Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Toda-type systems
aa r X i v : . [ n li n . S I] F e b MULTI-TIME LAGRANGIAN 1-FORMSFOR FAMILIES OF B ¨ACKLUND TRANSFORMATIONS.TODA-TYPE SYSTEMS
RAPHAEL BOLL, MATTEO PETRERA, YURI B. SURISInstitut f¨ur Mathematik, MA 7-2,Technische Universit¨at Berlin, Str. des 17. Juni 13610623 Berlin, GermanyE-mail: boll, petrera, [email protected]
Abstract.
General Lagrangian theory of discrete one-dimensional integrable systemsis illustrated by a detailed study of B¨acklund transformations for Toda-type systems.Commutativity of B¨acklund transformations is shown to be equivalent to consistencyof the system of discrete multi-time Euler-Lagrange equations. The precise meaning ofthe commutativity in the periodic case, when all maps are double-valued, is established.It is shown that gluing of different branches is governed by the so called superpositionformulas. The closure relation for the multi-time Lagrangian 1-form on solutions of thevariational equations is proved for all Toda-type systems. Superposition formulas areinstrumental for this proof. The closure relation was previously shown to be equivalent tothe spectrality property of B¨acklund transformations, i.e., to the fact that the derivativeof the Lagrangian with respect to the spectral parameter is a common integral of motionof the family of B¨acklund transformations. We relate this integral of motion to themonodromy matrix of the zero curvature representation which is derived directly fromequations of motion in an algorithmic way. This serves as a further evidence in favorof the idea that B¨acklund transformations serve as zero curvature representations forthemselves. Introduction
The present paper can be considered as an extended illustration of the general La-grangian theory of discrete integrable systems of classical mechanics, developed in [19].This development was prompted by an example of the discrete time Calogero-Moser sys-tem studied in [22], and belongs to the line of research on variational formulation of moregeneral discrete integrable systems, initiated by Lobb and Nijhoff in [12].The notion of integrability of discrete systems, lying at the basis of this development, isthat of the multidimensional consistency. This understanding of integrability of discretesystems has been a major breakthrough [6], [15], and stimulated an impressive activityboost in the area, cf. [7]. According to the concept of multi-dimensional consistency,integrable d -dimensional systems can be imposed in a consistent way on all d -dimensionalsublattices of a lattice Z m of arbitrary dimension. This means that the resulting multi-dimensional system possesses solutions whose restrictions to any d -dimensional sublatticeare generic solutions of the corresponding two-dimensional system. In the case d = 1the concept of multi-dimensional consistency is more or less synonymous with the idea ofintegrability as commutativity which has been advocated by Veselov [21]. In the important case of discrete integrable systems of dimension d = 2, this approach led to classificationresults [2] (ABS list) which turned out to be rather influential.The Lagrangian aspects of the theory were, as mentioned above, pushed forward in [12].They observed that the value of the action functional for ABS equations remains invariantunder local changes of the underlying quad-surface, and suggested to consider this asa defining feature of integrability. Their results, found on the case-by-case basis forsome equations of the ABS list, have been extended to the whole list and given a moreconceptual proof in [8], and have been subsequently generalized in various directions: formulti-field two-dimensional systems [13], [4], for asymmetric two-dimensional systems [10],for dKP, the fundamental three-dimensional discrete integrable system [14], and for theabove mentioned example of one-dimensional integrable systems [22].General Lagrangian theory of discrete one-dimensional integrable systems has beendeveloped in [19]. We give a short account of this theory in Section 2. It turns outthat the most significant class of examples is given by B¨acklund transformations , i.e.,one-parameter families of commuting symplectic maps. In the main body of this paper,Sections 4–9, we work out the relevant results for B¨acklund transformations for integrablesystems of the Toda type. The list of systems under consideration is given, for convenienceof the reader, in Section 3. Our main results are the following.(1) Although B¨acklund transformations for the Toda lattice are very well studied(cf. [11] and quotations therein), there is one aspect which was not sufficiently dealtwith in the existing literature. Usually one considers these systems under periodicboundary conditions, which yields multi-valuedness of the corresponding maps.A general discussion of commutativity of multi-valued maps (correspondences) iscontained in [21]. However, its applicability to B¨acklund transformations of Toda-like systems, in particular, the choice of branches ensuring commutativity of suchmaps seems to be a completely open problem. We give here a complete solution ofthis problem. The key ingredient are the so called superposition formulas , whichenable us to precisely describe the branching behavior on the level of local algebraicrelations.(2) The main feature of the Lagrangian theory of discrete integrable systems is the socalled closure relation , which expresses the fact that the Lagrangian form on themulti-dimensional space of independent variables is closed on solutions of varia-tional equations. In the case of B¨acklund transformations, it was shown in [19] thatthe closure relation is equivalent to the so called spectrality property introducedby Sklyanin and Kuznetsov in [11], which had, up to now, a somewhat mysteriousreputation. We prove spectrality (and thus the closure relation) for all systems ofthe Toda type. Superposition formulas turn out to be of a crucial importance alsofor this aim.(3) In [19], an idea was pushed forward that Lax representations for B¨acklund trans-formations are already encoded in the equations of motion themselves. This isa one-dimensional counterpart of an analogous idea for two-dimensional systems,which was one of the main breakthroughs of [6], [15]. Here, we support this idea byan algorithmic derivation of transition and monodromy matrices for all Toda-typeintegrable systems. We think that a great portion of a mystic flair still enjoyed byintegrable systems gets herewith a rational and ultimately simple explanation.
ULTI-TIME LAGRANGIAN 1-FORMS 3 General theory of discrete multi-time Euler-Lagrange equations
We will now recall the main positions of the Lagrangian theory of discrete one-dimen-sional integrable systems, developed in [19], in application to
B¨acklund transformations ,i.e., to one-parameter families of commuting symplectic maps. Suppose that such a sym-plectic map F λ : ( x, p ) ( e x, e p ), depending on the parameter λ , admits a generatingfunction Λ: F λ : p = − ∂ Λ( x, e x ; λ ) ∂x , e p = ∂ Λ( x, e x ; λ ) ∂ e x . (1)Here, the first equation should be (at least locally) solvable for e x , i.e., the matrices ofthe mixed second order partial derivatives of the Lagrange function Λ should be non-degenerate, det( ∂ Λ /∂x∂ e x ) = 0. When considering a second such map, say F µ , we willdenote its action by a hat: F µ : p = − ∂ Λ( x, b x ; µ ) ∂x , b p = ∂ Λ( x, b x ; µ ) ∂ b x . (2)We assume that F λ ◦ F µ = F µ ◦ F λ . As a consequence, the following equations, called corner equations, are obtained by elim-inating p from (1), (2) at the four vertices of a square on Fig. 1: ∂ Λ( x, e x ; λ ) ∂x − ∂ Λ( x, b x ; µ ) ∂x = 0 , ( E ) ∂ Λ( x, e x ; λ ) ∂ e x + ∂ Λ( e x, be x ; µ ) ∂ e x = 0 , ( E ) ∂ Λ( x, b x ; µ ) ∂ b x + ∂ Λ( b x, be x ; λ ) ∂ b x = 0 , ( E )and ∂ Λ( b x, be x ; λ ) ∂ be x − ∂ Λ( e x, be x ; µ ) ∂ be x = 0 . ( E )These equations admit the following variational interpretation. We define the discretemulti-time Lagrangian 1-form for a family of B¨acklund transformations as a discrete 1-form whose values on the (directed) edges of Z are given by Λ( x, e x ; λ ), resp. Λ( x, b x ; µ ). Ageneralization to Z m with any m ≥ x : Z m → R delivering critical points for the action along any discrete curve in Z m . Then equations( E )–( E ) are nothing but multi-time Euler-Lagrange equations for this variational prob-lem; see [19].Consistency of the system of multi-time Euler-Lagrange equations ( E )–( E ) shouldbe understood as follows: start with the fields x , e x , b x satisfying the corner equation( E ). Then each of the corner equations ( E ), ( E ) can be solved for be x . Thus, we obtaintwo alternative values for the latter field. Consistency takes place if these values coincideidentically (with respect to the initial data), and, moreover, if the resulting field be x satisfiesthe corner equation ( E ). This is equivalent to commutativity of F λ and F µ . See Fig. 1.We mention that the standard single-time Euler-Lagrange equations for the maps F λ , ∂ Λ( x e , x ; λ ) ∂x + ∂ Λ( x, e x ; λ ) ∂x = 0 , are a consequence of equation ( E ) and (the downshifted version of) equation ( E ). RAPHAEL BOLL, MATTEO PETRERA, YURI B. SURIS x ( E ) ( E ) e x ( E ) b x be x ( E ) ( a) ( x, p ) ( e x, e p )( b x, b p ) (cid:0)be x, be p (cid:1) F λ F µ F λ F µ ( b) Figure 1.
