Backlund transformations for the elliptic Gaudin model and a Clebsch system

B¨ ac klund transformations fo r t he ellipt ic Gaudin mo de l and a Clebs c h system F ederico Zullo Dipartimen to di Fis ica, Univ ersit` a di Roma T re Istituto Nazionale di F isica Nucleare, sezione di Roma T re Via V asca Nav ale 84, 00146 R oma, Italy E-mail: zullo@fis.uniroma3.it Abstract A t wo-paramete r s family of B¨ ac klund transformations for the classica l elliptic Gaudin m o del is constructed. The maps are exp licit, symplectic, preserv e the same in tegrals as for the con tin u ou s flows and are a time discretization of eac h of these flo ws. The transformations can m ap real v ariables in to real v ariables, sending p h ysical solutions of the equations of motion into ph ysical solutions. The starting p oin t of the analysis is the in tegrabilit y structure of the mo del. It is sho wn ho w the analogue trans f ormations for the rational and trigonometric Gaudin mo del are a limiting case of this one. An application to a particular case of the Clebsc h system is given. KEYW ORDS: B¨ ac klund T ransformations, Integrable maps, Ga udin mo dels, Clebsc h system, Lax represen tation, r -matrix. 1 In tro ducti o n The G audin models describ e completely inte g rable long r ange spin-spin systems, b o th at the classic a l and at the quan tum lev el. They were first in tro duced b y G audin [7], as the anisotro pic, in tegrable generalization of what to day is called the xxx , or isotropic, Gaudin mo del; the results of the analysis w ere the construction of the xxz mo del, for the partially anisotropic case, and of the xy z mo del, for the fully anisotropic case. The Lax matrices depend on the spectral parameter respectiv ely through rational, trigo no- metric and elliptic functions. The P oisson structure of the mo dels can b e sp ecified in the framew ork of the r - ma t r ix approach [2 2]; in [24] Sklyanin sho w ed how to obtain b oth the La x matr ices a nd the r -matr ices of the mo dels b y a limiting pro cedure on the lattice Heisen b erg mag net. Ob viously the in tegra bility structures of the trigonometric and rational cases can b e also obtained as particular realizations of the corresp onding structures of the more general elliptic mo del. F rom the p oin t of view of separation of 1 v ariables, f unctiona l Bethe ansatz and quantum in v erse scattering metho d, a lots of results app eared in the late 80s-early 90s [10] [9] [23] [28] [29]. On the other hand a great num b er of w orks hav e p oin ted up until now v ery in teresting connections b et w een the G audin mo dels and v arious branc hes of ph ysics. F or a recen t list of the sub jects in v olved t he reader can see for example [18]. Here, in order to giv e an idea o f t he heterogeneit y of the concerned issue s, w e wish only to men tion the connections with the BCS theory and small metallic gr a ins [1], with pairing mo dels in n uclear phys ics [6], with the Coulomb three-b o dy pro blem [12], with Lagrange and Kirchhoff tops [16]. In [8], [2 5],[26] the authors ga ve the B¨ ac klund transformations for t he r ational Gaudin mo del in the framew o rk of a larger researc h program started some y ears b efore b y Kuznetso v and Skly anin (see [14]) on the applications and prop erties of suc h transfor- mation to finite dimensional in tegrable systems. In par t icular in [8] it was shown ho w the t wo-parameters family of tra nsformations obtained can b e seen as a time discretiza- tions of a family of flo ws and that the interpolating Hamilto nian flow encompasses a ll the commuting flo ws of the mo del. The features of the tr a nsformations f o und here are notew orthy : in fact the maps are (i) explicit, (ii) symplectic, (iii) they preserv e all the in tegrals of the con tinuous mo del, (iv) they p ossess the so called sp e ctr ality prop ert y [13] [14]. F urthermore they commute , tha t is the comp osition of tw o differen t trans- formations do es not dep end on the order of a pplicatio n, and with a tw o - parameters transformation it is p ossible to send real v ariable in to real v ariables (so that t he B¨ ack- lund transformations send a real solution of the equations of motion into another real solution). In the w ake of these results in 2010 the analo gue constructions for the trigonometric case w ere found [19] [20]. All the ab ov e pro p erties hold true in this case to o. In [20] the authors ha v e also sho wn how , in the limit of small angles, the transformations obta ined giv e exactly t he results of the rational case as giv en in [8]. The aim of this work is to complete the picture finding the B¨ ac klund transformations and the corresp onding time-discretization for the elliptic Gaudin mo del. The pap er is organized as follows: in section (2) the main f eat ures and the integrabilit y structures of the elliptic Gaudin mo del are briefly revised, in section (3) the dr e ssing matrix is constructed; the