An algebraic approach to discrete time integrability
aa r X i v : . [ m a t h - ph ] S e p AN ALGEBRAIC APPROACH TO DISCRETE TIMEINTEGRABILITY
ANASTASIA DOIKOU AND IAIN FINDLAY
Abstract.
We propose the systematic construction of classical and quantumtwo dimensional space-time lattices primarily based on algebraic considera-tions, i.e. on the existence of associated r -matrices and underlying spatial andtemporal classical and quantum algebras. This is a novel construction thatleads to the derivation of fully discrete integrable systems governed by sets ofconsistent integrable non-linear space-time difference equations. To illustratethe proposed methodology, we derive two versions of the fully discrete non-linear Sch¨rodinger type system. The first one is based on the existence of arational r -matrix, whereas the second one is the fully discrete Ablowitz-Ladikmodel and is associated to a trigonometric r -matrix. The Darboux-dressingmethod is also applied for the first discretization scheme, mostly as a consis-tency check, and solitonic as well as general solutions, in terms of solutionsof the fully discrete heat equation, are also derived. The quantization ofthe fully discrete systems is then quite natural in this context and the twodimensional quantum lattice is thus also examined. Introduction
The fundamental paradigm in the frame of classical integrable systems is the AKNSscheme [1]. This offers the main non-relativistic set up, and is naturally associatedto the non-linear Schr¨odinger system (NLS), the mKdV and KdV equations, andcan be also mapped to typical examples of relativistic systems such as the sine-Gordon model. The AKNS scheme and NLS type hierarchies are among the mostwidely studied integrable prototypes (see for instance [1, 2, 3, 22, 38] and [35, 36,37]). Both continuum and discrete versions have been thoroughly investigated fromthe point of view of the inverse scattering method or the Darboux and Zakharov-Shabat (ZS) dressing methods [51, 52], [3], [39, 11, 40, 43, 54, 4, 13, 15], yieldingsolutions of hierarchies of integrable non-linear PDEs (ODEs) as well as hierarchiesof associated Lax pairs. Numerous studies from the Hamiltonian point of viewin the case of periodic (see for instance [19, 46, 45] and [23, 24]) and genericintegrable boundary conditions [47, 6] also exist. The Hamiltonian or algebraicframe offers the most systematic means for constructing and studying classicalintegrable systems. The potency of the algebraic approach relies on the existenceof a classical r -matrix that satisfies the classical Yang-Baxter equation. This thensignifies the presence of associated Poisson structures [46, 45] that naturally lead to sets of quantities in involution, i.e. integrals of motion. Quantization in thiscontext is then quite natural as the classical r -matrix is replaced by a quantum R -matrix that obeys the quantum YBE, and the classical Poisson algebra is replacedby a quantum algebra [18]. The existence of a classical (quantum) r -matrix allowsalso the computation of the time components of the Lax pairs of the hierarchyvia the fundamental Semenov-Tian-Shansky formula (STS) [45], that involves the r and L matrices. This universal formula has been extended to the case of openboundary conditions as well as at the quantum level [6, 12, 33].In the present investigation we are proposing the algebraic setting for con-structing “space-time” discrete integrable systems. The study of fully discretesystems has been a particularly active field in recent decades (see e.g [26] andreferences therein) and the main frame for describing such integrable systems andthe associated partial difference equations is the so-called consistency approach[41, 5, 27]. These studies have also produced various significant connections withYang-Baxter maps and the set theoretic Yang-Baxter equation (also linked to thenotion of Darboux-B¨acklund transformations) [42, 48], cluster algebras [21, 31, 28],and the concept of algebraic entropy (see e.g. [25, 49] and references therein), tomention a few. Our approach is mainly based on algebraic considerations and isgreatly inspired by earlier works on space-time dualities [7, 8, 13] and the existenceof underlying spatial and temporal Poisson structures. To illustrate the algebraicapproach we present two distinct fully discrete versions of the NLS-type hierarchybased on the existence of classical and quantum r -matrices and the underlying de-formed algebras: 1) the fully discrete version of the system introduced in [36, 37](fully DNLS), which is the more natural discretization of the NLS-type systems(AKNS scheme generally) from the algebraic point of view, and is associated toa rational r -matrix. 2) The fully discrete Ablowitz-Ladik (AL) model (see e.g.[2, 3, 35]) associated to a trigonometric r -matrix. Generalized local [39] trans-formations are then employed in order to identify solutions of the associated fullyDNLS nonlinear partial difference equations as well as to confirm the findings fromthe algebraic point of view. When discussing the solutions of the relevant partialdifference equations we are primarily focused on the discrete version of the DNLShierarchy associated to a rational r -matrix (see [37] and references therein). Notethat the DNLS model is a natural integrable version of the discrete-self-trapping(DST) equation introduced and studied in [17] to model the nonlinear dynamics ofsmall molecules, such as ammonia, acetylene, benzene, as well as large molecules,such as acetanilide. It is also related to various physical problems such as arraysof coupled nonlinear wave-guides in nonlinear optics and quasi-particle motion ona dimer among others.We stress that this is the first time to our knowledge that a systematic con-struction of fully discrete space-time integrable systems based on the existence of aclassical r -matrix is achieved. This fundamental idea is naturally extended to thequantum case and the two dimensional quantum lattice can then be constructed. N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 3
This derivation is based on the existence of copies of two distinct quantum alge-bras associated to spatial and temporal “quantum spaces” and it is in a mannerin the spirit of constructing higher dimensional quantum lattices via the solutionof the tetrahedron equation [53, 9], although in our construction there is a cleardistinction between spatial and temporal quantum algebras.Let us briefly outline what is achieved in the article: • In section 2 we present the spatial and temporal Poisson structures asso-ciated to discrete time integrable systems. In this frame the time com-ponents of Lax pairs, i.e. the V -operators are required to be represen-tations of a quadratic Poisson structure, whereas the space componentssatisfy linear Poisson structures in the semi-discrete time case, and qua-dratic classical algebras in the fully discrete case. We first examine thesemi discrete time case and we consider the time like approach, i.e. fora given V -operator we apply the corresponding STS formula [7] and de-rive the hierarchy of the space components of the Lax pairs. This partserves as a predecessor, providing the main frame to consistently formu-late the fully discrete case. After we provide the general algebraic set upfor fully discrete integrable systems we examine two prototypical systemsthat are discretizations of the NLS-type scheme. For both examples thetime components of the Lax pairs are constructed as representations ofthe quadratic temporal algebras. Having identified the Lax pairs we alsoderive the associated partial difference equations via the fully discrete zerocurvature condition. • In section 3 the Darboux-B¨acklund methodology is implemented for thefully DNLS system. The purpose of this section is two-fold : 1) We ex-tract the Lax pairs for the space-time discretization of NLS confirming thefindings of the algebraic approach. 2) We derive solutions via certain lo-cal Darboux transforms. More specifically, by employing the fundamentalDarboux transformation we perform the dressing process and we identifythe Lax pairs of the discrete hierarchy. Explicit expressions for the firstfew members are presented and the findings of the algebraic approach areconfirmed. Via the fundamental Darboux transform we also derive twotypes of discrete solitonic solutions that are the fully discrete analoguesof the solutions found in [15]. More importantly, with the use of a Todatype Darboux matrix we identify generic new solutions (i.e. not only soli-tonic) of the non-linear partial difference equations in terms of solutionsof the associated linear equations, i.e. the fully discrete heat equation,generalizing the findings of [15] to the fully discrete case. • In section 4 we present the two dimensional quantum lattice. In order tobe able to build the two dimensional quantum lattice along the space andtime directions, in analogy to the classical case as described in section 2, weintroduce the notion of spatial and temporal “quantum spaces”. Despite
ANASTASIA DOIKOU AND IAIN FINDLAY the slight abuse of language, we employ the notion of quantum spacesto describe copies of the underlying quantum algebras when constructingthe corresponding spin chain like systems with N ( M ) sites, or quantumspaces, along the space (time) direction. This construction is in exactanalogy to the classical description. The quantum discrete NLS model,associated to the Yangian R -matrix, as well as the quantum Ablowitz-Ladik model (or q bosons), associated to a trigonometric R -matrix, areconsidered as our prototypical quantum systems.2. Discrete time Integrability: algebraic formulation
In this section we suggest the algebraic formulation for the construction of dis-crete time integrable systems. Specifically, we present the space and time likePoisson structures associated to discrete time integrable systems. We first con-sider the semi-discrete time case, which basically serves as a predecessor of thefully discrete frame. The Lax pairs are perceived in this context as representa-tions of the underlying space-time Poisson structures. The discrete versions of thezero curvature condition provide compatibility conditions among the various fieldsinvolved and yield the associated difference/differential equations. To explicitlyillustrate the proposed methodology we examine two prototypical systems that arediscretizations of the NLS-type scheme and are associated to two distinct classical r -matrices (rational versus trigonometric). For both examples the Lax pairs areconstructed as representations of the spatial and temporal algebras. Having iden-tified the Lax pairs we derive the associated partial difference equations from thefully discrete zero curvature condition.2.1. The semi-discrete time setting.
