Orbit Approach to Separation of Variables in sl(3)-Related Integrable Systems
aa r X i v : . [ n li n . S I] A ug Orbit Approach to Separation of Variablesin sl (3) -Related Integrable Systems Julia Bernatska and Petro Holod
National University of ‘Kyiv Mohyla Academy’[email protected]
Abstract
Using the orbit method we attempt to reveal geometric and algebraic meaning of sepa-ration of variables for integrable systems on coadjoint orbits of an sl (3) loop algebra. Weconsider two types of generic orbits, embedded into a common manifold endowed with twononsingular Lie-Poisson brackets. We prove that separation of variables on orbits of bothtypes is realized by the same variables of separation. We also construct integrable systemson the orbits: a coupled 3-component nonlinear Schr¨odinger equation and an isotropic SU(3)Landau-Lifshitz equation. Separation of variables is a powerful technique for solving problems in mathematical physics,in particular in the theory of integrable Hamiltonian systems. There are well known separationof variables schemes for the integrable systems of KdV and MKdV equations, sin(sinh)-Gordonequations, nonlinear Sch¨odinger equation, isotropic Landau-Lifshitz equation, XXX and XYZmagnetic chains, XXX Gaudin model. Many authors contributed to progress of this technique,the reader can find the corresponding references in Sklyanin’s paper [1] giving, in particular, areview on separation of variables, and its applications to some of the mentioned systems.Among all the papers cited by Sklyanin we find a common effective algorithm linked to theLax representation of an integrable system. The algorithm fits well into the orbit method [2]used in [3, 4], and developed in the present paper. The orbit approach gives an opportunity tounderstand geometric and algebraic meaning of this separation of variables scheme.We mention as well another scheme [5–8] considering separation of variables within the bi-Hamiltonian geometry. The manifold where a system lives is called bi-Hamiltonian if endowedwith a pair of holomorphic Poisson brackets, at least one is nonsingular [9]. Such manifoldpossesses a set of Nijenhuis or Darboux-Nijenhuis coordinates, canonical with respect to the cor-responding symplectic form. Bi-Hamiltonian property gives the separability criterion: integralsof motion, defined as invariant functions, are in involution with respect to the both Poissonbrackets. The bi-Hamiltonian property gives also an algorithm of computing variables of sepa-ration and exhibits separation relations. Many, if not most, of the known integrable systems arebi-Hamiltonian.We distinguish between these two schemes due to different mathematical apparatus. Thefirst scheme substantially uses the apparatus of loop Lie algebras, but the second deals withsymplectic manifolds not considering them as subspaces of loop algebras. A connection betweenthese two schemes is shown in [10].In the present paper we develop the first scheme of separation of variables by means of theorbit method. This method deals with integrable systems constructed on coadjoint orbits inloop algebras. The orbit approach to separation of variables on orbits of sl (2) loop algebras ispresented in [11]. Here we develop this approach for systems on orbits of an sl (3) loop algebra,which we call sl (3) -related systems. Some of such systems are considered in the cited papers:the SL(3) magnetic chain in [12], the 3-particle Calogero-Moser model in [1], and a coupled2-component nonlinear Schr¨odinger equation in [3].1he main idea of Sklyanin’s paper [1] is the following: the poles of the properly normalizedBaker-Akhiezer function give variables of separation. This receipt is good for already solvedsystems, when a solution in the form of Baker-Akhiezer function is known. Otherwise, one hassome uncertainty caused by the proper normalization. However Sklyanin proposes a procedureof constructing variables of separation. This procedure is a brilliant idea, and is developedin [13, 14] for the sl ( n ) case. At the same time Sklyanin declared in [1] that ‘algebraic structuresunderlying the separation of variables’ remains unclear. Attempting to reveal geometric andalgebraic meaning of this separation of variables procedure we appeal to the orbit method. Thisapproach solves the problem of proper normalization because the relations producing variablesof separation are explicitly constructed from a loop algebra restricted to an orbit. Evidently,these variables serve as poles of the corresponding Baker-Akhiezer function.In [3, 4] Adams, Harnad and Hurtubise also solved the problem of constructing Darbouxcoordinates for integrable Hamiltonian systems on coadjoint orbits in loop algebras. Thesecoordinates was obtained from a divisor of a section of the dual eigenvector line bundle overthe invariant spectral curve of a system. This approach gives some geometric explanation of theseparation of variables scheme. In the present paper we come to similar results, but proposeanother explanation that looks intuitively obvious.In our paper we deal with bi-Hamiltonian systems. Both the systems in question evolvein a common symplectic manifold, endowed with two nonsingular Poisson brackets possessinga common spectral curve, and all integrals of motion are in involution with respect to boththe Poisson brackets. The integrals of motion divides into Casimir functions and Hamiltonians,Casimir functions with respect to the first Poisson tensor serve as Hamiltonians for the secondPoisson tensor, and vice verse [9]. Therefore, we have two transversal foliations of the manifoldinto orbits, every orbit serves as a symplectic leaf.The paper is organized as follows. After introducing some algebraic structures in Section 2we consider two types of orbits of the standard graded sl (3) loop algebra. These orbits serveas phase spaces for two integrable systems, as shown in Section 3 where Lie-Poisson bracketsand invariant functions are introduced. Section 4 is devoted to separation of variables on orbitsof the first type. The obtained variables of separation are proven to be canonically conjugate.Some words about connection to the well-known results are given in Section 5. Separation ofvariables on orbits of the second type is realized in Section 6. We obtain the same relationsproducing variables of separation, but the variables are quasi-canonically conjugate on orbitsof the second type. In Section 7 we construct integrable systems on orbits of the both types.This is a coupled 3-component nonlinear Schr¨odinger equation for the first type, and an isotropicSU(3) Landau-Lifshitz equation for the second one. Some conclusion and discussion are given inSection 8. At the beginning we construct a loop algebra based on the algebra g = sl (3 , C ) with the Cartan-Weyl basis X = , X = , X = , H = 13 diag(2 , − , − , Y = , Y = , Y = , H = 13 diag(1 , , − . We denote the set { X , Y , H , X , Y , X , Y , H } by { Z a : a = 1 , . . . , } . With respect to thebilinear form h A , B i = Tr AB we introduce the dual algebra g ∗ with the basis { Z ∗ a } : X ∗ j = Y j , Y ∗ j = X j , j = 1 , , , H ∗ = diag(1 , − , , H ∗ = diag(0 , , − . Let P ( λ, λ − ) be the algebra of Laurent polynomials in λ , and e g be the loop algebra sl (3 , C ) ⊗ ( λ, λ − ) . Then Z ma = λ m Z a is a basis in e g . The loop algebra is homogeneous or standard graded, this is easily checked bymeans of the operator d = d/dλ of homogeneous degree. The superscript of Z ma indicates a ho-mogeneous degree of the basis element. By g m , m ∈ Z we denote the eigenspace of homogeneousdegree m , that is g m = span C { X m , Y m , H m , X m , Y m , X m , Y m , H m } . According to the Kostant-Adler scheme [15] e g is decomposed into two subalgebras e g + = X m > g m , e g − = X m< g m , e g = e g + + e g − . Further, we introduce the ad -invariant bilinear forms h A ( λ ) , B ( λ ) i k = res λ =0 λ − k − Tr A ( λ ) B ( λ ) , A ( λ ) , B ( λ ) ∈ e g , k ∈ Z and use them to define the spaces dual to e g + and e g − . Example 1.
Let k = − . We have ( e g − ) ∗ = e g + , ( e g + ) ∗ = e g − , where ( e g − ) ∗ and ( e g + ) ∗ contain only the nonzero functionals on e g ± . Example 2.
