THE KP THEORY REVISITED. III. THE BIHAMILTONIAN ACTION AND GEL FAND DICKEY EQUATIONS

Transcription

1 THE KP THEORY REVISITED. III. THE BIHAMILTONIAN ACTION AND GEL FAND DICKEY EQUATIONS Paolo Casati 1, Gregorio Falqui 2, Franco Magri 1, and Marco Pedroni 3 1 Dipartimento di Matematica, Università di Milano Via C. Saldini 50, I Milano, Italy E mail: casati@vmimat.mat.unimi.it, magri@vmimat.mat.unimi.it 2 SISSA/ISAS, Via Beirut 2/4, I Trieste, Italy E mail: falqui@neumann.sissa.it 3 Dipartimento di Matematica, Università di Genova Via Dodecaneso 35, I Genova, Italy E mail: pedroni@dima.unige.it Abstract. In this paper we accomplish the study of the geometry of Gel fand Dickey manifolds. They are a special class of infinite dimensional bihamiltonian manifolds of relevance in the theory of soliton equations. We introduce the notion of generalized Casimir functions, and we prove that they are in involution. We compute these functions for the Gel fand Dickey manifolds by the method of dressing transformations. We show that the corresponding canonical equations are (a new formulation of) the celebrated Gel fand Dickey equations. Finally, we use Noether theorem to study their conservation laws. We identify two sets of conjugated conserved quantities, and we prove that they obey the reduced KP and dual KP equations. By this result we throw the bridge between the GD and the KP theories. RUNNING TITLE: The KP theory revisited. III. Gel fand Dickey equations. 1. Purposes and main results This is the third of four papers presenting a new formulation of the KP theory [SS] from the point of view of Hamiltonian mechanics. It differs from the currently accepted presentation in many respects. Its aim is to deduce the whole KP theory from the study of the geometry of a special class of infinite dimensional Poisson manifolds, called Gel fand Dickey manifolds. They have been presented in the second paper of this series. The purpose of this paper is to accomplish the study of the geometry of these manifolds, and to throw the bridge with the KP theory. This work has been supported by the Italian M.U.R.S.T. and by the G.N.F.M. of the Italian C.N.R 1 Typeset by AMS-TEX

2 We recall [CFMP2] that the GD manifolds are defined as quotient spaces of a symplectic manifold with respect to a suitable equivalence relation. The symplectic manifold is the affine hyperplane S of the matrices S = p p p n q q n 1 p 0, (1.1) whose entries are scalar valued functions defined on S 1. This affine hyperplane is endowed with the canonical Poisson bracket n 1 {F,G} S = a=0 S 1 ( δf δg δf ) δg dx. (1.2) δq a δp a δp a δq a The equivalence relation is motivated by the theory of Poisson reduction of Marsden and Ratiu [MR]. It can be defined as follows. Starting from g (0) := 1,0,...,0, (1.3) let us iteratively construct a sequence of row vectors g (k) in C n+1 according to the recursion relation where g (k+1) = g (k) x + g (k) (S + λa), (1.4) A = (1.5) and λ is a real parameter. It is possible to show that the first (n+1) vectors of this sequence form a basis of C n+1, called the Frobenius basis associated with the point S. By expanding the next vector g (n+1) on this basis, we obtain an equation of the form n 1 g (n+1) = u j (S) g (j) + λ g (0), (1.6) j=0 called the characteristic equation at the point S. Then we agree that two points S and S of the symplectic manifold S are equivalent if they have the same characteristic equation, or, otherwise, if u j (S) = u j (S ) j = 0,...,n 1. (1.7) The quotient space N of S with respect to this equivalence relation is the Gel fand Dickey manifold associated to the Lie algebra sl(n + 1). It is parametrized by n periodic functions u j (x), and is endowed with a pair of compatible Poisson brackets. They are the Adler Gel fand Dickey brackets described in [CFMP2]. For this reason N is called a bihamiltonian manifold. 2

3 According to the main result presented in this paper, on any GD manifold there exists an infinite family of functions which are in involution with respect to both AGD brackets. They may be computed algebraically by solving a generalized Riccati equation. This equation is constructed by introducing the Faà di Bruno polynomials h (k) iteratively defined by h (k+1) = h (k) x + hh (k), (1.8) starting from h (0) = 1. Each Faà di Bruno polynomial h (k) is a differential polynomial of order (k 1) in the given periodic function h(x). The generalized Riccati equation n 1 h (n+1) = u j h (j) + z n+1 (1.9) j=0 is obtained by replacing the row vector g (k) by the corresponding Faà di Bruno polynomial h (k) in the characteristic equation (1.6), and by setting λ = z n+1. It admits a unique solution in the form of a monic Laurent series h(z) = z + j 1 h j z j, (1.10) and the coefficients h j (x) can be computed algebraically by recurrence. They are the local densities of the GD Hamiltonians H j (u a ) := (n + 1) h j+n+1 (u,u x,...)dx. (1.11) S 1 Our aim is to prove the following result. Proposition 1.1. The GD Hamiltonians verify the Lenard recursion relations {,H j } N 1 = {,H n+1+j } N 0, j n, (1.12) with respect to the pair of AGD brackets defined on the GD manifold N.The first (n + 1) Hamiltonians (H n,...,h 0 ) are Casimir functions of the first Poisson bracket: {,H q } N 0 = 0 q = n,...,0. (1.13) Therefore the GD Hamiltonians are in involution with respect to both Poisson brackets. The corresponding bihamiltonian equations are the GD equations on N. u a = {u a,h j } N 1 = {u a,h n+1+j } N 0 (1.14) Our strategy to prove this result is anti reductionist. We renounce to attack the problem directly on the quotient manifold N, due to the great complication of the AGD brackets, and we go back to the symplectic leaf S, which generates the GD manifold. This approach requires some preparatory work, aiming to clarify the relation existing between the Lenard recursion relations on S and the ones defined on the quotient space N. However, in our opinion it has two main advantages: it provides a clear explanation of the appearence of the generalized Riccati equation, and it leads to discover a remarkable new form of the GD equations. As an intermediate step, we prove the following result. 3

