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A discrete time relativistic Toda lattice

arXiv:solv-int/9510007v1 23 Oct 1995

Yuri B. SURIS

Centre for Complex Systems and Visualization, Unversity of Bremen, Postfach 330 440, 28334 Bremen, Germany e-mail: suris mathematik.uni-bremen.de

Abstract. Four integrable symplectic maps approximating two Hamiltonian ?ows from the relativistic Toda hierarchy are introduced. They are demostrated to belong to the same hierarchy and to examplify the general scheme for symplectic maps on groups equiped with quadratic Poisson brackets. The initial value problem for the di?erence equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian ?ows are found for all the maps.

1

Introduction

Although the subject of integrable symplectic maps received in the recent years a considerable attention, the order in this area seems still to lack. Given an integrable system of ordinary di?erential equations with such attributes as Lax pair, r–matrix and so on, one would like to construct its di?erence approximation, desirably also with a (discrete–time analog of) Lax pair, r– matrix etc. Recent years brought us several successful examples of such a construction [1–8], but still not the general rools and recipes, not to say about algorithms. Recently there appeared for the ?rst time examples where the Lax matrix of the discrete–time approximation coincides with the Lax matrix of the continuous–time system, so that the discrete–time system belongs to the same integrable hierarchy as the underlying continuous–time one (systems of Calogero–Moser type [7,8]). We want to present here one more example of this type, which can be studied in full (and beautiful) details, – the discrete– time analog of the relativistic Toda lattice [9], see also [10–12]. The paper is organized as follows. In sect. 2,3 we recall some facts about the continuous–time relativistic Toda lattice, its r–matrix structure and the solution in terms of a factorization problem in a matrix group. The most part of these facts is by now well known, but it turned out to be rather di?cult or even impossible to ?nd them in literature in the form suitable for our present purposes. In sect. 4 we introduce the equations of motion of the discrete–time relativistic Toda lattice and discuss their symplectic structure. Sect. 5 contains the Lax pair representation for our system, and in sect. 6 we give the solution of the initial value problem for our system in terms of a factorization problem in a matrix group.

2

Relativistic Toda lattice

The relativistic Toda lattice with the coupling constant g 2 ∈ R is described by the Newtonian equations of motion xk = xk+1 xk ¨ ˙ ˙ g 2 exp(xk+1 ? xk ) g 2 exp(xk ? xk?1 ) ?xk xk?1 ˙ ˙ , 1 + g 2 exp(xk+1 ? xk ) 1 + g 2 exp(xk ? xk?1 ) 1 ≤ k ≤ N. (2.1)

1

with one of the two types of boundary conditions: open–end, x0 ≡ ∞, or periodic, x0 ≡ xN , xN +1 ≡ x1 . It is a known fact (although usually not stressed in the literature) that the equation (2.1) may be put into the Hamiltonian form in two di?erent ways, which lead to two di?erent Hamiltonian functions belonging, remarkably, to one and the same integrable hierarchy. The ?rst way to introduce the variables pk canonically conjugated to xk is: xk ˙ , (2.2) exp(pk ) = 2 exp(x 1+g k+1 ? xk ) which leads to the system xk = exp(pk )(1 + g 2 exp(xk+1 ? xk )), ˙ pk = g 2 exp(xk+1 ? xk + pk ) ? g 2 exp(xk ? xk?1 + pk?1), ˙ a Hamiltonian system with the Hamiltonian function

N

xN +1 ≡ ?∞,

J+ =

k=1

exp(pk )(1 + g 2 exp(xk+1 ? xk )).

(2.3)

The second way to introduce the momenta pk is: exp(pk ) = ? which leads to the system xk = ? exp(?pk )(1 + g 2 exp(xk ? xk?1 )), ˙ pk = ?g 2 exp(xk+1 ? xk ? pk+1 ) + g 2 exp(xk ? xk?1 ? pk ), ˙ a Hamiltonian system with the Hamiltonian function

N

1 + g 2 exp(xk ? xk?1 ) , xk ˙

(2.4)

J? =

k=1

exp(?pk )(1 + g 2 exp(xk ? xk?1 )). 2

(2.5)

The Lax representation and the integrability for the ?ows with the Hamiltonians (2.3),(2.5) are dealt with in the following statement. Introduce two N by N matrices depending on the phase space coordinates xk , pk and (in the periodic case) on the additional parameter λ:

N N

L =

k=1 N

exp(pk )Ekk + λ

k=1 N

Ek+1,k ,

U =

k=1

Ekk ? λ?1

k=1

g 2 exp(xk+1 ? xk + pk )Ek,k+1 .

Here Ejk stands for the matrix whose only nonzero entry on the intersection of the jth row and the kth column is equal to 1. In the periodic case we set EN +1,N = E1,N , EN,N +1 = EN,1 ; in the open–end case we set λ = 1, and EN +1,N = EN,N +1 = 0. Consider also following two matrices: T+ = LU ?1 , T? = U ?1 L. (2.6)

Theorem 1. The ?ow with the Hamiltonian (2.3) is equivalent to the following matrix di?erential equations: ˙ L = LB ? AL, which imply also where

N N

˙ U = UB ? AU, ˙ T? = [T? , B] ,

˙ T+ = [T+ , A] ,

A = B =

(exp(pk ) + g 2 exp(xk ? xk?1 + pk?1 ))Ekk + λ

k=1 N k=1 N

Ek+1,k , Ek+1,k .

(exp(pk ) + g 2 exp(xk+1 ? xk + pk ))Ekk + λ

k=1 k=1

The ?ow with the Hamiltonian (2.5) is equivalent to the following matrix di?erential equations: ˙ L = LD ? CL, which imply also ˙ T+ = [T+ , C] , 3 ˙ U = UD ? CU, ˙ T? = [T? , D] ,

where

N

C = ?λ?1

k=1 N

g 2 exp(xk+1 ? xk ? pk+1 + pk )Ek,k+1, g 2 exp(xk+1 ? xk )Ek,k+1.

k=1

D = ?λ?1

So we see that either of the matrices T± (they are in fact connected by means of a similarity transformation) serves as the Lax matrix for both the ?ows (2.3), (2.5). Note also that the Hamiltonians J± belong to the set of invariant functions of T± , as it is easy to check that J+ = tr(T± ),

?1 J? = tr(T± ).