Consistency of multi-time Euler-Lagrange equations: (a) Startwith data x , e x , b x related by corner equation ( E ); solve corner equations( E ) and ( E ) for be x ; consistency means that the two values of be x coincideidentically and satisfy corner equation ( E ). (b) Maps F λ and F µ commute.As shown in [19], on solutions of discrete multi-time Euler-Lagrange equations, wehave: Λ( x, e x ; λ ) + Λ( e x, be x ; µ ) − Λ( x, b x ; µ ) − Λ( b x, be x ; λ ) = ℓ ( λ, µ ) = const . (3)Moreover, ℓ ( λ, µ ) = 0, i.e., the discrete multi-time Lagrangian 1-form is closed on so-lutions, if and only if ∂ Λ( x, e x ; λ ) /∂λ is a common integral of motion for all F µ . Thelatter property is a re-formulation of the mysterious “spectrality property” of B¨acklundtransformations discovered by Kuznetsov and Sklyanin [11].3. Toda-type systems and their time discretizations
We will illustrate the above concepts with an important and representative set ofexamples, namely, B¨acklund transformations for Toda-type systems. The latter term isused to denote integrable lattice equations of the general form¨ x k = r ( ˙ x k ) (cid:0) f ( x k +1 − x k ) − f ( x k − x k − ) (cid:1) . The integrable discretizations [18] are of the form g ( e x k − x k ; h ) − g ( x k − x e k ; h ) = f ( x e k +1 − x k ; h ) − f ( x k − e x k − ; h ) , with h being an arbitrary parameter (time step). It is this parameter (or its inverse)which will play the role of the B¨acklund parameter λ in all our examples. The list ofexamples includes: • The original (exponential)
Toda lattice: ¨ x k = e x k +1 − x k − e x k − x k − , (4)with a discrete time counterpart e e x k − x k − e x k − x e k = h (cid:0) e x e k +1 − x k − e x k − e x k − (cid:1) . (5) • Dual Toda lattice: ¨ x k = ˙ x k ( x k +1 − x k + x k − ) , (6) ULTI-TIME LAGRANGIAN 1-FORMS 5 with a discrete time counterpart e x k − x k x k − x e k = 1 + h ( x e k +1 − x k )1 + h ( x k − e x k − ) . (7) • Modified Toda lattice: ¨ x k = ˙ x k (cid:0) e x k +1 − x k − e x k − x k − (cid:1) , (8)with a discrete time counterpart e e x k − x k − e x k − x e k − he x e k +1 − x k he x k − e x k − . (9) • Symmetric rational additive Toda-type system: ¨ x k = − ˙ x k (cid:18) x k +1 − x k − x k − x k − (cid:19) , (10)with a discrete time counterpart1 e x k − x k − x k − x e k = 1 x e k +1 − x k − x k − e x k − . (11) • Symmetric rational multiplicative Toda-type system: ¨ x k = − ( ˙ x k − (cid:18) x k +1 − x k − x k − x k − (cid:19) , (12)with a discrete time counterpart( e x k − x k + h )( e x k − x k − h ) · ( x k − x e k − h )( x k − x e k + h ) = ( x e k +1 − x k + h )( x e k +1 − x k − h ) · ( x k − e x k − − h )( x k − e x k − + h ) . (13) • Symmetric hyperbolic multiplicative Toda-type system: ¨ x k = − ( ˙ x k − (cid:0) coth( x k +1 − x k ) − coth( x k − x k − ) (cid:1) , (14)with a discrete time counterpartsinh( e x k − x k + h )sinh( e x k − x k − h ) · sinh( x k − x e k − h )sinh( x k − x e k + h ) = sinh( x e k +1 − x k + h )sinh( x e k +1 − x k − h ) · sinh( x k − e x k − − h )sinh( x k − e x k − + h ) . (15)(The names of the last three systems are justified by the appearance of their discrete timeversions.)We consider these systems with finitely many degrees of freedom (1 ≤ k ≤ N ). Thisrequires to specify certain boundary conditions. We will consider either periodic boundaryconditions (all indices taken mod N , so that x = x N , x N +1 = x ), or the so called open-end boundary conditions, which can be imposed in all cases except for the dual Todalattice and which can be formally achieved by setting x = ∞ and x N +1 = −∞ . Forthe dual Toda lattice (and for the modified Toda lattice, as well), a certain ersatz for theopen-end boundary conditions exists. It is achieved by considering the periodic systemwith N + 1 particles enumerated by 0 ≤ k ≤ N and by restricting it to x = x N +1 = 0.For such a reduction, all results of the present paper can be established, but we do notdeal with it in detail, since the resulting systems have a somewhat different flavor of theaffine root system C (1) N rather than the classical root system A N − . In particular, thesesystems do not depend on differences x k +1 − x k alone, due to the presence of the boundary RAPHAEL BOLL, MATTEO PETRERA, YURI B. SURIS terms depending on x and on x N , and therefore the total momentum P Nk =1 p k is not aconserved quantity for them.4. B¨acklund transformations for Toda lattice
Here we illustrate the main constructions by the well-known example of B¨acklundtransformations for the Toda lattice (4). The maps F λ : R N → R N are given byequations of the type (1): F λ : p k = 1 λ (cid:0) e e x k − x k − (cid:1) + λe x k − e x k − , e p k = 1 λ (cid:0) e e x k − x k − (cid:1) + λe x k +1 − e x k , (16)cf. [20], [11], [18]. The corresponding Lagrangian is given byΛ( x, e x ; λ ) = 1 λ N X k =1 (cid:0) e e x k − x k − − ( e x k − x k ) (cid:1) − λ N X k =1 e x k +1 − e x k , (17)and the standard single-time Euler-Lagrange equations coincide with (5) with h = λ .In the open-end case, we omit the term with e x − e x from the expression for p , theterm with e x N +1 − e x N from the expression for e p N , and we let the second sum in (17) extendover 1 ≤ k ≤ N − e x , e x , . . . , e x N (in this order), to give e e x − x = 1 + λp , e e x − x = 1 + λp − λ e x − x λp , · · · ,e e x N − x N = 1 + λp N − λ e x N − x N − λp N − − λ e x N − − x N − λp N − − . . . − λ e x − x λp . In the periodic case, all e e x k − x k can be expressed as analogous infinite periodic continuedfractions, and are, therefore, double-valued functions of ( x, p ).As discussed in the previous section, commutativity of the maps F λ , F µ (in the open-end case, when they are well-defined, i.e., single-valued) is equivalent to consistency ofthe system of corner equations:1 λ (cid:0) e e x k − x k − (cid:1) + λe x k − e x k − = 1 µ (cid:0) e b x k − x k − (cid:1) + µe x k − b x k − , ( E )1 λ (cid:0) e e x k − x k − (cid:1) + λe x k +1 − e x k = 1 µ (cid:16) e be x k − e x k − (cid:17) + µe e x k − be x k − , ( E )1 µ (cid:0) e b x k − x k − (cid:1) + µe x k +1 − b x k = 1 λ (cid:16) e be x k − b x k − (cid:17) + λe b x k − be x k − , ( E )1 λ (cid:16) e be x k − b x k − (cid:17) + λe b x k +1 − be x k = 1 µ (cid:16) e be x k − e x k − (cid:17) + µe e x k +1 − be x k . ( E )We have to clarify the meaning of the both notions (commutativity of F λ , F µ and con-sistency of corner equations) in the periodic case. To do this, we prove the followingstatement. ULTI-TIME LAGRANGIAN 1-FORMS 7
Theorem 1.
Suppose that the fields x , e x , b x satisfy corner equations ( E ). Define the fields be x by any of the following two formulas, which are equivalent by virtue of ( E ): λ (cid:16) e be x k − b x k − (cid:17) − µ (cid:16) e be x k − e x k − (cid:17) + λe x k +1 − e x k − µe x k +1 − b x k = 0 , ( S λ (cid:0) e e x k +1 − x k +1 − (cid:1) − µ (cid:0) e b x k +1 − x k +1 − (cid:1) + λe b x k +1 − be x k − µe e x k +1 − be x k = 0 , ( S called superposition formulas. Then the corner equations ( E )–( E ) are satisfied, as well. Proof.
