Integrable boundary conditions and modified Lax equations

In tegrable b oundary condit i o ns and mo dified Lax equations Jean Av an 1 a and Anastasi a Doikou 2 b a LPTM, Univ ersite de C e rgy-Po ntoise (CNRS UMR 8089), Saint- Martin 2 2 a v enue Adolphe Chauvin, F-95302 Cergy-Po ntoise Cedex, F rance b Univ ersit y of Bologna, Physic s Departmen t, I NFN Section Via Irnerio 46, Bologna 40126, Italy Abstract W e consider in tegrable b oundary conditions for b oth discrete and cont inuum clas- sical in tegrable m o dels. Lo cal in tegrals of motion generated by the corresp ondin g “transfer” matrices giv e rise to time evo lution equations for the initial L ax op erator. W e systematically id en tify the mo dified Lax pairs for b oth d iscrete and con tinuum b ound ary in tegrable mo dels, dep ending on the classical r -matrix and the b oundary matrix. 1 av an@ptm.u-cerg y .fr 2 doikou@bo.infn.it 1 In tro ductio n Lax represen tation of classical dynamical ev olution equations [1] is one k ey ing redien t in the mo dern theory of classical inte gra ble systems [2]–[7] together with the asso ciated not io n of classical r -matrix [8, 9]. It tak es the generic form of an isospectral ev olution equation: dL dt = [ L, A ], where L encapsulates the dynamical v a riables and A defines the time ev o- lution. W e shall generically consider the situatio n where L and A dep end on a complex (sp ectral) parameter. The sp ectrum of the Lax matrix or it s extension (transfer matrices), or equiv alently the in v a r ian t co efficien ts of the characteristic determinan t, thu s pro vide au- tomatically candidates to realize t he hierarc h y of Poiss on- commuting Hamiltonians required b y Liouville’s theorem [10 , 11]. Existence of t he classical r -ma t r ix then guarantee s Poiss on- comm utativit y of these natural dynamical quan tities tak en as generator s of the algebra of classical conserv ed charges. The question of finding an appropriate time ev olution matrix A f or a g iv en Lax matrix L therefore en tails the p ossibilit y of systematically constructing classically inte grable mo dels once a Lax mat r ix is defined with suitable prop erties. Depending on whic h prop erties are emphasized, this problem ma y b e approac hed in s ev eral w a ys. One approac h uses p o stulated P oisson algebra prop erties of L , sp ecifically the r -matrix structure, as a starting p oin t, and establishes a systematic construction of the time ev olution op erator A asso ciated to the Hamiltonian ev olution obtained from any function in the en v eloping a lgebra of the P oisson- comm uting traces of s uch an L matrix. Suc h a fo rm ulation w as prop osed long time ago [8, 9] for the bulk case, when the P oisson structure for L is a simple linear or quadratic r -matr ix structure. W e shall consider here more general situations when t he dynamical system also dep ends on supplemen ta ry parameters encapsulated into a matrix K . W e shall restrict ourselv es to the situation where these parameters are non-dynamical, i.e. the P oisson brac k ets with themselv es and with the initial “bulk” parameters is zero. F or situations where the extra pa- rameters are dynamical see [12, 13]. Note that inte rpretatio n of the ph ysical meaning of these c -n um b er parameters will come a posteriori when computing the ass o ciated Hamiltonians. In particular an y ph ysical interpre tatio n of the K matrix as a description of the “b oundary prop erties” (external fields, ...) may not b e appropriate in all cases as s hall app ear in our discussion o f examples. W e shall how eve r k eep this designation a s a b o ok-k eeping device throughout this pap er. Our central purpose will b e t w ofold. W e shall first of all define generic sets of sufficien t algebraic conditions on K also form ulated in terms of the “bulk” r -matrix. A “b oundary– mo dified” generating matrix of candidate conserv ed quan tities, hereafter denoted T , will b e accordingly constructed as a suitable combination of L and K matrices. The idea for suc h 1 a construction naturally a rises when considering the semi-classical limit of the w ell–kno wn Cherednik-Skly anin reflection algebras preserving bulk quan tum integrabilit y [1 4]. W e shall then redefine the time ev olution op erat o r A asso ciated to a giv en Hamiltonian constructed from T (modified mono dromy matrix) from the new basic elemen ts, i.e. the matrix T , the bulk r -matrix o r ( r , s ) pair [9, 15], and the reflection matrix K . Suc h a construction w as exemplified in [16]; what we prop ose here is ho w ev er a g eneric, systematic pro cedure to obtain mo dified “b oundary” or “folded” classical in tegrable systems from initial pure “bulk” systems. Note also tha t the example work ed o ut in [16] is precisely a case whic h the K -matrix do es describ e bo undar y effects . It m ust be emphasized here that an alternativ e, analytic approach to this qu estion, at least in the bulk case, was extensiv ely describ ed in [7] (c hapter 3 ). In this approa c h one uses instead the analytic prop erties (lo cation of p o les and algebraic structure o f residues) of the Lax matr ix L as a meromorphic function of the sp ectral parameter. They pro vide for a unique consisten t form of an asso ciated A matrix. The subsequen t La x equation is then dev elop ed into separate equations corresp onding to the p oles of L and A . The P oisson structure and r matrix structure a r e only then defined as pr oviding a consisten t Hamiltonian in terpretation of this Lax equation. Similar reformulation of o ur results m ust exist but we shall not consider them here. W e shall exp and here the r - matrix approach in three situations. W e start with the simpler case where the initia l bulk structure is a linear P oisson structure for a Lax ma- trix parametrized by a single r -matrix. W e then dev elop the cases of b o th discrete and con tin uous par a metrized Lax matrices relev an t to the description of systems on a latt ice or on a con tinuous line. In these last t w o situations the relev an t r -matr ix structure is a quadratic Skly anin-t yp e brack et [1 7]. W e shall in all three cases deriv e (or actually rederiv e, in some cases) the form of the generating functional for Poiss on- comm uting Hamiltonians, and establish the explicit general fo rm ula yielding the A op erator. Explicit examples shall b e dev elop ed in all cases , a lb eit restricting ourselv es to the simplest situations of non–dynamical non–constan t double–pole rational r - matrices. More complicated situations (trig o nometric and/or dynamical r -matrices, relev an t for e.g. sine-Gordo n mo dels o r affine T oda field the- ories) will b e left f o r f urther studies. 