It is often convenient to use instead of the canonically conjugated variables xk , pk another set of variables ck , dk de?ned as dk = exp(pk ), ck = g 2 exp(xk+1 ? xk + pk ), (2.7)

which satisfy the Poisson brackets {ck , ck+1 } = ?ck ck+1 , {ck , dk+1 } = ?ck dk+1, {ck , dk } = ck dk

(only the non-vanishing brackets are written down). In terms of these variables the Hamiltonians J± are expressed as

N

J+ =

(ck + dk ),

k=1

J? =

ck + d k , k=1 dk dk+1

N

and the corresponding Hamiltonian ?ows read: ck = ck (dk+1 + ck+1 ? dk ? ck?1 ), ˙ for the J+ Hamiltonian, and ck = ck ˙ 1 1 ? , dk dk+1 ˙ dk = dk ck?1 ck ? dk dk+1 dk?1 dk (2.9) ˙ dk = dk (ck ? ck?1) (2.8)

for the J? Hamiltonian. 4

The fundamental matrices L, U have in new coordinates the form

N N

L=

k=1 N

dk Ekk + λ

k=1 N

Ek+1,k ,

(2.10)

U=

k=1

Ekk ? λ?1

k=1

ck Ek,k+1.

(2.11)

For the further reference we give here also the expressions in the variables ck , dk for the matrices involved in the theorem 1:

N N

A(c, d, λ) = B(c, d, λ) =

(dk + ck?1 )Ekk + λ

k=1 N k=1 N

Ek+1,k , Ek+1,k ,

(2.12) (2.13) (2.14) (2.15)

(dk + ck )Ekk + λ

k=1 N k=1

C(c, d, λ) = ?λ?1 D(c, d, λ) = ?λ?1

ck Ek,k+1, k=1 dk+1 ck Ek,k+1. k=1 dk

N

3

Algebraic structure

Here we recall some of the results of [11,12] on the algebraic interpretation of the relativistic Toda lattice as a Hamiltonian system on a particular orbit of a certain Poisson bracket on a matrix group ([11] deals with a gauge transformed Lax matrix, which results in a di?erent Poisson bracket on a group). The results concerning the di?erence equations (part c of the Theorem 2 below) are, to my knowledge, new; however the similar results for less general Poisson brackets can be found in [13,14]. First of all, we de?ne the relevant algebras, groups and decompositions. 1) For the open–end case we set g = gl(N). As a linear space, g may be represented as a direct sum of two subspaces, which serve also as subalgebras: g = g+ ⊕ g? , where g+ (g? ) is a space of all lower triangular (resp. strictly upper triangular) N by N matrices. The corresponding groups are: G = GL(N); G+ (G? ) is a group of all nondegenerate lower triangular N by 5

N matrices (resp. upper triangular N by N matrices with unities on the diagonal). 2) For the periodic case g is a certain twisted loop algebra over gl(N): g = τ (λ) ∈ gl(N)[λ, λ?1] : ?τ (λ)??1 = τ (ωλ) , where ? = diag(1, ω, . . . , ω N ?1), ω = exp(2πi/N). Again, as a linear space g = g+ ⊕ g? , where g+ (g? ) is a subspace and subalgebra consisting of τ (λ) containing only non–negative (resp. only negative) powers of λ. The corresponding groups are: G, the twisted loop group, i.e. the group of GL(N)– valued functions T (λ) of the complex parameter λ, regular in CP 1 \{0, ∞} and satisfying ?T (λ)??1 = T (ωλ); G+ (G? ) is the subgroup consisting of T (λ) regular in the neighbourhood of λ = 0 (resp. regular in the neighbourhood of λ = ∞ and taking the value I in λ = ∞). For both the open–end and periodic cases every τ ∈ g admits a unique decomposition τ = l ? u, where l ∈ g+ , u ∈ g? . We denote l = π+ (τ ), u = π? (τ ). Analogously, for the both cases every T ∈ G from some neighbourhood of the group unity admits a unique factorization T = L U ?1 , where L ∈ G+ , U ∈ G? . We denote the factors as L = Π+ (T ), U = Π? (T ). Recall also that the derivative d?(T ) of the conjugation invariant function ? : G → C is de?ned by the relation tr(d?(T )u) = d d = , ?(T eεu ) ?(eεu T ) dε dε ε=0 ε=0 ?u ∈ g.

Theorem 2. a) Equip G × G with the quadratic Poisson bracket (38)–(41) from Ref. [12], and G with the quadratic Poisson bracket (33) from Ref. [12]. Then the set of pairs of matrices {(L(c, d, λ), U(c, d, λ))} forms a Poisson submanifold in G×G, the set of matrices {T± (c, d, λ)} forms a Poisson submanifold in G, and the maps (L, U) → T+ = LU ?1 and (L, U) → T? = U ?1 L are Poisson maps from G × G into G. b) Let ? : G → C be an invariant function on G. Then the Hamiltonian ?ow on G × G with the Hamiltonian function ?(LU ?1 ) = ?(U ?1 L) has the form ˙ L = Lπ± (d?(T? )) ? π± (d?(T+ ))L, ˙ U = Uπ± (d?(T? )) ? π± (d?(T+ ))U, 6

and the Hamiltonian ?ow on G with the Hamiltonian function ?(T ) has the form ˙ T = [T, π± (d?(T ))] , T = T+ or T? . These ?ows admit the following solution in terms of the factorization problem

?1 etd?(T± (0)) = L± (t) U± (t),

L± (t) ∈ G+ ,

U± (t) ∈ G?