First of all, we show that equations ( S
1) and ( S
2) are indeed equivalent byvirtue of ( E ). For this, we re-write these equations in algebraically equivalent forms: λ − µe be x k − x k +1 − λµ = λe x k +1 − e x k − µe x k +1 − b x k , (18)and ( λ − µ ) e x k +1 − be x k − λµe x k +1 − be x k = 1 µ (cid:0) e b x k +1 − x k +1 − (cid:1) − λ (cid:0) e e x k +1 − x k +1 − (cid:1) , (19)respectively. The left-hand sides of the latter two equations are equal. Thus, their differ-ence coincides with ( E ).Second, we show that equations ( S
1) and ( S
2) yield ( E ). (For ( E ) everything isabsolutely analogous.) For this aim, we re-write these equations in still other algebraicallyequivalent forms. Namely, ( S
1) is equivalent to e be x k − e x k = λµe x k +1 − e x k + µ − λµe e x k − b x k − λ , (20)while ( S
2) with k replaced by k − λµe e x k − be x k − = e e x k − x k + λ − µµ − λe b x k − e x k . (21)An obvious linear combination of these expressions leads to1 µ e be x k − e x k + µe e x k − be x k − = λe x k +1 − e x k + 1 λ e e x k − x k + λ − µλµ , which is nothing but ( E ).Third, we observe that the sum of equations ( E ), ( S
1) and ( S
2) is nothing but thecorner equation ( E ). (cid:4) Remark.
Observe that each of the equations ( S
1) and ( S
2) is a quad-equation withrespect to (cid:16) e x k +1 , e e x k , e b x k , e be x k (cid:17) , resp . (cid:16) e x k +1 , e e x k +1 , e b x k +1 , e be x k (cid:17) , i.e., can be formulated as vanishing of a multi-affine polynomial of the four specifiedvariables. Equations (18) and (19) are then interpreted as the three-leg forms of theoriginal equations, centered at x k +1 . Similarly, equations (20) and (21) are the three-legforms of the original equations, centered at e x k .This theorem allows us to achieve an exhaustive understanding of the consistency fordouble-valued B¨acklund transformations. First, suppose that we are given the fields x , e x , b x satisfying the corner equation ( E ). Each of equations ( E ), ( E ) produces two valuesfor be x . Then consistency is reflected in the following fact: one of the values for be x obtained RAPHAEL BOLL, MATTEO PETRERA, YURI B. SURIS x ( E ) ( E ) e x ( E ) b x be x ( E ) ( a) ( x, p ) ( e x, e p )( b x, b p ) (cid:0)be x, be p (cid:1) F λ F µ F λ F µ ( b) Figure 2.
Consistency of multi-time Euler-Lagrange equations in the caseof periodic boundary conditions: (a) Given the fields x , e x , b x satisfyingthe corner equation ( E ), each of the corner equations ( E ), ( E ) has twosolutions be x . One of the values for be x obtained from ( E ) coincides with oneof the values for be x obtained from ( E ). This common value of be x , togetherwith e x and b x , satisfies ( E ). (b) The four branches of F λ ◦ F µ and the fourbranches of F µ ◦ F λ coincide pairwise.from ( E ) coincides with one of the values for be x obtained from ( E ), see Fig. 2(a). Indeed,this common value is nothing but be x obtained from the superposition formulas ( S S F λ , F µ , i.e., by working with the variables ( x, p ) rather than with the variables x alone. Indeed, each of the compositions F λ ◦ F µ and F µ ◦ F λ is four-valued. It follows fromTheorem 1 that their branches must pairwise coincide, as shown on Fig. 2(b). Indeed,Theorem 1 delivers four possible values for ( e x, b x, be x ) satisfying all corner equations ( E )–( E ), namely one be x for each of the four possible combinations of ( e x, b x ). Theorem 2.
The discrete multi-time Lagrangian 1-form is closed on any solution of thecorner equations ( E )–( E ). Proof.
First of all, we show that the closure relation ℓ ( λ, µ ) = 0 is equivalent to N X k =1 ( be x k − e x k − b x k + x k ) = 0 ⇔ N Y k =1 e be x k − e x k − b x k + x k = 1 . (22)This can be done in two different ways. On one hand, combining ( S S
2) with ( E ), wearrive at the two formulas1 λ e e x k +1 − x k +1 − µ e b x k +1 − x k +1 − λ e be x k − b x k + 1 µ e be x k − e x k = 0 , (23) λe x k +1 − e x k − µe x k +1 − b x k − λe b x k +1 − be x k + µe e x k +1 − be x k = 0 . (24) ULTI-TIME LAGRANGIAN 1-FORMS 9
By virtue of these formulas, most of the terms on the left-hand side of (3) with theLagrange function (17) cancel, leaving us with ℓ ( λ, µ ) = (cid:18) λ − µ (cid:19) N X k =1 ( be x k − e x k − b x k + x k ) . For an alternative proof of the fact that ℓ ( λ, µ ) = 0 is equivalent to (22), we can refer tothe spectrality criterium stating that ℓ ( λ, µ ) = 0 is equivalent to ∂ Λ( x, e x ; λ ) /∂λ being anintegral of motion for F µ . One easily computes: ∂ Λ( x, e x ; λ ) ∂λ = − λ N X k =1 p k + 1 λ N X k =1 ( e x k − x k ) , the first sum on the right-hand side being an obvious integral of motion.Now the desired result (22) can be derived from the following form of the superpositionformula: e be x k − e x k − b x k + x k +1 = λe b x k +1 − µe e x k +1 λe b x k − µe e x k , (25)which is in fact equivalent to either of equations (23), (24). In the periodic case (22) followsdirectly by multiplying equations (25) for 1 ≤ k ≤ N , in the open-end case equation (25)holds true for 1 ≤ k ≤ N − e x = λe b x − µe e x λ − µ , e be x N − e x N − b x N = λ − µλe b x N − µe e x N , (26)which are equivalent to ( E ) for k = 1, resp. to ( E ) for k = N . (cid:4) Thus, we have proved that the quantity P Nk =1 ( e x k − x k ) is an integral of motion for F µ with an arbitrary µ . Actually, it is not difficult to give a matrix expression for this inte-gral. The participating matrices are nothing but transition matrices of the zero curvaturerepresentation for F λ , but the latter notion is not necessary for establishing the result. Theorem 3.
Set L k ( x, p ; λ ) = (cid:18) λp k − λ e x k e − x k (cid:19) , and T N ( x, p ; λ ) = L N ( x, p ; λ ) · · · L ( x, p ; λ ) L ( x, p ; λ ) . Then in the periodic case the quantity Q Nk =1 e e x k − x k is an eigenvalue of T N ( x, p ; λ ) , whilein the open-end case it is equal to the (11) -entry of T N ( x, p ; λ ) . Proof.
We use the following notation for the action of matrices from GL (2 , C ) on C by M¨obius transformations: (cid:18) a bc d (cid:19) [ z ] = az + bcz + d . With this notation, we can re-write the first equation in (16) as e e x k = e x k (cid:16) (1 + λp k ) − λ e x k − e x k − (cid:17) = L k ( x, p ; λ )[ e e x k − ] . This is equivalent to saying that L k ( x, p ; λ ) (cid:18) e e x k − (cid:19) ∼ (cid:18) e e x k (cid:19) . The proportionality coefficient is easily determined by comparing the second componentsof these vectors: L k ( x, p ; λ ) (cid:18) e e x k − (cid:19) = e e x k − − x k (cid:18) e e x k (cid:19) . (27)Now the claim in the periodic case follows immediately, with the corresponding eigenvectorof T N ( x, p ; λ ) being (cid:0) e e x N , (cid:1) T . In the open-end case, equation (27) holds true for 2 ≤ k ≤ N , and has to be supplemented by the following two relations: L ( x, p ; λ ) (cid:18) (cid:19) = e − x (cid:18) e e x (cid:19) , and (cid:0) (cid:1) (cid:18) e e x N (cid:19) = e e x N . As a consequence, (cid:0) (cid:1) T N ( x, p ; λ ) (cid:18) (cid:19) = N Y k =1 e e x k − x k . (cid:4) In the subsequent sections, we prove similar results for B¨acklund transformations forall the remaining Toda-type systems. For each of them, • we find superposition formulas which yield commutativity of B¨acklund transfor-mations, both in the single-valued case of open-end boundary conditions and inthe double-valued case of periodic boundary conditions; • we prove the spectrality property, so that the discrete 1-form L is closed on solu-tions of the Euler-Lagrange equations. The proof is based on superposition formu-las. This result provides also the existence of a big number of common integralsfor the whole family F λ . Actually, there are sufficiently many common integralsin involution to ensure complete integrability in the Liouville-Arnold sense; • and we give an expression of the corresponding conserved quantity ∂ Λ( x, e x ; λ ) /∂λ in terms of canonically conjugate variables ( x, p ). This is done with the help ofthe monodromy matrix of the corresponding discrete time zero curvature repre-sentation, which is derived directly from equations of motion in an unambiguousand algorithmic way. This gives a further support to the idea pushed forwardin [19], that B¨acklund transformations can serve as zero curvature representationsfor themselves.5. B¨acklund transformations for dual Toda lattice
Our second example constitute B¨acklund transformations for the dual Toda lattice (6)which are given by equations of the type (1): F λ : (cid:26) e p k = ( e x k − x k )( λ + x k − e x k − ) ,e e p k = ( e x k − x k )( λ + x k +1 − e x k ) . (28)The corresponding Lagrangian is given byΛ( x, e x ; λ ) = N X k =1 ψ ( e x k − x k ) − N X k =1 ψ ( λ + x k +1 − e x k ) , where ψ ( ξ ) = ξ log ξ − ξ . The standard single-time Euler-Lagrange equations coincidewith (7) with h = λ − . Recall that for the dual Toda lattice we only consider periodicboundary conditions. ULTI-TIME LAGRANGIAN 1-FORMS 11
To establish commutativity of the maps F λ , F µ , we consider the system of cornerequations: ( e x k − x k )( λ + x k − e x k − ) = ( b x k − x k )( µ + x k − b x k − ) , ( E )( e x k − x k )( λ + x k +1 − e x k ) = ( be x k − e x k )( µ + e x k − be x k − ) , ( E )( b x k − x k )( µ + x k +1 − b x k ) = ( be x k − b x k )( λ + b x k − be x k − ) , ( E )( be x k − b x k )( λ + b x k +1 − be x k ) = ( be x k − e x k )( µ + e x k +1 − be x k ) . ( E ) Theorem 4.