4 Proposition 1.2. The functions H S j (q a,p a ) := H j (u a (S)) (1.15) on the symplectic manifold S are in involution with respect to the canonical Poisson bracket on S. The corresponding Hamilton equations q a t j = δhs j δp a, p a t j = δhs j δq a, (1.16) are the extended GD equations on the symplectic manifold S. Their projections on N are the usual GD equations. Henceforth we consistently study this new, albeit classical, form of the equations. We recall that the symplectic manifold S is embedded into a bihamiltonian manifold M, and we study the GD equations from the point of view of this immersion. We denote by Ṡj the vector field defined by Eq.s (1.16) on S, and we prove that these vector fields are the generators of a bihamiltonian action of an abelian symmetry algebra on S. To construct this action by the method of dressing transformations we consider the special point B on S corresponding to q a (x) = p a (x) = 0, and we denote by Λ the matrix B + λa. Then we associate a basis { ψ a } in C n+1 (the eigenvectors basis ) with any point S of the symplectic manifold S. We call { l a } the basis at the point B, and we introduce the dressing matrix K connecting the basis at the point B to the basis at S: ψ a = K l a. (1.17) We note that K admits an expansion in powers of λ 1. We use K to introduce the matrices V j = KΛ j K 1. (1.18) We denote by V j = res λ V j the residue of V j, and by (V j ) + the positive part of its expansion in powers of λ. We remark that the residues V j are exact 1 forms on S, and that their potentials are the GD Hamiltonians (1.15): As a final result, we prove V j,ṡ = d dt HS j. (1.19) Proposition 1.3. The GD equations (1.16) coincide with the generators Ṡ j = [A,res λ V j ] = (V j ) +x + [(V j ) +,S + λa] (1.20) of a bihamiltonian action on S of the abelian algebra spanned by the matrices Λ j. This is the basic characterization of the GD equations suggested in this paper. It is used to investigate their conservation laws according to Noether theorem. From the conservation laws of the GD Hamiltonians we deduce that the monic Laurent series h obeys continuity equations of the form t j h = x H (j), (1.21) and we compute the conserved currents H (j) in term of (V j ) + : H (j) = g (0) (V j ) + ψ 0. (1.22) This allows to identify Eq.s (1.21) with a reduction of the KP equations studied in the fourth paper of this series, setting a bridge with the KP theory. 4

5 2. The generalized Casimir functions Since the ideas to be developped in this Section are indipendent of the specific feature of the GD example, we believe useful to adopt a purely geometric point of view, and to formulate the theory of generalized Casimir functions for a bihamiltonian manifold obtained by a MR reduction. We recall that a bihamiltonian manifold is a manifold M endowed with a pair of compatible Poisson brackets {, } 0 and {, } 1. Two Poisson brackets are compatible if the linear combinations {f,g} λ := {f,g} 1 λ{f,g} 0 (2.1) verify the Jacobi identity for any value of the real parameter λ. The bracket {, } λ is called the Poisson pencil defined on M. A formal Casimir of the Poisson pencil is a formal series f(λ) = j 0 f jλ j satisfying the equation {,f(λ)} λ = 0. (2.2) Therefore the first coefficient f 0 is a Casimir function of the first Poisson bracket {,f 0 } 0 = 0, (2.3) and the other coefficients verify the Lenard recursion relations {,f j } 1 = {,f j+1 } 0. (2.4) These coefficients are generalized Casimir functions in the sense of [CFMP1]. A systematic study of these functions requires to go deeply into the study of the geometry of a bihamiltonian manifold. In particular, one has to disentangle an intricate combination of nested distributions sitting on any bihamiltonian manifold, usually referred to as the Veronese web of the manifold [GZ]. A partial view of this plot has been shown in part I, dealing with the simplest example of GD manifold. Fortunately, we can dispense from studying in general the existence of these functions, since we shall give a direct proof of their existence for the GD manifolds. So, in this Section, we can restrict ourselves to display three simple properties of the generalized Casimir functions which are used henceforth. The first property explains why we are interested in such functions. Proposition 2.1. Let f(λ) and g(λ) be any pair of formal Casimirs of the Poisson pencil. Then the generalized Casimir functions {f j } j N and {g k } k N commute in pairs with respect to both Poisson brackets on M: {f j,g k } 0 = {f j,g k } 1 = 0 (2.5) Proof. By using repeatedly the Lenard recursion relations (2.4) we find {f j,g k } 0 = {f j,g k 1 } 1 = {f j+1,g k 1 } 0 = = {f j+k,g 0 } 0 = 0 (2.6) since g 0 is a Casimir function of {, } 0. 5

