On a Schwarzian PDE associated with the KdV hierarchy

13 March 2000

Physics Letters A 267 Ž2000. 147–156 www.elsevier.nlrlocaterphysleta

On a Schwarzian PDE associated with the KdV hierarchy Frank Nijhoff a , Andrew Hone b, Nalini Joshi b a

b

Department of Applied Mathematics, The UniÕersity of Leeds, Leeds LS2 9JT, UK Department of Pure Mathematics, The UniÕersity of Adelaide, Adelaide, 5005 Australia

Received 12 November 1999; received in revised form 20 January 2000; accepted 21 January 2000 Communicated by A.P. Fordy

Abstract We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under Mobius transformations, that is related to the Korteweg–de Vries hierarchy. In fact, this PDE can be considered as the ¨ generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDEs for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the full Painleve´ VI equation, i.e. with four arbitrary parameters. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction In this Letter we introduce a non-autonomous 1 q 1-dimensional nonlinear evolution equation for the variable z Ž t, s ., namely: z st ztt z st zss z st z s s z st z t t zss ztt z s st t s z s st q q z st t q y z st q 2 q 2 zs zt zt zs zs zt zs zt

ž

1

s

q syt t

ž

1 y

Ž syt. 1

/ ž

/ ž

2

z s st y n2

1

y 2 Ž syt.

3

z st z s s zs

s2 zs t

2

y

zt

ž

z st y

1 z st2 2 zt zs ztt zt

t

/

/ ž y

s

/

q m2

z st t y

t 2 zt 2

s zs

ž

z st z t t

y

zt

1 z st2 2 zs

z st y

zt zss zs

/

/

s t Ž 4 t y 3s . s z s Ž 4 s y 3t . t z t n2 z s 1 q y m2 z t 1 q 2 t zt s zs t s2

ž

/

ž

/

.

Ž 1.1 .

We will refer to this equation simply as the Schwarzian PDE ŽSPDE.. It is straightforward to verify that it is invariant under Mobius transformations ¨ z Ž t, s.

¨ ag zz ŽŽ tt ,, ss.. qqbd ,

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 0 6 3 - 3

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with constant a , b ,g , d . It is straightforward to verify also that the Eq. Ž1.1. arises as the Euler–Lagrange equations from the following Lagrangian density: L z Ž t, s. s

1 Ž s y t . z st2 2

st

1 q

zs zt

2Ž s y t .

ž

n2

s zs t

3

zt

q m2

t zt s3 zs

/

.

Ž 1.2 .

Note that the Lagrange density L depends explicitly on the independent variables s and t. We further assert the following properties of the SPDE: 1. 2.

3. 4. 5. 6. 7.

The SPDE Ž1.1. arises from a Žnon-isospectral. Lax pair and is, therefore, expected to be completely integrable in the soliton sense. The SPDE Ž1.1. is part of a compatible parameter-family of PDEs, meaning that the same dependent variable z obeys Eq. Ž1.1. simultaneously for all possible pairs of parameters n,m Žwith each parameter being associated with a different independent variable.. We will make this statement more precise in what follows. This infinite family of PDEs is equivalent to the Schwarzian KdV hierarchy via expansions on the independent variables. As indicated already, the system carries a Lagrangian structure. We expect that from this Lagrangian one can infer that the system is multi-Hamiltonian, although we will not discuss this point further here. There exists a Miura chain, i.e. a short sequence of differential relations connecting the SPDE with other generating PDEs sharing the main properties with the SPDE. The similarity reduction of the SPDE under scaling symmetry reduces the SPDE to the full sixth Painleve´ equation ŽPVI. with four arbitrary parameters. The SPDE has a fully discrete counterpart which is integrable in its own right.

2. Properties of the SPDE and related generating PDEs We now demonstrate or give arguments for the properties that we listed in the Introduction. 2.1. Lax pair The following overdetermined set of linear equations constitute a Lax pair for the SPDE: 2 twt s n

ž

2 s ws s m

1 yk n Ž 1 q a . 0 wq a k y t k n2 Ž 1 y a2 . r Ž 2 tz t .