Suppose that the fields x , e x , b x satisfy corner equations ( E ). Define thefields be x by any of the following two superposition formulas, which are equivalent by virtueof ( E ): ( be x k − b x k )( λ + x k +1 − e x k ) = ( be x k − e x k )( µ + x k +1 − b x k ) , ( S e x k +1 − x k +1 )( λ + b x k +1 − be x k ) = ( b x k +1 − x k +1 )( µ + e x k +1 − be x k ) . ( S Then the corner equations ( E )–( E ) are satisfied, as well. Proof.
Each of the equations ( S
1) and ( S
2) is a quad-equation with respect to (cid:0) x k +1 , e x k , b x k , be x k (cid:1) , resp . (cid:0) x k +1 , e x k +1 , b x k +1 , be x k (cid:1) . The three-leg forms of these equations, centered at x k +1 , are: λ + x k +1 − be x k µ + x k +1 − be x k = λ + x k +1 − e x k µ + x k +1 − b x k , and λ + x k +1 − be x k µ + x k +1 − be x k = b x k +1 − x k +1 e x k +1 − x k +1 , respectively. Their quotient coincides with ( E ).The three-leg forms of equations ( S
1) and ( S e x k , are: b x k − e x k λ − µ + b x k − e x k = be x k − e x k λ + x k +1 − e x k , and b x k +1 − e x k +1 λ − µ + b x k +1 − e x k +1 = e x k +1 − x k +1 µ + e x k +1 − be x k , respectively. The quotient of these two equations (the second with the downshifted index k ) coincides with ( E ).The three-leg forms of superposition formulas ( S
1) and ( S be x k , are: λ + x k +1 − be x k µ + x k +1 − be x k = be x k − e x k be x k − b x k , and λ + x k +1 − be x k µ + x k +1 − be x k = λ + b x k +1 − be x k µ + e x k +1 − be x k , respectively. The quotient of these two equations coincides with ( E ). (cid:4) Theorem 5.
The discrete multi-time Lagrangian 1-form is closed on any solution of thecorner equations ( E )–( E ). Proof.
With the help of the spectrality criterium, we see that the claim of the theoremis equivalent to N Y k =1 ( λ + b x k +1 − be x k ) = N Y k =1 ( λ + x k +1 − e x k ) . (29)To prove this relation, we observe that superposition formulas ( S
1) and ( S
2) admit thefollowing further equivalent formulations:( λ − µ )( λ + x k +1 − e x k ) = ( λ − µ + b x k − e x k )( λ + x k +1 − be x k ) , and ( λ − µ )( λ + b x k +1 − be x k ) = ( λ − µ + b x k +1 − e x k +1 )( λ + x k +1 − be x k ) , respectively. There follows: λ + b x k +1 − be x k λ + x k +1 − e x k = µ − λ + e x k +1 − b x k +1 µ − λ + e x k − b x k . Under the periodic boundary conditions, this formula yields (29). (cid:4)
Theorem 6.
Set L k ( x, p ; λ ) = (cid:18) − x k x k ( λ + x k ) + e p k − λ + x k (cid:19) , and T N ( x, p ; λ ) = L N ( x, p ; λ ) · · · L ( x, p ; λ ) L ( x, p ; λ ) . Then, under the periodic boundary conditions, the quantity Q Nk =1 ( λ + x k +1 − e x k ) is aneigenvalue of T N ( x, p ; λ ) . Proof.
We can re-write the first equation in (28) as e x k = x k + e p k λ + x k − e x k − = ( λ + x k ) x k + e p k − x k e x k − λ + x k − e x k − = L k ( x, p ; λ )[ e x k − ] . This is equivalent to L k ( x, p ; λ ) (cid:18)e x k − (cid:19) ∼ (cid:18)e x k (cid:19) . The proportionality coefficient is determined by comparing the second components ofthese two vectors: L k ( x, p ; λ ) (cid:18)e x k − (cid:19) = ( λ + x k − e x k − ) (cid:18)e x k (cid:19) . Now the claim follows immediately from the periodic boundary conditions, with the cor-responding eigenvector of T N ( x, p ; λ ) being (cid:0)e x N , (cid:1) T . (cid:4) ULTI-TIME LAGRANGIAN 1-FORMS 13 B¨acklund transformations for modified Toda lattice
Our third example constitute B¨acklund transformations for the modified Toda lattice(8) which are given by equations of the type (1): F λ : (cid:26) e p k = (cid:0) e e x k − x k − (cid:1) (cid:0) λ + e x k − e x k − (cid:1) ,e e p k = (cid:0) e e x k − x k − (cid:1) (cid:0) λ + e x k +1 − e x k (cid:1) . (30)Like for the standard Toda lattice, in the open-end case the first equations in (30) areuniquely solved for e x , e x , . . . , e x N (in this order), and in the periodic case e x k are double-valued functions of ( x, p ).The corresponding Lagrangian is given in the periodic case byΛ( x, e x ; λ ) = N X k =1 ψ ( e x k − x k ; − − N X k =1 ψ ( x k +1 − e x k ; λ ) , and in the open-end case byΛ( x, e x ; λ ) = N X k =1 ψ ( e x k − x k ; − − N − X k =1 ψ ( x k +1 − e x k ; λ ) + ( e x N − x ) log λ, where ψ ( ξ ; λ ) = Z ξ log( e η + λ ) dη. The standard single-time Euler-Lagrange equations are (9) with h = λ − .To establish commutativity of the maps F λ , F µ , we consider the system of cornerequations: (cid:0) e e x k − x k − (cid:1) (cid:0) λ + e x k − e x k − (cid:1) = (cid:0) e b x k − x k − (cid:1) (cid:0) µ + e x k − b x k − (cid:1) , ( E ) (cid:0) e e x k − x k − (cid:1) (cid:0) λ + e x k +1 − e x k (cid:1) = (cid:0) e be x k − e x k − (cid:1)(cid:0) µ + e e x k − be x k − (cid:1) , ( E ) (cid:0) e b x k − x k − (cid:1)(cid:0) µ + e x k +1 − b x k (cid:1) = (cid:0) e be x k − b x k − (cid:1)(cid:0) λ + e b x k − be x k − (cid:1) , ( E ) (cid:0) e be x k − b x k − (cid:1)(cid:0) λ + e b x k +1 − be x k (cid:1) = (cid:0) e be x k − e x k − (cid:1)(cid:0) µ + e e x k +1 − be x k (cid:1) . ( E ) Theorem 7.
Suppose that the fields x , e x , b x satisfy corner equations ( E ). Define thefields be x by any of the following two superposition formulas, which are equivalent by virtueof ( E ): (cid:0) e be x k − b x k − (cid:1)(cid:0) λ + e x k +1 − e x k (cid:1) = (cid:0) e be x k − e x k − (cid:1)(cid:0) µ + e x k +1 − b x k (cid:1) , ( S (cid:0) e e x k +1 − x k +1 − (cid:1)(cid:0) λ + e b x k +1 − be x k (cid:1) = (cid:0) e b x k +1 − x k +1 − (cid:1)(cid:0) µ + e e x k +1 − be x k (cid:1) . ( S Then the corner equations ( E )–( E ) are satisfied, as well. Proof.