We first examine the case of discretetime and continuous space classical integrable systems and we mainly focus on thetime-like algebraic picture. The main reason we consider this case first is the factthat the STS formula is available, and the hierarchy of associated U -operatorscan be thus systematically derived [7]. This will be achieved in the followingsubsection for a particular example, the semi discrete time NLS system. Such aconstruction will provide a first indication on the consistent forms of Lax pairsin the fully discrete scenario. Note that in this case we only consider integrablesystems associated to rational r -matrices, i.e. the Yangian [50].Let us first recall the space-like description and we then move on to the time-like picture as discussed in [7, 8, 13]. The starting point is the existence of a Laxpair (cid:16) U, V (cid:17) consisting of generic c -number d × d matrices (see e.g. [19]). TheLax pair matrices depend in general on some fields and a spectral parameter, andform the auxiliary linear problem: ∂ x Ψ( x, a, λ ) = U ( x, a, λ )Ψ a ( x, a, λ ) , Ψ( a + 1 , x, λ ) = V ( x, a, λ )Ψ( x, a, λ ) , (2.1) N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 5 where a is the discrete time index. Compatibility of the two equations above leadsto the discrete time zero curvature condition:(2.2) ∂ x V ( x, a, λ ) = U ( x, a + 1 , λ ) V ( x, a, λ, x ) − V ( x, a, λ, x ) U ( x, a, λ ) . Before we move on to the algebraic formulation of discrete time integrabilitylet us first introduce some useful objects. Let us define the space like monodromymatrix, which is a solution of the first of the equations of the auxiliary linearproblem (2.1)(2.3) T S ( x, y, a, λ ) = ←− P exp n Z xy U a ( ξ, λ ) dξ o , x > y, where ←− P denotes path ordered integration. We also define the periodic spacetransfer matrix as t S ( a, λ ) = trT S ( A, − A, a, λ ), then using the discrete time zerocurvature condition as well as assuming periodic space boundary conditions, orvanishing conditions at ± A , we conclude that t S ( a, λ ) is constant in the discretetime, i.e. t S ( a, λ ) = t S ( a +1 , λ ). A more detailed discussion on the latter statementis provided in the next subsection, where the fully discrete case is examined.The M -site time monodromy T T , which is a solution to the time part of (2.1),is defined as(2.4) T T ( x, b, a, λ ) = V ( x, b, λ ) · · · V ( x, a + 1 , λ ) V ( x, a, λ ) , b > a, and the time-like transfer matrix is given by t T ( λ ) = trT T ( x, M, , λ ). By meansof the zero curvature condition (2.2) and assuming periodic time-like boundaryconditions we conclude that d t T ( λ ) dx = 0From the algebraic point of view the fundamental statement is that the U -operator satisfies the linear Poisson structure [19, 37](2.5) n U ( x, a, λ ) ⊗ , U ( y, a, µ ) o S = h r ( λ − µ ) , U ( x, a, λ ) ⊗ I + I ⊗ U ( y, a, µ ) i δ ( x − y ) , where I is in general the d × d identity matrix, the subscript S denotes space-like Poisson structure, and the r -matrix is a solution of the classical Yang-Baxterequation [45],(2.6) (cid:2) r ( λ − λ ) , r ( λ ) (cid:3) + (cid:2) r ( λ − λ ) , r ( λ ) (cid:3) + (cid:2) r ( λ ) , r ( λ ) (cid:3) = 0 . The r -matrix acts on V ⊗ V , V is a d dimensional vector space in general, and inthe index notation r = P ij r ( ij | kl ) e ij ⊗ e kl ⊗ I , similarly for r and r , and e ij are in general d × d matrices with elements ( e ij ) kl = δ ik δ jl .In [7], where the continuum space-time scenario was examined, it was assumedthat V, as well as U satisfy linear Poisson structures (see also [8] on further empha-sis on the algebraic/ r -matrix description). Indeed, it was noticed that the time-likePoisson bracket could be constructed from a corresponding algebraic expressionregarding the time component of the Lax pair. ANASTASIA DOIKOU AND IAIN FINDLAY
Here we assume time-like discretization and introduce time-like indices a, b ,then the V -operator in (2.1) is required to satisfy the quadratic algebra:(2.7) n V ( x, a, λ ) ⊗ , V ( x, b, µ ) o T = h r ( λ − µ ) , V ( x, a, λ ) ⊗ V ( x, b, µ ) i δ ab , where r is the same classical r -matrix as in (2.5), and the subscript T denotes thetime-like Poisson structure. Both space and time Sklyanins bracket’s (2.5) and(2.7) are the typical Poisson structures on the LGL d loop group.The time like mondromy matrix (2.4) satisfies the quadratic algebra (we writefor simplicity T T ( x, M, , λ ) = T T ( λ ))(2.8) n T T ( λ ) ⊗ , T T ( µ ) o T = h r ( λ − µ ) , T T ( λ ) ⊗ T T ( µ ) i . Consequently, one obtains commuting operators, with respect to the time-likePoisson structure n trT T ( λ ) , trT T ( µ ) o T = 0 . Deriving V -operators. Our main objective now is to identify the form of thetime components of the Lax pairs, i.e. the V -operators for algebras associated tothe rational r -matrix [50],(2.9) r ( λ ) = 1 λ d X i,j =1 e ij ⊗ e ji . The quantity P i,j e ij ⊗ e ji is the so called permutation operator. We express the V -operator in the following generic form as a finite λ series expansion(2.10) V ( k ) ( λ ) = k X m =0 λ m Y ( m,k ) . In the case we examine here, i.e. the DNLS hierarchy we consider Y ( k,k ) = D =diag(1 , , . . . ,
0) ( d − Y ( m,k ) are in general d × d matrices to be identified algebraically. Note that for Y ( k ) = I ( I is the d × d identity matrix) we essentially deal with the classical version of the gl d Yangian(here we focus on d = 2). More generally in the Yangian case Y ( k,k ) can be aconstant non-singular matrix.We impose the following two fundamental assumptions in order to identify each V ( k ) of the generic form (2.10). • The basic assumptions (1) Each V ( k ) of the form (2.10) satisfies the quadratic algebra (2.7).(2) det V ( k ) = λ k + P k − n =0 a n λ n .From assumption 1 and the general form of the V ( k ) -operator (2.10), the followingPoisson relations emerge, being the classical analogues of the Yangian gl d ( Y ( k,k )N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 7 is a constant (non-dynamical) matrix) n Y ( m − ,k ) ⊗ , Y (0 ,k ) o T = h P , Y ( m,k ) ⊗ Y (0 ,k ) in Y ( m − ,k ) ⊗ , Y ( l,k ) o T − n Y ( m,k ) ⊗ Y ( l − ,k ) o T = h P , Y ( m,k ) ⊗ Y ( l,k ) i , (2.11)where m, l ∈ { , . . . , k } and P = P di,j =1 e ij ⊗ e ji is the permutation operator.In the language of dressing Darboux transform assumption 2 is equivalent to sayingthat the determinant of V ( k ) is independent of the fields, i.e. the determinant ofthe “dressed” V -operators should be equal to the one of the “bare” operators(free of fields). By employing the two fundamental assumptions above we canthen express all Y ( m,k ) in terms of some “fundamental” fields (see also [13] fora relevant discussion), that satisfy certain basic Poisson relations. The problemthen reduces into classifying representations of the classical algebra (2.7) of thegeneral structure (2.10).Given the operator V ( k ) we can then apply the time-like STS formula [45, 7],which is valid for continuum x [13], derive the hierarchy of associated U -operatorsand extract in turn the hierarchy of non linear integrable ODEs. This will beachieved in the next subsection. Notice that extra compatibility conditions emergefrom the discrete time zero curvature condition ensuring the consistency of our con-struction. Let us now focus on the first two members of the hierarchy, and identify V (1) , V (2) from the algebraic point of view. In the continuum time situation foreach time flow t k a corresponding V ( k ) ( x, t k , λ ) exists. In analogy, in the discretecase our notation will be V ( k ) ( x, a k , λ ) for each discrete time index a k . We shalldrop the sub-index k henceforth for brevity. The V (1) -operator. The first non-trivial V -operator is linear in λ and is associ-ated to a discrete time version of the transport equation, V (1) ( x, a, λ ) = λ D + Y (0 , = λ + N (1) a ( x ) ˆ u a ( x ) u a ( x ) 1 ! , (2.12)where from the condition det V (1) = λ + 1 we obtain(2.13) N (1) a ( x ) = 1 + ˆ u a ( x ) u a ( x ) . Note that the x dependence in the fields is always implied even if omitted forbrevity. Due to the fact that V (1) satisfies the quadratic algebra (2.7) we derivethe Poisson relations:(2.14) n Y (0 , ⊗ , Y (0 , o T = P (cid:16) D ⊗ Y (0 , − Y (0 , ⊗ D (cid:17) , ANASTASIA DOIKOU AND IAIN FINDLAY where P = P e ij ⊗ e ji is the permutation operator. Hence, the Poisson relationsfor the fields follow(2.15) n u a ( x ) , ˆ u b ( x ) o T = δ ab , n ˆ u a ( x ) , N (1) b o T = ˆ u a ( x ) δ ab , n u a ( x ) , N (1) b o T = u a ( x ) δ ab . The field N (1) (2.13) is apparently compatible with the classical algebra (2.15). (cid:3) The V (2) -operator. We now derive the V -operator quadratic in λ : V (2) ( x, a, λ ) = λ D + λ Y (1 , ( x, a ) + Y (0 , ( x, a )= λ + λ N (2) a + A a λ ˆ u a + B a λu a + C a D a ! . (2.16)Requiring that det V (2) = λ + 1 we conclude that the algebraic quantities N , A , D are expressed in terms of the fundamental fields u, ˆ u, B , C as N (2) a = u a B a + ˆ u a C a u a u a , A a = 1 + B a C a u a u a , D a = 1 + ˆ u a u a . (2.17)Requiring also that V (2) satisfies the time-like Poisson structure (2.7) we thenproduce the Poisson relations n Y (1 , ⊗ , Y (1 , o T = P (cid:16) D ⊗ Y (1 , − Y (1 , ⊗ D (cid:17) , (2.18) n Y (1 , ⊗ , Y (0 , o T = P (cid:16) D ⊗ Y (0 , − Y (0 , ⊗ D (cid:17) , (2.19) n Y (0 , ⊗ , Y (0 , o T = P (cid:16) Y (1 , ⊗ Y (0 , − Y (0 , ⊗ Y (1 , (cid:17) . (2.20)and hence the time-like algebra for the fields (we only write below the non zerocommutators for the fundamental fields, see also [7], see also Appendix B for thecorresponding quantum algebra relations): n u a ( x ) , B b ( x ) o T = − n ˆ u a ( x ) , C b ( x ) o T = (cid:16) u a ( x ) u a ( x ) (cid:17) δ ab , n B a ( x ) , C b ( x ) o T = − (cid:16) u a ( x ) B a ( x ) + ˆ u a ( x ) C a ( x ) (cid:17) δ ab . (2.21)The quantities defined in (2.17) are compatible with the algebra (2.21). (cid:3) It is worth noting that in the space-like formulation the U -matrix (2.1) is thestarting point and the conserved quantities as well as the hierarchy of V -operatorsemerge from it [45, 14]. In the time-like approach on the other hand the startingpoint is some V -operator, and from this the time-like conserved quantities as wellas the U -hierarchy are derived [7]. In the next subsection, we focus only on time-like Poisson structures thus we drop the subscript T whenever this applies. The discrete time-like Lax pair hierarchy.
