GENERALIZED r-matrix STRUCTURE AND ALGEBRO-GEOMETRIC SOLUTION FOR INTEGRABLE SYSTEM

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1 Reviews in Mathematical Physics, Vol. 13, No c World Scientific Publishing Company GENERALIZED r-matrix STRUCTURE AND ALGEBRO-GEOMETRIC SOLUTION FOR INTEGRABLE SYSTEM ZHIJUN QIAO Institute of Mathematics, Fudan University, Shanghai , P. R. China Fachbereich 17, Mathematik Informatik, Universität-GH Kassel, Heinrich-Plett-Str. 40, D Kassel, Germany and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA zhijun@uni-kassel.de, zjqino@yahoo.com Received 12 February 2000 Revised 9 May 2000 The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations NLEEs. Our starting point is a generalized Lax matrix instead of the usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Poisson bracket. It can be clearly seen that various nonlinear constrained c- and restricted r- systems, such as the c-akns, c-mkdv, c-toda, r-toda, c-levi, etc, are derived from the reductions of this structure. All these nonlinear systems have r-matrices, and are completely integrable in Liouville s sense. Furthermore, our generalized structure is developed to become an approach to obtain the algebro-geometric solutions of integrable NLEEs. Finally, the two typical examples are considered to illustrate this approach: the infinite or periodic Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary. Keywords: Lax matrix, r-matrix structure, integrable system, algebro-geometric solution. Mathematics Subject Classification 2000: 35Q53, 58F07, 35Q35 1. Introduction Completely integrable systems are widespreadly applied in fields theory, fluid mechanics, nonlinear optics and other fields of nonlinear sciences. The new development of integrability theory can be roughly divided into three stages. The first one was the direct use of Lax equations for some systems such as the Calogero Moser system [33] and the Euler rigid equation [32], which were allowed for integration. The second one was the so-called algebraization, i.e. the tools 545

2 546 Z. Qiao of Lie algebras, Kac Moody algebras were used to sysmetically construct a large class of soliton equations and integrable systems [3, 43], and simultaneously present the Lax representations of soliton equations and Hamiltonian structures. The third one is being developed and witnessed through the use of nonlinearization method [8] to generate finite dimensional integrable systems. These systems can be the Bargmann system [12, 35], the C. Neumann system [12, 35], the higher-order constrained flows or symmetric constrained flows [4, 5], and the stationary flows of soliton equations [51]. Indeed, with the help of this method, many new completely integrable systems were successively found [12, 35]. In this way, each integrable system is generated through making nonlinearized procedure for a concrete spectral problem or Lax pair, and has its own characteristic property. Then a natural question arises whether or not there is a unified structure such that it can contain those concrete integrable systems? Recently, the study of r-matrix for nonlinear integrable systems brings a great hope to dealing with this problem. Semenov Tian Shansky ever gave the definition of r-matrix [45], and used the r-matrix to construct Lie algebra and new Poisson bracket [44] in a given Lie algebra and corresponding coadjoint orbit. The main idea of Semenov Tian Shansky and Reyman was how to obtain the new Poisson bracket from a given r-matrix and an element of Lie algebra. Here our thought is how to present r-matrix structure from a given Lax matrix and the standard Poisson bracket: L, {, }? = r-matrix. In the present paper, we give a sure answer for the above question. We propose an approach to generate finite dimensional integrable systems by beginning with the so-called generalized Lax matrix instead of the usual Lax pair. Another main result of this paper is to deal with the algebro-geometric solutions of integrable nonlinear evolution equations NLEEs. It is well-known that the ideal aim for nonlinear equations is to obtain their explicit solutions. According to the nonlinearization method, solutions of integrable NLEEs can have the parametric representations [10] or involutive representations [11], and also have numeric representations in the discrete case [42]. However, these representations of solutions are not given in an explicit form. Thus, an open question is how to obtain their explicit forms. In the paper we would like to give solutions of integrable NLEEs in the form of algebro-geometric Θ-functions. The algebro-geometric solutions for some soliton equations with the periodic boundary value problems were known since the works of Lax [28], Dubrovin, Mateev and Novikov [20]. Similar results for the periodic Toda case were obtained slightly later by Date and Tanaka [14]. Afterwards, the relations between commutative rings and ordinary linear periodic differential operators and between algebraic curves and nonlinear periodic difference equations were discussed by Krichever [27]. The technique they used is the Bloch eigenfunctions, the spectral theory of linear periodic operators, and some analysis tools on Riemann surfaces.

3 Generalized r-matrix Structure and Algebro-Geometric Solution 547 In the first example of this paper, we give the algebro-geometric solution of the periodic or infinite Toda lattice equation. Our method is a constrained approach, connecting finite dimensional integrable systems with integrable NLEEs, instead of the usual spectral techniques and Bloch eigenfunctions which are often available to the periodic boundary problems. The results with the periodic boundary conditions are included in ours. In the second example, we consider the well-known AKNS equation. The Ablowitz Kaup Newell Segur AKNS equations are a very important hierarchy [1] of NLEEs in soliton theory. It can turn out that the KdV, MKdV, NLS, sine-gordon, sinh-gordon equations etc. All these equations can be solvable by the inverse scattering transform IST [24], and usually have N-soliton solutions [2]. But the algebro-geometric solutions of the AKNS equations seem not to be obtained. We shall deal with this problem by using our constrained procedure. The considered AKNS equation is under the case of decay at infinity or periodic boundary condition. The whole paper is organized as follows. We first introduce a generalized Lax matrix in the next section, then construct a generalized r-matrix structure and a generalized set of involutive Hamiltonian functions in Sec. 3. All those Hamiltonian systems have Lax matrices, r-matrices, and are therefore completely integrable in Liouville s sense. In Sec. 4 it can be clearly seen that various nonlinear constrained c- and restricted r- integrable flows, such as the c-akns, c-mkdv, c-toda, r-toda, c-wki, c-levi, etc, can be derived from the reductions of this structure. Moreover, the following interesting facts are given in Secs. 5, 6, 7, respectively: Several pairs of different integrable systems share the same r-matrices with the good property of being non-dynamical i.e. constant. In particular, a discrete and a continuous dynamical system possess the common Lax matrix, r-matrix, and even completely same involutive set. Additionally, on a symplectic submanifold integrability of the restricted Hamiltonian flow for continuous case and symplectic map for discrete case are described by introducing the Dirac- Poisson bracket. They also have the same r-matrix but being dynamical. A pair of constrained integrable systems, produced by two gauge equivalent spectral problems, possesses different r-matrices being non-dynamical. New integrable systems are generated through choosing new r-matrices from our structure, and the associated spectral problems are also new. In the last section, as a development of the generalized structure, through considering the relation lifting finite dimensional system to infinite dimensional system and using the algebro-geometric tools we present an approach for obtaining the algebro-geometric solution of integrable NLEEs. To illustrate the procedure we take the periodic or infinite Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary as the examples. Before displaying our main results, let us first give some necessary notation: dp dq stands for the standard symplectic structure in Euclidean space R 2N = {p, q p =p 1,...,p N,q =q 1,...,q N };, is the standard inner product in