Let k = N − > . Then ( e g − ) ∗ = X m > N g m , ( e g + ) ∗ = X m< N g m . sl (3 , C ) ⊗ P ( λ, λ − ) as phase spaces Fixing N > we introduce the variables { γ ( m )1 , β ( m )1 , α ( m )1 , γ ( m )3 , β ( m )3 , γ ( m )2 , β ( m )2 , α ( m )2 : m = 0 , , . . . , N } denoted all together by { L ( m ) a : a = 1 , . . . , } . Consider the space M ∈ e g ∗ of theelements L ( λ ) = N X m =0 dim g X a =1 L ( m ) a (cid:0) Z ma (cid:1) ∗ = α ( λ ) β ( λ ) β ( λ ) γ ( λ ) α ( λ ) − α ( λ ) β ( λ ) γ ( λ ) γ ( λ ) − α ( λ ) , (1)where α ( λ ) = N X m =0 λ m α ( m )1 , β ( λ ) = N X m =0 λ m β ( m )1 , γ ( λ ) = N X m =0 λ m γ ( m )1 ,α ( λ ) = N X m =0 λ m α ( m )2 , β ( λ ) = N X m =0 λ m β ( m )2 , γ ( λ ) = N X m =0 λ m γ ( m )2 ,β ( λ ) = N X m =0 λ m β ( m )3 , γ ( λ ) = N X m =0 λ m γ ( m )3 . Let C ( M ) be the space of smooth functions on M . For all f , f ∈ C ( M ) we define the firstLie-Poisson bracket by the formula { f , f } f = N X m,n =0 dim g X a,b =1 P mnab ( − ∂f ∂L ( m ) a ∂f ∂L ( n ) b , (2) P mnab ( −
1) = h L ( λ ) , [ Z − m − a , Z − n − b ] i − . { L ( N ) a } annihilate this Lie-Poisson bracket, thus they are constant. To make thebracket nonsingular we restrict it to the subspace M of M defined by the constraints L ( N ) a = const , a = 1 , . . . , . On M we define the second Lie-Poisson bracket by the formula { f , f } s = N X m,n =0 dim g X a,b =1 P mnab ( N − ∂f ∂L ( m ) a ∂f ∂L ( n ) b , (3) P mnab ( N −
1) = h L ( λ ) , [ Z − m + N − a , Z − n + N − b ] i N − . Remark 1.
In addition to the brackets (2) and (3), one can define intermediate brackets withthe Poisson tensors P mnab ( k ) = h L ( λ ) , [ Z − m + ka , Z − n + kb ] i k , k = 0 , . . . , N − . (4)In what follows we consider the space M , and use the set { γ ( m )1 , γ ( m )2 , γ ( m )3 , β ( m )1 , β ( m )2 , β ( m )3 , α ( m )1 , α ( m )2 : m = 0 , , . . . , N − } as dynamic variables in it. We call M a finite gap sectorof e g , more precisely the N -gap sector. With respect to the bilinear form h· , ·i − we introduce thecoadjoint action of e g − on M . Indeed the factor-algebra e g − / P l< −N g l acts effectively on M ,that is M ⊂ ( e g − ) ∗ , see Example 1. The first Lie-Poisson bracket {· , ·} f arises from the bilinearform h· , ·i − . On the other hand, we introduce the coadjoint action of e g + on M with respectto the bilinear form h· , ·i N − : the factor-algebra e g + / P l > N g l acts effectively on M , that is M ⊂ ( e g + ) ∗ , see Example 2. The second Lie-Poisson bracket {· , ·} s arises from h· , ·i N − .Next, we introduce the ad ∗ -invariant functions I ( λ ) ≡ Tr L ( λ ) = (cid:2) α ( λ ) (cid:3) + (cid:2) α ( λ ) (cid:3) − α ( λ ) α ( λ ) + β ( λ ) γ ( λ )++ β ( λ ) γ ( λ ) + β ( λ ) γ ( λ ) = (5) = − (cid:12)(cid:12)(cid:12)(cid:12) α ( λ ) β ( λ ) γ ( λ ) α ( λ ) − α ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) α ( λ ) − α ( λ ) β ( λ ) γ ( λ ) − α ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) α ( λ ) β ( λ ) γ ( λ ) − α ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) ,I ( λ ) ≡ Tr L ( λ ) = α ( λ ) (cid:2) α ( λ ) (cid:3) − α ( λ ) (cid:2) α ( λ ) (cid:3) + β ( λ ) γ ( λ ) α ( λ )++ β ( λ ) (cid:2) β ( λ ) γ ( λ ) − γ ( λ ) α ( λ ) (cid:3) ++ β ( λ ) (cid:2) γ ( λ ) γ ( λ ) − [ α ( λ ) − α ( λ )] γ ( λ ) (cid:3) = det L ( λ ) . Every function I k is a sum of the diagonal k th minors with an accuracy of the sign. Thefunctions I , I are polynomials in the spectral parameter λ , and their coefficients serve asinvariant functions in dynamic variables, namely: I ( λ ) = h + h λ + · · · + h N λ N , I ( λ ) = f + f λ + · · · + f N λ N ,h ν = − X m + n = ν (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) α ( m )1 β ( n )1 γ ( m )1 α ( n )2 − α ( n )1 (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) α ( m )2 − α ( m )1 β ( n )2 γ ( m )2 − α ( n )2 (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) α ( m )1 β ( n )3 γ ( m )3 − α ( n )2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ,ν = 0 , , . . . , N ; (6) f ν = X m + n + k = ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ( m )1 β ( n )1 β ( k )3 γ ( m )1 α ( n )2 − α ( n )1 β ( k )2 γ ( m )3 γ ( n )2 − α ( k )2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ν = 0 , , . . . , N . Evidently, h N , f N are constant, for they do not contain dynamic variables.The following assertions are immediately derived from the Kostant-Adler scheme [15]. Proposition 3.1.
All functions h ν , ν = 0 , , . . . , N − and f ν , ν = 0 , , . . . , N − definedby (6) are in involution with respect to the brackets (2) and (3) . That is, these functions serveas integrals of motion. roposition 3.2. The functions h ν , f ν + N , ν = N , . . . , N − are functionally independent on M and annihilate the first Lie-Poisson bracket (2) , namely they are Casimir functions withrespect to the first Lie-Poisson bracket. The rest of integrals of motion: h ν , ν = 0 , . . . , N − ,and f ν , ν = 0 , . . . , N − serve as Hamiltonians with respect to the first Lie-Poisson bracket. Let O f ⊂ M be the algebraic manifold defined by h ν = c ν , f ν + N = d ν + N , ν = N , . . . , N − , (7)where all c ν , d ν + N are fixed complex numbers. The manifold O f is a generic orbit of coadjointaction of the subalgebra e g − on M , dim O f = 6 N . Variation of the constants c ν , d ν + N givesa foliation of M into orbits of the first type. Every orbit serves as a symplectic leaf in thesymplectic manifold M . Proposition 3.3.
