Commun. Math. Phys. 192, 67 – 76 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1998

Self-Duality and Schlesinger Chains for the Asymmetric d-PII and q -PIII Equations A. Ramani1 , Y. Ohta2 , J. Satsuma3 , B. Grammaticos4 1

CPT, Ecole Polytechnique, CNRS, UPR 14, 91128 Palaiseau, France Department of Applied Mathematics, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, 739 Japan 3 Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153, Japan 4 GMPIB (ex LPN), Universit´ e Paris VII, Tour 24-14, 5e e´ tage, 75251 Paris, France

2

Received: 15 February 1997 / Accepted: 19 June 1997

Abstract: We analyse two asymmetric discrete Painlev´e equations, namely d-PII and q-PIII which are known to be discrete forms of PIII and PVI respectively. We show that both equations are self-dual. This means that the same equation governs the evolution along the discrete independent variable and the transformations under the action of the Schlesinger transforms along the parameters of the discrete Painlev´e. A bilinear formulation of the self-dual equation is given as a system of nonautonomous Hirota– Miwa equations.

1. Introduction Since the discovery of the discrete Painlev´e (dP) equations [1] one question has been present in the minds of all practitioners: “up to which point can one push the analogy between discrete and continuous Painlev´e’s?” The answer to this question depends on the attitudes one can have with respect to discrete systems. Those who believe that continuous systems are more fundamental (merely because they are more familiar with them) try to establish the discrete analog of the known properties of the continuous Painlev´e equations through discretisation. However, discretisation is a delicate procedure (in particular in the domain of integrable systems) and despite multiple efforts no systematic approach seems to exist to date. The converse attitude consists in considering the discrete systems as the most fundamental ones. Thus one does use the properties of the continuous systems only as a guide and tries to develop techniques which are specific to the discrete systems (the comparison with the continuous system being straightforward through the continuous limit implemented in a systematic way). The advantage of this point of view is clear: discrete systems may and, in fact, do possess properties with no continuous analog or rather properties the continuous avatar of which does not allow one to guess what the discrete property should be. Thus when one sticks too closely to the continuous systems one can easily miss these important discrete properties.

68

A. Ramani, Y. Ohta, J. Satsuma, B. Grammaticos

The main theme in this paper is such a purely discrete property: the self-duality of dP’s. Let us illustrate what we mean by self-duality in the example where this notion first appeared: the alternate d-PII Eq. [2]: zn−1 1 zn + = −xn + + zn + a, xn+1 xn + 1 xn xn−1 + 1 xn

(1.1)

where zn = δn + z0 and a is a parameter. The Schlesinger transforms of (1.1) were presented in [3]. By denoting by x and x˜ the solutions of alt-d-PII corresponding to ˜ parameters a − δ and a + δ respectively, we have: 1 a(1 + xn xn−1 ) + xn = xn 1 + xn xn−1 − zn−1 xn ˜ and

 x˜ n =

(a + δ)(1 + xn xn−1 ) xn − 1 + xn xn−1 − zn−1 xn−1

(1.2)

−1 .

(1.3)

Eliminating xn−1 between (1.2) and (1.3) we obtain the dual equation of alt-d-PII i.e. the equation where the parameter a is now the independent variable. We find: a 1 a+δ + = x + − a − z, xx˜ − 1 xx − 1 x ˜

(1.4)

where we have dropped the index n, and z(≡ zn ) is now just a parameter. We remark that (1.4) is essentially alt-d-PII itself. The only, minor, change is the fact that the x of the dual equation is multiplied by i with respect to the initial one. This is the self-duality property: the evolution equation in the discrete independent variable and in the space of the parameters is the same. The interesting question is whether this property of self-duality is a general property of discrete Painlev´e equations. The answer to this question is “yes” provided we do not restrict their freedom unnecessarily. As a matter of fact, when we obtained d-P’s through the singularity confinement approach [4], it turned out that several d-P’s possess nonautonomous parts having an even-odd asymmetry. This parity dependence was quite often neglected since it does not survive as such at the continuous limit. (Let us point out here that there is no way one can guess such parity-dependent terms if one tries to “discretize” [5] a given Painlev´e equation.) It is our experience that these terms are crucial for self-duality. Indeed, the symmetrical forms of the d-P equations resulting from the omission of these parity-dependent terms are not self-dual. On the contrary once the full, asymmetric, forms are considered, self-duality is recovered. In what follows, we are going to use the terminology: “asymmetric” d-Pn . By that we mean that when this d-P is symmetrized (dropping the (−1)n terms) the continuous limit is Pn . The “asymmetric” equation is the one where the full freedom is maintained. The equations we are going to analyse in this paper are the asymmetric d-PII and q-PIII . The first has already been identified in [6] as a d-PIII , quite distinct from the well-known (symmetric) q-PIII , while the asymmetric form of the latter was shown in [7] to be in fact a form of q-PVI .

