Mathematics Physics
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Computer Science
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The Discrete and Continuous Painleve VI Hierarchy and the Garnier Systems

We present a general scheme to derive higher-order members of the Painleve VI (PVI) hierarchy of ODE's as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.


Introduction
In recent years there has been a growing interest in discrete analogues of the famous Painlevé equations, i.e. nonlinear nonautonomous ordinary difference equations tending to the continuous Painlevé equations in a well-defined limit and which are integrable in their own right, cf. [1]. Even though the qualitative features of the solutions of these systems are not yet fully understood, nonetheless in most of the known examples the main ingredients of their integrability have been exhibited. Recently, a classification of continuous as well as discrete Painlevé equations in terms of the root systems associated with affine Weyl groups, has been proposed on the basis of the singularities of the rational surfaces of their initial conditions and their blowings-up, cf. [2].
In a recent paper, [3], we established a connection between the continuous Painlevé VI (P VI ) equation and a non-autonomous ordinary difference equation depending on four arbitrary parameters. This novel example of a discrete Painlevé equation arises on the one hand as the nonlinear addition formula for the P VI transcendents, in fact what is effectively a superposition formula for its Bäcklund-Schlesinger transforms, on the other hand from the similarity reduction on the lattice (cf. [4,5]), of a system of partial difference equations associated with the lattice KdV family. In subsequent papers, [6,7], some more results on these systems were established, namely the existence of the Miura chain and the discovery of a novel Schwarzian PDE generating the entire (Schwarzian) KdV hierarchy of nonlinear evolution equations and whose similarity reduction is exactly the P VI equation, this being the first example of an integrable scalar PDE that reduces to full P VI with arbitrary parameters.
In the present note we extend these results to multi-dimensional systems associated with higher-order generalisations of the P VI equation. Already in [3] we noted that the similarity reduction of the lattice KdV system could be generalised in a natural way to higher-order differential and difference equations, without, however, clarifying in detail the nature of such equations. What we will argue here is that, in fact, such equations constitute what one could call the Painlevé VI hierarchy and its discrete counterpart. Whilst the idea of constructing hierarchies of Painlevé equations by exploiting the similarity reductions of hierarchies of nonlinear evolution equations of KdV type is at least two decades old, cf. [8], the issue has gained renewed interest in recent years, cf. e.g. [9]- [13], because of the hypothetical possibility that these hierarchies of higher-order Painlevé equations yield new transcendents. Evidence to that effect might be given by the asymptotic analysis of the higher-order equations, since they seem to be governed by hyper-elliptic functions rather than elliptic ones as is the case for the original Painlevé equations, [14].  [3,6,7] we are now well-equipped to tackle this problem, and in the present paper we outline the basic construction of the equations in the discrete as well as continuous P VI hierarchy. In fact, we shall demonstrate that the lattice KdV system can be naturally embedded in a multidimensional lattice system achieving the higher-order reductions by including more terms in the relevant similarity constraint which provokes the coupling between the various lattice directions.

Most
It should be noted that in a sense higher-order P VI systems already were constructed by R. Garnier in his celebrated paper of 1912, [15], extending the original approach of R.
Fuchs who was the first in [16] to find P VI arising from the isomonodromic deformation of a second-order linear differential equation. We will conclude our paper with a discussion of these Garnier systems, which in view of the recent interest in algebraic solutions of P VI , cf. e.g. [17]- [20], deserve in our opinion some renewed attention.