The functions h ν , f ν , ν = 0 , . . . , N − are functionally independent on M and annihilate the second Lie-Poisson bracket (3) , namely they are Casimir functions with respectto the second Lie-Poisson bracket. The rest of integrals of motion: h ν , ν = N , . . . , N − , and f ν , ν = N , . . . , N − serve as Hamiltonians with respect to the second Lie-Poisson bracket. The algebraic manifold O s ⊂ M defined by h ν = c ν , f ν = d ν , ν = 0 , . . . , N − , (8)where all c ν , d ν are fixed complex numbers, is a generic orbit of coadjoint action of the subal-gebra e g + on M , dim O s = 6 N . Variation of the constants c ν , d ν gives another foliation of M into orbits of the second type. In what follows we call O f and O s simply orbits , and call (7), (8)orbit equations. O f The orbit O f with the first Lie-Poisson bracket (2) has the following Poisson structure: { L ( m ) ij , L ( n ) kl } f = L ( m + n +1) kj δ il − L ( m + n +1) il δ kj , (9)also written in terms of the r -matrix r ( u − v ) = 1 u − v X a,b h Z a , Z b i Z ∗ a ⊗ Z ∗ b (10) { L ( u ) ⊗ , L ( v ) } f = [ r ( u − v ) , L ( u ) + L ( v )] with L ( u ) = L ( u ) ⊗ I , L ( v ) = I ⊗ L ( v ) , where I is the identity matrix.We parameterize the orbit O f by the dynamic variables { γ ( m )1 , γ ( m )2 , γ ( m )3 , β ( m )1 , α ( m )1 , α ( m )2 : m = 0 , , . . . , N − } , that is we eliminate the set { β ( m )2 , β ( m )3 } . One can choose another set toeliminate, requiring all the invariant functions are linear in this set of variables. Thus, the otherpossible sets of eliminated variables are the following: { β ( m )1 , β ( m )3 } , { γ ( m )2 , γ ( m )3 } , or { γ ( m )1 , γ ( m )3 } .Using linearity of the orbit equations (7) in the variables { β ( m )2 , β ( m )3 : m = 0 , , . . . , N } one canwrite them in the matrix form c f = F + β + η + f , (11a) F + = F N F N − . . . F F F N . . . F F ... ... . . . ... ... . . . F N F N − . . . F N , β = β (0) β (1) ... β ( N − β ( N ) , c f = c f N c f N +1 ... c f N − c f N , η + f = η f N η f N +1 ... η f N − η f N , F j = (cid:20) γ ( j )2 γ ( j )3 Γ ( j + N )2 Γ ( j + N )3 (cid:21) , β ( j ) = (cid:20) β ( j )2 β ( j )3 (cid:21) , c f j = (cid:20) c j d j + N (cid:21) , η f j = (cid:20) η j H j + N (cid:21) , (11b)5 j = − A ( j ) + X m + n = j α ( m )2 α ( n )2 , H j = − X m + n = j α ( m )2 A ( n ) , (11c) Γ ( j )2 = − X m + n = j (cid:12)(cid:12)(cid:12)(cid:12) α ( m )1 β ( n )1 γ ( m )3 γ ( n )2 (cid:12)(cid:12)(cid:12)(cid:12) , Γ ( j )3 = X m + n = j (cid:12)(cid:12)(cid:12)(cid:12) γ ( m )1 α ( n )2 − α ( n )1 γ ( m )3 γ ( n )2 (cid:12)(cid:12)(cid:12)(cid:12) , (11d) A ( j ) = X m + n = j (cid:12)(cid:12)(cid:12)(cid:12) α ( m )1 β ( n )1 γ ( m )1 α ( n )2 − α ( n )1 (cid:12)(cid:12)(cid:12)(cid:12) . (11e)Supposing F N is nonsingular, one easily eliminates the variables ββ = ( F + ) − ( c f − η + f ) , or β β ... β N − β N = F − N e F N − . . . e F e F F − N . . . e F e F ... ... . . . ... ... . . . F − N e F N − . . . F − N c f N − η f N c f N +1 − η f N +1 ... c f N − − η f N − c f N − η f N , e F N − n = F − N n X k =1 (cid:0) − F N − n − k F − N (cid:1) k , n = 1 , . . . , N . Next, substitute β into the Hamiltonians h , h , . . . , h N − , f , f , . . . , f N − h f = F − β + η − f = F − ( F + ) − c f + η − f − F − ( F + ) − η + f , (12)where F − = g . . . g g . . . ... ... . . . ... ... g N − g N − . . . g F g N − . . . g g F F . . . g g ... ... . . . ... ... F N − F N − . . . F g N − , h f = f f ... f N − h f h f ... h f N − , η − f = H H ... H N − η f η f ... η f N − , g j = (cid:2) Γ ( j )2 Γ ( j )3 (cid:3) , g j = (cid:20) ( j )2 Γ ( j )3 (cid:21) , h f j = (cid:20) h j f j + N (cid:21) . Note that the expressions (12) are linear in { c ν , d ν + N : ν = N , . . . , N } .To proceed we need to define the characteristic polynomial P ( w, λ ) = det (cid:0) L ( λ ) − w I (cid:1) . (13)It defines the spectral curve R : w − I ( λ ) w − I ( λ ) = 0 , (14)which is a curve of genus N − in general. The spectral curve is common for integrable systemson orbits of both the types: O f and O s . Restriction to an orbit is realized by implementation ofthe orbit equations (7) or (8), which fix some coefficients in (14). The rest of coefficients serveas Hamiltonians on the orbit and also remain constant during the evolution of a system.Consider the spectral curve restricted to the orbit O f . Denoting its points by { ( λ k , w k ) } wewrite the following set of equations w k = w k (cid:16) h + h λ k + · · · h N − λ N − k + c N λ N k + c N +1 λ N +1 k + · · · + c N λ N k (cid:17) ++ (cid:16) f + f λ k + · · · f N − λ N − k + d N λ N k + d N +1 λ N +1 k + · · · + d N λ N k (cid:17) , (15)6 = 1 , . . . , N , or in the matrix form W − f h f + W + f c f = w cubed , W − f = λ . . . λ N − W λ W . . . λ N − W λ . . . λ N − W λ W . . . λ N − W ... ... . . . ... ... ... . . . ... λ N . . . λ N − N W N λ N W N . . . λ N − N W N , W k = [ w k λ N k ] , W + f = λ N W λ N +11 W . . . λ N W λ N W λ N +12 W . . . λ N W ... ... . . . ... λ N N W N λ N +13 N W N . . . λ N N W N , w cubed = w w ... w N . and solve them for N Hamiltonians. Suppose that all pairs { ( λ k , w k ) : k = 1 , . . . , N } aredistinct points and W − f is nonsingular, then the Hamiltonians can be expressed by the formula h f = − ( W − f ) − W + f c f + ( W − f ) − w cubed . (16)On the orbit O f the formulas (12) and (16) define the same set of functions, moreover, both ofthem are linear in { c ν , d ν + N : ν = N , . . . , N } . As { c ν , d ν + N } are independent parameters onecan equate the corresponding terms, that is F − ( F + ) − = − ( W − f ) − W + f , η − f − F − ( F + ) − η + f = ( W − f ) − w cubed ⇒ W − f F − + W + f F + = 0 , W − f η − f + W + f η + f = w cubed . (17)The first matrix equation (17) gives the following Γ (0)2 + Γ (1)2 λ + · · · + Γ (2 N )2 λ N k + w k (cid:0) γ (0)2 + · · · + γ ( N )2 λ N k (cid:1) = 0 , (18a) Γ (0)3 + Γ (1)3 λ + · · · + Γ (2 N )3 λ N k + w k (cid:0) γ (0)3 + · · · + γ ( N )3 λ N k (cid:1) = 0 (18b)or more concisely Γ ( λ k ) + w k γ ( λ k ) = 0 , Γ ( λ k ) + w k γ ( λ k ) = 0 , (18c)where Γ and Γ are polynomials of degree N in general, and at least Γ (2 N )2 , γ ( N )3 or Γ (2 N )3 , γ ( N )2 are nonzero. The N equations (18c) are consistent if (cid:12)(cid:12)(cid:12)(cid:12) γ ( λ k ) γ ( λ k )Γ ( λ k ) Γ ( λ k ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (19)The second matrix equation (17) gives w k = − α ( λ k ) A ( λ k ) + w k (cid:0) α ( λ k ) − A ( λ k ) (cid:1) , or (20) (cid:2) w k + α ( λ k ) (cid:3)(cid:2) w k − α ( λ k ) w k + A ( λ k ) (cid:3) = 0 , which is a simplification of the spectral curve equation (14) realized at every point ( λ k , w k ) . Here A ( λ ) = A (0) + A (1) λ + · · · + A (2 N ) λ N . The set { λ , w } ≡ { ( λ k , w k ) : k = 1 , . . . , N } of variables defined by (19), (18c) are points ofthe spectral curve R obtained from its restriction (15) to the orbit O f . The variables { λ , w } give another parametrization of O f , we call them spectral variables , they serve as variables ofseparation as shown below.Now we trace the sequence of changes of variables on the orbit. First of all it is suitable tochange the dynamic variables { γ ( m )1 , β ( m )1 , α ( m )1 , γ ( m )3 , γ ( m )2 , α ( m )2 : m = 0 , , . . . , N − } into thefollowing: { γ , Γ , α , A } that is n γ (0)2 , γ (1)2 , . . . , γ ( N − , Γ (0)2 , Γ (1)2 , . . . , Γ (2 N − , (0)2 , α (1)2 , . . . , α ( N − , A (0) , A (1) , . . . , A (2 N − o , according to (11d), (11e). Equally one can use the set n γ (0)3 , γ (1)3 , . . . , γ ( N − , Γ (0)3 , Γ (1)3 , . . . , Γ (2 N − ,α (0)2 , α (1)2 , . . . , α ( N − , A (0) , A (1) , . . . , A (2 N − o . Then from (12) the variables { α , A } are replaced by the Hamiltonians h f = { h , h , . . . , h N − , f , f , . . . , f N − } . In this way { γ , Γ , α , A } is changed into { γ , Γ , h f } . At last (18a), (15)connect the latter to the spectral variables { λ , w } . The equations (18c) give direct relationsbetween the dynamic and the spectral variables; they are used for computation.Using the conventional notations we formulate Conjecture 1 from [12] for our purpose Separation of variables theorem 1.