(this problem has solutions at least for su?ciently small t):

?1 L(t) = L?1 (t)L(0)L? (t) = U+ (t)L(0) U? (t), + ?1 U(t) = L?1 (t)U(0)L? (t) = U+ (t)U(0) U? (t), +

so that

?1 T± (t) = L?1 (t)T± (0)L± (t) = U± (t)T± (0) U± (t). ±

c) If f : G → G is the derivative of an invariant function on G, then the system of di?erence equations (t ∈ hZ) L(t + h) = Π?1 f (T+ (t)) L(t)Π± f (T? (t)) , ± U(t + h) = Π?1 f (T+ (t)) U(t)Π± f (T? (t)) ± de?nes a Poisson map G × G → G × G, and the di?erence equation T (t + h) = Π?1 f (T (t)) T (t)Π± f (T (t)) , ± T = T+ or T?

de?nes a Poisson map G → G. These di?erence equations admit following solution in terms of the factorization problem

?1 f n (T± (0)) = L± (nh) U± (nh),

L± (nh) ∈ G+ ,

U± (nh) ∈ G?

(this problem has solutions for a given n at least if f (T± (0)) is su?ciently close to the group unity I):

?1 L(nh) = L?1 (nh)L(0)L? (nh) = U+ (nh)L(0) U? (nh), + ?1 U(nh) = L?1 (nh)U(0)L? (nh) = U+ (nh)U(0) U? (nh), +

so that

?1 T± (nh) = L?1 (nh)T± (0)L± (nh) = U± (nh)T± (0) U± (nh). ±

7

d) The solutions of the di?erence equations of the part c) are interpolated by the ?ows of the part b) with the Hamiltonian function ?(T ) de?ned by d?(T ) = h?1 log(f (T )). The part b) of the last theorem explains, in particular, the theorem 1, as for J+ (T ) = tr(T ), J? (T ) = tr(T ?1 ) we have dJ+ (T ) = T, and it is not hard to check that A = π+ (T? ), B = π+ (T+ ),

?1 C = π? (?T? ), ?1 D = π? (?T+ ).

dJ? (T ) = ?T ?1 ,

4

Discrete equations of motion and symplectic structure

We propose two systems of di?erence equations as discrete–time versions of the relativistic Toda latti ce. The equations of motion of the ?rst system read: exp(xk (t + h) ? xk (t)) ? 1 = exp(xk (t) ? xk (t ? h)) ? 1 1 + g 2 exp(xk+1 (t) ? xk (t)) 1 + g 2 exp(xk+1 (t ? h) ? xk (t)) 1 + g 2 exp(xk (t) ? xk?1 (t + h)) 1 + g 2 exp(xk (t) ? xk?1 (t)) , 1 ≤ k ≤ N. (4.1) The equations of motion of the second one read: exp(?xk (t + h) + xk (t)) ? 1 = exp(?xk (t) + xk (t ? h)) ? 1 1 + g 2 exp(xk+1 (t + h) ? xk (t)) 1 + g 2 exp(xk+1 (t) ? xk (t)) 1 + g 2 exp(xk (t) ? xk?1 (t)) 1 + g 2 exp(xk (t) ? xk?1 (t ? h)) , 1 ≤ k ≤ N, (4.2)

8

The both systems are considered under one of the two types of boundary conditions, either open–end: x0 (t) ≡ ∞, or periodic: x0 (t) ≡ xN (t), xN +1 (t) ≡ x1 (t) for all t ∈ hZ. xN +1 (t) ≡ ?∞ for all t ∈ hZ,

The functions xk (t) in (4.1), (4.2) are supposed to be de?ned for t ∈ hZ, h > 0 (note that the equations themselves do not depend on h explicitly). However, it is convenient to consider (4.1), (4.2) as ?nite–di?erence approximations to (2.1), and in this context xk (t) are to be considered as the smooth functions xk ¨ of t ∈ R. Then the left–hand sides are expanded in powers of h as 1 + h + xk ˙ O(h2 ), and the right–hand sides as 1 + h xk+1 ˙ g 2 exp(xk+1 ? xk ) g 2 exp(xk ? xk?1 ) + O(h2 ), ? xk?1 ˙ 2 exp(x 2 exp(x ? x 1+g 1+g k+1 ? xk ) k k?1 )

so that we recover in the continuous limit the equation (2.1). Of course, the both systems are closely related, namely by means of the time reversion operation. (Note that the underlying continuous–time system (2.1) is invariant with respect to this operation). We would like to remark that the equations (4.1), (4.2) admit a simple non–relativistic limit: set xk (t) = qk (t) + ct in (4.1) (resp. xk (t) = qk (t) ? ct in (4.2)) with c > 0 playing the role of the speed of light; then in the limit c → ∞ the both equations tend to one and the same system: exp(qk (t+ h) ?2qk (t) + qk (t?h)) = 1 + g 2 exp(qk+1 (t) ? qk (t)) , 1 + g 2 exp(qk (t) ? qk?1 (t)) 1 ≤ k ≤ N,

i.e. to the equations of motion of the discrete–time Toda lattice from [3]. In the following we will adopt the notations from [7,8]: if xk = xk (t), then xk = xk (t + h), xk = xk (t ? h), so that (4.1), (4.2) take the form (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) exp(xk ? xk ) ? 1 = , exp(xk ? xk ) ? 1 (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) 1 ≤ k ≤ N; (4.3) 9

exp(?xk + xk ) ? 1 (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) = , exp(?xk + xk ) ? 1 (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 ))

1 ≤ k ≤ N,

(4.4) respectively. Obviously, (4.3), (4.4) are systems of nonlinear algebraic equations for xk , 1 ≤ k ≤ N of the form Fk (x, x, x ) = 0, 1 ≤ k ≤ N. (4.5)

It will be convenient to discuss the solvability of these systems simultaneously with the symplectic structure. The general recipe to derive the invariant symplectic structure for a discrete evolutionary equation of the type (4.5) was given in [1,15]: represent the equations of motion in the Lagrangian form ? Λ(x, x) + Λ(x, x ) /?xk = 0, then the momenta pk canonically conjugate to xk are de?ned as pk = ?Λ(x, x )/?xk . The map (x, p) → (x, p) preserves the standard symplectic 2–form dpk . Note that for the x we have the equation pk = ??Λ(x, x)/?xk , and then p may be computed as pk = ?Λ(x, x)/? xk . (4.9)