The Discrete PVI Hierarchy
In [3], following earlier work e.g. [4,5], cf. also [1], a coherent framework was developed in which the similarity reduction of both discrete as well as continuous equations associated with the lattice KdV family were treated. Surprisingly, from these reductions the full P VI equation for arbitrary parameters emerged together with a four-parameter discrete equation, i.e. a discrete Painlevé equation. From the treatment of [3] it was evident how to to extend the lattice system of partial difference equations and their similarity constraints leading to the reductions to P VI . Here we describe explicitely this higher-dimensional lattice system and discuss their explicit reductions.
The lattice KdV family of equations contains many related equations such as the lattice Schwarzian KdV, the lattice modified KdV (mKdV) and the actual lattice KdV equations. We concentrate here on one member of this family only, namely the lattice mKdV equation: cf. e.g. [21], with discrete independent variables n, m and depending on additional parameters of the equation p, q, i.e. the lattice parameters. As was pointed out earlier, cf.
[21], the lattice equation (2.1) actually represents a compatible parameter-family of partial difference equations: namely, we can embed the equation (2.1) into a multidimensional lattice by imposing a copy of (2.1) with different parameters on any two-dimensional sublattice, identifying each lattice direction with a corresponding lattice parameter p i ∈ C in which direction the sites are labelled by discrete variables n i (noting that these are not necessarily integers, but shift by units, i.e. n i ∈ θ i + Z, θ i ∈ C). Thus, combining two different lattice directions, labelled by (i, j) we can write the lattice equation (2.1) on the corresponding sublattice as in which we use the right superscripts i, j to denote the shifts in the corresponding directions, whereas we will use left subscripts i, j denote shifts in the reverse direction, 2)) the special property that these three different ways of calculating this point actually lead to one and the same value. It is indeed at this point that the consistency of the embedding of the lattice MKdV into the multidimensional lattice is tested. In fact, by a straightforward calculation we find that this value is given (which is clearly invariant for any permutation of the labels ijk), independent of the way in which we calculate this value! Thus, the equation ( As a consequence of this compatibility we will call the system (2.2) a holonomic system of partial difference equations. Now, we turn to the issue of the symmetry reduction of the multidimensional lattice in the sense of [5]. It follows from the general framework of [3] that the similarity constraint for the multidimensional lattice MKdV system is as follows:

3)
µ and λ being constants, and in which the variables a i are given by The sum in (2.3) is over all the i labelling the lattice directions, the choice of which decides the order of the reduction. To analyse the reduction we need a number of relations for the objects a i which follow from (2.2), namely in terms of the following variables: (2.8) The variables x ij = x ji and X ij = 1/X ji are not independent, but related via: as well as We note that since the left-hand side of (2.6) depends only on the label i but not on j, for fixed i this represents a set of N − 2 coupled first-order ordinary difference equations with respect to the shift in the discrete variable n i between the N − 1 variables X ij , j = i.
Furthermore, the relations (2.5), for the same fixed label i, provide us with a set of N − 1 first-order relations between the variables a j , j = i, and thus together with the similarity constraint (2.3) where a i is substituted by (2.6) we obtain a set of 2(N − 1) first-order nonlinear ordinary difference equations for the 2(N − 1) variables X ij , a j , j = i, which together form our higher-order discrete system. In the next section we will explicitely disentangle this coupled system in the cases N = 2 and N = 3.
The continuous equation for the PVI hierarchy derive from the differential equations with respect to the lattice parameters p i , which read: It can be shown that the differential relations (2.11) are actually compatible not only amongst themselves, but also with the the discrete equations on the lattice (2.2), i.e. the discrete and continuous flows are commuting: This can actually be demonstrated by explicit calculation exploiting the discrete relations (2.5), but we will not give the details here (which follow closely the pattern of calculations of [3]). Thus, we have a coherent framework of a large multidimensional system of equations with discrete (in terms of the variables n i ) as well as continuous (in terms of the parameters p i ) commuting flows, in terms of which compatible equations of three different types (partial difference, differential-difference and partial differential) figure in one and the same framework: the partial difference equations are precisely the lattice equations (2.2), the differential-difference equations are the relations (2.11), whilst for the partial differential equations in the scheme we refer to our recent paper [7]. Here we will focus now on the reductions under the symmetry constraint (2.3) in order to derive closed-from ODE's in terms of the lattice parameter p i . To make this reduction explicit we use (2.11) in combination with (2.5)-(2.10) to obtain differential relations for the a i , namely as well as the following relations for the reduced variables X ij in which we have abbreviated (2.14) Using (2.12) in conjunction with (2.13) and using the similarity constraint (2.3) to eliminate the a i , we obtain a coupled first-order system of differential equations w.r.t. the independent variable t i = p 2 i in terms of the 2N − 2 variables a k , X ik , (k = i). Solving the variables a k from the linear system given by the equations (2.13) and inserting them into (2.12) we obtain a coupled set of second-order nonlinear differential equations for the variables X ik .

