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Since by definition the discrete Painlev´e equations are second-order mappings, these multicomponent systems include equations which are local.. This feature is in contrast with the cont

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EQUATIONS AND THEIR GEOMETRICAL DERIVATION

B GRAMMATICOS, A RAMANI, AND T TAKENAWA

Received 9 October 2005; Revised 5 January 2006; Accepted 5 January 2006

We show that two recently discoveredq-discrete Painlev´e equations are one and the same

system Moreover we provide a novel derivation of thisq-discrete system based on

trans-formations obtained with the help of aﬃne Weyl groups

Copyright © 2006 B Grammaticos et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

One of the characteristics of discrete Painlevé equations is that they may possess more than one canonical form Indeed we often encounter equations which are written as a system involving several dependent variables Since by definition the discrete Painlevé equations are second-order mappings, these multicomponent systems include equations which are local It is then straightforward, if some equation is linear in one of the vari-ables, to solve for this variable and eliminate it from the final system One thus obtains two perfectly equivalent forms which may have totally different aspects

This feature is in contrast with the continuous Painlevé case where the latitude left by the transformations which preserve the Painlevé property is minimal The fact that there exist just 6 continuous Painlevé equations at second order while the number of possible second-order discrete Painlevé equations is in principle infinite may play a role The pos-sible existence of an unlimited number of discrete Painlevé equations has been explicitly pointed out in [2] In that paper we have, in fact, presented a novel definition for the dis-crete Painlevé equations The traditional definition of a disdis-crete Painlevé equation is that

of an integrable, nonautonomous, second-order mapping, the continuous limit of which

is a continuous Painlev´e equation This definition turned out to be severely limitative since it binds the discrete systems to the continuous ones through the continuous limit However, as was shown repeatedly, the discrete systems are more fundamental than their continuous counterparts, and in the case of discrete Painlev´e equations much richer, as far as the degrees of freedom are concerned We were thus naturally led in [2] to propose

Hindawi Publishing Corporation

Advances in Di ﬀerence Equations

Volume 2006, Article ID 36397, Pages 1 11

DOI 10.1155/ADE/2006/36397

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a novel definition of discrete Painlev´e equations, which is now a discrete system defined

by a periodic repetition of a given nonclosed pattern on the weight lattice of the aﬃne Weyl group E(1)8 or one of its degenerations

This proliferation of discrete Painlev´e equations raises the question of the indepen-dence of the various forms Indeed, while one can, in principle, construct an unlimited number of such systems, there exists no a priori guarantee that they are all diﬀerent This

is something to be assessed for each case at hand In this paper we will concentrate on two recently discovered discrete Painlev´e equations and show that they are one and the same equation Moreover we will provide a novel derivation of thisq-discrete system based on

the geometrical approach we have developed in [10]

2 The two discrete Painlev´e systems

In a recent paper, Kajiwara et al [6] have introduced the following system:

¯f0= a0a1f11 +a2f2+a2a3f2f3+a2a3a0f2f3f0

1 +a0f0+a0a1f0f1+a0a1a2f0f1f2,

¯f1= a1a2f2

1 +a3f3+a3a0f3f0+a3a0a1f3f0f1

(2.1)

with

and the “bar” indicates the evolution along the independent discrete variable The latter was introduced by taking

whereupon one finds thatz is of the form λ nand thus the system is aq-discrete equation.

The inverse evolution of (2.1) is given by

f0= f3

a0a1

a2a1a0+a1a0f2+a0f2f1+f2f1f0

a0a3a2+a3a2f0+a2f0f3+f0f2f3,

f

1= f0

a1a2

a1a0a3+a0a3f1+a3f1f0+f1f0f3,

f

2= f1

a2a3

,

f3= f0

a a

a a a +a a f +a f f +f f f .

(2.4)

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This system was studied in detail by Masuda [7] who has shown that its continuous limit

is the Painlev´e V equation

From (2.1) we find that the dependent variables satisfy the relations

¯f0¯f2= λ f1f3, ¯f1¯f3= λ f0f2. (2.5) However the constraint f0f2= f1f3is unwarranted One can perfectly relinquish it and obtain a valid discrete Painlev´e equation Thus it is possible to assume

f0f2= γz, f1f3= δz, ¯f0¯f2= δ ¯z, ¯f1¯f3= γ ¯z, (2.6) where ¯z = λz, whereupon the equation acquires one more degree of freedom This

ex-tension was introduced by one of us (T Takenawa) in [13] where it was shown that the geometry of the evolution of this extended equation, together with its Schlesinger trans-formations, can be described by the aﬃne Weyl group D(1)

5 By using the freedom of the origin ofz, we can define z =γδλ nand find, finally,

f0f2= kz, f1f3= z

wherek =γ/δ Of course from (2.5) we find that ¯f0¯f2= ¯z/k and similarly ¯f1¯f3= k ¯z.

