Backlund transformations for discretePainleve equations1 : Discrete PII - PV

A. Sakka †, U. Mugan ‡

† Department of Mathematics, Islamic University of GazaP.O. Box 108 Rimal, Gaza, PalestineE-mail: asakka@mail.iugaza.edu

‡ Department of Mathematics, Bilkent University06800 Bilkent, Ankara, TurkeyE-mail: mugan@fen.bilkent.edu.tr

30 September 2004

Abstract

Transformation properties of discrete Painleve equations are investigated byusing an algorithmic method. This method yields explicit transformationswhich relates the solutions of discrete Painleve equations, discrete PII-PV,with different values of parameters. The particular solutions which are ex-pressible in terms of the discrete analogue of the classical special functions ofdiscrete Painleve equations can also be obtained from these transformations.

1 Introduction

Painleve and his school classified the integrable second order equation of the formy′′ = f(x, y, y′) where f is rational in y and y′ and analytic in x, whose solutionshave no movable critical points, and discovered six transcendental equations thatare called Painleve equations, PI - PVI [1, 2, 3]. Their general solutions can notbe expressed in terms of the known elementary functions and can be regarded asnonlinear analogues of the classical special functions. However, PII - PVI have ra-tional solutions and solutions expressible in terms of the classical special functionsfor certain values of parameters. PII - PVI also possess Backlund transformationswhich relate solutions of the same equation with different values of parameters, or tosolution of another equation of Painleve type [4, 5, 6]. Although, Painleve equationswere first discovered from strictly mathematical considerations, they have appearedin physical applications. For example, PIII arises in the Ising model [7], and PIV

appears in quantum gravity [8].Discrete analogues of the Painleve equations are nonautonomous mappings that

are integrable in the same sense as the continues Painleve equations [9, 10, 11], and

1Accepted for publication by ”CHAOS, SOLITONS and FRACTALS”, April 2005

2 A Sakka, U Mugan

recently have attracted much attention. The discrete Painleve equations, dPI-dPVI,which have the form

xn+1 =f1(xn; n) + xn−1f2(xn; n)

f3(xn; n) + xn−1f4(xn; n)(1.1)

where fj(xn; n) are polynomials of degree at most four in xn [12]. In continuous limit,the discrete Painleve equations yield a Painleve equation, though some of the discretePainleve equations have limits more then one Painleve equation. More over, discretePainleve equations possess properties similar to the ones of the continuous Painleveequations. For example, discrete Painleve equations also form coalescence cascades[9, 13, 14, 15], possess Lax’s pairs [16, 17, 18], have rational solutions for certainvalues of parameters [19, 20, 21], have particular solutions for certain parametervalues expressible in terms of the discrete analogue of the special functions [16, 19, 22,23, 24], and have Backlund transformations [19, 22, 24]. Discrete Painleve equationsalso appear in physics, for example, the computation of a certain partition function ina model of two-dimensional quantum gravity led discrete PI [25]. The only differencebetween continuous and discrete Painleve equations is that the continuous Painleveequations have unique canonical form up to a Mobius transformation, but there ismore then one inequivalent discrete equation which has the Painleve equation as itscontinuous limit.

Recently, Sakai [26] characterized the Painleve equations in the frame work of thetheory of rational surfaces, and showed that the translation part of the affine Weylgroups give rise to discrete Painleve equations whereas the whole group acts as theirgroups of symmetries, Backlund transformations. The six continues Painleve equa-tions appear as degenerate cases of this construction. The geometrical description inthe framework of the affine Weyl group E

(1)6 of the asymmetric q-PV and asymmetric

d-PIV which are known as discrete analogues of the Painleve VI equation, was givenin the recent work of Grammaticos, Ramani and Ohta [27].

