) − s − 2 X i = − 1 h i h − 1 J ( s − i ) + J s − 1 J (2) . (14) Since ( ∂ x + h ) F ⊂ F and J (2) ∈ F , then one can wr ite r ecursiv ely all the curr en ts using elements o f F .  In the lig ht of this pro position, we can rephrase the fir st of eq uations (13) saying that we consider only the c a se J (2) = z 2 . The s tud y of mor e ge ne r al choices of the c urren t J (2) is under cons ide r ation. The basic reason for this c hoice of ours is that the space J + defined by (13) contains the curr en ts of the CH hier arc hy (see Prop osition 2). Moreover, it turns o ut that J + is par ameterized by three fields, namely h − 1 , h 0 , and h 1 . This ca n b e seen as f ollows. Since z 2 J + ⊂ J + and J (2) = z 2 , w e get that J (4) = z 4 . The recursion relations (14) allow us to write all the cur r en ts, a nd namely J (4) , as differential po lynomials in the c omponents h k of the for mal Laurent series h . Thus we ar riv e at z 2 h − 1 2 ( h x + h 2 ) − z 2  h − 1 x h − 1 3 + 2 h 0 h − 1 2  h − z 2  h 0 x h − 1 2 − h 0 2 h − 1 2 + 2 h 1 h − 1 − h 0 ( h − 1 x ) h − 1 3  = z 4 . (15) 2 Another consistent solution to this problem is giv en by requiri ng that ( ∂ x + h ) h ∈ J + . The resulting system of comm uting PDEs leads to a 2 + 1 dimensional extension of the HD hierarch y [13, 16]. 4 It is straight forward to c heck that this rela tion e na bles one to reco ver h 2 , h 3 , . . . as diff erential p olynomials in h − 1 , h 0 , h 1 . So the system (12), deter mines a hierar c hy of 1+1 evolutionary PDEs in the three fields (dependent v ariables) h − 1 , h 0 , h 1 . F or instance, the first non trivial flow is [15]: ∂ 3 h − 1 = − h − 1 x h 1 h − 1 2 + h 1 x h − 1 ∂ 3 h 0 = 3 2 h 1 x h − 1 x h − 1 3 − 3 2 h 1 ( h − 1 x ) 2 h − 1 4 − 1 2 h 1 xx h − 1 2 + 1 2 h − 1 xx h 1 h − 1 3 ∂ 3 h 1 = − 3 2 h − 1 x h 1 xx h − 1 4 + 5 2 h − 1 x h − 1 xx h 1 h − 1 5 + 15 4 ( h − 1 x ) 2 h 1 x h − 1 5 − 15 4 h 1 ( h − 1 x ) 3 h − 1 6 + h 1 2 h − 1 x h − 1 3 + 1 4 h 1 xxx h − 1 3 − 1 4 h − 1 xxx h 1 h − 1 4 − h 1 x h − 1 xx h − 1 4 − h 1 x h 1 h − 1 2 . (16) W e notice that the field h 0 do es not a ff ect the dynamics . Actually , this is true for all the times of the hierarch y we are consider ing. This is a co ns equence of the fac t that no currents dep ends on h 0 , as one can se e by recur s ion using (14). Therefore the constr ain t given by (15) do not depend o n h 0 as well, and s o we can limit ourselves to the s tu dy of the system in the tw o dep enden t v a riables h − 1 , h 1 . W e sha ll prov e that this system is bi–Hamiltonian a nd a dmits an iter able Casimir, that is, a Ca simir of the pencil that generates, via the L enard recur sion rela tions, the commuting flows. Our pro of will b e done in a sequence of steps as follows. First we no tice that, if we p e rform the c hange of v ariables h − 1 = α and h 1 = γ α the first and third of equations (16) b ecome: ∂ 3 α =  γ α 2  x ∂ 3 γ = α 4  1 α  1 α  γ α 2  x  x  x . (17) F rom the genera l theor y , and namely from the r epresen tation (5) of the P D Es, we see tha t this system has an infinite seq uence o f conserved quantities, whose densities are giv en by the co e fficien ts of the for mal Laurent series (15 ) with h 0 = 0, i.e.: 1 α 2 ( h x + h 2 ) − α x α 3 h − 2 γ α 2 = z 2 . (18) It is worthwhile to remar k aga in that this equa tion determines a ll the co efficien ts h i , i ≥ 0 as differ en tial po lynomials in α, γ . F or insta nce we have, apar t form the