Analysing negative resonances in the Painlevé test

Physics Letters A 160 (1991) 347-354 North-Holland

PHYSICS LETTERS A

Analysing negative resonances in the Painlevé test Allan Fordy and Andrew Pickering Depart men! ofApplied Mathematical Studies and Centrefor Nonlinear Studies, University ofLeeds, Leeds, LS2 9JT, UK Received 25 June 1991; revised manuscript received 19 September 1991; accepted for publication 23 September 1991 Communicated by A.R. Bishop

We improve the Painlevé test in such a way that negative resonances can be treated. To this end we demand that the general solution of both the given nonlinear equation and itslinearisation be single valued. This gives rise to compatibility conditions for every resonance. For equations with no principal branch, such as Chazy’s equation, but with enough integer resonances, we (formally) build the general solution with an essential singularity, reducing to the Painlevé (finite pole) solutions for special choices ofarbitrary constant. In the Context ofintegrable hierarchies, our approach gives a clear relationship between negative resonances and lower order commuting flows.

1.

Introduction

It is well known that there is a close connection between soliton equations and the “Painlevé property” [1,2]. Indeed, there is a conjecture [2] that every reduction (travelling wave, similarity, etc.) of a nonlinear evolution equation solvable to 1ST will be of Painlevé type. This is the basis ofthe “Painlevé test”, used to isolate integrable equations. To dispense with checking “every” reduction, Weiss et al. [31 introduced an improved Painlevé test (WTC method), with which one directly attacks the PDE. Both of these methods only test for necessary conditions, so any equation thus isolated must be further analysed to provide proofof complete integrability. Nevertheless, these methods have proved to be very effective. The WTC method, in fact, goes some way towards providing sufficient conditions for integrability in that truncation of the Laurent series may provide information regarding Lax pairs, Backlund transformations and other devices associated with soliton equations. As remarked above, the Painlevé test has been a very successful tool for isolating integrable differential equations (both ODEs and PDEs). Nevertheless, this method (as currently used) is incapable of decisively testing certain equations, such as those possessing several negative resonances. One such

equation is that of Chazy (see (3.1 a) below), which has only negative resonances. This equation has a movable natural boundary, but its general solution can be written down in terms of hypergeometric functions (see, for instance, refs. [4,5]) and is single valued in its domain of definition. Even integrable equations, such as the members of the KdV hierarchy, have “secondary” Painlevé branches which possess several negative resonances [6]. There has been some discussion of negative resonances [5], but many people seem to be confident that they will not affect the equation’s integrability and thus choose to ignore them (arguing that they violate the leading order hypothesis). In this note we show that negative resonances can contain important (sometimes decisive) information regarding the integrability of an equation. To this end we give a method of deriving compatibility conditions for negative resonances and show that it can be extremely hazardous to ignore them! Furthermore, we show that “secondary” branches are by no means secondary; for many equations our method extracts the same information from all branches, even though the conventional Painlevé analysis could only derive full information from the principal branch. Our approach is to simultaneously test a nonlinear equation and its linearisation, treated as a (rather weakly) coupled system. Applied to the linearised

0375-9601 /91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

347

Volume 160, number 4

PHYSICS LETTERS A

equation, the Painlevé analysis reduces to a Fuchsian analysis about a regular singularity. The roots of the indicial equation of the linearisation are just (up to a constant integer shift) the resonances of the nonlinear equation. We demand that the solutions to both the nonlinear and linear equations be single valued. We therefore require that all roots of the indicial equation (and thus all resonances of the nonlinear equation) be distinct integers, whether positive or negative. Since the roots of the indicial equation differ by integers, compatibility conditions arise, Our conditions at negative resonances (when they arise) are thus in addition to those of the standard Painlevé analysis. We present examples of equations for which the standard Painlevé test is inadequate but for which our test (with its additional conditions) is decisive. For integrable hierarchies our analysis clearly relates negative resonances to the presence oflower order commuting flows. The recursion operator can be used to build the resonance polynomials for the hierarchy. This simplifies the calculations of Newell et al. [6,7] andenablesustogiveaclosedformforthe resonance polynomial of any branch for each member of the KdV hierarchy. In this Letter we just give a basic discussion of our method, together with a series of examples which illustrate the difference between the conventional Painlevé test and ours. Further explanation and developments ofour test are presented in ref. [8] while a more complete account of our analysis of integrable hierarchies can be found in ref. [9].