Here we exclusively discuss the time-like case, and extract the associated charges in involution as well as the hierarchy
N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 9 of U -operators. The generating function of the hierarchy of the local conservedquantities associated to the system is given by: G ( λ ) = ln (cid:0) tr (cid:0) T ( λ ) (cid:1)(cid:1) , where T ( λ ) = T T ( M, , λ ) the time-like monodromy (2.4) ( x dependence is im-plied).We may also derive the generating function that provides the hierarchy of U -operators associated to each one of the time-like Hamiltonians. Indeed, taking intoconsideration the zero curvature condition as well the time-like Poisson structuresatisfied by V one can show that the generating function of the U -components ofthe Lax pairs is given by the time-like analogue of the STS formula (see also [7]for a more detailed derivation) U ( a, λ, µ ) = t − ( λ ) tr (cid:16) T ( M, a, λ ) r ( λ − µ ) T ( a − , , λ ) (cid:17) , (2.22)where recall the time-like monodromy matrix defined in (2.4) for b > a . We alsointroduce the index notation: A = A ⊗ I and A = I ⊗ A for any d × d matrix A , I is the d × d identity matrix, and r acts on V ⊗ V ( V is the d dimensional vectorspace). In the case where the r -matrix is the Yangian (2.9) the latter expression(2.22) reduces to U ( a, λ, µ ) = t − ( λ ) λ − µ T ( a − , , λ ) T ( M, a, λ ) . (2.23)We restrict our attention now on the hierarchy associated to V (2) (2.16). Indeed,expanding the monodromy matrix (2.4) constructed by the V -operator (2.16), inpowers of λ , we obtain the associated charges in involution. We report below thefirst couple of conserved quantities:(2.24) H (1) = M X a =1 u a B a + ˆ u a C a u a ˆ u a ,H (2) = M X a =1 (cid:16) ˆ u a u a − + 1 + B a C a u a ˆ u a − (cid:0) u a B a + ˆ u a C a u a ˆ u a (cid:1) (cid:17) . . . In fact, H (2) is the Hamiltonian of the semi discrete time NLS system.In addition to the derivation of the time-like charges in involution above wecan also compute the corresponding U -operators of the time-like hierarchy via theexpansion in powers of λ of (2.23). The pair (cid:16) U ( k ) , V (2) (cid:17) gives rise to the sameequations of motion as Hamilton’s equations with the Hamiltonian H ( k ) associatedto the x k flow. “Conserved” with respect with respect to spatial variations for the monodromy matrix builtusing V . We provide below the first few members of the series expansion of U corre-sponding to the charges (2.24)(2.25) U (1) ( λ ) = ! , U (2) ( x, a, λ ) = λ ˆ u a ( x ) u a − ( x ) 0 ! , . . . We focus on the second member of the hierarchy, which is going to give an in-tegrable time discertization of the NLS model. Note that the U (2) -operator ofthe system under study, satisfies the algebra (2.5), thus the space-like Poissonstructure for the fields is given by:(2.26) n u a − ( x ) , ˆ u a ( y ) o S = δ ( x − y ) . Having identified both the charges in involution as well as the various U -operators, we focus on the second member of the hierarchy. In particular, let usobtain via the Hamiltonian H (2) (and the time-like Poisson relations) and/or theLax pair (cid:16) U (2) , V (2) (cid:17) the corresponding equations of motion. Equations (A.1),(A.3), via the definition of N (2) (2.17), lead to(2.27) B a = ∂ x ˆ u a − ˆ u a ∂ x u a − u a ˆ u a , C a = u a ∂ x ˆ u a − ∂ x u a − u a ˆ u a . Also, from the zero curvature condition we obtain the following constraints(2.28) ∂ x B a = ˆ u a +1 D a − A a ˆ u a , ∂ x C a = u a A a − D a u a − . Given that A , D , N (2) (2.17), and B , C (2.27) are expressed in terms of the fun-damental fields u a , ˆ u a and their x -derivatives, equations (2.28) are the equationsof motion for the fundamental fields fields u a , ˆ u a .As shown above the Lax pair (cid:16) U (2) , V (2) (cid:17) produces the discrete time analogueof the NLS equation, whereas the Lax pair (cid:16) U (2) , V (3) (cid:17) is expected to yield thediscrete time complex mKdV equations in analogy to the findings of [13, 15] (seealso [10] on the mKdV Lax pair). The algebraic derivation of V (3) is not includedin our computations here as it is quite involved and will be presented elsewhere.2.2. The fully discrete setting.
We come now to our primary objective, whichis the derivation and study of fully discrete integrable systems. In this frame thenotion of space-time duality will be more natural given that space and time are atequal footing, in exact analogy to the continuous space-time picture [7, 13]. We aregoing to describe the problem algebraically, whereas in the subsequent subsectionwe apply the fully discrete dressing process as a further consistency check on thederivation of the associated Lax pairs.Consider the fully discrete Lax pair (cid:16)
L, V (cid:17) that depends on the fields and somespectral parameter. Let also n denote a discrete space index, and a a discrete timeone, then the fully discrete auxiliary linear problem takes the form:Ψ( n + 1 , a, λ ) = L ( n, a, λ )Ψ( n, a, λ )(2.29) Ψ( n, a + 1 , λ ) = V ( n, a, λ )Ψ( n, a, λ ) . (2.30) N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 11
Consistency of the two equation of the auxiliary linear problem lead to the fullydiscrete equations of motion (the fully discrete analogue of the zero curvaturecondition):(2.31) V ( n + 1 , a, λ ) L ( n, a, λ ) = L ( n, a + 1 , λ ) V ( n, a, λ ) . In this context both discrete space and time are at equal footing as in the contin-uous case [7].We consider the space-like monodromy matrix defined at some discrete time a as(2.32) T S ( n, m, a, λ ) = L ( n, a, λ ) · · · L ( m + 1 , a, λ ) L ( m, a, λ ) , n > m, and the space-like transfer matrix is defined as t S ( a, λ ) = trT S ( N, , a, λ ). Recallthat in general L, V are d × d matrices and the trace above is defined with respect tothe d dimensional (auxiliary) space. By virtue of the fully discrete zero curvaturecondition we show that t S ( a, λ ) = t S ( a +1 , λ ), i.e. the transfer matrix is a constantwith respect to the discrete time. Indeed, consider t S ( a + 1 , λ ), also from (2.31)we have that L ( n, a + 1) = V ( n + 1 , a ) L ( n, a ) V − ( n, a ), then(2.33) t S ( a + 1 , λ ) = tr (cid:16) V ( N + 1 , a, λ ) L ( N, a, λ ) · · · L (1 , a, λ ) V − (1 , a, λ ) (cid:17) . Assuming periodic space like boundary conditions, i.e. V ( N + 1 , a, λ ) = V (1 , a, λ ),and recalling (2.33) and the definition of the space-like transfer matrix we con-clude that t S ( a + 1 , λ ) = t S ( a, λ ). The λ -series expansion of the transfer pro-vides naturally the conserved quantities of the system with the respect to thediscrete time denoted by the index a . Note that the continuous limit of (2.32),( L ( n, λ ) → I + δU ( n, λ )), provides the solution of the space part of the discretetime auxiliary linear problem of the previous section given by (2.3).Similarly, let us consider the time-like transfer matrix defined for any spaceindex n as t T ( n, λ ) = trT T ( n, M, , λ ), where the time monodromy matrix T T ingiven by (2.4). Through (2.31) V ( n + 1 , a, λ ) = L ( n, a + 1 , λ ) V ( n, a, λ ) L − ( n, a, λ )and assuming time like boundary conditions L ( n, M + 1 , λ ) = L ( n, , λ ) we con-clude that the transfer matrix is invariant with respect to the discrete space in-dexed by n , i.e. t T ( n + 1 , λ ) = t T ( n, λ ), i.e suitable expansion in powers of λ produces the hierarchy of associated invariants for the system with respect to thediscrete space characterized by the index n . In the continuous space limit thelatter reduces to d t T ( x,λ ) dx = 0 (see also comments at the beginning of section 2).We graphically represent the Lax pair (cid:16) L, V (cid:17) in our fully discrete set up as:... n { a } L ( n, a ) . . . { n } aV ( n, a ) The set of time indices { a } ≡ { a, a − , . . . , a − k + 1 } denotes discrete timedependence in L , which is usually implicit. The integer k depends on the form ofthe L operator. Similarly, the set of space like indices { n } ≡ { n, n − , . . . , n − l + 1 } denotes discrete space dependence in V , which is usually implicit. In thecases considered here k = l = 2. The dashed line represents the d dimensional“auxiliary” space of the Lax pairs (in the examples that follow d = 2.)Next we graphically represent the space-like monodromy:... · · · · · · N N − { a } The space-like monodromy T S ( a ), corresponds to a one dimensional N -cite space-like lattice at a given discrete time a . The space transfer matrix is defined aftertaking the trace over the auxiliary space resulting in periodic space boundaryconditions, that is the space transfer matrix is graphically depicted by a cylinder,i.e. consider the first and N th site in the figure above to coincide. The time-like monodromy is the vertical analogue of the above figure and represents anone dimensional time M -site time-like lattice, for a given space index n . Webasically consider a 90 degrees rotation of the figure above and replace the spatialindices { , N } with temporal ones { , M } , and the fixed index a with n (also thecolors are interchanged accordingly: green ↔ purple, i.e. horizontal lines greenand vertical lines purple).The auxiliary space does not appear in the two dimensional lattice that isgraphically depicted below for a given lax pair (cid:16) L, V (cid:17) :......1 aM − M · · · · · · nN − N The figure above should be carefully interpreted, especially when referring to mon-odromies and transfer matrices. More specifically, for a fixed time index a we focus N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 13 on the space-like monodromy/transfer matrix (2.32) (horizontally), and the respec-tive space-like discrete system whereas, in the time-like situation the space index n is fixed and we focus on the time-like monodromy/transfer matrix (2.4) (ver-tically) (see also relevant comments on the “conservation” laws discussed earlierin this section). The space and time monodromies can be seen as horizontal andvertical “stripes” respectively in the two dimensional lattice above. The latterinterpretation applies also in the continuum scenario on the x − t plane [7, 8] whenconsidering the corresponding continuous monodromies. The clear distinction be-tween space and time indices becomes more transparent below when presentingthe algebraic formulation of the problem. When considering a given Lax pair andthe fully discrete zero curvature condition in order to extract the space time dif-ference equations the two dimensional lattice is interpreted in the usual sense asthe discretization of the x − t plane.Let us now focus on the algebraic formulation of fully discrete integrable sys-tems. The key object in describing the space-like discrete picture is the L operator,which satisfies the quadratic Poisson structure(2.34) n L ( n, a, λ ) ⊗ , L ( m, a, µ ) o S = h r ( λ − µ ) , L ( n, a, λ ) ⊗ L ( m, a, µ ) i δ nm λ, µ are spectral parameters, and the r -matrix satisfies the classical Yang-Baxterequation.Similarly to the semi-discrete time case described in the preceding subsection werequire that the time component V of the Lax pair satisfies the time-like Poissonstructure:(2.35) n V ( n, a, λ ) ⊗ , V ( n, b, µ ) o T = h r ( λ − µ ) , V ( n, a, λ ) ⊗ V ( n, b, µ ) i δ ab . The classical r -matrix is the same as the one of the space-like algebra (2.34).The involution of the charges produced by the space and time-like transfer ma-trices is guaranteed by the existence of the Poisson structures (2.34) and (2.35).Indeed, the monodromies (2.32) and (2.4) satisfy (2.34) and (2.35) respectively,and thus the corresponding transfer matrices are in involution for different spectralparameters: (cid:8) t S ( λ ) , t S ( λ ′ ) (cid:9) S = (cid:8) t T ( λ ) , t T ( λ ′ ) (cid:9) T = 0. This fact stipulates the ex-istence of extra continuous dynamical parameters (underlying continuous “time”)in accordance to Hamilton’s equations. The associated hierarchies of the continu-ous time components of Lax pairs can then be obtained via the STS formula forboth discrete space-like or time-like systems constructed as described above.We focus now on two distinct versions of the fully discrete NLS model associatedto rational and trigonometric classical r -matrices respectively.2.2.1. The fully discrete NLS model.