1 0

ž

ž

/

1 yk m Ž 1 q b . 0 wq 2 b k y s k m Ž 1 y b 2 . r Ž 2 sz s .

1 0

ž

/

2 t 2 zt ynt Ž 1 y a .

/

w,

2 s2 zs yms Ž 1 y b .

/

Ž 2.1a . w.

Ž 2.1b .

In the above a s aŽ t, s . and b s bŽ t, s . are auxiliary variables. The compatibility conditions arising from the consistency of cross-differentiating Ž2.1a. and Ž2.1b. are nsa s s mtbt s

stz st s

1 2Ž s y t .

mt 2 z t b y ns 2 z s a syt

n

.

2

s2 zs t zt

2

Ž1ya . ym

2

t 2 zt s zs

Ž1 y b2 . ,

Ž 2.2a . Ž 2.2b .

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The SPDE Ž1.1. follows immediately from the relations Ž2.2a. and Ž2.2b.. In fact, differentiating Ž2.2b. with respect to s and with respect to t yields relations for a t and bs , whilst Ž2.2a. gives a s and simultaneously bt . Thus, with these expressions we can cross-differentiate, e.g. E t a s s Es a t , to obtain an equation where we can use the previous obtained expressions to to eliminate all a’s and b’s Žmiraculously, it turns out that the single remaining a or b drops out entirely at the end.. We note that the Lax pair Ž2.1. falls in the general class of Lax pairs of the non-isospectral type1 that were treated in Ref. w1x. However, although general families of Lax equations were postulated in that paper, the precise reduction which leads to Ž2.1. was not investigated there. 2.2. Consistency and integrability The SPDE Ž1.1. is compatible with itself in the following sense: if one considers for one and the same function z s z Ž t 1 ,t 2 ,t 3 , . . . . of infinitely many independent variables t 1 ,t 2 ,t 3 , . . . copies of the Eq. Ž1.1. in terms of any two distinct variables t i ,t j replacing t and s inŽ1.1., each associated with its own parameters n i ,n j replacing n and m in Ž1.1., then all these copies of the SPDE on one and the same function z are consistent. Thus, we have effectively an infinite family of commuting flows all given by the same equation, but with different parameters: in other words the SPDE represents an infinite-parameter family of consistent PDEs. This statement can be verified by direct calculation, but since Ž1.1. is not in evolution form the verification is most easily done using the Lax pair. In fact, suppose we augment the system Ž2.1. with a third part, i.e. an equation similar to Ž2.1a. and Ž2.1b. but now in terms of a third independent variable Ž u say. associated with a new parameter l Žinstead of n or m. and a third auxiliary variable c. Obviously the compatibility relations with the original members of the Lax pair, namely with Ž2.1a. or Ž2.1b. lead to extra relations of the same type as Ž2.2. involving now relations for a u , bu , c s , c t as well as z su and z t u . We have to check the consistency of these relations, i.e. the validity of expressions such as Eu a s s Es a u and Eu z st s E t z su s Es z t u . This follows by tedious but straightforward direct calculation. As is clear from the variational equations from the Lagrangian Ž1.2., namely

dL dz

E2 s

EL

E

EL

E

EL

Et

E zt

ž / ž / ž / y

E sE t E z st

Es

E zs

y

s0 ,

Ž 2.3 .

the SPDE can be written as a conservation law. Since we know from the above argument that the SPDE can be imposed on z s z Ž t, s,u, . . . . in as many independent variables as we want Žeach associated with its own parameter n,m, l , . . . ., we derive in this way an infinity of conservation laws all for the same object z. What is potentially nontrivial is to rewrite these conservation laws in terms of only one pair of preferred variables, s and t say. It is only at this point that possible nonlocalities in the explicit expressions for the conservation laws may occur. We will not discuss this point here in detail any further, but postpone this discussion to future investigations of the SPDE. 2.3. Connection with SKdV hierarchy Using the expansion of the independent variables:

E Et

1

sy t

n

`

1r2

Ý js1

1 j

E

t E xj

E Es

m sy s

1r2

`

Ý js1

1 j

E

s E xj

,

Ž 2.4 .