K’ [u] w =

For simplicity, we restrict attention to nonlinear evolution equations of the form u1=K[u] (2.la) where K[u] is a polynomial function ofu, u~, UN~. The case of ODEs is then easily incorporated by taking u~=0.The linearisation of (2.la) is

-~-

th

K[u + w ]I.

=

Eq. (2.lb) is the equation satisfied by generalised symmetries (commuting flows) of (2.1 a), when they exist. Remark. Our examples of section 3 are, in fact, all ODEs, but the method is applicable to all cases. In this Letter PDEs arise in section 4 in our discussion of integrable hierarchies. We first carry out a standard Painlevé expansion of the nonlinear equation (2.la), using Conte’s modification [10] of the WTC method. This “invariant” approach simplifies many of the expressions, hiding the complications in the definition of S, C and ~ (see the appendix). We seek a solution of the form u(P)=X_a

~

u,~’

(2.2)

.

A leading order analysis gives a number of possible choices of a, depending upon the nonlinearities. Each a corresponds to a possible choice of dominant terms .k~[u] of K[u] and this, in turn, leads to a number of possible starting terms u0 as solutions of the a!gebraic equation ~[uox~] I~_’=~=~ =Pa(uo)X~=0, (2.3a) where fi is the weight of the dominant expression 1 and Pa a polynomial. The coefficients u, are determined recursively by

~x

—ai

[K’ [uo~ =

2. The method

25 November 1991

~‘_~=~‘~=o]U1

expressions involving u0

u,_~,S, C. (2.3b)

Remark. The appearance of the Fréchet derivative in (2.3b) is an indication of the importance of the linearised equation (2.lb). The coefficient ofu1 on the left vanishes for certain values of i, called resonances, which must be integer for an “integrable” equation:

,

...,

= K’ [u] w~ where

348

(2. lb)

~={ri r~} r~~
Volume 160, number 4

re

PHYSICS LETTERS A

No conditions arise at negative resonances. To pass the conventional Painlevé test an equation should have the following properties: (i) Each possible choice of a and all corresponding r must be integer, with r, being distinct. ~.

(ii) All branches should be such that at any positive resonance, the compatibility conditions are identically satisfied. (iii) Thereexistsaprincipalbranchwiththenumber of resonances n = N, the order of thefull operator K[u], with all r• (except r~ = —1) being nonnegative. This gives a Painleve expansion containing a full set of arbitrary functions. Remark. Negative resonances of “secondary” branches are allowed since they give no conditions which contradict integrability. There are equations such as that of Chazy (3.1 a) which do not possess a principal branch in the above sense, and yet do possess a general solution which is single valued. However, this cannot be determined by a conventional Painlevé analysis. To retain such equations we slightly weaken the above requirements. To pass the “nonlinear part” of our test, an equation must possess the following properties: (P1) Each possible choice of a and all corresponding r• must be integer, with r, being distinct. (P11) All branches should be such that at any positive resonance the compatibility conditions are identically satisfied, (P11) There exists a branch with the number of resonances n=N, the orderofthefulloperatorK[u]. Remark. We have dropped the requirement of a principal branch with the consequence that it may not be possible to build the general solution as a finite pole Laurent expansion. However, we still require the existence of a branch with the “correct” number of resonances. This weakening of condition (iii) allows some “bad” equations through our net, but these are caught at step 2 below. Our second step is to consider the linearised equation (2.1 b), with u = u the Painlevé expansion (2.2), for each of the branches. For each branch we can write (2.lb) as ~,

f~’[u~]w=I’ where

~‘

[u~]w

(2.4)

and I’ are respectively the linearisations of

25 November 1991

the dominant and inferior parts of K— u,. K’ [u ~] is scaled in such a way that x= 0 is a regular singularity in the Fuchsian sense. We seek an expansion

w=x~~ w.x’, w00, where ae{a1 ‘,

—a

(2.5a)

a,,} is a root of the indicial equation a

K [uoX ]x I~=~=o=o• Comparing this with (2.3), we see that

(2.5b)

a•=r—a

(2.5c)

forr1E3~

i1,... n.