obvious r elations h − 1 = − α, h 1 = − γ /α , the expressions h 2 =  γ 2 α 2  x , h 3 = γ 2 2 α 3 −  1 α  γ α 2  x  x , h 4 = total deri vativ e, h 5 = γ 3 2 α 5 − 1 12 γ 2 α xx α 6 + 1 8 γ γ xx α 5 − 7 24 γ γ x α x α 6 + total der ivati ve, . . . (19) and so on and s o for th. The motiv ation for t he c hange of v ariables, as w ell as further hints for our pr o gram come fr o m considering of the disp ersionless limit of (17), that is, ∂ 3 α =  γ α 2  x ∂ 3 γ = 0 . (20) This equation is bi–Hamiltonian w.r.t. to the Poisson tensor s P disp 0 =   0 ∂ x α α∂ x γ ∂ x + ∂ x γ   P disp 1 =   ∂ x 0 0 0   (21) 5 with Hamiltonian dens ities h 3 = γ 2 / 2 α 3 , h 1 = − γ / α . This prop ert y sug gests that the full dispersive hierarch y can be obtained by suitably defor ming the p encil of Poisson tensors (21). As a first step in this direction, one notices that the flow (17) can b e o bt ained in a ”Hamiltonian” wa y , via the a ction o f the antisymmetric tensors P 0 =   0 ∂ x α α∂ x γ ∂ x + ∂ x γ + α 4 ∂ x T 2 α α   , P 1 =   ∂ x 1 4 ∂ x T 2 α α α 4 ∂ x T 2 α α 16 ∂ x T 4 α α   , (22) where T α is the op erator 3 1 α ∂ x , as  ∂ 3 α ∂ 3 γ  = P 0 d Z h 3 dx = P 1 Z h 1 dx , where h 1 and h 3 are the densities (19). F urthermo r e, a direct computation shows that h 1 is the density of a Casimir of P 0 . Actually , our use of this terminolog y is justified by the following prop osition, who se pro of, that ca n be dir ectly obtained via a straightforward alb eit tedious co mputation, will b e a pparen t from the sequel. Prop osition 4 The tensors (22) ar e a p air of c omp atible Poisson tensors. T o push our analysis fur ther , the following observ ation is imp ortant. W e notice that the member P 1 of the pair (22) is greatly degenerate. Indeed one sees that vector fields ( ˙ α, ˙ γ ) b elong to its imag e if a nd only if the relation ˙ γ = α 4 ∂ x 1 α ∂ x 1 α ˙ α (= α 4  T † α  2 ˙ α ) . (23) This entail that the system (17), as well as any bi–Hamiltonian vector field asso ciated with the pair (22) admits as a n inv ar ia n t submanifold the one de fined by γ − 1 4 ∂ 2 x ln α + 1 8 ( ∂ x ln α ) 2  ≡ γ − 1 8 ( α ( T α − T † α ) T α ( α )  = const. (24) This fact (tog ether with the particularly simple dep endence on γ of the relation (24)) prompts us to consider the dep enden t v a riable u = y − 1 4 ∂ 2 x ln α + 1 8 ( ∂ x ln α ) 2 . In the co ordinates ( α, u ) the tensors of (22) b ecome P 0 =    0 ∂ x α α∂ x u∂ x + ∂ x u − 1 4 ∂ 3 x .    , P 1 =   ∂ x 0 0 0   . (25) The fact that the antisymmetric tensors we are co nsidering indeed make up a Poisson pair is no w apparent from the theory of affine Poisso n structure s on duals of Lie algebra s. This new form of the p encil will also allow us to state tha t the hierarch y of co mm uting vector fields starting with (17) is indeed a bi– Hamiltonian hier arc hy . According to the Gel’fand–Za kharevich bi–Hamiltonian scheme, we loo k for a Casimir of the p encil (25). This a moun ts to finding an exact one-form Ω( λ ) = ( X ( λ ) , Y ( λ )) that satisfies the equation ( P 1 − λP 0 )Ω = 0 , with as ymptotics Ω ( λ ) = Ω 0 + Ω 1 λ + · · · , whose fir st element is the differe n tial of the Ca simir of P 0 (in pa rticular, with obvious meaning of the notation, Y 0 ≃ 1 α ). So we can trade the a b ov e eq uation fo r the system X ( λ ) = λα Y ( λ ); λ α 2 Y ( λ ) 2 + 2 uY ( λ ) 2 − 1 2 Y xx ( λ ) Y ( λ ) + 1 4 ( Y x ( λ )) 2 = λ. (26) 3 Oper ator comp osition is here and in the following, understoo