In another recent paper [11] two of the present authors (A Ramani and B Grammati-cos), in collaboration with Willox et al., have examined the limits of theq-PVIequation [1]

x n x n+1 − z n z n+1

x n x n −1 − z n z n −1

x n x n+1 −1

x n x n −1 −1 =

x n − az n

x n − z n /a

x n − bz n

x n − z n /b

x n − c

x n −1/c

x n − d

x n −1/d , (2.8) wherez n = z0λ nanda, b, c, d are the parameters of the equation By letting a → ∞and

c → ∞simultaneously, we found the equation

x n x n+1 − z n z n+1

x n x n −1− z n z n −1

x n x n+1 −1

x n x n −1−1 = f z n

x n − bz n

x n − z n /b

x n − d

x n −1/d , (2.9) where f stands for the ratio a/c As we have shown the equation has PVas a continuous limit Again, the form (2.9) does not encapsulate the full freedom of the equation and an extension is possible This can be obtained either by starting from (2.9) and extending

it with the help of some discrete integrability criterion [3], [4] or by starting from the

“asymmetric” form of (2.8) which incorporates the maximal number of parameters To make a long story short the extended form of (2.9) turns out to be

x n x n+1 − z n z n+1

x n x n −1− z n z n −1

x n x n+1 −1

x n x n −1−1 = f z n

θ n

x n − θ n bz n

x n − θ n z n /b

x n − d

x n −1/d (2.10) with logθ n = α(−1) n As we have pointed out in [11], the geometry of the transformations

of this equation is related to the aﬃne Weyl group D(1)

, just as in the case of (2.1)

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This result is not a coincidence As we will show, the two equations are identical In order to show this we introduce the variablesx ≡ f0, y ≡ f1and use the relations f2= kz/x, f3= z/(k y) The evolution equations for x can now be written as

x n x n+1 = z n z n+1

a0x n+ 1

+y n kz n+1 /a3+a0a1x n y n

1 +a0x n+k y n z n+1 /a3+a0a1x n y n ,

x n x n −1 = k y n z n z n −1

x + a0

+x n z n /a3+kz n z n −1 a0a1

k y n

x n+a0

+x n z n /a3+kz n z n −1 a0a1

.

(2.11)

Next we eliminatey between the two equations and reorganise the result We find

x n x n+1 − z n z n+1

x n x n −1− z n z n −1

x n x n+1 −1

x n x n −1−1 = a1z n

ka3

x n+kz n a2

x n+kz n /a2

x n+a0

x n+ 1/a0 (2.12) which is exactly (2.10) withθ n = k This specific choice is due to the fact that we have

written the equation aroundx n Had we written the equation aroundx n ±1, we would have found (2.10) withθ n ±1 =1/k Thus (2.1) is perfectly equivalent to (2.10)

3 A derivation using discrete Miura transformations

In [6] the derivation of (2.1) was based on the analysis of discrete dynamical systems associated to extended aﬃne Weyl groups of type A(1)

m ×A(1)

n The derivation of (2.7), on the other hand, as mentioned above, was based on the limits of equations related to the

E(1)7 aﬃne Weyl group However as explained in [11,13], (2.1) and (2.7) can be connected

to the aﬃne Weyl group D(1)

5 It is thus natural to present a derivation of these systems (and here we choose (2.9) for simplicity reasons) based on the Miura transformations obtained from the geometry of D(1)5

In [10] we have studied in detail the geometry of the “asymmetric”q-PIII[9], which was shown by Jimbo and Sakai [5] to be a discrete form of PVI, and we have found that it is described by the aﬃne Weyl group D(1)

5 This equation was the first for which the property

of self-duality was established What we mean by self-duality is that the same equation describes the evolution along the independent variable or among any of the parameters

of the equation (the latter evolution being mediated by the Schlesinger transformations)

In this sense all the parameters, including the independent variable, play the same role The form of the “asymmetric”q-PIIIwe are going to use in what follows is

y y

ˆ =

x + ap/q

x + 1/(apq)

1 +xa/(pq)

x x=

y + r/ q

y + 1/(r q)

1 +y/(s q)

where the “hat” symbol is used in order to indicate evolution along theq direction, that

is,q= λq The form of (3.1) is chosen so as to indicate that thex variable exists only

on “even” lattice sites while the y variable exists only on “odd” sites with respect to the

evolution ofq.