We first examine the fully discrete version ofthe NLS model associated to the Yangian r -matrix. The L operator of the discreteNLS-type hierarchy (2.36) is given as [36, 37](2.36) L ( n, a, λ ) = λ + N na X na Y na − ! , where N na = θ + X na Y na − , θ is an arbitrary constant. The Lax operator satisfiesSklyanin’s bracket (2.34). Notice that the discrete time dependence in L is fullyjustified by the time-like derivation of U (2) via the STS formula in the previoussubsection ( U (2) is the continuum space limit of L ). The Poisson structure (2.34)leads to the following Poisson relations among the fields n X na , Y ma − o S = − δ nm , n X na , X ma o S = n Y na , Y ma o S = 0 . (2.37)In analogy to the semi-discrete time case described in the previous section weconsider the following generic form for the V -hierarchy(2.38) V ( k ) ( n, a, λ ) = λ k D + k − X l =0 λ l Y ( l,k ) ( n, a ) , where D = diag(1 , V ( k ) ( n, a , a , ..., a k .. ), however for simplicity we suppress the time-like indices a l , l = k and we instead write V ( k ) ( n, a ). We also require that det V ( k ) = λ k + 1,and all V ( k ) ( n, a ) satisfy the quadratic algebra (2.35), with the same r -matrix asin (2.34). Then all Y ( l,k ) can be expressed in terms of some “fundamental” fields(see also [7, 13]), that satisfy the basic Poisson relations. Let us focus on the firsttwo members of the hierarchy, and identify V (1) , V (2) and the corresponding spacetime difference equations. The V (1) -operator. The first non-trivial V (1) of the general form is linear andis associated to a discrete time version of the transport equation, V (1) ( n, a, λ ) = λ + N (1) na X na Y n − a ! , (2.39)where from the condition det V (1) = λ + 1 we obtain(2.40) N (1) na = 1 + X na Y n − a . Due to the fact that V (1) satisfies the quadratic algebra (2.35) and hence (2.14), wederive the Poisson relations for the fundamental fields (i.e. the time like analogueof (2.37)):(2.41) n Y na , X n − b o T = δ ab . Having identified the Lax pair we may now extract the equations of motionassociated to (cid:16)
L, V (1) (cid:17) . These are linear difference equations in analogy to thecontinuous case, i.e. they are the discrete analogues of the linear transport equa-tion:(2.42) F n +1 a = F na +1 , F ∈ (cid:8) X , Y (cid:9) . (cid:3) N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 15
The V (2) -operator. The V (2) operator, quadratic in λ , reads as V (2) ( n, a, λ ) = λ + λ N (2) na + A na λ X na + B na λ Y n − a + C na D na ! , (2.43)where as in the semi discrete case requiring det V (2) = λ + 1 we obtain the fields N , A , D expressed in terms of the fundamental fields X , Y B , C : N (2) na = Y n − a B na + X na C na na Y n − a , A na = 1 + B na C na na Y n − a , D na = 1 + X na Y n − a . (2.44)Requiring also that V satisfies the time-like Poisson structure (2.35) and hence(2.18)-(2.20) we produce the time-like algebra for the fields, which reads as (weonly write below the fundamental commutators, see also [7], and Appendix B forthe time-like quantum algebra): n Y n − a , B nb o T = − n X na , C nb o T = (cid:16) na Y n − a (cid:17) δ ab , n B na , C nb o T = − (cid:16) Y n − a B na + X na C na (cid:17) δ ab . (2.45)We now consider the fully discrete version of the NLS like model with a Laxpair (cid:16) L, V (2) (cid:17) given in (2.36) and (2.43), and we employ the fully discrete zerocurvature condition (see all the associated relations in Appendix A). Then as inthe semi-discrete time case studied in the previous subsection we can identify thefields B , C in terms of X , Y using equations (A.10), (A.12) and the definition for N (2) (2.44): B na = X n +1 a − N na X na + X na (cid:0) Y n − a − N n − a +1 Y n − a (cid:1) − X na Y n − a (2.46) C na = Y n − a (cid:0) X n +1 a − N na X na (cid:1) + Y n − a − N n − a +1 Y n − a − X na Y n − a (2.47)Substituting the above expressions in (A.11) and (A.13) of the Appendix A weobtain the quite involved space-time non-linear difference equations for the fieldsX , Y B n +1 a = N na +1 B na + X na +1 D na − A n +1 a X na (2.48) C na = C n +1 a N na +1 + D n +1 a Y n − a − Y na A na . (2.49)These equations are the fully discrete analogues of NLS type equations. Comparing(2.46)-(2.49) with (2.27), (2.28) we conclude that (2.46)-(2.49) are the discretespace analogues of (2.27), (2.28). Indeed, in the continuous space limit N → F n +1 a − F na → ∂ x F a ( x ), so equations (2.46)-(2.49) reduce to (2.27), (2.28). (cid:3) The fully discrete Ablowitz-Ladik model.
We now examine an alternativeversion of the fully discrete NLS model associated to a trigonometric r -matrix.Specifically, we examine the fully discrete version of the AL model. Indeed, the r -matrix for the AL model is a trigonometric one, a variation of the classicalsine-Gordon r -matrix [35, 19]:(2.50) r ( λ ) = 12sinh ( λ ) (cid:16) cosh ( λ ) X j =1 e jj ⊗ e jj + X i = j =1 e ij ⊗ e ji +sinh ( λ ) X i = j =1 ( − j − i e ii ⊗ e jj (cid:17) . We also recall that the classical Lax operator for the AL model is given by (seee.g. [2, 30, 32]) L ( n, z ) = z ˆ b n b n z − ! , (2.51)where z = e λ is the multiplicative spectral parameter. The L -operator satisfies(2.34), with r being the trigonometric matrix (2.50). This leads to the classicalalgebra for the fields (see also [30]): n b n , ˆ b m o S = δ nm (cid:0) − b n ˆ b m (cid:1) , n b n , b m o S = n ˆ b n , ˆ b m o S = 0 . (2.52)The AL model is thus associated to a deformed harmonic oscillator classical algebra( q -bosons at the quantum level [35]). Note that n, m denote space like indices.Dependence of a continuum time-like parameter t or a discrete time dependencecharacterized by some time index a is implied, but is not explicitly stated for now.From the space-like transfer matrix we obtain the following space-like conservedquantities, after expanding suitably in powers of z ± (2.53) H + S = N X n =1 ˆ β n +1 β n , H − S = N X n =1 β n +1 ˆ β n . Let us also introduce realizations of the time-like algebra (2.35) with the r -matrix given in (2.50). These realizations will play the role of the discrete timecomponents of the fully discrete AL Lax pairs: V − ( a, z ) = z ˆ B a B a − z A a + z − ! , V + ( a, z ) = z − z − A a ˆ B a B a z − ! . (2.54)Note that here a, b denote time indices, whereas space dependence is implied,but is not explicitly stated for now. Requiring that both V ± satisfy the time-likealgebra we obtain the associated time-like Poisson relations for the fields: n B a , ˆ B b o T = δ ab (cid:0) − B a ˆ B a (cid:1) , n B a , B b o T = n ˆ B a , ˆ B b o T = 0 , n ˆ B a , A b o T = A a ˆ B a δ ab , n B a , A b o T = − A a B a δ ab , (2.55)where A a = − B a B a and is compatible with the Poisson structure above . A can be defined up to an overall multiplicative constant. N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 17
From the time-like transfer matrix (2.4) we obtain the following time-like con-served quantities corresponding to V ± :(2.56) H + T = M X a =1 (cid:16) ˆ B a +1 B a − ˆ B a B a (cid:17) , H − T = M X a =1 (cid:16) ˆ B a B a +1 − ˆ B a B a (cid:17) . We shall use suitable Lax pairs to produce space and time diescretizations ofthe AL model by considering three distinct cases: A. We first consider the Lax pair (cid:16)
L, V − (cid:17) :(2.57) L ( n, a, z ) = z ˆ β na − β na z − ! , V − ( n, a, z ) = z ˆ β n − a β na − z A − na + z − ! where A − na = − β na ˆ β n − a . From the fully discrete compatibility condition(2.31) the equations of motion arise (from the anti-diagonal entries):ˆ β na − = ˆ β n − a + ˆ β na − ˆ β na β na ˆ β n − a β na +1 = β n +1 a + β na − β n +1 a ˆ β na β na . (2.58) B. We next consider the Lax pair (cid:16)
L, V + (cid:17) :(2.59) L ( n, a, z ) = z ˆ β na − β na z − ! , V + ( n, a, z ) = z − z − A + na ˆ β na − β n − a +1 z − ! where A + na = −
1+ ˆ β na − β n − a +13 . From the fully discrete compatibility condition(2.31) we obtain the partial difference equations:ˆ β na = ˆ β n +1 a − + ˆ β na − − ˆ β n +1 a − β na +1 ˆ β na − β na = β n − a +1 + β na +1 − β na +1 ˆ β na − β n − a +1 . (2.60) Remark 2.1.