6 The second property concerns the generalized Casimir functions defined on an exact bihamiltonian manifold. We recall that such a manifold is a bihamiltonian manifold endowed with a vector field X such that {f,g} 0 = {X(f),g} λ + {f,x(g)} λ X({f,g} λ ). (2.7) This vector field is called the characteristic vector field of M. If we set we get f (λ) = X(f(λ)), (2.8) {f(λ), } 0 = {f (λ), } λ (2.9) by using the assumption that f(λ) is a formal Casimir of the Poisson pencil. This property tells us that, if we use the formal Casimirs of the Poisson pencil as Hamiltonians for the first Poisson bracket {, } 0, we get vector fields which are bihamiltonian, the second Hamiltonian being simply the derivative of f(λ) along the characteristic vector field X. The third property concerns the reduced bihamiltonian manifolds obtained by a Marsden Ratiu reduction process. We recall that this process considers a symplectic leaf S of the first Poisson bracket {, } 0 and a foliation E of S. The foliation E is defined by the intersections of S with the integral leaves of the distribution D = P 1 df : f Casimir function of P 0 (2.10) spanned by the vector fields which are Hamiltonian with respect to the second bracket {, } 1, and whose Hamiltonians are the Casimir functions of the first bracket {, } 0. We have denoted by P 0 and P 1 the Poisson tensors associated with the given Poisson brackets, to be considered as linear skew symmetric maps from T M to T M. The reduction theorem claims that the quotient space N = S/E is still a bihamiltonian manifold. For any function f on N, we denote by f S the function on S defined by f S = f π, where π : S N is the canonical projection of S onto N. Furthermore, we denote by Xf S the Hamiltonian vector field on S associated to f S with respect to the symplectic form on S. Finally, we denote by V f any 1 form on M, which need not to be defined outside S, such that S df V f,ṡ = dt (2.11) V f,d = 0 (2.12) for any vector field Ṡ tangent to S. This 1 form, which is not unique, is said to be a lifting of f into M. Let now {f k } k N be the coefficients of a formal Casimir of the reduced Poisson pencil on N, and let {V k } k N be any lifting of {f k } k N into M. The third property we are concerned claims that the 1 forms V k may be chosen in such a way that there exists an additional 1 form V 1 such that the following result holds. Proposition 2.2. For any formal Casimir {f k } k N on the quotient space N, there exists a Lenard sequence of 1 forms {V p } p 1 on T S M such that 1. V 1 belongs to the kernel of P 0 ; 6

7 2. {V k } k N are liftings of the functions {f k } k N ; 3. they obey the Lenard recurrence relations P 1 V k = P 0 V k+1, k 1. (2.13) 4. they are orthogonal with respect to both P 0 and P 1 : V k,p 0 V j = V k,p 1 V j = 0. (2.14) Proof. Let P N 0 and P N 1 be the reduced Poisson tensors on N; hence we have that P N 0 df 0 = 0 P N 1 df k = P N 0 df k+1. By the Marsden Ratiu reduction process, there exist liftings {W k } k 0 of {f k } k 0 such that Pi N df k = dπ P i W k for i = 0,1. Therefore P 1 W k P 0 W k+1 belongs to the kernel of dπ, that is, P 1 W k P 0 W k+1 = P 1 W k with W k Ker P 0. Setting V k = W k W k we get the Lenard recurrence relations (2.13) for k 0. Moreover, we note that dπ P 0 V 0 = P0 N df 0 = 0, so that there exists V 1 Ker P 0 such that P 1 V 1 = P 0 V 0. Finally, Eq.(2.14) is proved by the same argument used in Proposition 2.1. Conversely, a sequence {V p } p 1 with the properties described in Proposition 2.2 can be projected on the quotient manifold N, as stated in Proposition 2.3. Let {V k } k 1 be a sequence of covectors in TS M, S S, such that Then: 1. There exist functions F k on S such that df k = V k T S ; 2. V 1 belongs to the kernel of P 0 ; 3. P 1 V k = P 0 V k+1 for k 1. a) The functions F k are constant along the leaves of the distribution E, and therefore they define functions f k on N; b) The functions f k are generalized Casimirs for the reduced pencil. Proof. It suffices to show a), since b) easily follows from the reduction theorem. We note that V k annihilates the distribution D, since V k,p 1 Ker P 0 = Ker P 0,P 1 V k = Ker P 0,P 0 V k+1 = 0. Therefore df k,e = 0, and the proof is complete. 3. The method of dressing transformations We return to the study of GD manifolds. Our purpose is to apply the method of Lenard sequences of 1 forms V (λ) to find the generalized Casimir functions of the AGD brackets. We recall that the GD manifolds are reduced exact bihamiltonian manifolds. The manifold M, the symplectic leaf S, and the foliation E pertinent to the GD theory have been studied in detail in [CFMP2]. We recall that M is the loop algebra of C functions from the S 1 to sl(n + 1). The Poisson tensors are (P 0 ) S V = [A,V ] (3.1) (P 1 ) S V = V x + [V,S], (3.2) 7