pa rametrization obtained g iv es directly the dressing mat r ix of the trigonometric case [20] when the elliptic mo dulus k of the Jacobi elliptic function is set equal t o zero. In section (4) the exp licit form of the t r ansformations is found b y using the sp ectrality prop erty . The symplecticit y of the maps is also discus sed. F or an y fixed set of initial conditions the t r a nsformations turn out to b e birat io nal maps. In section (5) it is shown how it is p ossible to obtain a t w o- parameters family o f phy sical transformations, that is tr ansformations sending real v ariables in to real v ariables. In section (6) the con tinuous limit of the discrete dynamics is f ound: the in terp olating Hamiltonian flow aga in encompasses all the comm uting contin uous flows of the mo del. Lastly , as an application of the results found, w e construct the B¨ ack lund tra nsfor- mations for a particular case of the Clebsc h system [4] through a pro cedure of p ole coalescence on t he Lax matrix of t he Gaudin mo del and taking adv an tage of the fact that this pro cedure preserv es the r -matrix structure. The results are a generalization of the B¨ ac klund transformations for the Kirc hhoff top as giv en in [21]. 2 2 The ellipt i c Gaud in magnet The elliptic Gaudin mo del is defined b y the follow ing Lax matrix: L ( λ ) =  A ( λ ) B ( λ ) C ( λ ) − A ( λ )  (1) A ( λ ) = N X j =1 cn( λ − λ j ) sn( λ − λ j ) s z j , B ( λ ) = N X j =1 s x j − i s y j dn( λ − λ j ) sn( λ − λ j ) (2) C ( λ ) = N X j =1 s x j + i s y j dn( λ − λ j ) sn( λ − λ j ) . In (1) and (2) λ ∈ C is the spectral parameter, λ j are a set of N arbitrar y real parameters of the mo del, while  s x j , s y j , s z j  , j = 1 , . . . , N , are the dynamical v ariables (spins) of the system. In the r -matrix for ma lism the P oisson structure is fixed b y the follo wing equiv alence [28]:  L ( λ ) , L ( µ )  =  r e ( λ − µ ) , L ( λ ) ⊗ 1 + 1 ⊗ L ( µ )  , (3) where r e is the elliptic solution t o the classic a l Y ang-Baxter equation of Bela vin and Drinfel’d [5],[3],[28]: r e ( λ ) = i sn( λ )      cn( λ ) 0 0 1 − dn( λ ) 2 0 0 1+dn( λ ) 2 0 0 1+dn( λ ) 2 0 0 1 − dn( λ ) 2 0 0 cn( λ )      , (4) The r -matrix structure (3) en tails t he follow ing P oisson brac kets for the functions ( 2): { A ( λ ) , A ( µ ) } = 0 , { B ( λ ) , B ( µ ) } = i ( A ( λ ) + A ( µ )) dn( λ − µ ) − 1 sn( λ − µ ) { C ( λ ) , C ( µ ) } = i ( A ( λ ) + A ( µ )) 1 − dn( λ − µ ) sn( λ − µ ) { A ( λ ) , B ( µ ) } = i C ( λ )(1 − dn( λ − µ )) − B ( λ )(1 + dn( λ − µ )) + 2 B ( µ )cn( λ − µ ) 2sn( λ − µ ) { A ( λ ) , C ( µ ) } = i B ( λ )(dn( λ − µ ) − 1) + C ( λ )(1 + dn( λ − µ )) − 2 C ( µ )cn( λ − µ ) 2sn( λ − µ ) { B ( λ ) , C ( µ ) } = i ( A ( µ ) − A ( λ ))(1 + dn( λ − µ )) sin( λ − µ ) . (5) Equiv alen tly , the spin v ariables  s x j , s y j , s z j  , j = 1 ..N , ha ve to ob ey to the corresp onding algebra:  s x j , s y k  = δ j k s z k ,  s y j , s z k  = δ j k s x k ,  s z j , s x k  = δ j k s y k (6) 3 Due to the direct sum structure o f the P oisson brack et (6), the square length o f eac h spin is a Casimir f unction fo r the elliptic Gaudin mo del, so w e ha v e N Casimirs, giv en b y ( s x j ) 2 + ( s y j ) 2 + ( s z j ) 2 . = s 2 j j = 1 ..N F urthermore the mo del p ossesse s N integrals of motion in inv o lutions w.r.t. the Pois- son bra ck ets (6): the determinan t of the Lax matrix is a generating function of suc h in tegrals (see A): − det( L ) = A 2 ( λ ) + B ( λ ) C ( λ ) = N X i =1  s 2 i sn 2 ( λ − λ i ) + 2  H i ζ (  ( λ − λ i ))  − H 0 (7) where ζ is the W eierstraß zeta function,  = ( e 1 − e 2 ) − 1 2 , e i = ℘ ( w i 2 ) and ( w 1 , w 2 ) are the p erio ds of the W eierstraß ℘ function (see A). The N Ha milto nians H i are giv en b y: H i = N X k 6 = i s z i s z k cn( λ i − λ k ) + s y i s y k dn( λ i − λ k ) + s x i s x k sn( λ i − λ k ) (8) Note that only N − 1 among these Hamiltonians are indep enden t, b ecause of P i H i = 0. The other in tegra l H 0 is giv en by the form ula (see A): H 0 = N X i,k ( s z i s z k dn( λ i − λ k ) + k 2 s y i s y k cn( λ i − λ k ))+ + N X k 6 = i a ( λ i − λ k )  s z i s z k cn( λ i − λ k ) + s y i s y k dn( λ i − λ k ) + s x i s x k  sn( λ i − λ k ) , a ( λ ) . =   ζ (  λ ) − ζ (2  λ )  − 1 sn(2 λ ) (9) Due t o t he existe nce of an r -matrix, the Hamiltonians H i are in in v olution for the P oisson bra c k et (6): { H i , H j } = 0 i, j = 0 , . . . , N − 1 (10) The corresp onding Hamilto nian flow s a r e then giv en b y: ds 3 j dt i = { H i , s 3 j } ds ± j dt i = { H i , s ± j } (11) 3 The dress i ng matrix By an Hamiltonian p oint of view, B¨ ack lund T ransformations fo r finite-dimensional in tegrable systems ar e (f a milies of ) symplectic maps preserving the integrals of motion [13]: so we are searching for an expression relating the dynamical v ariables  s x j , s y j , s z j  to the new set of v ariables  ˜ s x j , ˜ s y j , ˜ s z j  and suc h that the brac