Interestingly, by adding equations (2.58) and (2.60) we obtain aspace-time discrete analogue of an mKdV like equation, provided that β n +1 a → ˆ β n +1 a − : ˆ β na − ˆ β na − = 12 (cid:0) − ˆ β na ˆ β na − (cid:1)(cid:0) ˆ β n +1 a − − ˆ β n − a (cid:1) (2.61) C. Finally we consider the Lax Pair (cid:16) L + , V − (cid:17) :(2.62) L + ( n, a, z ) = z − z − ˆ A na ˆ β na β na − z − ! , V − ( n, a, z ) = z ˆ β n − a β na − z A − na + z − , ! where ˆ A na = − β na β na − and A − na = − β na ˆ β n − a , also L is structurallysimilar to V + , but the time and space indices are interchanged, i.e. L satisfies the To emphasize the notion of “ultra-locality”, and also for our notation to be compatible withthe rest of the examples, we may introduce a new fundamental field γ na − := β na +1 , so thefields that appear in L in this case are ˆ β na − , γ na − , and the fields in V + are ˆ β na − , γ n − na − . space-like algebra and the fields then satisfy the ultra-local Poisson relations forfixed time a : n β na − , ˆ β ma o S = δ nm (cid:16) − β na − ˆ β na (cid:17)n β na − , β ma − o S = n ˆ β na , ˆ β ma o S = 0 . (2.63)The space time difference equations arising from the fully discrete zero curvaturecondition read as: ˆ β na +1 + ˆ β n − a − ˆ β na = ˆ β n − a β na ˆ β na +1 β n +1 a + β na − − β na = β n +1 a ˆ β na β na − . (2.64)Consistency checks have been also performed by comparing the diagonal termsin the compatibility condition (2.31) for the three distinct Lax pairs presentedabove. 3. Darboux-dressing formulation & solutions The most efficient way to derive the continuous time components of Lax pairs,i.e. the V -operators is the use of the STS formula. This formula can be derivedprovided that an associated Poisson structure is available, then use of the zerocurvature condition and Hamilton’s equations leads to STS formula [45]. However,in the discrete time set up the analogue of the STS formula is not available,thus alternative ways to construct the V –hierarchy are required. In the precedingsection we were able to construct the V -operators by requiring that they satisfythe time-like quadratic Poisson structure. In what follows, mostly as a consistencycheck on the findings of the previous section, we implement the discrete timeDarboux-dressing formulation to identify the V -hierarchy, and confirm the findingsof the algebraic approach. This process also offers a systematic means to derivesolutions of the associated integrable non-linear difference equations as discussedin subsection 3.3.3.1. The semi-discrete time NLS hierarchy.
We first examine the semi-discretetime scenario and consider the Lax pair (cid:16)
U, V (cid:17) , where U is given by U (2) in (2.25)and the hierarchy of V -operators will be derived through the dressing process, i.e.we are considering now the space-like description as opposed to the time-like con-sideration of subsection 2.1.1. In particular, we are going to explicitly derive thefirst two members of the discrete time hierarchy, V (1) and V (2) confirming thealgebraic findings of subsection 2.1.Consider the associated auxiliary linear problem (2.1), and let M be the Dar-boux transform such that:(3.1) Ψ( x, a, λ ) = M ( x, a, λ ) ˆΨ( x, a, λ ) , where both Ψ , ˆΨ are solutions of associated linear problems with Lax pairs (cid:16) U, V (cid:17) and (cid:16) ˆ U , ˆ V (cid:17) respectively. Let us focus on the x -part of the linear auxiliary problem N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 19 to derive the x -part of the Darboux-B¨acklund relations:(3.2) ∂ x M ( x, a, λ ) = U ( x, a, λ ) M ( x, a, λ ) − M ( x, a, λ ) ˆ U ( x, a, λ ) . We consider here the fundamental Darboux transform for the NLS hierarchy (seealso recent relevant results for the NLS model and generalizations [15], [13])(3.3) M ( x, a, λ ) = λ + A a ( x ) B a ( x ) C a ( x ) λ + D a ( x ) ! . Also, recall that U is given by U (2) in (2.25), and ˆ U = λ
00 0 ! . Using the fun-damental Darboux matrix above and solving the x -part of the Darboux-B¨acklundtransformation (BT) relations (3.2) we obtain the following sets of constraints (seealso e.g. [13]):(3.4) B a = − ˆ u a , C a = u a − , D a = Θ − A a , where Θ is an arbitrary constant, and the extra constraints ∂ x ˆ u a = ˆ u a A a , ∂ x u a − = u a − A a , ∂ x A a = ˆ u a u a . (3.5)We shall also perform the discrete time dressing to obtain the V -operators andconfirm the expressions for V derived algebraically in the previous section. Let usconsider the general form of the V -operator associated to a certain discrete timecharacterized by an index a ,(3.6) V ( m ) ( λ ) = λ m D + m − X k =0 λ k Y ( k,m ) , recall D = diag (1 , M = λI + K , where I is the 2 × K reads from (3.3). Also, ˆ V ( m ) ( λ ) = λ m D + I . From thediscrete time part of the Darboux relations(3.7) M ( x, a + 1 , λ ) ˆ V ( x, a, λ ) = V ( x, a, λ ) M ( x, a, λ ) , the following recursion relations emerge for the generic object V ( m ) (3.6) ( x de-pendence on the expression below is always implied, but omitted for brevity): Y ( m − ,m ) ( a ) = K ( a + 1) D − D K ( a ) Y ( k − ,m ) ( a ) = −Y ( k,m ) ( a ) K ( a ) , k ∈ (cid:8) , . . . , m − (cid:9) Y (0 ,m ) ( a ) − −Y (1 ,m ) ( a ) K ( a ) K ( a + 1) = Y (0 ,m ) ( a ) K ( a ) . (3.8)We now focus on the explicit derivation of the first two members of the discretetime hierarchy, V (1) and V (2) .Let us first identify the V (1) operator and the corresponding non-linear ODEs.The constraints emerging from (3.8) associated to V (1) are summarized below: Y (0 , ( a ) = − B a , Y (0 , ( a ) = C a +1 , Y (0 , ( a ) − A a +1 − A a . (3.9) which lead to Y (0 , ( a ) = ˆ u a , Y (0 , ( a ) = u a . Also, A a +1 − A a = Y (0 , ( a ) A a + ˆ u a u a − , A a +1 − A a = u a ˆ u a u a − u a − = u a A a , ˆ u a +1 − ˆ u a = ˆ u a A a + ˆ u a u a . (3.10)Combining the constraints (3.9) and (3.10), and recalling the x -part of the Darbouxtransform (3.5) we conclude: Y (0 , ( a ) = 1 + u a ˆ u a , and the nonlinear ODEsˆ u a +1 − ˆ u a = ˆ u a u a + ∂ x ˆ u a u a − u a − = − u a ˆ u a + ∂ x u a . (3.11)Analogous expressions were obtained in [15], where the semi-discrete space casewas studied. In this particular case we observe a simple exchange of the roleof space and time. The latter equations can be seen as non-linear versions of thetransport equation. We have thus reproduced expression (2.12) for V (1) confirmingthe algebraic approach of the previous section.We move on to derive V (2) via the dressing formulation. Let us introduce thefollowing notation compatible with the expression (2.16) derived in the previoussubsection: Y (0 , ( a ) = A a , Y (0 , ( a ) = D a , Y (0 , ( a ) = B a , Y (0 , ( a ) = C a . and Y (1 , ( a ) = 0, Y (1 , ( a ) = N (2) a . Then from equations (3.8) for V (2) we obtainfor the off diagonal entries(3.12) Y (1 , ( a ) = ˆ u a , Y (1 , ( a ) = u a as well as the following set of constraints B a = ˆ u a A a + N (2) a ˆ u a , (3.13) B a A a = ˆ u a +1 − A a ˆ u a (3.14) C a = − u a A a , (3.15) C a A a = u a − D a u a − . (3.16)The diagonal entries of (3.8) lead to: A a +1 − A a = C a ˆ u a + u a ˆ u a A a , (3.17) A a +1 − A a = N (2) a (3.18) A a +1 = A a A a + B a u a − , (3.19) D a = 1 + u a ˆ u a , A a = 1 − N (2) a A a − ˆ u a u a − . (3.20)Combining equations (3.17), (3.18) and (3.13) we conclude the N (2) a as expectedis given by expressions (2.17). Similarly, D a given in (3.20) agrees with expres-sion (2.17) from the algebraic viewpoint. Also, for A a we conclude via (3.20) and(3.15), (3.16) and the definition of D a (3.20) to the expression given by (2.17). Thedressing process yields exactly the same expression for V (2) as the algebraic formu-lation of subsection 2.1 and this is indeed a strong consistency check. Moreover,the equations of motion derived in subsection 2.2 via the zero curvature condition N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 21 (see also (A.5), (A.6)), are recovered via equations (3.13)–(3.20) and using the x -part of the Darboux-BT relations (3.5). Indeed, expressions (A.5) are immediatelyrecovered by combining (3.13), (3.15) and recalling (3.5).3.2. The fully discrete time NLS hierarchy.
