Two discretisations of the Ermakov-Pinney equation

24 November 1997 PHYSICS

LETTERS

A

Physics Letters A 235 f 1997) 574-580

ELSEVIER

Two discretisations of the Ermakov-Pinney A.K. Corona,

equation

M. Musette b

a institute of Mathematics and Statistics, Cornwallis Building, University of Kent. Canterbury, Kent Cl? 7NE UK b Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium Received 24 February 1997; revised manuscript received 15 July 1997; accepted for publication 6 August 1997

Communicatedby A.P. Fordy

Abstract We propose two candidates for discrete analogues to the nonlinear Ermakov-Pinney equation. The first one based on an association with a two-dimensional conformal mapping defines a second-degree difference scheme. It possesses the same features as in the continuum: a nonlinear supe~osition principle relating its general solution to a second-order linear difference equation and by direct Iinearisation a relationship with a third-order difference equation. The second form, which is new, is obtained from a slight improvement of the superposition principle. It has the advantage of leading to a first degree difference scheme and preserves all the nice properties of its linearisation. @ 1997 Elsevier Science B.V.

1. Introduction

It is well known that the E~~ov-Pinney 0,x, +

f(x>u

+

cu-3= 0,

[ 1,2] equation,

c constant,

(1)

is related to the linear second-order equation

*xx+ f(x)@

=0

by a nonlinear supe~sition u=

(2) formula

Q@(X) + 2PrcII(x)rclz(x)

f &(x)

(3)

with cry - p2 = -cW-‘/2, W representing the Wronskian of a fund~ent~ set (+I, $2) of solutions to the linear equation (2). Instead of Eq. ( I), where the general solution does not possess the PainlevC property, we will consider the equation

for its square y = u2:

YYXX -

&2fy2-2c=o,

0375-9601/97/$17.~ @ 1997 Elsevicr Science B.V. All rights reserved. PII SO375-9601(97)00649-X

AK, Common, M. Musette/Ph~sics Letters A 235 (1997) 574580

575

where Y now does have the Painleve property. This equation is related to Eq. (G.22) of the Gambier [ 3,4] classification, which concerns algebraic differential equations of second order and first degree possessing the Painleve property. In the present Letter, we address the problem of discretising Eq. (4) and keeping its properties of linearisation, i.e. (i) its relationship by a quadratic expression to the solutions of the second-order linear equation (Z), Y = QIcI:(x) f

+w$l(x)@2(~>

ii) its relationship

by derivation

+r$&a

(5)

with respect to x to the third-order

linear equation, (6)

Y.rXX+ 4fY.X + 2&Y = 0.

Moreover, we will end up with a discretisation scheme of second order and first degree, corresponding to the degree in the highest derivative of the differential equation (4) with the goal of adding support to the rule stated in Ref. [5]. In a first attempt, we obtained a discrete analogue of (4) by using an association with a particular conformal mapping which coincides in the continuum limit with its analogous two-dimensional Riccati linearisable system. The resulting equation of degree two possesses the same nonlinear superposition principle (5) as its continuous counterpart, relating its general solution to a second-order linear difference equation. Its linearisability into a third-order difference equation may also be easily proved. By proceeding in a different way, the same discrete form was recently derived by Schief [ 61. We already reported our result in Ref. [ 71 but reproduce it here in order to clarify the procedure we follow to derive an alternative new discretisation of first degree, possessing relationships with both a second-order and a third-order difference equation. This second discretisation also enables us to obtain the discrete analogue of the equivalence existing in the continuum limit between the third-order Schwarzian equation, the Riccati equation and the second-order linear equation [ 8,9].

2. Link of the Ermakov-Pinney

equation

with the conformal

Riccati system

Let us first recall that the two main classes of linearisable coupled Riccati systems (1) Projective Riccati equations: They have the matrix form

[IO] are, respectively:

O.x=a+Bu,+w(c,w),

(7)

where the elements of the N x N matrix B and N-dimensional variable X. (2) Conformal Riccati equations: w, =pi-Eo+au+o(y,w)

vectors a, c are functions

- &$w,o)

of the independent

(8)

with the N x N matrix E satisfying Ef+fET=O,

(9)

where P is such that {S, Cr) C STf& = stcr, + &CC2+ *. . + sp(w, - ~p+lcyp+l -.