2 Linear P oisson structur e W e consider here the origina l situation [1] where the full dynamical system under study is represen t ed by a single Lax mat r ix, living in a represen tatio n of a finite-dimensional Lie alge- bra o r a lo op algebra. In this last case the Lax matrix also dep ends on a complex parameter 2 λ known as “sp ectral parameter”. In a ll cases the requiremen t of P oisson-comm utativity for candidate Hamiltonians T r L n is equiv alen t [18] to the existence of a classical r -matrix [8, 9] realizing a linear P oisson structure f o r L . Indeed, conside r the Lax pair ( L, A ) satisfying ∂ L ∂ t = h A, L i (2.1) the asso ciated sp ectral problem L ( λ ) ψ = u ψ , det( L ( λ ) − u ) = 0 (2.2) pro vides the integrals of motion, obtained through the expansion in p o we rs of the spectral parameter λ of tr L ( λ ). Alternativ ely , in part icular when no sp ectral para meter exists, one should consider the traces o f p o we rs of the La x matrix tr L n ( λ ) as natural Hamiltonians. Assuming that the L mat r ix satisfies the fundamental relation n L a ( λ ) , L b ( µ ) o = h r ab ( λ − µ ) , L a ( λ ) + L b ( µ ) i (2.3) it is sho wn using (2.3) that fo r a ny integer n, m : n tr L n ( λ ) , tr L m ( µ ) o = 0 . (2.4) The recipro cal pro p ert y was sho wn in [18 ]. Let us recall [8, 9, 18] how one may iden tify the A -op erator asso ciated to the v arious charges in inv olution. Bearing in mind the fundamen tal relation (2.3) it is sho wn that n tr a L n a ( λ ) , L b ( µ ) o = n tr a  L n − 1 a ( λ ) r ab ( λ − µ )  L b ( µ ) − n L b ( µ ) tr a  L n − 1 a ( λ ) r ab ( λ − µ )  . (2.5) F rom (2.5) one extracts A : A n ( λ, µ ) = n tr a  L n − 1 a ( λ ) r ab ( λ − µ )  . (2.6) In the case of the simplest ra t io nal non-dynamical r -mat r ices [19 ] r ( λ ) = P λ where P = N X i,j =1 E ij ⊗ E j i (2.7) P is t he permutation op erato r , and ( E ij ) k l = δ ik δ j l , we end up with a simple form for A n : A n ( λ, µ ) = n λ − µ L n − 1 ( λ ) (2.8) and as usual to obtain the Lax pair associated to eac h lo cal integral of motion one has to expand A n = P i A ( i ) n λ i . Note t ha t generically , using the dual form ulation of the classical r -matrix [9] one also has: A ( λ, µ ) = T r ( r ( λ, µ ) dH ) (2.9) 3 where H is the Hamiltonian expressed as an y function in t he en v eloping algebra g enerated b y T r L n . Ultimately w e w ould lik e t o consider an extended classical algebra in a nalogy to the quan- tum boundary algebras arising in integrable systems with non-trivial b oundary conditions that preserv e in tegrabilit y . Subsequen tly w e shall deal with t w o t yp es of algebras whic h ma y b e asso ciated with the tw o ty p es of known quantum b oundary conditions. These b oundary conditions a re kn own as soliton preserving (SP), traditionally studied in the fra mew ork of in tegrable quantum spin ch ains (see e.g. [14], [20]–[23]), a nd soliton non- preserving (SNP) originally in tro duced in the con text of classical in tegrable field t heories [16], and further in v estigated in [24, 25]. SNP b oundary conditions hav e b een also in tro duced and studied f or in tegrable quan tum lattice systems [26]–[30]. F rom the a lgebraic p ersp ectiv e the t w o t yp es of b oundary conditions are asso ciat ed with tw o distinct algebras, i.e. the reflection algebra [14] and the twis ted Y angian resp ectiv ely [3 1, 32] (see a lso [25, 29, 30, 33, 34]). The classical v ersions of b oth algebras will be defined subsequen t ly in the text (se e section 3.2). It will b e con ve nien t for our purp oses here to in tro duce some useful no t ation: ˆ r ab ( λ ) = r ba ( λ ) for SP , ˆ r ab ( λ ) = r t a t b ba ( λ ) for SNP r ∗ ab ( λ ) = r ab ( λ ) for SP , r ∗ ab ( λ ) = r t b ba ( − λ ) for SNP ˆ r ∗ ab ( λ ) = r ba ( λ ) for SP , ˆ r ∗ ab ( λ ) = r t a ab ( − λ ) for SNP (2.10) together with: ˆ L ( λ ) = − L ( − λ ) for SP , ˆ L ( λ ) = L t ( − λ ) for SNP . (2.11) In addition the “b oundary conditions”, to b e in terpreted on sp ecific examples, are parametrized b y a single non-dynamical matrix k( λ ). W e prop ose here the following set of algebraic rela- tions: n T 1 ( λ 1 ) , T 2 ( λ 2 ) o = r − 12 ( λ 1 , λ 2 ) T 1 ( λ 1 ) − T 1 ( λ 1 ) ˜ r − 12 ( λ 1 , λ 2 ) +r + 12 ( λ 1 , λ 2 ) T 2 ( λ 2 ) − T 2 ( λ 2 ) ˜ r + 12 ( λ 1 , λ 2 ) (2.12) where w e define: r − 12 ( λ 1 , λ 2 ) = r 12 ( λ 1 − λ 2 ) k 2 ( λ 2 ) − k 2 ( λ 2 ) r ∗ 12 ( λ 1 + λ 2 ) ˜ r − 12 ( λ 1 , λ 2 ) = k 2 ( λ 2 ) ˆ r 12 ( λ 1 − λ 2 ) − ˆ r ∗ 12 ( λ 1 + λ 2 ) k 2 ( λ 2 ) r + 12 ( λ 1 , λ 2 ) = r 12 ( λ 1 − λ 2 ) k 1 ( λ 1 ) + k 1 ( λ 1 ) ˆ r ∗ 12 ( λ 1 + λ 2 ) ˜ r + 12 ( λ 1 , λ 2 ) = k 1 ( λ 1 ) ˆ r 12 ( λ 1 − λ 2 ) + r ∗ 12 ( λ 1 + λ 2 ) k 1 ( λ 1 ) , (2.13) for a k matrix satisfying: n k 1 ( λ ) , k 2 ( µ ) o = 0 , n k 1 ( λ ) , L 2 ( µ ) o = 0 , (2.14) 4 r 12 ( λ 1 − λ 2 ) k 1 ( λ ) k 2 ( λ 2 ) + k 1 ( λ 1 ) ˆ r ∗ 12 ( λ 1 + λ 2 )k 2 ( λ 2 ) = k 1 ( λ 1 ) k 2 ( λ 2 ) ˆ r 12 ( λ 1 − λ 2 ) + k 2 ( λ 2 ) r ∗ 12 ( λ 1 + λ 2 ) k 1 ( λ 1 ) . (2.15) W e then sho w: The o r em 2.1. : The quantit y T ( λ ) = L ( λ ) k( λ ) + k( λ ) ˆ L ( λ ) (2.16) is a represen t a tion of t he algebra defined by (2 .12), with k ob eying (2 .1 4), (2.15). Pr o of : W e shall need in addition to (2.3) the following exc hange relations: n ˆ L a ( λ ) , L b ( µ ) o = ˆ L a ( λ ) ˆ r ∗ ab ( λ + µ ) + ˆ r ∗ ab ( λ + µ ) L b ( µ ) − ˆ r ∗ ab λ + µ ) ˆ L a ( λ ) − L b ( µ ) ˆ r ∗ ab ( λ + µ ) n L a ( λ ) , ˆ L b ( µ ) o = L a ( λ ) r ∗ ab ( λ + µ ) + r ∗ ab ( λ + µ ) ˆ L b ( µ ) − r ∗ ab ( λ + µ ) L a ( λ ) − ˆ L b ( µ ) r ∗ ab ( λ + µ ) n ˆ L a ( λ ) , ˆ L b ( µ ) o = h ˆ r ab ( λ − µ ) , ˆ L a ( λ ) + ˆ L b ( µ ) i . (2.17) By explicit use of the alg ebraic relations (2 .1 5) and (2 .17) w e o bta in: n L a ( λ )k a ( λ ) + ˆ L a ( λ )k a ( λ ) , L b ( µ )k b ( µ ) + k b ( µ ) ˆ L b ( µ ) o = . . . = r − ab ( λ, µ )  L a ( λ )k a ( λ ) + k a ( λ ) ˆ L a ( λ )  −  L a ( λ )k a ( λ ) + k a ( λ ) ˆ L a ( λ )  ˜ r − ab ( λ, µ ) +r + ab ( λ, µ )  L b ( µ )k b ( µ ) + k b ( µ ) ˆ L b ( µ )  −  L b ( µ )k b ( µ ) + k b ( µ ) ˆ L b ( µ )  ˜ r + ab ( λ, µ ) +  L a ( λ ) + L b ( µ )  − r ab ( λ − µ )k a ( λ )k b ( µ ) − k a ( λ ) ˆ r ∗ ab ( λ + µ )k b ( µ ) +k b ( µ ) r ∗ 12 ( λ + µ )k a ( λ ) + k a ( λ )k b ( µ ) ˆ r 12 ( λ − µ )  +  − r ab ( λ − µ )k a ( λ )k b ( µ ) − k a ( λ ) ˆ r ∗ ab ( λ + µ )k b ( µ ) +k b ( µ ) r ∗ 12 ( λ + µ )k a ( λ ) + k a ( λ )k b ( µ ) ˆ r 12 ( λ − µ )  ˆ L a ( λ ) + ˆ L b ( µ )  . (2.18) Bearing ho w ev er in mind that the k-matrix ob eys (2.15) w e conclude that the last four lines of the equation ab ov e disapp ear, whic h show s that (2.16) satisfies (2.12), and this concludes our pr o of.  Define T ( λ ) = k − 1 ( λ ) T ( λ ), w e no w pro ve : The o r em 2.2. : n tr a T N a ( λ ) , tr b T M b ( µ ) o = 0 . (2.19) 5 Pr o of : n tr a T N a ( λ ) , tr b T M b ( µ ) o = X n,m tr ab T N − n a ( λ ) T M − m b ( µ ) n T a ( λ ) , T b ( µ ) o T n − 1 a ( λ ) T m − 1 b ( µ ) (2.20) emplo ying (2.1 2) the preceding expression b ecomes . . . ∝ tr ab T N − 1 a ( λ ) T M − 1 b ( µ )k − 1 a k − 1 b  r − ab ( λ, µ ) T a ( λ ) − T a ( λ ) ˜ r − ab ( λ, µ ) + r + ab ( λ, µ ) T b ( µ ) − T b ( µ ) ˜ r + ab ( λ, µ )  = . . . = 0 . (2.21) Note that in o r der to show that the latter expression is zero w e mo v ed appropriately t he factors in the pro ducts within the tra ce a nd w e used (2.1 5).  