We come now to the applica-tion of the fully discrete Darboux-dressing process in order to construct the fullydiscrete NLS hierarchy. Let M be the local Darboux transformation such that(3.21) Ψ( n, a, λ ) = M ( n, a, λ ) ˆΨ( n, a, λ ) , where both Ψ , ˆΨ satisfy the auxiliary linear problem (2.30) with (cid:16) L, V (cid:17) and (cid:16) ˆ L, ˆ V (cid:17) respectively, then it follows:(3.22) M ( n + 1 , a, λ ) ˆ L ( n, a, λ ) = L ( n, a, λ ) M ( n, a, λ )where L is given by (2.36) and ˆ L is in general of the same form, but with fieldsˆ X, ˆ Y , and here we consider the simple case where ˆX = ˆY = 0. Similarly for thediscrete time components of the Lax pair the transformation (3.21) leads to(3.23) V ( n, a, λ ) M ( n, a, λ ) = M ( n, a + 1 , λ ) ˆ V ( n, a, λ ) . We consider for now the fundamental Darboux matrix given in (3.3), but G a ( x ) →G na , where G ∈ {
A, B, C, D } .From the discrete space part of the Darboux-BT relations (3.22) we obtain, B na = − X na , C na = Y n − a − , (3.24)We also derive, as expected that D na = 1 − A na and N na = 1 + X na Y na − . Thediscrete space dressing has been performed in [15]), and detailed computations canbe found there.Let us first derive V (1) , being of the form (2.39). From (3.23), we obtainrelations (3.9)-(3.10) provided that u a ( x ) → Y n − a , ˆ u a ( x ) → X na and N a ( x ) → N na , conforming also that N na = 1 + X na Y na − . Similarly, for the derivationof V (2) (2.43) we obtain via (3.23) equations (3.13)-(3.16) and (3.17)-(3.20), but u a ( x ) → Y n − a , ˆ u a ( x ) → X na and F a ( x ) → F na , where F ∈ (cid:8) N (2) , B , C , A , D (cid:9) ,confirming also equations (2.44) coming from the algebraic approach.3.3. Solutions.
Having derived the V -operators of the discrete time NLS systemsvia the dressing methodology we come now to the derivation of solutions of theassociated integrable non-linear difference equations.Solitonic solutions can be obtained from the fundamental Darboux matrix asin the continuous and the semi-discrete space case (see e.g [15] and referencestherein). In fact, by solving the constraints from the space part of the Darbouxtransform we obtain such solutions. We do not provide the detailed computationshere, however for a more detailed analysis on the derivation of these expressionwe refer the interested reader to [15]. In any case, such expressions will be alsoidentified in the subsequent section in a more straightforward manner using adifferent Darboux matrix, which provides not only solitonic, but generic solutions for both the semi-discrete time scenario and the fully discrete case. We reportbelow the expressions of the stationary solutions found in the semi-discrete spacecase [15], which are also valid in the fully discrete case:(1) Solitons of type IX n = ξ n − ( ξ − ξ n − ( ξ − ) − d , Y n = ξ − n ( ξ − − a )y ξ − n ( ξ − ) − a , (3.25) where x , y , a , d are constants. Periodic boundary conditions are validfor all the associated fields and this can be easily checked by inspection pro-vided that ξ N = 1. Note that in the stationary solutions above the discretetime dependence is naturally introduced: (cid:0) X n , Y n (cid:1) → (cid:0) X na , Y na − (cid:1) and ξ n → ξ n ζ a , where ζ − ξ − , (see also next section, where a detaileddiscussion on the related dispersion relations is presented). The soliton Isolutions ˆ u a ( x ) , u a − ( x ) for the discrete time and continuum space casestudied in the proceeding section have the same form as in (3.25), but ξ n → e − kx and the dispersion relation becomes ζ = k + 1 (see also nextsection).(2) Solitons of type IIX n = ( ¯ ξ − ( ¯ ξ − κ ˆd ) η − n +1 − ¯ κ ˆd ǫ − n +1 , Y n = η ( ˜ ξ − − ˜ κ ˆa )y ( ¯ ξ − κ ˆa ) η n − ¯ κ ˆa ǫ n , where ¯ ξ = ǫη − , ¯ κ = η − , η = 1 + c, ǫ = 1 − c , and ˜ ξ = ¯ ξ − , ˜ κ = − ¯ κ ¯ ξ − (see also [15]), and x , y , ˆa , ˆd . As in the case 1 above thetime dependence is easily implemented: (cid:0) X n , Y n (cid:1) → (cid:0) X na , Y na − (cid:1) and η n → η n ζ a , ǫ n → ǫ n ˆ ζ a , where ˆ ζ − ǫ − , ζ − η − , (see alsonext subsection). Similarly to case 1 the soliton II solutions ˆ u a ( x ) , u a − ( x )for the discrete time and continuum space case studied in the proceedingsection have the same form as in (3.26), but η n → e − k x , ǫ n → e − k x , and the dispersion relation becomes ζ = k + 1 , ˆ ζ = k + 1, (see also nextsubsection on the issue of dispersion relations).Note that 2-soliton solutions can be obtained by repeatedly applying the funda-mental Darboux and using Bianchi’s permutability theorem. Detailed computa-tions and explicit expressions of such solutions can be found in [15] for the semi-discrete space NLS model. Solutions from the Toda type Darboux.
We consider in what follows adifferent type of Darboux transformation, the Toda type Darboux (see also e.g.[15]). We shall employ this transformation to identify generic solutions for boththe semi-discrete time and the fully discrete NLS systems generalizing the findingsof [15].
Let us first discuss the semi-discrete time NLS caseand derive solution via the Toda type Darboux transformation repeating some of
N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 23 the fundamental computations of [15]. Recall the U -operator of the continuousspace and discrete time Lax pair (cid:16) U, V (cid:17) , where U is given by U (2) in (2.25). Aswas shown in [15] in order to derive general solutions of the non-linear ODEs/PDEsin the simplest possible way we use the Toda type Darboux transform:(3.26) M ( x, a, λ ) = λ + A a B a C a ! . The x -part of the Darboux transform gives:(3.27) ∂ x M ( x, a, λ ) = U ( x, a, λ ) M ( x, a, λ ) − M ( x, a, λ ) U ( x, a, λ ) , where U is also given by U (2) (2.25), but u a → u (0) a , ˆ u a → ˆ u (0) a . If u (0) a = 0, thenˆ u (0) a satisfies the linear equation (we consider the example of the NLS-like equation(2.27)-(2.28))(3.28) ˆ u (0) a +1 − ˆ u (0) a = ∂ x ˆ u (0) a . The equation above is nothing but the discrete time version of the heat equation.The solution of the linear equation ˆ u (0) a can be expressed as(3.29) ˆ u = S X s =1 c s e − k s x +Λ s a , and/or ˆ u = Z R dλc ( λ ) e iλx +Λ λ a with dispersion relations given as(3.30) Λ s = ln (cid:0) k s + 1 (cid:1) , Λ λ = ln (cid:0) − λ + 1 (cid:1) . From the Darboux relations (3.27) we obtain: B a = ˆ u (0) a , C a = u a − ∂ x ˆ u (0) a = − A a ˆ u (0) a , ∂ x u a − = A a u a − , ∂ x A a = u a − ˆ u a . (3.31)Solving the equations above leads to:(3.32) u a − = g ˆ u (0) a and ˆ u a = − g − ˆ u (0) a ∂ x (ˆ u (0) a ) − ( ∂ x ˆ u (0) a ) ˆ u (0) a . Choosing for instance the simple linear solutions: ˆ u = c e − k x +Λ a + c or ˆ u = c e − k x +Λ a + c e − k x +Λ a we obtain one soliton solutions, respectively: u a − = gc e − k x +Λ a + c type I soliton(3.33) u a − = gc e − k x +Λ a + c e − k x +Λ a type II soliton , (3.34)and similarly for ˆ u a .
2. The fully discrete NLS.
We focus now on the derivation of the fully discreteNLS solutions by means of the Toda type Darboux.(3.35) M ( n + 1 , a, λ ) L ( n, a, λ ) = L ( n, a, λ ) M ( n, a, λ )where the L operator is give by (2.36) and L is given by the same expression as L , but with X na → X (0) na and Y na → Y (0) na . As in the semi discrete case above weare considering the case where Y (0) na = 0, then it follows from the set of equations of motion for the fields (A.10)-(A.17) that X (0) na satisfy the set of linear differenceequations:(3.36) X (0) n +2 a − (0) n +1 a + X (0) na = X (0) na +1 − X (0) na , which is the fully discrete analogue of heat equation. The solutions of the lineardifference equations above are of the generic form(3.37) X (0) na = S X s =1 c s ξ ns ζ as , and/or X (0) na = Z | ξ | =1 dξ c ( ξ ) ξ n ζ aξ , and the associated dispersion relations are easily extracted in this setting and readas(3.38) ζ s − ξ s − . After solving the set of equations provided by (3.35) for (3.26) we conclude(recall we have set Y (0) na = 0, see also [15]),Y na − − Y n − a − = Y na − A na , (3.39) X (0) n +1 a (X (0) na ) − − na Y na − − A n +1 a (3.40) A n +1 a − A na = X na Y na − . (3.41)Via (3.39)–(3.41) we obtainY na − = n Y m =2 (cid:0) − A ma (cid:1) − Y , X na = ( A n +1 a − A na ) n Y m =2 (cid:0) − A ma (cid:1) Y − . Having the solution X (0) n at our disposal we can immediately solve for(3.42) 1 − A na = X (0) n +1 a (X (0) na ) − ⇒ n Y m =2 (cid:0) − A ma (cid:1) = X (0) n +1 a X (0)2 , and hence obtain the explicit expressions for both fields:Y na = X (0)2 X (0) n +1 a Y , X na = − (X (0)2 ) − Y − X (0) n +2 a X (0) na − (X (0) n +1 a ) X (0) na . (3.43)Periodic boundary conditions (X N +1 = X , Y N +1 = Y ) are valid provided thatX (0) N +1 = X (0)1 . Also, X (0)2 and Y (boundary terms) in the expressions above aretreated as constants. Expressions (3.43) are general new solutions of the non-linearpartial differential equations for the fully DNLS hierarchy in terms of solutions ofthe fully discrete heat equation. We describe below two simple solutions of thetype (3.43), which reproduce the two types of discrete solitons.As in the discrete time case examined in the previous subsection we considerthe following simple linear solutions:(1) We first choose X (0) na = c + c ξ n ζ a (3.44) N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 25 where recall ζ − ξ − ,( see also (3.38)). We substitute (3.44) in(3.43), and we obtain the discrete analogues of type I solitons:X na = − (X (0)2 ) − Y − c c ( ξ − c + c ξ − n ζ − a , Y na = X (0)2 Y c + c ξ n +1 ζ a . (3.45)(2) The second simple choice isX (0) n = c η n ζ aη + c ǫ n ζ aǫ (3.46) where ζ η,ǫ are given by (3.38). After substituting the above in (3.43) weobtain the type II discrete solitons:X na = − (X (0)2 ) − Y − c c ( η − ǫ ) c ǫ − n η − aǫ + c η − n ζ − aη , Y n = X (0)2 Y c η n +1 ζ aη + c ǫ n ζ aǫ . (3.47)With this we conclude our explicit computation of the two types of discrete solitonsolutions for the fully discrete NLS model. For generic Fourier transforms of thesolutions of the linear problem we obtain distinct solutions of the fully discreteNLS. 4. The two dimensional quantum lattice