Each of the equations ( S
1) and ( S
2) is a quad-equation with respect to (cid:0) e x k +1 , e e x k , e b x k , e be x k (cid:1) , resp . (cid:0) e x k +1 , e e x k +1 , e b x k +1 , e be x k (cid:1) . The three-leg forms of equations ( S
1) and ( S x k +1 , are: λ + e x k +1 − be x k µ + e x k +1 − be x k = λ + e x k +1 − e x k µ + e x k +1 − b x k , and λ + e x k +1 − be x k µ + e x k +1 − be x k = e b x k +1 − x k +1 − e e x k +1 − x k +1 − , respectively. Their quotient coincides with ( E ).The three-leg forms of equations ( S
1) and ( S e x k , are: e b x k − e x k − λ − µe b x k − e x k = e be x k − e x k − λ + e x k +1 − e x k , and e b x k +1 − e x k +1 − λ − µe b x k +1 − e x k +1 = e e x k +1 − x k +1 − µ + e e x k +1 − be x k , respectively. The quotient of these two equations (the second with the downshifted index k ) coincides with ( E ).The three-leg forms of superposition formulas ( S
1) and ( S be x k , are: λ + e x k +1 − be x k µ + e x k +1 − be x k = e be x k − e x k − e be x k − b x k − , and λ + e x k +1 − be x k µ + e x k +1 − be x k = λ + e b x k +1 − be x k µ + e e x k +1 − be x k , respectively. The quotient of these two equations coincides with ( E ). (cid:4) Theorem 8.
The discrete multi-time Lagrangian 1-form is closed on any solution of thecorner equations ( E )–( E ). Proof.
With the help of the spectrality criterium, we see that the claim of the theoremin the periodic case is equivalent to P ( x, e x ) = P ( b x, be x ) , (31)where in the periodic case P ( x, e x ) = N Y k =1 (cid:0) λe e x k − x k +1 (cid:1) , while in the open-end case P ( x, e x ) = e e x N − x N − Y k =1 (cid:0) λe e x k − x k +1 (cid:1) . To prove this relation, we observe that superposition formulas ( S
1) and ( S
2) admit thefollowing further equivalent formulations:( λ − µ ) (cid:0) λe e x k − x k +1 (cid:1) = ( λe e x k − b x k − µ (cid:1)(cid:0) λe be x k − x k +1 (cid:1) , and ( λ − µ ) (cid:0) λe be x k − b x k +1 (cid:1) = ( λe e x k +1 − b x k +1 − µ (cid:1)(cid:0) λe be x k − x k +1 (cid:1) , ULTI-TIME LAGRANGIAN 1-FORMS 15 respectively. There follows: 1 + λe be x k − b x k +1 λe e x k − x k +1 = λe e x k +1 − b x k +1 − µλe e x k − b x k − µ . (32)In the periodic case this formula yields (31). In the open-end case equation (32) holdstrue for 1 ≤ k ≤ N −
1, and has to be supplemented with λe e x − b x − µλ − µ = e x − b x , λ − µλe e x N − b x N − µ = e be x N − e x N , (33)which are equivalent to ( E ) for k = 1, resp. to ( E ) for k = N . Product of equations (32)with 1 ≤ k ≤ N − (cid:4) Theorem 9.
Set L k ( x, p ; λ ) = (cid:18) λ + e p k e x k λe − x k (cid:19) , and T N ( x, p ; λ ) = L N ( x, p ; λ ) · · · L ( x, p ; λ ) L ( x, p ; λ ) . Then in the periodic case the quantity P ( x, e x ) from (6) is an eigenvalue of T N ( x, p ; λ ) ,while in the open-end case its counterpart from (6) is equal, up to the factor λ , to the (11) -entry of T N ( x, p ; λ ) . Proof.
We can re-write the first equation in (30) as e e x k = e x k (cid:18) e p k λ + e x k − e x k − (cid:19) = (cid:0) λ + e p k (cid:1) e e x k − + e x k λe − x k + e x k − + 1 = L k ( x, p ; λ )[ e e x k − ] . This is equivalent to L k ( x, p ; λ ) (cid:18) e e x k − (cid:19) ∼ (cid:18) e e x k (cid:19) . The proportionality coefficient is determined by comparing the second components ofthese two vectors: L k ( x, p ; λ ) (cid:18) e e x k − (cid:19) = (cid:0) λe e x k − − x k (cid:1) (cid:18) e e x k (cid:19) . Now the claim in the periodic case follows immediately, with the corresponding eigenvectorof T N ( x, p ; λ ) being (cid:0) e e x N , (cid:1) T . In the open-end case, we have to use additionally thefollowing relations: L ( x, p ; λ ) (cid:18) (cid:19) = λe − x (cid:18) e e x (cid:19) , (cid:0) (cid:1) (cid:18) e e x N (cid:19) = e e x N . As a consequence, (cid:0) (cid:1) T N ( x, p ; λ ) (cid:18) (cid:19) = λe e x N − x N − Y k =1 (cid:0) λe e x k − x k +1 (cid:1) . (cid:4) B¨acklund transformations for symmetric rational additive Toda-typesystem
The next example constitute B¨acklund transformations for the symmetric rationaladditive Toda-type system (10) which are given by equations of the type (1): F λ : p k = λ e x k − x k + λx k − e x k − , e p k = λ e x k − x k + λx k +1 − e x k . (34)The corresponding Lagrangian is given byΛ( x, e x ; λ ) = λ N X k =1 log | e x k − x k | − λ N X k =1 log | x k +1 − e x k | . (35)The standard single-time Euler-Lagrange equations are (11) with h = λ . In the open-endcase, all terms with x − e x and with x N +1 − e x N should be omitted both from the equationsof motion (34) and from the Lagrangian (35). As usual, in the open-end case the firstequations in (34) are uniquely solved for e x , e x , . . . , e x N (in this order) in terms of ( x, p ),while in the periodic case all e x k can be expressed as infinite periodic continued fractionsand are, therefore, double-valued functions of ( x, p ).To establish commutativity of the maps F λ , F µ , we consider the system of cornerequations: λ e x k − x k + λx k − e x k − = µ b x k − x k + µx k − b x k − , ( E ) λ e x k − x k + λx k +1 − e x k = µ be x k − e x k + µ e x k − be x k − , ( E ) µ b x k − x k + µx k +1 − b x k = λ be x k − b x k + λ b x k − be x k − , ( E ) λ be x k − b x k + λ b x k +1 − be x k = µ be x k − e x k + µ e x k +1 − be x k . ( E ) Theorem 10.
Suppose that the fields x , e x , b x satisfy corner equations ( E ). Define thefields be x by any of the following two superposition formulas, which are equivalent by virtueof ( E ): µ ( be x k − b x k )( x k +1 − e x k ) = λ ( be x k − e x k )( x k +1 − b x k ) , ( S µ ( e x k +1 − x k +1 )( b x k +1 − be x k ) = λ ( b x k +1 − x k +1 )( e x k +1 − be x k ) . ( S Then the corner equations ( E )–( E ) are satisfied, as well. Proof.
Observe that equations ( S
1) and ( S
2) are quad-equations with respect to (cid:0) x k +1 , e x k , b x k , be x k (cid:1) , resp . (cid:0) x k +1 , e x k +1 , b x k +1 , be x k (cid:1) , namely the cross-ratio equations (Q1 δ =0 in the notation of ABS list [2]).The three-leg forms of these equations, centered at x k +1 , are: λ − µ be x k − x k +1 = λ e x k − x k +1 − µ b x k − x k +1 , ULTI-TIME LAGRANGIAN 1-FORMS 17 and λ − µ be x k − x k +1 = λ e x k +1 − x k +1 − µ b x k +1 − x k +1 , respectively. Their difference coincides with ( E ).The three-leg forms of equations ( S
1) and ( S e x k , resp. at e x k +1 , are: λ − µ b x k − e x k = λx k +1 − e x k − µ be x k − e x k , and λ − µ b x k +1 − e x k +1 = λx k +1 − e x k +1 − µ be x k − e x k +1 , respectively. The difference of these two equations (the second one with k replaced by k −
1) coincides with ( E ).The three-leg forms of equations ( S
1) and ( S be x k , are: λ − µx k +1 − be x k = λ b x k − be x k − µ e x k − be x k , and λ − µx k +1 − be x k = λ b x k +1 − be x k − µ e x k +1 − be x k , respectively. The difference of these two equations coincides with ( E ). (cid:4) Theorem 11.
The discrete multi-time Lagrangian 1-form is closed on any solution of thecorner equations ( E )–( E ). Proof.