N k=1

(4.6)

(4.7) dxk ∧ (4.8)

Remark. Note that if the system (4.5) may be represented in the Lagrangian form (4.6) with the Lagrangian function Λ(x, x ), then the system Fk (x , x, x) = 0, 1 ≤ k ≤ N may be represented in the Lagrangian form with the Lagrangian function ?Λ(x , x). Now it is easy to see from (4.7)–(4.9) 10

that the symplectic maps (x, p) → (x, p) corresponding to these two systems are mutually inverse. This is just the case for (4.1), (4.2), up to minor modi?cations due to the ”arti?cial” parameter h introduced in the de?nitions of momenta below. We prefer, however, to consider all the maps separately, and will return to the above–mentioned circumstance at the end of the section 6. The Lagrangian functions for the equations (4.3), (4.4) are expressed with the help of two functions φ1 (ξ), φ2 (ξ) which are de?ned by φ′1 (ξ) = log exp(ξ) ? 1 , h φ′2 (ξ) = log(1 + g 2 exp(ξ)).

It turns out that, just as in the continuous–time case, the Lagrangian functions and hence the momenta pk may be choosen in two di?erent ways. They lead to two pairs of di?erent symplectic maps (one pair for each of (4.1), (4.2)) belonging, remarkably, to one and the same integrable hierarchy. Still more remarkable, however, is that this hierarchy is the same as in the continuous– time case (see sections 5, 6)!

4.1

System (4.3), the ?rst choice of momenta

N N N

It is easy to see that (4.3) is equivalent to (4.6) with Λ(x, x ) =

k=1

φ1 (xk ? xk ) +

k=1

φ2 (xk ? xk?1 ) ?

k=1

φ2 (xk ? xk?1 ).

Then the de?nition (4.7) of momenta pk takes the form (exp(xk ? xk ) ? 1) exp(pk ) = h(1 + g 2 exp(xk+1 ? xk )) , (4.10)

and the equation (4.8) for x takes the form exp(pk ) = (exp(xk ? xk ) ? 1) (1 + g 2 exp(xk ? xk?1 )) , h(1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) (4.11)

(formulas (4.10), (4.11) are to be compared with (2.2)). The equation (4.9) for p together with (4.11) implies : exp(pk ) = exp(pk ) (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) . (4.12) (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) 11

To solve (4.3) for x is now equivalent to solving (4.11) for x. It may be directly veri?ed that this last equation may be rewritten as: h exp(pk ) 1 + g 2 exp(xk+1 ? xk ) = 1 ? exp(xk ? xk ) 1 ? exp(xk?1 ? xk?1 ) . 1 + g 2 exp(xk ? xk?1 )

= 1 + h exp(pk ) + g 2 exp(xk ? xk?1 )

In terms of the coordinates ck , dk this means: if we denote

ak

= h exp(pk )

1 + g 2 exp(xk+1 ? xk ) , 1 ? exp(xk ? xk ) hck?1

ak?1

(4.13)

then

ak

= 1 + hdk +

,

1 ≤ k ≤ N.

(4.14)

Suppose for a moment that these recurrent relations de?ne ak as certain functions of ck , dk . Then, according to (4.13), this means that the equations for x are solved. Indeed, it follows from (4.13) that exp(xk ? xk ) = + hck ak ? hdk

ak

To express now the resulting map in terms of the variables ck , dk alone, we derive from (4.13) 1 + g 2 exp(xk ? xk?1 ) = ak ? hdk , 1 + g 2 exp(xk ? xk?1 ) which together with the previous expression and (4.12) implies: ck = ck

ak+1

+ hck+1 , ak + hck

dk = dk

ak+1 ? hdk+1 ak

? hdk

.

(4.15)

Returning to the relations (4.14), we note that in the open–end case c0 = 0, hence we obtain from (4.11) the following ?nite continued fractions expressions for ak ’s: a1 = 1 + hd1 ; hc1 a2 = 1 + hd2 + ; 1 + hd1 12

···

aN

= 1 + hdN + 1 + hdN ?1 +

hcN ?1 hcN ?2 1 + hdN ?2 + . . . + hc1 1 + hd1

.

Obviously, we have:

ak

= 1 + h(dk + ck?1 ) + O(h2 ).

(4.16)

In the periodic case the recurrent relations (4.14) uniquely de?ne the ak ’s as the N-periodic in?nite continued fractions. It can be proved that these continued fractions converge and their values satisfy (4.16). Because of (4.16) it is obvious that the map (4.15) is a di?erence approximation to the ?ow (2.8).

4.2

System (4.3), the second choice of momenta

N N N

The equation (4.3) can be treated also as a Lagrangian equation (4.6) with Λ(x, x ) = ?

k=1

φ1 (xk ? xk ) ?

k=1

φ2 (xk ? xk?1 ) +

k=1

φ2 (xk ? xk?1 ).

Then the de?nition (4.7) takes the form: exp(pk ) = h(1 + g exp(xk ? xk?1 )) . (1 ? exp(xk ? xk )) (1 + g 2 exp(xk+1 ? xk ))

2

(1 + g 2 exp(xk+1 ? xk ))

(4.17)

and the equation (4.8) for x takes the form: exp(pk ) = h(1 + g 2 exp(xk ? xk?1 )) , 1 ? exp(xk ? xk ) (4.18)

(Equations (4.17), (4.18) are to be compared with (2.4)). Equation (4.9) together with (4.18) implies:: exp(pk ) = exp(pk ) (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) . (4.19) (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) 13

As before, to solve (4.3) for x is equivalent to solving (4.18) for x. The formula (4.18) may be rewritten as: exp(xk ? xk + pk ) = exp(pk ) ? h ? hg 2 exp(xk ? xk?1 ). In terms of the coordinates ck , dk this means: if we denote

dk

= g 2 exp(xk+1 ? xk ), 1 ≤ k ≤ N.