Suppose the orbit O f is parameterized by the variables { γ ( m )1 , β ( m )1 , α ( m )1 , γ ( m )3 , γ ( m )2 , α ( m )2 : m = 0 , . . . , N − } as above. Then the new variables { ( λ k , w k ) : k = 1 , . . . , N } defined by the formulas B ( λ k ) = 0 , w k = A ( λ k ) , (21) where B is the polynomial of degree N and A is the algebraic function such that B ( λ ) = (cid:2) α ( λ ) − α ( λ ) (cid:3) γ ( λ ) γ ( λ ) − β ( λ ) γ ( λ ) + γ ( λ ) γ ( λ ) = (cid:12)(cid:12)(cid:12)(cid:12) γ ( λ ) γ ( λ )Γ ( λ ) Γ ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) , (22a) A ( λ ) = α ( λ ) − β ( λ ) γ ( λ ) γ ( λ ) ≡ − Γ ( λ ) γ ( λ ) or (22b) = α ( λ ) − α ( λ ) − γ ( λ ) γ ( λ ) γ ( λ ) ≡ − Γ ( λ ) γ ( λ ) (22c) have the following properties: (i) a pair ( λ k , w k ) is a root of the characteristic polynomial (13) ; (ii) a pair ( λ k , w k ) is canonically conjugate with respect to the first Lie-Poisson bracket (2) : { λ k , λ l } f = 0 , { λ k , w l } f = δ kl , { w k , w l } f = 0; (23)(iii) the corresponding Liouville 1-form is Ω f = X k w k dλ k . Proof. (i) The equation (22a) is equivalent to the consistent condition (19). This implies thattwo expressions (22b), (22c) for A coincide at all zeros { λ k } of B . The characteristic polynomial P defined by (13) has B ( λ k ) as a factor, vanishing at every point ( λ k , w k ) , for example with(22b) one can easily compute: P ( w k , λ k ) = 1 γ ( λ k ) (cid:16)(cid:2) α ( λ k ) + α ( λ k ) (cid:3) β ( λ k ) γ ( λ k )++ β ( λ k ) γ ( λ k ) − β ( λ k ) γ ( λ k ) (cid:17) B ( λ k ) ≡ . Both the expressions for A give the same eigenvalue of the L -matrix (1).(ii) The assertion follows from the lemmas below. Similar lemmas for a quadratic Poissonbracket can be found in [12, 14]. 8 onjugate variable lemma 1. If B and A satisfy the following identities with respect to thefirst Lie-Poisson bracket (9) {B ( u ) , B ( v ) } f = 0 , {A ( u ) , A ( v ) } f = 0 , {A ( u ) , B ( v ) } f = f ( u, v ) B ( u ) − B ( v ) u − v , where f is an arbitrary function such that lim v → u f ( u, v ) = 1 , then the variables { ( λ k , w k ) : k = 1 , . . . , N } defined by B ( λ k ) = 0 , w k = A ( λ k ) are canonically conjugate with respect to {· , ·} f : { λ k , λ l } f = 0 , { λ k , w l } f = δ kl , { w k , w l } f = 0 . Proof.
The equations B ( λ k ) = 0 , w k = A ( λ k ) imply ∂λ k ∂L ( m ) a = − lim u → λ k B ′ ( u ) ∂ B ( u ) ∂L ( m ) a , (24) ∂w k ∂L ( m ) a = lim u → λ k ∂ A ( u ) ∂L ( m ) a + A ′ ( u ) ∂u∂L ( m ) a ! = lim u → λ k ∂ A ( u ) ∂L ( m ) a − A ′ ( u ) B ′ ( u ) ∂ B ( u ) ∂L ( m ) a ! . Taking into account that
N − X m,n =0 X a,b ∂ F ( u ) ∂L ( m ) a ∂ G ( v ) ∂L ( n ) b { L ( m ) a , L ( n ) b } f = {F ( u ) , G ( v ) } f for any functions F and G , one obtains the following { λ k , λ l } f = {B ( λ k ) , B ( λ l ) } f B ′ ( λ k ) B ′ ( λ l ) = 0 , { w k , λ l } f = lim u → λ k v → λ l (cid:18) − B ′ ( v ) {A ( u ) , B ( v ) } f + A ′ ( u ) B ′ ( u ) B ′ ( v ) {B ( u ) , B ( v ) } f (cid:19) == − f ( λ k , λ l ) B ( λ k ) − B ( λ l )( λ k − λ l ) B ′ ( λ l ) = − δ kl , { w k , w l } f = lim u → λ k v → λ l (cid:18) A ′ ( u )[ f ( v, u ) B ( v ) − B ( u )] B ′ ( u )( v − u ) − A ′ ( v )[ f ( u, v ) B ( u ) − B ( v )] B ′ ( v )( u − v ) (cid:19) == (cid:18) A ′ ( λ k ) B ′ ( λ k ) − A ′ ( λ l ) B ′ ( λ l ) (cid:19) δ kl = 0 , as required. A - B bracket lemma 1. For B and A defined by (22) the following identities are true withrespect to the first Lie-Poisson bracket (9) : {B ( u ) , B ( v ) } f = 0 , {A ( u ) , A ( v ) } f = 0 , {A ( u ) , B ( v ) } f = f ( u, v ) B ( u ) − B ( v ) u − v , where f ( u, v ) = γ ( v ) /γ ( u ) for (22b) , and f ( u, v ) = γ ( v ) /γ ( u ) for (22c) .Proof. It is realized by the direct computation. From (9) written for polynomials as { L ij ( u ) , L kl ( v ) } f = L kj ( u ) − L kj ( v ) u − v δ il − L il ( u ) − L il ( v ) u − v δ kj , one obtains { γ ( u ) , γ ( v ) } = { γ ( u ) , Γ ( v ) } = { γ ( u ) , Γ ( v ) } = 0 , γ ( u ) , Γ ( v ) } = −{ γ ( u ) , Γ ( v ) } = 1 u − v (cid:12)(cid:12)(cid:12)(cid:12) γ ( u ) γ ( u ) γ ( v ) γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) , { Γ ( u ) , Γ ( v ) } = 1 u − v (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) γ ( u ) γ ( u )Γ ( v ) Γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) γ ( v ) γ ( v )Γ ( u ) Γ ( u ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) , and then { γ ( u ) , B ( v ) } = γ ( v ) u − v (cid:12)(cid:12)(cid:12)(cid:12) γ ( u ) γ ( u ) γ ( v ) γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) , { Γ ( u ) , B ( v ) } = 1 u − v (cid:18) γ ( u ) B ( v ) − γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) γ ( v ) γ ( v )Γ ( u ) Γ ( u ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . Γ ( u ) { γ ( u ) , B ( v ) } − γ ( u ) { Γ ( u ) , B ( v ) } = 1 u − v (cid:2) γ ( v ) B ( u ) − γ ( u ) B ( v ) (cid:3) . From Leibniz’s rule for a Poisson bracket: −{ Γ ( u ) , B ( v ) } = {A ( u ) γ ( u ) , B ( v ) } = γ ( u ) {A ( u ) , B ( v ) } − Γ ( u ) γ ( u ) { γ ( u ) , B ( v ) } , where A ( u ) = − Γ ( u ) /γ ( u ) , one easily gets {A ( u ) , B ( v ) } = 1 u − v (cid:16) γ ( v ) γ ( u ) B ( u ) − B ( v ) (cid:17) . Other identities from the lemma statement are easily computed in the similar way.