W e finally iden tify the mo dified Lax for m ulation asso ciated to t he generalized alg ebra (2.14)–(2.12) as: The o r em 2.3. : D efining Hamiltonians as: tr a T n ( λ ) = P i H ( i ) n λ i the classic al equations of mo- tion for T : ˙ T ( µ ) = n H ( i ) n , T ( µ ) o (2.22) tak e a zero curv ature form ˙ T ( µ ) = h A ( λ, µ ) , T ( µ ) i , (2.23) where A n is iden tified as: A n ( λ, µ ) = n tr a  T n − 1 a ( λ )k − 1 a ( λ ) ˜ r + ab ( λ, µ )  . (2.24) Pr o of : n tr a T n a ( λ ) , T b ( µ ) o = . . . = n tr a  T n − 1 a ( λ )k − 1 a ( λ )k − 1 b ( µ )(r − ab ( λ, µ ) T a ( λ ) − T a ( λ ) ˜ r − ab ( λ, µ ) + r + ab ( λ, µ ) T b ( µ ) − T b ( µ ) ˜ r + ab ( λ, µ ))  = n tr a  T n − 1 a k − 1 a ( λ )k b ( µ )(r − ab ( λ, µ ) T a ( λ ) − T a ( λ ) ˜ r − ab ( λ, µ ))  + n tr a  T n − 1 a k − 1 a ( λ )k b ( µ )(r + ab ( λ, µ ) T b ( µ ) − T b ( µ ) ˜ r + ab ( λ, µ ))  . (2.25) T aking in to accoun t (2.15) w e see that the first term of RHS of the equalit y ab ov e disapp ears and the last term may b e a ppropriately rewritten suc h as: n tr a T n a ( λ ) , T b ( µ ) o = n tr a  T n − 1 ( λ )k − 1 a ( λ ) ˜ r + ab ( λ, µ )  T b ( µ ) − n T b ( µ ) tr a  T n − 1 ( λ )k − 1 a ( λ ) ˜ r + ab ( λ, µ )  . (2.26) 6 F rom the latter f o rm ula (2.24) is deduced.  Finally , let T ( λ ) , T ′ ( λ ) b e tw o represen tations of (2.12), and let also n T a ( λ ) , T ′ b ( µ ) o = 0 . (2.27) It is then straightforw ar d to show , based solely on the f act that T , T ′ satisfy (2.12) and (2.27) , that the sum T ( λ ) + T ′ ( λ ) is also a represen tation of (2.12) . 2.1 Examples W e shall pres ent here a simple example, starting from the classical rational Gaudin mo del [35]. Details on the so called “dual” description of the T o da c hain [36, 37] a nd the DST mo del [38, 39], b o th asso ciated to the A (1) N − 1 r -matrix [40], will b e presen ted in a forth- coming publication. Before w e pro ceed with the particular example let us rewrite A n . W e consider a situation where the P o isson brac ke ts are parametrized b y the simplest rational non-dynamical r -matrix [19]. After subs tituting the rational r -matrix in (2.24) w e get for b oth t yp es of b oundary conditions: A n ( λ, µ ) = n  T n − 1 ( λ ) λ − µ + k( λ ) T n − 1 ( λ )k − 1 ( λ ) λ + µ  for SP , A n ( λ, µ ) = n  T n − 1 ( λ ) λ − µ − (k( λ ) T n − 1 ( λ )k − 1 ( λ )) t λ + µ ) for SNP . (2.28) The L -matrix asso ciated to the classical g l N Gaudin mo del, and satisfying the algebraic relation (2.3) is L ( λ ) = N X α,β =1 N X i =1 P ( i ) αβ λ − z i E αβ (2.29) where P αβ ∈ g l N . Recall t ha t in the “bulk” case the in tegrals of motion are obtained through tr L n ( λ ). The first non- trivial example reads t (2) ( λ ) = tr L 2 ( λ ) = N X α,β =1 N X i,j =1 P ( i ) αβ P ( j ) β α ( λ − z i )( λ − z j ) (2.30) and b y t aking the residue of the latter expres sion for λ → z i w e obtain the Gaudin Hamil- tonian: H (2) = N X i 6 = j =1 N X α,β =1 P ( i ) αβ P ( j ) β α z i − z j . (2.31) 7 Let us now come to the generic algebra (2.12), (2.14 ), (2.15) considering b oth SP and SNP cases. Based on the definitions (2.11) we hav e: ˆ L ( λ ) = N X α,β =1 N X i =1 Q ( i ) αβ λ + z i E α where Q αβ = P αβ for SP , Q αβ = −P β α for SNP . (2.32) Let us take as a represen tat ion T ( λ ) = L ( λ )k( λ ) + k( λ ) ˆ L ( λ ) + K ( λ ) where k is a solution of (2.14), (2 .1 5), a nd K is a c -n um b er represen tation o f (2 .1 2) with zero P oisson brac ke t. T o obtain the relev a n t Hamiltonian w e now fo r m ulate tr a T 2 ( λ ) (where T = k − 1 T and w e also define ˜ K = k − 1 K ), and the Hamiltonia n ar ises as the residue of the latt er expression at λ = z i : H (2) = N X i 6 = j =1 N X α,β =1 P ( i ) αβ P ( j ) β α z i − z j + N X i,j =1 N X α,β ,γ ,δ,ǫ =1 k αγ k δǫ P ( i ) γ δ Q ( j ) ǫα z i + z j + N X i =1 N X α,β ,γ ,δ,ǫ =1 ˜ K ǫα k − 1 αγ k δǫ P ( i ) γ δ (2.33) A sp ecial case o f the generic algebra (2.12) is discuss ed in [41, 42]. Note a lso that expression (2.33) ma y be also o btained as the se miclassical limit of the quantum g l N inhomogeneous op en spin c hain for sp ecial b o undary conditions, (see e.g. [4 3]), z j b eing the inhomogeneities attac hed to each site j . Ho w ev er, here the Hamiltonians are obtained directly from the represen t ations o f o ur new classical algebra (2.1 2), (2.14), (2.15). It app ears in this example that the parameters k( λ ) play the r ole of “coupling constants” consisten t with the integrable “folding” of a 2 N site Gaudin mo del, and not the role o f b oundary parameters. 3 Quadratic P oisso n st ructure s : t h e discrete case Quadratic P oisson structures first app eared as the w ell-know n Skly anin brac k et [4 4]. A more general fo r m, characterize d by a pair o f resp ectiv ely sk ew symmetric and symmetric matrices ( r, s ) app eared in [1 5] in the fo r mulation of consisten t P oisson structures for non- ultralo cal classical in tegrable field theories. Fina lly it was sho wn [45] that this was the natural quadratic fo r m a la Skly anin for a no n-sk ew-symmetric r -matrix, reading: n L 1 , L 2 o = h r − r π , L 1 L 2 i + L 1 ( r + r π ) L 2 − L 2 ( r + r π ) L 1 . (3.1) A t ypical situation when one considers naturally a quadratic Pois son structure for the Lax matrix occurs when considering discrete (on a lattice) or con tinuous (on a line) in tegrable systems where t he Lax mat r ix dep ends on either a discrete or a contin uo us v ariable; the Lax pair is th us asso ciated to a p oint on the space-lik e lattice or con tin uous line [6, 17]. Let us first examine the discrete case where one considers a finite set of Lax matrices L n lab elled b y n ∈ N . 8 3.1 P erio dic b oundary conditions In tro duce the Lax pair ( L, A ) for discrete integrable mo dels [4] (see a lso [46] for statistical systems ), and the asso ciated auxiliary problem (see e.g. [6]) ψ n +1 = L n ψ n ˙ ψ n = A n ψ n . (3.2) F rom the latter equations one may immediately obtain the discrete zero curv ature condition: ˙ L n = A n +1 L n − L n A n . (3.3) The mono dro my matrix arises from the first equation (3.2) (see e.g. [6]) T a ( λ ) = L aN ( λ ) . . . L a 1 ( λ ) (3.4) where index a denotes the auxiliary space, and the indices 1 , . . . , N denote the sites of the one dimensional classical discrete mo del. Consider now a sk ew symmetric classical r -matrix whic h is a solution of the classical Y ang-Baxter equation [8, 9] h r 12 ( λ 1 − λ 2 ) , r 13 ( λ 1 ) + r 23 ( λ 2 ) i + h r 13 ( λ 1 ) , r 23 ( λ 2 ) i = 0 , (3.5) and let L satisfy the asso ciated Sklyanin brac ke t n L a ( λ ) , L b ( µ ) o = h r ab ( λ − µ ) , L a ( λ ) L b ( µ ) i . (3.6) It is then immediate t ha t that (3.4) also satisfies (3.6). Use of the latter equation show s that the quan tities tr T ( λ ) n pro vide charges in in v olution, tha t is n tr T n ( λ ) , tr T m ( µ ) o = 0 (3.7) whic h again is trivial b y virtue of (3 .6). In the simple sl 2 case the only non tr ivial quan tity is tr T ( λ ) = t ( λ ), that is the usual “ bulk” transfer matr ix. Let us no w briefly review ho w the classical A -op erator and the corresp onding classical equations of mo t ion a r e obtained from the expansion of the mono dro my matrix in the simple case o f p erio dic b o undary conditions. Let us first in tro duce some useful nota tion T a ( n, m ; λ ) = L an ( λ ) L an − 1 ( λ ) . . . L am ( λ ) , n > m. (3.8) T o form ulate { t ( λ ) , L ( µ ) } or ev en b etter { ln t ( λ ) , L ( µ ) } , giv en that usually ln tr T ( λ ) gives rise t o lo cal integrals of motion, w e first deriv e the quantit y { T a ( λ ) , L bn ( µ ) } . 