In this article, we investigate the transformation properties of the discrete Painleveequations by using an algorithmic method which is similar to the method developedby Fokas and Ablowitz [5] for investigating the transformation properties of thecontinuous equations of the Painleve type. In [5], for given continuous Painleveequations

v′′ = P2 (v′)2 + P1 v′ + P3 (1.2)

where P1, P2, P3 depend on v, independent variable z and set of parameters α, thetransformation of type

u(z; α) =v′ + av2 + bv + c

dv2 + ev + f(1.3)

where a, b, ..., d depend on z only and u(z; α) solves some second order equation ofthe Painleve type with set of parameters α , was considered. If we solve (1.3) for v′,

Backlund transformations for discrete equations : Discrete PII - PV 3

we obtain

v′ = (du− a)v2 + (eu− b)v + (fu− c). (1.4)

That is, solution v of the given equation of Painleve type also satisfies a Riccatiequation with the coefficients depending linearly on the solution u of related Painlevetype equation. By following the similar argument, for a given discrete Painleveequation (1.1) with parameter set α, we consider discrete Riccati equation, that is

xn+1 =Anxn + Bn

Cnxn + Dn

(1.5)

where An = A1,nyn + A0,n, Bn = B1,nyn + B0,n, Cn = C1,nyn + C0,n, andDn = D1,nyn + D0,n such that yn solves discrete equation of Painleve type withparameter set α. The aim is to determine Aj,n, ..., Dj,n, j = 0, 1 requiring that(1.5) defines a one-to-one invertible map between the solutions xn of a given dis-crete Painleve equation, and solutions yn of some second order discrete equation ofPainleve type. This method yields explicit transformations between a given discretePainleve equation and the same discrete Painleve equation but with different valuesof its parameters, and between two different discrete equations of Painleve type. Asan application of the method, we obtain particular solutions of discrete Painleveequations in terms of discrete analogue of the classical special functions.

The method can be summarized as follows: From equation (1.5), one writes

xn−1 = −Dn−1xn −Bn−1

Cn−1xn − An−1

. (1.6)

Substituting xn+1 and xn−1 given in (1.5) and (1.6) respectively into given discretePainleve equation (1.1) gives an equation which is polynomial for xn with the coeffi-cients depending on Aj,n, ..., Dj,n, j = 0, 1, yn and yn−1. Now, we choose Aj,n, ..., Dj,n

such that the polynomial for xn reduces to a polynomial of degree one or of degreetwo. That is,

E(yn, yn−1 ; n)xn + F (yn, yn−1 ; n) = 0, (1.7)

or

E(yn, yn−1 ; n)x2n + F (yn, yn−1 ; n)xn + G(yn, yn−1 ; n) = 0. (1.8)

If one solves (1.7) or (1.8) for xn and substitutes into (1.5), (1.5) yields a discreteequation of Painleve type for yn. It turns out that, similar to the case of continuousPainleve equations, discrete PII - PV admit transformations of both types (1.7) and(1.8). However, discrete PVI does not admit a transformation of type (1.7). Inthis article we will consider the transformation of type (1.7), the type (1.8) will beconsidered later and published elsewhere.

4 A Sakka, U Mugan

2 Discrete Painleve II Equation

In this section, we consider d-PII equation:

xn+1 + xn−1 =znxn + a

1− x2n

, (2.1)

where zn = αn + β, and α, β, a are constants. Substituting xn+1 and xn−1 given in(1.5) and (1.6) respectively into (2.1) gives the following polynomial for xn,

(znxn + a)(Cnxn + Dn)(Cn−1xn − An−1) =

(1− x2n)

[(Anxn + Bn)(Cn−1xn − An−1)− (Cnxn + Dn)(Dn−1xn −Bn−1)

](2.2)

Our goal now is to choose An, ...Dn in such a way that (2.2) becomes a linear equationfor xn. (2.2) reduces to a linear equation for xn,

(Bn + Bn−1 + zn − 2)xn = Bn −Bn−1 − a (2.3)

with the choice of An = Cn = Dn = 1. Without loss of generality, we may chooseBn = yn − 1

2zn − 1

4α + 1. Hence, equations (1.5) and (2.3) respectively give

yn = (xn + 1)(xn+1 − 1) +1

2zn +

1

4α (2.4)

and

xn =yn − yn−1 + ν

yn + yn−1

, (2.5)

where ν = −12α−a. Eliminating xn between (2.4) and (2.5) leads to a discrete form

of PXXXIV [22]:

(yn + yn−1)(yn + yn+1) =ν2 − 4y2

n

yn − 12zn − 1

4α

. (2.6)

Miura transformation (2.5) was also given in [22].