. . - &%v

and 1 < p < N. Defining standard forms [ 1 I], which are equivalent after some changes of variables systems, one can associate with them linearisable nonlinear differential equations belonging

I 101 to these first-order to the classification

A.K. Common, M. Musetre/Physics

S76

of Gambier and Chazy [4,12]. In particular, in the following way. Take N = 2,

with .s2 = fl

and consider

L.eiiers A 235 (1997) 574-580

the two-dimensional

the standard form associated

conformal

system can be related to Eq. (4)

with

a=EzO,

(11)

where Al, 02 are constant 51,x = AI + 62,.x = A2(x)

(02

# 0) and A2( x) is an analytic function

of x. Then (8) may be written as

~~D25152, -

$2
(12) -&2&t

(13)

or setting 53 = ST + c’s, one obtains a three-dimensional can be linearised by defining the transformation 52 = $2/$4

51 = @II*49

3

first-order

coupled system of projective

53 = *319+4

type which

(14)

into 0

*1,x +2.x

=

*3,x

L

ti4,l

)i

0 0 0 2A, 2e2A2(x) 0 -ENDS

In the general case (Al equation for 5 = 51, &$Xx- ;[;

0 +2

Al

*1

A2(x)

+2

0

0 0

0

*3

# 0), the elimination

+ 2A15, - cs2A2(x)D2t2

*4

j(

1

of 52 between

+ ;e2D;t4

(15)

.

(12) and ( 13) yields the second-order

- ;A; = 0,

nonlinear

(16)

which is identical in the variable y = 5-l to the Gambier equation (G.27) in the particular case of n = 2. This is related by the transformation y = z,/( A1 z ) to a third-order bilinear equation which by derivation yields the fourth-order linear equation zxI-x.r+ 2c2 DzA2z.r.r + e2 DzAz,xzx - c2 ( DzA, ) 2z = 0. In the variable

c = Y’/~, one gets a generalisation

uxl + ;c2D2A2(x)u

+ 2A,u2u,

+ ;A:43

(17)

[ 131 of the Pinney equation

- ;E~D;u-’

= 0.

( 1) , (18)

In the particular case of A1 s 0, the linear system (15) degenerates into three coupled equations because @i 3 K = constant. The linear equation in the variable fl4 corresponds to the third-order equation (6)) which linearises (4).

3. Discretisations

of second and first degree

We first consider the approach based on the connection with discrete conformal transformation which yields a discrete form of the Pinney equation whose general solution is connected to a linear second-order difference equation by the same nonlinear superposition formula as in the continuum.

AK. Common, h4. ~usette/P~ysics Letters A 235 (1997) 574-580

577

The discrete mapping considered here is 61 (xl

& = ~AI +

1-

he2D252(x)

+

$2&2D;[g(x>

+&2&x>]

(19)

’

where ti = ti( .x + h), x = nh and one can easily recover the standard form ( 12), ( 13) in the continuum limit h --+ 0.

Ail the details are given in Ref. [ 71 con~ming the line~isation of the system ( 19), (20) into a set of four linear recurrence relations and its degeneration into three relations in the case of Al E 0. A discrete form of Rq. (4) may then be obtained by eliminating l*(x) between the set of the two equations, (f9) and (20), for A: = 0. From the first of these we have a quadratic equation for 5;(n) and we choose the root (-1 +j/mj

icz=-j-$

(21)

to get the correct continuum Iimit. Substituting in the second one with the identification A2( x) = 2f(x), D2 = 1, we obtain for y(x) = u(x)-’ the equation y(x+h)[2-h’f(x)]=

y(xfh)y(x)-$h2cz+

y(x+2h)y(xth)-$h2e2.

(22)

The square roots may be eliminated by squaring twice. Introducing the compact notation ~=y(x++h),

F=ytx+h), g = Y(X - 2h),

r=y(x-h), we

(23)

get the discrete equation [r - y - A2(x)y12

with A2(x)

- A2(x)(47y

(24)

c = E2h2. Setting y(x) =+(x)+(x),

= [2 - h2f(x)12,

==

($ q5 - &j - A’(x)&

- c) = 0

7)’

-- 4A2(x)~~~~

Eq. (24) becomes

= -cA2(x).