(4.20) (4.21)

then

ck

dk

= dk ? h ? hdk?1 ,

Supposing that thes relations de?ne dk ’s as certain functions on ck , dk , we see that (4.20) allows to solve the equation for x. In order to express the resulting map in terms of the variables ck , dk alone, we derive from (4.18), (4.20) the relations exp(xk ? xk ) = dk dk , ck 1 + g 2 exp(xk+1 ? xk ) hdk =1? . 2 exp(x 1+g dk+1 k+1 ? xk )

We use them and (4.21) to derive from (4.19): ck = ck+1 ck + hdk , ck+1 + hdk+1 dk = dk+1 dk ? hdk?1 . dk+1 ? hdk (4.22)

Returning to (4.21), we immediately obtain in the open–end case d0 = 0, and the ?nite continued fraction expressions for dk ’s follow:

d1

=

c1 , d1 ? h c2

dN ?1

, hc1 d2 ? h ? d1 ? h ··· cN ?1 = hcN ?2 dN ?1 ? h ?

d2

=

.

dN ?2 ? h ? . .

. ? hc1 d1 ? h

14

Obviously, we have

dk

=

ck + O(h), dk

1 ≤ k ≤ N.

(4.23)

In the periodic case the recurrent relations (4.21) uniquely de?ne dk ’s as the N–periodic in?nite continued fractions, that again converge and whose values satisfy the relation (4.22). In view of (4.22) it is obvious that the map (4.21) is a di?erence approximation to the ?ow (2.9).

4.3

System (4.4), the ?rst choice of momenta

N N N

Turning now to the system (4.2), we see that it is equivalent to (4.6) with Λ(x, x ) = ?

k=1

φ1 (?xk + xk ) ?

k=1

φ2 (xk ? xk?1 ) +

k=1

φ2 (xk ? xk?1 ).

Then the de?nition (4.7) of momenta pk takes the form (1 + g 2 exp(xk ? xk?1 )) , h(1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) (1 ? exp(?xk + xk ))

exp(pk ) =

(4.24)

and the equation (4.8) for x takes the form exp(pk ) = 1 ? exp(?xk + xk ) , h(1 + g 2 exp(xk+1 ? xk )) (4.25)

(formulas (4.24), (4.25) are to be compared with (2.2)). The equation (4.9) for p together with (4.25) implies : exp(pk ) = exp(pk ) (1 + g 2 exp(xk ? xk?1 )) (1 + g 2 exp(xk+1 ? xk )) . (4.26) (1 + g 2 exp(xk ? xk?1 )) (1 + g 2 exp(xk+1 ? xk ))

To solve (4.4) for x is now equivalent to solving (4.25) for x. This last equation may be rewritten as: exp(?xk + xk ) = 1 ? h exp(pk ) ? 15 hg 2 exp(xk+1 ? xk + pk ) . exp(?xk+1 + xk+1 )

In terms of the coordinates ck , dk this means: if we denote βk = exp(?xk + xk ), then βk = 1 ? hdk ? hck , βk+1 1 ≤ k ≤ N. (4.27) (4.28)

Relations (4.28) imply that βk as certain functions (continued fractions, see below) on ck , dk . According to (4.27), this means that the equations for x are solved. To express now the resulting map in terms of the variables ck , dk alone, we derive from (4.25),(4.27), and (4.28) the equality hck 1 + g 2 exp(xk+1 ? xk ) = βk + hdk , = 1? 2 exp(x 1+g βk+1 k+1 ? xk ) so that (4.26) implies ck = ck βk ? hck?1 , βk+1 ? hck dk = dk βk?1 + hdk?1 . βk + hdk (4.29)

As for the continued fractions expressions for βk ’s, we have in the open– end case: cN = 0, hence βN = 1 ? hdN ; hcN ?1 βN ?1 = 1 ? hdN ?1 ? ; 1 ? hdN ··· hc1 β1 = 1 ? hd1 ? . hc2 1 ? hd2 ? 1 ? hd3 ? . . . Obviously, βk = 1 ? h(ck + dk ) + O(h2 ). (4.30) In the periodic case the recurrent relations (4.28) uniquely de?ne the βk ’s as the N-periodic in?nite continued fractions. It can be proved that these continued fractions converge and their values satisfy (4.30). Because of (4.30) it is obvious that the map (4.29) is a di?erence approximation to the ?ow (2.8). 16 ? hcN ?1 1 ? hdN

4.4

System (4.4), the second choice of momenta

N N N

The equation (4.4) can be treated also as a Lagrangian equation (4.6) with Λ(x, x ) =

k=1

φ1 (?xk + xk ) +

k=1

φ2 (xk ? xk?1 ) ?

k=1

φ2 (xk ? xk?1 ).

Then the de?nition (4.7) takes the form: h(1 + g 2 exp(xk ? xk?1 )) exp(pk ) = exp(?xk + xk ) ? 1 , (4.31)

and the equation (4.8) for x takes the form: exp(pk ) = h(1 + g 2 exp(xk ? xk?1 )) (1 + g 2 exp(xk+1 ? xk )) . (exp(?xk + xk ) ? 1) (1 + g 2 exp(xk+1 ? xk )) (4.32)

(Equations (4.32), (4.33) are to be compared with (2.4)). Equation (4.9) together with (4.32) implies:: (1 + g 2 exp(xk ? xk?1 )) (1 + g 2 exp(xk+1 ? xk )) . (4.33) exp(pk ) = exp(pk ) (1 + g 2 exp(xk ? xk?1 )) (1 + g 2 exp(xk+1 ? xk )) As before, to solve (4.4) for x is equivalent to solving (4.32) for x. The formula (4.32) may be rewritten as: h(1 + g 2 exp(xk ? xk?1 )) = 1 ? exp(xk ? xk ) = h + exp(pk ) + g 2 exp(xk+1 ? xk + pk ) 1 ? exp(xk+1 ? xk+1 ) . 1 + g 2 exp(xk+1 ? xk )

In terms of the coordinates ck , dk this means: if we denote hγk = ck then 1 ? exp(xk+1 ? xk+1 ) , 1 + g 2 exp(xk+1 ? xk ) 1 ≤ k ≤ N. (4.34)

ck?1 = h + dk + hγk , γk?1 17

(4.35)