4 548 Z. Qiao R N ; in the symplectic space R 2N,dp dq the Poisson bracket of two Hamiltonian functions F, G is defined by [6] N F G {F, G} = F G F = q i=1 i p i p i q i q, G F p p, G ; 1.1 q I and stand for the 2 2 unit matrix and the tensor product of matrix, respectively; λ 1,...,λ N are N arbitrarily given distinct constants; λ, µ are the two different spectral parameters; Λ = diagλ 1,...,λ N, I 0 = q, q, J 0 = p, q, K 0 = p, p, I 1 = Λp, p Λq, q, J 1 = Λp, q, a 0, a 1 =const.. Denote all infinitely times differentiable functions on real field R by C R. 2. A Generalized Lax Matrix Consider the following matrix called Lax matrix Aλ Bλ Lλ = Cλ Aλ where Aλ =a 2 I 1,J 1 λ 2 + a 1 J 0 λ 1 + a 0 + a 1 λ + Bλ =b 1 I 0,J 0 λ 1 + b 0 J 0 Cλ =c 1 J 0,K 0 λ 1 + c 0 J 0 + N N N 2.1 p j q j λ λ j, 2.2 q 2 j λ λ j, 2.3 p 2 j λ λ j, 2.4 with some undetermined functions a 2, a 1, b 1, c 1, b 0, c 0 C R. Now, in order to produce finite dimensional integrable systems directly from the Lax matrix 2.1, we need an inevitable assumption. Assumption A: {Aλ,Aµ}, {Aλ,Bµ}, {Aλ,Cµ}, {Bλ,Bµ}, {Bλ, Cµ}, and{cλ,cµ} are expressed as some linear combinations of Aλ, Aµ, Bλ, Bµ, Cλ, Cµ with the cofficients in C R. Then we have the following lemma. Lemma 2.1. Under Assumption A, Lλ only contains the following cases: 1 If a 2 const., a 0 = b 0 = c 0 = a 1 = 0, a 1 = J 0, b 1 = I 0, and c 1 = K 0, then a 2 satisfies the relation I 1 =J 1 + a fa 2 ; if a 2 =const. 0and a 0 = b 0 = c 0 = a 1 =0, then a 1 = J 0, b 1 = I 0, c 1 = K 0,ora 1 =const., b 1 = I 0 + f 1 J 0,c 1 = K 0 + g 1 J 0, where f 1, g 1 satisfy the relation f 1 g 1 = J 2 0 2a 1J 0 +const..

5 Generalized r-matrix Structure and Algebro-Geometric Solution a 2 = a 1 = b 1 = c 1 = b 0 = c 0 = a 1 =0, and a 0 =const. 3 a 2 = b 1 = a 0 = b 0 = c 0 = a 1 =0, c 1 = K 0, and a 1 satisfies a a 2 = a 1 = b 1 = c 1 = b 0 = a 1 =0, a 0 =const., and c a 2 = c 1 = b 0 = a 1 =0,a 1,a 0 =const., b 1 = I 0 + gj 0, and c 0 satisfies c 0 g = 2a 0. 6 a 2 = c 1 = a 0 = b 0 = c 0 = a 1 =0, a 1 = J 0 +const., and b 1 = I 0. 7 If a 2 = a 0 = b 0 = c 0 = a 1 =0, then there are the following five subcases: i a 1 =const., c 1 = K 0 + f 2 J 0, and b 1 = I 0 + g 2 J 0 ; ii a 1 = J 0, b 1 = I 0, and c 1 = K 0 ; iii a 1 = J 0 +const., and b 1 = b 1 J 0,c 1 = c 1 J 0 satisfy b 1 c 1 =2a 1 ; iv a 1 = J 0 +const., b 1 = I 0, and c 1 = c 1 J 0 ; v a 1 = J 0 +const., c 1 = K 0, and b 1 = b 1 J 0. 8 a 2 = a 1 = b 1 = c 1 =0,a 0, a 1 =const., b 0 0, c 0 0, and b 0,c 0 satisfy the relation b 0 c 0 = 2a 1. 9 a 2 = a 1 = b 1 = c 1 =0,c 0, a 1,a 0 =const., and b a 2 = b 1 = c 0 = a 1 =0,a 1, a 0 =const., c 1 = K 0 + hj 0, and b 0 satisfies the relation b 0 h = 2a 0. The above all functions f, g, h, f i, g i i =1, 2 are in C R, and means d dj 0. Proof. Through some calculations we have {Aλ,Aµ} =2 a 2 λ Λp, p I 1 µ 2 Bλ µ λ 2 Bµ +2 a 2 λ Λq, q I 1 µ 2 Cλ µ λ 2 Cµ +2 a 2 I 1 +2 a 2 I 1 1 λ 2 1 µ 2 µ λ 2 λ µ 2 {Bλ,Bµ} =2 b 1 1 J 0 µ Bλ 1 λ Bµ 1 +2 λ 1 µ {Cλ,Cµ} =2 c 1 J 0 +2 Λp, p b 1 I 0 Λq, q c 1 + K 0 Λp, p b 0 + Λq, q c 0, +2 db 0 dj 0 Bλ Bµ b 1 I 0 db 0 dj 0 I 0 + b 0 b 1 J 0 b 1 db 0 dj 0 1µ Cλ+ 1λ Cµ +2 dc 0 1 λ + 1 µ, dj 0 Cλ+Cµ c 1 K 0 dc 0 dj 0 K 0 + c 0 c 1 J 0 c 1 dc 0 dj 0,

6 550 Z. Qiao {Aλ,Bµ} = 2 λ µ Bµ+Bλ 2 λ + 2 b 1 µ Bλ 2µ I 0 λ 2 2 b 1 I 0 2 Λq, q +2 b 1 I 0 2 a 2 da 1 dj 0 Bµ Bµ+2 a 2 Λq, q Aµ J 1 I 1 a 2 J 1 + b 1 a 2 a 2 +2a 2 I 1 I 0 J 1 I 1 da 1 b 1 da 1 I 0 b 1 + b 1 + b 1 dj 0 I 0 dj 0 2 Λq, q a 2 I 1 a 2 2 b 0 +2a 0 Λq, q a 2 J 1 I 1 J 0 + a 1 + a 2 J 1 I 0 b 1 λ 2 µ λ 2 µ 1 λ 1 µ 1 λ 2 +2b 0 da 1 dj 0 λ 1 2b 0 b 1 I 0 µ 1 4a 1 Λq, q a 2 I 1 λ 2 µ 2, {Aλ,Cµ} = 2 λ µ Cµ Cλ + 2 λ + 2 c 1 Cλ+ 2µ µ K 0 λ 2 +2 Λp, p 2 c 1 c a 2 a 2 K 0 I 1 da 1 dj 0 Cµ Cµ+2 a 2 Λp, p Aµ J 1 I 1 J 1 + c 1 a 2 a 2 2a 2 K 0 J 1 I 1 da 1 c 1 da 1 K 0 c 1 c 1 c 1 K 0 dj 0 K 0 dj 0 2 Λp, p a 2 I 1 a 2 2 c 0 +2a 0 Λp, p a 2 J 1 I 1 J 0 + a 1 + a 2 J 1 K 0 + c 1 λ 2 µ λ 2 µ 1 λ 1 µ 1 λ 2 2c 0 da 1 dj 0 λ 1 2c 0 c 1 K 0 µ 1 4a 1 Λp, p a 2 I 1 λ 2 µ 2, b 1 I 0 Aµ {Bλ,Cµ} = 4 λ µ Aµ+Aλ + 4 λ 4 c 1 1 c 1 Aλ+2 + dc 0 µ K 0 µ J 0 dj 0 +4 b 1 +1 a 2 µ 2 λ 1 +4 I 0 1 Bλ+2 λ c 1 K 0 +1 b 1 + db 0 Cµ J 0 dj 0 a 2 λ 2 µ 1