With the help of the spectrality criterium, we see that the claim of the theoremis equivalent to P ( x, e x ) = P ( b x, be x ) , (36)where in the periodic case P ( x, e x ) = Q Nk =1 ( x k +1 − e x k ) Q Nk =1 ( e x k − x k ) , while in the open-end case the product in the numerator of the latter formula is over1 ≤ k ≤ N − S S
2) in the following equivalent forms:( λ − µ )( be x k − b x k )( x k +1 − e x k ) = λ ( e x k − b x k )( x k +1 − be x k ) , resp. ( λ − µ )( b x k +1 − be x k )( e x k +1 − x k +1 ) = λ ( e x k +1 − b x k +1 )( x k +1 − be x k ) . As a consequence, we arrive at the following superposition formula: b x k +1 − be x k be x k − b x k = x k +1 − e x k e x k +1 − x k +1 · e x k +1 − b x k +1 e x k − b x k . (37) In the periodic case this formula yields (36). In the open-end case equation (37) holdstrue for 1 ≤ k ≤ N −
1, and has to be supplemented with λ be x N − b x N = λ − µ e x N − b x N , λ e x − x = λ − µ e x − b x . (38)Product of equations (37) with 1 ≤ k ≤ N − (cid:4) Theorem 12.
Set L k ( x, p ; λ ) = I + λ − (cid:18) p k x k − p k x k p k − p k x k (cid:19) , and T N ( x, p ; λ ) = L N ( x, p ; λ ) · · · L ( x, p ; λ ) L ( x, p ; λ ) . Then in the periodic case the quantity Q Nk =1 ( e x k − x k +1 ) Q Nk =1 ( e x k − x k ) is an eigenvalue of T N ( x, p ; λ ) , while in the open-end case its counterpart, Q N − k =1 ( e x k − x k +1 ) Q Nk =1 ( e x k − x k ) , is equal to the (21) -entry of T N ( x, p ; λ ) . Proof.
We can re-write the first equation in (34) as e x k = x k + λp k − λx k − e x k − = p k x k ( x k − e x k − ) − λ e x k − p k ( x k − e x k − ) − λ = L k ( x, p ; λ )[ e x k − ] . This is equivalent to L k ( x, p ; λ ) (cid:18)e x k − (cid:19) ∼ (cid:18)e x k (cid:19) . The proportionality coefficient is determined by comparing the second components ofthese two vectors and is equal to1 − λ − p k ( x k − e x k − ) = 1 − ( x k − e x k − ) (cid:18) e x k − x k + 1 x k − e x k − (cid:19) = e x k − − x k e x k − x k . Thus, L k ( x, p ; λ ) (cid:18)e x k − (cid:19) = e x k − − x k e x k − x k (cid:18)e x k (cid:19) . (39)Now the claim in the periodic case follows immediately, with the corresponding eigenvectorof T N ( x, p ; λ ) being (cid:0)e x N , (cid:1) T . In the open-end case, equation (39) holds true for 2 ≤ k ≤ N , and has to be supplemented by the following two relations: L ( x, p ; λ ) (cid:18) (cid:19) = 1 e x − x (cid:18)e x (cid:19) , and (cid:0) (cid:1) (cid:18)e x N (cid:19) = 1 . ULTI-TIME LAGRANGIAN 1-FORMS 19
As a consequence, (cid:0) (cid:1) T N ( x, p ; λ ) (cid:18) (cid:19) = Q N − k =1 ( e x k − x k +1 ) Q Nk =1 ( e x k − x k ) . (cid:4) B¨acklund transformations for symmetric rational multiplicativeToda-type system
The next example constitute B¨acklund transformations for the symmetric rationalmultiplicative Toda-type system (12) which are given by equations of the type (1): F λ : e p k = e x k − x k + λ e x k − x k − λ · x k − e x k − + λx k − e x k − − λ ,e e p k = e x k − x k + λ e x k − x k − λ · x k +1 − e x k + λx k +1 − e x k − λ . (40)The corresponding Lagrangian is given byΛ( x, e x ; λ ) = N X k =1 ψ ( e x k − x k ; λ ) − N X k =1 ψ ( x k +1 − e x k ; λ ) , (41)where ψ ( ξ ; λ ) = 12 Z ξ + λξ − λ log ηdη. The standard single-time Euler-Lagrange equations are (13) with h = λ . In the open-endcase, all terms with x − e x and with x N +1 − e x N should be omitted both from the equationsof motion (40) and from the Lagrangian (41).To establish commutativity of the maps F λ , F µ , we consider the system of cornerequations: e x k − x k + λ e x k − x k − λ · x k − e x k − + λx k − e x k − − λ = b x k − x k + µ b x k − x k − µ · x k − b x k − + µx k − b x k − − µ , ( E ) e x k − x k + λ e x k − x k − λ · x k +1 − e x k + λx k +1 − e x k − λ = be x k − e x k + µ be x k − e x k − µ · e x k − be x k − + µ e x k − be x k − − µ , ( E ) b x k − x k + µ b x k − x k − µ · x k +1 − b x k + µx k +1 − b x k − µ = be x k − b x k + λ be x k − b x k − λ · b x k − be x k − + λ b x k − be x k − − λ , ( E ) be x k − b x k + λ be x k − b x k − λ · b x k +1 − be x k + λ b x k +1 − be x k − λ = be x k − e x k + µ be x k − e x k − µ · e x k +1 − be x k + µ e x k +1 − be x k − µ . ( E ) Theorem 13.
Suppose that the fields x , e x , b x satisfy corner equations ( E ). Define thefields be x by any of the following two superposition formulas, which are equivalent by virtueof ( E ): µ ( be x k − b x k )( x k +1 − e x k ) − λ ( be x k − e x k )( x k +1 − b x k ) + λµ ( λ − µ ) = 0 , ( S µ ( e x k +1 − x k +1 )( b x k +1 − be x k ) − λ ( b x k +1 − x k +1 )( e x k +1 − be x k ) + λµ ( λ − µ ) = 0 . ( S Then the corner equations ( E )–( E ) are satisfied, as well. Proof.
Observe that equations ( S
1) and ( S
2) are quad-equations with respect to (cid:0) x k +1 , e x k , b x k , be x k (cid:1) , resp . (cid:0) x k +1 , e x k +1 , b x k +1 , be x k (cid:1) , namely of the type Q1 δ =1 from the ABS list [2].The three-leg forms of these equations, centered at x k +1 , are: x k +1 − e x k + λx k +1 − e x k − λ = x k +1 − b x k + µx k +1 − b x k − µ · be x k − x k +1 − λ + µ be x k − x k +1 + λ − µ , and e x k +1 − x k +1 + λ e x k +1 − x k +1 − λ = b x k +1 − x k +1 + µ b x k +1 − x k +1 − µ · be x k − x k +1 + λ − µ be x k − x k +1 − λ + µ respectively. Their product coincides with ( E ) with k replaced by k + 1.The three-leg forms of equations ( S
1) and ( S e x k , resp. at e x k +1 , are: x k +1 − e x k + λx k +1 − e x k − λ = be x k − e x k + µ be x k − e x k − µ · b x k − e x k + λ − µ b x k − e x k − λ + µ , and e x k +1 − x k +1 + λ e x k +1 − x k +1 − λ = e x k +1 − be x k + µ e x k +1 − be x k − µ · b x k +1 − e x k +1 − λ + µ b x k +1 − e x k +1 + λ − µ , respectively. The product of these two equations (the second one with k replaced by k − E ).The three-leg forms of equations ( S
1) and ( S be x k , are: be x k − b x k + λ be x k − b x k − λ = be x k − e x k + µ be x k − e x k − µ · be x k − x k +1 + λ − µ be x k − x k +1 − λ + µ , and b x k +1 − be x k + λ b x k +1 − be x k − λ = e x k +1 − be x k + µ e x k +1 − be x k − µ · be x k − x k +1 − λ + µ be x k − x k +1 + λ − µ , respectively. The product of these two equations coincides with ( E ). (cid:4) Theorem 14.
The discrete multi-time Lagrangian 1-form is closed on any solution of thecorner equations ( E )–( E ). Proof.