This de?nes γk ’s as certain functions on ck , dk , which, according to (4.34), allows to solve the equation for x. It is not hard to derive from (4.34) the relations hγk ck , exp(xk+1 ? xk+1 ) = hγk 1+ dk 1? 1 + g 2 exp(xk+1 ? xk ) hγk =1+ . 2 exp(x 1+g dk k+1 ? xk )

We use them and (4.33) to express the resulting map in terms of the variables ck , dk alone: ck = ck?1 ck ? hγk , ck?1 ? hγk?1 dk = dk?1 dk + hγk . dk?1 + hγk?1 (4.36)

The expressions for γk ’s in the open–end case follow from (4.35) and γN = 0: γN ?1 = γN ?2 = cN ?1 , dN + h cN ?2 hcN ?1 dN ?1 + h + dN + h ··· c1 hc2 ,

γ1 = d2 + h +

.

d3 + h + . . . + hcN ?1 dN + h Obviously, we have γk = ck + O(h), dk+1 1 ≤ k ≤ N. (4.37)

In the periodic case the recurrent relations (4.35) uniquely de?ne γk ’s as the N–periodic in?nite continued fractions, that again converge and whose values satisfy the relation (4.37). In view of (4.37) it is obvious that the map (4.36) is a di?erence approximation to the ?ow (2.9).

18

5

Lax representations

We show in this section that our discrete–time systems admit the discrete analogs of Lax representations with the same matrices L, U, T± (see (2.10), (2.11), (2.6)) as the continuous–time relativistic Toda lattice. In the following theorems we adopt the conventions described before the formula (2.6). The dependence of all the matrices below on the discrete time t ∈ hZ is supposed to appear through the dependence of ck , dk on t. Theorem 3. The symplectic map de?ned by (4.14), (4.15) admits a representation in the form of matrix equations A(t)L(t + h) = L(t)B(t), so that T+ (t + h) = A?1 (t)T+ (t)A(t), with the matrices

N N

A(t)U(t + h) = U(t)B(t),

(5.1)

T? (t + h) = B?1 (t)T? (t)B(t)

(5.2)

A(c, d, λ) =

k=1 N

ak Ekk

+ hλ

k=1 N

Ek+1,k ,

(5.3)

B(c, d, λ) =

k=1

bk Ekk

+ hλ

k=1

Ek+1,k ,

(5.4)

where ak ’s are de?ned by (4.14), and bk ’s by (5.7) or (5.8) below. Proof. It is staightforward to check that the matrix equations (5.1) are equivalent to the following ones:

a k dk a k ck

= dk b k ,

hdk + ak+1 = hdk+1 + bk , hck ? ak+1 = hck+1 ? bk+1.

(5.5) (5.6)

= ck bk+1 ,

Now it would be not hard to check directly that the equations (5.5), (5.6) are satis?ed with the following identi?cations: (4.11), (4.12) for dk , dk ; ck = g 2 exp(xk+1 ? xk )dk and similarly for ck ; ,

ak

= exp(xk ? xk )

(1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) , (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) 19

as in (4.13), and bk = exp(xk ? xk ) We prefer, however, to work directly with (5.5), (5.6), not using the expressions in terms of the xk variables. Namely, (5.5) is equivalent to dk = dk

ak+1 ? hdk+1 ak

? hdk

,

bk

= ak

ak+1

? hdk+1 , ak ? hdk

(5.7)

and (5.6) is equivalent to ck = ck

ak+1

+ hck+1 , ak + hck

bk+1

= ak

ak+1 + hck+1 ak

+ hck

.

(5.8)

The ?rst equations in (5.7), (5.8) coincide with (4.15), and the compatibility of the second ones is equivalent to

ak (ak+1

? hdk+1 ) ak?1 (ak ? hdk ) = , ak + hck ak?1 + hck?1

which is a direct consequence of (4.14). This completes the proof. We would like to mention here that the matrices (5.3), (5.4) when compared with (2.12), (2.13) satisfy A(c, d, λ) = I + hA(c, d, λ) + O(h2 ), B(c, d, λ) = I + hB(c, d, λ) + O(h2 ),

as it follows from (4.16). Theorem 4. The symplectic map de?ned by (4.21), (4.22) admits a representation in the form of matrix equations L(t + h)D(t) = C(t)L(t), so that T+ (t + h) = C(t)T+ (t)C?1 (t), with the matrices

N N

U(t + h)D(t) = C(t)U(t),

(5.9)

T? (t + h) = D(t)T? (t)D?1 (t)

(5.10)

C(c, d, λ) =

k=1 N

Ekk + hλ?1

k=1 N

ck Ek,k+1 ,

(5.11)

D(c, d, λ) =

k=1

Ekk + hλ?1

k=1

dk Ek,k+1 ,

(5.12)

20

where dk ’s are de?ned by (4.21), and ck ’s by (5.15) or (5.16) below. Proof. It is staightforward to check that the matrix equations (5.9) are equivalent to the following ones: dk dk = ck dk+1, ck dk+1 = ck ck+1 , dk + hdk?1 = dk + hck , ck ? hdk = ck ? hck . (5.13) (5.14)

Now it could be checked by means of direct calculation that the equations (5.13), (5.14) are satis?ed with the following identi?cations: (4.18), (4.19) for dk , dk ; ck = g 2 exp(xk+1 ? xk )dk and similarly for ck ; dk = g 2 exp(xk+1 ? xk ) as in (4.2), and

ck

= g 2 exp(xk+1 ? xk )

(1 ? exp(xk+1 ? xk+1 )) (1 + g 2 exp(xk ? xk?1 )) , (1 ? exp(xk + ?xk )) (1 + g 2 exp(xk+1 ? xk ))

. Working, however, again directly with (5.13), (5.14), without turning to the representation through the xk variables, we note that (5.13) is equivalent to dk ? hdk?1 dk ? hdk?1 dk = dk+1 , ck = dk , (5.15) dk+1 ? hdk dk+1 ? hdk and (5.14) is equivalent to ck = ck+1 ck + hdk , ck+1 + hdk+1

ck

= dk+1

ck + hdk . ck+1 + hdk+1

(5.16)

