B"acklund Transformations for the Trigonometric Gaudin Magnet

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 6 (2010), 012, 6 pages B¨ ac klund T ransformations for the T rigonometri c Gaudin Magnet ⋆ Orlando RAG NISCO and F e deric o ZULLO Dip artimento di Fisic a Uni v ersit´ a R oma T r e and Istituto Nazionale di Fisic a Nucle ar e, Sezione di R oma, I-00146 R oma, Italy E-mail: r agnisc o @fis.unir oma3.it , zul lo@fis.unir oma3.it URL: http://webuser s.fis.un iroma3/ ~ ragnisco / Received Decem b er 12, 20 09, in f inal form January 27, 2010; P ublished o nline January 29, 201 0 doi:10.38 42/SIGMA.20 10.012 Abstract. W e co nstruct a B¨ acklund transformation for the trigono metric classical Ga udin magnet starting from the Lax representation of the model. The Darb oux dressing matrix obtained depe nds just on one set of v ariables because of the so-called sp e ctr ality prop erty in- tro duced by E. Sklyanin and V. K uz ne ts ov. In the end we mention s ome po ssibly int eresting op en pro blems. Key wor ds: B¨ acklund tra nsformations; integrable ma ps; Gaudin systems 2010 Mathematics Su bje ct Classific ation: 37J35 ; 70H06; 70 H15 1 In tro duction B¨ ac klund transf ormations are a pr ominen t tool in the theory of int egrable systems and soliton theory . Historically they ap p eared f irst in the w orks of Bianc hi [1] and B¨ ac klund [2] on sur faces of constant curv ature and allo w ed them to pass from a surface of constan t curv atur e to a new one, or from a solution of a giv en PDE to a new one. By this p oin t of view B¨ ac klund transformations ha v e b een extensivel y exploited [3, 4, 5, 6]. In the f ield of f inite-dimensional systems th ey can b e seen as in tegrable P oi sson maps that discretize a family of con tin uous f lo ws; one of th e earliest accoun t of this sub ject is in [7] where th e te rm inte gr able L agr ange c or r esp on denc es is used for inte gr able maps . This p oin t of view has b een widely exp lored b y Sur is [8], Sklya nin [9], Sklya nin and Kuznetsov [10], Kuznetso v and V anhaec k e [11]. Numerous r elev an t resu lts ap p eared in the 90’s and at the b eginning of the present cent ury on exac t time discretizatio ns of man y b o dy systems. Our pap er is an ideal contin u ation, almost 10 ye ars later, of a joint pap er b y our dear friend V adim, Andy Hone and O.R. [12], where the same problem has been stud ied and solv ed for the rational Gaudin c hain. The k ey observ ation we make (see also [13]) is that the trigonometric Gaud in mo del with N sites is just the r atio nal Gaudin mo d el with 2 N sites with an extra r e fle ction symmetry (“inner automorphism”), en tailing the follo win g inv olution on the corresp ondin g Lax m atrix: L ( z ) = σ 3 L ( − z ) σ 3 , (1) where z is the sp ectral parameter, and σ 3 is the usual P auli matrix diag (1 , − 1). In the follo wing section w e will deriv e (1 ) from the standard form of the trigonometric Lax matrix. Here w e can alrea dy argue that, to preserv e th e ref lection symmetry , the elemen tary d ressing matrix, that we will call D after Darb oux, h as to enjo y a s imilar pr op ert y (up to an inessen tial scalar ⋆ This p aper is a contribution to t he Proceedings of the XVI I Ith International Colloqu ium on Integrable Sys- tems and Quantum Symmetries (June 18–20, 2009, Prague, Czec h Republic). The full collection is av ailable at http://w ww.emis.de/j ournals/SIGMA/ISQS2009.html 2 O. Ragnisco and F. Zullo factor), and consequ ently it has to exhibit pairs of