This does not mean that the spectral parameter k depends on s,t: k is constant w.r.t. the independent variables. However, the poles in the Lax matrices depend explicitly on s and t.

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where x 1 , x 2 , . . . is the infinite sequence of higher times associated with the KdV hierarchy, we can expand the Lagrangian Ž1.2. in powers of s and t as follows: 1 Ls

mn

2 Ž st . 3r2

sqt

½

q syt

ž

1

1 y

s

t

/

zx2

y

z x21 x 1

z x1

z x21

q

ž

1 s2

1 y

t2

/

z x3 z x1

y2

z x1 x1 z x1 x 2 z x21

q

z x21 x 1 z x 2 z x31

5

q... .

Ž 2.7 . Thus order by order we obtain a sequence of Lagrangians which turns out to be exactly the Lagrangians for the SKdV hierarchy, the first equations in the sequence reading zx2

s  z , x1 4 '

z x1

z x1 x1 x1 z x1

y

3 z x21 x 1 2 z x21

,

Ž 2.6a .

which is the Schwarzian KdV equation w2–4x, and z x3 z x1

E2 s2

E x 12

zx2

ž /

q

z x1

3 z x2 2 2 z x21

,

Ž 2.6b .

which is the first higher-order SKdV equation. ŽStrictly speaking the sequence of Lagrangians appearing in Ž2.6. yield the x 1 derivatives of this hierarchy of equations.. The expansions Ž2.4. amount exactly to the transition from the higher time variables in the hierarchy to so-called Miwa coordinates w5x. We mention that the notion of a generating PDE for a hierarchy of integrable nonlinear evolution equations, and their corresponding Lagrangians, is in spirit the same as the idea of ‘compounding hierarchies’ that was put forward some years ago by one of the authors w6x. We refer also to w7x for related issues concerning the Baker–Akhiezer function for the KP hierarchy in terms of Miwa coordinates.

2.4. Miura chain The different PDEs in the Miura chain are related by a sequence: SchwarzianPDE

™ modifiedPDE ™ regularPDE

The mPDE Žmodified PDE. is the equation for a variable Õ Ž t, s . that is connected to the auxiliary variables aŽ t, s . and bŽ t, s . via na s y2 tE t log Õ , mb s y2 sEs log Õ .

Ž 2.7 .

It is helpful to introduce the quantity

Y'

m tz t Ž 1 y b . n sz s Ž 1 y a .

,

Ž 2.8 .

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so that from Ž2.2a. we obtain

Es E t log Õ s

nm 4 st Ž s y t .

t Ž 1 y a. Ž 1 q b . Y y s Ž 1 q a. Ž 1 y b .

1 Y

.

Ž 2.9a .

Differentiating Ž2.8. with respect to s and t and using Ž2.2b. yields 2 stEs E t logY s nsEs Ž 1 y Y .

2 tY q Ž s y tY . Ž 1 q a .

Ž t ys. Y

y mtE t Ž 1 y Y .

2 s y Ž s y tY . Ž 1 q b .

Ž t ys. Y

.

Ž 2.9b .

The mPDE Žas an equation for Õ alone. can be obtained by solving from the quadratic Eq. Ž2.9a. for Y in terms of Õ and its derivatives Žwhich appear in a and b via Ž2.7.. and substituting the result into Eq. Ž2.9b.. The resulting complicated-looking PDE for Õ, Žwhich we omit because of its length., is of second degree in the highest derivative Õs st t . Finally to derive the regular PDE, the last part of the Miura chain, it is necessary to apply the gauge transformation

ws

S krÕ

ž

Õ f 0

/

Ž 2.10 .

to the Lax representation Ž2.1., where the function S remains to be specified. This gauging yields the non-isospectral Lax pair

ft s

ž

0 P

1 0 fy 0 kyt

fs s

ž

0 Q

1 m y BUs 0 fy 0 k y s B Ž m y BUs .