2.1 Backlund Transformation for d-PII

Since, (2.6) is quadratic in ν, thus yn(ν) = yn(−ν). But then, from equation (2.5)

xn(−ν) =yn(−ν)− yn−1(−ν)− ν

yn(−ν) + yn−1(−ν)

=yn(ν)− yn−1(ν)− ν

yn(ν) + yn−1(ν)

= xn(ν)− 2ν

yn(ν) + yn−1(ν)(2.7)

Backlund transformations for discrete equations : Discrete PII - PV 5

Hence, expressing yn in terms of xn and using ν = −12α − a give the following

Backlund transformation for d-PII

xn = xn − 2ν(xn + 1)

2(xn+1 − 1)(xn + 1)− znxn − a; a = −a− α. (2.8)

Backlund Transformation for d-PII was also given in [22, 24, 28].

2.2 Special Solution

The transformation (2.5) breaks down if

yn + yn−1 = 0 (2.9)

and

yn − yn−1 + ν = 0. (2.10)

By solving (2.9) and (2.10), we find that yn = ν = 0. Substituting yn = ν = 0 into(2.4) yields the following discrete Riccati equation:

xn+1 =2xn − zn + a + 2

2(xn + 1). (2.11)

Therefore, particular solution of dPII is characterized by (2.11), iff a = −12α. (2.11)

can be linearized by a Cole-Hopf transformation xn =wn

wn−1

− 1, and we thus obtain

the discrete analogue of the Airy equation:

2wn+1 + (zn − a)wn−1 − 4wn = 0. (2.12)

Special solution in terms of discrete Airy functions of d-PII was also given in [22,23, 24].

3 Discrete Painleve III Equation

In this section, we consider the following discrete Painleve III equation, q-PIII [14]:

xn+1xn−1 =ab(xn − cλn)(xn − dλn)

(xn − a)(xn − b), (3.1)

where a, b, c, d are constants. Using the method introduced in the introduction,we find

(Anxn + Bn)(Dn−1xn −Bn−1)

(Cnxn + Dn)(Cn−1xn − An−1)= −ab(xn − cλn)(xn − dλn)

(xn − a)(xn − b). (3.2)

6 A Sakka, U Mugan

With the choice of Dn = −aCn and An = bCn, equation (3.2) can be reduced to thefollowing linear equation for xn:[bCnBn−1 + aCn−1Bn + ab(c + d)λ2nCnCn−1

]xn = abcdλ2nCnCn−1−BnBn−1. (3.3)

Without loss of generality we may let, Bn = µλn+ 12 yn, Cn = 1, where µ2 = abcd.

Then equations (1.5) and (3.3) become

yn =xn+1(xn − a)− bxn

µλn+ 12

(3.4)

and

xn =µ2λn+ 1

2 (1− ynyn−1)

aµλyn + bµyn−1 + abλ12 (c + d)

(3.5)

respectively. Eliminating xn between the equations (3.4) and (3.5) leads to thefollowing discrete equation for yn

(ynyn+1 − 1)(ynyn−1 − 1) = −λ

n(yn − α)(yn − 1

α)(yn − β)(yn − 1

β)

(γyn − λn), (3.6)

where

α = − µ

acλ12

, β = − µ

adλ12

, γ = −µλ12

ab, λ = λ−1 . (3.7)

Equation (3.6) is the special case, δ = 0, of q-PV [14]:

(ynyn+1 − 1)(ynyn−1 − 1) =λ

2n(yn − α)(yn − 1

α)(yn − β)(yn − 1

β)

(γyn − λn)(δyn − λn). (3.8)

Thus there exists the one to one correspondence given by (3.4) and (3.5) betweenthe solutions of q-PIII and (3.6).

3.1 Backlund Transformation for discrete PIII

Backlund transformation can be obtained by finding two sets of {α, β, γ} such that(3.6) is invariant. It should be noted that equation (3.6) is invariant under thechange of parameters

α =1

α, β =

1

β, γ = γ. (3.9)

By using (3.9) and following the same procedure given in section 2.1, we obtain thefollowing Backlund transformation for q-PIII

xn =dλxn[axn+1(xn − a) + bxn−1(xn − b)− 2abxn + ab(c + d)λn]

c[bxn+1(xn − a) + aλ2xn−1(xn − b)− (a2λ2 + b2)xn + ab(c + d)λn+1];

a =bd

λc, b =

aλd

c, c =

dd

c.