Self-Duality and Schlesinger Chains for Asymmetric Equations

69

2. The asymmetric d-PII The singularity confinement analysis of (the standard form of) d-PII has been performed in [4] and led to the following form compatible with integrability: up−1 + up+1 =

(2pδ + α)up + β + γ(−1)p . u2p − 1

(2.1)

The usual approach consists in dropping the (−1)p term since, at first sight, this term does not appear to have a continuous limit. The equation then goes over to PII at the continuous limit. The Miura and B¨acklund transformations of the symmetric d-PII have been studied in detail in [8]. Let us summarize here briefly the derivation of the autoB¨acklund in the symmetric case (γ = 0). We start from (2.1) and introduce the following Miura transformation: (2.2a) vp = (up − 1)(up+1 + 1), up =

β + vp−1 − vp , vp−1 + vp − ζp

(2.2b)

where ζp = 2δp + α. Eliminating v between the two equations leads to d-PII (2.1). Eliminating u and with the substitution w = v − ζ/2 − δ/2 we find the equation: (wp + ζp /2 + δ/2)(wp + wp+1 )(wp + wp−1 ) = (β − δ)2 − 4wp2 .

(2.3)

Equation (2.3) is the discrete form of equation P34 which plays the role of a “modified” PII . The important remark is that d-P34 depends on β through the square (β − δ)2 . The construction of the auto-B¨acklund becomes now straightforward. We start with a u(β), i.e. a solution of (2.1) with parameter β. We transform through (2.2) to v(β − δ), and since (2.3) is invariant under a sign change of its parameter we can come back through the Miura to u(−(β − δ) + δ) = u(2δ − β). Since (2.1) is invariant under a simultaneous sign change of u and β we have u(2δ − 2β) = −u(β − 2δ). Performing this chain of transformations we find: up (β − 2δ) = −up (β) +

(β − δ)(up (β) + 1) , (up (β) − 1)(up−1 (β) + 1) + (β − ζp )/2

(2.4)

which is the auto-B¨acklund of the symmetric d-PII . Let us now turn to the asymmetric d-PII . The interpretation of the (−1)p term is, in fact, straightforward. Separating even and odd terms in (2.1) one can transform it into a system: (4mδ + α)u2m + β + γ , (2.5a) u2m−1 + u2m+1 = u22m − 1 u2m + u2m+2 =

((4m + 2)δ + α)u2m+1 + β − γ . u22m+1 − 1

(2.5b)

Putting u2m = xn , u2m+2 = xn+2 , u2m−1 = yn−1 , u2m+1 = yn+1 , so that the indices of the x’s are even, those of the y’s are odd, and introducing zn = nδ +α/2, a = −(β +γ)/2, b = (γ − β)/2 we find: 2zn xn − 2a , (2.6a) yn−1 + yn+1 = x2n − 1

70

A. Ramani, Y. Ohta, J. Satsuma, B. Grammaticos

xn + xn+2 =

2zn+1 yn+1 − 2b . 2 yn+1 −1

(2.6b)