/

ž

n y AUt

Ut

A Ž n y AUt .

AUt

ž

/

/

Us BUs

f

Ž 2.11a .

/

Ž 2.11b .

f

where A, B, P and Q are some auxiliary variables that can be expressed in terms of the earlier variables a, b, Õ, z and S. The main new variable U enters through Ut s

n2 Ž a2 y 1. Õ 2 4 t 2 zt

Us s

m2 Ž b 2 y 1. Õ 2 4 s2 zs

,

Ž 2.12 .

the two expressions being consistent through cross-differentiation as can be easily verified using the relations Ž2.2. and Ž2.7.. The explicit expressions for the variables P and Q in terms of S, z and Õ reveal that Ps s Q t , but they are otherwise arbitrary Žfixing a choice for P and Q, for instance P s nrŽ2 t 1r2 . and Q s mrŽ2 s 1r2 . to be consistent with the formalism of Ref. w10x, yields conditions on the choice of the auxiliary variable S .. The compatibility conditions between Ž2.11a. and Ž2.11b. yields, apart from the relation Ps s Q t , the set of coupled relations Ust s

1

mUt y nUs q 2 Ž A y B . Us Ut

syt

As s Q q Bt s P q

1 tys 1

syt

Ž 2.13a .

2 Ž A y B . Us q m Ž A y B .

Ž 2.13b .

2 Ž A y B . Ut y n Ž A y B . .

Ž 2.13c .

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Differentiating Ž2.13b. with respect to t and Ž2.13c. with respect to s and combining them we obtain a coupled set of equations for A y B and U. By solving for A y B in Ž2.13a. and substituting the result into this coupled equation we find the PDE Us st t s Us st

ž

1 q syt

q Us s

ž

Ust

q

Us m2

Ut 2

Ž syt.

m2

2

Ut

q

3

2 Ž s y t . Us 1 q 2Ž s y t .

Ust2

Us2

Ut t Ut

/ ž

y

Ust2

q Ust t

Us2

1 q tys 1

s y t Us

1

1 y

Us

Ut

/

Us s Us

/ ž q Ut t

Ž Us q Ut q 2 Ž t y s . Ust . y

ž

q

Ut

Ust

y

Ust

/

y Ust

n2

Us s Ut t

Us2 2

Ž s y t . Ut

n2

Us Ut

Us 3

2 Ž s y t . Ut

2

y

Ust2 Ut

2

1 q

Ust

s y t Ut

/

Ž Us q Ut q 2 Ž s y t . Ust .

.

Ž 2.14 .

The Eq. Ž2.14. is the regular PDE, by which we mean the PDE that upon using the expansion Ž2.4. yields the entire hierarchy of KdV equations, and it may be derived from the following Lagrangian: L UŽ t , s. s

1 2

Ž syt.

Ust2

1 q

Us Ut

2Ž s y t .

ž

n2

Us Ut

q m2

Ut Us

/

.

Ž 2.15 .

It is remarkable that the only apparent difference between the Lagrangians Ž1.2. and Ž2.15. resides in the way the variables s and t enter. The Miura transformation between the regular PDE Ž2.14. and the mPDE given by the system Ž2.9. is

Ž t y s . Ust q mUt q nUs Ž n q 2 twt . Us q Ž m y 2 sws . Ut s , Ž s y t . Ust q mUt q nUs Ž n y 2 twt . Us q Ž m q 2 sws . Ut

Ž 2.16 .

where w ' log Õ. To derive this implicit Miura relation it suffices to use Ž2.13. together with Ž2.7. and the identity AyBs

1 2

ž

n Ž 1 q a. Ut

y

mŽ 1 q b . Us

/

.