Remark 2. A - B bracket lemma 1 shows that (22b) , (22c) give good expressions for w ( λ ) becauseof canonical conjugation of A and B when the point ( λ, w ) tends to ( λ k , w k ) from the vicinity.Explicit computation shows that solutions of (20) taken as the A -function do not possess thisgood property. The reason is that (20) is true only at the points { ( λ k , w k ) } but not in theirvicinities. In other words, the A -function defined by (22) conserves canonical conjugation withthe polynomial B when the dynamic variables evolve, but the solutions of (20) do not. (iii) The Liouville 1-form on O f is implied by (23): Ω f = X k w k dλ k . Fixing values of the Hamiltonians h , h , . . . , h N − , f , f , . . . , f N − we obtain a Liouville torus.On the torus every variable w k becomes an algebraic function of the corresponding conjugatevariable λ k due to (15), and the form Ω f becomes a sum of meromorphic differentials on theRiemann surface P ( w, λ ) = 0 .This completes the proof of Separation of variables theorem 1.In order to obtain the required N points { ( λ k , w k ) } we need the polynomial B of degree N , this is provided by the maximal degrees of the polynomials Γ and γ , or Γ and γ . If theleading coefficient L ( N ) of the Lax matrix L defined by (1) does not provide the maximal degrees,one can apply a proper similarity transformation to L .Inversely, given a set of pairs { ( λ k , w k ) : k = 1 , . . . , N } one computes the dynamic variables { γ (1) m , β (1) m , α (1) m , γ (3) m , γ (2) m , α (2) m : m = 0 , , . . . , N } such that the equations (21) are satisfied.Thus, one defines a homomorphism C N → O f (25)that maps { ( λ k , w k ) } to a point of O f . When all the Hamiltonians are fixed the homomorphism(25) turns into the map from the symmetrized product of N Riemann surfaces R defined by(15) to the Liouville torus: Sym {R × R × · · · × R} 7→ T N . emark 3. One can observe a mnemonic rule: the expressions (22) are easily obtained from (5) written in the matrix form (cid:18) I ( λ ) I ( λ ) (cid:19) = (cid:18) γ ( λ ) γ ( λ )Γ ( λ ) Γ ( λ ) (cid:19) (cid:18) β ( λ ) β ( λ ) (cid:19) + (cid:18) α ( λ ) − A ( λ ) − α ( λ ) A ( λ ) (cid:19) . The consistent equation (19) provides singularity of the matrix ( γ γ Γ Γ ) at every point ( λ k , w k ) ,thus the spectral curve (14) is reduced to the form (20) . Considering separation of variables for integrable systems on orbits in loop algebras we refer thepapers [1, 3, 4, 12–14].We start from Sklyanin’s paper [12] where the representation (21) for variables of separationwas declared firstly as a conjecture, and proven for the classical SL(3) magnetic chain. In [14]this assertion was extended to the classical SL( n ) magnetic chain. The system was consideredin the phase space with a quadratic Poisson bracket, but separation of variables was realized bythe expressions similar to (22). This is presumably true for any integrable system on coadjointorbits of the sl (3) loop algebra. As shown below the variables of separation on orbits of thesecond type are defined by the same expressions.For further explanation we introduce the matrix N ( λ, w ) ≡ L ( λ ) − w I with the L -matrix (1),and denote by e N its adjoint matrix whose entries e N ij are cofactors of N ji . One can easily seethat (18c) is equivalent to e N ( λ k , w k ) ≡ (cid:12)(cid:12)(cid:12)(cid:12) γ ( λ k ) α ( λ k ) − α ( λ k ) − w k γ ( λ k ) γ ( λ k ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , e N ( λ k , w k ) ≡ − (cid:12)(cid:12)(cid:12)(cid:12) α ( λ k ) − w k β ( λ k ) γ ( λ k ) γ ( λ k ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , which gives expressions for the A -function. At the same time elimination of { w k } leads to thepolynomial B , giving the consistent condition (19). In [1] Sklyanin presented this result as is;however it naturally follows from the orbit method. Considering the manifold M as a foliationinto generic orbits we obtain expressions for A and B from the relations between dynamic andspectral variables.In addition we compute the eigenvector of L ( λ k ) corresponding to the eigenvalue w k definedby (22b) or (22c). Using (18c) and supposing λ k does not coincide with any root of γ and γ ,by Gauss’ method one reduces the matrix N ( λ k , w k ) to the form − γ ( λ k ) e N ( λ k ) 0 γ ( λ k ) e N ( λ k )0 − γ ( λ k ) e N ( λ k ) γ ( λ k ) e N ( λ k ) γ ( λ k ) γ ( λ k ) − α ( λ k ) − w k with the vanishing left upper × block, and the nonvanishing last column. The correspondingeigenvector has the form Ω T = (cid:0) Ω , Ω , (cid:1) such that γ ( λ k )Ω + γ ( λ k )Ω = 0 , (26)then Ω T = Ω (cid:0) , − γ ( λ k ) /γ ( λ k ) , (cid:1) or Ω (cid:0) − γ ( λ k ) /γ ( λ k ) , , (cid:1) . (27)One can also observe that the polynomial B has the form of (26), and the vector Γ ( λ k ) − Γ ( λ k )0 = w k − γ ( λ k ) γ ( λ k )0 at every root of B serves as an eigenvector for L ( λ k ) , here we use the relations (18c). If B has themaximal degree N then there exist N values of { λ k } satisfying the consistent condition (19),11nd the corresponding values of { w k } . Note that we take only one eigenvalue w k for every λ k .Recall that coefficients of Γ , Γ , γ , γ are not constant but serve as dynamic variables of thesystem in question. Their evolution implies an evolution of the the spectral variables ( λ, w ) ,and it is convenient to shift the focus from the dynamic variables onto the spectral ones becausethe latter are canonically conjugate. The relations (18c) fix an unambiguous connection betweendynamic and spectral variables. It means we take a sufficient number of points ( λ k , w k ) such thatthe spectral curve has a simple form: among N branch points N pairwise coincide. Conservingthis property, the curve evolves together with the dynamic variables.Now we obtain the matrix K , introduced in [12], which realizes the similarity transformationreducing the L -matrix to a block-triangular form. A certain entry of this block-triangular formis used to obtain the polynomial B . In [13] this idea of constructing the polynomial B wasdeveloped for the SL( n ) case. Using (27) one immediately writes the transformation matrix K and the corresponding transformation of L : K = − γ ( λ k ) /γ ( λ k ) 1 00 0 1 ! , K LK − = α − β γ γ β β B /γ α − α + β γ γ β + β γ γ γ − α or K = − γ ( λ k ) /γ ( λ k ) 00 1 00 0 1 ! , K LK − = α + γ γ γ −B /γ β + β γ γ γ α − α − γ γ γ β γ − α . It is easy to extract the polynomial B whose vanishing detects the eigenvalue given by (22b) or(22c) respectively. These transformations of L show that all points { ( λ k , w k ) } belong to the samesheet of the trigonal curve (14).In [3] and more detailed in [4] Adams, Harnad and Hurtubise showed that variables of sep-aration, called spectral Darboux coordinates, are zeros of e N ( λ, w ) v with an arbitrary vector v usually chosen as (1 , , . . . , T . The variables form the divisor of a section of the eigenvectorline