This system has been analyzed in detail in [6] where it was shown that at the continuous limit it goes over to PIII . We put y = u/, z = t/, x = t/u, a = α/, b = β/, and find at the limit  → 0 the equation u00 = u02 /u − u0 /t + u3 /t2 − αu2 /t2 + β/t − 1/u, i.e. PIII (although in noncanonical form). In order to simplify the presentation, we shall introduce a shorthand notation for each of the three directions of evolution (along z and along each of the parameters a and b, the latter evolutions being induced by the Schlesinger transforms). Thus we denote ¯ vn−1 = v , etc., where v is any of the x, y, (and the variables to be vn =v, vn+1 = v, introduced later, w, τ ). The¯ evolution along the a axis will be represented by a tilde, i.e, v(a + δ) = v, ˜ v(a − δ) = v , while that of b will be represented by a hat, i.e. v(b + δ) = v, ˆ v(b − δ) = v . Using these˜ notations, we can transcribe (2.6) into: ˆ 2zx − 2a , (2.7a) y + y¯ = x2 − 1 ¯ x + x¯ =

2(z + δ)y¯ − 2b . y¯ 2 − 1

(2.7b)

Next, we consider the Miura transformations introducing the auxiliary variable w: w˜ = y¯ −

z+a z−a = −y + . x−1 x +1 ¯

(2.8)

Similarly

z+a z−a w = −y¯ + =y− . (2.9) x + 1 x −1 ˜ ¯ The dual equations to (2.7) can be obtained in a straightforward way. Combining (2.8) with (2.9) we get: 2ax − 2z . (2.10a) w + w˜ = x2 − 1 ˜ The second equation requires the knowledge of x. ˜ This in turn is based on the implementation of the Schlesinger of asymmetric d-PII . Its derivation follows the one for the symmetric d-PII . We can thus compute x˜ and it turns out that the resulting equation is x + x˜ =

2(a + δ)w˜ − 2b . w˜ 2 − 1

(2.10b)

The self-duality is apparent. It suffices to compare (2.7) to (2.10): z and a have exchanged roles. Similarly we can introduce the Miura along the b direction. We have: wˆ¯ = x¯ −

(z + δ) + b (z + δ) − b = −x + y¯ − 1 y¯ + 1

(2.11)

(z + δ) − b (z + δ) + b =x− . y¯ + 1 y¯ − 1

(2.12)

and w¯ = −x¯ + ˆ

Combining (2.11) and (2.12) we obtain:

Self-Duality and Schlesinger Chains for Asymmetric Equations

w¯ + wˆ¯ = ˆ

71

2by¯ − 2(z + δ) , y¯ 2 − 1

(2.13a)

and using the Schlesinger for y we have: y¯ + yˆˆ¯ =

2(b + δ)wˆ¯ − 2a . wˆ¯ 2 − 1

(2.13b)

Again self-duality is evident. Now b has exchanged its role with z.

Fig. 1. Two consecutive planes (corresponding at the values of parameter b−δ and b) in the cubic lattice covered by the solution of the asymmetric d-PII under the action of the corresponding Schlesinger transformations. The nonlinear variables x, y are shown, together with the τ -function, at their corresponding vertices.

Before proceeding to the bilinear formulation of the asymmetric d-PII and its Schlesinger’s we must introduce a most important ingredient of our analysis, namely the geometry of these transformations. The z, a, b can be thought of as defining a cubic lattice. We present in Fig. 1 two consecutive planes. In order to simplify the treatment of the bilinear equations we have assumed that the plane where the initial x, y live corresponds to a value (b − δ), which explains why these variables appear in the figure as “down-hatted” x, y¯ . (This is in fact the reason for the convention adopted: the b plane ˆ ˆ contains one τ -function that is not shifted in any direction). With these notations we remark readily that the only τ ’s that exist have all symbols (¯,˜,ˆ) appearing in even numbers, or all in odd numbers. Similar, although more complicated, rules can be formulated for the x, y and w. For x, the number of shifts in the ¯ and ˜ directions have the same parity, which is opposite to the parity of the number of shifts in theˆdirection. For y, the privileged direction is ˜, while for w, it is the¯direction. From Fig. 1 (complemented mentally by the other horizontal planes corresponding to b + δ, b ± 2δ, etc.) we remark that each of the x, y, w, has two nearest neighbouring τ ’s. They lie in the z direction for w, the a direction for y and the b direction for x. There exist also for each of the x, y, w, four next-nearest neighbouring τ ’s in the diagonal along the two other directions. The ansatz for the bilinearization consists in writing 1 + x and 1 − x (similarly for y, w) as a ratio of two products of two τ -functions. The denominator is the product of the two nearest neighbouring τ -functions. Thus for x we have τ τ . In ˆ and since there ˆ the numerator two of the next nearest neighbouring τ -functions appear are four of them it is clear that some choice must be made (resulting into an unavoidable asymmetry). We have thus: τ˜¯ τ x = ˆ ˆ˜¯ − 1 = 1 − ˆ ττ ˆ