Backlund transformations for discrete equations : Discrete PII - PV 7

(3.10)

3.2 Special Solution

The transformation (3.5) breaks down if

ynyn−1 − 1 = 0, (3.11)

and

aµλyn + bµyn−1 + abλ12 (c + d) = 0. (3.12)

By solving equations (3.11) and (3.12), we find that yn = − bc

µ√

λ, and λµ2 = b2c2.

Then, (3.4) leads to the following discrete Riccati equation

xn+1 =b(xn − cλn)

xn − a. (3.13)

Therefore, particular solution of q-PIII is characterized by (3.13), iff λad = bc. (2.11)

can be linearized by a Cole-Hopf transformation xn = a+wn

wn−1

, and we thus obtain

the discrete analogue of the Bessel equation [16]:

wn+1 − b(a− cλn)wn−1 + (a− b)wn = 0. (3.14)

The linearisability condition for q-PIII and the particular solution expressible interms of the discrete analogue of the Bessel functions were also obtained in [15, 16,29]

4 Discrete Painleve IV Equation

In this section, we consider the discrete Painleve IV equation, d-PIV :

(xn+1 + xn)(xn−1 + xn) =(x2

n − a2)(x2n − b2)

(xn − zn)2 − c2, (4.1)

where zn = αn+β, and a, b, α, β are constants. Equation (4.1) gives the followingequation after substituting xn+1 and xn−1 respectively given in (1.5) and (1.6),

[Cnx2n + (Dn + An)xn + Bn][Cn−1x

2n − (Dn−1 + An−1)xn + Bn−1]

(Cnxn + Dn)(Cn−1xn − An−1)=

(x2n − a2)(x2

n − b2)

(xn − zn)2 − c2. (4.2)

8 A Sakka, U Mugan

(4.2) can be reduced to a linear equation for xn

xn =−(yn−1 − zn + µ + α)(yn − zn − µ− α) + z2

n − c2

yn + yn−1

, (4.3)

with the choice of An = −(yn−zn +µ), Bn = ab, Cn = 1, and Dn = yn−zn−µ−α,where µ = −1

2(a + b + α). With these choices, the equation (1.5) yields

xn+1 =−(yn − zn + µ)xn + ab

xn + yn − zn − µ− α. (4.4)

By eliminating xn between (4.3) and (4.4), we obtain d-PIV

(yn+1 + yn)(yn−1 + yn) =(y2

n − a2)(y2n − b2)

(yn − zn)2 − c2, (4.5)

where

zn = zn+1

2α, a2 = [c−1

2(a+b−α)]2, b2 = [c+

1

2(a+b−α)]2, c2 =

1

4(a−b)2. (4.6)

If we replace yn with xn in (4.4), we thus obtain the following Backlund transfor-mation for d-PIV

xn = − [xn+1(xn − zn − µ− α)− xn(zn − µ)− ab]

(xn+1 + xn), (4.7)

such that xn solves d-PIV with the parameters a, b, c given by (4.6). The Backlundtransformation (4.7) for discrete Painleve IV equation was first given in [19].

4.1 Special Solution

The transformation (4.3) breaks down if

yn + yn−1 = 0, (4.8)

and

−(yn−1 − z + µ + α)(yn − zn − µ− α) + z2 − c2 = 0. (4.9)

yn = µ + α + c = 0 solve the equations (4.8) and (4.9). Equation (4.4) yields thefollowing discrete Riccati equation,

xn+1 =(a + b− c + zn)xn + ab

xn + c− zn

. (4.10)

after substituting yn = µ + α + c = 0 [15, 19, 29]. Therefore, particular solution ofd-PIV satisfies (4.10), iff a + b − 2c = α. Cole-Hopf transformation xn = zn − c +(wn/wn−1) transforms the Riccati equation (4.10) into the following linear equationfor wn:

wn+1 − (zn − c + a)(zn − c + b)wn−1 − 2cwn = 0. (4.11)

Equation (4.11) has been shown to be solvable in terms of the discrete analogues ofHermite functions [19].