9 n T a ( λ ) , L bn ( µ ) o = T a ( N , n + 1; λ ) r ab ( λ − µ ) T a ( n, 1 ; λ ) L bn ( µ ) − L bn ( µ ) T a ( N , n ; λ ) r ab ( λ − µ ) T a ( n − 1 , 1; λ ) . (3.9) T aking the trace ov er the auxiliary space a , and then the loga rithm we conclude n ln t ( λ ) , L ( µ ) o = t − 1 ( λ )  tr a { T a ( N , n + 1; λ ) r ab ( λ − µ ) T a ( n, 1 ; λ ) } L ( µ ) − L ( µ ) tr a { T a ( N , n ; λ ) r ab ( λ − µ ) T a ( n − 1 , 1; λ ) }  (3.10) the auxiliary index b is suppress ed from the latter expression for simplic ity . Then plugging the latter forms in to the classical equations of motio ns for all inte gra ls of motion ˙ L n ( µ ) = n t ( λ ) , L n ( µ ) o (3.11) w e conclude tha t t he zero curv a ture condition (3.3) is realized b y: A n ( λ, µ ) = t − 1 ( λ ) tr a { T a ( N , n ; λ ) r ab ( λ − µ ) T a ( n − 1 , 1; λ ) } . (3.12) Let us fo cus on the classical r a tional r -matrix (2.7). In this case A n tak es the simple form: A n ( λ, µ ) = t − 1 ( λ ) λ − µ T ( n − 1 , 1; λ ) T ( N , n ; λ ) . (3.13) In the op en case, as w e shall see in the subsequen t section, it is sufficien t to consider the expansion of t ( λ ) in order to obta in the lo cal integrals o f mo t ion. 3.2 Op en b oundary cond itions W e no w generalize the pro cedure desc rib ed in the preceding section to the case of generic in tegrable “b oundary conditions”. W e prop ose a construction of tw o t yp es of mono drom y and transfer matrices, and asso ciated Lax-type ev olution equations, of the form (3.6)–(3.12), alb eit incorp o r a ting a suppleme ntary set of non-dynamical parameters encapsulated into a “reflection” matrix K ( λ ). In some examples they ma y indeed be inte rpreted as b oundary effects consisten t with integrabilit y of an op en ch ain- lik e system. W e should stress that this is the first time to our knowle dge that suc h an inv estigation is systematically undertak en. There are related studies regarding particular examples of open spin c hains such as XXZ, XYZ and 1D Hubbard models [47, 48]. Ho w ev er, the deriv ation of the corresp onding Lax pair is restricted to the Hamiltonian only and not to all asso ciated in tegrals of motio n of the op en c hain. In t his study w e presen t a generic description indep enden t of the c hoice of 10 mo del, and we deriv e t he Lax pair f o r eac h one of the en tailed b oundary in tegrals of motion. P articular examples are also presen ted. The tw o types of mono drom y matrices will resp ectiv ely represe nt the classical v ersion of the reflection a lgebra R , and the t wisted Y angian T written in the compact form: (see e.g. [14, 15, 49]): n T 1 ( λ 1 ) , T 2 ( λ 2 ) o = r 12 ( λ 1 − λ 2 ) T 1 ( λ 1 ) T 2 ( λ 2 ) − T 1 ( λ 1 ) T 2 ( λ 2 ) ˆ r 12 ( λ 1 − λ 2 ) + T 1 ( λ 1 ) ˆ r ∗ 12 ( λ 1 + λ 2 ) T 2 ( λ 2 ) − T 2 ( λ 2 ) r ∗ 12 ( λ 1 + λ 2 ) T 1 ( λ 1 ) (3.14) where ˆ r , r ∗ , ˆ r ∗ are defined in (2.10). The latter equation ma y b e though t of as the se mi- classical limit of the reflection equation [14]. In most w ell kno wn ph ysical cases, suc h as t he A (1) N − 1 r -matrices r t 1 t 2 12 = r 21 implying that in the SNP case r ∗ ab = ˆ r ∗ ab . In the case of the Y angian r -matrix r 12 = r 21 , hence all the express ions ab ov e ma y b e written in a more sym- metric f o rm. These t w o distinct algebras are resp ectiv ely asso ciated with the SP b oundary conditions ( R ) and the SNP b oundary conditions ( T ). In order to construct r epresen tations of (3.14) yielding the g enerating function of the in tegrals of motion one now in tro duces c -num b er represen tations ( K ± ) of the algebra R ( T ) satisfying (3.14) for SP a nd SNP resp ectiv ely , and also the non-dynamical condition: n K ± 1 ( λ 1 ) , K ± 2 ( λ 2 ) o = 0 . (3.15) T aking now as T ( λ ) an y bulk mono dromy matrix ( 3 .4) built fro m lo cal L matrices o b eying (3.6) and defining in additio n ˆ T ( λ ) = T − 1 ( − λ ) for SP , ˆ T ( λ ) = T t ( − λ ) for SNP . (3.16) one gets: The o r em 3.1. : R epresen t a tions of t he corresp onding algebras R , T , are giv en b y the fol- lo wing expression see e.g. [14, 13]: T ( λ ) = T ( λ ) K − ( λ ) ˆ T ( λ ) . (3.17) F or a detailed pro of see e.g. [13].  Define no w as generating function o f the inv o lutiv e quan tities t ( λ ) = tr { K + ( λ ) T ( λ ) } . (3.18) The o r em 3.2. : Due to (3.1 4) it is sho wn that [14, 13] n t ( λ 1 ) , t ( λ 2 ) o = 0 , λ 1 , λ 2 ∈ C . (3.19) 11  The expansion of t ( λ ) naturally g ives rise to the integrals of motions i.e. t ( λ ) = X i H ( i ) λ i . (3.20) Usually one considers the quantit y ln t ( λ ) to get lo c al in tegrals of motion, ho we v er for t he examples w e are going to examine here the expansion of t ( λ ) is enough to provide the asso- ciated lo cal quantities as will be t ransparen t in the subsequen t section. Finally one has: The o r em 3.3. : Time evolution o f the lo cal Lax matrix L n under generating Hamiltonian action of t ( λ ) is giv en b y: ˙ L n ( µ ) = A n +1 ( λ, µ ) L n ( µ ) − A n ( λ, µ ) L n ( µ ) , (3.21) where A n is the mo dified (bo undary) quan tity , A n ( λ, µ ) = tr a  K + a ( λ ) T a ( N , n ; λ ) r ab ( λ − µ ) T a ( n − 1 , 1; λ ) K − a ( λ ) ˆ T a ( λ ) + K + a ( λ ) T a ( λ ) K − a ( λ ) ˆ T a (1 , n − 1; λ ) ˆ r ∗ ab ( λ + µ ) ˆ T a ( n, N ; λ )  . (3.22) Pr o of : W e need in addition to (3.6) one more fundamen tal relation i.e. n ˆ L a ( λ ) , L b ( µ ) o = ˆ L a ( λ ) ˆ r ∗ ab ( λ ) L b ( µ ) − L b ( µ ) ˆ r ∗ ab ( λ + µ ) ˆ L a ( λ ) . (3.23) where the notation ˆ L is self-explanatory f r o m (3.16). T aking in to accoun t the latt er expres- sions w e deriv e for ˆ T n ˆ T a ( λ ) , L bn ( µ ) o = ˆ T a (1 , n ; λ ) ˆ r ∗ ab ( λ + µ ) ˆ T a ( n + 1 , N ; λ ) L bn ( µ ) − L bn ( µ ) ˆ T a (1 , n − 1; λ ) ˆ r ∗ ab ( λ + µ ) ˆ T a ( n, N ; λ ) . (3.24) The next step is to fo rm ulate n t ( λ ) , L bn ( µ ) o ; indeed recalling (3 .9), (3.24), (3.17) and (3.18) w e conclude: n t ( λ ) , L bn ( µ ) o = tr a  K + a ( λ ) T a ( N , n + 1; λ ) r ab ( λ − µ ) T a ( n, 1 ; λ ) K − a ( λ ) ˆ T ( λ ) + K + a ( λ ) T a ( λ ) K − a ( λ ) ˆ T a (1 , n ; λ ) ˆ r ∗ ab ( λ + µ ) ˆ T a ( n + 1 , N ; λ )  L bn ( µ ) − L bn ( µ ) tr a  K + a ( λ ) T a ( N , n ; λ ) r ab ( λ − µ ) T a ( n − 1 , 1; λ ) K − a ( λ ) ˆ T ( λ ) + K + a ( λ ) T a ( λ ) K − a ( λ ) ˆ T a (1 , n − 1; λ ) ˆ r ∗ ab ( λ + µ ) ˆ T a ( n, N ; λ )  . (3.25 ) Expression (3.22) is readily extracted from (3.25 ).  