τ¯ τ˜ ˆ˜ ˆ¯ . ττ ˆ

(2.14)

72

A. Ramani, Y. Ohta, J. Satsuma, B. Grammaticos

Similarly we have: τ¯ τ y¯ = ˆ − 1 = 1 − ˆ ˆτ˜¯ τ˜¯ ˆ

and

τ τ¯ ˆ τ˜¯ τ¯ ˆ ˆ˜

(2.15)

τ τ˜ τ˜ τ (2.16) w ˜ = ˆ −1=1− ˆ. τ˜¯ τ˜ ˆ τ˜¯ τ˜ ˆ ˆ¯ ˆ ˆ¯ A first equation can be obtained by equating the two expressions for x (and two more ˆ starting from y¯ and w ˜ ). We have ˆ ˆ τ˜¯ τ + τ¯ τ˜ − 2τ τ = 0. (2.17) ˆ ˆ˜¯ ˆ˜ ˆ¯ ˆ Note that this equation is of Hirota–Miwa form. Moreover it is autonomous: none of z, a, b appears explicitly. It is most probable that by now even the most dedicated reader has trouble keeping track of the symbols (¯,˜,ˆ). So we introduce a notation that will simplify the situation. We associate the directions z, a, b, to the indices 0,1,2. Moreover an up-shift will be represented by an upper index (appearing a number of times equal to the number of up-shifts) and a down-shift by a lower index. Following this rule we can rewrite (2.17) as {01} {1} {0} (2.18a) τ{2} τ{012} + τ{02} τ{12} − 2τ τ{22} = 0. The two remaining autonomous Hirota–Miwa equations can be easily written {00}

{01} {0}

{11}

{01} {1}

τ {00} τ{22} + τ τ{22} − 2τ{2} τ{12} = 0 and

τ {11} τ{22} + τ τ{22} − 2τ{2} τ{02} = 0

(2.18b) (2.18c)

The duality of Eqs. (2.18abc) is not easy to perceive because of the various up- and down-shifts due to the fact that the τ -functions exist only at certain points of the lattice. Had we written objects that do not exist the self-duality would have become transparent. Shifting Eqs. (2.18) in various directions we can formally rewrite them in explicitly self-dual form: {1} {0} (2.19a) τ {01} τ{01} + τ{0} τ{1} − 2τ {2} τ{2} = 0, {2} {0}

τ {02} τ{02} + τ{0} τ{2} − 2τ {1} τ{1} = 0, {1} {2}

τ {12} τ{12} + τ{2} τ{1} − 2τ {0} τ{0} = 0.

(2.19b) (2.19c)

The Hirota–Miwa Eqs. (2.19) are not sufficient in order to characterize the asymmetric d-PII . They must be complemented by the equation resulting from the Miura transform (2.8–9, 11–12). Starting from the first part of (2.8), for instance, we find: τ˜ τ¯ − τ¯ τ˜ = (z − a)τ τ˜¯ , ˆ ˆ˜ ˆ¯

(2.20)

or, in the notation with indices: {0}

{1}

{01}

τ {11} τ{12} − τ {00} τ{02} = (z0 − z1 )τ τ{2} .

(2.21)

Here z0 , z1 and z2 denote the independent variables in the directions z, a, b, i.e. z0 ≡ z = nδ + α/2, z1 ≡ a = mδ − (β + γ)/2, z2 ≡ b = kδ + (γ − β)/2 with integer m and k. Similarly we can obtain two more equations that will be the duals of (2.21). However,

Self-Duality and Schlesinger Chains for Asymmetric Equations

73

(2.21) and its duals are not the only equations one can write. Indeed using the second part of (2.8) we have: {01}

{1}

τ {11} τ{012} − τ{00} τ{2} = (z0 + z1 )τ τ{02}

(2.22)