In order that the general solution of the linear equation (2.4) e single valued we require, in addition to (PI)—(PIII), (Fl) that the indicial equation has n distinct, integer solutions (a1, (Fil) that the compatibility conditions arising at each a, 1 ~ i ~ n, be identically satisfied. Remarks. (i) (Fl) is implied by (P1) and (2.5c), but is here for emphasis. A double root gives rise to logarithms in w, justifying the common practice of insisting that negative r• be integer and distinct. (ii) FlI gives n conditions, corresponding to compatibility conditions for r,, regardless of whether they are positive or negative. We are demanding much more than (2.4) being just Fuchsian when we ask for w to be single valued. This is a strong constraint which, together with the above weakened Painlevé property of (2.la) enables us to distinguish integrable cases in a wide variety of equations hitherto untestable by the Painlevé method. A feature of equations which pass our test, but have no principal branch, is the presence of an essential singularity. We build a general solution by an infinite negative expansion and the negative resonances give rise to the arbitrary constants. The usual Painlevé (pole) expansions arise as particular solutions. These results are, of course, formal since we nowhere consider convergence. ...,

3. Examples We now present a variety of examples which iilustrate the equations power of our when applied to awkward suchmethod as Chazy s equation. Example 1. Chazy’s equation [11] 349

Volume 160, number 4

PHYSICS LETTERS A

~

(3.la)

25 November 1991

~ K~’=0, 0

K’ ~

—2u~~w=0. (3.lb)

There are two branches: (i) a=l,

u 0=—6,

/3= —4,

K’[u]w~~7 ~ l1x1=O~ where

(3.6a)

1Z~[u]=K[u]

~={—3, —2,— l}

,

(3.2a)

K0=—u0(u0+6)

,

K~I~0,~6=24u~

K2I~0+o~o=60(u2+S) .

(ii) a=2,

(3.6b)

u0arbitrary, 1~[u]=—2uu~~+3u~,

/3= —6,

~={— 1, 0}

(3.2b)

.

Neither branch is principal and no resonance can be testedbythePainlevemethod. However,branch (i) has three resonances as required by (Pill). Corresponding to each branch we write the linearised equation (2.4): (i) ~ 2uw~~ + 6u~w~ 2u~w = 0, (ii) 2uw~+ 6u~w~2u,~~w=~ —

—

—

—

—

(3. 3a) (3.3b)

For branch (ii) we have put the “inferior” part on the right. in each case the corresponding branch of the Painlevé expansion must be inserted for u, so, noting the respective leading behaviour given 0 is aorder regular singularity in in (3.1), we see that x= each case. The Fuchsian analysis ofbranch (ii) treats eq. (3.3b) as effectively second order, since the third order part is not dominant. Thus, the indicial equation of this branch is only quadratic, giving rise to only two resonances. In fact, the indicial equations are respectively

Since there are no positive resonances, no obstructions arise, so that all the u, can be recursively constructed, the first three being (with u0 ~ 0) u0=—6, u~=0, u2=—S.

(3.6c)

With this expansion for u inserted into (3.3a) we can try to construct the expansion (3.5a) for win the same way. However, this time there are possible obstructions. The higher roots of the indicial equation are positive resonances (as s1= a,+ 4) for this expansion, but correspond to negative resonances for u. The following compatibility conditions arise (w0 0): s1=0: ~2

s

=

1:

10=—20w0(u0+6)=0, 1 i~=—6 = —40w0u1 =0,

3=2: /2I~0÷6=~1~0=—64w0(u2+S)=0, (3.6d) which are identically satisfied when (3.6c) hold. No further conditions arise. Similarly, for branch (ii), substituting u of (3.5b) into (3.1) leads to

(i) (a+4)(a+3)(a+2)=0,

(3.4a)

u0arbitrary,

(ii) (a+3)(a+2)=0.

(3.4b)

while the compatibility conditions for w at s,=a,+3 are

The expansions for u and w take the form 4 ~ wa’, (i) u=x~ i=0 ~ u~’, u0~0, w=~ 10

ui =—6—u0~,

(3.7a)

w0 arbitrary, ~2 = 1: 10w0(u1 +u0,, +6)=0, (3.7b) which are identically satisfied. —

w 0 arbitrary,

2 ~ u,x’, (ii) u—x i=o

(3.5a)

Thus we see that in both cases the compatibility condition at resonances of the Fuchsian expansion are identically satisfied as a consequence of the definitions of u,, given by the corresponding Painlevé

(3.51,)

expansions. Thus, Chazy’s equation passes our test, as it should. Indeed, one can construct the general solution to Chazy’s equation if one allows an essential singular-

w=~3 ~ w,,~’, i=0

w 0 arbitrary.