singular p oin ts in the sp ectral complex plane. Those singular p oint s can b e (opp osite) zero es and/or (opp osite) p oles, due to the symm etric role p la y ed by D and D − 1 . As the B¨ ac klund transformation b etw een the “old” Lax matrix L and the up d ated ˜ L has to preserve the sp ectral inv arian ts of L , it has to b e def ined through a sim ilarity map: ˜ L ( z ) = D ( z ) L ( z )[ D ( z )] − 1 . (2) Ob viously we should require that the rational structure of the Lax matrix b e preserved, i.e. that the up dated matrix has the same num b er of p oles and zero es as th e old one. In the sequel we will f o cus our attent ion on elementary B¨ ac klu n d transformations, where the corresp onding D has ju st one pair of (o pp osite) sin gular p oin ts. 2 The trigonometric Gaudin magnet As it is well kno wn the trigonometric Gaudin mo d el is go v erned by the follo wing Lax matrix: L ( λ ) =  A ( λ ) B ( λ ) C ( λ ) − A ( λ )  , (3) A ( λ ) = N X j =1 cot( λ − λ j ) s 3 j , B ( λ ) = N X j =1 s − j sin( λ − λ j ) , C ( λ ) = N X j =1 s + j sin( λ − λ j ) . (4) The dynamical v ariables  s + j , s − j , s 3 j  , j = 1 , . . . , N , ob ey to the P oisson structure giv en b y the brac k ets:  s 3 j , s ± j  = ∓ iδ j k s ± k ,  s + j , s − j  = − 2 iδ j k s 3 k , with the N Casimirs giv en by s 2 j =  s 3 j  2 + s + j s − j . This structure corresp ond s to the trigonometric r t matrix, given by r t ( λ ) = 1 sin( λ )     cos( λ ) 0 0 0 0 0 1 0 0 1 0 0 0 0 0 cos( λ )     , with the Lax matrix satisfying the line ar r -matrix P oisson algebra,  L 1 ( λ ) , L 2 ( µ )  =  r t ( λ − µ ) , L 1 ( λ ) + L 2 ( µ )  , (5) where, as usually , the s up ers cripts on th e mat rices denote tensor p ro du cts: L 1 = L ⊗ I , L 2 = I ⊗ L. The equatio n (5) is equiv alen t to the follo wing P oisson brac k ets for the elemen ts A ( u ), B ( u ) and C ( u ): { A ( λ ) , A ( µ ) } = { B ( λ ) , B ( µ ) } = { C ( λ ) , C ( µ ) } = 0 , { A ( λ ) , B ( µ ) } = cos( λ − µ ) B ( µ ) − B ( λ ) sin( λ − µ ) , B¨ ac klund T rans formations for the T rigonometric Ga udin Magnet 3 { A ( λ ) , C ( µ ) } = C ( λ ) − cos( λ − µ ) C ( µ ) sin( λ − µ ) , { B ( λ ) , C ( µ ) } = 2( A ( µ ) − A ( λ )) sin( λ − µ ) . Through th e “uniformization” mapp ing: λ → z = e iλ the Lax matrix (3) acquires a rational f orm in z : − iL ( z ) = N X j =1 s 3 j σ 3 + N X j =1 L j 1 z − z j − σ 3 L j 1 z + z j σ 3 ! , (6) where th e m atrices L j 1 , j = 1 , . . . , N , ha v e the s imple form: L j 1 = z j s 3 j s − j s + j − s 3 j ! . The equ ation (6) leads to th e ref lection symmetry (1): L ( z ) = σ 3 L ( − z ) σ 3 . 3 The Darb oux matrix The simplest c hoice for the sp ectral structure of the Darb oux-dr essin g matrix requir es that it ob eys the ref lection symmetry (1) and conta ins only one pair of opp osite simple p oles. Then , it reads: D = D ∞ + D 1 z − ξ − σ 3 D 1 σ 3 z + ξ . (7) The matrix D ∞ , i.e. lim z →∞ D ( z ) def ines the normalizatio n of th e p roblem. The equation (2), rewritten in the form : ˜ L ( z ) D ( z ) = D ( z ) L ( z ) (8) in the limit z → ∞ yields: ( ˜ S z ) σ 3 D ∞ = D ∞ ( S z ) σ 3 , where by S z w e ha v e denoted the z -comp onent of the total “spin” S . As S z P oisson comm u tes with tr L 2 , the generating fu nction of the complete family of in v olutiv e Hamiltonians, it has to b e preserv ed by our B¨ ac klund transformation, whic h is a symmetry for the w hole h ierarc h y . This implies D ∞ to b e diagonal. As for b ound ed v alues of z , equation (8) implies that b oth sides