d. 6 In tur n, the second of these equations is equiv alent to the following sys tem h x + h 2 = λα 2 + 2 u, h = z Y ( λ ) + 1 2 Y x ( λ ) Y ( λ ) , (27) where z 2 = λ, and h = h − 1 z + h 0 + h 1 z + · · · . It can b e easily shown that the ser ies h(z) solving the first of these equations is, in the sense of formal Laurent serie s, indeed the po ten tials o f the one-for m Ω( λ ). Also, the co efficien ts h i can b e a lgebraically co mputed in a recur siv e way . The co mparison of this Riccati equation with the Riccati equa tion asso ciated with the lo cal CH hierarch y sug gests a further mino r co ordinate change, namely to set m = α 2 . Indeed in the co ordinates ( m, u ) the Poisson p encil P 1 − λP 0 is (25)   2( ∂ x m + m∂ x ) 0 0 0   − λ    0 ∂ x m + m∂ x ∂ x m + m∂ x − 1 4 ∂ 3 x + ∂ x u + u∂ x    , (28) and the co rrespo nding Riccati equation is h x + h 2 = 2 u + mz 2 , z = √ λ. (29) The v ector field (17) bec o mes s imply ∂ 3 m = (2 u∂ x + 2 ∂ x u − 1 2 ∂ 3 x ) 1 √ m ∂ 3 u = 0 , (30) Summing up,the search for a Cas imir of the p encil (28) is r educed to the problem of solving - in the space of for ma l Laurent series - the Riccati equation for h ( z ) = h − 1 z + ∞ X i =1 h i z i . This pro blem can b e itera tiv ely solved, and is equiv a len t, up to the to tal deriv ative h 0 , to (18) written in the u, m v ariables. Remarks . 1) On u = 1 2 the fir st of the equatio ns (30) b ecomes the first nontrivial lo cal CH flow (2). 2) In the co ordinates ( m, u ) (as well as in the co ordinates ( α, u )), all vector fields of this hierar c hy are somewhat trivial, since they read ∂ t i m = ∂ x ( F i ( m, u )) , ∂ t i u = 0 . (31) This fact can b e, in a se nse, under stoo d also in the framew ork of th e theory of r e c iprocal transformatio ns. F or instance, tra nsforming the system (17) under the r eciproca l transformation induced b y its fir st element (seen as a conserv ation law) yield the tria ngular system ∂ 3 U = 1 2 ( U V ) z ∂ 3 V = 1 4 ( V z zz + 6 V V z ) where dx = U dz + 1 2 U V dt 3 , U = 1 α , and V = − 2 γ α 2 . T o fully ex a mine these equations in the light of the theory of recipr o cal trans f ormations , how ever, is outside the a im of the pr esen t pap e r [9]. 3) As a final c heck of the bi–Hamiltonian analysis w e per formed, w e notice th e following W e exchange the role of the Poisson tensors P 0 and P 1 and consider the Casimir function K = R ( u + m ) dx of P 1 . Clearly enough, the vector field P 0 dK is just x -translation. This Ca simir do es not giv e rise to a new Lena rd sequence, since P 0 dK 0 do es not lie in the image of P 1 . How ever from the fac t that x =tra nslation is the image under P 0 of a Casimir of P 1 confirms that it co mm utes with all the vector field of the hiera rc hy , as it should be . 7 4 Bac k to th e CH hierarc hy: its bi–Hamiltonian structure and its Lax represen tation As we have seen, the bi–Hamiltonia n geometry of the manifold we a r e considering is particularly s imp le: indeed, it is stra tifi ed by the submanifolds g iv en by u = κ for so me cons ta n t κ , a nd these submanifolds are left inv aria n t by a ll vector fields that are Hamiltonian w.r.t P 1 , and thus b y all bi– Hamiltonian v ector fields. Also, o n the inv ariant submanifold u = 1 2 we have that the first flow of our hierarch y co incides with the fir st lo cal CH flow, and the Riccati equation (29) reduces to the Riccati e quation ass ociated with the CH hierarchy (1). These facts sugges t the oppo rtunit y to consider the Dirac reduction of the p encil (28). Prop osition 5 The Dir ac re duction of (28) on t h e c onstr aint u = κ gives a Po isson p encil for the Camassa Holm. The hier ar chy re stricts to this su bma nifold as a bi–Hamiltonian hier ar chy. Pro of . T o prov e the ass ertion, we find it mor e conv enient to use the no tation of Poisson bra c kets ra th er than that o f Poisson tensors . According with Dirac’s theor y , the reduction o n u = const of the Poisson brack ets asso ciated with our p encil is given by { m ( x ) , m ( y ) } D 0 := { m ( x ) , m ( y ) }| u = κ − Z dw Z dz { m ( x ) , u ( w ) } ( { u ( w ) , u ( z ) } ) − 1 { u ( z ) , m ( y ) }| u = κ where { u i ( x ) , u j ( y ) } 0 := R dz δu i ( x ) δu k ( x ) ( P λ ) kl δu j ( y ) δu l ( x ) . A s imple computatio n shows that P D λ | u = κ = 2( ∂ x m + m∂ x ) − λ ( ∂ x m + m∂ x )  2 κ ∂ x − 1 4 ∂ 3 x  − 1 ( ∂ x m + m∂ x ) . It is easy to recognize in the above for m ula (one o f ) the Poisson p encils of the CH hierarch y , namely the one given by the sta nda rd Lie Poisson tensor and the first nonlo cal tens o r with the suitable c hoice κ = 1 2 4 . The Dira c reduction of the Poisson structure (28) generates exactly the lo cal part of the CH hierarch y . This follows from the fact that the Dirac defor mation o f the Poisson bracket ass ociated with P 0 is achiev ed by means of Cas imir functions of the other brackets. This entails that Lenar d relatio ns P 0 dH = P 1 dK hold also fo r the cor responding Dirac r eductions. On the manifold u = κ (e.g., u = 1 2 ) we can recov er the standard nonlo cal part o f CH hierarch y using the so lution o f (29) who se asymptotic behavior is 1 + O ( z ) a s in [4], v ia the usual CH substitution m = 4 v − v xx . In this pictur e, the flows of the po sitiv e CH hierarch y (and so, the CH equation as well) play the role of “ additional” (co mm uting) symmetries of these flows, which are r estrictions to u = 1 2 of the linear flows de fined by (7).  A further outcome the previous cons tr uction is to provide a Lax repr esen tatio n of the (extended) lo cal CH hierar c hy as a suita ble flow in the space of pseudo differen tia l op erators. W e w ill basically follow a construction presented in [2] for the KdV–K P case . The Ricca ti constr a in t (13) can be read a s the requirement that the function ψ = exp ( R hdx ) be an eigenfunction o f the op erator L = 1 m ∂ 2 x − 2 u m with eigenv alue z 2 . Also , the equations of mo tio n imply ∂ s J ( r ) = ∂ r J ( s ) and ∂ s h = ∂ x J ( s ) . Therefore, from the compatibility of equations L ψ = z 2 ψ and ∂ s ψ = J ( s ) ψ , we get ∂ 2 s +1 L = h J (2 s +1) , L i s ≥ 1 , (32) while times and currents with even lab el 2 s are trivial, as implied by the co nstrain t (13). In order to obtain a n op eratorial version o f the equations o f motion we relate the curr e n ts J ( s ) with L . First of all we need the following technical Lemma 6 Under the c onstr aint (13 ) it holds J ( s ) = Π J + ( z s ) . 4 Indeed, κ can be r e scaled to 1 2 without loss of generalit y . F or κ = 0, we get a Poisson pencil of HD. 8 Pro of The spac e J + = Π J + ( J ) is, by definition, the linea r span o f J ( i ) . Therefore there is a unique wa y to write the element Π J + ( z s ) by means o f the currents J ( s ) . Since the le a ding term of J ( s ) is exa ctly z s , the a ssertion is tr ue .  