Substituting u of (3.5a) into (3.1) gives, for branch (i): 350

Volume 160, number 4

PHYSICS LETTERS A

ity, which is single valued. It should be recalled that Painlevé did not restrict to “movable poles” but also allowed essential singularities (see ref. [5] for a discussion of this). Wedo not use the WTC approach here, but set x=x (the “classical” approach). Thus u takes the form (3.8a)

U=X_l ~U1X~,

where u• are constants. We find that u0=—6; u_1, u_2, u_3 arbitrary,

(3.8b)

with 1+!

(l+3)(l+2)(l+ l)u,—2 ~

(i—2)(i— 1 )u1u1_1

1= —

25 November 1991 /3=

(ii) a =

~,

—

(I—i—I )(i— 1 )u1u1_1=0

(3.8c)

.

We see that, when u_2=u_3=0, then u,=0 Vi~<—2, ‘giving 6

u_~ 2

(3.9a) x With x=x, this is the Painlevé expansion for branch (ii). Setting u_ 1 = 0 gives — —

x

+

6 u=——, x

(3.9b)

which is the Painlevé expansion for branch (i) when x=x. Thus, we can represent the general solution of Chazy’s equation as (3.8a), meaning that x=0 is an essential singularity. The two branches of the (classical) Painlevé expansion arise as particular solutions of this general solution. Example 2. Bureau’s equation (7.14). This example is very similar to Chazy’s equation, and is considered by Bureau [12]: ~

(3.l0a)

K’ ~

—

(3.lOb) ,

~={—l,0}.

(3.llb)

In this case there are no obstructions to constructing the Painlevé expansion, since there are no positive resonances. There is no principal branch, but it is not possible to determine integrability of (3.lOa) by the standard Painlevé test. However, a double resonance at —1 means that the indicial equation of (3.lob) has a double root of —2, which means that w contains a logarithm in its solution, thus failing our test. Indeed, we can exhibit a logarithmic element to the general solution of (3.lOa) by carrying out an inverse power expansion, analogous to (3.8a), but with logarithmic terms: u=x’

>,

(~

uu(lnxY)x_i,

i=O

(3.12)

=0

where u00 = —2; u10, u11, u60 are arbitrary constants. Remark. The second branch does, of course, have ample to illustrate effectbut of awedouble rootthis to the an algebraic branchthepoint, include exindicial equation. Example 3. Bureau’s equation (20.4). This equation was studied by Bureau [12]: ~ 3+2uu~)—d 2—d 2u 1 u—d0 =0,

—d3(u ~

(3.l3a)

2+6uu~—2d +(3u~~—3d3u

3u~—2d2u—d1)w

=0.

(3.l3b)

Here, we have used Bureau’s notation for the coefficients c0, d1, which are functions ofx. There are two branches: (i) a=l

,

u0 =1

~[u]_uxxx+3uu~x+3u~+3u2ux,

/3~z—4,

=0. There are two branches: (i) a= 1, u 1[u]=K[u} 0= —2,

—

fl=—’f,

1+!

+3 ~

—4, E3~={—6,—1, l}. (3.1 la) u0 arbitrary, 1~[ u] = 7uu~~ + 11 u~,

(ii)a=1

,

~

1, 1, 3}.

u0=2,

(3.l4a) 2u~,

~[u] = u~~+3uu~~+ 3u~+3u fl —4, .~3~{—2, —1, 3}.

(3.l4b) 351

Volume 160, number 4

PHYSICS LETTERS A

25 November 1991

A standard Painlevé analysis on branch (i):

a=2,

(ii)

u0arbitrary,

4~K u=x~ujxl,

K[u]=~

1~’,

(3.15a)

~

leads to the following compatibility conditions at the positive resonances: r2=l:

/3= —8, ~ = { —3, —2, (iii) a=3, u0 arbitrary,

—

1, 0},

(3.1 9b)

d3=0; r3=3: d2=d1—co~=0. (3.15b)

Thus, the equation 3 c (ut. + 3uu~+ u 0 u), d0

/3= —10,

3~={—12, —1,0, l}. (3.l9c)

—

(3.16)