The ?rst equations in (5.15), (5.16) coincide with (4.22), and the compatibility of the second ones is equivalent to

dk+1 (dk+1 dk (dk ? hdk?1 ) ? hdk ) = , ck+1 + hdk+1 ck + hdk

which is a direct consequence of (4.21). The proof is complete. Note that the matrices (5.11), (5.12) when compared with (2.14), (2.15) satisfy C(c, d, λ) = I ? hC(c, d, λ) + O(h2 ), as it follows from (4.23). 21 D(c, d, λ) = I ? hD(c, d, λ) + O(h2),

Theorem 5. The symplectic map de?ned by (4.28), (4.29) admits a representation in the form of matrix equations L(t + h)B(t) = A(t)L(t), so that T+ (t + h) = A(t)T+ (t)A?1 (t), with the matrices

N N

U(t + h)B(t) = A(t)U(t),

(5.17)

T? (t + h) = B(t)T? (t)B?1 (t)

(5.18)

A(c, d, λ) =

k=1 N

αk Ekk ? hλ

k=1 N

Ek+1,k ,

(5.19)

B(c, d, λ) =

k=1

βk Ekk ? hλ

k=1

Ek+1,k ,

(5.20)

where βk ’s are de?ned by (4.28), and αk ’s by (5.23) or (5.24) below. Proof. It is staightforward to check that the matrix equations (5.17) are equivalent to the following ones: dk βk = αk dk , ck βk+1 = αk ck , hdk ? βk?1 = hdk?1 ? αk , hck + βk = hck?1 + αk . (5.21) (5.22)

Now it would be not hard to check directly that the equations (5.5), (5.6) are satis?ed with the following identi?cations: (4.25), (4.26) for dk , dk ; ck = g 2 exp(xk+1 ?xk )dk and similarly for ck ; βk = exp(?xk +xk ) as in (4.27), and αk = exp(?xk + xk ) (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 )) . (1 + g 2 exp(xk+1 ? xk )) (1 + g 2 exp(xk ? xk?1 ))

We prefer, however, to work directly with (5.21), (5.22), not using the expressions in terms of the xk variables. Namely, (5.21) is equivalent to dk = dk βk?1 + hdk?1 , βk + hdk αk = βk βk?1 + hdk?1 , βk + hdk (5.23)

and (5.22) is equivalent to ck = ck βk ? hck?1 , βk+1 ? hck αk = βk+1 22 βk ? hck?1 . βk+1 ? hck (5.24)

The ?rst equations in (5.23), (5.24) coincide with (4.29), and the compatibility of the second ones is equivalent to βk (βk?1 + hdk?1) βk+1 (βk + hdk ) = , βk+1 ? hck βk ? hck?1 which is a direct consequence of (4.28). This completes the proof. We would like to mention here that the matrices (5.19), (5.20) when compared with (2.12), (2.13) satisfy A(c, d, λ) = I ? hA(c, d, λ) + O(h2), B(c, d, λ) = I ? hB(c, d, λ) + O(h2 ),

as it follows from (4.30). Theorem 6. The symplectic map de?ned by (4.35), (4.36) admits a representation in the form of matrix equations C(t)L(t + h) = L(t)D(t), so that T+ (t + h) = C ?1 (t)T+ (t) C(t), with the matrices

N N

C(t)U(t + h) = U(t)D(t),

(5.25)

T? (t + h) = D ?1 (t)T? (t)D(t)

(5.26)

C(c, d, λ) =

k=1 N

Ekk ? hλ?1

k=1 N

γk Ek,k+1 ,

(5.27)

D(c, d, λ) =

k=1

Ekk ? hλ?1

k=1

δk Ek,k+1 ,

(5.28)

where γk ’s are de?ned by (4.35), and δk ’s by (5.31) or (5.32) below. Proof. It is staightforward to check that the matrix equations (5.25) are equivalent to the following ones: γk?1 dk = dk?1 δk?1 , γk?1ck = ck?1 δk , dk ? hγk = dk ? hδk?1 , ck + hγk = ck + hδk . (5.29) (5.30)

Now it could be checked by means of direct calculation that the equations (5.29), (5.30) are satis?ed with the following identi?cations: (4.32), (4.33) for 23

dk , dk ; ck = g 2 exp(xk+1 ? xk )dk and similarly for ck ; δk = g 2 exp(xk+1 ? xk ), and γk = g 2 exp(xk+1 ? xk ) . Working, however, again directly with (5.29), (5.30), without turning to the representation through the xk variables, we note that (5.29) is equivalent to dk + hγk dk + hγk dk = dk?1 , δk?1 = γk?1 , (5.31) dk?1 + hγk?1 dk?1 + hγk?1 and (5.30) is equivalent to ck = ck?1 ck ? hγk , ck?1 ? hγk?1 δk = γk?1 ck ? hγk . ck?1 ? hγk?1 (5.32) (exp(?xk+1 + xk+1 ) ? 1) (1 + g 2 exp(xk ? xk?1 )) , (exp(?xk + xk ) ? 1) (1 + g 2 exp(xk+1 ? xk ))

The ?rst equations in (5.31), (5.32) coincide with (4.36), and the compatibility of the second ones is equivalent to γk?1(dk + hγk ) γk (dk+1 + hγk+1 ) = , ck ? hγk ck?1 ? hγk?1 which is a direct consequence of (4.35). The proof is complete. Note that the matrices (5.27), (5.28) when compared with (2.14), (2.15) satisfy C(c, d, λ) = I + hC(c, d, λ) + O(h2 ), as it follows from (4.37). D(c, d, λ) = I + hD(c, d, λ) + O(h2 ),

6

Factorization problems and interpolating Hamiltonians

It is very remarkable that the matrices A, B, C, D and A, B, C, D from the previous section may be identi?ed with the certain factors Π± (f (T± )), as in the Theorem 2.

24

Theorem 7. There hold following relations: A(c, d, λ) = Π+ (I + hT+ (c, d, λ)) , B(c, d, λ) = Π+ (I + hT? (c, d, λ)) ,

?1 C(c, d, λ) = Π?1 I ? hT+ (c, d, λ) , ? ?1 D(c, d, λ) = Π?1 I ? hT? (c, d, λ) . ?