Our goal now is to generalize the fully discrete description in the quantum case byconstructing the two dimensional quantum lattice. Out basis for such a construc-tion will be the fundamental RTT scheme for deriving quantum algebras (see e.g.[18, 33]). We first briefly review this formulation and then we use it for the con-struction of the two dimensional quantum lattice. For a given R -matrix, solutionof the Yang-Baxter equation, associated quantum algebras emerge from the corerelation:(4.1) R ( λ − λ ) (cid:0) L ( λ ) ⊗ I (cid:1) (cid:0) I ⊗ L ( λ ) (cid:1) = (cid:0) I ⊗ L ( λ ) (cid:1) (cid:0) L ( λ ) ⊗ I (cid:1) R ( λ − λ )As in the classical frame the L -operator is the fundamental object and encodesthe key algebraic information.Before we proceed with our construction let us first introduce the “double quan-tum space” notation, which is suitable for the description of the two dimensionalquantum lattice. Let A S and A T denote the spatial and temporal quantum alge-bras respectively, and let us also distinguish two types of L operator: space-likeLax operators L ∈ End( C d ) ⊗ A S ⊗ A ⊗ kT , versus time-like operators, i.e. the quan-tum analogues of V -operators, V ∈ End( C d ) ⊗ A ⊗ lS ⊗ A T , both satisfying (4.1).In the examples we are considering here k = l = 2. In the double quantum indexnotation for L ( n, a ) ( n space index and a times index), A S occupies the n th sitein the space-like tensor product, whereas A ⊗ kT occupy the sites a − k + 1 to a inthe time-like tensor product. An analogous interpretation holds for V ( n, a ).In the space-like description, precisely as in the classical case, we “freeze” thetime index and we construct the one dimensional space monodromy T S ( N, , a, λ ) ∈ End( C d ) ⊗ A ⊗ NS ⊗ A ⊗ kT as in (2.32), whereas in the time-like description we freezespace indices and construct the time-like mondromy T T ( n, M, , λ ) ∈ End( C d ) ⊗ A ⊗ lS ⊗ A ⊗ MT as in (2.4). Naturally T S and T T satisfy (4.1) and consequently tracesover the auxiliary space lead to commuting transfer matrices: t S ∈ A ⊗ NS ⊗ A ⊗ kT and t T ∈ A ⊗ lS ⊗ A ⊗ MT .It is worth noting that in the space transfer matrix the discrete time dependenceis considered to be implicit, and similarly in the time transfer matrix the spacedependence is implicit. The term “quantum spaces”, albeit slightly misleading,refers in general to copies of the spatial and temporal quantum algebras (thatmight be also represented). The double quantum index notation for the quan-tum Lax pair (cid:16) L ( n, a ) , V ( n, a ) (cid:17) is also compatible with the classical notationof section 2. The figures in pages 11-12 as well as relevant comments on spaceand time monodromies, and the two dimensional lattice apply in the quantumcase as well. Specifically, the purple and green lines in these figures correspond asexpected to spatial and temporal quantum spaces respectively. A concrete framethat describes two dimensional quantum integrable lattices is provided by the socalled tetrahedron equation [53, 9]. Our construction is more straightforward inthe sense that the partial quantum algebras we are interested in are independentof each other, and they both emerge from the fundamental relation (4.1) as arguedabove. In fact, both space and time algebras can be embedded in a bigger algebra,which however simply decomposes into two independent parts ruled by (4.1).We next examine the quantum versions of the two main examples consideredin the classical case, i.e. the fully discrete NLS and AL models.
1. The discrete NLS model.
We first examine the quantum DNLS system. Inspiredby the classical expressions we consider the generic algebraic objects of the form(4.2) L ( m ) ( λ ) = m X k =0 λ k Y ( k,m ) , where Y ( m,m ) = diag (1 , • The basic assumptions (1) The L ( m ) -operators satisfy the quantum algebra (4.1), where R ( λ ) = λ + P is the Yangian R -matrix, and recall P = P di,j =1 e ij ⊗ e ji is the permuta-tion operator for the general gl d case.(2) We require the existence of an inverse:(4.3) L ( m ) ( λ ) ¯ L ( m ) ( − λ ) = f ( m ) ( λ ) I where f ( m ) ( λ ) = λ m + P m − k =0 a k λ k , and we define(4.4) ¯ L ( λ ) = (cid:0) U ⊗ id (cid:1) L t a ( − λ − (cid:0) U ⊗ id (cid:1) , U = antidiag ( i, − i ) and t a denotes transposition with respect to thetwo dimensional in our case ( d dimensional in general),“auxiliary” space.Specifically, let L ( λ ) = P i,j e ij ⊗ L ij ( λ ) then L t a ( λ ) = P i,j e ji ⊗ L ij ( λ ). N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 27
The condition (4.3) is equivalent to the requirement that the quantum determinantof L ( m ) is proportional to the identity, in analogy to the classical case. The problemthus reduces into deriving realizations of the quantum algebra of the form (4.1)subject to the constraint (4.3).Let us focus on the first two elements of the algebraic hierarchy L (1) and L (2) assuming that they provide realizations of the quantum algebra (4.1). Let usexpress L (1) , L (2) as follows L (1) ( λ ) = λ + N XY 1 ! , L (2) ( λ ) = λ + λ N (2) + A λ X + B λ Y +
C D ! . (4.5) The L (1) - operator. For m = 1 and f (1) = λ + 1 we recover the DNLS model[36, 37]. Indeed, solving condition (4.3) we conclude that N = 1 + XY and dueto the fact that L (1) satisfies the quantum algebra (4.1) we obtain the familiarcanonical relations for the fields: (cid:2) X , Y (cid:3) = 1 , and the extra relations (cid:2) X , N (cid:3) = Xcompatible with the definition of N from (4.3). A familiar representation of thecanonical fields is given in terms of differential operators as X xξ, Y x − ∂ ξ , where x commutes with both ξ, ∂ξ . (cid:3) The L (2) -operator. For m = 2 and f ( λ ) = λ + a λ + a , then condition (4.3)gives rise to the following identities (we choose a = 1) (4.6) D = 1 + XY , N (2) = (cid:16) X C + B Y (cid:17) D − and also(4.7) A = D − (cid:16) a − X C + BC (cid:17) The above expressions (4.6) and (4.7) are the quantum analogues of (2.45). Thealgebraic relations between the fundamental fields are dictated by (4.1) and aregiven as follows, we first give the exchange relations among the fundamental fields: (cid:2) X , Y (cid:3) = 0 , (cid:2) B , C (cid:3) = N (2) D , (cid:2) X , C (cid:3) = D , (cid:2) Y , B (cid:3) = − D (4.8)All the exchange relations among the various fields emerging from (4.1) are pre-sented in Appendix B. The semi-classical limit of the quantum algebraic relationsabove indeed lead to the Poisson relations (2.18)-(2.20) and (2.45), provided that − (cid:2) , (cid:3) → (cid:8) , (cid:9) A representation of the algebra (4.8) in terms of differential operators is givenbelow(4.9) X f x, Y gy, B g − (cid:0) f gxy (cid:1) ∂ y , C
7→ − f − (cid:0) f gxy (cid:1) ∂ x , where f, g commute with each other and also commute with x, y, ∂ x ∂ y . Also,as is well known typical realizations of the algebra (4.8) are obtained as tensorproducts of the algebra, i.e. we define L (2) ( { j } , λ ) = L (1) ( j + 1 , λ ) L (1) ( j, λ ), where There is a freedom on the derivation of the fields up to constant and/or a shift dependingon the choice of the constants a k . here we use the index notation and j can be either space or time index. (cid:3) Given the form of the L –operators we derived above we can now identify thequantum Lax pair for the discrete space-time NLS system expressed in the dou-ble index notation. The space component is given by L (1) → L ( n, a ) : X → X na , Y → Y na − , and the time component: L (2) → V (2) ( n, a ) : X → X na , Y → Y n − a , F → F na , where F ∈ (cid:8) B , C , N (2) , A , D (cid:9) .The elements of the temporal quantum algebra, for a fixed n , can be expressedin the double quantum index notation as (4.9)X na f n x a , Y n − a g n − y a , B na g − n − (cid:0) f n g n − x a y a (cid:1) ∂ y a , C na
7→ − f − n (cid:0) f n g n − x a y a (cid:1) ∂ x a , (4.10)where in the expressions above a representation for f n , g n , compatible with thespace like algebra, can be given as f n ξ n , g n ∂ ξ n .