The relation between the SKdV and mKdV equations is given by the famous Cole–Hopf transformation. For the SPDE Ž1.1. and the mPDE Ž2.9. this relation has an analogue in the form of the implicit transformation 4 st Ž s y t . wst s

t 2 zt sz s

Ž m2 y 4 s 2 ws2 . y

s2 zs tz t

Ž n2 y 4 t 2 wt2 . ,

which can be straightforwardly inferred from Ž2.8. in combination with Ž2.9a. and Ž2.7.. 2.5. Connection with PVI The similarity reduction of the SPDE is obtained via the constraint

m z q tz t q sz s s 0 .

Ž 2.17 .

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Using the constraint to eliminate the derivatives with respect to one of the independent variables Ž s, say. and introducing the new variable 2 t zt W Ž t. s1q m z we obtain from the SPDE a third-order ODE for W which reads: W XXX s 2

1

ž

1 q

W

y

Wy1

/

W X W XX y 2

ž

1

1 q

t

tys

/

W XX y

1 2

W

1

q

Ž W y 1.

2

q W Ž W y 1.

/

W X3

Ž 11t y 5s . W q 2 s y 5t X 2 W 2 Ž s y t . tW Ž W y 1 . 3

q

ž

1

2

3

n 2 s 2 W 3 y m 2 t 2 Ž W y 1 . q m2 Ž s y t . W 3 Ž W y 1 . q 2 Ž s y t . tW 2 Ž W y 1 . 2 2

2

Ž s y t . t W Ž W y 1. 2 2

n s W

2

q

2

2

WX

2 Ž s y t . W y t y m2 t 2 Ž W y 1. Ž s y t . W q t y 2 s 3

2 Ž s y t . t 3 W Ž W y 1. y

m2 W Ž W y 1. Ž s q t . W y t

,

2Ž s y t . t 3

Ž 2.18 .

where the prime denotes differentiation with respect to t. Eq. Ž2.18. can be integrated once using the integrating factor t Ž t ys. Ž t ys. W yt W Ž W y 1. leading to the following second-order equation for W: 1 1 1 tys 1 1 Wy1 W XX s q q W X2 y q y WX 2 W Wy1 t tys Ž t ys. Wyt Ž t ys. Wyt

ž

q

/

W Ž W y 1. Ž t y s . W y t 2 t Ž t ys.

ž

ž

m2

/

m2 y

t

n2 s

Ž t y s. W 2

n 2s

q t Ž t y s . Ž W y 1.

2

y

Ž Ž t ys. W yt .

2

/

,

Ž 2.19 . where the integration constant is conveniently chosen as n 2 s with n independent of t, s Žnoting that W depends on t, s through the combination srt .. It is not hard to see that Eq. Ž2.19. is equivalent to the Painleve´ VI equation ŽPVI . after the trivial change of variables trŽ t y s . t . Thus, we obtain PVI in the standard form 2

d w dt

2

1 s

ž

1

2 w q

1 q

1

dw

q

w Ž w y 1. Ž w y t . 2t 2 Ž t y 1 .

2

/ž / ž

1

y

wyt

wy1

¨

2

ž

dt

2

m ym

2

t

t w2

1 q

qn

2

1 q

ty1 ty1

Ž w y 1.

wyt 2

yn

2

/

dw

dt t Ž t y 1.

Ž wyt .

2

PVI

/

.

for w Žt . s W Ž t .. PVI , which incidentally was first found by R. Fuchs in Refs. w8,9x in 1905, Žand not by either Painleve´ or Gambier as is ocasionally claimed in the recent literature., was found to be connected to the lattice 2

Alternatively, the similarity reduction can be obtained by explicitly solving the constraint Ž2.17. as z Ž t, s . s Ž st .ym r2 ZŽ sr t . . This leads to a third order equation for ZŽ sr t ., namely the Schwarzian PVI which we found in Ref. w13x via a different approach.