7 Generalized r-matrix Structure and Algebro-Geometric Solution 551 c 1 b 1 +2 I 0 +2 b 1 c 1 J 0 + b 1 c 1 K 0 J 0 I 0 I 0 K 0 J 0 K 0 b 1 c 1 c 1 b 1 2 b 1 a 1 +2 c 1 a 1 +2a 1 λ 1 µ 1 J 0 J 0 I 0 K 0 +2 b 1 +2 c 1 c 0 + b 1 dc 0 I 0 dc 0 b 1 2 b 1 a 0 J 0 I 0 dj 0 dj 0 I 0 b 0 + c 1 db 0 K 0 db 0 c 1 +2 c 1 a 0 J 0 K 0 dj 0 dj 0 K 0 db0 2 c 0 + dc 0 b 0 +2a 1 dj 0 dj 0 λ 1 µ 1 +4 c 1 K 0 a 1 µ 1 λ 4 b 1 I 0 a 1 λ 1 µ. According to Assumption A, the terms that do not contain Aλ, Aµ, Bλ, Bµ, Cλ, Cµ in the above six equalities, are zero. After discussing these terms, we can obtain every result in Lemma Generalized r-matrix Structure and Integrable Hamiltonian Systems Let L 1 λ =Lλ I, L 2 µ =I Lµ. In the following, we search for a general 4 4 r-matrix structure r 12 λ, µ such that the fundamental Poisson bracket [21]: {Lλ,Lµ} =[r 12 λ, µ,l 1 λ] [r 21 µ, λ,l 2 µ] 3.1 holds, where r 21 λ, µ =Pr 12 λ, µp, P = i=0 σ i σ i,andσ i s are the standard Pauli matrices. For the given Lax matrix 2.1 and the Poisson bracket 1.1, we have the following theorem. Theorem 3.1. Under Assumption A, r 12 λ, µ = 2 µ λ P + S 3.2 is an r-matrix structure satisfying 3.1, where 2λ a 2 µ da 1 2 b 1 J 1 µ dj 0 µ J 0 2 dc 0 0 S = dj 0 2λ µ 2 Λp, p a 2 2 I 1 µ 0 2λ µ 2 Λq, q a 2 0 I 1 2 c 1 µ K 0 2λ µ 2 Λq, q a 2 I 1 b db 0 I 0 dj 0 2λ µ 2 Λp, p a 2 I 1 2 µ Proof. Under Assumption A, we have c 1 J 0 2λ µ 2 a 2 J µ da 1 dj 0.

8 552 Z. Qiao {Aλ,Aµ} =2 a 2 λ Λp, p I 1 µ 2 Bλ µ λ 2 Bµ +2 a 2 λ Λq, q I 1 µ 2 Cλ µ λ 2 Cµ {Bλ,Bµ} =2 b 1 J 0 {Cλ,Cµ} =2 c 1 J 0 1 µ Bλ 1 λ Bµ, +2 db 0 dj 0 Bλ Bµ, 1µ Cλ+ 1λ Cµ +2 dc 0 dj 0 Cλ+Cµ, da 1 {Aλ,Bµ} = 2 λ µ Bµ+Bλ 2 Bµ+ 2 λ dj 0 µ 2µ a 2 λ 2 Bµ 2 a 2 Λq, q Aµ, J 1 I 1 da 1 b 1 I 0 Bλ c 1 {Aλ,Cµ} = 2 λ µ Cµ Cλ + 2 Cµ+ 2 Cλ λ dj 0 µ K 0 + 2µ a 2 λ 2 Cµ+2 a 2 Λp, p Aµ, J 1 I 1 b 1 c 1 {Bλ,Cµ} = 4 λ µ Aµ+Aλ + 4 Aµ 4 Aλ λ I 0 µ K 0 1 c dc 0 1 b 1 Bλ+2 + db 0 Cµ, µ J 0 dj 0 λ J 0 dj 0 which complete the proof. In general, Eq. 3.2 is a dynamical r-matrix structure, i.e. dependent on canonical variables p i,q i [7]. Now, we turn to consider the determinant of Lλ where det Lλ = 1 2 Tr L2 λ =A 2 λ+bλcλ = 2 N H i λ i E j +, 3.3 λ λ j i= 4 H 4 = a 2 2, 3.4 H 3 =2a 2 a 1, 3.5 H 2 = a a 2 a 0 + b 1 c 1 2a 2 Λ 1 p, q, 3.6

9 Generalized r-matrix Structure and Algebro-Geometric Solution 553 H 1 =2a 2 a 1 +2a 1 a 0 + b 1 c 0 + b 0 c 1 2a 2 Λ 2 p, q 2a 1 Λ 1 p, q b 1 Λ 1 p, p + c 1 Λ 1 q, q, 3.7 H 0 = a a 1 a 1 + b 0 c 0 +2a 1 p, q, 3.8 H 1 =2a 0 a 1, 3.9 H 2 = a 2 1, 3.10 E j =2a 2 λ 2 j Γ j = +b 1 λ 1 j N k=1,k j +2a 1 λ 1 j +2a 0 +2a 1 λ j p j q j + b 0 p 2 j c 1λ 1 j + c 0 q 2 j Γ j, 3.11 p j q k p k q j 2 λ j λ k, j =1, 2,...,N Let Eq. 3.3 be multiplied by a fixed multiplier λ k k Z, then it leads to 1 2 k 1 N 2 λk Tr L 2 λ = H l λ l+k + F i λ k 1 i λ k j + E j λ λ j where F m = = l= 4 1 l=k 4 l=k i=0 k 1 H l k λ l + H l k + F k 1 l λ l l=0 k+2 N + H l k λ l λ k j + E j, 3.13 λ λ j N λ m j E j, m =0, 1, 2,..., 3.14 which read F m =2a 2 Λ m 2 p, q +2a 1 Λ m 1 p, q +2a 0 Λ m p, q +2a 1 Λ m+1 p, q + b 1 λ m 1 p, p + b 0 Λ m p, p c 1 Λ m 1 q, q c 0 Λ m q, q Λ i p, p Λ j q, q Λ i p, q Λ j p, q i+j=m 1 Because there is an r-matrix structure satisfying Eq. 3.1, one can obtain {L 2 λ,l 2 µ} =[ r 12 λ, µ,l 1 λ] [ r 21 µ, λ,l 2 µ], 3.16 where 1 1 r ij λ, µ = L 1 k 1 λl 1 l 2 µ r ij λ, µ L k 1 λll 2 µ, k=0 l=0 i =12,j =

10 554 Z. Qiao Thus, 4{Tr L 2 λ, Tr L 2 µ} =Tr{L 2 λ,l 2 µ} = So, by Eq we immediately obtain Theorem 3.2. Under Assumption A, the following equalities {E i,e j } =0, {H l,e j } =0, {F m,e j } =0, 3.19 i, j =1, 2,...,N, l= 4,...,2, m =0, 1, 2,..., hold. Hence, the Hamiltonian systems H l and F m H l :q x = H l p, p x = H l q, l = 4,...,2, F m :q tm = F m p, p t m = F m, m =0, 1, 2,..., q are completely integrable in Liouville s sense Corollary 3.1. All composition functions fh l,f m,f C R are completely integrable Hamiltonians in Liouville s sense. 4. Reductions For the various cases of Lemma 2.1, we give the corresponding reductions of r-matrix structure r 12 λ, µ in this section. The following numbers of title coincide with the ones in Lemma 2.1, i.e. the corresponding conditions are coincidental. Before giving our reductions, we d like to re-stress the two terminologies used usually in the theory of integrable systems in order to avoid some confusions: one is constrained system, which means the finite dimensional Hamiltonian system or symplectic map in R 2N under the Bargmann-type constraint; the other restricted system, which means the finite dimensional Hamiltonian system or symplectic map on some symplectic submanifold in R 2N under the Neumann-type constraint. In the future we shall follow this principle. 1 r 12 λ, µ = 2λ µµ λ P +2 a 2 J 1 λ µ 2 S +2 a 2 I 1 λ Q, 4.1 µ Λq, q S = , Q = Λq, q Λp, p Λp, p 0 0

11 Generalized r-matrix Structure and Algebro-Geometric Solution 555 Particularly, with fa 2 = 1, Eq. 4.1 exactly reads as the r-matrix of the constrained WKI c-wki system. With a 2 =const. 0, Eq. 3.2 reads as the r-matrix r 12 λ, µ = P of ellipsoid geodesic flow [26], or reads 2λ µµ λ r 12 λ, µ = 2 µ λ P + 2 S, S= µ 0 f , 0 0 g 1 0 which is a new r-matrix structure. For simplicity, below write = 2 d dj 0. r 12 λ, µ = 2 P. 4.2 µ λ This is nothing but the r-matrix of the well-known constrained AKNS c-akns system [8]. 3 a r 12 λ, µ = 2 µ λ P + 2 S, S= µ ,a a 1 In particular, with a 1 = J 0, Eq. 4.3 reads as the r-matrix of the constrained LZ c-lz system [12] r 12 = 2 µ λ P + c 0S, S= With c 0 = 2 J 0, Eq. 4.4 reads as the r-matrix of the constrained Hu c-h system [12]. 5 r 12 λ, µ = 2 P + S, S= µ λ 1 0 µ g 0 0 c µ µ g 0 0 c 0 0