2. The quantum AL model.
Let us now focus on the case of a trigonometric R -matrix and the quantum versions of the AL model. The quantum AL model.Consider now various solutions of the RTT relations (4.1) in the case we choosethe trigonometric matrix [35]:(4.11) R ( λ ) = a ( λ ) X j =1 e jj ⊗ e jj + c X i = j =1 e ij ⊗ e ji + b ( λ ) X i = j =1 q sgn ( j − i ) e ii ⊗ e jj , where q = e µ and a ( λ ) = sinh ( λ + µ ) , b ( λ ) = sinh ( λ ) , c = sinh ( µ ) . We consider below the quantum analogues of the three distinct cases discussedin the classical case:(4.12) L ( z ) = z ˆ bb z − ! , and the associated quantum algebra (4.1) is given as (see also [35]) (recall z = e λ )(4.13) q ˆ bb − q − b ˆ b = q − q − We also consider the following L -operators, solutions of (4.1)(4.14) L − ( z ) = z ˆ BB − z A + z − ! , L + ( z ) = z − z − A ˆ BB z − ! , where A = − BB (defined up to an overall multiplicative constant). Thecorresponding quantum algebra: q ˆ BB − q − B ˆ B = q − q − , ˆ BA = q − A ˆ B , BA = q AB (4.15)The semi-classical limit of the quantum algebraic relations (4.13), (4.15) lead tothe Poisson relations (2.52), (2.55), provided that − µ (cid:2) , (cid:3) → (cid:8) , (cid:9) . N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 29
Representations of the algebras (4.13), (4.15) are provided as follows (see also[33, 16] and relevant references therein). Let X , Y : XY = q YX , then(4.16) ˆ b := ( qξ X + 1) Y ζ, b := Y − ζ − where ξ, ζ commute with X , Y and they commute with each other, similarly forˆ B , B . Typical realizations of the elements X , Y are given as X := e ˆx , Y := e ˆy provided that (cid:2) ˆx , ˆy (cid:3) = 2 µ ( q = e µ ). X , Y can be represented in terms of differentialoperators: ˆx
7→ − µx, ˆy ∂ x , or by matrices, for example X P pk =1 q − k e kk and Y P p − k =1 e kk +1 + e p . The latter p dimensional representation is called thecyclic representation and is valid for µ = πp .The operators L , L ± will be now used for realizing the quantum discrete ALmodel. Below, we express the quantum Lax pairs in the double quantum indexnotation. A. We first consider the Lax pair (cid:16)
L, V − (cid:17) : L → L ( n, a ) : ˆ b → ˆ β na − , b → β na and L − → V − ( n, a ) : ˆ B → ˆ β n − a , B → β na . B. We also consider the Lax pair (cid:16)
L, V + (cid:17) : L → L ( n, a ) : ˆ b → ˆ β na − , b → γ na − and L + → V + ( n, a ) : ˆ B → ˆ β na − , B → γ n − a − . C. Finally we consider the Lax Pair (cid:16) L + , V − (cid:17) : L + → L + ( n, a ) : ˆ B → ˆ β na , B → β na − and V − ( n, a ) is defined as in case A.When defining the object A = − BB appearing in L ± we consistently keepa specific order for the fields involved, which of course is irrelevant at the classicallevel. Similarly, the order in the non-linear terms in the partial difference equations(2.58), (2.60), (2.64) is important in the quantum case. Our quantum descriptionis also compatible with the notion of the quantum auxiliary linear problem and thequantum Darboux-B¨acklund transformation as discussed in [12, 34]. The variousquantum equations of motion of (2.58), (2.60) and (2.64) are as expected preciselyof the form of the quantum Darboux-B¨acklund relations appearing in [12, 34], dueto the form of the discrete zero curvature condition (2.31) (cf. (3.22)). Moreover,the algebraic content of the quantum Darboux matrix as suggested in [12] is fullyjustified by the existence of space-time quantum algebras, and in particular thefact that the V -operator, which plays the role of the Darboux matrix, satisfies thetemporal quantum algebra. Remark 4.1.
The trigonometric R -matrix (4.11) as well as the various expres-sions for the L -operators can be associated to the more familiar U q ( sl ) R -matrix [29] (see for instance the use of the various versions in [35, 12, 34] ) via suitabletransformations. Indeed, let L ( λ ) = (cid:0) G − ⊗ V − (cid:1) ˆ L ( λ ) (cid:0) G ⊗ id (cid:1) and R ( λ ) = (cid:0) G ⊗ G − (cid:1) ˆ R ( λ ) (cid:0) G ⊗ G − (cid:1) ,where G = diag ( q , q − ) and ˆ R is the XXZ (or sine-Gordon) R matrix is given as [29](4.17) ˆ R ( λ ) = a ( λ ) X j =1 e jj ⊗ e jj + c X i = j =1 e ij ⊗ e ji + b ( λ ) X i = j =1 e ii ⊗ e jj , where recall, q = e µ and a ( λ ) = sinh ( λ + µ ) , b ( λ ) = sinh ( λ ) , c = sinh ( µ ) . Also, the algebraic object V is such that, A = V − , equivalently, ˆ B V = qV ˆ B and B V = q − V B . Also, det q ˆ L ∝ id or equivalently (4.3) is valid for ˆ L . It can beshown by direct computation for (4.12) and (4.14) that (cid:0) G − ⊗ V − (cid:1) ˆ L ( λ ) (cid:0) G ⊗ id (cid:1) = (cid:0) G ⊗ id) ˆ L ( λ ) (cid:0) G − ⊗ V − (cid:1) . ˆ L satisfies relation (4.1) with the ˆ R -matrix (4.17). (cid:3) Remark 4.2.
The L and ˆ L operators are associated to the same quantum algebra,however they provide distinct co-products. Let us use L ± as our examples to illustrate this. Let(4.18) ˆ L − ( z ) = zV ˆ CC − zV − + z − V ! , ˆ L + ( z ) = zV − z − V − ˆ CC z − V ! , where C = q − V B , ˆ C = q V ˆ B . We also multiply ˆ L and L with σ z (the diagonalPauli matrix), (( σ z ⊗ σ z ) R ( σ z ⊗ σ z ) = R , same for ˆ R ). The quantum algebrasemerging form (4.1) are Hopf algebras equipped with a co-product ∆( L ( λ )) = L (2 , λ ) L (1 , λ ), where 1 , L − we obtain(recall A = V − )(4.19) ∆( B ) = B ⊗ id , ∆(ˆ B ) = ˆ B ⊗ id + V − ⊗ ˆ B , whereas from ˆ L − :(4.20) ∆( C ) = C ⊗ V, ∆( ˆ C ) = ˆ C ⊗ V + V − ⊗ ˆ C Similarly, from L + ( ˆ L + ) with B , ˆ B ( C , ˆ C ) being interchanged in the co productsabove. (cid:3) We have only considered here periodic boundary conditions at both classical andquantum level. The significant point then is the implementation of integrable spaceand time integrable boundary conditions [47, 6, 13] in the discrete systems, andthe effect of these boundary conditions on the behavior of the solutions. As a finalremark we note that although the fully discrete case represented various technicaland conceptual difficulties, we were able to achieve the consistent simultaneousdiscretizations of both space and time directions in such a way that integrabiltywas ensured, based on the concurrent existence of temporal and spatial classicaland quantum algebras.
Acknowledgments.
AD acknowledges support from the EPSRC research grantEP/R009465/1.
N ALGEBRAIC APPROACH TO DISCRETE TIME INTEGABILITY 31
Appendix A. Discrete time NLS equations: consistency
1. Semi-discrete time NLS.
We obtain the following set of constraints from thezero curvature condition, by focusing first on the off diagonal elements: B a = ∂ x ˆ u a + N (2) a ˆ u a (A.1) ∂ x B a = ˆ u a +1 D a − A a ˆ u a (A.2) C a = − ∂ x u a + u a N (2) a (A.3) ∂ x C a = u a A a − D a u a − (A.4)Dependence of the fields on x is in all the always implied, but omitted for brevity.Equations (A.1), (A.3), via the definition of N (2) (2.17), lead to(A.5) B a = ∂ x ˆ u a − ˆ u a ∂ x u a − u a ˆ u a , C a = u a ∂ x ˆ u a − ∂ x u a − u a ˆ u a . Also, from the zero curvature condition we obtain the following constraints(A.6) ∂ x B a = ˆ u a +1 D a − A a ˆ u a , ∂ x C a = u a A a − D a u a − . Given that that A , D , N (2) (2.17), and B , C (A.5) are expressed in terms of thefundamental fields u a , ˆ u a and their x -derivatives, equations (A.6) are the equationsof motion for the fundamental fields fields u a , ˆ u a .Consistency checks are also performed for the diagonal entries of the matrixfrom the zero curvature condition: ∂ x D a = u a B a − C a ˆ u a (A.7) ∂ x N (2) = ˆ u a +1 u a − ˆ u a u a − (A.8) ∂ x A a = ˆ u a +1 C a − B a u a − . (A.9)Indeed, using the fundamental equations (A.1)-(A.2) and the definitions (2.17) theabove equations are confirmed.
2. Fully discrete NLS.
Having at our disposal the Lax pair we can now write downthe set of equations merging for the fully discrete version of the zero curvaturecondition (2.31) B na = X n +1 a + (cid:0) N (2) n +1 a − N na +1 (cid:1) X na (A.10) B n +1 a = N na +1 B na + X na +1 D na − A n +1 a X na (A.11) C n +1 a = Y n − a − Y na (cid:0) N na − N (2) na (cid:1) (A.12) C na = C n +1 a N na +1 + D n +1 a Y n − a − Y na A na (A.13)The first equations above come form the off diagonal elements of the zero curvaturecondition. Equations (A.10)-(A.13) together with (2.44) provide the fully discreteanalogues of the equations of motion (A.1)-(A.4). The diagonal entries provide extra consistency constraints N (2) n +1 a − N na +1 = N (2) na − N na (A.14) D n +1 a − D na = Y na B na − C n +1 a X na (A.15) A n +1 − A n = N na +1 N (2) na − N (2) n +1 a N na + X na +1 Y n − a − X n +1 a Y na − (A.16) A n +1 a N na − A na N na +1 = X na +1 C na − B n +1 a Y na − . (A.17)Equations (A.15)-(A.17) are the discrete space analogues of the semi-discrete case(A.7)-(A.9). However in the fully discrete case an extra constraint arising andcorresponds to equation (A.14). Appendix B. Algebraic relations & compatibility By means of (4.1) we obtain the following algebraic relations involving all the fields(4.6), (4.7): (cid:2) C , D (cid:3) = − Y N (2) , (cid:2) B , D (cid:3) = N (2) X , (cid:2) X , D (cid:3) = (cid:2) Y , D (cid:3) = 0(B.1) (cid:2) N (2) , B (cid:3) = − B , (cid:2) N (2) , C (cid:3) = C , (cid:2) N (2) , A (cid:3) = (cid:2) N (2) , D (cid:3) = 0(B.2) (cid:2) X , A (cid:3) = B , (cid:2) Y , A (cid:3) = − C , (cid:2) X , N (2) (cid:3) = X , (cid:2) Y , N (2) (cid:3) = − Y(B.3) (cid:2) A , B (cid:3) = A X − N (2) B , (cid:2) A , C (cid:3) = CN (2) − Y A (B.4) (cid:2) A , D (cid:3) = C X − Y B , (cid:2) X , D (cid:3) = (cid:2) Y , D (cid:3) = 0 . (B.5)All the relations above are compatible with the fields as defined in (4.6), (4.7) and(4.8). References [1] M.J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Phys. Rev. Lett. 31 (1973) 125.[2] M.J. Ablowitz and J.F. Ladik, J. Math. Phys. 16 (1975) 598.[3] M.J. Ablowitz, B. Prinari and A.D. Trubatch,
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