k ets (6) and the in tegrals (8) are preserv ed. The generating function of the integrals is giv en b y the determinan t 4 of the Lax matrix (1), so the dressed Lax matrix ˜ L ( λ ), that is the Lax matrix of the tilded dynamical v ariables, has to ha v e t he same determinan t of L ( λ ). This means that the tw o L a x matr ices are related by a similarity transformation b y means of a dr essing matrix D ( λ ): ˜ L ( λ ) D ( λ − λ 0 ) = D ( λ − λ 0 ) L ( λ ) (12) Here λ 0 will b e one of the tw o B¨ ac klund parameters: actually we will construct a parametric family of transformations and this will b e a crucial p oint when w e will request to the maps to b e “ph ysical”, that is to send real v ariables in real v a riables. The rational and tr ig onometric G audin mo dels are limiting case of the elliptic one. The dressing matrices for the Lax matrices of these mo dels are giv en, resp ectiv ely , b y the elemen tary La x matrix of the xxx and xxz Heisen b erg spin chain on the lat tice [8], [20]. It is obvious, for the dressing matrix o f the elliptic Gaudin mo del, to mak e the ansatz of the elemen tary La x matrix of the xy z Heisen b erg spin chain. Note ho wev er that this matrix has to enjoy the same symmetry prop erties with the Lax matrix (1). In fact, the quasi-p erio dicity (66) of the Jacobi elliptic functions en tails the follo wing form ulae f or L ( λ ) (see also [29]): L ( λ + 2 K ) = σ 3 L ( λ ) σ 3 L ( λ + 2 iK ′ ) = σ 1 L ( λ ) σ 1 (13) where with σ i , i = 1 , 2 , 3 w e indicate the P auli matrices, a nd K and K ′ are resp ectiv ely the complete elliptic integral of the first kind and the complemen tary integral (67). The ab ov e p oin ts suggest to mak e the follo wing a nsatz for D ( λ ) D ( λ ) = S 0 1 + i sn( λ )  S 1 σ 1 + dn ( λ ) S 2 σ 2 + cn ( λ ) S 3 σ 3  (14) This is exactly t he one-site Lax matrix for the xy z Heisen b erg spin c hain on the lattice [5]. The symmetries (13) are also preserv ed: D ( λ + 2 K ) = σ 3 D ( λ ) σ 3 D ( λ + 2 iK ′ ) = σ 1 D ( λ ) σ 1 (15) So far, S i , i = 0 .. 3 are four undetermined v ariables, but w e are fr ee to fix one of them b ecause of the homogeneity of t he equation (12). The La x matrix (1) has simple p oles at the p oints λ = λ j (mo d 2 K, 2 iK ′ ), j = 1 ..N ; t he relation (12) is an equiv alence b et we en meromorphic functions (the elemen ts of the matrices), so tha t w e hav e to equate the residue a t the p oles on b ot h sides. Because of the symmetries (13) and (15) w e can lo ok only at the p oles in λ = λ j , j = 1 ..N , that is: ˜ L j D j = D j L j (16) where L j =  s z j s x j − is y j s x j + is y j − s z j  , D j = D ( λ = λ j ) (17) In principle equation (16) g ives an implicit relationship b et w een the old (un tilded) v ariables and the new (tilded) ones. It is how ev er p ossible to get an explicit relationship b y recurring to the so-called spectrality prop erty [13] [14]. T o this aim, one need to 5 force the determinan t of the Darb oux matrix D ( λ ) to ha ve t w o nondynamic al zero es for t w o arbitrar y v alue of the sp ectral parameter λ , say f or λ = λ 0 ± µ . This lea ve s us with only two undetermined v ariables in (14). As w e will see, the sp ectralit y prop ert y will fixes these t w o v ariables, that w e will call P and Q , as functions of only the un tilded dynamical v ariables, so that the maps defined b y ( 1 6) will b e explicit. Summarizing, b y fixing for simplicit y S 0 = 1 and imp osing the constrain ts det ( D ( λ − λ 0 ))    λ = λ 0 ± µ = 0 w e a re left with tw o undetermined parameters, that we denote with P and Q : b y c ho osing a particular parametrization of the constrain ts w e can write: D ( λ ) = 1 + i S 3 cn( λ ) sn( λ ) i S 1 + S 2 dn( λ ) sn( λ ) i S 1 − S 2 dn( λ ) sn( λ ) 1 − i S 3 cn( λ ) sn( λ ) ! with          i S 3 = P Q − sn( µ ) cn( µ ) i S 1 + S 2 dn( µ ) = P i S 1 − S 2 dn( µ ) = Q (2 sn( µ ) − P Q ) (18) W e recall again that P and Q a re undetermined dynamical v aria ble and that λ 0 and µ are constan ts: they are parameters for the B¨ ac klund transformations. Note also that with this parametrization, in the limit k → 0, one obtains the dressing matrix for the trigonometric Gaudin mo del [19] (up to a trivial m ultiplicativ e factor inessen tial for the form of the B¨ ack lund transformation, as explained b efore). 