8 where V x denotes the derivative of the loop V with respect to the coordinate x on S 1. The Poisson pencil P λ is The characteristic vector field is the constant field (P λ ) S V = V x + [V,S + λa]. (3.3) Ṡ = X(S) = A. (3.4) The matrix A, the symplectic leaf S, and the foliation E have been already described in Section 1. Our problem is to find formal series of 1 forms V (λ) = k 1 V kλ k which solve the equation V x + [V,S + λa] = 0 (3.5) at the point of the canonical symplectic leaf (1.1), and which are exact when restricted to S. This problem is solved by a geometrical version of the method of dressing transformations [ZS] The spectral analysis of V (λ). The basic strategy to solve Eq.(3.5) is to look at the eigenvalues and eigenvectors of the matrix V. The eigenvalues of V may be chosen arbitrarily, provided that they are independent of x. The eigenvectors must solve an auxiliary linear problem. The argument rests on the remark that, by Eq.(3.5), the matrix V has to commute with the first order matrix differential operator x +(S +λa). Suppose that this differential operator has a set of eigenvectors ψ a, obeying the equation ψ a x + (S + λa) ψ a = h a ψ a (3.6) for some suitable functions h a, and suppose that these eigenvectors form a basis in C n+1. Then any matrix V having constant eigenvalues and eigenvectors ψ a, solves Eq.(3.5) The auxiliary eigenvalue problem. The idea is thus to study the auxiliary linear problem V ψ a = c a ψ a, (3.7) ψ x + (S + λa) ψ = h ψ (3.8) to characterize the eigenvectors of V. We project this equation on the Frobenius basis g (k) at the point S, to find After an integration by parts, we find g (k) ψ x + g (k) S + λa ψ = h g (k) ψ. g (k+1) ψ = g (k) ψ x + h g (k) ψ. Then, by imposing the normalization condition g (0) ψ = 1 (3.9) 8

9 we finally get g (k) ψ = h (k), (3.10) where h (k) is the Faà di Bruno polynomial of order k of the eigenvalue h. This equation completely characterizes the normalized eigenvector associated with the eigenvalue h. To find h, we recall that the Frobenius basis verifies the characteristic equation n 1 g (n+1) u j g (j) = λ g (0). By projecting on ψ, we get or j=0 n 1 g (n+1) ψ u j g (j) ψ = λ g (0) ψ j=0 n 1 h (n+1) u j h (j) = λ. (3.11) j=0 This is the generalized Riccati equation for the eigenvalues of the auxiliary linear problem. Lemma 3.1. Set λ = z n+1. The characteristic equation (3.11) admits a unique solution of the form h(z) = z + j 1 h j z j. (3.12) Its coefficients can be computed algebraically by recurrence from the equation. Proof. We write n 1 h (n+1) z n+1 = u j h (j). j=0 But the very definition of the Faà di Bruno polynomials tells us that h (k) = (h) k + P k, Pk being a differential polynomial in h of degree k 1. Then the equation (3.11) yields (h) n+1 z n+1 = Diff. pol. in h of degree n which proves that the solutions to our problem can be computed algebraically by recurrence and have the desired form Eigenvalues and eigenvectors. Other solutions are easily obtained from h(z) by the change of variable h a (z) = h(ω a z), where a = 0,1,...,n and ω is the (n + 1) th root of unit ω = exp( 2πi ). (3.13) n + 1 9

10 Let ψ a be the normalized eigenvector associated with the eigenvalue h a (x). If we expand in powers of z the determinant of the matrix g (k) ψ a, at the highest order we find the Vandermonde determinant of the roots of unit. Therefore, the eigenvectors ψ a are linearly independent, and they form a new basis in C n+1 associated with the point S. In particular, at the point B = , (3.14) this basis coincides with the basis of the eigenvectors l a of the matrix Λ = B+λA: Λ l a = ω a z l a. (3.15) The basis { ψ a } at S, { l a } at B, and the dominant eigenvalue h(z) are the basic tools to solve the problem of the bihamiltonian iteration Dressing transformations. We introduce the matrices C, J, and K according to: C l a = c a l a J l a = h a l a K l a = ψ a. (3.16) We assume that the coefficients c a are constant Laurent series in z (i.e., independent of x). We call C the generator of the dressing transformation, K the dressing matrix, and J the momentum map. Finally we define on S the scalar function the 1 form and the vector field H C = J,C, (3.17) V C = KCK 1, (3.18) Ṡ C = [A,V C ]. (3.19) We restrict the generators C to have the form C = j N c jλ j, with c j independent of z, and we call this infinite dimensional abelian algebra the symmetry algebra g S of the symplectic leaf S. Proposition 3.2. Suppose that C = j N c jλ j, with c j independent of z. Write N = (n + 1)r + q with 0 q n. Then: i) The matrix V C admits the expansion λ r p 1 V pλ p in powers of λ. ii) It is a solution of the equation (3.5) for any choice of the generator C. iii) Its Hamiltonian on S is the function H C. iv) The Hamiltonian functions commute with respect to the symplectic form on S: {H C1,H C2 } S = 0. v) The vector fields ṠC define a symplectic action of g S on S. 10