12 556 Z. Qiao With b 1 = I 0,c 0 =const., Eq. 4.5 reads as the r-matrix of the constrained Qiao c-q system [38]. 6 This is a new r-matrix. 7i r 12 λ, µ = 2 µ λ P + 2 S, S= µ r 12 λ, µ = 2 µ λ P + 2 S, S= µ This is also a new r-matrix. ii r 12 λ, µ = g f λ P. 4.8 µµ λ This is the r-matrix of the constrained Heisenberg spin chain c-hsc system [39]. iii r 12 λ, µ = 2 µ λ P S, S= µ b c 1 1 With b 1 = 1, c 1 =1, Eq. 4.9 becomes the r-matrix of the constrained Levi c-l system [35]. iv r 12 λ, µ = 2 µ λ P 2 S, S= µ c 1 1

13 Generalized r-matrix Structure and Algebro-Geometric Solution 557 This is a new r-matrix. v r 12 λ, µ = 2 µ λ P S, S= µ b This is also a new r-matrix. 8 r 12 λ, µ = 2 P + S, S= µ λ 0 2b c With b 0 = c 0 = J 0,a 1 = 1 2, Eq reads as the r-matrix of the constrained Tu c-t system [12] r 12 λ, µ = 2 µ λ P + S, S= b With b 0 = J 0, Eq reads as the common r-matrix of the constrained Toda c-toda system a discrete system and the constrained CKdV c-ckdv system a continuous system, which will be seen in Sec b r 12 λ, µ = 2 1 P + S, S= µ h µ µ λ b µ h 0 With h =const., b 0 =0, Eq reads as the r-matrix of the constrained MKdV c-mkdv system [37]. Proof. For simplicity, we only present the proof in Cases 1 and 10, other cases are similar.

14 558 Z. Qiao Case 1. With a 2 const., the matrix S becomes 2λ a 2 µ 2 2 2λ 0 J 1 µ µ 2 Λq, q a 2 0 I S = µ 2λ µ 2 Λp, p a I 1 µ 2λ 0 µ 2 Λp, p a 2 0 I 1 2λ µ 2 Λq, q a 2 I 1 2λ µ 2 a 2 J 1 2 µ. Substituting S into Eq. 3.2 and sorting it, we can obtain Eq. 4.1, where a 2 satisfies the relation I 1 =J 1 + a fa 2, for any fa 2 C R. Particularly, choosing fa 2 = 1 yields a 2 = 1+ Λp, p Λq, q Λp, q. Thus Eq. 4.1 reads r 12 λ, µ = 2λ µµ λ P 2λ µ 2 S + λ 1 µ 2 Q, 1+ Λp, p Λq, q while the corresponding Lax matrix Lλ becomes l 11 q, q λ 1 N 1 pj q j qj 2 Lλ = +, p, p λ 1 l 11 λ λ j p j q j where l 11 = 1+ Λp, p Λq, q Λp, q λ 2 p, q λ 1. p 2 j r WKI Set an auxiliary matrix M 1 as follows λ M 1 = M 1 λ = Λp, p λ 1+ Λp, p Λq, q Λq, q λ 1+ Λp, p Λq, q. λ Then the Lax equation L x =[M 1,L] is equivalent to the following finite dimensional Hamilton system H 4 : Λq, q q x = Λq + Λp = H 4, 1+ Λp, p Λq, q p H 4 : Λp, p p x =Λp Λq = H 4, 1+ Λp, p Λq, q q 4.15 with H 4 = a 2 = Λp, q + 1+ Λp, p Λq, q,

15 Generalized r-matrix Structure and Algebro-Geometric Solution 559 which is obviously integrable by Theorem 3.2. Let Λq, q Λp, p u =, v = Λp, p Λq, q 1+ Λp, p Λq, q Then H 4 is nothing but the WKI spectral problem [49] λ λu y x = y λv λ with the above two constraints 4.16, λ = λ j,andy =q j,p j T, j =1,...,N. That means that r WKIisther-matrix of the integrable constrained WKI c-wki system Other subcases in Case 1 can be similarly proven. The proof of Case 10 can be found in Ref. [37]. In this case, the corresponding constrained system is reduced to the well-known MKdV spectral problem [48]. Remark 4.1. From the above formulae , the r-matrices of Cases 2, 6, and 7ii are non-dynamical. But in fact, for other cases we can also obtain non-dynamical r-matrices if choosing some special functions, for instance, in Eq. 4.3 setting a 1 such that a 1 =const. leads to a non-dynamical one. Of course, we can also get dynamical r-matrices, for instance, in Eq. 4.4 choosing c 0 = 2 J 0 yields a dynamical one. Remark 4.2. Equations cover most r-matrices of 2 2 constrained systems. But among them there are also some new r-matrices and finite dimensional integrable systems like Cases 6, 7i, 7iv, and 7v also see Sec. 7. Their Lax matrices are altogether unified in Eq So, quite a large number of finite dimensional integrable systems are classified or reduced from the viewpoint of Lax matrix and r-matrix structure. 5. Different Systems Sharing the Same r-matrices In the above r-matrices, we find some pairs of different integrable systems sharing the common r-matrices. Now, we present these results as follows The constrained Toda and CKdV flows Let us consider the following 2 2 traceless Lax matrix [36] corresponding to Case 9 in Sec. 4 1 λ p, q L TC = L TC λ, p, q = 2 + L λ ATC λ B TC λ, 5.1 C TC λ A TC λ

16 560 Z. Qiao where N 1 pj q j p 2 j L 0 = L 0 λ, p, q =. 5.2 λ λ j qj 2 p j q j The determinant of Eq. 5.1 leads to 1 2 λ TrLTC 2 λ = 1 2 λ3 + p, q λ +2H C + N Ej TC, 5.3 λ λ j Ej TC = λ j p j q j p 2 j p, q q2 j Γ j, j =1, 2,...,N, 5.4 where Γ j is defined by Eq and the Hamiltonian function H C is H C = 1 2 p, p Λq, p 1 q, q p, q Viewing the variables q and p as the functions of continuous variables x, wehave the following Hamiltonian canonical equation H C : = 1 2 Λp + 1 q, q p + p, q q, 2 q x = H C p = p Λq q, q q, which is nothing but the coupled KdV CKdV spectral problem [29] 1 2 λ u v ψ x = λ 1 ψ u p x = H C q with the two constraints Bargmann-type u = q, q, v = p, q, 5.8 λ=λ j and ψ =p j,q j T.So,H C coincides with the constrained CKdV c-ckdv flow. Let us consider endowing with an auxiliary 2 2 matrix M T as follows 0 g M T = 1 λ q, q, g 2 = Λq, q p, q q, q g g Then, we have the following theorem. Theorem 5.1. The discrete Lax equation L TC M T = M T L TC, L TC = L TC λ, p,q 5.10 is equivalent to a finite dimensional symplectic map H T : R 2N R 2N, p, q p,q, which is called the constrained Toda c-toda flow: p = gq, q Λq p q, q q 5.11 =. g