ha v e equal residues at the simple p oles ± z j , ± ξ . Ho w ev er, in view of the symmetry prop ert y (1), (7) it will b e enough to lo ok a t half of them, sa y z j , ξ . The corresp onding equations will b e: ˜ L ( j ) 1 D ( z j ) = D ( z j ) L ( j ) 1 , (9) ˜ L ( ξ ) D 1 = D 1 L ( ξ ) . (10) 4 O. Ragnisco and F. Zullo The crucial problem to solv e no w is to ensure that (9), (10) pr o vide an explicit (and symplectic) mapping betw een the old and the new s pin v ariables. In other w ords, to get a Darb oux matrix that dep ends just on one set of v ariables, say the old ones. As it has b een sho wn for instance in [10, 12], this can b e done thanks to th e so-called sp e ctr ality prop erty . In the p resen t con text, this amoun ts to require that det D p ossess, in addition to the t w o opp osite p oles ± ξ , t w o opp osite nondynamic al zero es, say ± η and that D 1 is, up to a f actor, a pr o jector. Again, by symmetry it w ill b e enough to lo ok at one of the zeroes, sa y η . By setting z = η in (8) w e get ˜ L ( η ) D ( η ) = D ( η ) L ( η ) . But D ( η ) is a rank one matrix, h aving a one dimensional Kern el | K ( η ) i , whence: 0 = D ( η ) L ( η ) | K ( η ) i en tailing L ( η ) | K ( η ) i = µ ( η ) | K ( η ) i , (11) i.e. the p oin ts ± η , ± µ ( η ) b elong to the sp e ctr al curve det( L ( z ) − µI ) = 0. | K ( η ) i is th en fully determined in terms of the old dynamical v ariables. The equation (10) giv e us another one dimensional Kernel K ( ξ ) b ecause also D 1 is a rank 1 matrix, so (10) en tails: L ( ξ ) | Ω( ξ ) i = µ ( ξ ) | Ω( ξ ) i . (12) The t w o sp ectralit y conditions (11), (12) allo w to write D in terms of the old dynamical v ariables and of the tw o B¨ ac klund parameters ξ and η , so that the B¨ ac klund equations (9) yield an explicit map b etw een the n ew (tilded) an d the old (u n tilded) dyn amical v ariables. In order to clarify the p oin t ab ov e let us mak e some observ ations. First of all note that r equ iring D 1 to b e a rank one matrix amounts to requ ire that the d eterminan t of ( z 2 − ξ 2 ) D ( z ) b e zero for z = ξ or, b y symmetry , for z = − ξ . In fact :  z 2 − ξ 2  D ( z ) | z = ξ = 2 ξ D 1 ,  z 2 − ξ 2  D ( z ) | z = − ξ = 2 ξ σ 3 D 1 σ 3 . Since t w o Darb oux matrices dif f ering just b y a m ultiplicativ e scalar factor def ine the same BT, w e can c ho ose to w ork with a mo dif ied Darb oux m atrix D ′ ( z ) def ined by the relatio n: D ′ ( z ) ≡ z 2 − ξ 2 z D ( z ) . Hence, to ensur e that th e sp ectralit y prop erty holds true we hav e to requ ir e det D ′ ( z ) to v anish at z = ξ and z = η . The form tak en b y the Darb oux m atrix D ′ ( z ) can b e fu rther simplif ied by writing: D ′ ( z ) = z − 1 ˆ A + ˆ B + ˆ C z . (13) The matrix ˆ C is immediately seen to be a diagonal one b y lo oking at the b eha vior for la rge v alues of z and r equiring ˜ S z = S z . On the other hand, L (0) as w ell as its dressed v ersion ˜ L (0) are d iagonal m atrices: L (0) = N X j =1 S ( j ) z σ 3 − N X j =1 L ( j ) 1 + σ 3 L ( j ) 1 σ 3 z j . This readily imp lies th at ˆ A in (13 ) is diagonal . In tu rn, (1) implies that if ev en p o w ers of z are d iagonal, o dd p o w ers must b e of f-diagonal, en tailing that ˆ B is an of f-diagonal matrix. The B¨ ac klund T rans formations for the T rigonometric Ga udin Magnet 5 t w o matrices ˆ A and ˆ C are then giv en resp ectiv ely b y diag ( a 1 , a 2 ), diag ( c 1 , c 2 ), whereas the of f-diagonal m atrix ˆ B is giv en b y  0 b 1 b 2 0  . W e get a deep er insigh t on the p