12 Sp ecial care should b e tak en at the b oundary p oints n = 1 and n = N + 1. Indeed go ing bac k to formula (3.22) restricting ourselv es to n = 1 and n = N + 1 and taking in to accoun t that T ( N , N + 1 , λ ) = T (0 , 1 , λ ) = ˆ T (1 , 0 , λ ) = ˆ T ( N + 1 , N , λ ) = I w e obtain the explicit form for A 1 , A n +1 . W e should stress that the deriv ation of the b oundary Lax pair is univ ersal, namely the expressions (3.22) are generic and indep enden t of the c hoice of L, r . Note that expansion of the latter expres sions of A n ( λ, µ ) = P A ( i ) n λ i pro vides all the A ( i ) n asso ciated to the corresponding integrals of motion H ( i ) , whic h follo w from the ex pansion of t ( λ ). This will b ecome quite transparent in the examples presen ted in the subsequen t section. Remark : A differen t construction of (3.14) w as already giv en in a ve ry general setting in [49]. It is related to the form ulation of non-ultralo cal in tegrable field theories o n a lattice and extends the analysis of [15]. The essen tial difference with our construction is tha t the k matrix (denoted γ ) in [49], is sandw ich ed b etw een the “lo cal” mono drom y matrices T n,n − 1 so as to obtain an o ve rall Poisson brac k et of the form (3.14) for the “dressed” mono dro my matrix ... T n +1 ,n γ T n,n − 1 γ .... The matrix γ allows t o tak e in to a ccoun t the effects of the non- ultralo cal part δ ′ ( x − y ) of the P oisson brack et structure. In this resp ect γ also correspo nds to “b oundary” effects, although multiple and in ternal. 3.3 Examples W e shall now examine a simple example, i.e. the op en generalized D ST mo del, whic h ma y b e seen as a lat t ice vers ion of the generalized (v ector) NLS mo del, (see also [39, 50, 5 1, 52, 13] for further details). The op en T o da c hain will b e a lso discuss ed as a limit of the DST mo del. W e shall explicitly ev aluate the “b oundar y” Lax pairs for the first in tegrals of motion. If w e fo cus on the sp ecial case (2.7), whic h will be our main interes t here, the latter expression reduces to the fo llo wing express ions for SP and SNP b oundary conditions. In particular, A n for SP b oundary conditions reads: A n ( λ, µ ) = 1 λ − µ T ( n − 1 , 1; λ ) K − ( λ ) ˆ T ( λ ) K + ( λ ) T ( N , n ; λ ) + 1 λ + µ ˆ T ( n, N ; λ ) K + a ( λ ) T ( λ ) K − ( λ ) ˆ T (1 , n − 1; λ ) for SP (3.26) and for SNP b oundary conditions: A n ( λ, µ ) = 1 λ − µ T ( n − 1 , 1; λ ) K − ( λ ) ˆ T ( λ ) K + ( λ ) T ( N , n ; λ ) − 1 λ + µ ˆ T t (1 , n − 1; λ ) K − t a ( λ ) T t ( λ ) K + t ( λ ) ˆ T t ( n, N ; λ ) for SNP , (3.27) where w e recall that for the special p oin ts n = 1 , N + 1 we take into account t hat: T ( N , N + 1 , λ ) = T (0 , 1 , λ ) = ˆ T (1 , 0 , λ ) = ˆ T ( N + 1 , N , λ ) = I . 13 The Lax op erator of the g l ( N ) DST mo del has the followin g form: L ( λ ) = ( λ − N − 1 X j =1 x ( j ) X ( j ) ) E 11 + b N X j =2 E j j + b N X j =2 x ( j − 1) E 1 j − N X j =2 X ( j − 1) E j 1 (3.28) with x ( j ) n , X ( j ) n b eing canonical v ariables n x ( i ) n , x ( j ) m o = n X ( i ) n , X ( j ) m o , n x ( i ) n , X ( j ) m o = δ nm δ ij i, j ∈ { 1 , . . . , N } , n, m ∈ { 1 , . . . , N } . (3.2 9) In [13] the first non-trivial in tegral of motion for the SNP case, choosing t he simplest con- sisten t v alue K ± = I w as explicitly computed: H = − 1 2 N X n =1 N 2 n − b N X n =1 N − 1 X j =1 X ( j ) n x ( j ) n +1 − 1 2 N − 1 X j =1 ( X ( j ) N X ( j ) N + b 2 x ( j ) 1 ) where N n = N − 1 X j =1 x ( j ) n X ( j ) n . (3.30) Our aim is no w to determine the mo dified Lax pair induced by the non-tr ivial in tegrable b oundary conditions. W e shall fo cus here on the case o f SNP b oundary conditions, basically b ecause in the particular example w e consider here suc h b oundary conditions a re tec hnically easier to study . Moreov er, the SNP b oundary conditions ha v e not b een so m uc h analyzed in the con text of lattice in t egrable mo dels, whic h pro vides an extra motiv ation to in ves tiga t e them. T aking in to accoun t (3.27) w e explicitly derive the mo dified Lax pair for the general- ized DST mo del with SNP b o undary conditions. Indeed, after expanding (3.27) in p ow ers of λ − 1 w e obtain the quantit y asso ciated to the Hamiltonian (3.30) 3 : A (2) n = λE 11 − X j 6 =1 X ( j − 1) n − 1 E j 1 + b X j 6 =1 x ( j − 1) n E 1 j , n ∈ { 2 , . . . N } A (2) 1 = λE 11 − b X j 6 =1 x ( j − 1) 1 E j 1 + b X j 6 =1 x ( j − 1) 1 E 1 j , A (2) N +1 = λE 11 − X j 6 =1 X ( j − 1) N E j 1 + X j 6 =1 X ( j − 1) N E 1 j . (3.31) Let us now consider the simplest p ossible case, i.e. the sl 2 DST mo del. It is w orth stressing that in this case the SP and SNP b o undary coincide giv en that L − 1 ( − λ ) = V L t ( − λ ) V , V = antid(1 , . . . , 1) . (3.32) 3 T o obtain b o th H (2) and A (2) n we divided the original expressions by a facto r tw o. 14 In this particular case the Hamiltonian (3.30) r educes into: H (2) = − 1 2 N X n =1 x 2 n X 2 n − b N − 1 X n =1 x n +1 X n − b 2 2 x 2 1 − 1 2 X 2 N . (3.33) The equations of motion asso ciated to the latt er Hamiltonian ma y b e readily extracted by virtue of 4 ˙ L = n H (2) , L o . (3.35) It is deduced from (3.31) for the sl 2 case: A (2) n ( λ ) = λ bx n − X n − 1 0 ! , n ∈ { 2 , . . . , N } A (2) 1 ( λ ) = λ bx 1 − bx 1 0 ! , A (2) N +1 ( λ ) = λ X N − X N 0 ! . (3.36) Alternativ ely the equations of motion may b e deriv ed from the zero curv ature condition ∂ L n ∂ t = A ( i ) n +1 L n − L n A ( i ) n (3.37) whic h the mo dified Lax pair satisfies. It is clear that to eac h o ne of the hig her lo cal c harges a differen t quantify A ( i ) n is asso ciated. Bot h equations (3.35), (3.37) lead natur a lly to the same equations of motio n, whic h fo r t his pa rticular example read a s: ˙ x n = x 2 n X n + bx n +1 , ˙ X n = − x n X 2 n − bX n − 1 , n ∈ { 2 , . . . N − 1 } ˙ x 1 = x 2 1 X 1 + bx 2 , ˙ X 1 = − x 1 X 2 1 − bx 1 ˙ x N = x 2 N X N + X N , ˙ X N = − x N X 2 N − bX N − 1 . (3.38) The T o da mo del ma y b e seen as an appropriate limit of the DST mo del (see also [53]). Indeed consider the following limiting pro cess as b → 0: X n → e − q n , x n → e q n ( b − 1 + p n ) (3.39) It is clear that the harmonic oscillator algebra defined b y ( x n , X n , x n X n ) reduces to the Euclidian Lie algebra ( e ± q n , p n ), a nd conseq uen tly the La x op erator reduces to: L n ( λ ) = λ − p n e q n − e − q n 0 ! (3.40) 4 The asso ciated Poisso n bra ck ets for both DST and T o da mo dels are defined as: n A, B o = X n  ∂ A ∂ x n ∂ B ∂ X n − ∂ A ∂ X n ∂ B ∂ x n  , DST mo del n A, B o = X n  ∂ A ∂ q n ∂ B ∂ p n − ∂ A ∂ p n ∂ B ∂ q n  , T o da chain . (3.34) 15 where q n , p n are canonical v ariables. In this case the corresp onding Hamiltonian may be readily extracted and takes the fo rm H (2) = − 1 2 N X n =1 p 2 n − N − 1 X n =1 e q n +1 − q n − 1 2 e 2 q 1 − 1 2 e − 2 q N (3.41) and the corresp onding A (2) n are expressed as A (2) n ( λ ) = λ e q n − e q n − 1 0 ! , n ∈ { 2 , . . . , N } A (2) 1 ( λ ) = λ e q 1 − e q 1 0 ! , A (2) N +1 ( λ ) = λ e − q N − e − q N 0 ! . (3.42) In this case as w ell b oth form ulas (3 .3 7) and ( 3 .35) lead to the follo wing set of equations of motions: p n = ˙ q n , ¨ q n = e q n +1 − q n − e q n − q n − 1 , n ∈ { 2 , . . . , N − 1 } p 1 = ˙ q 1 , ¨ q 1 = e q 2 − q 1 − e 2 q 1 p N = ˙ q N , ¨ q N = e − 2 q N − q q N − q N − 1 . (3.43) 4 The con tin uous case 4.1 P erio dic b oundary conditions Let us no w recall the basic notions rega rding t he Lax pair and the zero curv ature condition for a contin uous in tegrable mo del follo wing es sen tially [6]. D efine Ψ as b eing a solution of the follow ing set o f equations (see e.g. [6]) ∂ Ψ ∂ x = U ( x, t, λ )Ψ (4.1) ∂ Ψ ∂ t = V ( x, t, λ )Ψ (4.2) U , V b eing in general n × n matrices with en tries defined as functions of complex v alued dynamical fields, their deriv ativ es, and the sp ectral parameter λ . Compatibilit y conditions of the t w o differen tia l equation (4.1), ( 4 .2) lead to the zero curv a t ure condition [3 ]– [5] ˙ U − V ′ + h U , V i = 0 , (4.3) giving rise to the corresp o nding classical equations o f motio n of the system under consider- ation. The mono drom y matrix may b e written from (4.1) as: T ( x, y , λ ) = P exp n Z x y U ( x ′ , t, λ ) dx ′ o , ( 4 .4) 16 where apparently T ( x, x, λ ) = 1 . The fact tha t the mono drom y matrix satisfies equation (4.1) is extensiv ely used to get the relev ant in tegrals of mo t io n and the a sso ciated Lax pairs. Hamiltonian formulation of the equations of motion is av ailable again under the r -matrix approac h. In this picture the underlying classical algebra is manifestly a na logous to the quan tum case. Let us first recall this metho d for a general classical in tegrable system on the full line. The existence of the Poiss on structure for U realized by the classical r-matrix, satisfying t he class ical Y ang-Baxter equation (3.5), guarantee s the integrabilit y of the clas- sical system. Indeed assuming tha t the op erator U satisfies the follo wing ult r a lo cal form of P oisson brack ets n U a ( x, λ ) , U b ( y , µ ) o = h r ab ( λ − µ ) , U a ( x, λ ) + U b ( y , µ ) i δ ( x − y ) , (4.5) T ( x, y , λ ) satisfies: n T a ( x, y , t, λ 1 ) , T b ( x, y , t, λ 2 ) o = h r ab ( λ 1 − λ 2 ) , T a ( x, y , t, λ 1 ) T b ( x, y , t, λ 2 ) i . (4.6) Making use of the latter equation one may readily show for a system on the full line: n ln tr { T ( x, y , λ 1 ) } , ln tr { T ( x, y , λ 2 ) } o = 0 (4.7) i.e. the system is integrable, and the charges in in volution –lo cal integrals of motion– are obtained b y expansion of the generating function ln tr { T ( x, y , λ ) } , based essen tia lly o n the fact t ha t T satisfies (4.1). Let us now recall ho w one constructs the V -op erator asso ciated to giv en lo cal in tegrals of motion. One easily pro ve s the follow ing identit y using (4.5) n T a ( L, − L, λ ) , U b ( x, µ ) o = ∂ M ( x, λ, µ ) ∂ x + h M ( x, L, − L, λ, µ ) , U b ( x, µ ) i (4.8) where w e define M ( x, λ, µ ) = T a ( L, x, λ ) r ab ( λ − µ ) T a ( x, L, λ ) . (4.9) F or mo r e details on the pro of of the fo rm ula ab o v e w e refer the in terested reader to [6 ]; (4.8) ma y be seen as the con tin uum v ersion of relatio n (4.6). Recall that t ( λ ) = tr T ( λ ) then it naturally follows from (4.8), a nd (4.3) , that n ln t ( λ ) , U ( x, λ ) o = ∂ V ( x, λ, µ ) ∂ x + h V ( x, λ, µ ) , U ( x, λ ) i (4.10) with V ( x, λ, µ ) = t − 1 ( λ ) tr a  T a ( L, x, λ ) r ab ( λ, µ ) T a ( x, − L, λ )  (4.11) and in the rational case (4.1 1) reduces to V ( x, λ, µ ) = t − 1 ( λ ) λ − µ T ( x, − L, λ ) T ( L, x, λ ) . (4.12) 17 4.2 General in tegrable b oundary conditions Our aim here is to consider in tegrable mo dels on the in terv al with consisten t “b oundary conditions”, and deriv e rigorously the Lax pairs asso ciated to the en tailed b oundar y lo cal in tegrals of motion as a con tin uous extension of theorems 3.1.–3.3. F or this purp ose w e follo w the line of action describ ed in [6], using now Skly anin’s for mulation for the sys tem on the in t erv al or on the ha lf line. W e briefly describ e this pro cess below for any classic al in tegrable syste m on the interv a l. In this case o ne constructs a mo dified ‘mono dromy’ matrix T , based on Skly anin’s form ulation and satisfying ag ain the P oisson brack et algebras R or T . T o construct the generating function of the integrals of motio n one also needs c -n um b er represen t ations of the a lg ebra R or T satisfying (3.14 ) for SP and SNP resp ectiv ely , suc h that: n K ± 1 ( λ 1 ) , K ± 2 ( λ 2 ) o = 0 . (4.13) The o r em 4.1. : The mo dified ‘mono drom y’ matrices, realizing the cor r esp o nding algebras R , T , are giv en by t he follo wing expressions [14] ( ˆ T b eing defined in (3.1 6)): T ( x, y , t, λ ) = T ( x, y , t, λ ) K − ( λ ) ˆ T ( x, y , t, λ ) . (4.14)  The generating function o f the in v olutiv e quan tities is defined as t ( x, y , t, λ ) = tr { K + ( λ ) T ( x, y , t, λ ) } . (4.15) Indeed one sho ws: The o r em 4.2. : n t ( x, y , t, λ 1 ) , t ( x, y , t, λ 2 ) o = 0 , λ 1 , λ 2 ∈ C . (4.16)  In the case of o p en b oundary conditio ns, exactly as in t he discrete in tegrable mo dels, and taking in to a ccoun t (3 .2 4) w e pro v e n T a (0 , − L, λ ) , U b ( x, µ ) o = M ′ a ( x, λ, µ ) + h M a ( x, λ, µ ) , U b ( x, µ ) i (4.17) where w e define M ( x, λ, µ ) = T (0 , x, λ ) r ab ( λ − µ ) T ( x, − L, λ ) K − ( λ ) ˆ T (0 , − L, λ ) + T (0 , − L, λ ) K − ( λ ) ˆ T ( x, − L, λ ) ˆ r ∗ ab ( λ + µ ) ˆ T (0 , x, λ ) . (4.18) 18 Finally b earing in mind the definition of t ( λ ) a nd (4.1 7) w e conclude with: The o r em 4.3. : n ln t ( λ ) , U ( x, µ ) o = ∂ V ( x, λ, µ ) ∂ x + h V ( x, λ, µ ) , U ( x, µ ) i (4.19) where V ( x, λ, µ ) = t − 1 ( λ ) tr a  K + ( λ ) M a ( x, λ, µ )  . (4.2 0 )  As in the discrete case particular atten tion should b e paid to the b oundary p oints x = 0 , − L . Indeed, for these tw o p oints one has to simply ta k e into a ccoun t that T ( x, x, λ ) = ˆ T ( x, x, λ ) = I . M oreov er, the expressions deriv ed in (4.18), (4.20) a re univ ersal, that is indep enden t of the c hoice of mo del. As w as remarke d up on when discussing t he discrete case, a quadratic alg ebra of the form (3.1 4) was initially obtained in the con tin uous case [15] when extending the deriv ation of 4.1 to situations where the P oisson brack ets (4.5) are non-ultralo cal, exhibiting δ ′ ( x − y ) terms. Connection to “b oundary” effects w as discussed previously (see Section 3 .2 ). 