bundle over the invariant spectral curve corresponding to a system. Applying this idea tothe above system on O f we get the equations e N ( λ, w ) ! = e N ( λ, w ) e N ( λ, w ) e N ( λ, w ) = 0 , which take place if one eliminates the set { β ( m )1 , β ( m )3 : m = 0 , , . . . , N } of the dynamic variables.The equations e N ( λ, w ) = 0 , e N ( λ, w ) = 0 give expressions for A like (22b), (22c). The firstequation is a simplification of a spectral curve equation like (20), true only for the set { ( λ k , w k ) } satisfying both the other two equations. Note that the first equation e N ( λ, w ) = 0 can notbe used to define w ( λ ) , though this is a characteristic equation. The reason is the absence ofdesirable property of canonical conjugation (see Remark 2). Only expressions (22b), (22c) meetthis requirement as A - B bracket lemma 1 shows.To complete the comparison with the results from [4] we note that the proposed procedureof separation of variables gives the equations (18c), (20) equivalent to e N T ( λ, w ) ! = 0 . The reader can see that such section of the dual eigenvector line bundle over the spectral curveis also acceptable.Solvability of the equations for the spectral coordinates { ( λ k , w k ) } is a delicate questionrelated to the leading coefficient L ( N ) of the L -matrix. In [3, 4] it was formulated in terms of thevector v : if v is an eigenvector of L ( N ) the equations e N ( λ, w ) v = 0 give only N − spectralpoints { ( λ k , w k ) } , this number coincides with the genus of the spectral curve. The two missingpoints lie over λ = ∞ . Adams, Harnad, Hurtubise gave a rule how to construct the complete set12f variables of separation in this special case. In the proposed procedure of separation of variablesthis question arises if the degree of B is less then N , then a proper similarity transformation ofthe L -matrix solves the problem.We see that all ideas of constructing variables of separation receive simple and obvious ex-planations by means of the orbit method. It allows to obtain the relations producing variablesof separation in a natural way by restriction to an orbit and changing variables from dynamic tospectral. O s The orbit O s equipped with the second Lie-Poisson bracket (3) has the Poisson structure: { L ( m ) ij , L ( n ) kl } s = L ( m + n +1 −N ) kj δ il − L ( m + n +1 −N ) il δ kj , (28)or in terms of the r -matrix (10) { L ( u ) ⊗ , L ( v ) } s = − [ r ( u − v ) , v N L ( u ) + u N L ( v )] . We parameterize the orbit O s by the same dynamic variables { γ ( m )1 , γ ( m )2 , γ ( m )3 , β ( m )1 , α ( m )1 , α ( m )2 : m = 0 , , . . . , N − } , namely: we again eliminate the set { β ( m )2 , β ( m )3 } . Due to linearityof the orbit equations (8) in the eliminated variables we write them in the matrix form c s = S − β + η − s , (29)where S − = S . . . S S . . . ... ... . . . ... ... S N − S N − . . . S
00 0 . . . F N , β = β β ... β N − β N , c s = c s c s ... c s N − c f N , η − s = η s η s ... η s N − η f N , S j = (cid:20) γ ( j )2 γ ( j )3 Γ ( j )2 Γ ( j )3 (cid:21) , β ( j ) = (cid:20) β ( j )2 β ( j )3 (cid:21) , c s j = (cid:20) c j d j (cid:21) , η s j = (cid:20) η j H j (cid:21) . Supposing S is nonsingular, we eliminate the variables ββ = ( S − ) − ( c s − η − s ) , or β β ... β N − β N = S − . . . e S S − . . . ... ... . . . ... ... e S N − e S N − . . . S −
00 0 . . . F − N c s − η s c s − η s ... c s N − − η s N − c f N − η f N , e S n = S − n X k =1 (cid:0) − S n +1 − k S − (cid:1) k , n = 1 , . . . , N − . Then substitute β into the Hamiltonians h N , h N +1 , . . . , h N − , f N , . . . , f N − h s = S + β + η + s = S + ( S − ) − c s + η + s − S + ( S − ) − η − s , (30)where S + = S N S N − . . . S S g N +1 S N . . . S S ... ... . . . ... ... g N − g N − . . . S N S N − g N g N − . . . g N +1 g N g N . . . g N +2 g N +1 ... ... . . . ... ... . . . g N g N − , h s = h s N h s N +1 ... h s N − f N f N +1 ... f N − , η + s = η s N η s N +1 ... η s N − H N H N +1 ... H N − , h s j = (cid:20) h j f j (cid:21) . { c ν , d ν : ν = 0 , . . . , N − } .Now consider the spectral curve restricted to the orbit O s defined by (8). We write thefollowing set of equations for N points { ( λ k , w k ) } on the orbit w k = w k (cid:16) c + c λ k + · · · + c N − λ N − k + h N λ N k + · · · + h N − λ N − k + c N λ N (cid:17) ++ (cid:16) d + d λ k + · · · + d N − λ N − k + f N λ N k + · · · + f N − λ N − k + d N λ N k (cid:17) (31)or in the matrix form W − s c s + W + s h s = w cubed , W − s = W λ W . . . λ N − W λ N W W λ W . . . λ N − W λ N W ... ... . . . ... ... W N λ N W N . . . λ N − N W N λ N N W N , W k = [ w k ] , W + s = λ N W λ N +11 W . . . λ N − W λ N λ N +11 . . . λ N − λ N W λ N +12 W . . . λ N − W λ N λ N +12 . . . λ N − ... ... . . . ... ... ... . . . ... λ N N W N λ N +13 N W N . . . λ N − N W N λ N N λ N +13 N . . . λ N − N . Suppose all pairs { ( λ k , w k ) } are distinct points and the matrix W + s is nonsingular, then theHamiltonians can be computed by the formula h s = − ( W + s ) − W − s c s + ( W + s ) − w cubed . (32)On an orbit O s of the second type the formulas (30) and (32) define the same set of functions,and both of them are linear in { c ν , d ν : ν = 0 , . . . , N } . As { c ν , d ν } are independent parametersone can equate the corresponding terms, that is S + ( S − ) − = − ( W + s ) − W − s , η + s − S + ( S − ) − η − s = ( W + s ) − w cubed ⇒ W + s S + + W − s S − = 0 , W + s η + s + W − s η − s = w cubed . (33)The matrix equations (33) give the equations (18c), connecting the dynamic variables { γ , Γ , γ , Γ } with the spectral variables { λ , w } , and the simplification (20) of the spectral curveequation at the points { ( λ k , w k ) } . Evidently, we obtain the consistent equation (19), definingthe polynomial B . Separation of variables theorem 2.
Suppose the orbit O s is parameterized by the variables { γ ( m )1 , β ( m )1 , α ( m )1 , γ ( m )3 , γ ( m )2 , α ( m )2 : m = 0 , . . . , N − } as above. Then the new variables { ( λ k , w k ) : k = 1 , . . . , N } defined by the formulas B ( λ k ) = 0 , w k = A ( λ k ) , (34) where B is the polynomial of degree N and A is the algebraic function given by the expressions (22) , have the following properties: (i) a pair ( λ k , w k ) is a root of the characteristic polynomial (13) . (ii) a pair ( λ k , w k ) is quasi-canonically conjugate with respect to the second Lie-Poisson bracket (3) : { λ k , λ l } s = 0 { λ k , w l } s = − λ N k δ kl , { w k , w l } s = 0; (35)(iii) the corresponding Liouville 1-form is Ω s = − X k λ −N k w k dλ k . roof. (i) The proof repeats one for Separation of variables theorem 1.(ii) The assertion follows from the lemmas below. Conjugate variable lemma 2. If B and A satisfy the following identities with respect to thesecond Lie-Poisson bracket (28) {B ( u ) , B ( v ) } s = 0 , {A ( u ) , A ( v ) } s = 0 , {A ( u ) , B ( v ) } s = − f ( u, v ) v N B ( u ) − u N B ( v ) u − v , where f is an arbitrary function such that lim v → u f ( u, v ) = 1 , then the variables { ( λ k , w k ) } definedby B ( λ k ) = 0 , w k = A ( λ k ) are quasi-canonically conjugate with respect to {· , ·} s : { λ k , λ l } s = 0 , { λ k , w l } s = − λ N k δ kl , { w k , w l } s = 0 . Proof.