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Next we consider an evolution along the p variable and use the “tilde” symbol for it,

that is, p= λp, while in the derivation that follows the parameters a, r, and s remain

constant A new dependent variablew is introduced through the Miura transformations

w

y =

ax + 1/(pq)

1 +ax/(pq) = y

y w= a x + 1/

p q

1 +a x/

We now solve (3.2) and (3.3) forx and x and use ( 3.1b) in order to obtain an equation involving justy, w

, andw:

w

y −1/(pq)

w

y/(pq) −1

⎛

⎝ y w−1/(pq)

y w/(p q) −1

⎞

⎠ = a12(y + r/ q)y + 1/(r q)

1 +y/(s q)

(1 +sy/ q) . (3.4)

In order to bring (3.4) under canonical form we introduce formallyY = y/q and W = w/ p By this generic notation we mean that one has to use the local value of p or q We

remind at this point thatp is invariant under the “hat” evolution and similarly q does not

change when we follow the “tilde” evolution We find thus

⎛

⎝W Y−1/

p2q2

W

Y−1

⎞

⎠

⎛

⎜YW −1/

p2q2

Y W −1

⎞

⎟

⎠ = a21p2 Y + r/ q2 Y + 1/

r q 2

(1 +Y /s)(1 + s Y ) . (3.5)

As can be assessed by inspection, (3.5) describes an evolution along an “oblique” direction where a single step is a combination of two steps, one in each of the “hat” and “tilde” directions We are thus led into introducing formally the new independent variable

Z = 1

whereupon the quantity 1/(p2q2) on the left-hand side of (3.5) can be consistently rewrit-ten asZZ

Similarly we have 1/(p

2q2)= Z Z Moreover introducing the auxiliary quantity

g = p/ q we can give finally ( 3.5) into a form which, with the appropriate interpretation,

is identical to (2.12):

⎛

⎝W Y− ZZ

W

Y−1

⎞

⎠

⎛

⎜YW − Z Z

Y W −1

⎞

⎟

⎠ = Z

a2g

(Y + rg Z)( Y + g Z/r) (1 +Y /s)(1 + s Y ) . (3.7)

We proceed now, along similar lines, to derive the second equation of the system First we

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write the dual equations of (3.1):

ww

=

x + 1/(apq)

(x + aq/ p)

1 +xa/(pq)

xx

=

w

+r/ p

w

+ 1/(r p)

1 +sw

/ p

1 +w

/

sp

Next we solve forx from the leftmost equality of (3.2) and downshifting the rightmost equality of (3.2) twice along the tilde direction we solve it forx

Using (3.8b) we can now obtain an equation involving justw

,y, and y We find

w

y −1/(pq)

w

y/(pq) −1

⎛

⎜

y

w−1/

qp

y

w/

qp

−1

⎞

⎟

⎠ =

1

a2

(w

+r/ p)

w

+ 1/(r p)

1 +w

/(sp)

1 +sw

/ p

Again in order to bring the equation to canonical form we use the variablesY and W.

Without entering into all the tedious but straightforward manipulations, we give the form

of the final equation

W

Y− ZZ

W

Y−1

⎛

⎜WW Y − ZZ

Y −1

⎞

⎟

⎠ = gZ a2 (W +rZ/g)

W

+Z/(rg)

(1 +W

/s)(1 + sW )

This equation complements (3.7) We must point out here that the parameter g has

shifted position, with respect to (3.7), from numerator to denominator and vice versa,

in perfect agreement with (2.10), where logθ n = α(−1) n

4 Relation to the Weyl group

In this section, we present explicit relations of these discrete Painlev´e equations to the extended Weyl group of type D(1)5 , and discuss their space of initial conditions in the spirit of the Okamoto-Sakai approach [8,12] Let us define the transformationsw i(i=

0, 1, , 5), σ01, andπ on the space (x, y; a, p, q, r, s, λ) ∈ C2× (the parameter space) as follows:w0maps (x, y; a, p, q, r, s, λ) to

x, y; 1

p,

1

a,q, r, s, λ

w1maps it to

w2maps it to

x, ay(x + pq/a) q(x + ap/q) ;q, p, a, r, s, λ

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w3maps it to

λ √ r/s x(y + qs/λ) q(y + rλ/q) ,y; a, p, λ

r

s,

q √ rs

λ ,

λ √ rs

q ,λ

w4maps it to

x, y; a, p, q,1

r,s, λ

w5maps it to

x, y; a, p, q, r,1

s,λ

σ01maps it to

x,1

y;