Because of Lemma 3, J + is also the linear span of the { ( ∂ x + h ) i z 2 } i ≥ 0 . Mo r eo ver, extending b y rec ur sion the definition of ( ∂ x + h ) i z 2 to nega tiv e p o wers, the set { ( ∂ x + h ) i z 2 } i ∈ Z is a basis of a ll the space J . The map φ : J → Ψ D O ( ∂ x + h ) i z 2 → ∂ i x · L by means of the basis ( ∂ x + h ) n z 2 with n ∈ Z of the space J , gives the op eratorial action of an element J on ψ Prop osition 7 Under the c onstr aint (13) it holds J ( s ) ψ =  L s/ 2 − 1  + Lψ Pro of T he map φ intert wines b et ween Π J + and the op erator ( · L − 1 ) + L on the ΨDO spac e where ( · ) + is the standard pro jection on the differe n tial part of a ΨDO op erator. This pr operty can be ea sily pr o ved remarking that it holds for a n y element ( ∂ x + h ) i z 2 of the J basis. Therefore J ( s ) ψ = φ ( J ( s ) ) ψ = φ (Π J + ( z s )) ψ = ( L s/ 2 − 1 ) + Lψ .  The equations (3 2 ) b ecome then ∂ 2 s +1 L = h ( L s − 1 / 2 ) + L, L i . (33) W e finally notice that in the CH case that is, under the constraint u = 1 2 , the Lax oper a tor is L = 1 m ∂ 2 x − 1 m ; therefore ( L 1 / 2 ) + = m − 1 / 2 ∂ x − 1 2 ( m − 1 / 2 ) x and the previous equation g iv es ∂ 3 1 m = − 2 m − 2  ∂ x − 1 4 ∂ 3 x  m − 1 / 2 which is equiv a len t to the lo cal CH (2). W e end this Section noticing that the integrability of the system constructed starting fr om the La x op erator for loc al CH can be proven also by means o f a dir e c t computation. Indeed it holds: Prop osition 8 L et D O 2 b e the sp ac e of se c ond or der differ ential op er ators of the form λ = a∂ 2 + b ∂ + c , and let () + b e t h e pr oje ction op er ator fr om Ψ DO to D O . The e qu a tions ∂ s λ =  ( λ s 2 ) + λ, λ  define a family of c ommuting flows on D O 2 , that is, ∂ r ∂ s λ = ∂ s ∂ r λ . Pro of . W e sta r t expa nding ∂ r ∂ s λ = ∂ r  ( λ s 2 ) + λ, λ  =  ( ∂ r λ s 2 ) + λ, λ  +  ( λ s 2 ) + ∂ r λ, λ  +  ( λ s 2 ) + λ, ∂ r λ  = h  ( λ r 2 ) + λ, λ s 2  + λ, λ i +  ( λ s 2 ) +  ( λ r 2 ) + λ, λ  , λ  +  ( λ s 2 ) + λ,  ( λ r 2 ) + λ, λ  , as well as ∂ s ∂ r λ . Then the asser tion follows using standar d tec hniques in the Ψ D O a pproac h to KP-type equations (see e.g. [7]), with the crucial rema r ks that, since we a re cons idering degree 2 op erators,  ( λ r 2 ) − λ, ( λ s 2 ) −  + =  ( λ r 2 ) −  ( λ s 2 ) − , λ  + = 0 , bec ause the de g rees of the op erators a ppearing in these expres sion is less than zer o . 9  Ac knowledgemen ts The authors would lik e to thank Marco Pedroni, Bor is Dubrovin, Andre w Hone, Paolo Lo renzoni, a nd F ra nc o Ma g ri for useful discuss ions and remarks. G.O a lso b enefited fr om discussions with Paolo Casati and Boris Konop elchenk o, and thanks the Universit y of Mila no B icocca for the kind hospitality , as w ell a s the orga niz e r of the NEEDS 200 7 Conference. This pap er was par tially suppo rted by the Europ ean Comm unity through the FP6 Mar ie Curie R TN ENIGMA (Co n tract n umber MR TN-CT-200 4-5652), by the Euro pean Scie nc e F o undation pro ject MISGAM , and by the Italian MIUR Cofin2006 pro ject “Geo metrical metho ds in the theor y o f nonlinear wa ves and a pplica tions”. References [1] R. Ca massa, D.D. Holm, An inte gr able shal low water e quation with p e ake d solitons. Phys. Rev. Lett. 71 (19 93), no. 11 , 1661–1 664. [2] P . Cas ati, G. F alq ui, F. Magri, M. Pedroni, Soliton e quations, bi–Hamiltonian manifolds and inte- gr ability 21 o Col´ oquio Br a sileiro de Matem´ atica. 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