4 B¨ ac klund transformations No w w e mak e use of the sp ectrality prop ert y to find P and Q in terms of only one set of v ariables, the un tilded ones. The matrices ( D ( λ − λ 0 ))    λ = λ 0 + µ and ( D ( λ − λ 0 ))    λ = λ 0 − µ are of r a nk one. W e call | Ω + i a nd | Ω − i t heir resp ectiv e k ernels. By acting with these k ernels on the equiv alence defining the B¨ a c klund t r a nsformations, w e see that they are also the eigen v ectors o f L ( λ 0 + µ ) and L ( λ 0 − µ ): ˜ L ( λ 0 ± µ ) D ( ± µ ) | Ω ± i = 0 = D ( ± µ ) [ L ( λ 0 ± µ ) | Ω ± i ] = ⇒ L ( λ 0 ± µ ) | Ω ± i = γ ± | Ω ± i (19) By viewing the generating function of the inte g rals (7) as a function of λ , w e define: γ 2 ( λ ) . = − det ( L ( λ )) = A ( λ ) 2 + B ( λ ) C ( λ ) (20) where A ( λ ), B ( λ ) and C ( λ ) are giv en by (2). The tw o eigen v alues are then giv en by γ ± = γ ( λ )    λ = λ 0 ± µ . The tw o k ernels | Ω ± i a re expressions of t he v ariables P and Q , so that the eigen v ectors relations ( 1 9) fo r L ( λ 0 ± µ ) link these v ariables with the elemen ts of the Lax matrix o f the un tilded v ariables. Explicitly , the tw o k ernels are giv en by: | Ω + i =  1 − Q  | Ω − i =  P 2 sn( µ ) − P Q  (21) 6 and these expressions in turns lead to the formulae: Q = Q ( λ 0 + µ ) = A ( λ ) − γ ( λ ) B ( λ )     λ = λ 0 + µ 1 P = Q ( λ 0 + µ ) − Q ( λ 0 − µ ) 2 sn( µ ) (22) Note tha t, for arbitrary n um b er N of in teracting spins of the mo del, P and Q contain al l the dynamical v a r ia bles so that the B¨ ac klund maps touc h all the spin sites. These maps asso ciate to a give n solution of the equations of motion (11) a new solution. By fixing the initial conditions, the generating function (7), and therefore the function γ ( λ ), is a constan t independen t of time. As will b e clear in the next lines, if γ ( λ ) is constan t, then the B¨ ac klund transformat io ns are actually rat io nal maps (or, b etter to sa y , birational maps). The equation (1 6) allo w us to write the explicit transformations as follo ws: ˜ s x k = (( α 2 k + ς 2 k − β 2 k − δ 2 k ) s x k + i ( δ 2 k + ς 2 k − β 2 k − α 2 k ) s y k − 2( α k β k − ς k δ k ) s z k ) 2∆ k ˜ s y k = ( i ( α 2 k + δ 2 k − ς 2 k − β 2 k ) s x k + ( β 2 k + ς 2 k + α 2 k + δ 2 k ) s y k − 2 i ( β k α k + ς k δ k ) s z k ) 2∆ k ˜ s z k = (( β k ς k − α k δ k ) s x k + i ( β k ς k + α k δ k ) s y k + ( α k ς k + β k δ k ) s z k ) ∆ k (23) where f or brevit y of notation w e hav e introduced the functions ( α k , β k , δ k , ∆ k , ς k ) de- fined b y the follow ing for mulae: α k = sn( λ k − λ 0 ) + P Q − sn( µ ) cn( µ ) cn( λ k − λ 0 ) β k = ( P + Q (2 sn( µ ) − P Q )) 2 + ( P − Q (2 sn( µ ) − P Q )) 2 dn( µ ) dn( λ k − λ 0 ) δ k = ( P + Q (2 sn( µ ) − P Q )) 2 − ( P − Q (2 sn( µ ) − P Q )) 2 dn( µ ) dn( λ k − λ 0 ) ς k = sn( λ k − λ 0 ) − P Q − sn( µ ) cn( µ ) cn( λ k − λ 0 ) ∆ k = α k ς k − β k δ k (24) When the elliptic mo dulus k of the Jacobi elliptic functions is zero, the tr a nsformations (23) coincide with those for the trigonometric Ga udin magnet as giv en in [19]. A t this p oin t w e hav e to deal with the symplec ticity o f our maps. As the transformatio ns are explicit, the direct path to prov e their symplec ticity migh t b e to use (6) in order t o sho w that indeed the P oisson structure is preserv ed. But this path is presumably not so pla in b ecause of t he h uge calculations. As in [19], w e will follow a finer argumen t due to Sklyanin [26]. C onsider the relation (12) in an extended phase space, whose co ordinate a r e giv en b y ( s x k , s y k , s z k , P , Q ) and supp ose that D ( λ ) ob eys to the quadr a tic P oisson bra c k et, as follows : { D ( λ ) ⊗ 1 , 1 ⊗ D ( τ ) } = [ r e ( λ − τ ) , D ( λ ) ⊗ D ( τ )] (25) 7 In the extended space we hav e to re-define (12) as: ˜ L ( λ ) ˜ D ( λ − λ 0 ) = D ( λ − λ 0 ) L ( λ ) (26) In fact in the left hand side of the previous one has to use tilded v ariables also for D ( λ ) b ecause (26) defines the B¨ acklund transformation in the extended pha se space, where there is also a ˜ P and a ˜ Q . Note that in the new phase space the en tries of D P oisson comm utes with those of L . The k ey observ atio n is that if b oth L and D hav e the same P oisson structure, giv en b y equation (25), then this prop ert y holds true for LD and D L as we ll, b ecause of the P oisson commutativit y of the en tries of L and D . This means that the transformatio n (26) defines a “canonical” transformation. Skly anin sho wed [26] that if one now restricts the v ariables on the constrain t manifold ˜ P = P and ˜ Q = Q the symplecticit y is preserv ed; ho we ver this constrain t leads to a dep endence of P and Q on the en tries of L , that for consistency must b e the same as the o ne giv en b y the equation (26) on this constrained manifold. But there (26) reduce to (12), so that the map preserv es the sp ectrum of L ( λ ) and is canonical. What remains to sho w is t hat indeed (2 5) is fulfilled b y our D ( λ ). F or the B¨ acklun d tr ansformations o f the rational Gaudin magnet the dressing matrix has the quadratic P oisson structure imposed b y the rational r -matrix pro vided P a nd Q are canonically conjugated in the extended space [26]. In the trigonometric case one need to hav e a non trivial brack et betw een P and Q in the extende d space in order to gua r a n tee the simplecticit y of B¨ acklun d transformations [19]. As w e will show in the next lines, in the elliptic case w e found a non trivial brac ke t that, in the limit k → 0 go es to the trigonometric result a s given in [19]. By a direct insp ection it is p ossible to sho w that (25) en tails the follow ing brac k ets b et we en the elemen ts S i , i = 0 .. 