It is similar to the proof of Conjugate variable lemma 1. Using (24) one can easily compute { λ k , λ l } s = {B ( λ k ) , B ( λ l ) } s B ′ ( λ k ) B ′ ( λ l ) = 0 , { w k , λ l } s = lim u → λ k v → λ l (cid:18) − B ′ ( v ) {A ( u ) , B ( v ) } s + A ′ ( u ) B ′ ( u ) B ′ ( v ) {B ( u ) , B ( v ) } s (cid:19) == f ( λ k , λ l ) λ N l B ( λ k ) − λ N k B ( λ l )( λ k − λ l ) B ′ ( λ l ) = λ N k δ kl , { w k , w l } s = lim u → λ k v → λ l (cid:18) − A ′ ( u )[ f ( v, u ) u N B ( v ) − v N B ( u )] B ′ ( u )( v − u ) ++ A ′ ( v )[ f ( u, v ) v N B ( u ) − u N B ( v )] B ′ ( v )( u − v ) (cid:19) = − (cid:18) A ′ ( λ k ) B ′ ( λ k ) − A ′ ( λ l ) B ′ ( λ l ) (cid:19) λ N k δ kl = 0 , as required. A - B bracket lemma 2. For B and A defined by (22) the following identities are true withrespect to the second Lie-Poisson bracket (28) {B ( u ) , B ( v ) } s = 0 , {A ( u ) , A ( v ) } s = 0 , {A ( u ) , B ( v ) } s = − f ( u, v ) v N B ( u ) − u N B ( v ) u − v , where f ( u, v ) = γ ( v ) /γ ( u ) for (22b) , and f ( u, v ) = γ ( v ) /γ ( u ) for (22c) .Proof. It repeats the proof of A - B bracket lemma 1.Using the second Lie-Poisson bracket in the form { L ij ( u ) , L kl ( v ) } f = − v N L kj ( u ) − u N L kj ( v ) u − v δ il + v N L il ( u ) − u N L il ( v ) u − v δ kj , one obtains { γ ( u ) , Γ ( v ) } = −{ γ ( u ) , Γ ( v ) } = v N u − v (cid:12)(cid:12)(cid:12)(cid:12) γ ( u ) γ ( u ) γ ( v ) γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) , { Γ ( u ) , γ ( v ) } = −{ Γ ( u ) , γ ( v ) } = − u N u − v (cid:12)(cid:12)(cid:12)(cid:12) γ ( u ) γ ( u ) γ ( v ) γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) , { Γ ( u ) , Γ ( v ) } = 1 u − v (cid:18) − u N (cid:12)(cid:12)(cid:12)(cid:12) γ ( u ) γ ( u )Γ ( v ) Γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) + v N (cid:12)(cid:12)(cid:12)(cid:12) γ ( v ) γ ( v )Γ ( u ) Γ ( u ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) , { γ ( u ) , B ( v ) } = − v N γ ( v ) u − v (cid:12)(cid:12)(cid:12)(cid:12) γ ( u ) γ ( u ) γ ( v ) γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) , { Γ ( u ) , B ( v ) } = 1 u − v (cid:18) − u N γ ( u ) B ( v ) + v N γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) γ ( v ) γ ( v )Γ ( u ) Γ ( u ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . Γ ( u ) { γ ( u ) , B ( v ) } − γ ( u ) { Γ ( u ) , B ( v ) } = 1 u − v (cid:2) γ ( u ) u N B ( v ) − γ ( v ) v N B ( u ) (cid:3) . From Leibniz’s rule for a Poisson bracket with A ( u ) = − Γ ( u ) /γ ( u ) one gets {A ( u ) , B ( v ) } = − u − v (cid:16) γ ( v ) γ ( u ) v N B ( u ) − u N B ( v ) (cid:17) . Other identities from the lemma statement are computed in the similar way.(iii) The Liouville 1-form on O s is implied by (35): Ω s = − X k λ −N w k dλ k . Reduction to a Liouville torus is realized by fixing values of the Hamiltonians h N , h N +1 , . . . , h N − , f N , f N +1 , . . . , f N − . On the torus every w k is an algebraic function of the conjugatevariable λ k due to (31). After this reduction the form Ω s becomes a sum of meromorphicdifferentials on the Riemann surface P ( w, λ ) = 0 .This completes the proof of Separation of variables theorem 2.Further, we construct integrable systems on the orbits O f and O s : a coupled 3-componentnonlinear Schr¨odinger equation and an isotropic SU(3) Landau-Lifshitz equation. O f and O s This equation as an integrable system arises on the orbit O f from the Hamiltonian flows generatedby h N − , h N − . In general, every Hamiltonian with respect to the first bracket (9) gives rise toa nontrivial flow on M : ∂L ( m ) a ∂τ = { L ( m ) a , H} f , (36)where H runs over the set { h , h , . . . , h N − , f , f , . . . , f N − } . We write the flows generatedby h N − , h N − in the Lax form ∂ L ( λ ) ∂x = [ L ( λ ) , ∇ − h N − ] = [ ∇ N − h N − , L ( λ )] , (37a) ∂ L ( λ ) ∂t = [ L ( λ ) , ∇ − h N − ] = [ ∇ N − h N − , L ( λ )] , (37b)where ∇ k denotes the matrix gradient with respect to the bilinear form h· , ·i k : ∇ k H = N − X m =0 dim g X a =1 ∂ H ∂L ( m ) a Z k − ma , where L ( m ) a = h L ( λ ) , Z k − ma i k . The matrix gradients ∇ N − h N − , ∇ N − h N − are used instead of ∇ − h N − , ∇ − h N − due totheir simplicity: ∇ N − h N − = λ L ( N ) + L ( N − , ∇ N − h N − = λ L ( N ) + λ L ( N − + L ( N − , where L ( m ) denotes the matrix coefficient of L of power m .16he chosen Hamiltonian flows commute, that can be expressed through the compatibilitycondition which is the zero curvature equation ∂ ∇ − h N − ∂t − ∂ ∇ − h N − ∂x + [ ∇ − h N − , ∇ − h N − ] = 0 . Now we show how to obtain a coupled 3-component nonlinear Schr¨odinger equation form(37). The coefficients of power N in λ displays that L ( N ) is constant in x and t , we assign L ( N ) = diag (cid:0) α ( N )1 , α ( N )2 − α ( N )1 , − α ( N )2 (cid:1) . The coefficient of power N − from (37a) gives ex-pressions for the dynamic variables { L ( N − a } in terms of { L ( N − a , ∂ x L ( N − a } : β ( N − = ∂ x β ( N − α ( N )1 − α ( N )2 γ ( N − = − ∂ x γ ( N − α ( N )1 − α ( N )2 β ( N − = ∂ x β ( N − α ( N )2 − α ( N )1 γ ( N − = − ∂ x γ ( N − α ( N )2 − α ( N )1 β ( N − = ∂ x β ( N − α ( N )1 + α ( N )2 γ ( N − = − ∂ x γ ( N − α ( N )1 + α ( N )1 . The rest of equations show that α ( N − , α ( N − are constant, we compute them from the orbitequations h N − = c N − , f N − = d N − . The coefficient of power N − from (37a) allows toexpress { L ( N − a } in terms of { L ( N − a , ∂ x L ( N − a , ∂ xx L ( N − a } , in particular: (cid:0) α ( N )1 − α ( N )2 (cid:1) β ( N − = ∂ xx β ( N − α ( N )1 − α ( N )2 − (cid:0) α ( N − − α ( N − (cid:1) ∂ x β ( N − α ( N )1 − α ( N )2 ++ β ( N − ∂ x γ ( N − α ( N )2 − α ( N )1 + γ ( N − ∂ x β ( N − α ( N )1 + α ( N )2 + (cid:0) α ( N − − α ( N − (cid:1) β ( N − , (cid:0) α ( N )2 − α ( N )1 (cid:1) β ( N − = ∂ xx β ( N − α ( N )2 − α ( N )1 − (cid:0) α ( N − − α ( N − (cid:1) ∂ x β ( N − α ( N )2 − α ( N )1 −− β ( N − ∂ x γ ( N − α ( N )1 − α ( N )2 − γ ( N − ∂ x β ( N − α ( N )1 + α ( N )2 + (cid:0) α ( N − − α ( N − (cid:1) β ( N − , (cid:0) α ( N )1 + α ( N )2 (cid:1) β ( N − = ∂ xx β ( N − α ( N )1 + α ( N )2 − (cid:0) α ( N − + α ( N − (cid:1) ∂ x β ( N − α ( N )1 + α ( N )2 −− β ( N − ∂ x β ( N − α ( N )2 − α ( N )1 + β ( N − ∂ x β ( N − α ( N )1 − α ( N )2 + (cid:0) α ( N − + α ( N − (cid:1) β ( N − . (38)The equations for α ( N − and α ( N − are easily integrated and give α ( N − = − β ( N − γ ( N − α ( N )1 − α ( N )2 − β ( N − γ ( N − α ( N )1 + α ( N )2 + C ,α ( N − = − β ( N − γ ( N − α ( N )2 − α ( N )1 − β ( N − γ ( N − α ( N )1 + α ( N )2 + C . The constants C , C are computed from the orbit equations h N − = c N − , f N − = d N − . Remark 4.