Backlund transformations for discrete equations : Discrete PII - PV 9

5 Discrete Painleve V Equation

In this section, we consider the q-PV equation

(xnxn+1 − 1)(xnxn−1 − 1) =λ2n(xn − a)(xn − b)(xn − 1

a)(xn − 1

b)

(cxn − λn)(dxn − λn), (5.1)

where a, b, c, d are constants. Applying the method introduced in the introduction,we find

[Anx2n + (Bn − Cn)xn −Dn][−Dn−1x

2n + (Bn−1 − cn−1)xn + An−1]

(Cnxn + Dn)(Cn−1xn − An−1)=

λ2n(xn − a)(xn − b)(xn − 1a)(xn − 1

b)

(cxn − λn)(dxn − λn).

(5.2)

Equation (5.2) can be reduced to a linear equation for xn with the choice of

An = µλn, Bn = yn − (a + b)µλn, Cn = yn and Dn = −abAn, where µ2 =λ

abcd.

Then the equations (5.2) and (1.5) yield

xn =µλn−1(abλyn−1 + yn)− (1

c+ 1

d)λn

ynyn−1 − 1(5.3)

and

xn+1 =µλn(xn − a− b) + yn

ynxn − abµλn. (5.4)

respectively. We obtain the following q-PV for yn by eliminating xn between theequations (5.3) and (5.4):

(ynyn+1 − 1)(ynyn−1 − 1) =λ2n(yn − α)(yn − 1

α)(yn − β)(yn − 1

β)

(γyn − λn)(δyn − λn), (5.5)

where γ = 1aµ

, δ = 1bµ

, α = µcλ

, and β = µdλ

. Therefore, we have the followingBacklund transformation for q-PV

xn =µλn[abxn+1 + xn − a− b]

xnxn+1 − 1;

a =µc

λ, b =

µd

λ, c =

1

aµ, d =

1

bµ.

(5.6)

5.1 Special Solution

The transformation (5.3) breaks down if

ynyn−1 − 1 = 0, (5.7)

10 A Sakka, U Mugan

and

µλn−1(abλyn−1 + yn)− (1

c+

1

d)λn = 0. (5.8)

If we substitute the solutions yn =λ

µc, and λ2 = µ2c2 of (5.7) and (5.8) into (5.4),

we obtain the following discrete Riccati equation [29],

xn+1 =λn(xn − a− b) + abd

ab(dxn − λn). (5.9)

Therefore, particular solution of q-PV is characterized by (5.9), iff c = λabd. Eq.

(2.11) can be linearized by a Cole-Hopf transformation xn =λn

d

(1− wn

wn−1

), and

we thus obtain the following linear equation for wn

wn+1 − 1

λ

(1

a− d

λn

)(1

b− d

λn

)wn−1 +

(1

abλ− 1

)wn = 0. (5.10)

The linearization condition of q-PV was also given in [30], and shown that xn canbe expressed in terms of discrete analogue of confluent hypergeometric functions.

6 Conclusion

In this article, we have presented an algorithm which is similar to the algorithmintroduced in [5] for continuous Painleve equations, to obtain the Backlund trans-formations for discrete PII-PV. The algorithm is simple and based on the investiga-tion of discrete Riccati equation (1.5) for the solution of a given discrete Painleveequation, with the coefficients depending linearly on the solution of another discreteequation of Painleve type. The Miura transformation for d-PII and d-PXXXIV, andthe Backlund transformation for d-PII and d-PIV that we have presented, were pre-viously known. The special-function solutions for the discrete Painleve equationsare extensively covered in the literature. But the transformations for q-PIII andq-PV were not discussed in the literature before. Moreover, as an application of thealgorithm, we have presented the special solutions which are discrete analogue ofthe classical special functions, for discrete PII-PV whenever the parameters satisfycertain conditions (the linearisability conditions).

Acknowledgment

This work was partially supported by the Scientific and Technical Research Councilof Turkey (TUBITAK).