3 [5]: { S i , S 0 } = J j k S j S k { S i , S j } = − S 0 S k (27) where ( i, j, k ) is a cyclic p erm utatio n of (1 , 2 , 3) with J 12 = k 2 , J 23 = 1 − k 2 , J 31 = − 1. With the following p o sitions:                    ( S 0 ) 2 = cn( µ )dn( µ ) sn( µ )  1 − 2 sn( µ ) P Q + P 2 Q 2 − k 2 h ( P Q − sn( µ )) 2 + cn ( µ ) 2 ( Q ( P Q − 2 sn( µ ) ) − P ) 2 4 i i S 3 = P Q − sn( µ ) cn( µ ) S 0 i S 1 + S 2 dn( µ ) = P S 0 i S 1 − S 2 dn( µ ) = Q (2 sn( µ ) − P Q ) S 0 (28) after some calculations one can sho w that indeed (27) are fullfilled pro vided that : { Q, P } = i  1 − 2 sn( µ ) P Q + P 2 Q 2 − k 2 h ( P Q − sn( µ )) 2 + cn ( µ ) 2 ( Q ( P Q − 2 sn( µ )) − P ) 2 4 i cn( µ )dn( µ ) (29) 8 So the symplecticit y of our maps is prov ed. A t this p oin t let us to stress some remarks. The dressing matrix defined by the relations (28) is, except for the m ultiplicativ e factor S 0 , completely equiv alen t to the dressing matr ix as giv en b y the equation (18). As explained befo r e, given the homogeneit y of the equation (12), a f a ctor of prop ortionality b et we en tw o dressing matrices is inessen tia l as r ega rds B¨ a cklund transformatio ns, so b y this p oin t of view the definitions (28) are compatible with (18). Secondly , the Pois son brac k et (29 ) b etw een P and Q reduces exactly , in the limit k → 0, to the brack et of the corresp onding v ariables in the trig onometric case. 5 Ph ysical B¨ ac klund transformatio ns In this section w e will sho w that with an appropriat e c hoice of the parameters λ 0 and µ in (23), the B¨ ack lund transformations map real solutions in real solutions, so in this sense the transformations can b e considered “physic al” . The c hoice amounts to require λ 0 to b e a real n umber and µ to b e a purely imaginary num b er. So, hereafter in this section, w e put: µ = iǫ ( λ 0 , ǫ ) ∈ R 2 (30) The matrices L j , j = 1 ..N defined in (17) and corresp onding to the real solutions  s x j , s y j , s z j  of the equations of motio n are Hermitian. The request for ph ysical B¨ ac klund transformations is equiv a len t t o the request for Hermitian dressed ma t r ices ˜ L j . By (16) w e see that this means to ha v e dressing matrices D j prop ortional to unitary matrices. W e claim t hat indeed when (30) a re fulfilled then D j are of the for m: D j =  α j β j − ¯ β j ¯ α j  (31) where the ba r means complex conjugation. F or clarity let us make the following p osi- tions: λ + = λ 0 + iǫ, λ − = ¯ λ + (32) W e observ e that, for t he functions A, B , C , as defined in (2), one has: A ( λ + ) = ¯ A ( λ − ) , B ( λ + ) = ¯ C ( λ − ) , C ( λ + ) = ¯ B ( λ − ) . (33) These r elations entails γ 2 ( λ + ) = ¯ γ 2 ( λ − ). Note tha t this last relation implies t ha t the co efficien ts of the series o f γ 2 ( λ ) with resp ect to λ are real, consisten tly with the expansion (7). W e recall that the matrices D j are written in terms of P a nd Q , that are defined b y the relations Q = Q ( λ + ) = A ( λ + ) − γ ( λ + ) B ( λ + ) = − C ( λ + ) A ( λ + ) + B ( λ + ) P = 2 sn( iǫ ) Q ( λ + ) − Q ( λ − ) (34) By sp ecifying the sign of the function γ on the Riemann surface b y γ ( λ + ) = − ¯ γ ( λ − ), one has: ¯ Q ( λ + ) = − 1 Q ( λ − ) 9 and t his equation in turns implies that the matrices D j = D ( λ )    λ = λ j , with D ( λ ) g iv en b y (18) , ar e of the form (31), with α j and β j giv en b y the follo wing formulae: α j = 1 + sn( iǫ ) cn( λ j ) cn( iǫ ) sn( λ j ) | Q | 2 − 1 | Q | 2 + 1 β j = sn( iǫ ) sn( λ j )  ¯ Q + Q | Q | 2 + 1 + dn( λ j ) dn( iǫ ) ¯ Q − Q | Q | 2 + 1  (35) So, under the giv en assumptions, the matrices D j are prop ortio na l to unitary matrices. 6 In terp olatin g Hamiltonian flo w No w w e wan t to get the interpolating flow of the discrete dynamics generated b y the maps (2 3). As w e will see the B¨ ac klund transformat io n can b e seen as a time discretization o f a o ne-parameter ( λ 0 ) family of Hamiltonian flo ws with the difference 2 ǫ play ing the r o le o f the time-step and with the Hamiltonian defining the in terp olating flo w given b y γ ( λ 0 ), where γ ( λ ) is defined in (20). First of all let we ta k e the limit ǫ → 0 . One has: Q = A ( λ 0 ) − γ ( λ 0 ) B ( λ 0 ) + O ( ǫ ) , (36) P = − iǫ B ( λ 0 ) γ ( λ 0 ) + O ( ǫ 2 ) . (37) One can carefully insert these expres sions in the dressing matrix (1 8) to find: D ( λ − λ 0 ) = 1 − iǫ γ ( λ 0 )sn( λ − λ 0 ) D 0 ( λ, λ 0 ) , (38) where D 0 ( λ, λ 0 ) . = A ( λ 0 )cn( λ − λ 0 ) B ( λ 0 )+ C ( λ 0 ) 2 + B ( λ 0 ) − C ( λ 0 ) 2 dn( λ − λ 0 ) B ( λ 0 )+ C ( λ 0 ) 2 − B ( λ 0 ) − C ( λ 0 ) 2 dn( λ − λ 0 ) − A ( λ 0 )cn( λ − λ 0 ) ! . (39) In the limit ǫ → 0 the equation of the map ˜ LD = D L turns in to the Lax equation for a con tinu o us flow : ˙ L ( λ ) = [ L ( λ ) , M ( λ, λ 0 )] . (40) where the time deriv ativ e is defined as: ˙ L = lim ǫ → 0 ˜ L − L 2 ǫ (41) and the matrix M ( λ, λ 0 ) is give n b y: M ( λ, λ 0 ) = i 2 γ ( λ 0 )sn( λ − λ 0 ) D 0 ( λ, λ 0 ) . (42) 10 With the help of the P oisson brack ets b etw een the elemen ts of the Lax mat rix (5), the dynamical system (40) can b e cast in Hamiltonian fo rm: ˙ L ij ( λ ) = {H ( λ 0 ) , L ij ( λ ) } , i, j ∈ { 1 , 2 } , (43) with the Hamilton’s function g iv en by: H ( λ 0 ) = γ ( λ 0 ) = p A 2 ( λ 0 ) + B ( λ 0 ) C ( λ 0 ) . (44) So the Hamiltonian (44) c haracterizing the in terp olating flow is (the square ro ot of ) the generating function (7) of the whole set of conserv ed quantities . By choo sing the parameter λ 0 to b e equal to an y of the p oles ( λ i ) of the Lax matrix, the map leads to N differen t maps { B T ( i ) } i =1 ..N , where B T ( i ) discretizes the flo w corresp onding to the Hamiltonian H i , given b y equation (8). In fact, b y p osing λ 0 = δ + λ i and ta king the limit δ → 0, the Hamilton’s f unction (44) giv es: γ ( λ 0 ) = s i δ + H i s i + O ( δ ) . (45) and the equations of motion tak e the f o rm: ˙ L ij ( λ ) = 1 s i { H i , L ij ( λ ) } , i, j ∈ { 1 , 2 } . (46) Note t hat the corr esp o nding interpola ting Hamiltonian flo ws of the B¨ ack lund trans- formations for the trigono metric and rat ional Gaudin mo dels can b e obtained a s the corresp onding limiting case of this one. 7 An application to the Cle bsc h mo del The Clebsc h mo del [4] is an in tegrable case of the Kirc hhoff equations [11] describing the motion of a solid in an infinite incompressible fluid. If the solid has three p erp endicular planes of symmetry and there are no external forces then the Kirc hhoff system can b e describ ed in terms o f an Hamiltonian (the kinetic energy of the system solid+fluid) quadratic and diagonal in the impulsive for c e p and impulsive p air J v ectors, represen ting respective ly the sum of the impulse and a ngular momen tum of the solid a nd those applied by the solid to t he b oundary of the fluid in con tact with it [15]. The Hamiltonian then reads: T = 1 2  α 1 ( p x ) 2 + α 2 ( p y ) 2 + α 3 ( p z ) 2  + 1 2  β 1 ( J x ) 2 + β 2 ( J y ) 2 + β 3 ( J z ) 2  , (47) where α i and β i are a set of constan ts depending on the shap e of the solid. Clebsc h [4] disco v ered that if the follo wing constraint on these quan tities holds: α 1 − α 2 β 3 + α 2 − α 3 β 1 + α 3 − α 1 β 2 = 0 , (48) 11 then the corresp onding equations of motion are integrable. Since the dynamical v ariables are impulse and angular momentum v ectors, the Lie- P oisson structure is defined by the e (3) algebra: { J i , J j } = ǫ ij k J k , { J i , p j } = ǫ ij k p k , { p i , p j } = 0 . (49) where i, j, k b elong to t he set { x, y , z } . These brack ets ha v e t wo Casimirs : p · J . = c 1 , p 2 . = c 2 . (50) No w w e will sho w ho w, by the means of a pro cedure of p o le c o alesc enc e on the Lax matrix of the t w o-site elliptic Gaudin mo del, it is po ssible to obtain the Lax matrix for the Clebsc h mo del gov erned b y the f ollo wing Ha miltonian: 2 H =  C − k 2 ( A + B )  ( p x ) 2 + C ( p y ) 2 +  C + B (1 − k 2 )  ( p z ) 2 + B ( J x ) 2 + +  Ak 2 + B  ( J y ) 2 + ( A + B ) ( J z ) 2 . (51) The construction of the integrabilit y structure of the mo del is not new ( cf. with [17]), but here is briefly rep orted for completeness. F or more details ab out the p ole coalescence pro cedure see [1 6], [17]. Note that the Clebsc h’s constraint (48) holds for (51) , but here we hav e only fo ur ( no t fiv e) a rbitrary constan ts, so actually w e will obtain a sp ecial realizatio n of (47). Note also that in the case k = 0 one obtains the Kirc hhoff top [21]. First let us in tro duce a con traction parameter, sa y ǫ , a nd ta k e in the Lax matrix (1) λ 1 → ǫλ 1 and λ 2 → ǫλ 2 , where we recall that λ 1 and λ 2 are the t w o arbitrary parameters of the Gaudin mo del. By setting: J . = s 1 + s 2 , p . = ǫ ( λ 1 s 1 + λ 2 s 2 ) (52) and letting ǫ → 0 in (1) af ter this iden tification, one obtains the follo wing expression: sn( λ ) L ( λ ) = = cn( λ ) J z + p z dn( λ ) sn( λ ) J x − i dn( λ ) J y + cn( λ )dn( λ ) sn( λ ) p x − i cn( λ ) sn( λ ) p y J x + i dn ( λ ) J y + cn( λ )dn( λ ) sn( λ ) p x + i cn( λ ) sn( λ ) p y − cn( λ ) J z − p z dn( λ ) sn( λ ) ! (53) The ov erall term sn ( λ ) mu lt iplying L ( λ ) on the l.h.s. of the previous equation can b e ob viously skipped, so in the fo llo wing w e assume that the Lax matrix of the mo del. that w e will call L c , is give n only b y the r.h.s. term of (53). Note that by using (6), it is readily seen that the v a r ia bles J and p (52) ob ey the Lie- Poisson algebra e (3 ) (49). The determinant of this matrix is the generating function of the integrals of motions. One has: − det ( L c ( λ )) . = Υ 2 ( λ ) = H 1 − H 0 sn 2 ( λ ) + 2 c 1 cn( λ )dn( λ ) sn( λ ) + c 2 cn 2 ( λ ) sn 2 ( λ ) (54) where c 1 and c 2 are the Casimirs (50), while H 0 and H 1 are the t wo Poiss on comm uting in tegrals: H 0 = ( J z ) 2 + k 2  ( J y ) 2 − ( p x ) 2  , H 1 = J 2 + ( p z ) 2 − k 2  ( p x ) 2 + ( p z ) 2  . (55) The ph ysical hamiltonian (51) is now obtained b y the f o llo wing linear com bination: 2 H . = A H 0 + B H 1 + C c 1 12 7.1 B¨ ac klund transformations The ansatz for the dressing matrix for the mo del just describ ed is inherited f r om that of the elliptic Gaudin mo del since the r-matrix structure is preserv ed by the p ole coalescence pro cedure. So for the general form of D ( λ ) w e again refer to (1 8): D ( λ ) = 1 + i S 3 cn( λ ) sn( λ ) i S 1 + S 2 dn( λ ) sn( λ ) i S 1 − S 2 dn( λ ) sn( λ ) 1 − i S 3 cn( λ ) sn( λ ) ! with          i S 3 = pq − sn( µ ) cn( µ ) i S 1 + S 2 dn( µ ) = p i S 1 − S 2 dn( µ ) = q (2 sn( µ ) − pq ) (56) Exactly as for the Gaudin mo del, the v a r ia bles p and q can b e determined in terms o f only one set of v ariables thanks to the spectrality prop erty . The result is: q = q ( λ 0 + µ ) = L c 11 ( λ ) − Υ( λ ) L c 12 ( λ )     λ = λ 0 + µ 1 p = q ( λ 0 + µ ) − q ( λ 0 − µ ) 2 sn( µ ) (57) Again b y c ho osing λ 0 real and µ purely imaginary one obtains transformations sending real v ariables in to real v aria bles. T o write explicitly the ma ps one can, fo r example, tak e the residue at the p ole in λ = 0 in the equiv alence ˜ L c ( λ ) D ( λ − λ 0 ) = D ( λ − λ 0 ) L c ( λ ) and its v alue at λ = K ( k ), where K ( k ) is the complete elliptic integral of the first kind (67). W e write directly the real transformations by p osing: µ = iǫ ( λ 0 , ǫ ) ∈ R 2 . (58) They reads: ˜ p x = a 2 0 + ¯ a 2 0 − b 2 0 − ¯ b 2 0 2 h 0 p x + i ¯ a 2 0 − a 2 0 + ¯ b 2 0 − b 2 0 2 h 0 p y − a 0 b 0 + ¯ a 0 ¯ b 0 h 0 p z ˜ p y = a 2 0 + ¯ a 2 0 + b 2 0 + ¯ b 2 0 2 h 0 p y + i a 2 0 − ¯ a 2 0 + ¯ b 2 0 − b 2 0 2 h 0 p x − i a 0 b 0 − ¯ a 0 ¯ b 0 h 0 p z ˜ p z = | a 0 | 2 − | b 0 | 2 h 0 p z + a 0 ¯ b 0 + b 0 ¯ a 0 h 0 p x + i b 0 ¯ a 0 − a 0 ¯ b 0 h 0 p y ˜ J x = a 2 1 + ¯ a 2 1 − b 2 1 − ¯ b 2 1 2 h 1 J x + i k ′ ¯ a 2 1 − a 2 1 − ¯ b 2 1 + b 2 1 2 h 1 J y − k ′ a 1 b 1 + ¯ a 1 ¯ b 1 h 1 p z ˜ J y = a 2 1 + ¯ a 2 1 + b 2 1 + ¯ b 2 1 2 h 1 J y + i a 2 1 − ¯ a 2 1 + ¯ b 2 1 − b 2 1 2 k ′ h 1 J x − i a 1 b 1 − ¯ a 1 ¯ b 1 h 1 p z (59) where the functions ( a i , b i , h i ), i ∈ { 0 , 1 } , are defined b y the followin g form ulae: a 0 . = sn( iǫ )cn ( λ 0 )  | q | 2 − 1  − sn( λ 0 )  | q | 2 + 1  b 0 . = sn( iǫ ) ( q + ¯ q ) − sn( iǫ ) dn( iǫ ) dn( λ 0 ) ( q − ¯ q ) a 1 . = cn( λ 0 ) dn( λ 0 )  | q | 2 + 1  + k ′ sn( iǫ )sn( λ 0 ) cn( iǫ )cn( λ 0 )  | q | 2 − 1  b 1 . = sn( iǫ ) ( q + ¯ q ) − k ′ sn( iǫ ) dn( iǫ )dn( λ 0 ) ( q − ¯ q ) h i . = | a i | 2 + | b i | 2 i ∈ { 0 , 1 } (60) 13 In the previous formulae k ′ is the complemen tary mo dulus of the Jacobi elliptic func- tions, k ′ = √ 1 − k 2 ; the bar means complex conjug a tion. The expression for ˜ J z follo ws for example b y the constraint ˜ J · ˜ p = J · p , but a cleaner expression can b e fo und taking the v alue λ = K ( k ) + i K ′ ( k ) in ˜ L c ( λ ) D ( λ − λ 0 ) = D ( λ − λ 0 ) L c ( λ ), where K ′ ( k ) is the complemen tary in tegral (67). It r eads: ˜ J z = a 2 2 − b 2 2 − a 2 3 + b 2 3 h J z − 2 k a 3 b 2 + a 2 b 3 h J 1 − 2 k a 2 a 3 + b 2 b 3 h p y where        a 2 . = dn( λ 0 ) cn( λ 0 )  | q | 2 + 1  b 2 . = i k ′ sn(i ǫ ) cn(i ǫ )cn( λ 0 )  | q | 2 − 1           a 3 . = − i k sn(i ǫ ) ( q + ¯ q ) b 3 . = k k ′ sn(i ǫ )sn( λ 0 ) dn(i ǫ )cn( λ 0 ) ( q − ¯ q ) h . = a 2 2 + a 2 3 − b 2 2 − b 2 3 (61) Note that if in these transformations one p oses k = 0, then the B¨ ack lund tra nsforma - tions for the Kirchhoff top as giv en in [21] are obtained. No w let us consider the interpolating Ha miltonian flo w. W e recall tha t the r - matrix structure of the mo del is that of G audin, so the P oisson brac k ets (5) w orks again by substituting A = L c 11 , B = L c 12 and C = L c 21 . This is enough to ensure that the in- terp olating Hamiltonian is giv en, as for the Gaudin mo dels, by the square ro ot of the generating function of the in tegrals of the system (54) for λ = λ 0 : H ( λ 0 ) = Υ( λ 0 ) = s H 1 − H 0 sn 2 ( λ 0 ) + 2 c 1 cn( λ 0 )dn( λ 0 ) sn( λ 0 ) + c 2 cn 2 ( λ 0 ) sn 2 ( λ 0 ) . (62) With a c hoice of the pa rameter λ 0 it is p ossible to o btain a discretization of the con- tin uous flo w corresponding to eac h linear com bination of the Hamiltonians H 0 and H 1 . A last remark: the symplecticit y of these maps simply follow s from the symplec ticity of the maps fo r the a ncestor Gaudin mo del, again thanks to the preserv ation of the r -matrix structure. 