17 Generalized r-matrix Structure and Algebro-Geometric Solution 561 Proof. Through direct calculations, one can readily show and H T dp dq =dp dq. When we understand the above two matrices L TC and M T in the following sense: L TC L TC M Tn i.e. q q n,p p n,herenis the discrete n+1, M T variable, and set { un = ± Λqn,qn pn,qn qn,qn 2 1 2, 5.12 v n = q n,q n, the constrained Toda flow 5.11 is none other than the well-known Toda spectral problem Lψ n E 1 u n + v n + u n Eψ n = λψ n, Ef n = f n+1, E 1 f n = f n with the above constraint 5.12, λ = λ j and ψ n = q n,j. Theorem 5.1 shows that the constrained Toda flow H T has the discrete Lax representation Equation 5.12 is a kind of discrete Bargmann constraint [42] of the Toda spectral problem The Hamiltonian systems H T andh C share the common Lax matrix 5.1. Thus, they have the following same r-matrix: r 12 λ, µ = 2 µ λ P S, S = , which is proven to satisfy the classical Yang Baxter equation YBE [r ij,r ik ]+[r ij,r jk ]+[r kj,r ik ]=0, i,j,k =1, 2, The restricted Toda and CKdV flows Let us now consider the case on a symplectic manifold. We restrict the Toda and CKdV flows on the following symplectic submanifold M in R 2N M = {q, p R 2N F q, q 1=0,G q, p 12 } = Let us first introduce the Dirac bracket {f,g} D = {f,g} + 1 {f,f}{g, g} {f,g}{f, g}, which is easily proven to be a Poisson bracket on M. According to the idea of Ref. [36], the following Lax matrix 1 0 L TC R = L TC R λ, p, q = 2 AR λ B R λ + L C R λ A R λ

18 562 Z. Qiao yields where λ 2 det L TC R = 1 2 λ2 Tr L TC R 2 = 1 4 λ2 + p, q λ +2H C R + N λ 2 j ETC R,j λ λ j, 5.19 H C R = 1 2 Λp, q 1 2 q, q p, p p, q 2, 5.20 ER,j TC = ETC R,j p, q =p jq j Γ j, i =1,...,N, 5.21 and L 0 is defined by Eq An important observation is: if we consider the Hamiltonian canonical equation produced by Eq in R 2N, then this equation is exactly the well-known constrained AKNS flow, which will be discussed in the next subsection. Now, we first consider the Hamiltonian canonical equation restricted on M : HR C :q x = {q, HR C } D, p x = {p, HR C } D, 5.22 which reads as the following finite dimensional system: p x = 1 2 Λp + 1 Λq, q 1p + p, p q, 2 q x = p Λq 1 Λq, q 1q, q, q =1, q, p = 1 2. This is actually the CKdV spectral problem 5.7 with the two constraints Neumann-type [13] u = Λq, q 1, v = p, p, 5.24 and λ = λ j, ψ =p j,q j, j =1, 2,...,N.So,thefinite dimensional system 5.23 coincides with the restricted CKdV r-ckdv flow. Let us return to the Lax matrix After endowing with an auxiliary matrix M T,R as follows 0 a M T,R = 1 a λ b a a 2 = Λq p, Λq p + Λq, q Λq, q 2,, 5.25 we have the following theorem. b = Λq, q 1, Theorem 5.2. The discrete Lax equation L TC R M T,R = M T,R L TC R, LTC R = L TC R λ, p,q 5.26

19 Generalized r-matrix Structure and Algebro-Geometric Solution 563 is equivalent to a discrete Neumann type of finite dimensional symplectic map H T : p, q T p,q T p = aq, q = a 1 Λq p bq, q, q =1, q, p = , which is called the restricted Toda r-toda flow. Remark 5.1. If we understand the above two matrices L TC R and M T,R in the following sense: L TC R L TC R n+1, M T,R M T,R n i.e. q q n,p p n,a a n,b b n,herenis the discrete variable, then the restricted Toda flow 5.27 on the symplectic submanifold M = {q, p R 2N q, q =1, q, p = 1 2 } is nothing but the discrete Neumann system studied by Ragnisco [41]. Let L TC R1 = LTC R λ, p, q I and LTC R2 = I LTC R µ, p, q. Then, under the Dirac bracket 5.17 we obtain the following theorem. Theorem 5.3. The Lax matrix L TC R λ, p, q defined by Eq satisfies the following fundamental Dirac Poisson bracket {L TC R λ,l TC R µ} D =[r 12 λ, µ,l TC R1 λ] [r 21 µ, λ,l TC R2 µ] 5.28 with a dynamical r-matrix where r 12 λ, µ = 2 µ λ P S 12λ, µ, r 21 µ, λ =Pr 12 µ, λp, 5.29 CR µ 0 S 12 =E 11 E 22 E 12 + E BR µ 0 2AR µ + E 12 + E C R µ 0 0 C R µ and P = 1 2 I + 3 σ j σ j is the permutation matrix, σ j j =1, 2, 3 are the Pauli matrices, and E ij stands for the 2 2 matrix with the i-th line and jth column element 1 and other elements 0. This theorem ensures that Eq satisfies {ER,i TC,ETC R,j } D =0,i,,...,N For the r-toda flow 5.27, we have ER,i TCp,q = ER,i TC p, q as well as n i=1 ETC R,i = p, q = 1 2 from the discrete Lax equation Thus, in the set {ER,j TC}N, only ETC R,1, ETC R,2,...,ETC R,N 1 are independent on M. Theorem 5.4. The restricted Toda flow H T is completely integrable, and its independent and invariant N 1-involutive system is {ER,i TC}N 1 i=1.

20 564 Z. Qiao For the restricted CKdV flow on M, we have HR C = 1 N λ j ER,j TC which implies {HR C,ER,j TC } D =0, j =1, 2,...,N Thus, the following theorem holds. Theorem 5.5. The r-ckdv flow HR C is completely integrable, and its independent N 1-involutive system is also {ER,k TC}N 1 k=1. Remark 5.2. As shown in this and last subsection, the r-toda i.e. Neumanntype and the r-ckdv flows, and the c-toda i.e. Bargmann-type and the c-ckdv flows respectively share the completely same Lax matrix, r-matrix and involutive conserved integrals. Thus, we say that the finite dimensional integrable CKdV flow both restricted and constrained is the interpolating Hamiltonian flow of invariant of the corresponding Toda integrable symplectic map The constrained AKNS and Dirac D flows Fromnowonweassume: N 1 pj q j qj 2 L 0 = L 0 λ, p, q = λ λ j p 2 j p j q j Let us again consider Eq. 5.18, and rewrite it as the following version: 1 0 L AKNS = L AKNS λ, p, q = + L 0, while we introduce 0 1 L D = L D λ, p, q = + L Then we have 1 2 λ2 TrL AKNS 2 λ =λ 2 +2λ p, q + p, q 2 +2H AKNS λ2 TrL D 2 λ =λ 2 + λ q, q + p, p N 1 4 p, p + q, q 2 +2H D + λ 2 j EAKNS j λ λ j, 5.37 N λ 2 j ED j λ λ j, 5.38

21 Generalized r-matrix Structure and Algebro-Geometric Solution 565 where H AKNS = Λp, q 1 q, q p, p, H D = 1 2 Λq, q + Λp, p p, q 2 q, q p, p p, p + q, q 2, 5.40 E AKNS j =2p j q j Γ j, j =1,...,N, 5.41 Ej D = p 2 j + qj 2 Γ j,,...,n Thus, H AKNS and H D generate the following two Hamiltonian systems q x = H AKNS = q, q p +Λq, p H AKNS : p x = H 5.43 AKNS = p, p q Λp ; q q x = H D p = p, q q + 1 p, p q, q p +Λp, 2 H D : p x = H D = p, q p q 2 p, p q, q q Λq. It can be easily seen that H AKNS andh D are changed to the well-known Zakharov Shabat AKNS spectral problem [52] λ u y x = y 5.45 v λ and the Dirac spectral problem [30] v λ u y x = y 5.46 λ u v with the constraints u = q, q, v = p, p, λ = λ j, y =q j,p j T,andtheconstraints u = 1 2 p, p q, q, v = p, q, λ = λ j,y=q j,p j T, respectively. Therefore H AKNS andh D coincide with the constrained AKNS c-akns system and the constrained Dirac c-d system, respectively. Let L J 1 λ =L J λ I, L J 2 µ =I L J µ J = AKNS, D. Then we have the following theorem. Theorem 5.6. The Lax matrices L J λ J = AKNS, D defined by Eq and Eq satisfy the fundamental Poisson bracket {L J λ,l J µ} =[r 12 λ, µ,l J 1 λ] [r 21µ, λ,l J 2 µ] Here the r-matrices r 12 λ, µ, r 21 µ, λ are exactly given by the following standard r-matrix r 12 λ, µ = 2 µ λ P, r 21µ, λ =Pr 12 µ, λp, 5.48