arameterizatio n of matrices ˆ A , ˆ B , ˆ C resorting ag ain to the sp ectralit y p rop erty: w e stress once more that this amo unts to requiring D ′ ( ξ ) and D ′ ( η ) to b e rank one matrices. Th is means that there exists a function of one v ariable, sa y p , suc h that: ( c 1 ξ + a 1 /ξ + b 1 p ( ξ ) = 0 , b 2 + p ( ξ )( c 2 ξ + a 2 /ξ ) = 0; (14) ( c 1 η + a 1 /η + b 2 p ( η ) = 0 , b 2 + p ( η )( c 2 η + a 2 /η ) = 0 . (15) The four equations (14), (15) lea v e us with tw o und etermined parameters, one of which is a global m ultiplicativ e fact or for D ′ ( z ), sa y β . The other is denoted by γ . The parameteriza tion of D ′ reads as follo ws: D ′ ( z ) = β   z ( p ( η ) η − p ( ξ ) ξ ) γ + ( p ( ξ ) η − p ( η ) ξ ) η ξ γ z ξ 2 − η 2 γ γ p ( ξ ) p ( η ) ( ξ 2 − η 2 ) ηξ γ ( p ( η ) η − p ( ξ ) ξ ) z + γ z ( p ( ξ ) η − p ( η ) ξ ) ηξ   . The k ernel of D ( ξ ) (resp. D ( η ))) is sim p ly give n b y the row | Ω( ξ ) i = (1 , p ( ξ )) T (resp. | Ω( η ) i = (1 , p ( η )) T ). It is an eige n v ectors of L ( ξ ) (resp. L ( η )). Hence p ( ξ ) can b e written as: p ( ξ ) = µ ( ξ ) − A ( ξ ) B ( ξ ) , where we recall that µ ( z ) is such that µ 2 ( z ) = A 2 ( z ) + B ( z ) C ( z ) and A ( z ), B ( z ), C ( z ) are giv en b y (4). In terms of p ( η ), p ( ξ ), the matrices D ∞ and D 1 in (7) tak e th e form: D ∞ = β p ( η ) η − p ( ξ ) ξ γ 0 0 γ p ( ξ ) η − p ( η ) ξ ηξ ! , D 1 = β  η 2 − ξ 2  p ( ξ ) γ − 1 γ − γ p ( ξ ) p ( η ) ηξ γ p ( η ) ηξ ! . Since the Darb oux matrix D ( z ) is completely kno wn in terms of one set of dynamical v ariables, equation (9) yields an explicit B¨ ac klund transformation for the trigonometric Gaudin magnet. In a forthcoming pap er we will pro v e that (9) pro vides indeed a symp lectic map b et w een old and new d ynamical v ariables, and moreo ver that, according to a Skly anin conjecture, the Darb oux matrix (13) is in fact iden tical to Lax matrix of the elementa ry trigonometric Heisen b erg magnet. Th e inte rp olating Hamiltonian f lo w will b e also deriv ed and some examples of discrete dynamics w ill b e d ispla y ed and discussed. Ac kno wledgmen ts This pap er is in tended to b e a con tribution to the Pr o ceedings of the In ternational Conference “In tegrable Sy s tems and Quan tum Sy m metries 2009”, organized b y Professor ˇ C. Bu rd ´ ık and held in Prague, June 18–20 , 2009. One of the authors (O.R.) wan ts to warmly thank for his hospitalit y the Newton In s titute for Mathematical Sciences, and all the organizers and the participan ts to the Program “Discrete In tegrable S y s tems”. It w as in fact dur in g his s tay in Cam bridge that the main ideas presented in the pap er hav e b een made precise. Also, he ac kno wledges enligh tenin g discussions with A. Levine (ITEF) at the w orkshop “Einstein at S I SSA 2009”, partially fu nded b y the Russian F oundation for Basic Researc h w ith in the pr o ject “The Th eory of Nonlinear In tegrable Systems”. 6 O. Ragnisco and F. Zullo References [1] Bianchi L., Ricerc he sulle su p erf icie elicoidali e sulle sup erf icie a curv atura costan te, Ann . Sc. Norm. Sup er . Pisa Cl. Sci. (1) 2 (1879), 285–341. [2] B¨ ac klund A .V., Einiges ¨ ub er Curven- und Fl¨ ac hen-T ransformatio nen, Lunds Univ. ˚ Arsskr. 10 (1874), 1–12. [3] Rogers C., B¨ ac klund transformatio ns in soliton theory , in Soliton Theory: a Su rvey of Results, Editor A. F ordy , Nonline ar Sci. The ory Appl. , Manchester Univ . Press, Manchester, 1990, 97–130. 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