11 Proof. i) We remark that a Laurent series L in z contains only the coefficients of z (n+1)p if and only if L(ωz) = L(z) for all z, where ω = e 2πi n+1. Then from the definition of K and from the fact that l a (ωz) = l a+1 (z) and ψ a (ωz) = ψ a+1 (z), we deduce that K(ωz) = K(z), so that K depends only on λ. Now we prove that K = K 0 + K 1 λ Indeed if K = K m λ m +... with m 1, then h (k) a = g k K l a = λ m g k K m l a +... But a comparison of the degrees in z shows that g k K m l a = 0 for all k, a, and this implies that K m = 0. From the development K = K 0 + K 1 λ it follows that K 1 = K0 1 + K 1 λ Finally we have that V C = KCK 1 = c N K 0 Λ N K0 1 + = λ ( r+1)c N K 0 A q K , where Λ q = λa q + B q. ii) The matrix V C has constant eigenvalues and the same eigenvectors of the operator + (S + λa). Hence the two operators commute. iii) We note that J and K satisfy the equation J = K 1 (S + λa)k K 1 K x, (3.20) as easily checked by evaluating it on the vectors l a. Let Ṡ be any tangent vector to S. Taking into account that J depends on S S both explicitly and through K, we get: d ( ) dt H C = J,C = K 1 ṠK + [J,K 1 K],C dx, S 1 where we have set K = d dt K(S + tṡ) t=0. Since J commutes with C, we get dh ( ) C = K 1 ṠK,C dx = (Ṡ,KΛK ) 1 dx = V C dt,ṡ. S 1 iv) By definition of the Poisson bracket on S, {H C1,H C2 } S = V C1,[A,V C2 ] = A,[V C2,V C1 ] = 0, since the matrices V C1 and V C2 commute. v) Let ω be the symplectic form on S. The result follows from: ω(ṡc,ṡ) = ω([a,v C],Ṡ) = V C,Ṡ = d dt H C The second Hamiltonian. Since the GD manifold is exact, we expect that the previous action is bihamiltonian, and that the function H C = V C,A (3.21) is the second Hamiltonian of Ṡ C. To show this result, we recall that the entries of V C are differential polynomials of the coordinates (q a (x),p a (x)), and we introduce the matrix V C = V C q 0 (3.22) whose entries are the partial derivatives of the entries of V C with respect to the coordinate q S 1

12 Lemma 3.4. (1) The 1 form V C, restricted to S, is exact and H C is its Hamiltonian on S: dh C dt = VC,Ṡ. (3.23) (2) The generator ṠC is a bihamiltonian vector field on S, and the conjugated functions H C and H C are its Hamiltonians with respect to P 0 and P λ respectively: Ṡ C = [A,V C ] = V Cx + [V C,S + λa]. (3.24) Proof. By deriving the equation V Cx + [V C,S + λa] = 0 with respect to q 0, and recalling that S q 0 = A, we immediately get V Cx + [V C,S + λa] + [V C,A] = 0, proving Eq.(3.24). Then we note that V C can be written in the form V C = V C q 0 = [ K q 0 K 1,V C ]. Finally, we compute the derivative of H C along Ṡ: dh C dt = V C,A = [ KK 1,V C ],A = [A,V C ], KK 1 = V Cx + [V C,S + λa], KK 1 = V C,( KK 1 ) x + [ KK 1,S + λa] = V C,Ṡ V C,K JK 1 = V = V C,Ṡ, since from (3.20) it is easily seen that C,Ṡ [ K K 1,V C ],K 1 JK q 0 Ṡ = ( KK 1 ) x + [ KK 1,S + λa] + K JK 1, and the matrices V C and K 1 JK commute. Thus the method of dressing transformations provides a way of constructing a bihamiltonian action of the symmetry algebra g S on the symplectic leaf S. The formulas H C = J,C HC (3.25) = V C,A define the associated momentum mappings. They will be the basic objects of the abstract KP theory dealt with in the fourth paper [CFMP4]. 12

13 4. The GD Hamiltonians and the GD equations We complete the analysis of the previous Section by computing explicitly the coefficients of the expansion of H C and V C in powers of λ. We fix C = Λ j (4.1) and we denote by H j and V j the corresponding series. We write j in the form j = (n + 1)r + q with 0 q n. For further convenience, we formally extend the definition (1.11) of the GD Hamiltonians to negative integers: H j := (n + 1) h n+1+j dx, j Z, (4.2) S 1 by setting h 0 = 0, h 1 = 1, and h j = 0 for j 2. Furthermore, we recall that we have denoted by V j the residue of the matrix V j. Proposition 4.1. The series H j and V j have the following expansions in λ: H j = λ r p 2 V j = λ r p 2 Their coefficients H j and V j are related by 1 λ p+1 H (n+1)p+q (4.3) 1 λ p+1 V (n+1)p+q. (4.4) H j t = V j,ṡ. (4.5) Therefore, the restrictions to T S of the residues V j are the differentials of the GD Hamiltonians H j. Proof. By using the definitions of H C, J, Λ, the properties of the roots of unity, and the identity Λ n+1 = λi, we get H j = J,Λ j = λ r J,Λ q n = λ r h a (z)(ω a z) q dx = λ r S 1 S 1 = λ r l 1 = λ r p 1 a=0 n h(ω a z)(ω a z) q dx a=0 n z q l (ω a ) S q l h l dx 1 a=0 z (n+1)p S 1 h (n+1)p+q dx, since n a=0 (ωa ) q l does not vanish if and only if l = q + p(n + 1). The limitation on p follows from l 1. 13