(and, quite expectedly, two more dual equations). However, (2.22) is not an essentially new equation. It can be deduced from (2.21) based on a general parity property of asymmetric d-PII . In fact, if we reverse the signs of a, b, but not of z, then x, y change sign but not w. Under this change, the index {0} stays where it is while {1, 2} move from upper position to lower and vice versa. Implementing these changes into (2.22) we find {12} {02} {2} (2.23) τ{11} τ{0} − τ{00} τ{1} = (z0 − z1 )τ τ{01} . Upshifting this equation once in directions 0 and 1 and downshifting it once in direction 2 (note that (z0 − z1 ) does not change) we find exactly (2.21). Thus asymmetric d-PII can be expressed as a system of three self-dual Hirota–Miwa equations in three dimensions. Two of these Hirota-Miwa equations are non autonomous, and are related through a parity transformation. 3. The asymmetric q-PIII The asymmetric form of q-PIII , although obtained in the initial derivation of the discrete analog of PIII , did not receive much attention until Jimbo and Sakai [7] showed that it is in fact a discrete form of PVI . Let us start directly with the asymmetric form written as: (x + a)(x + b) y y¯ = , (1 + x/c)(1 + x/d) ¯

(3.1a)

(y¯ + p)(y¯ + r) , (1 + y/s)(1 ¯ + y/t) ¯

(3.1b)

xx¯ =

where a, b, p, r ∝ λ−n and c, d, s, t ∝ λn . The parameters a, b, . . ., t are not all free. In the gauge we are using in this paper we have the constraints cd = λ−2 st and ab = λ2 pr. Moreover a scaling transformation on x and y can be used in order to ensure that the common value of abcd and prst is unity, further reducing the number of degrees of freedom to five. (We must point out here that our notation, using λ in the independent variable is at odds with the usual one where q appears. Still we prefer to use this notation, to avoid ambiguities, since q will enter later in the unified description of the evolution and the Schlesinger’s.) The Schlesinger transformations of asymmetric q-PIII have been studied in [9]. Let us for instance consider the Miura: √ x+b (1 + x/d) ad 1 √ = y¯ , (3.2) w˜ = y (1 + x/c) bc x+a ¯ √ 1 (1 + x/d) ad x+b √ w= . (3.3) =y x+a ˜ y¯ (1 + x/c) bc ¯ (We recall that abcd = 1.) Let us point out that if in (3.2) we replace b by a and c by d we recover 1/w. So this would not be a new Miura. On the other hand replacing just c ˜ b we do obtain a Miura in a new direction. Moreover around y¯ one can by d but keeping define two more directions by coupling, for instance, r to s or r to t.

74

A. Ramani, Y. Ohta, J. Satsuma, B. Grammaticos

It is straightforward to obtain the dual equation for the w variable: (x + b)(x + d) . ww˜ = (1 + x/c)(1 + x/a) ˜

(3.4)

However, the second equation for x, relating the product xx˜ to w˜ involves more complicated parameters. Thus it is time to dispense with the notation involving symbols (¯,˜) and introduce the analog to the one we employed in Sect. 2, based on indices. The basic geometry in the present case is a hypercubic five-dimensional lattice. We associate the index 0 to the direction n, and the indices 1, 2, 3, 4 to the four directions related to the Schlesinger’s of asymmetric q-PIII . The expressions of the parameters a, b,. . ., t are now: a = q0−1 q1 q2 , b = q0−1 q1−1 q2−1 ,

p = (λq0 )−1 (q3 /λ)(q4 /λ), r = (λq0 )−1 (q3 /λ)−1 (q4 /λ)−1 ,

c = q0 q1 q2−1 ,

s = (λq0 )(q3 /λ)(q4 /λ)−1 ,

d = q0 q1−1 q2 ,

t = (λq0 )(q3 /λ)−1 (q4 /λ).