Special Cases: N=2, N=3
We will now analyse the basic relations of the general framework presented in the previous section in the cases N = 2, 3 only in order to arrive at slightly more explicit equations, demonstrating that the reduction leads to ordinary difference equations (in the discrete case) or to ordinary differential equations (in the continuous case).

N=2:
We will be very brief about the two-dimensional case N = 2 which was the main subject of study in the earlier paper [3]. There the compatibility of the similarity constraint and the lattice equation was stated, and the various relations resulting from (2.5) were already written down. Using in this case the slightly simpler notation: a 1 = a , a 2 = b , x 12 = x , X 12 = X and using the discrete independent variables n 1 = n, n 2 = m, as well as the lattice parameters p 1 = p, p 2 = q, we derived the second order nonlinear non-autonomous difference equation: 2(n + 1) 1 − y n+1 y n + 2n 1 − y n y n−1 = µ + λ(−1) n + 2n + 1 + + (µ − λ(−1) n )(r 2 − 1)y n + r(1 − y 2 n ) (n + 1 2 ) − (m + 1 2 )(−1) n (r + y n )(1 + ry n ) , (3.1) (using a slightly different notation from the one of [3]), where r = p/q and where the variables y n are related to the X and x by the prescription: y 2n = x(2n) for the even sites, whilst y 2n+1 = −1/X(2n + 1) for the odd lattice sites (the latter choice being mainly motivated by the wish to cast the equation into a convenient shape). It was pointed out in [3] that whilst a continuum limit of (3.1) yields the P V equation, its general solution can be expressed in terms of P VI transcendents (noting its dependence on four arbitrary parameters, µ, λ, r and m).
The continuous equation for the variable X in terms of the lattice parameter p as independent variable in this case reads: and it is not difficult to show that this is actually the P VI equation through the identification w(t) = pX(p), where t = p 2 , and setting q = 1, leading to with the identification of the parameters α,β,γ,δ as follows: Eq. (3.2) is interesting in its own right since it provides us with a covariant way of writing P VI , noting its invariance under the transformations: n ↔ m , p ↔ q , X ↔ 1/X .

N=3:
This first higher-order case deals with the first genuinely multidimensional situation of three two-dimensional sublattices, on each of which a copy of the lattice MKdV equation 2) is defined. In addition there is also the similarity constraint (2.3) which couples the three lattice directions. Thus, for the three-dimensional case we have a coupled system of equations whose symbolical representation is shown in figure 2.
In the previous section we have already demonstrated the consistency of the three copies This has been carried out for this three-dimensional case using MAPLE. Obviously, the iteration involves too many steps and the expressions soon become too large to reproduce here. In order to analyse the explicit reduction in this case, we redefine the following objects a 1 = a , a 2 = b , a 3 = c X 12 = X , X 13 = Y , X 12 = X , X 13 = Y using also n 1 = n, n 2 = m, n 3 = h, as well as p 1 = p, p 2 = q and p 3 = r to simplify the notation. To start with the continuous equations, fixing the independent variable to be p we obtain the following linear system for the quantities b and c from eq. (2.13)  where we have used the similarity constraint to elininate the quantity a. Furthermore, from (2.12) we obtain the differential relations (3.5b) Solving b and c from the linear system (3.4), and substituting the results in the differential relations (3.5a) and (3.5b), we obtain two coupled second-order nonlinear ODE's depending effectively on six free parameters, namely µ, ν, n, m, h and q/r.
Alternatively, we can derive a system of second-order ordinary difference equations by fixing one of the discrete variables, say n = n 1 , and using the relations (2.5) to obtain the where the tilde denotes the shift in the lattice direction associated with the variable n.
Using the similarity constraint to eliminate the variables c, we obtain the following linear system in terms ofb and b  where the a andã can be expressed in terms of X and Y by (where the undertilde denotes the backward shift with respect to the discrete variable).
The system of equations (3.6), (3.7) and (3.9) -or, equivalently, (3.8) together with (3.9) leads in principle to a fourth order ordinary difference equation in one variable. In fact, solving b andb from (3.8) and then eliminating b altogether by a shift in the independent variable n we get a coupled system containing one equation in terms of X,X,X, X and Y , and the equation (3.9) which is first order in the both X and Y with respect to the shift in the variable n. This system of equations depends effectively on six free parameters, namely µ, ν, m, h, q/p and r/p.