(6.1) (6.2) (6.3) (6.4)

Proof. De?ne following two matrices:

N N

Q? (c, d, λ) =

k=1 N

Ekk ? λ?1 ck

dk

k=1 N

ck

ak

Ek,k+1 ∈ G? ,

Q+ (c, d, λ) =

k=1

Ekk + λ

k=1

Ek+1,k ∈ G+ .

Note now that the recurrent relation (4.14) is just equivalent to the matrix equality U(c, d, λ) + hL(c, d, λ) = A(c, d, λ)Q? (c, d, λ), (6.5) and the recurrent relation (4.21) is just equivalent to the matrix equality L(c, d, λ) ? hU(c, d, λ) = Q+ (c, d, λ)D(c, d, λ). Multiplying (6.5) from the right by U ?1 , we obtain: I + hT+ = AQ? U ?1 , and since Q? U ?1 ∈ G? , we obtain (6.1). From the previous equation with the help of (5.1) we derive also I + hT? = U ?1 AQ? = BU ?1 Q? , which proves (6.2) in view of U ?1 Q? ∈ G? . Next, multiplying (6.6) from the left by L?1 , we obtain:

?1 I ? hT? = L?1 Q+ D,

(6.6)

which just implies (6.4) because of L?1 Q+ ∈ G+ . Finally, from the previous equation we derive with the help of (5.9):

?1 I ? hT+ = Q+ DL?1 = Q+ L?1 C,

25

which means the validity of (6.3) because of Q+ L?1 ∈ G+ . The theorem is proved. Theorem 8. There hold following relations: A(c, d, λ) = Π?1 (I ? hT+ (c, d, λ))?1 , + B(c, d, λ) = Π?1 (I ? hT? (c, d, λ))?1 , +

?1 C(c, d, λ) = Π? (I + hT+ (c, d, λ))?1 , ?1 D(c, d, λ) = Π? (I + hT? (c, d, λ))?1 .

(6.7) (6.8) (6.9) (6.10)

Proof. De?ne following two matrices:

N

P? (c, d, λ) =

k=1 N

Ekk ? λ?1

ck Ek,k+1 ∈ G? , k=1 βk+1

N

P+ (c, d, λ) =

N ck?1 Ekk + λ Ek+1,k ∈ G+ . k=1 k=1 γk?1

Note now that the recurrent relation (4.28) is just equivalent to the matrix equality U(c, d, λ) ? hL(c, d, λ) = P? (c, d, λ)B(c, d, λ), (6.11) and the recurrent relation (4.35) is just equivalent to the matrix equality L(c, d, λ) + hU(c, d, λ) = C(c, d, λ)P+ (c, d, λ). Multiplying (6.11) from the left by U ?1 , we obtain: I ? hT? = U ?1 P? B, and since U ?1 P? ∈ G? , we obtain (6.8). From the previous equation with the help of (5.17) we derive also I ? hT+ = P? BU ?1 = P? U ?1 A, which proves (6.7) in view of P? U ?1 ∈ G? . Next, multiplying (6.12) from the right by L?1 , we obtain:

?1 I + hT+ = CP+ L?1 ,

(6.12)

26

which just implies (6.9) because of P+ L?1 ∈ G+ . Finally, from the previous equation we derive with the help of (5.25):

?1 I + hT? = L?1 CP+ = D L?1 P+ ,

which means the validity of (6.10) because of L?1 P+ ∈ G+ . The theorem is proved. Substitute now the expressions (6.1)–(6.4) into the discrete Lax equations (5.1), (5.2), (5.9), (5.10), and (6.7)–(6.10) into (5.17), (5.18), (5.25), (5.26). One recognizes immediately the di?erence equations from the part c) of the Theorem 2. This gives us the solution of the initial v alue problem for the dynamical system (4.14), (4.15) in terms of the factorization of the matrices ((I + hT± (0)))n , the solution of the initial value problem for the dynamical system (4.21), (4.22) in terms of the factorization of the matrices

?1 (I ? hT± (0)) n

.

, the solution of the initial value problem for the dynamical system (4.28), (4.29) in terms of the factorization of the matrices (I ? hT± (0))?1

n

.

, and the solution of the initial value problem for the dynamical system (4.35), (4.36) in terms of the factorization of the matrices

?1 (I + hT± (0))?1 n

.

, The part d) of the Theorem 2 may be formulated in our case in the following lines. Corollary. The interpolating Hamiltonian for the map ((4.14), (4.15) is given by ?+ (T ) = tr(Φ+ (T )) = J+ (T )+O(h), where Φ+ (ξ) = h?1

0 ξ

dη log(1+hη), η

27

for the map (4.21), (4.22) – by

∞

?? (T ) = tr(Φ? (T )) = J? (T )+O(h), for the map (4.28), (4.29) – by ψ+ (T ) = tr(Ψ+ (T )) = J+ (T )+O(h), and for the map (4.35), (4.36) – by

where

Φ? (ξ) = h?1

ξ

1 dη log( ), η 1 ? hη ?1

where

Ψ+ (ξ) = h?1

0

ξ

dη 1 log , η 1 ? hη

∞

ψ? (T ) = tr(Ψ? (T )) = J? (T )+O(h),

where

Ψ? (ξ) = h?1

ξ

dη log(1+hη ?1 ). η

7

Conclusion

We have introduced the di?erence approximations for two Hamiltonian ?ows from the relativistic Toda hierarchy. They turned out to belong to the same hierarchy. The inclusion in the general scheme of symplectic maps on groups equiped with quadratic Poisson brackets allowed to solve the di?erence equations in terms of factorization problem in the group and to ?nd the interpolating Hamiltonians.

28

8

Acknowledgements

The research of the author is ?nancially supported by the Deutsche Forschungsgemeinschaft. The substantial part of this work was done during my one–week visit to the University of Rome III. I cordially thank Professor Orlando Ragnisco for hosting this visit, as well as for the most helpful encouragement and collaboration.

29

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