(3.5)

The reason why we have written two quantities as (q3 /λ) and (q4 /λ) rather than just q3 , q4 , is the same as why in the previous section the plane where the variables x and y lived was indexed by (b−δ) rather than b. When the bilinear form will be considered the τ ’s will have more symmetrical indices. Indeed with this choice the number of shifts appearing in the indices of the τ ’s have all the same parity. On the other hand the variables x, y and w are shifted in such a way that three shifts have one parity and the two remaining have the other one. Thus we have x = X{34} , i.e. X(n0 , n1 , n2 , n3 − 1, n4 − 1), y = X{034} , ¯ {0} {1} y¯ = X{34} , w˜ = X{34} , etc. The asymmetric q-PIII can now be rewritten as: {0}

X{034} X{34} = {0}

{00}

X{34} X{34} =

(X{34} + q0−1 q1 q2 )(X{34} + q0−1 q1−1 q2−1 ) (X{34} q0−1 q1−1 q2 + 1)(X{34} q0−1 q1 q2−1 + 1)

,

(3.6a)

{0}

(X{34} + (λq0 )−1 (q3 /λ)(q4 /λ))(X{34} + (λq0 )−1 (q3 /λ)−1 (q4 /λ)−1 )

. {0} {0} (X{34} (λq0 )−1 (q3 /λ)−1 (q4 /λ) + 1)(X{34} (λq0 )−1 (q3 /λ)(q4 /λ)−1 + 1) (3.6b) Similarly (3.4) becomes now: {1}

X{134} X{34} =

(X{34} + q1−1 q0 q2 )(X{34} + q1−1 q0−1 q2−1 ) (X{34} q1−1 q0−1 q2 + 1)(X{34} q1−1 q0 q2−1 + 1)

.

(3.7a)

˜ It is now straightforward to write the second equation in the “1” direction, for x, x: {11} X{34} X{34}

{1}

{1}

(X{34} + (λq1 )−1 (q3 /λ)(q4 /λ))(X{34} + (λq1 )−1 (q3 /λ)−1 (q4 /λ)−1 )

. {1} {1} (X{34} (λq1 )−1 (q3 /λ)−1 (q4 /λ) + 1)(X{34} (λq1 )−1 (q3 /λ)(q4 /λ)−1 + 1) (3.7b) The self-duality is evident: Eqs. (3.7) are obtained from (3.6) simply by exchanging 0 and 1. Similarly one can introduce the evolution equation in the direction 2. For directions =

Self-Duality and Schlesinger Chains for Asymmetric Equations

75

3 and 4 we must use as a central point y¯ instead of x. Self-duality is, of course present in all directions. Let us now turn to the bilinear formulation of asymmetric q-PIII and its Schlesinger’s. Now, the geometry is such that each x-site (and similarly for y, w) has four nearestneighbouring τ ’s. Consider for instance the variable x = X{34} . Its nearest neighbouring sites correspond to τ -functions τ , τ{3344} , τ{33} and τ{44} . (Any τ -function with odd shifts will be among the next nearest-neighbours at best.) With respect to the positions of X{34} two of the τ -functions (namely τ , τ{3344} ) correspond to shifts of both indices either upwards or downwards, the remaining two (τ{33} and τ{44} ) correspond to shifts of one index upward and the other one downwards. The choice for the representation of the X in terms of τ -functions will be to write it as a ratio with the product of the first two τ -functions in the numerator while the product of the remaining two appear at the denominator: τ τ{3344} (3.8) X{34} = τ{33} τ{44} (but the reverse would have been just as acceptable). We must point out here that the choice of the expression of X is self-duality preserving. Thus we expect the bilinear equations for the τ ’s to be also self-dual. In order to obtain the bilinear equations associated to asymmetric q-PIII we start with {1} the Miura (3.2). With the notations that we introduced, w˜ = X{34} and y = X{034} . The ¯ expressions of these variables in terms of τ -functions read: {1} X{34}

{1}

=

{012}

τ{0234} τ{34} {12}

{01}

τ{034} τ{234}

,

(3.9)

{12}

X{034} =

τ{01234} τ{034}

.