1

a,p,

1

q,

1

r,

1

s,

1

λ

π maps it to

y, x;

r

s,

√

rs, q

λ,ap,

p

a,

1

λ

From Sakai’s theory on the relation to rational surfaces, the transformations we intro-duced here act on the root basis (α0,α1,α2,α3,α4,α5) as

w i

α j

= α j+

α i,αj

where the bilinear form (αi,αj) is given by the Cartan matrix of negative sign of type D(1)5 ;

σ01:

α0,α1,α2,α3,α4,α5

−→α1,α0,α2,α3,α4,α5

,

π :

α0,α1,α2,α3,α4,α5

−→α4,α5,α3,α2,α0,α1

They generate the extended aﬃne Weyl group of type D(1)

5 Below we give the list of mappings and their actions on the parameter space and on the root basis The dependent variables that appear below correspond to the following diagram:

.

··· y

ˆ x y x ···

w w

ˆ x y x ···

w

.

(4.11)

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(i) The mapw (3.1a) : (x, y

ˆ)→(x,y) defined by (3.1a) is described by the generators as

w(3.1a)= w5◦ w4◦ w1◦ w0◦ w2◦ w1◦ w0◦ σ01◦ w2:

(x, y

ˆ;a, p, q, r, s, λ)−→

x, (x + ap/q)

x + 1/(apq)

y

ˆ

1 +xa/(pq)

1 +x p/(aq);a, p, q, r, s,1

λ

,

α −→− α0,−α1,−α2,α0+α1+ 2α2+α3+α4+α5,−α4,−α5

(4.12)

(1)w (3.1b) : (x,y) →(x,y),

w(3.1b)= w0◦ w1◦ w4◦ w5◦ w3◦ w4◦ w5◦ π ◦ σ01◦ π ◦ w3:

x, y; a, p, q, r, s,1

λ

−→ y + r/(qλ)

y + 1/(qrλ)

x

1 +y/(qsλ)

1 +ys/(qλ), y; a, p, qλ2,r, s, λ

,

α −→− α0,−α1,α0+α1+α2+ 2α3+α4+α5,−α3,−α4,−α5

.

(4.13)

(ii) Miura transformationw (3.2) : (x,y) →(x, w

),

w(3.2)= w5◦ w4◦ σ01◦ w0◦ w1◦ w2◦ w0: (x,y; a, p, q, r, s, 1/λ) −→

x, ax + 1/(pq)

y

1 +ax/(pq);q, a, p, r, s, λ

,

α −→− α0− α2,α1+α2,−α1,α0+α1+α2+α3+α4+α5,−α4,−α5

.

(4.14)

(iii) The other Miuraw (3.3) : (x,y) →(x, w) is the same as w ( 3.2),

w(3.3)= w5◦ w4◦ σ01◦ w0◦ w1◦ w2◦ w0:

x, y; a, p, qλ 2,r, s, λ

−→

x, a x + 1/

pqλ2

y

1 +a x/

pqλ2 ;qλ2,a, p, r, s, λ

The action on the root basis is the same as that ofw (3.2)

(2)w (3.8a) : (x, w

)→(x,w) is the same as w ( 3.1a),

w5◦ w4◦ w1◦ w0◦ w2◦ w1◦ w0◦ σ01◦ w2:

(x, w

;q, a, p, r, s, λ) −→

x, (x + aq/ p)

x + 1/(apq)

w

1 +xq/(ap)

1 +xa/(pq);q, a, p, r, s,1

λ

. (4.16)

The action on the root basis is the same as that ofw (3.1a)

(3)w (3.8b) : (x

,w)→(x, w) is the same asw (3.1b),

w0◦ w1◦ w4◦ w5◦ w3◦ w4◦ w5◦ π ◦ σ01◦ π ◦ w3:

x

,w;q, a,

p

λ2,r, s, λ

−→

⎛

⎝ (w+rλ/ p)

w

+λ/(pr)

x

1 +w

λ/(ps)

(1 +w

sλ/ p)

,w

;q, a, p, r, s, λ

⎞

⎠. (4.17)

The action on the root basis is the same as that ofw (3.1b)

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(4)w (3.1b)◦ w(3.1a)= w(3.8b)◦ w(3.8a) acts on the root basis as

α −→α0,α1,α2− δ, α3+δ, α4,α5

whereδ = α0+α1+ 2α2+ 2α3+α4+α5is the null vector orthogonal to any basis; thus this mapping is a translation of the Weyl group