Backlund transformations for discrete equations : Discrete PII - PV 11

References

[1] P. Painleve, Acta Math., 25 (1902) 1.

[2] B. Gambier, Acta Math., 33 (1909) 1.

[3] E.L. Ince, Ordinary Differential Equations, Dover, New York (1956).

[4] H. Airault, Stud. Appl. Math., 61 (1979) 31-53.

[5] A.S. Fokas, M.J. Ablowitz, J. Math. Phys., 23 (1982) 2033-2042.

[6] V.I. Gromak, I. Laine, and S. Shimomura, Painleve Differential equations inthe Complex Plane, Walter de Gruyter, Berlin-New York (2002).

[7] B.M. McCoy, J.H.H. Perk, and R.E. Shrock, Nuclear Phys. B, 220 (1983) 35-47.

[8] A.S. Fokas, A.R. Its, and A.V. Kitaev, Commun. Math. Phys., 142 (1991)313-344.

[9] A. Ramani, B. Grammaticos and J. Hietarinta, Phys. Rev. Lett., 67 (1991)1829-1832

[10] B. Grammaticos, F.W. Nijhoff, A. Ramani, ”Discrete Painleve equations”, in:The Painleve Propert, One century Later, Ed. R. Conte, CRM Series in Math.Phys., (Springer, New York, 1999) pp. 413-516.

[11] B. Grammaticos, A. Ramani, Meth. and Appl. of Ann. 4 (1991) 1829.

[12] G.R.W. Quispel, J.A.G. Roberts and C.J. Thompson, Physica D, 34 (1989)183.

[13] B. Grammaticos, A, Ramani, ”Discrete Painleve equations: Derivations andProperties”, NATO ASI C413 (1993), pp.299.

[14] A. Ramani and B. Grammaticos, Physica A, 228 (1996) 160-171, solv-int/9510011.

[15] K.M. Tamizhmani, A. Ramani, B. Grammaticos and K.Kajiwara, J. Phys. A:Math. Gen., 31 (1998) 5799-5810.

[16] B. Grammaticos, F.W. Nijhoff, V. Papageorgiou, A. Ramani and J. Satsuma,Phys. Lett. A, 185 (1994) 446-452, solv-int/9310003.

[17] V.G. Papageorgiou, F.W. Nijhoff, B. Grammaticos and A. Ramani, Phys. Lett.A, 164 (1992) 57-64.

[18] N. Joshi, D. Bartoclay and R.G. Halburd, Lett. Math. Phys., 26 (1992) 123.

12 A Sakka, U Mugan

[19] K.M. Tamizhmani, B. Grammaticos and A. Ramani, Lett. Math. Phys., 29(1993) 49-54.

[20] J. Hietarinta, K. Kajiwara, ”Rational solutions to d-PIV”, solv-int/9705002.

[21] K. Kajiwara, K. Yamamoto and Y. Ohta, Phys. Lett. A, 232 (1997) 189-199.

[22] A. Ramani, B. Grammaticos, J. Phys. A: Math. Gen., 25 (1992), L633-L637,

[23] K. Kajiwara, Y. Ohta, J. Satsuma, B. Grammaticos and A. Ramani, J. Phys.A: Math. Gen., 27 (1994) 915-922.

[24] A.S. Carstea, A. Ramani, R. Willox and B. Grammaticos, J. Phys. A: Math.Gen., 36 (2003) 8419-8431.

[25] E. Brezin, V.Kazakov, Phys. Lett. B, 236 (1990) 144-150.

[26] H. Sakai, Commun. Math. Phys., 220 (2001) 165-229

[27] B. Grammaticos, A. Ramani and Y. Ohta, J. Non. Math. Phys., 10 (2003)215-228.

[28] U. Mugan, A. Sakka and P. M. Santini, ”Schlesinger transformations for discretesecond Painleve equation”, preprint, (submitted )

[29] T. Tamizhmani, B. Grammaticos, A. Ramani, and K. M. Tamizhmani, PhysicaA, 295 (2001) 359.

[30] K.M. Tamizhmani, B. Grammaticos, A. Ramani and Y. Ohta, Lett. Math.Phys., 38 (1996) 289.