14 Eq.(4.4) follows from the identity: res λ (λ p r V j ) = res λ K(λ p Λ q )K 1 = res λ KΛ (n+1)p+q K 1 = res λ V (n+1)p+q. Finally, Eq.(4.5) is proved by V j,ṡ = res λ V j,ṡ = d dt res λ J,Λ j = d dt res λ (λ p J,Λ q ) = d dt res λ (λ p H q ) = d dt H j. By the previous results we can easily prove all the statements made in Section 1. We note that these residues form an infinite sequence bounded from below, since V j = 0 for j < (2n + 1), according to Eq.(4.4). Since the series V j obey Eq.(3.5), we conclude that the first n+1 residues V (2n+1),...,V (n+2) belong to the kernel of the first Poisson tensor P 0 : [A,V (2n+1)+q ] = 0 q = 0,1,...,n. (4.6) The others verify the Lenard recursion relations Therefore, they also verify the orthogonality relations [A,V n+1+j ] = V j x + [V j,s]. (4.7) [A,V j ],V k = A,[V j,v k ] = 0 (4.8) according to Proposition 2.2. This means that the functions H j are in involution with respect to the symplectic 2 form on S, proving Proposition 1.2. Proposition 1.1 easily follows from the recurrence relations (4.7) according to the Marsden Ratiu reduction scheme. To prove Proposition 1.3 we pass to the study of GD equations. We define these equations as the canonical equations (1.16) on the symplectic leaf S. Let Ṡj denote the corresponding vector fields on S. According to Eq.(4.5), we can write Then, if we remark that Ṡ j = [A,V j ]. (4.9) ((V j ) + ) x + [(V j ) +,S + λa] = ((V j ) ) x [(V j ),S + λa] (4.10) since V j is a solution of the stationary equation (3.5), we can conclude that [A,V j ] = ( (V j ) + )x + [(V j ) +,S + λa], (4.11) 14

15 since only the coefficient of λ 0 can be not zero in both sides of Eq.(4.10), and from the right hand side one sees that it is [A,res λ V j ]. Therefore, we have proved that the GD vector fields Ṡj admit the bihamiltonian representation Ṡ j = [A,V j ] = ( (V j ) + )x + [(V j ) +,S + λa]. (4.12) For sake of brevity, we omit to detail the study of the Hamiltonian functions associated on S to the positive parts (V j ) +. If we combine the results on the residues V j, we arrive at the final picture of the GD hierarchy shown in Fig. 1. In this picture we have divided the infinite sequence I the sequence {V (n+1)p } is not considered.) To each residue V j we have associated the corresponding Hamiltonian density h n+1+j and the vector field Ṡj. Two consecutive n tuples are related by the Poisson tensors P 0 and P 1 as shown in Fig. 1, where ր denotes the action of P 0, while ց represents the action of P 1. Finally, the horizontal arrows stemming from the rightmost column represent the action of the Poisson pencil P λ. of residues V j into classes of n contiguous elements. (Since V (n+1)p = δ 1 p 15

16 . h 1 h 2. h n. h n+2 h n+3. h 2n+1 V (2n+1) V 2n.. V (n+2). V n V n+1. V 1 V 1 V 2. V n Fig ր 0 ց ր. Ṡ n Ṡ n+1.ṡ 1 (V n ) + (V n+1 ) +. (V 1 ) + ց Ṡ 1 (V 1 ) + Ṡ2 (V 2 ) +....Ṡn ր (V n ) + ց. 16

17 5. Noether theorem We now investigate the conservation laws associated to the GD equations. Since these equations are Hamiltonian and their Hamiltonian functions are in involution, we expect that the latter are conserved along the flows of the GD hierarchy. For convenience, we collect all the Hamiltonians H j into the single Laurent series H GD = j (n+2) H j, (5.1) zj+1 which we call the first GD Hamiltonian, and we write all the conservation laws in the concise form H GD t j = 0. (5.2) Since H GD = (n + 1)z n S 1 h(z) dx, (5.3) we expect that the dominant solution h(z) of the generalized Riccati equation n 1 h (n+1) u j h (j) = z n+1 (5.4) j=0 verify local conservation laws of the form We now compute the current densities H (j). h t j = x H (j). (5.5) Proposition 5.1. The dominant solution h(z) of the generalized Riccati equation evolves according to the local conservation laws (5.5) defined by current densities H (j) = g (0) (V j ) + ψ 0. (5.6) Proof. Let us set V = (V j ) +, t = t j, and H = g (0) (V j ) + ψ 0 for simplicity of notations. Then we have the relations ψ 0 x + (S + λa) ψ 0 = h ψ 0 [ x + (S + λa), t + V ] = 0. They entail that where ( x + (S + λa)) φ = h φ h t ψ 0, (5.7) φ = ( t + V ) ψ 0. (5.8) Now we develop φ on the basis { ψ a } of eigenvectors of x + (S + λa), to get φ = n a=0 c a ψ a, where the c a are Laurent series in z. Then (5.7) implies n ( c ax ψ a + c a h a ψ a ) = a=0 17 n hc a ψ a h t ψ 0, (5.9) a=0