(251

The left-hand side of this relation can be identified with a perfect square if --

(26) or == Ifi‘4 = [AC++

91 tAW’;t; + 41.

(27)

Thus, the condition (26) introduced in (25) implies that (p# -&k)

= fihe

(28)

while the condition (27) leads to the separation

F

A(x)+ + J,

(29)

A.K. Common, M. Musette/Physics

578

or equivalently

Letters A 235 (1997)

to the relation

==_ @ 4 - 4 ICI= A(x)[k(x)

- k(x)-‘l~$+

k(x)@+

Ux)-‘d$,

(30)

which is compatible with (28) only if k(x) = -1. Therefore, the functions solutions of the second-order difference equation 1V(x+2h)

+A(x)p(x+h)

with the Wronskian

the solution

of the difference

withcuy-P2=--$.

+p(x)

equal to fibs.

Y(X) = a$(X)2

A(x)

574-580

Applying

equation

+ 2PNx)rNx)

$ and C$are two linear independent

=0,

(31)

the linear transformation

(24) becomes + r+(X)*

(32)

Making in (3 1) the identification

= -2 + h2f(.x),

(33)

we recover, in the limit k --) 0, the linear differential equation (2). Thus the general solution of the discrete equation (24) satisfies the same nonlinear superposition principle as in the continuum. It is also linearisable into a third-order difference equation. Indeed, the elimination of the constant c from the expression (24) yields the following factorisation in two linear third-order difference equations, [A$+(1

--_ -AA)Ay-(1

In the continuous linear equation

---AA)Ay-;;iy][AF-(l+AA)AT-(l+AA)Ay+xy]

limit k ---) 0, the first factor, with A(x)

=O.

related to f(x)

by (33))

tends to the differential

y”’ + 4fy’ + 2f’y = 0,

(35)

which comes from the elimination we obtain yYIX - ;y,2 - 2f(x>y2

(34)

of c in Eq. (4). Finally, expanding

5 and 7 up to the terms 0( k*) in (24)

- i&2 = 0.

(36)

Thus, the difference equation (24) deserves the name of “discrete Ermakov-Pinney equation” for three reasons: (i) it is linearisable into a third-order difference equation possessing the correct continuous limit; (ii) it tends, in the limit h -+ 0, to the nonlinear differential equation (4); (iii) its general solution, related to the second-order linear difference equation (31)) satisfies the same nonlinear superposition principle as in the continuous case. However, as conjectured in Ref. [ 51, there should exist a discretisation of the Pinney equation of degree one, and not two as in (24). To obtain such an equation we will proceed as Schief [6] did for deriving (24) but instead of imposing the nonlinear superposition principle (5) we build the discrete equation possessing the general solution Y(X) = a+& + P(@

+ 99)

+ r@

with cuy - p2 = KI and 4, fi two linear independent p+AAW+W=O,

A(x)

= -2 + k2f(x).

(37) solutions

of the second-order

equation (38)

A.K. Common, M. ~usetfe/Physics Letters A 235 (1997) 574-580

579

equivalent to (31). The elimination of the particuIar solutions F, 9, $, I/J, 6, 4, 4, 56 between 7, y, -y by using the linear -= -I= equation (38) and the constant value of the Wronskian equal to h& leads to the following discrete equation of the first degree in 7 and -y, (B+.Y)(Y+Y)

K=

=AA(y2+h210,

KlK&

(39)

Expanding -jr, y up to terms O(h2) one recovers in the continuous limit h --f 0 the differential equation (4). The elimina%on of K from the expression (39) yields the linear third-order difference equation

T+y Y+Y L---=-y+y=o,

_

(40)

AA

AA

which by expanding 2, 4, 7, 7 and y up to terms 0( h3) yields the correct continuous limit (6). Eq. (39) is the second discretisation of the Ern&ov-Pinney equation related to the second-order linear equation by the superposition principle (37) and linearisable into the third-order equation (40). Moreover, we can derive the discrete analogue of the relationship linking in the continuum the Schwarzian equation, the Riccati equation and the linear equation (2). Indeed, making the substitution Y=

$w’-(x+U/(x-2&

(41)

Eq. (39) becomes

(x-&:)(x-m (x-&:)(x-m

= (AA)-'.