(3.10)

√ (τ τ{3344} + bτ{33} τ{44} )/ b √ , = (τ{33} τ{44} + τ τ{3344} /c) c

(3.11)

{2}

{1}

τ{0134} τ{0234}

Substituting into (3.2) we find: {1} X{034} X{34}

{012}

≡

τ{01234} τ{34} {2}

{01}

τ{0134} τ{234}

where we have used expression (3.2) for x = X{34} . Using the expressions of b and c, we √ equate the numerators and denominators of both sides of (3.11) (with the factor 1/ b being considered as being part of the numerator). We finally obtain the bilinear nonautonomous Hirota–Miwa equations: {012}

1/2 1/2 1/2

−1/2 −1/2 −1/2 q1 q2 τ{33} τ{44} ,

τ{01234} τ{34} = q0 q1 q2 τ τ{3344} + q0 {2}

{01}

−1/2 −1/2 1/2 q1 q2 τ τ{3344}

τ{0134} τ{234} = q0

1/2 1/2 −1/2

+ q0 q1 q2

τ{33} τ{44} .

(3.12a) (3.12b)

Similarly to the asymmetric d-PII case these two equations are related through a updownward symmetry. All the remaining equations can be obtained by duality transformations from these two nonautonomous Hirota–Miwa equations. Both can be written in a compact way as:

76

A. Ramani, Y. Ohta, J. Satsuma, B. Grammaticos

τ (ni + σi , nj + σj , nk + σk )τ (ni − σi , nj − σj , nk − σk ) = σ σk /2 σi σk /2 σi σj /2 qj qk τ (nl

qi j

− 1, nm − 1)τ (nl + 1, nm + 1) +

−σ σ /2 −σ σ /2 −σ σ /2 qi j k qj i k qk i j τ (nl

(3.13)

− 1, nm + 1)τ (nl + 1, nm − 1),

where (i, j, k, l, m) is any permutation of (0,1,2,3,4), all the σ’s are ±1, and the indices not explicitly mentioned are assumed to be unshifted. This expression is an explicitly self-dual nonautonomous Hirota-Miwa which represents q-PIII and its Schlesinger transformations. 4. Conclusion In this paper we have presented a property of the discrete Painlev´e equations that sets them apart from the continuous ones: self-duality. First obtained in the case of the alternate d-PII , self-duality did not appear at the time to be characteristic of all dP’s. The present work shows that when one considers the most general form of a given dP (without unnecessary assumptions leading to symmetrisation) this form is naturally self-dual. This allows for a unified description of a given dP and its Schlesinger transformation, what we have dubbed [10] the “Grand Scheme”. In fact, self-duality means that the same equation governs the evolution in the direction of the independent variable and in the direction of the parameters. Written in bilinear form this equation turns out to be the Hirota–Miwa, discrete Toda equation. Thus the dP’s can be represented as systems of, in general nonautonomous, Hirota–Miwa equations governing the evolution in any direction, be it the independent variable or the parameters. Two equations have been treated in this paper: the asymmetric d-PII and q-PIII which are known to be discrete forms of PIII and PVI . One problem that we plan to address in the near future is that of d-PI . This equation is the only one among the dP’s that does not possess a bilinear but, rather, a trilinear form. It would be interesting to investigate whether the notion of self-duality applies to the asymmetric d-PI leading to a description in terms of the Hirota–Miwa equation. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Ramani, A., Grammaticos, B. and Hietarinta, J.: Phys. Rev. Lett. 67, 1829 (1991) Fokas, A.S., Grammaticos, B. and Ramani, A.: J. Math. An. and Appl. 180, 342 (1993) Nijhoff, F., Satsuma, J., Kajiwara, K., Grammaticos, B., Ramani, A.: Inv. Probl. 12, 697–716 (1996) Grammaticos, B., Ramani, A. and Papageorgiou, V.: Phys. Rev. Lett. 67, 1825 (1991) Conte, R. and Musette, M.: Phys. Lett. A 223, 43 (1996) Grammaticos, B., Nijhoff, F.W., Papageorgiou, V., Ramani, A. and Satsuma, J.: Phys. Lett. A185, 446– 452 (1994) Jimbo, M. and Sakai, H.: Lett. Math. Phys. 38, 145 (1996) Ramani, A. and Grammaticos, B.: J. Phys. A25, L633 (1992) Jimbo, M., Sakai, H., Ramani, A. and Grammaticos, B.: Phys. Lett. A 217, 111 (1996) Ramani, A. and Grammaticos, B.: The Grand Scheme for discrete Painlev´e equations. Lecture at the Toda symposium (1996)

Communicated by T. Miwa