18 where h 0 = h, and therefore c ax + c a h a = c a h for all a = 1,...,n. Then the Laurent series c a, for a 0, must fulfill the condition c ax = c a (h a h), and, since h a h has degree 1 in z, this implies that c a = 0. Hence φ = c 0 ψ 0, and c 0 = H for (5.8). Therefore we have the relation φ = H ψ 0, that, plugged in (5.7), implies h t = H x. We now elaborate on the properties of the currents H (j), to show how they can be computed directly from the dominant solution h(z), avoiding the construction of the matrix (V j ) +. The idea is to compare the expansions of the current densities H (j) on the fixed basis {z j } and on the Faà di Bruno basis {h (j) }. We denote by H + = h (0),h (1),h (2),... (5.10) the module over C (S 1 ) spanned by the Faà di Bruno polynomials associated with the series h(z). Proposition 5.2. The current densities H (j) vanish for j 1. For j 0, the current density H (j) has a Laurent expansion of the form H (j) = z j + l 1 H j l z l (5.11) and a truncated Faà di Bruno expansion j 1 H (j) = h (j) + c j ah (a), (5.12) where the coefficients c j a are independent of z. Therefore the currents H (j) are the projections H (j) = π + (z j ) (5.13) of z j onto H +, along the splitting of the space of Laurent series in the direct sum of H + and of the subspace of strictly negative Laurent series. Proof. By definition, H (j) = g (0) (V j ) + ψ 0 = g (0) V j ψ 0 g (0) (V j ) ψ 0 = z j g (0) (V j ) ψ 0, since V 1 ψ 0 = z ψ 0. Moreover, we note that (V j ) = V j 1 λ ; this implies that a=0 g (0) (V j ) ψ 0 = Hj 1 z +... (5.14) since g (k) ψ 0 = h (k) and λ = z n+1. Hence H (j) has the form (5.11). If we put g (0) (V j ) + = n i=0 aj i g(i), then we obtain H (j) = n a j i h(i), (5.15) i=0 where the a j i are polynomials in λ. Now we observe that for all k,i 0, λk h (i) H +, as a consequence of the characteristic equation n 1 λ = h (n+1) u j h (j). (5.16) 18 j=0

19 Indeed, acting on Eq.(5.16) by the operator x + h, we get λh = h (n+2) n u 1 jh (j). (5.17) This shows that λh H +, and so on. Therefore from Eq.(5.15) it is easily seen that H (j) H +, proving Eq.s (5.12) and (5.13). The assertion concerning H (j), for j 1, is a consequence of the previous result. Indeed, suppose that j < 0; then, since H (j) H +, we have H (j) = M i=1 a ih (m i), with m i 0. But from (5.11) we see that H (j) has no term in z k with k 0, and therefore a i = 0 i = 1,...,m. To see how these properties actually determine the currents H (j), let us try to compute, for example, H (2). We start from the expansion and we compute j=0 H (2) = h (2) + ah (1) + bh (0) (5.18) h (2) + ah (1) + bh (0) = (z 2 + 2h 1 ) + az + b + O(z 1 ) (5.19) by using the definition of the Faà di Bruno polynomials. Therefore we have to set so that a = 0, b = 2h 1 (5.20) H (2) = h (2) 2h 1 h (2). To give the general form of the currents H (j), we introduce the infinite matrix h (0) h (1) h (2)... res h(0) z res h(1) z res h(2) z... 0 res h(1) z res h(2) 2 z... 2 H = 0 res h(2) z (5.21) Lemma 5.3. The currents H (j) are (up to a sign) the principal minors of the matrix H. Proof. It is enough to develop the principal minors along the elements of the first row, and to remark that resh (j) z k = 0 for k = 1,2,...,j. This formula coincides with Eq.(5.6) when h(z) is a solution of the generalized Riccati equation (5.4). However, it defines the current densities H (j) as differential 19

20 polynomials in h(z), independently of the assumption that h(z) is a solution or not of the Riccati equation. This means that henceforth we can drop this assumption, and consider the equations h t j = x H (j) (5.21) for a general monic Laurent series h(z). Within this generic situation the equations (5.21) will be called the full set of KP equations.the will be extensively studied in the fourth paper [CFMP4]. 6. The second GD Hamiltonian Now we introduce a second GD Hamiltonian H according to the results on the pairing of the Hamiltonians H C and HC discussed in Section 3. We consider the Laurent series 1 V GD = z j+1 res λ V j, (6.1) j (2n+1) we note that and therefore we define We easily get with h = H GD t = V GD,Ṡ, (6.2) H GD = V GD,A. (6.3) HGD = h dx, S 1 (6.4) j (2n+1) 1 z j+1 (res λ V j,a). (6.5) Just like h(z), the function h (z) verifies a system of conservation laws called the reduced dual KP equations. The following remark, showing how to compute h directly from h rather than from Eq.(6.5), is basic. Lemma 6.1. The dual density h (z) admits the expansion h (z) = 1 j 1 res H (j) z j+1. (6.6) Proof. From the definition of H (j) it follows immediately that res z H (j) = res z g (0) (V j ) + ψ 0 ; (6.7) moreover, recall that H (j) = 0 for j < 0. Hence we have to prove that { resz g (0) (V j ) + ψ 0 for j 1 (V j,a) = 1 for j = 1 20