From (37a) one obtains expressions for all dynamic variables in terms of { L ( N − a } and their derivatives with respect to x . Next, we write the coefficient of power
N − from (37b): ∂β ( N − ∂t = (cid:0) α ( N )1 − α ( N )2 (cid:1) β ( N − ∂γ ( N − ∂t = − (cid:0) α ( N )1 − α ( N )2 (cid:1) γ ( N − β ( N − ∂t = (cid:0) α ( N )2 − α ( N )1 (cid:1) β ( N − ∂γ ( N − ∂t = − (cid:0) α ( N )2 − α ( N )1 (cid:1) γ ( N − ∂β ( N − ∂t = (cid:0) α ( N )1 + α ( N )2 (cid:1) β ( N − ∂γ ( N − ∂t = − (cid:0) α ( N )1 + α ( N )2 (cid:1) γ ( N − and substitute the expressions from (38) instead of { L ( N − a } .Assigning α ( m )1 , = i a ( m )1 , , γ ( m )1 , , = − (cid:0) β ( m )1 , , (cid:1) ∗ we restrict the system to an su (3) loop algebra,and put c N − = d N − = c N − = d N − = 0 implying a ( N − = a ( N − = 0 and C = C = 0 . Thefinal equations for β ( N − , , (the superscripts are omitted, namely: β ( N − , , = β , , , a ( N )1 , = a , ) i ∂β ∂t = ∂ xx β a − a − β ∂ x β ∗ (cid:0) a − a (cid:1) − β ∗ ∂ x β (cid:0) a + a (cid:1) ++ (cid:18) | β | a − a + | β | a + a − | β | a − a (cid:19) β , i ∂β ∂t = ∂ xx β a − a + β ∂ x β ∗ (cid:0) a − a (cid:1) + β ∗ ∂ x β (cid:0) a + a (cid:1) ++ (cid:18) | β | a − a + | β | a + a − | β | a − a (cid:19) β , i ∂β ∂t = ∂ xx β a + a − β ∂ x β (cid:0) a − a (cid:1) + β ∂ x β (cid:0) a − a (cid:1) ++ (cid:18) | β | a − a + | β | a − a + 2 | β | a + a (cid:19) β (39)are the same as presented in [16], called there the ‘3-wave hierarchy generalization of the nonlinearSchr¨odinger equation’. Here we call them a coupled 3-component nonlinear Schr¨odinger equation . This equation arises on the orbit O s from the Hamiltonian flows generated by h N , h N +1 . Withrespect to the second bracket (28) every Hamiltonian H from the set { h N , h N +1 , . . . , h N − , f N , f N +1 , . . . , f N − } gives rise to a nontrivial flow on M : ∂L ( m ) a ∂τ = { L ( m ) a , H} s . (40)We write the flows of h N , h N +1 in the Lax form: ∂ L ( λ ) ∂x = [ ∇ N − h N , L ( λ )] = [ L ( λ ) , ∇ − h ] , (41a) ∂ L ( λ ) ∂t = [ ∇ N − h N +1 , L ( λ )] = [ L ( λ ) , ∇ − h ] , (41b)where for the sake of simplicity we use expressions with the matrix gradients ∇ − h = λ − L (0) , ∇ − h = λ − L (1) + λ − L (0) . This system is also constructed in the loop algebra su (3) , and we change the basis { Z a } intothe Gell-Mann basis { X a : a = 1 , . . . , } (see [17]): Tr X a X b = − δ ab , [ X a , X b ] = f abc X c , X a X b + X b X a = − δ ab I − d abc X c , f abc = − X c [ X a , X b ] , d abc = Tr X c (cid:0) X a X b + X b X a (cid:1) . As above for dynamic variables we use the coordinates corresponding to the basis elements withrespect to the bilinear form, that is { µ ( m ) a = h L ( λ ) , X N − − ma i N − } serve as dynamic variables forthe system on the orbit O s . The matrix coefficient of L of power m has the form L ( m ) = i µ ( m )3 + √ µ ( m )8 µ ( m )1 − i µ ( m )2 µ ( m )4 − i µ ( m )5 µ ( m )1 + i µ ( m )2 − µ ( m )3 + √ µ ( m )8 µ ( m )6 − i µ ( m )7 µ ( m )4 + i µ ( m )5 µ ( m )6 + i µ ( m )7 − √ µ ( m )8 , { µ ( m ) a } and { α ( m )1 , , β ( m )1 , , , γ ( m )1 , , } . The Poisson structure interms of the new dynamic variables is given by { µ ( m ) a , µ ( n ) b } s = − f abc µ ( m + n +1 −N ) c . The orbit O s is defined by the equations (in matrix and vector notations) Tr (cid:0) L (0) (cid:1) = − µ (0) a µ (0) a = c Tr (cid:0) L (0) (cid:1) = − d abc µ (0) a µ (0) b µ (0) c = d Tr L (0) L (1) = − µ (0) a µ (1) a = c Tr L (0) L (0) L (1) = − d abc µ (0) a µ (0) b µ (1) c = d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tr P m + n = N L ( m ) L ( n ) = Tr P m + n + k = N L ( m ) L ( n ) L ( k ) == − P m + n = N µ ( m ) a µ ( n ) a = c N , = − P m + n + k = N d abc µ ( m ) a µ ( n ) b µ ( k ) c = d N , where indices appearing twice imply summation. The Hamiltonian equations obtained from (41)have the following vector and matrix forms: ∂µ ( m ) a ∂x = − f abc µ ( m +1) b µ (0) c , ∂µ ( m ) a ∂t = − f abc (cid:0) µ ( m +1) b µ (1) c + µ ( m +2) b µ (0) c (cid:1) (42a) ∂ L ( m ) ∂x = [ L ( m +1) , L (0) ] , ∂ L ( m ) ∂t = [ L ( m +1) , L (1) ] + [ L ( m +2) , L (0) ] . (42b)We denote the variables { µ (0) a } by { µ a } , and L (0) by M , and introduce the variables { T a = d abc µ b µ c } or in the matrix form T = M − c I . The stationary equation at m = 0 looks like ∂ M ∂x = ad L (1) M , M , L (1) ∈ g ∗ , where ad is the adjoint action in the Lie algebra g . This equation can be easily solved on theorbit O s , where M = c M + d I or in the vector form d abc d cpq µ b µ p µ q = c µ a , namely: µ (1) a = c − d f abc (cid:16) c T b T c,x − d T b µ c,x + c µ b µ c,x (cid:17) ++ c c − d d c − d µ a +
92 2 d c − c d c − d T a or L (1) = − c − d (cid:16) c [ T , T x ] − d [ T , M x ] + c [ M , M x ] (cid:17) ++ c c − d d c − d M +
92 2 d c − c d c − d T . Commutativity of the chosen Hamiltonian flows implies the compatibility condition in the zerocurvature form ∂ ∇ N − h N ∂t − ∂ ∇ N − h N +1 ∂x + [ ∇ N − h N , ∇ N − h N +1 ] = 0 , that gives in particular ∂µ a ∂t = ∂µ (1) a ∂x or ∂ M ∂t = ∂ L (1) ∂x . Then we get the equation ∂µ a ∂t = c − d f abc (cid:16) c T b T c,xx − d [ T b µ c,xx + µ b T c,xx ] + c µ b µ c,xx (cid:17) ++ c c − d d c − d µ a,x +
92 2 d c − c d c − d T a,x or (43a) ∂ M ∂t = − c − d (cid:16) c [ T , T xx ] − d ([ T x , M x ] + [ T , M xx ]) + c [ M , M xx ] (cid:17) + c c − d d c − d M x +
92 2 d c − c d c − d T x . (43b)The obtained equation is similar to the Landau-Lifshitz equation for an isotropic SU(2) magnet,and so we call it a generalized Landau-Lifshitz equation for an isotropic SU(3) magnet , for moredetails see [18]. 19 Conclusion and Discussion
We briefly summarize the proposed separation of variables procedure. Recall that we deal withan integrable system constructed on a coadjoint orbit of a loop Lie algebra, and we use theCartan-Weyl basis. The key point of the proposed procedure is restriction to an orbit locatedin the dual space to the loop algebra. We realize this restriction by eliminating a subset ofdynamic variables corresponding to nilpotent commuting basis elements. On the other hand, weparameterize the orbit by a sufficient number of points of the spectral curve det (cid:0) L ( λ ) − w I (cid:1) = 0 ,where L is the Lax matrix of the system. Thus, we obtain two representations for every point ofthe orbit: in the dynamic and the spectral variables. It is possible to introduce these variablesso that the map between them is biunique. The spectral variables are proven to be variables ofseparation.In our opinion, the orbit approach allows to ‘elucidate the geometric and algebraic meaningof the construction’, that was declared as an unsolved problem in [12]. Moreover, the procedurecan be easily extended to generic orbits of sl ( n ) loop algebras, the only problem is cumbersomecomputations. Another open question is an extension of the orbit approach to degenerate orbits.Though the latter have a simpler geometry than generic orbits, it is difficult to define them byexplicit equations, that causes problems with applying the proposed procedure. References [1] Sklyanin E. K., Progr. Theor. Phys. Suppl. , (1995) 35–60.[2] Kirillov. A. A., Bull. Amer. Math. Soc. , (1999) 433–488.[3] Adams M. R., Harnad J., Hurtubise J., Commun. Math. Phys. , (1993) 385–413.[4] Adams M. R., Harnad J., Hurtubise J., Lett. Math. Phys. , (1997) 41–57.[5] Blaszak M., J. Math. Phys. , (1998) 3213.[6] Falqui G., Magri F., Pedroni M. and Zubelli J.-P., Reg. and Chaotic. Dyn. , (2000) 33–51.[7] Falqui G., Magri F. and Tondo G., Theor. Math. Phys. , (2000) 176–192.[8] Falqui G., Mathematical Physics, Analysis and Geometry , (2003) 139–179.[9] Magri F., Lect Notes in Phys. , (1997) 256-296.[10] Harnad J., Hurtubise J., J. Math. Phys. , (2008) 062903.[11] Bernatska J., Holod P., JNMP , (2007) 353–374[12] Sklyanin E. K., Commun. Math. Phys. , (1992) 181–191.[13] Scott D. R. D., J. Math. Phys. , (1994) 5831–5843.[14] Gekhtman M. I., Commun. Math. Phys. , (1995) 593–605.[15] Adler M. and van Moerbeke P., Adv. Math. , (1980) 318–379.[16] Fordy A. P., Kulish P. P., Commun. Math. Phys. , (1983), 427–443[17] Macfarlane A. J., Sudbery A., Weisz P. H., Commun. Math. Phys. , (1968) 77–90.[18] Bernatska J., Holod P., J. Phys. A: Math. Theor.42