Spectrality criterium requires to prove that the following quantity is an integralof motion for F µ : P ( x, e x ) = exp( − ∂ Λ( x, e x ; λ ) /∂λ ) = Q Nk =1 (( x k +1 − e x k ) − λ ) Q Nk =1 (( e x k − x k ) − λ )(in the periodic case; in the open-end case the product in the numerator of the latterformula is over 1 ≤ k ≤ N − N Y k =1 e p k = N Y k =1 e x k − x k + λ e x k − x k − λ · N Y k =1 x k +1 − e x k + λx k +1 − e x k − λ , we see that we have to prove the property P ( x, e x ) = P ( b x, be x ) (42) ULTI-TIME LAGRANGIAN 1-FORMS 21 for the quantity for either of the following two quantities: P ( x, e x ) = Q Nk =1 ( x k +1 − e x k + λ ) Q Nk =1 ( e x k − x k − λ ) , (43) P ( x, e x ) = Q Nk =1 ( x k +1 − e x k − λ ) Q Nk =1 ( e x k − x k + λ ) . (44)To prove this, we re-write superposition formulas ( S S
2) in the following equivalentforms:( λ − µ )( be x k − b x k ∓ λ )( x k +1 − e x k ± λ ) = λ ( e x k − b x k ∓ λ ± µ )( x k +1 − be x k ± λ ∓ µ ) , and( λ − µ )( b x k +1 − be x k ± λ )( e x k +1 − x k +1 ∓ λ ) = λ ( e x k +1 − b x k +1 ∓ λ ± µ )( x k +1 − be x k ± λ ∓ µ ) . As a consequence, we arrive at the following superposition formula: b x k +1 − be x k ± λ be x k − b x k ∓ λ = x k +1 − e x k ± λ e x k +1 − x k +1 ∓ λ · e x k +1 − b x k +1 ∓ λ ± µ e x k − b x k ∓ λ ± µ . (45)In the periodic case, the latter formula yields (42) for both quantities (43), (44). In theopen-end case equation (45) holds true for 1 ≤ k ≤ N −
1, and has to be supplementedwith the following two relations: λ be x N − b x N ∓ λ = λ − µ e x N − b x N ∓ λ ± µ , λ e x − x ∓ λ = λ − µ e x − b x ∓ λ ± µ , which are equivalent to equation ( E ) for k = N , resp. to equation ( E ) for k = 1. (cid:4) Theorem 15.
Set L k ( x, p ; λ ) = (cid:18) λ ( e p k + 1) + x k ( e p k −
1) ( λ − x k )( e p k − e p k − λ ( e p k + 1) − x k ( e p k − (cid:19) , and T N ( x, p ; λ ) = L N ( x, p ; λ ) · · · L ( x, p ; λ ) L ( x, p ; λ ) . Then in the periodic case quantity (2 λ ) N Q Nk =1 ( e x k − x k +1 − λ ) Q Nk =1 ( e x k − x k − λ ) , is an eigenvalue of T N ( x, p ; λ ) , while in the open-end case its counterpart, (2 λ ) N Q N − k =1 ( e x k − x k +1 − λ ) Q Nk =1 ( e x k − x k − λ ) , is equal to the (21) -entry of T N ( x, p ; λ ) . Proof.
We can re-write the first equation in (40) as e x k = e p k ( λ + x k )( x k − e x k − − λ ) + ( λ − x k )( x k − e x k − + λ ) e p k ( x k − e x k − − λ ) − ( x k − e x k − + λ )= ( λ ( e p k + 1) + x k ( e p k − e x k − + ( λ − x k )( e p k − e p k − e x k − + λ ( e p k + 1) − x k ( e p k − L k ( x, p ; λ )[ e x k − ] . This is equivalent to L k ( x, p ; λ ) (cid:18)e x k − (cid:19) ∼ (cid:18)e x k (cid:19) . The proportionality coefficient is determined by comparing the second components ofthese two vectors and is equal to( e p k − e x k − − x k ) + λ ( e p k + 1)= ( x k − e x k − + λ ) − e p k ( x k − e x k − − λ ) = − λ x k − e x k − + λ e x k − x k − λ . Thus, L k ( x, p ; λ ) (cid:18)e x k − (cid:19) = 2 λ e x k − − x k − λ e x k − x k − λ (cid:18)e x k (cid:19) . (46)Now the claim in the periodic case follows immediately, with the corresponding eigenvectorof T N ( x, p ; λ ) being (cid:0)e x N , (cid:1) T . In the open-end case, equation (46) holds true for 2 ≤ k ≤ N , and has to be supplemented by the following two relations: L ( x, p ; λ ) (cid:18) (cid:19) = 2 λ e x − x − λ (cid:18)e x (cid:19) , and (cid:0) (cid:1) (cid:18)e x N (cid:19) = 1 . As a consequence, (cid:0) (cid:1) T N ( x, p ; λ ) (cid:18) (cid:19) = (2 λ ) N Q N − k =1 ( e x k − x k +1 − λ ) Q Nk =1 ( e x k − x k − λ ) . (cid:4) B¨acklund transformations for symmetric hyperbolic multiplicativeToda-type system
Our last example constitutes B¨acklund transformations for the symmetric hyperbolicmultiplicative Toda-type system (14) which are given by equations of the type (1): F λ : e p k = sinh( e x k − x k + λ )sinh( e x k − x k − λ ) · sinh( x k − e x k − + λ )sinh( x k − e x k − − λ ) ,e e p k = sinh( e x k − x k + λ )sinh( e x k − x k − λ ) · sinh( x k +1 − e x k + λ )sinh( x k +1 − e x k − λ ) . (47)The corresponding Lagrangian is given byΛ( x, e x ; λ ) = N X k =1 ψ ( e x k − x k ; λ ) − N X k =1 ψ ( x k +1 − e x k ; λ ) , (48)where ψ ( ξ ; λ ) = 12 Z ξ + λξ − λ log sinh( η ) dη. The standard single-time Euler-Lagrange equations are (15) with h = λ . In the open-endcase, all terms with x − e x and with x N +1 − e x N should be omitted both from the equationsof motion (47) and from the Lagrangian (48).To establish commutativity of the maps F λ , F µ , we consider the system of cornerequations:sinh( e x k − x k + λ )sinh( e x k − x k − λ ) · sinh( x k − e x k − + λ )sinh( x k − e x k − − λ ) = sinh( b x k − x k + µ )sinh( b x k − x k − µ ) · sinh( x k − b x k − + µ )sinh( x k − b x k − − µ ) , ( E ) ULTI-TIME LAGRANGIAN 1-FORMS 23 sinh( e x k − x k + λ )sinh( e x k − x k − λ ) · sinh( x k +1 − e x k + λ )sinh( x k +1 − e x k − λ ) = sinh( be x k − e x k + µ )sinh( be x k − e x k − µ ) , · sinh( e x k − be x k − + µ )sinh( e x k − be x k − − µ ) , ( E )sinh( b x k − x k + µ )sinh( b x k − x k − µ ) · sinh( x k +1 − b x k + µ )sinh( x k +1 − b x k − µ ) = sinh( be x k − b x k + λ )sinh( be x k − b x k − λ ) · sinh( b x k − be x k − + λ )sinh( b x k − be x k − − λ ) , ( E )sinh( be x k − b x k + λ )sinh( be x k − b x k − λ ) · sinh( b x k +1 − be x k + λ )sinh( b x k +1 − be x k − λ ) = sinh( be x k − e x k + µ )sinh( be x k − e x k − µ ) · sinh( e x k +1 − be x k + µ )sinh( e x k +1 − be x k − µ ) . ( E ) Theorem 16.
Suppose that the fields x , e x , b x satisfy corner equations ( E ). Define thefields be x by any of the following two superposition formulas, which are equivalent by virtueof ( E ): ( e λ − e µ )( e b x k e e x k + e x k +1 e be x k ) + e µ (1 − e λ )( e b x k e be x k + e x k +1 e e x k )+ e λ ( e µ − e e x k e be x k + e x k +1 e b x k ) = 0 , ( S e λ − e µ )( e b x k +1 e e x k +1 + e x k +1 e be x k ) + e µ (1 − e λ )( e b x k +1 e be x k + e x k +1 e e x k +1 )+ e λ ( e µ − e e x k +1 e be x k + e x k +1 e b x k +1 ) = 0 . ( S Then corner equations ( E )–( E ) are satisfied, as well. Proof.