(42)

The left-hand side corresponds to the expression of the discrete Schwarzian given by Faddeev and Takhtajan [ 141. This one is very interestingly related to the cross-ratio of four adjacent values of X(x) and thus explicitly invariant under the homographic transformation X(x> -+ [aX(x> +

bl/[cX(x)

+ dJ .

(43)

Setting D=-(x-ZX+&‘),‘(%X)h

(44)

one obtains (1-

hR)(l+

=4(A&

hg)

(45)

and by the transformation h&B=-

%!+l ( A@

(46) >

one recovers the second-order linear difference equation (38) Finally, let us remark that in Ref. [S] Conte and Musette obtained as discretisations of the differential equation (4) in the particular case f(x) = c G 0 the two following forms, (i)

(~-1-44~)~

(ii)

(Y+Y)(Y+~)

- 16yy=O -4Y2=0

withy = (QX -t ~2)~)

(47)

with y = (CIX_ + ~2) (clx+ + ~2)

(48)

580

A. R Common, M. ~usette/Physic~

krters A 235 (19971574-580

(xi = x & ih), in complete agreement with the discrete forms (24), (39) and the expressions (5), (37) of their general solutions. In the case of a first degree discretisation scheme, expression (39) shows us that the function A*(x) = [ -2 + h*f( X) ] 2 must be discretised in two points.

4. ~oncl~i~n In Ref. [ 71 we presented a second-order nonlinear difference equation associated with the generic conformal mapping ( 19)) (20) linearisable into a fourth-order difference equation. This could be considered as a discrete form of the generalised Pinney equation ( 18). A group theory approach similar to ours was also applied by Winternitz [ 151 to derive a discretisation of the Pinney equation ( 1). Acknowledgement

Part of this work was done during a stay of MM in Saclay. She thanks R. Conte for his kind hospitality. A.K.C. and M.M. would like to thank the British Council and the Nationaal Fonds voor Wetenschappelijk Onderzoek for financial support for exchange visits during which much of this work was carried out. M.M. acknowledges financial support extended within the framework of the IUAP Contract nr P4/08 funded by the Belgian government. References [ I 1 VP Ermakov, Second-order differential equations. Conditions of complete integrability, Univ. Izv. Kiev Series III, 9 ( 1880) pp. I-25. 121 E. Pinney, The nonlinear equation v”(x) +p(x)y(x) + c?-“(x) = 0, Proc. Am. Math. Sot. 1 (1950) 681. (31 B. Gambier, Comptes Rendus 142 (1906) 1403. (41 B. Gambier, Acta Math. 33 (1910) 1. \ 51 R. Conte, M. Musette, Phys. Len. A 223 f 1996) 439. 161 WK. Schief, Appl. Math. Lett. IO (1997) 13. 171 A.K. Common, E. Hesameddini, M. Musette, Discrete linearisable Gambier equations, Proc. of the Side II Conf.. to be published. [ 81 E. Hille. Ordinary Differential Equations in the Complex Domain (Wiley, New York, 1976). 19 1 R. Conte, Unification of PDE and ODE versions of Painleve analysis into a single invariant version, in: Painleve Transcendents eds. D. Levi. P. Wintemitz (Plenum, New York, 1992) pp. 125-144. 1101 R.L. Anderson, J. Hamad, I? Wintemitz, Physica D 4 (1982) 164. I I 11 E. Hesameddini, Matrix Riccati equations and their applications, Thesis, University of Kent ( 1996). I 121 J. Chazy, Sur les equations diff~rentielles du troisieme ordre et d’ordre superieur dont I’integrale generale a ses points critiques fixes, These, Paris (1910); Acta Math. 34 (1911) 317. I 131 C. Rogers, W.K. Schief, P Wintemitz, Lie-theoretical generalization and discretization of the Pinney equation, Preprint CRM-2463, March 1997. I141 L.D. Faddeev, L.A. Takhtajan, Liouville model on the lattice, Springer Lect. Notes Phys. 246 (1986) 166. [ I5 1 P.Wintemitz, Nonlinear difference equations with superposition formulas, Proc. of the Side II Conf,, to be published. [ 161 AK. Common. E. Hes~eddini, M. Musette, J. Phys. A 29 ( 1996) 6343.