Isomonodromic Deformation Problem
The isomonodromic deformation problem for the multidimensional lattice system is of Schlesinger type, [22]. In the two-dimensional case it was already presented in [4] for special values of the parameters µ, ν, cf. also [3] for the general parameter case. The extension from the two-dimensional to the multidimensional lattice is immediate: one only needs to introduce additional terms of similar form for each additional lattice direction.
Thus, the Lax representation consists on the one hand of the linear shifts on the lattice of the form in which κ is a spectral parameter, and where the Lax matrices L i are given by leading to the Lax equations which lead to a copy of the lattice MKdV equation on each two-dimensional sublattice labelled by the indices (i, j). On the other hand we have the linear differential equation for ψ(κ) with respect to its dependence on the spectral variable κ the compatibility of which with (4.1) leads to the similarity constraint (2.3). In addition, we have differential equations for ψ in terms of its dependence on the lattice parameters p i which are of the form for each of the variable p i . It is the variables t i = p 2 i that play the role as independent variables in the continuous PVI hierarchy.
The elimination of the back-shifted vectors T −1 i ψ by using the inverse of the Lax relations (4.1) lead to the following linear differential equation for ψ thus leading to the problem in the Schlesinger form, with regular singularities at 0, ∞, {t i }.
The matrices A 0 and A i are given by The continuous isomonodromic deformation is provided by the linear differential equations in terms of the lattice parameters, namely where Eq. (4.7) is not quite in standard form, and we need to apply a gauge transformation of the form to remove the term with P i , where the auxiliary variable U obeys an interesting set of equations by itself (in fact this is the object obeying the lattice KdV system of equations) , cf. [3], but we will not give any details here. With this gauge, the continuous isomonodromic deformation (4.7) adopts the standard form whilst the discrete isomonodromic is readily obtained from the Lax representation of (4.1).
is completely integrable 1 , and that it degenerates (under the autonomous limit) to a system living on the Jacobian of a hypereliptic curve, ii) that the symmetric combinations of the dependent variables of the system, as functions of each one of the essential singularities Subsequent work on the Garnier systems was done mostly by K. Okamoto and his school, cf. e.g. [23,24]. However, it seems that in most of these works these systems were treated rather as an underdetermined system of PDE's rather than (as Garnier himself clearly had in mind) as a consist system of ODE's. Although it is not easy to find the explicit transformation of the lattice system exposed in sections 2 and 3 to the systems that Garnier wrote down, in particular to find the explicit relation between the above system (5.1) and the system consisting of (3.4) and (3.5), it is to be expected that such a mapping exist. The identification is probably easiest to obtain via the transformation of the corresponding Schlesinger type of system as given in section 4 and the linear system that Garnier exploited in [15]. However, the search for such an identification will be left to a future study.
Let us finish with some remarks on the relevance of these results for work that is been done in recent years. One of the most exciting developments is the way in which the issue of algebraic solutions of P VI have arisen in recent years, e.g. in connection with WDVV equations, Frobenius manifolds and quantum cohomology, cf. e.g. the review [25]. Such algebraic solutions were already known to Picard, Painlevé and Chazy. In fact, in his early paper [16] R. Fuchs obtained a realisation of P VI in terms of an elliptic integral, and this realisation was subsequently used by Painlevé in [26], to derive an elliptic form for the P VI equation, a form of the equation that was recently recovered by Manin in [18] 3 . The assertions of Garnier in [15] on his generalisation of the Fuchs' approach might form a starting point for extending this elliptic connection to the Garnier systems, in which case we would expect to be able to find a realisation of those systems in terms of hyperelliptic integrals rather than elliptic ones. This might eventually lead to the construction of algebraic solutions of those systems, possibly in the spirit of the recent papers [27,28]. It would be of interest to further investigate the role of the discrete systems in connection with the Garnier systems: we expect them to constitute the superposition formulae for the underlying higher root systems of the corresponding affine Weyl groups. Thus, eventually, a geometric interpretation of the Garnier systems and their discrete analogues in the sense of the blowings-up of the corresponding rational surfaces of their initial conditions, along the lines of the recent paper [2], might be anticipated.