4.3 Example W e shall no w examine a particular example ass o ciated to t he rational r -mat rix (2.7), that is the g l N NLS model. Although in [13] an exten siv e analysis for b oth t yp es of b oundary conditions is presen ted, here w e shall fo cus on the simplest diagonal ( K ± = I ) b oundary conditions. The Lax pair is giv en b y the following expressions [6, 54 ]: U = U 0 + λ U 1 , V = V 0 + λ V 1 + λ 2 V 2 (4.21) where U 1 = 1 2 i ( N − 1 X i =1 E ii − E N N ) , U 0 = N − 1 X i =1 ( ¯ ψ i E i N + ψ i E N i ) V 0 = i N − 1 X i, j =1 ( ¯ ψ i ψ j E ij − | ψ i | 2 E N N ) − i N − 1 X i =1 ( ¯ ψ ′ i E i N − ψ ′ i E N i ) , V 1 = − U 0 , V 2 = − U 1 (4.22) 19 and ψ i , ¯ ψ j satisfy 5 : n ψ i ( x ) , ψ j ( y ) o = n ¯ ψ i ( x ) , ¯ ψ j ( y ) o = 0 , n ψ i ( x ) , ¯ ψ j ( y ) o = δ ij δ ( x − y ) . (4.24) Note that w e hav e suppressed the constan t κ from (4 .2 2) compared e.g. to [13 ] by rescaling the fields ( ψ i , ¯ ψ i ) → √ κ ( ψ i , ¯ ψ i ). F rom the zero curv ature condition (4.3) the classical equations of motion for the gener- alized NLS mo del with p erio dic b o undary conditions are en tailed i.e. i ∂ ψ i ( x, t ) ∂ t = − ∂ 2 ψ i ( x, t ) ∂ 2 x + 2 κ X j | ψ j ( x, t ) | 2 ψ i ( x, t ) , i, j ∈ { 1 , . . . , N − 1 } . (4.2 5 ) It is clear that for N = 2 the equations of motion of the usual NLS mo del are r ecov ered. The b oundary Hamiltonian f or the generalized NLS mo del ma y be expressed as H = Z 0 − L dx N − 1 X i =1  κ | ψ i ( x ) | 2 N − 1 X j =1 | ψ j ( x ) | 2 + ψ ′ i ( x ) ¯ ψ ′ i ( x )  − N − 1 X i =1  ψ ′ i (0) ¯ ψ i (0) + ψ i (0) ¯ ψ ′ i (0)  + N − 1 X i =1  ψ ′ i ( − L ) ¯ ψ i ( − L ) + ψ i ( − L ) ¯ ψ ′ i ( − L )  . (4.2 6) One sees here t hat the K - ma t rix indeed con tributes as a genuine b oundary effect. The Hamiltonian, obt a ined as one of the c harges in inv o lut io n (see e.g. [13] for further details) pro vides the classical equations of motio n b y virtue of: ∂ ψ i ( x, t ) ∂ t = n H (0 , − L ) , ψ i ( x, t ) o , ∂ ¯ ψ i ( x, t ) ∂ t = n H (0 , − L ) , ¯ ψ i ( x, t ) o , − L ≤ x ≤ 0 . (4.27) Indeed considering the Hamiltonian H , w e end up with the follo wing set of equations with Diric hlet type boundar y conditions i ∂ ψ i ( x, t ) ∂ t = − ∂ 2 ψ i ( x, t ) ∂ 2 x + 2 κ N − 1 X j =1 | ψ j ( x, t ) | 2 ψ i ( x, t ) ψ i (0) = ψ i ( − L ) = 0 i ∈ { 1 , . . . , N − 1 } . (4.28) F or a detailed and quite exhaustiv e ana lysis of the v arious in t egrable b o undary conditions of the NLS mo del see [1 3]. Note also that the N = 2 case w as in v estigated classically on the half line in [55], whereas t he NLS equation o n the interv a l w as studied in [5 6]. 5 The Poisson structure for the gener alized NLS mo de l is defined as: n A, B o = i X i Z L − L dx  δ A δ ψ i ( x ) δ B δ ¯ ψ i ( x ) − δ A δ ¯ ψ i ( x ) δ B δ ψ i ( x )  (4.23) 20 As men tioned our ultimate goal here is to derive the b oundary Lax pair, in particular the V op erator. In general for an y g l N r -matrix w e may express (4 .20), taking also in to a ccoun t (4.19), for the tw o t yp es of b oundary conditions a lready describ ed in the first section, i.e. V ( x, λ, µ ) = t − 1 ( λ ) λ − µ T ( x, − L, λ ) K − ( λ ) ˆ T (0 , − L, λ ) K + ( λ ) T (0 , x, λ ) + t − 1 ( λ ) λ + µ ˆ T (0 , x, λ ) K + ( λ ) T (0 , − L, λ ) K − ( λ ) ˆ T ( x, − L, λ ) for SP (4.29 ) and for the SNP b oundar y conditions w e obt a in: V ( x, λ, µ ) = t − 1 ( λ ) λ − µ T ( x, − L, λ ) K − ( λ ) ˆ T (0 , − L, λ ) K + ( λ ) T (0 , x, λ ) + t − 1 ( λ ) λ + µ ˆ T t ( x, − L, λ ) K − t ( λ ) T t (0 , − L, λ ) K + t ( λ ) ˆ T t (0 , x, λ ) for SNP (4.30) Again for the b oundary p oin ts x = 0 , − L w e should b ear in mind that T ( x, x, λ ) = ˆ T ( x, x, λ ) = I . Ultimat ely w e wish to expand T ( λ ) , ˆ T ( λ ) in p ow ers of λ − 1 in order t o determine the L a x pair for eac h one o f the integrals of motion. F or a detailed description of the deriv a tion o f the b o undar y in tegrals o f motion for the generalized NLS mo dels see [13]. Hereafter w e shall fo cus on the SP case with the simplest b oundary conditions i.e. K ± = I . Expanding t he expression (4.29 ) in p ow ers of λ − 1 (w e refer the interes ted reader to App endix C for tec hnical details) w e conclude tha t V (3) ( x, λ ) –the bulk part– coincides with V defined in (4.21), (4.22), a nd for the b oundary p oints x b ∈ { 0 , − L } in particular: V (3) ( x b , λ ) = − λ 2 2 i  N − 1 X i =1 E ii − E N N  + i N − 1 X i,j =1 ¯ ψ i ( x b ) ψ j ( x b ) E ij − i N − 1 X i,j =1  ¯ ψ ′ i ( x b ) E i N − ψ ′ i ( x b ) E N i  . (4.31) W e ma y alternatively rewrite the latter for mula as: V (3) ( x b , λ ) = V ( x b , λ ) + i N − 1 X i =1 | ψ i ( x b ) | 2 E N N + λ N − 1 X i =1 ( ¯ ψ i ( x b ) E i N + ψ i ( x b ) E N i ) . (4.32) The last t w o terms additional to V (4.21), (4.2 2) are due to the non-trivial b oundary condi- tions; of course more complicated b oundary conditions w ould lead to more in tricate mo difi- cations of the Lax pair V , how eve r suc h an exhaustiv e analysis is b ey ond the in tended scop e of t he pr esen t inv estigation. It can b e sho wn tha t the mo dified Lax pair ( U , V (3) ) gives rise to the classical equations o f motion (4.28). It is clear tha t the ‘bulk’ quantit y V in the case o f SP b oundary conditions remains intact. In the SNP case on the other hand w e may see that ev en the bulk part of the Lax pair is drastically mo dified, due to the fact that the bulk part of the corresp onding inte gra ls of motions is also dramatically altered. W e shall not f urther commen t o n this p oint, whic h will b e an yw ay treated in full detail elsewhere. 21 Ac kno wledgmen ts: This w ork w as supp orted by INFN, Bolog na section, throug h gra n t TO12. J.A. wishes to thank INFN and Univ ersit y of Bologna fo r hospitality . A App endix W e presen t here tec hnical details o n the deriv ation of the conserv ed quan tities for the gener- alized NLS mo del on the full line (see also [13 ]). Recall that for λ → ± i ∞ one ma y express T a s [6] T ( x, y , λ ) = ( I + W ( x, λ )) exp [ Z ( x, y , λ )] ( I + W ( y , λ )) − 1 (A.1) where W is an off diagonal mat rix i.e. W = P i 6 = j W ij E ij , and Z is purely diagonal Z = P N i =1 Z ii E ii . Also Z ii ( λ ) = ∞ X n = − 1 Z ( n ) ii λ n , W ij = ∞ X n =1 W ( n ) ij λ n . (A.2) The first step is to insert the ansatz (A.1) in equation (4.1). Then we separate the diagonal and off diagonal part and obta in the following expressions: Z ′ = λ U 1 + ( U 0 W ) ( D ) W ′ + W Z ′ = U 0 + ( U 0 W ) ( O ) + λ U 1 W ( A.3 ) where the sup erscripts ( D ) , ( O ) denote the diag onal a nd off diagonal part of the pro duct U 0 W . Recall that W = P i 6 = j W ij E ij , Z = P i Z ii E ii then it is straightforw a rd to obtain: ( U 0 W ) ( D ) = √ κ N − 1 X i =1  ¯ ψ i W N i E ii + ψ i W i N E N N  ( U 0 W ) ( O ) = √ κ X i 6 = j, i 6 = N , j 6 = N  ¯ ψ i W N j E ij + ψ i W ij E N j  . (A.4) Substituting the latt er expressions (A.4) in (A.3), w e obtain Z ( L, − L, λ ) = − iλL  N − 1 X i =1 E ii − E N N  + √ κ N − 1 X i =1 Z L − L dx  ¯ ψ i W N i E ii + ψ i W i N E N N  . (A.5) The leading contribution in the expansion of (ln tr T ), ( where T is given in (A.1)) for iλ → ∞ comes from Z N N , with a leading term iλL (all other Z ii , i 6 = N hav e a − iλL leading term, so when exp o nen tiating suc h con tributions v anish as iλ → ∞ ), indeed Z N N ( L, − L, λ ) = iλL + √ κ N − 1 X i =1 Z L − L dx ψ i ( x ) W i N ( x ) . (A.6) 22 Due to (A.6) it is obv ious that in this case it is sufficien t t o deriv e the co efficien ts W i N . In an y case one can show that the co efficien ts W ij satisfy the following equations: X i 6 = j W ′ ij E ij − iλ X i 6 = N  W N i E N i − W i N E i N  + √ κ X i 6 = N  ¯ ψ i W 2 N i E N i + ψ i W 2 i N E i N  = √ κ X i 6 = N  ¯ ψ i E i N + ψ i E N i  + √ κ X i 6 = j, i 6 = N , j 6 = N  ¯ ψ i W N j E ij + ψ i W ij E N j  − √ κ X i 6 = j, i 6 = N , j 6 = N  ¯ ψ j W N j W ij E ij + ψ i W i N W