The standard Painlevé test shows that the com-

has the Painlevé property, with branch (i) being principal. The compatibility conditions at the resonances of the linearised equation (3.1 3b) are identically satisfied. Branch (ii) only has one positive resonance, so a standard Painlevé analysis only gives one condition: d1+c0d3—2d2d3—c0~+2d2~=0, (3.17)

patibility conditions of all nonnegative resonances are satisfied. Furthermore, our Fuchsian analysis on (3.1 8b) gives all resonances compatible, so that eq. (3.18) passes our test. As with Chazy’s equation, the general solution of (3.18) can be constructed with a negative power series in x:

—

=

0

I

u=x which is identically satisfied if (3.lsb) hold. However, it is seen that a Painlevé analysis ofthis branch does not provide enough information to imply (3.l5b). On the other hand, a Fuchsian analysis of the linear equation (3.13b) does give rise to two nontrivial compatibility conditions which, when taken with (3.17), give precisely (3.15b)! Thus, we see that the secondary branch, whose Painlevé expansion does not give the general solution and does not give enough conditions to determine integrability, does contain the same amount of information as the principal branch. However, this can only be extracted through the Fuchsian part of our analysis. The integrable equation (3. 16) is just the stationary, third order Burgers equation when c0=d0=0. Example 4. This example is fifth order but has several features in common with Chazy’s equation: ~

(3.l8a)

~

~ u1x

(3.20a)

,

where u are constants. We find u0=30;

U_i, U_2, u_3, u_4,

3 20b with (l+l)(l+2)(l+3)(l+4)(l+5)u1 /+1

+

—

~

(i4)(i3)(i2)(i

10

10u3~w~+u4~w=0,

(I—i 1) (i —3) (i —2) (i —

352

I ) u, u1_,

/+ I

+

10 ~

(l—i—2)(l—i— 1 )(i—2) (i—I )u1u1_1

=0.

(3.20c)

Wenotethat u,=0

Vl~—3, (3.21a)

so that we have the truncated solution (3.21b) 2 x3 x x Onceagainx=Oisanessentialsingularityofthegen~

-.

a=l, /36, u0=30, ~{_5,K[u]=K[u], _4, 3,

—

(3.l8b)

where u,,~denotes the nth x-derivative of u. This equation has three branches: (i)

1)u1u1_1

1=—I

u_3=u_4=u5=0 —

u_5arbitrary,

2, _1}, (3.19a)

eral solution and the three (classical) Painlevé so-

Volume 160, number 4

PHYSICS LETTERS A

lutions arise as particular cases of the general solution (3.20).

25 November 1991

w52,,+1=K2~±1[u}w.

(4.3b) We note that the leading order part of (4.3a) is 2IK

4. Integrable hierarchies

{K’2n+i[U0X

2] 2m+i[U0X

_K’~m+I[uoX2]K2n+i[uoX2]}Ix_I=O.

In this section we are not concerned with testing equations for integrability, but assume that we have a hierarchy of integrable equations, meaning that we have a family of commuting flows. Our aim is to briefly describe the relationship between negative resonances and lower order commuting flows in these circumstances. For this simplicity we usethe thedetails KdV and hierarchy to illustrate point, leaving further examples to ref. [9]. The KdV hierarchy,

(44) For a given branch with u 0=—k(k+l)

(ii)

R=a

analysed from the Painlevé point of view [6,7]. The nth flow (4.1) possesses n branches, with u0 given by u0=—k(k+l),

k=l,...,n.

(4.2a)

Ix,=I

=0,

0~mz~k_l.

(4.7a)

Comparing (4.7a) with (2.3b), with a=2, and the definition of a resonance, we see that the k negative resonances 2m—l 0(m~k—l (4.7b)

P(r; n, k)= (r—2n--2) k

JJ (r+2i—1)(r—2i—2n—2) X fl (r—2j— 1) (r+2j—2n—2)

—2,n—3

where Ykm is a nonzero constant. It should be noticed that K 2~+ [u] itself falls into category (i). For type (i), eq. leads to (4.4) is trivially satisfied, while type (ii)

The resonances for the kth branch are solutions of the polynomial P(r; n, k), given by

x

YkmX

(4.6b)

(4.1)

where R is the recursion operator, has recently been

(4.5)

0~mz~k— I for which K2m+I[uoX2] I~=I

,

2,~ =R~u~=K2~+1[u] ,

l~k~n,

the operators K2m+i [u] fall into two categories: 2]Iz,,,i=0, (4.6a) (i) m>~k, forwhichK2m+i[uoX

2+4u+2u~a~

u,

forfixedk,

,

i= 1 n

,

rk.m=—

(4.2b)

(see (4.2c)), arise as a direct result of the first kflows in the hierarchy (4.1). This gives rise to table 1. We can construct the general solution of (4.3b) by

(4.2c)

Tablel

Remark. In refs. [6,7] a recursion relation is given

Flow: n

.

jk+I

The first k of these are negative: r,_——2(k—i)—l,

i=l,...,k.