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KdV systems in Ref. w10x. It is known that PVI is related to the Toda lattice, cf. w11,12x. for example. In a recent paper w13x we presented the full Miura chain associated with PVI, consisting of ordinary differential as well as ordinary difference equations, showing that in fact the parameter-family of ODEs that constitute PVI has a natural description in terms of both discrete as well as continuous equations. The Miura chain for PVI is just the reduction of the Miura chain for the SPDE under the similarity constraint Ž2.17., although we implimented this constraint differently in Ref. w13x. Thus the SPDE Ž1.1. yields PVI as similarity reduction with four arbitrary parameters. As far as we are aware this is the first case of an integrable scalar PDE that reduces to the full PVI equation. In the literature w14–17x PVI was obtained in several circumstances, either from the reduction of the three-wave resonant interaction system or in the context of the Einstein equations. However, in these instances only special parameter-cases of PVI were obtained. In Ref. w18x a connection between PVI and so-called generalised Ernst equations was studied, which seems to give a connection with full PVI, but the starting point is a complicated system of equations. What we have demonstrated here is that the full PVI equation is directly connected to the KdV hierarchy. 2.6. Connection with equations on the lattice We finish by pointing out that the SPDE Ž1.1. as well as the other generating PDEs in the Miura chain are not only compatible as infinite-parameter families of PDEs in the sense pointed out above, but also are compatible with a system of discrete equations. In fact, the actual derivation of the SPDE was based on this underlying structure which involves continuous as well discrete components. The following statement holds: Proposition 1. The SPDE (1.1) is consistent with the following set of differential-difference equations for z(t,s) s z n ,m(t,s): yt

ys

E zn,m Et E zn,m Es

sn

Ž z nq1 , m y z n , m . Ž z n , m y z ny1, m .

sm

Ž 2.20a .

z nq 1, m y z ny1, m

Ž z n , mq1 y z n , m . Ž z n , m y z n , my1 . z n , mq1 y z n , my1

,

Ž 2.20b .

in which the parameters of the SPDE play the role of independent discrete Õariables. In fact, one can show by direct calculation that the operations of shifting in the variables n and m governed by the Eqs. Ž2.20. commute with the continuous evolution according to the flows in terms of the variables s and t. Thus, in the above proposition the quantity z nq 1, mŽ t, s . is understood to solve the SPDE Ž1.1. with the parameter n replaced by n q 1, whilst z n, mq1Ž t, s . solves the SPDE with the parameter m replaced by m q 1, and so on. Obviously, a similar statement holds for the other generating PDEs, in which case the differential-difference equations for Õ Ž t, s . s Õn, mŽ t, s . read

E y2 t

Et

E log Õn , m s na n , m ,

y2 s

Es

log Õn , m s nbn , m ,

Ž 2.21 .

with the auxiliary variables aŽ t, s . s a n, mŽ t, s . and bŽ t, s . s bn, mŽ t, s . which appeared in the Lax representation Ž2.1. given explicitly in terms of Õn, m by the relations Õnq 1, m y Õny1, m Õn , mq1 y Õn , my1 an , m ' , bn , m ' . Ž 2.22 . Õnq 1, m q Õny1, m Õn , mq1 q Õn , my1

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Also for the object UŽ t, s . s Un, mŽ t, s . we have a set of differential-difference equations compatible with the generating PDE Ž2.14., namely

E Un , m Et

n s Unq 1, m y Uny1, m

,

E Un , m Es

m s Un , mq1 y Un , my1

.

Ž 2.23 .

Finally, to complete the picture we mention the associated fully discrete equation for the variable z n, mŽ t, s ., i.e. the partial difference equation ŽPDE.

Ž z n , m y z nq1, m . Ž z n , mq1 y z nq1, mq1 . s s . Ž z n , m y z n , mq1 . Ž z nq1, m y z nq1, mq1 . t

Ž 2.24 .