22 566 Z. Qiao P = = 1 I σ i σ i So, the c-akns and c-d flows share the same standard r-matrix 5.48, which is obviously non-dynamical. However, the two constrained flows, produced by Eqs s and 5.46 s extensive spectral problems 6.1 and 6.2 they are gauge equivalent, have different r-matrices see Sec. 6. Remark 5.3. In fact, the r-matrix r 12 λ, µ inthecaseofthec-aknsandc-d flows can be also chosen as r 12 λ, µ = 2 a b µ λ P + I S, S = 5.50 c d where the elements a, b, c, d can be arbitrary C -functions aλ, µ, p, q,bλ, µ, p, q, cλ, µ, p, q,dλ, µ, p, q with respect to the spectral parametres λ, µ and the dynamical variables p, q. This shows that for a given Lax matrix, the associated r-matrix is not uniquely defined there are even infinitely many r-matrices possible. Here we give the simplest case: a = b = c = d = 0, i.e. the standard r-matrix i= The constrained Harry Dym HD and Heisenberg spin chain HSC flows The constrained Harry Dym system describes the geodesic flow on an ellipsoid and shares the same r-matrix with the constrained Heisenberg spin chain HSC. To prove this, we consider the following Lax matrices: p, q λ 1 L HD = L HD λ 2 + q, q λ 1 λ, p, q = + L p, p λ 1 p, q λ 1 0, 5.51 p, q λ 1 L HSC = L HSC q, q λ 1 λ, p, q = + L p, p λ 1 p, q λ Here L HSC is included in the generalized Lax matrix 2.1, but L HD is not. We need two associated auxiliary matrices 0 1 M HD = Λp, p Λ 2 q, q λ 0, 5.53 iλ Λp, q iλ Λq, q M HSC =, iλ Λp, p iλ Λp, q i 2 =

23 Generalized r-matrix Structure and Algebro-Geometric Solution 567 Theorem 5.7. The Lax representations L HD x =[M HD,L HD ], 5.55 L HSC x =[M HSC,L HSC ] 5.56 respectively give the following fininte dimensional Hamiltonian flows: q x = p = H HD p, TQ N 1 H HD : Λp, p p x = Λ 2 q, q Λq = H HD q, TQ N 1 Λq, q =1; 5.57 H HSC : with the Hamiltonian functions q x = i Λq, q Λp i Λp, q Λq = H HSC p p x = i Λp, q Λp i Λp, p Λq = H HSC q,, 5.58 H HD = 1 Λp, p p, p 2 2 Λ 2 Λq, q 1, 5.59 q, q H HSC = 1 2 i Λp, p Λq, q 1 2 i Λp, q In Eq TQ N 1 is a tangent bundle in R 2N : TQ N 1 = {p, q R 2N F Λq, q 1=0,G Λp, q =0} Obviously, 5.57 is equivalent to q xx + Λq x,q x Λ 2 Λq =0, Λq, q =1, 5.62 q, q which is nothing but the equation of the geodesic flow [26] on the surface Λq, q =1 in the space R N and also coincides with the constrained HD c-hd flow [9]. In addition, 5.58 becomes the Heisenberg spin chain spectral problem [47] iλw iλu y x = y, i 2 = 1, 5.63 iλv iλw with the constraints u = Λq, q, v = Λp, p, w = Λp, q, λ = λ j, y =q j,p j T. Thus, Eq reads as the constrained Heisenberg spin chain c-hsc flow [39]. Their Lax matrices 5.51 and 5.52 share all elements except one, namely 0 λ

24 568 Z. Qiao This element does not affect the calculations concerning the fundamental Poisson bracket, one can readily deduce that the c-hd flow and the c-hsc flow possess the same non-dynamical r-matrix 2λ r 12 λ, µ = µµ λ P, r 21µ, λ =Pr 12 µ, λp Remark 5.4. The r-matrix 5.64 of the c-hd and c-hsc flows can be also chosen as 2λ r 12 λ, µ = µµ λ P + I S Evidently, 5.64 is the simplest case: S = 0 of The constrained G and Q flows In this subsection, we introduce the following Lax matrices: 1 + p, q λ 1 q, q λ 1 L G = L G λ, p, q = p, q 2 λ 1 + L 0, 5.66 λ 1 q, q λ 1 L Q = L Q λ, p, q = + L 0 λ If we set 1 M G = α λ 1 p, p q, q 1 α, α p, p q, q +1λ 1 α λ λ + 1 M Q = 2β 2 Λq, q p, p 1 Λq, q β p, p λ λ Λq, q p, p β 2β2 with α = p, p Λq, q 2 4 Λq, p, β =1 p, q, 5.70 then, by a lengthy and straightforward calculation we obtain the following theorem. Theorem 5.8. The following Lax representations and L G x =[M G,L G ] 5.71 L Q x =[M Q,L Q ], 5.72

25 Generalized r-matrix Structure and Algebro-Geometric Solution 569 where the first one is restricted to the surface M 1 = {p, q R 2N p, q = 0, Λq, q p, p + Λq, p = 0} in the space R 2N, respectively produce the finitedimensional systems: q x = 1 Λq + p, p Λq, q p p, α p x = α Λp + p, p Λq, q Λq+Λq, and q x =Λq Λq, q p + p, p Λq, q q, β 2β2 p x = Λp p, p Λq β 2β 2 p, p Λq, q p. Equations 5.73 and 5.74 turn out to be the spectral problem studied by Geng simply called G-spectral problem [22] with the constraint condition λu v 1 y x = y 5.75 λv +1 λu u = 1 α = 1 p, p Λq, q 2 4 Λq, p, v = p, p Λq, q α = p, p Λq, q p, p Λq, q 2 4 Λq, p, λ = λ j, y =q j,p j T, and the spectral problem proposed by Qiao simply called Q-spectral problem [38] with the constraint condition λ 1 y x = 2 uv λv u λ + 1 y uv u = v = Λq, q β p, p β = Λq, q 1 q, p, p, p = 1 q, p, λ = λ j, y =q j,p j T, respectively.

26 570 Z. Qiao So, Eqs and 5.74 are the constrained Geng c-g flow and the constrained Qiao c-q flow, and they have the same non-dynamical r-matrix: r 12 λ, µ = 2 µ λ P 2 µ S, S = = σ σ Here, the r-matrix r 12 λ, µ can be also chosen as r 12 λ, µ = 2 µ λ P 2 µ S + I S Equation 5.77 is the simplest case: S = 0 of We have already seen that the r-matrix r 12 λ, µ satisfying the fundamental Poisson bracket is not unique in fact, infinitely many and is usually composed of two parts, the first one being their main term, and the second one being the common term I S. Usually, to prove the integrability we choose their main term as the simplest r-matrix. 6. An Equivalent Pair with Different r-matrices This section reveals the following interesting fact: a pair of constrained systems, produced by two gauge equivalent spectral problems, possesses different r-matrices. In 1992, Geng introduced the following spectral problem [23] iλ iβuv u φ x = Mφ, M =, i 2 = v iλ + iβuv where u and v are two scalar potentials, λ is a spectral parameter and β is a constant, and discussed its evolution equations and Hamiltonian structure. Equation 6.1 is apparently an extension of the AKNS spectral problem Two years later the author considered an extension of the Dirac spectral problem 5.46 [40] ψ x = Mψ, is λ + r + βs M 2 r 2 =, 6.2 λ + r βs 2 r 2 is where r, s are two potentials, and obtained a finite dimensional involutive system being not equivalent to that one in Ref. [23]. But, the spectral problems 6.1 and 6.2 are gauge equivalent via the following transformation [50] 1 1 ψ = Gφ, G =, 6.3 i i v = ir s, u = ir + s. In Ref. [50], Wadati and Sogo discussed the gauge transformations of some spectral problems like Eq