j N E j N  . (A.7) Finally setting W ij = P ∞ n =1 W ( n ) ij λ n and using (A.7) we find expressions for W ( n ) i N i.e. W (1) i N ( x ) = − i √ κ ¯ ψ i ( x ) , W (2) i N ( x ) = √ κ ¯ ψ ′ i ( x ) W (3) i N ( x ) = i √ κ ¯ ψ ′′ i ( x ) − iκ 3 2 X k | ψ k ( x ) | 2 ¯ ψ i ( x ) , . . . . (A.8) In the b oundary case we shall need in a dditio n to (A.8) the fo llo wing ob jects: W (1) N i = i √ κψ i , W (2) N i = − iW ′ (1) N i + X i 6 = j, i 6 = N , j 6 = N W (1) N j W (1) j i , W ′ (1) j i = iW (1) j N W (1) N i W (3) N i = − iW ′ (2) N i + W (1) i N W (1) N i W (1) N i + X i 6 = j, i 6 = N ,j 6 = N W (1) N j W (2) j i W ′ (2) ij = iW (1) i N W (2) N j − iW j N (1) W (1) N j W (1) ij . (A.9) Based on the lat ter formu las, the expression (A.1), and defining ˆ W ( x, λ ) = W ( x, − λ ) w e ma y rewrite (4 .2 9) as: V ( x, λ, µ ) = 1 λ − µ (1 + W ( x )) E N N (1 + W ( x )) − 1 + 1 λ + µ (1 + ˆ W ( x )) E N N (1 + ˆ W ( x )) − 1 V (0 , λ, µ ) = ( X + 0 ) − 1  1 λ − µ X + K + ( λ ) + 1 λ − µ K + ( λ ) X +  V (0 , λ, µ ) = ( X − 0 ) − 1  1 λ − µ K − ( λ ) X − + 1 λ − µ X − K − ( λ )  (A.10) where w e define: X + = (1 + W ( 0 )) E N N (1 + ˆ W (0)) − 1 , X − = (1 + ˆ W ( − L )) E N N (1 + W ( − L )) − 1 X + 0 = h (1 + ˆ W (0 , λ )) − 1 K + ( λ )(1 + W (0 , λ )) i N N X − 0 = h (1 + W ( − L, λ )) − 1 K − ( λ )(1 + ˆ W ( − L, λ )) i N N . (A.11) References [1] P .D. L a x, Comm. Pure Appl. Math. 21 (19 68) 467. 23 [2] C.S. Gardner, J.M. Greene, M.D. Krusk a l and M.D. Miura, Ph ys. Rev. Lett. 19 (1967 ) 1095. [3] M.J. Ablowitz, D.J. Kaup, A.C. New ell and H. Segur, Stud. Appl. Math. 53 (1974) 249. [4] M.J. Ablowitz a nd J.F. Ladik, J. Math. Phy s. 17 (1976) 1011. [5] V.E. Za kharo v and A.B. Shabat, Anal. Appl. 13 (19 79) 13. [6] L.D. F addeev, In t egrable mo dels in 1 + 1 dimensional quen t um field t heory in ‘Recen t Adv ances in Field Theory and Statistical Mec ha nics’ (J.B. Z ub er and R. Stora, Ed.), Les Ho uc hes Lectures 198 2 , Nor t h Holla nd (1984) 561; L.D. F a ddeev and L.A. T akhtak a jan, Hamiltonian Metho ds in the The ory of Solitons , (1987) Springer-V erlag. [7] O. Bab elon, D. Bernard and M. T a lon, Intr o duction to classic al Inte gr able systems , (2003) Cam bridge monog raphs in Mathematical Ph ysics. [8] E.G. Skly anin, Preprin t LOMI E-3- 97, Leningrad, 1979 . [9] M.A. Semeno v-Tian-Shansky , F unct. Anal. Appl. 17 ( 1 983) 2 5 9. [10] J. Liouville, Journal de Mathematiques XX (1885) 13 7. [11] V.I. Arnold, Metho ds of Classi c al Me ch anics , Gra duate T exts in Mathematics 60, (1 978) Springer. [12] P . Baseilhac and G.W. D elius, J. Ph ys. A34 (200 1) 8259 . [13] A. Do ikou, D . Fio ra v an ti and F. Ra v anini, Nucl. Phy s. B790 [PM] ( 2 008) 4 6 5. [14] I. Cherednik, Theor. Math. Phy s. 61 ( 1 984) 977; E.K. Skly anin, J. Ph ys. A 21 ( 1 988) 2375. [15] J.-M. Maillet, Ph ys. Lett. B162 (1985 ) 1 37. [16] P . Bow co ck, E. Corr igan, P .E. D orey and R.H. Rietdijk, Nucl. Ph ys. B445 (19 9 5) 469; P . Bow co c k, E. Corrig an a nd R.H. Rietdijk, Nucl. Phy s. B465 (1996) 350. [17] E.K. Skly anin; Zap. Nauc h. Seminaro v LOMI 95 (1980) , 55 ; P .P . Kulish and E.K. Skly anin, in Tv arminne Lectures, edited by J. Hietar inta a nd C. Mon tonen, Springer Lectures in Ph ysics 151 (1 9 82). [18] O. Bab elon and C.M. Viallet, Phy s. Lett. B237 (1990 ) 411. [19] C.N. Y ang, Ph ys. R ev. Lett. 19 (1967) 1312 . [20] H.J. de V ega and A. Gonzalez–Ruiz, Nucl. Ph ys. B 417 (1994) 553; H.J. de V ega a nd A. Gonzalez–Ruiz, Ph ys. Lett. B332 (1994) 123. [21] L. Mezincescu and R.I. Nep o mechie , Nucl. Ph ys. B372 (19 9 2) 597 ; S. Artz, L . Mezinc escu and R.I. Nep omec hie, J. Phy s. A28 (1995) 5131. [22] A. Do ikou and R.I. Nep omec hie, Nucl. Phys . B521 (1998) 54 7; A. D o ik ou and R.I. Nep omechie , Nucl. Ph ys. B530 (199 8) 641. [23] W. Galleas a nd M.J. Martins, Ph ys. Lett. A335 (2005) 167; 24 R. Mala ra and A. Lima-Santos, J. Stat. Mech . 0609 (2006) P013; W.-L. Y ang and Y.-Z. Zha ng , JHEP 0412 ( 2 004) 019. [24] G.M. Ga nden b erger, hep-th /9911178 . [25] G.W. D elius a nd N. Mac k ay , Comm un. Math. Phys . 233 (2003) 173. [26] A. Do ikou, J. Phys . A33 (2 000) 8797. [27] D. Arnaudon, N. Cramp ´ e, A. Doikou, L. F rappat and E. Ragoucy , In t . J. Mo d. Ph ys. A21 (2006) 1537. [28] D. Arnaudon, J. Av an, N. Cramp´ e, A. Doikou, L. F ra ppat and E. Rag oucy , J. Stat. Mec h. 0408 (2004) P005 ; D. Arnaudon, N. Cramp ´ e, A. Do ik ou, L. F r a ppat and E. Ragoucy , J. Stat. Mec h. 0502 (2005) P007. [29] A. Do ikou, J. Math. Ph ys. 46 05350 4 (2005 ) . [30] N. Cramp´ e and A. Doikou, J. Math. Ph ys. 48 023511 (2007 ) . [31] G.I. Olshanski, Twisted Y angia ns and infinite-dimensional classical Lie algebras in ‘Quan tum Gro ups’ (P .P . Kulish, Ed.), Lecture notes in Math. 1510 , Springer (1992) 103; A.I. Molev, M. Nazaro v and G.I. Olshanski, Russ. Math. Surve ys 51 (1996) 206. [32] A.I. Molev, E. R agoucy and P . Sorba, Rev. Math. Phy s. 15 (2003) 789; A.I. Molev, Handb o ok of Algebr a , V ol. 3 , (M. Hazewink el, Ed.), Elsevier, (2003 ), pp. 907. [33] G.W. Delius, N.J. MacKa y and B.J. Short, Ph ys. Lett. B522 (200 1) 335; Erratum- ibid. B524 (2002) 401. [34] A. Do ikou, Nucl. Phy s. B725 (2005) 493. [35] M. G audin, L a F onction d’Ond e de Bethe , Masson (198 3); M. Gaudin, J. Ph ys. (Paris) 37 (1 9 76) 1078. [36] P . v an Mo erb eke , In v en t. Math 37 (1976) 45. [37] M. T o da, The ory of non-lin e ar lattic es , (198 1) Springer-V erlag. [38] E.K. Sklyanin, nlin.SI/000900 9 , “In tegra ble Systems: from classical to quan tum”; CRM Pro ceedings Lecture Notes 28 (20 00) 227. [39] A. Kundu and B. Basu Mallick, Mo d. Ph ys. Lett. 7 (1992 ) 61 . [40] M. Jimbo, Comm un. Math. Ph ys. 102 (1986) 53 [41] K. Hik ami, solv-int/950 6001 . [42] N. Cramp´ e and C.A.S. Y o ung, J. Ph ys. A40 (2007 ) 5491 . [43] K. Hik ami, J. Ph ys. A2 8 (19 95) 4997; W.-L. Y ang, Y.-Z. Zhang and M.D . Gould, Nucl. Phy s. B698 (2004) 503; W.-L. Y ang, Y.-Z. Zhang and R . Sasaki, Nucl. Phy s. B729 (2005) 594. 25 [44] E. Sklyanin, F unct. Anal. Appl. 16 (198 2) 263. [45] L.C. Li and S. Parmen tier, Comm un. Math. Phys . 125 (1989) 545. [46] B.M. McCoy and T.T. W u, Nuov o Cimen to B 56 (1 9 66) 31 1 . [47] H.Q. Zhou, J. Ph ys. A29 (19 96) L489; X.W. Guan, D.M. T ong and H.Q. Zho u, J. Ph ys. So c. Jap. 65 (1996 ) 2807 ; X.W. Guan, M.S. W ang a nd S.D. Y ang, Nucl. Phys . B485 (1997) 685 . [48] A. Lima-Sa n tos, solv-int/9912009 . [49] L. F reidel and J.M. Maillet, Ph ys. Lett. B263 (1991) 403. [50] V.S. Gerdjik ov and M.I. Iv anov, Theor. Math. Ph ys. 52 (1982) 676 [51] E. D ate, M. Jim b o and T. Miw a, J. Phys . So c. Japan 52 (1983 ) 761. [52] I. Merola, O. Ragnisco and G . Z hang T u, Inv erse Problems 10 (1994) 1315; A. Kundu a nd O. Ragnisco, J. Phy s. A27 (1994) 6335. [53] V.B. Kuznetso v, M. Salerno and E.K. Skly anin, J. Ph ys. A33 (20 0 0) 171. [54] A. F ordy and P .P . Kulish, Comm un. Math. Ph ys. 89 (19 83) 427. [55] V.O. T ar a so v Inv. Probl. 7 (1 991) 4 3 5; R.F. Bikbaev and V.O. T arasov, J. Phy s. A24 (1991) 2507. [56] A.S. F ok as and A.R. Its, nlin.SI/0412008 . 26