The Eq. Ž2.24., which was first given in Ref. w19x, is possibly the most fundamental equation in the entire structure, and it has been studied at length in the context of discrete conformal maps Žsee the recent monograph w20x.. Note that in Ž2.24. the independent variables t, s of the SPDE now enter as parameters of the discrete equation, which can be thought of as the parameter measuring the width of the discrete grid in the direction associated with the discrete variable variables n, respectively m. Taking into account the earlier statement that the SPDE represents a compatible parameter-family of equations, a similar statement holds for the discrete Eq. Ž2.24. as well: it represents a parameter-family of compatible partial difference equations, i.e. in terms of each pair of discrete variables n i , n j Žreplacing n respectively m in Ž2.24. . associated with the continuous parameters t i ,t j Žreplacing t respectively s . we have a copy of Eq. Ž2.24. for one and the same function z s z n1 , n 2 , . . . Ž t 1 ,t 2 , . . . . and these equations are not only compatible amongst themselves but also with the corresponding copies of the SPDE. In Ref. w21x Žsee also w22x. the similarity reduction on the lattice was formulated, namely by imposing the following constraint which is compatible on the lattice with the Eq. Ž2.24.: n

Ž z nq 1, m y z n , m . Ž z n , m y z ny1, m . z nq 1 , m y z ny1, m

qm

Ž z n , mq1 y z n , m . Ž z n , m y z n , my1 . z n , mq1 y z n , my1

ž

s mq

1 2

/

zn,m ,

Ž 2.25 .

It was, however only in the recent paper w13x that a closed-form third-order ordinary difference equation ŽO D E. was found that represents the similarity reduced equation, namely 2 Ž r 2 y 1 . Ž z nq 1 y z n . s 2 r 2

m z nq 1 Ž z nq2 y z n . y Ž n q 1 . Ž z nq2 y z nq1 . Ž z nq1 y z n .

Ž m y m y n . Ž z nq 2 y z n . q Ž n q 1 . Ž z nq2 y 2 z nq1 q z n .

= 2r2

m z n Ž z nq1 y z ny1 . y n Ž z nq1 y z n . Ž z n y z ny1 .

Ž m y m q n . Ž z nq 1 y z ny1 . q n Ž z nq1 y 2 z n q z ny1 .

q z nq1 y z n

q z n y z nq1 ,

Ž 2.26 . Žwhere we have omitted the label associated with the discrete variable m, since every z in Ž2.26. carries the same m, the latter entering only as a parameter in the equation.. Eq. Ž2.26. is related to the discrete PainleÕe´ equation that was studied at length in Ref. w10x. Discrete Painleve´ form an exciting class of ordinary difference equations which is increasingly becoming an area of intense activity Žsee Ref. w23x for a recent review.. In Ref. w13x we presented the discrete and continuous Miura chains of both discrete and continuous equations of Painleve´ type that are associated with the third-order O D E Ž2.26.. As is obvious from what we said above concerning the similarity reduction of the SPDE to the full PVI equation, these discrete and continuous equations are directly associated with PVI.

F. Nijhoff et al.r Physics Letters A 267 (2000) 147–156

156

3. Conclusions and outlook It has been a long-standing conjecture whether all differential equations of Painleve´ type could be derived as reductions from a larger integrable nonlinear equation or system of PDEs by similarity reduction. This is, in a sense, the converse of the famous ARS hypothesis w24,25x. The programme developed in Refs. w10,13,21,22x has demonstrated that there is an intimate interplay between discrete and continuous structures behind the Painleve´ equations and that notably PVI and its discrete counterparts are obtainable from either a system of PDEs on the full lattice or of a very rich parameter-family of PDEs in the continuum, which we have exhibited in the present paper. An obvious conjecture is that these connections also extend to discrete Painleve´ equations which are of a possibly richer type, in particular the so-called q-Painleve´ equations, which seemingly are ‘beyond’ the PVI equation. In that case, we may expect that there exist partial difference equations of q-type, generalizing the SPDE Ž1.1., whose similarity reduction leads to those discrete Painleve´ equations. The ultimate aim would be to find the ‘big equation’ that sits above the recently found most general discrete Painleve´ equation by Sakai w26x, which would be a partial difference equation of the elliptic type.