21 In order to do this let us calculate g (0) (V j ) + ψ 0 = g (0) V j ψ 0 g (0) (V j ) ψ 0 = = z j g (0) (V j ) ψ 0. (6.8) Therefore if j 1 we have res z g (0) (V j ) + ψ 0 = res z g (0) (V j ) ψ 0 = res z g (0) λ 1 res λ (V j ) ψ 0 = = res z g (0) λ 1 V j ψ 0. If we take into account that ψ 0 = e n z n +..., we conclude that res z g (0) λ 1 V j ψ 0 = g (0) V j e n = (V j,a) j 1, since A = e n g (0). We are left with the case j = 1. Eq.(6.8) implies that res z H ( 1) = res z g (0) (V 1 ) + ψ 0 = res z (z 1 g (0) (V 1 ) ψ 0 ) = 1 (V 1,A). But since H ( 1) = 0 we obtain that (V 1,A) = 1. We note once again that the new expression (6.6) of h (z) is formally independent on any constraint on h(z), and therefore it defines the dual density h (z) for any monic Laurent series h(z). The corresponding local conservation laws are called dual KP equations. They will be studied in [CFMP4]. 7. Conclusions In this paper we have defined the GD equations as canonical equations dq a dt j = δh j δp a, dp a dt j = δh j δq a (7.1) on the symplectic manifold (1.1). Their Hamiltonians are defined by the dominant solution H j = (n + 1) h n+1+j dx S 1 (7.2) h(z) = z + j 1 h j z j (7.3) of the generalized Riccati equation n 1 h (n+1) u j (p a,q a )h (j) = z n+1. (7.4) j=0 The coefficients u j (p a,q a ) of this equation are computed according to the Marsden Ratiu reduction scheme explained in Section 1. We have then remarked that the GD equations define a bihamiltonian action of the abelian algebra spanned by the 21

22 powers Λ j on the symplectic manifold (1.1). We have accordingly written the GD equations in the bihamiltonian form Ṡ j = [A,V j ] = ( (V j ) + )x + [(V j ) +,S + λa]. (7.5) Finally, we have used this form to study the conservation laws of the GD equations according to Noether theorem. We have shown that the dominant solution h(z) of the Riccati equation satisfies the conservation laws and that the current densities are given by h t j = x H (j) (7.6) H (j) = g (0) (V j ) + ψ 0. (7.7) We have also shown that these current densities can be computed directly as differential polynomials in h(z), according to the formula H (j) = π + (z j ), (7.8) independently of the assumption that h(z) is a solution of the Riccati equation (7.4). This remark provides a natural bridge with the abstract KP theory dealt with in the next paper [CFMP4]. Finally, we have considered the dual Hamiltonian density h 1 = z j+1 (res λ V j,a). (7.9) j (2n+1) We have shown that it can be computed as a differential polynomial in h(z), h = 1 j 1 1 z j+1 res z H (j), (7.10) and we have argued that it verifies a set of conjugated local conservation laws h t j = x H (j). (7.11) The associated dual current densities H(j) this series. will be computed in the fourth paper of References [CFMP1] P. Casati, G. Falqui, F. Magri, M. Pedroni, The KP theory revisited. I. A new approach based on the principles of Hamiltonian mechanics, submitted to Comm. Math. Phys. (1995). [CFMP2] P. Casati, G. Falqui, F. Magri, M. Pedroni, The KP theory revisited. II. The reduction theory and Adler Gel fand Dickey brackets, submitted to Comm. Math. Phys. (1995). [CFMP4] P. Casati, G. Falqui, F. Magri, M. Pedroni, The KP theory revisited. IV. KP equations, dual KP equations, Baker Akhiezer and τ functions, submitted to Comm. Math. Phys. (1995). 22

23 [D] L. A. Dickey, Soliton Equations and Hamiltonian Systems, Adv. Series in Math. Phys, vol. 12, World Scientific, Singapore, [GD] I. M. Gelfand, L. A. Dickey, Fractional Powers of Operators and Hamiltonian Systems, Funct. Anal. Appl. 10 (1976), [GZ] I. M. Gelfand, I. Zakharevitch, Webs, Veronese curves and bihamiltonian systems, Funct. Anal. Appl. 99 (1991), [MR] J. E. Marsden, T. Ratiu, Reduction of Poisson Manifolds, Lett. Math. Phys. 11 (1986), [SS] M. Sato, Y. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, Nonlinear PDEs in Applied Sciences (US-Japan Seminar, Tokyo) (P. Lax and H. Fujita, eds.), North-Holland, Amsterdam, 1982, pp [ZS] V. E. Zakharov, A. B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, Funct. Anal. Appl. 8 (1974),