27 Generalized r-matrix Structure and Algebro-Geometric Solution 571 Now, we discuss their r-matrices. Let us consider the following two Lax matrices: 1+2iβ p, q 0 L GX = L GX λ, p, q = il 0, iβ p, q L QZ = L QZ λ, p, q 1 0 β p, p + q, q = L β p, p + q, q 0 2 Then calculating their determinants leads to the following Hamiltonian systems q x = H GX p, p q, q q, q =Λq + iβ q p 1 + 2iβ p, q 2 1+2iβ p, p x = H 6.6 GX p, p q, q p, p = Λp iβ p + q 1 + 2iβ p, q 2 1+2iβ q, and q x = H QZ p =Λp β 4 p, q 2 + p, p q, q 2 2 p, q q + p, p q, q p p, 1 2β p, p + q, q 2 1 2β q, q + p, p p x = H 6.7 QZ q = Λq + β 4 p, q 2 + p, p q, q 2 2 p, q p p, p q, q q q +, 1 2β p, p + q, q 2 1 2β q, q + p, p with the Hamiltonian functions and H GX = i Λq, p p, p q, q iβ p, q 6.8 H QZ = 1 2 Λp, p Λq, q 4 p, q 2 + p, p q, q β p, p + q, q Obviously, Eqs. 6.6 and 6.7 become Eqs. 6.1 and 6.2 with the constrants q, q u = 1+2iβ p, q,v= p, p 1+2iβ p, q, 6.10 λ = λ j, φ =q j,p j T, j =1,...,N; and the constraints s = 2i p, q 1 2β q, q + p, p, r = p, p + q, q 1 2β q, q + p, p, 6.11 λ = λ j, ψ =q j,p j T, j =1,...,N, respectively. Thus, the finite dimensional Hamiltonian systems 6.6 and 6.7 are respectively the constrained flows of the

28 572 Z. Qiao spectral problems 6.1 and 6.2. Since they have Lax matrices 6.4 and 6.5, then the r-matrices of Eqs. 6.6 and 6.7 are respectively: r 12 λ, µ = 2 µ λ P +4iβS, S = , and r 12 λ, µ = 2 µ λ P +2βS, S = which are apparently different , New Integrable Systems In this section, three new integrable systems are generated as the representatives from our generalized r-matrix structure. 1 The first system is given by Case 6 in Sec. 4. The corresponding r-matrix and involutive systems are respectively r 12 λ, µ = 2 µ λ P + 2 µ S, S = , and Ej 1 =2 p, q + cλ 1 j p j q j + q, q λ 1 j p 2 j Γ j,,...,n, 7.2 where c R. Thus, the finite dimesional Hamiltonian systems Fm 1 defined by Fm 1 = N λm j E1 j, m =0,..., i.e. F 1 m =2 p, q + c λm 1 j p, q + q, q λ m 1 j p, p Λ i q, q Λ j p, p Λ i q, p Λ j p, q 7.3 i+j=m 1 are completely integrable. Particularly, with m = 2 the Hamiltonian system F2 1: q x = F1 2 =2cΛq 2 Λq, q p +4 Λp, q q +4 p, q Λq, p 7.4 p x = F1 2 = 2cΛp +2 p, p Λq 4 Λp, q p 4 p, q Λp, q

29 Generalized r-matrix Structure and Algebro-Geometric Solution 573 is a new integrable system, which becomes the following spectral problem 2c +4vλ +4u 2w φ x = φ 7.5 2sλ 2c +4vλ 4u with the constraint conditions u = Λp, q, v = p, q, w = Λq, q, s = p, p, and λ = λ j, φ =q j,p j T, j =1,...,N. Apparently, the spectral problem 7.5 is new. 2 The second system is produced by Case 7i in Sec 4. The corresponding r-matrix and involutive systems are respectively 0 g r 12 λ, µ = 2 µ λ P + 2 µ S, S = f , and E 2 j =2cλ 1 j p j q j + q, q + g 2 λ 1 + p, p f 2 λ 1 j qj 2 Γ j, j =1,...,N, 7.7 where c R, f 2 = f 2 p, q, g 2 = g 2 p, q C R. Hence, the Hamiltonian system F2 2 defined by F2 2 = N λ2 j E2 j, i.e. j p 2 j F 2 2 =2c Λp, q +2 Λp, q p, q + g 2 Λp, p f 2 Λq, q 7.8 is completely integrable. Meanwhile the Hamiltonian system F 2 2 : q x = F2 2 p =2cΛq +2 Λq, p q +2 p, q Λq +2g 2 Λp + Λp, p g 2q Λq, q f 2q, 7.9 p x = F2 2 q = 2cΛp 2 Λq, p p 2 p, q Λp Λp, p g 2p +2f 2 Λq Λq, q f 2p, 7.10 can be also related to a new 2 2 spectral problem with some constraint conditions. 3 The third system is derived by Case 7iv in Sec. 4. The corresponding r-matrix and involutive systems are respectively r 12 λ, µ = 2 µ λ P 2 µ S, S = , c 1 1

30 574 Z. Qiao and E 3 j =2 p, q + cλ 1 j p j q j + q, q λ 1 j p 2 j c 1λ 1 j qj 2 Γ j 7.12 where c R and c 1 = c 1 p, q C R. The following Hamiltonian system F2 3 : q x = F3 2 p =2cΛq Λq, q c 1 q +2p, 7.13 p x = F3 2 q = 2cΛp +2c 1 + p, p Λq + c 1 Λq, q p, is one of their products, where F2 3 =2c Λp, q Λq, q c 1 + p, p In general, with any c 1 Eq can t be changed to a 2 2 spectral problem with some constraints. But with two special c 1 : c 1 =0andc 1 = p, q, Eq can respectively become the spectral problem [31] 2cλ 2v φ x = φ uλ 2cλ with the constraint conditions u = p, p, v = Λq, q, and the spectral problem 2cλ v 2v φ x = φ uλ 2cλ + v with the constraint conditions u = p, q + p, v = Λq, q. Here in Eqs and 7.16 λ = λ j,φ=q j,p j T,,...,N are set. Equation 7.16 is a new spectral problem. We can consider further new integrable systems generated by Theorem 3.1. The above procedure actually gives an approach how to connect an r-matrix of finite dimensional system with a spectral problem, which is closely associated with integrable NLEEs. 8. Algebro-Geometric Solutions In this section, we connect the integrable NLEEs with the finite dimesional integrable flows, and solve them with a form of algebro-geometric solutions. Here, we take two examples: one being the periodic or infinite Toda lattice equation, the other the AKNS equation with the condition of decay at infinity or periodic boundary Toda lattice equation The Toda hierarchy associated with Eq is derived as follows: un v n = JG n j, j =0, 1, 2, t j