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x

S. P Burtsev, A.V. Mikhailov, V.E. Zakharov, Theor. Math. Phys. 70 Ž1987. 227. J. Weiss, J. Math. Phys. 24 Ž1983. 1405. J. Weiss, Phys. Lett. A 105 Ž1984. 387. O. Mokhov, Symplectic and Poisson Geometry on Loop Spaces of Manifolds and Nonlinear Equations, hep-thr9504076. T. Miwa, Proc. Jpn. Acad. Ser. A 58 Ž1982. 9. F.W. Nijhoff, Integrable Hierarchies, Lagrangian Structures and Non-commuting Flows, in: M. Ablowitz, B. Fuchssteiner, M. Kruskal ŽEds.., Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, 1987, pp. 150-181. B. Konopelchenko, L. Martinez Alonso, Phys. Lett. A 258 Ž1999. 272. R. Fuchs, C.R. Acad. Sci. ŽParis. 141 Ž1905. 555. R. Fuchs, Math. Ann. 63 Ž1907. 301. F.W. Nijhoff, A. Ramani, B. Grammaticos, Y. Ohta, On discrete Painleve´ equations associated with the lattice KdV systems and the Painleve´ VI equation, solv-intr9812011, submitted to Stud. Appl. Math. K. Okamoto, Ann. Mat. Pura Appl. 146 Ž1986. 337. K. Okamoto, J. Fac. Sci. Univ. Tokyo 34 Ž1987. 709–740. F.W. Nijhoff, N. Joshi, A. Hone, On the discrete and continuous Miura chain associated with the sixth Painleve´ equation, preprint solv-intr9906006. A.S. Fokas, R.A. Leo, L. Martina, G. Soliani, Phys. Lett. A 115 Ž1986. 329–332. V.I. Gromak, V.V. Tsegel’nik, Theor. Math. Phys. 78 Ž1989. 14–23. L. Martina, P. Winternitz, Ann. Phys. 196 Ž1989. 231–277. K.P. Tod, Phys. Lett. A 190 Ž1994. 221–224. R. Halburd, PhD thesis, University of New South Wales, 1995. F.W. Nijhoff, H.W. Capel, Acta Appl. Mat. 39 Ž1995. 133–158. A.I. Bobenko, R. Seiler, Discrete Integrable Geometry and Physics, Oxford University Press, 1999. F.W. Nijhoff, On some Schwarzian Equations and their Discrete Analogues, in: A.S. Fokas, I.M. Gel’fand ŽEds.., Algebraic Aspects of Integrable Systems; In Memory of Irene Dorfman, Birkhauser, 1997, pp. 237-260. ¨ F.W. Nijhoff, Discrete Painleve´ Equations and Symmetry Reduction on the Lattice, in: A.I. Bobenko, R. Seiler ŽEds.., Discrete Integrable Geometry and Physics, Oxford Univ. Press, 1999, pp. 209-234. B. Grammaticos, F.W. Nijhoff, A. Ramani, Discrete Painleve´ Equations, in: R. Conte ŽEd.., Lectures presented at the School: The Painleve´ Property, One Century Later, CRM Series in Mathematical Physics, Springer, Berlin, 1999, pp. 413-516. M.J. Ablowitz, A. Ramani, H. Segur, J. Math. Phys. 21 Ž1980. 715. M.J. Ablowitz, A. Ramani, H. Segur, J. Math. Phys. 21 Ž1980. 1006. H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painleve´ equations, preprint Kyoto University, May 1999.