8 Commen ts First of all w e hav e to men tion t he lac ks of our construction. Unlik e t he rational case (cf. with [8]), w e are not able to give the generating f unction of the cano nical transfor- mations defined b y the maps (2 3). Ho we ver this isn’t only a matter o f tec hnical difficul- ties; indeed for the xxx Gaudin mo del the tw o - parameters Ba cklund transformatio ns can b e written as the comp osition o f tw o simpler one-para meter transformations: the same prop ert y holds true f or the generating functions; y et in t he trigonometric case a factorizatio n of the dressing matrix cannot lead to a one parameter dressing matrix preserving all the symmetries of the problem ([20 ]). There are t w o p ossibilities: or one is able to find directly the t w o para meters generating function, or one should lo ok for a symmetry-violating generating function suc h that their comp osition restores the symmetries . These details can b e interesting b y a quan tum p oin t of view for their p oten tial connections with the Baxter’s Q op erator [27]. 14 In [21] it w a s sho wn how the discrete orbits defined by the B¨ ack lund transformations for the K ir c hhoff top exactly in terp o la te, in some sp ecial cases and in general as a conjecture, the contin uous orbits of the corresp onding ph ysical flow , indicating that indeed t he B¨ ac klund transformations can b e considered an integrator (n umerical or analytical) of the corresp onding con tin uous differen tial pro blem. It could b e intere st- ing to understand how man y of these results remain true for the tra nsformations here giv en, b o t h for Gaudin and Clebsc h mo dels. W orks are in progress in this direction. A Notations and for m ul ae Let w 1 , w 2 b e complex num b ers such that their r a tio is not real and consider the latt ice Λ generated b y these n umbers: Λ = { w ∈ C : w = n 1 w 1 + n 2 w 2 , n 1 , n 2 ∈ Z 2 } The W eierstraß zeta function is given by [2]: ζ ( u ) = 1 u + X w 6 =0  1 u − w + 1 w + u w 2  (63) The W eierstraß ℘ function is min us the deriv ative of ζ : ℘ ( u ) = − ζ ′ ( u ) = 1 u 2 + X w 6 =0  1 ( u − w ) 2 − 1 w 2  (64) By denoting the p erio d w 3 suc h t ha t w 1 + w 2 + w 3 = 0 and defining t he set e i , i = 1 .. 3 b y e i = ℘ ( w i 2 ), then holds the r elat io n [2]: ℘ ( u ) = e 2 + 1  2 sn( u, k ) (65) where  = ( e 1 − e 2 ) − 1 2 and the elliptic mo dulus for the Jacobi “sn” function is giv en by k 2 =  2 ( e 3 − e 2 ). The Jacobi elliptic functions sn( u , k ), cn( u, k ) a nd dn( u, k ) satisfies the follow ing quasi-p erio dic relatio ns [2]: sn( u + 2 mK + 2 inK ′ , k ) = ( − 1) m sn( u, k ) cn( u + 2 mK + 2 inK ′ , k ) = ( − 1) m + n cn( u, k ) dn( u + 2 mK + 2 inK ′ , k ) = ( − 1) n dn( u, k ) (66) where K and K ′ are resp ectiv ely the complete elliptic integral of the first kind and the complemen tary in tegral: K ( k ) = Z 1 0 dt p (1 − t 2 )(1 − k 2 t 2 ) K ′ ( k ) = Z 1 0 dt p (1 − t 2 )(1 − (1 − k 2 ) t 2 ) (67) The follo wing form ulae are useful in pro ving (7): cn( x ± y ) = cn( x )cn ( y ) ∓ dn ( x ± y )sn( x )sn( y ) (68) 15 dn( x ± y ) = dn( x )dn( y ) ∓ k 2 cn( x ± y )sn( x )sn( y ) (69) (  ( ζ (  x ) − ζ (  y )) − sn( y − x ) sn( x )sn( y ) = a ( y − x ) a ( x ) . =  ( ζ (  ( x )) − ζ (2  x )) − 1 sn(2 x ) (70) Equations (68) and (69) are only a consequenc e of addition formulae f o r the Jacobi ellip- tic functions, (70) can b e pro v ed in few lines. In fact suppose that x and y v ary while y − x remains constant and equal to b . Differentiating f ( x ) =  ( ζ (  x ) − ζ (  ( x + b ))) − sn( b ) sn( x )sn( x + b ) with resp ect to x we see that this function is indep endent of x . In fact f ′ ( x ) =  2  ℘ (( x + b )  ) − ℘ ( x )  − sn( b )  sn( x )sn( x + b )  ′  sn( x )sn( x + b )  2 By using t he relation (65) and again the addition formulas f or the Ja cobi elliptic f unc- tions it is readily sho wn t ha t f ′ ( x ) = 0, so f ( x ) is a constan t, that w e can take a s a function of b . This implies the relation  ( ζ (  x ) − ζ (  y )) − sn( y − x ) sn( x )sn( y ) = a ( y − x ). By p osing y = 2 x in this equation w e obtain the function a ( x ) as in (7 0 ). Now let us consider closely the formula for − det ( L ( λ )). F or brevit y w e p ose in t he follo wing v i = λ − λ i and v ij = λ i − λ j . F rom (2) we ha v e: − det ( L ( λ )) = X i,j cn( v i )cn( v j ) s z i s z j + s x i s x j + dn( v i )dn( v j ) s y i s y j sn( v i )sn( v j ) = = X i,j i 6 = j cn( v i )cn( v j ) s z i s z j + s x i s x j + dn ( v i )dn( v j ) s y i s y j sn( v i )sn( v j ) + + X i s 2 i sn( v i ) 2 − X i  ( s z i ) 2 + k 2 ( s y i ) 2  (71) By adding and subtracting the quantities P i 6 = j  dn( v ij ) s z i s z j + k 2 cn( v ij ) s y i s y j  in the last equation and using (68) and (69), w e find: − det ( L ( λ )) = X i s 2 i sn( v i ) 2 − X i,j  s z i s z j dn( v ij ) + k 2 s y i s y j cn( v ij )  + + X i,j i 6 = j cn( v ij ) s z i s z j + s x i s x j + dn ( v ij ) s y i s y j sn( v i )sn( v j ) (72) No w, using form ula (70) on the denominator o f the last sum of equation (72) and defining H i = N X j 6 = i s z i s z j cn( v ij ) + s y i s y k dn( v ij ) + s x i s x k sn( v ij ) one reac hes the result (7). 16 References [1] Amico L., F alci G., F azio R., The BCS mo del a nd the off-shell Bethe ansatz for v ertex mo dels, J. 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