Observe that equations ( S
1) and ( S
2) are quad-equations with respect to (cid:0) e x k +1 , e e x k , e b x k , e be x k (cid:1) , resp . (cid:0) e x k +1 , e e x k +1 , e b x k +1 , e be x k (cid:1) , namely of the type Q3 δ =0 from the ABS list [2].The three-leg forms of these equations, centered at x k +1 , are:sinh( x k +1 − e x k + λ )sinh( x k +1 − e x k − λ ) = sinh( x k +1 − b x k + µ )sinh( x k +1 − b x k − µ ) · sinh( be x k − x k +1 − λ + µ )sinh( be x k − x k +1 + λ − µ ) , and sinh( e x k +1 − x k +1 + λ )sinh( e x k +1 − x k +1 − λ ) = sinh( b x k +1 − x k +1 + µ )sinh( b x k +1 − x k +1 − µ ) · sinh( be x k − x k +1 + λ − µ )sinh( be x k − x k +1 − λ + µ ) , respectively. Their product coincides with ( E ) with k replaced by k + 1.The three-leg forms of equations ( S
1) and ( S e x k , resp. at e x k +1 , are:sinh( x k +1 − e x k + λ )sinh( x k +1 − e x k − λ ) = sinh( be x k − e x k + µ )sinh( be x k − e x k − µ ) · sinh( b x k − e x k + λ − µ )sinh( b x k − e x k − λ + µ ) , and sinh( e x k +1 − x k +1 + λ )sinh( e x k +1 − x k +1 − λ ) = sinh( e x k +1 − be x k + µ )sinh( e x k +1 − be x k − µ ) · sinh( b x k +1 − e x k +1 − λ + µ )sinh( b x k +1 − e x k +1 + λ − µ ) , respectively. The product of these two equations (the second one with k replaced by k − E ).The three-leg forms of equations ( S
1) and ( S be x k , are:sinh( be x k − b x k + λ )sinh( be x k − b x k − λ ) = sinh( be x k − e x k + µ )sinh( be x k − e x k − µ ) · sinh( be x k − x k +1 + λ − µ )sinh( be x k − x k +1 − λ + µ ) , and sinh( b x k +1 − be x k + λ )sinh( b x k +1 − be x k − λ ) = sinh( e x k +1 − be x k + µ )sinh( e x k +1 − be x k − µ ) · sinh( be x k − x k +1 − λ + µ )sinh( be x k − x k +1 + λ − µ ) , respectively. The product of these two equations coincides with ( E ). (cid:4) Theorem 17.
The discrete multi-time Lagrangian 1-form is closed on any solution of thecorner equations ( E )–( E ). Proof.
With the help of the spectrality criterium, we see that the claim of the theoremis equivalent to P ( x, e x ) = P ( b x, be x ) (49)for the quantity P ( x, e x ) = Q Nk =1 sinh( x k +1 − e x k − λ ) sinh( x k +1 − e x k + λ ) Q Nk =1 sinh( e x k − x k − λ ) sinh( e x k − x k + λ ) , (in the periodic case; in the open-end case the product in the numerator of the latterformula is over 1 ≤ k ≤ N − N Y k =1 e p k = N Y k =1 sinh( e x k − x k + λ )sinh( e x k − x k − λ ) · N Y k =1 sinh( x k +1 − e x k + λ )sinh( x k +1 − e x k − λ ) , we see that we have to prove the property (49) for either of the following two quantities: P ( x, e x ) = Q Nk =1 sinh ( x k +1 − e x k + λ ) Q Nk =1 sinh ( e x k − x k − λ ) , (50) P ( x, e x ) = Q Nk =1 sinh ( x k +1 − e x k − λ ) Q Nk =1 sinh ( e x k − x k + λ ) . (51)To prove this relation, we re-write superposition formulas ( S S
2) in the followingequivalent forms:sinh( be x k − b x k ∓ λ ) sinh( x k +1 − e x k ± λ ) = c sinh( e x k − b x k ∓ λ ± µ ) sinh( x k +1 − be x k ± λ ∓ µ ) , resp.sinh( b x k +1 − be x k ± λ ) sinh( e x k +1 − x k +1 ∓ λ ) = c sinh( e x k +1 − b x k +1 ∓ λ ± µ ) sinh( x k +1 − be x k ± λ ∓ µ ) , where c = ( e λ − / ( e λ − e µ ). As a consequence, we arrive at the following superpositionformula:sinh( b x k +1 − be x k ± λ )sinh( be x k − b x k ∓ λ ) = sinh( x k +1 − e x k ± λ )sinh( e x k +1 − x k +1 ∓ λ ) · sinh( e x k +1 − b x k +1 ∓ λ ± µ )sinh( e x k − b x k ∓ λ ± µ ) . (52)In the periodic case, the latter formula yields (49) for both quantities (50), (51). In theopen-end case, equation (52) holds true for 1 ≤ k ≤ N −
1, and has to be supplementedwith the following two relations: c sinh( be x N − b x N ∓ λ ) = 1sinh( e x N − b x N ∓ λ ± µ ) ,c sinh( e x − x ∓ λ ) = 1sinh( e x − b x ∓ λ ± µ ) , ULTI-TIME LAGRANGIAN 1-FORMS 25 which are equivalent to equation ( E ) for k = N , resp. to equation ( E ) for k = 1. (cid:4) Theorem 18.
Set L k ( x, p ; λ ) = (cid:18) e λ e p k − e λ e x k (1 − e p k ) e λ e − x k ( e p k − e λ − e p k (cid:19) , and T N ( x, p ; λ ) = L N ( x, p ; λ ) · · · L ( x, p ; λ ) L ( x, p ; λ ) . Then in the periodic case the quantity (1 − e λ ) N Q Nk =1 sinh( x k +1 − e x k + λ ) Q Nk =1 sinh( e x k − x k − λ ) , is an eigenvalue of T N ( x, p ; λ ) , while in the open-end case its counterpart, (1 − e λ ) N Q N − k =1 sinh( x k +1 − e x k + λ ) Q Nk =1 sinh( e x k − x k − λ ) , is equal to the (21) -entry of T N ( x, p ; λ ) . Proof.
We can re-write the first equation in (47) as e e x k = e x k ( e λ e p k − e e x k − − e λ e x k (1 − e p k ) e λ ( e p k − e e x k − + e x k ( e λ − e p k ) = L k ( x, p ; λ )[ e e x k − ] . This is equivalent to L k ( x, p ; λ ) (cid:18) e e x k − (cid:19) ∼ (cid:18) e e x k (cid:19) . The proportionality coefficient is determined by comparing the second components ofthese two vectors and is equal to e λ ( e p k − e e x k − − x k + e λ − e p k = (1 − e λ ) sinh( x k − e x k − + λ )sinh( e x k − x k − λ ) . Thus, L k ( x, p ; λ ) (cid:18) e e x k − (cid:19) = (1 − e λ ) sinh( x k − e x k − + λ )sinh( e x k − x k − λ ) (cid:18) e e x k (cid:19) . (53)Now the claim in the periodic case follows immediately, with the corresponding eigenvectorof T N ( x, p ; λ ) being (cid:0) e e x N , (cid:1) T . In the open-end case, equation (53) holds true for 2 ≤ k ≤ N , and has to be supplemented by the following two relations: L ( x, p ; λ ) (cid:18) (cid:19) = 1 − e λ sinh( e x − x − λ ) (cid:18) e e x (cid:19) , and (cid:0) (cid:1) (cid:18) e e x N (cid:19) = 1 . As a consequence, (cid:0) (cid:1) T N ( x, p ; λ ) (cid:18) (cid:19) = (1 − e λ ) N Q N − k =1 sinh( x k +1 − e x k + λ ) Q Nk =1 sinh( e x k − x k − λ ) . (cid:4) Conclusions
In a forthcoming paper, we will present results on the multi-time Lagrangian one-forms for a more general class of B¨acklund transformations, namely for systems of therelativistic Toda type. This will give us an opportunity to present an alternative approachto this theory, namely an approach from the point of view of two-dimensional integrablesystems. Indeed, it is well known since [1] that discrete time relativistic Toda systems arebest interpreted as systems on the regular triangular lattice. A general theory of Toda-type systems on graphs and their relation to quad-graph equations has been developedin [6], [3], [9]. A blend of both approaches, one- and two-dimensional, turns out to befruitful for both ones.Another point we plan to investigate is the quantum counterpart of the results presentedhere. It is well known that B¨acklund transformations for the standard Toda lattice admita natural quantum analog, the Baxter’s Q-operator [16]. The Lagrangian of the B¨acklundtransformation is a quasi-classical limit of the kernel of the integral Q-operator. Thespectrality property of the B¨acklund transformation is the quasi-classical limit of theBaxter’s equation relating the monodromy matrix and the Q-operator [17]. At the sametime, the quantum counterpart of the whole multi-time Lagrangian theory, in particular,of the closure relation, is not yet clear. Also here, the two-dimensional point of view willbe fruitful, as indicated by the work [5] treating a solvable model of statistical mechanics,for which the thermodynamical limit of the partition function is nothing but the actionfunctional of a certain discrete Toda-type model. In this framework, the closure relationobtains its interpretation as the thermodynamical limit of the Z -invariance of the partitionfunction. A blend of both approaches will likely deliver a rather universal and simplepicture.This research is supported by the DFG Collaborative Research Center TRR 109 “Dis-cretization in Geometry and Dynamics”. References [1] V.E. Adler. Legendre transforms on a triangular lattice.
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