(iv) The mapping w(3.2)−1◦ w(3.8b)◦ w(3.8a)◦ w(3.2) : (x,y) →(x,y) acts on the

root basis as

α −→α0+δ, α1− δ, α2,α3,α4,α5

thus, this sequence defines a translation of the Weyl group in another direction

Next, we consider the “diagonal mappings” w(3.7) : (y, w

)→(y, w) and w ( 3.10) : ((y

,w)→(y, w)) from the Weyl group theoretical point of view However, these map-pings do not belong to the same representation of the Weyl group The above mapmap-pings

w (3.1a), and so forth can be lifted to the automorphism of a family of rational surfaces,

which are obtained from p1(C)×p(C) (x, y) by 2 times blowing-up on each line x=0,

x = ∞, y =0, ory = ∞ These rational surfaces are called “space of initial conditions” in

the sense of Okamoto-Sakai [8,12] For example,w (3.1a) can be lifted to an isomor-phism from a rational surface obtained by blowups at

(x, y

ˆ)=

− ap q , 0

,

− apq1 , 0

,

− pq a ,∞

,

− aq p ,∞

,

0,− rλ q

,

0,− λ qr

,

∞,− sq λ

,

0,−q

sλ

to a rational surface obtained by blowups at

(x,y) =

− ap q , 0

,

− apq1 , 0

,

− pq a ,∞

,

− aq p ,∞

,

0,− r

qλ

,

0,− 1 qrλ

,

∞,−qsλ

,

0,−qλ

s

.

(4.21)

All elements of the above Weyl group preserve these parametrization, but the space of initial conditions forw (3.7) andw (3.10) has diﬀerent parametrization Actually, w (3.7): (y, w

)→(y, w) is lifted to the isomorphism from a rational surface obtained by blowups at

(y, w

)=(∞, 0), (0,∞),

− λ

qr,−rλ

p

,

− r

qλ,− λ

pr

,

− qλ

s ,−ps

λ

,

− qsλ, − p

sλ

,

1

y,yw

=

0,p

1− a2q2

q

p2− a2

,

y, 1

yw

=

0, p

q2− a2

q

1− a2p2

(4.22)

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to a rational surface obtained by blowups at

y, w

=(∞, 0), (0,∞),

− r

qλ,− 1

prλ

,

− 1 qrλ,− r

pλ

,

−qsλ,− pλ

s

,

− qλ

s ,−psλ

,

1

y,y w

=

0,p

1− a2q2λ4

qλ2

p2− a2

,

y, 1

y w

=

0, p

q2λ4− a2

qλ2

1− a2p2

.

(4.23)

These two diﬀerent parametrizations are connected by Miura transformations, for exam-ple, a mappingw d (3.2) : (y, w

)→(x,y), which is not an element of the Weyl group In order to avoid these complications, it is suﬃcient to consider parallelograms instead of diagonal lines, that is, for example, the mappingw (3.2)−1◦ w(3.8b)◦ w(3.3)◦ w(3.1b) : (x,y) →(x,y) is equivalent to the mapping w(3.10)◦ w(3.7) : (y, w

)→(y, w) through Miura transformationsw d(3.2), and so forth and it acts on the root basis as

α −→α0+δ, α1− δ, α2+δ, α3− δ, α4,α5

This diﬀerence can be explained at the level of the diagram (4.11) by the fact that only vertically or horizontally adjoined pairs of dependent variables are mapped to each other

by our D(1)5 Weyl group

5 Conclusions

In this paper we have examined two diﬀerent q-discrete Painlev´e equations The first one

was derived by Kajiwara et al [6] in the form of a system of four depenent variables subject to two constraints Under this form, the equation was shown by Masuda [7] to be

aq-discrete analogue of PV The constraints were shown [13] to be too restrictive and the equation was extended accordingly The second system was obtained [11] from a special limit ofq-PVI In the present paper we have shown that, despite their radically diﬀerent forms, the two systems are in fact the same mapping The geometry of both equations (since it was not known that they coincided) was given in [11,13] as being that of the aﬃne Weyl group D(1)

5 , but no explicit construction was oﬀered In the present paper we have provided the missing link and have explicitly derived the equation at hand from the elementary Miura transformations of D(1)5 Further, we have clarified these relations from the point of view of Weyl group theory and of rational surfaces

References

[1] B Grammaticos and A Ramani, On a novel q-discrete analogue of the Painlev´e VI equation,

Physics Letters A 257 (1999), no 5-6, 288–292.

[2] , Generating discrete Painlev´e equations from aﬃne Weyl groups, Regular & Chaotic

Dy-namics 10 (2005), no 2, 145–152.

[3] B Grammaticos, A Ramani, and V Papageorgiou, Do integrable mappings have the Painlev´e

property?, Physical Review Letters 67 (1991), no 14, 1825–1828.

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