31 Generalized r-matrix Structure and Algebro-Geometric Solution 575 where {G n j = J 1 KG n j 1 } j=0 is the Lenard sequence, Gn 1 =αu 1 n,βt Ker J, for all α = αt j, β = βt j C R, the two symmetric operators K, J are 1 K = 2 u ne E 1 u n u n E 1v n, v n 1 E 1 u n 2u 2 n E E 1 u 2 n 0 u n E 1 J = E 1 u n 0 In particular, with j =0,β = 1, Eq. 8.1 reads as the Toda lattice which can be changed to via the following transformation u n = u n v n+1 v n, v n =2u 2 n u 2 n 1, 8.3 ẍ n =2e 2xn+1 xn e 2xn xn It is easy to prove the following theorem. Theorem Let Ĝn =Ĝ1 n, Ĝ2 n possesses the operator solution u n = e xn+1 xn, v n =ẋ n. 8.5 T, Ĝ1 n, Ĝ2 n C R. Then the operator equation [V Ĝn,L]=L KĜn L JĜnL V Ĝn = E 1 u n Ĝ2 n E E 1 u n Ĝ 1 n unĝ1 n +u nĝ2 n E 8.6 where [, ] is the usual commutator; the operator L is defined by Eq. 5.13; L ξ =E 1 ξ 1 + ξ 2 + ξ 1 E, ξ =ξ 1,ξ 2 T,ξ 1,ξ 2 C R. 2 Let us choose the special Ĝ n = G n j, j = 1, 0, 1,..., then the Toda hierarchy 8.1 has the following Lax representation of operator form L tj =[W G n j,l], j =0, 1, 2,..., 8.7 where the operator W G n j = j k=0 V Gn k 1 Lj k. Particularly, the standard Toda Equation 8.4 possesses the Lax representation of operator form L t =[W G n 0,L], where the operator W Gn 0 E =exn+1 xn e xn xn 1 E 1,andu n,v n in L are substituted by Eq We have shown that the c-toda flow and the c-ckdv flow share a common nondynamical r-matrix, and in particular, this ensures the integrability of their flows. A calculation of determinant yields their common N-involutive systems E α = λ α p α q α p 2 α p, q q 2 α N β α,β=1 q α p β p α q β 2 λ α λ β, α =1,...,N, 8.8

32 576 Z. Qiao which are independent and invariant i.e. E α λ, p, q =E α λ, p,q. Apparently, the functions F s = N α=1 λs αe α, s =0, 1, 2,..., are given by F s = Λ s+1 p, q Λ s p, p p, q Λ s q, q Λ j p, p Λ k q, q Λ j p, q Λ k q, p 8.9 j+k=s 1 and {F m,f l } =0, m, l Z + implies the Hamiltonian systems F s are completely integrable. Let p 0 t s,q 0 t s T be a solution of the initial problem p Fs / q p p0 =, = t s q F s / p q q t 0 s=0 Set pn t s p0 = H n t s T 8.11 q n t s q 0 t s where H T is defined by Eq Now, we rewrite Eq as a map f : R 2N R 2 defined by f :p n,q n T u n,v n T Then, we have the following theorem. Theorem 8.2. u n t s,v n t s T = fp n t s,q n t s satisfies the Toda hierarchy d un = JG n s, s =0, 1, dt s v n Particularly, with s = 0 the following calculable method p0 F 0 p0 t H n pn t f un t 8.14 q 0 q 0 t q n t v n t produces a solution of the Toda lattice Equation 8.3. Thus, the standard Toda Equation 8.4 has the following formal solution x n t = q n t,q n t dt We shall concretely give the expression q n t,q n t. Let us rewrite the element C TC λ ofeq.5.1as C TC λ Qλ N Kλ, Kλ = λ λ α, 8.16 α=1 and choose N distinct real zero points µ 1,...,µ N of Qλ. Then, we have N N N Qλ = λ µ j, q, q = λ α µ j α=1

33 Generalized r-matrix Structure and Algebro-Geometric Solution 577 Let π j = A TC µ j, 8.18 then it is easy to prove the following proposition. Proposition 8.1. {µ i,µ j } = {π i,π j } =0, {π j,µ i } = δ ij, i,j =1, 2,...,N, 8.19 i.e. π j,µ j are conjugated, and thus they are the seperated variables [46]. Write det L TC λ = A 2 TC λ B TCλC TC λ = 1 4 λ2 N α=1 E α λ λ α = P λ Kλ,whereE α is defined by Eq. 8.8, and P λ isann + 2 order polynomial of λ whose first term s coefficient is 1 4,thenπ2 j = P µj Kµ j, j =1,...,N. Now, we choose the generating function N N µj W = W j µ j, {E α } N n P λ α=1 = dλ 8.20 Kλ µ j 0 where µ j 0 is an arbitrarily given constant. Let us view E α α =1,...,Nas actional variables, then angle-coordinates Q α are chosen as Q α = W, α =1,...,N, E α i.e. N µk n N k α,k=1 Q α = ω α, ω α = λ λ k 2 dλ, α =1,...,N KλP λ k=1 µ k 0 Hence, on the symplectic manifold R 2N,dE α dq α the Hamiltonian function F 0 = N α=1 E α produces a linearized flow Thus Q α = F 0 E α, Ė α =0. Q α n =Q 0 α + t + c αn, E α n =E α n 1, c α = N µk n+1 k=1 µ k ω n α, where c α are dependent on actional variables {E α } N α=1, and independent of t; Q0 α is an arbitrarily fixed constant. Choose a basic system of closed paths α i,β i, i =1,...,N of Riemann surface Γ: µ 2 = P λkλ with N handles. ω j j = 1,...,N are exactly N linearly independent holomorphic differentials of the first kind on this Riemann surface Γ. ω j are normalized as ω j = N l=1 r j,l ω l, i.e. ω j satisfy ω j = δ ij, ω j = B ij, α i β i

34 578 Z. Qiao where B =B ij N N is symmetric and the imaginary part Im B of B is a positive definite matrix. By Riemann Theorem [25] we know: µ k n satisfies N µk n k=1 µ k 0 ω j = φ j,φ j = φ j n, t = N l=1 r j,lq 0 l + t + c l n, j = 1,...,N, if and only if µ k n are the zero points of the Riemann-Theta function ΘP = ΘAP φ K which has exactly N zero points, where AP = P P 0 ω 1,..., P P 0 ω N T, φ = φn, t = φ 1 n, t,...,φ N n, t T, K C N is the Riemann constant vector, P 0 is an arbitrarily given point on Riemann surface Γ. Because of [18] 1 λd ln 2πi ΘP =C 1 Γ 8.24 γ where the constant C 1 Γ has nothing to do with φ; γ is the boundary of simple connected domain obtained through cutting the Riemann surface Γ along closed paths α i,β i. Thus, we have a key equality N µ k n =C 1 Γ Res λ= 1 λd ln ΘP Res λ= 2 λd ln ΘP 8.25 k=1 where 1 := 0, P z 1 Kz 1 z=0, 2 := 0, P z 1 Kz 1 z=0. Through a lengthy careful calculation and combining Eq. 8.17, we obtain N q n t,q n t = λ α C 1 Γ + d ln Θφn, t+k + η dt Θφn, t+k + η 2 α=1 where the jth component of η i i =1, 2 is η i,j = P 0 i ω j. By the Riemann surface properties, we can also have N l=1 r j,lc l = 1 2 ω j = N l=1 r 1 j,l 2 ω l which implies c l = 1 2 ω l = N 1 i l,i=1 λ λi P 0 dλ. So, the standard Toda Equation 8.4 has the P λkλ following explicit solution, called algebro-geometric solution ΘUn+ Vt+ Z x n t =ln + Cn + Rt +const ΘUn +1+Vt+ Z where U = ˆRĈ, V = ˆRĴ, Z = ˆRQ 0 + K + η 1 with Ĉ =c 1,...,c N T, Ĵ = 1,...,1 T, Q 0 = Q 0 1,...,Q 0 N T, R = N α=1 λ α C 1 Γ, while matrix ˆR = r j,l N N is determined by the relation N l=1 r j,l α i ω l = δ ij,andc is certain constant which can be determined by the algebro-geometric properties on the Riemann surface Γ [16]. The symmetric matrix B =B ij N N in Θ function is determined by N l=1 r j,l β i ω l = B ij. Hence, the algebro-geometric solution of Toda lattice Equation 8.3 is u n t =e xn+1 xn = e C Θ 2 Un +1+Vt+ Z ΘUn +2+Vt+ ZΘUn+ Vt+ Z, v n t =ẋ n = R + d 8.28 dt ln ΘUn+ Vt+ Z ΘUn +1+Vt+ Z.