Branch: k

Resonances: r 1

Commuting flow

for P( r, n, k) but not the closed form solution (4.2b), which appears to be new. We prove this in ref. [9] The flows (4.1) mutually commute, so that

K3 K5

1 I 2

r1 = — I r1=—l r1 = —3

K1 K1 K3

K~n+i[u]K2m+i[u]—K~m+i[u]K2n+i[u]0 Vn,m. (4.3a)

K7

I 2

r2=—l r~=—l r1=—3 r2=—l r1=—5 r2 = —3 r3=—l

K1 K1 K3

This is equivalent to saying that for fixed n, w=K2m+i [u] is a solution ofthe symmetry equation (linearisation) of (4.1):

K5 K3 K1

353

Volume 160, number 4

PHYSICS LETTERS A

Fuchsian expansion, with u given by one of the Painlevé branches. Particular solutions of (4.3b) are the commuting flows (4.1), and expressions for these can be obtained by special choices of arbitrary coefficients. The details of the work of this section will be presented elsewhere [9].

5. Conclusions In this Letter we have presented an extension of the Painlevé test which enables us to analyse negative resonances. The four examples illustrate various novel features, including the equivalence, in some cases, between principal and “secondary branches”. We have also given a clear relationship between negative resonances and lower order commuting flows in the context of integrable hierarchies. There is a well known relationship between resonances and Kowalevskaya exponents [13]. Furthermore, there is a similarity between our Fuchs—Painlevé method and the Kowalevskaya exponent approach, both of which study the linearised equation. However, the Kowalevskaya approach, which gives results on first integrals, seems to have more in common with our section 4 than with our test of seetions 2 and 3, but the exact relationship deserves further study. The linearised equation (2.lb) is the first order term in a perturbation expansion. In refs. [8,9] we have generalised our Fuchs—Painlevé test to one involving higher order perturbations. In ref. [8] we use this approach to isolate an integrable extension of the Chazy equation, with coefficients depending upon one arbitrary function.

Acknowledgement APF would like to thank Martin Kruskal for several conversations which focused our attention on negative resonances. We thank Robert Conte and his colleagues at CEA, for inviting us to Saclay, where the later stages of this work were carried out. AP was funded by an SERC Earmarked Studentship.

25 November 1991

Appendix In this appendix we state some formulae from Conte’s invariant analysis [10]. The functions x, in this paper, and the usual WTC ~, are related by (A.!) This x satisfies the two Ricatti equations 2, x~=l+~Sx2,x5=—C+C~X—~(C~~+CS)x (A.2) with integrability condition 5, + C~ + 2CXS+ Cs~ = 0,

(A.3)

where S and C are the unique differential invariants of the Möbius group, given by s= ~ ~ (~/~) 2 C= q~/co~. (A.4) —

—

The practical advantage of using this approach is to reduce the size of many expressions and thus simplify many of the calculations. References [1] M.J. Ablowitz and H. Segur, Phys. Rev. Lett. 38 (1977) 1103. [21 M.J. Ablowitz, A. Ramani and H. Segur, Lett. Nuovo Cimento 23 (1978) 333. [3]J. Weiss, M. Tabor and G. Carnevale, J. Math. Phys. 24 (1983) 522. [4] S. Chakravarty, M.J. Ablowitz and P.A. Clarkson, Phys. Rev. Lett. 65 (1990)1085.

151

M.D. Kruskal and P.A. Clarkson, The Painlevé—Kowalevski and poly-Painlevé tests for integrability, preprint (1990). [61 A.C. Newell, M. Tabor and Y.B. Zeng, Physica D 29 (1987) I. [71M. Tabor, in: Soliton theory: a survey of results, ed. A.P. Fordy (Manchester Univ. Press, Manchester, 1990) pp. 427—446. [8] R. Conte, A.P. Fordy and A. Pickering, A Fuchs extension to the Painlevé test, in: Proc. III Potsdam—V Kiev mt. Workshop on Nonlinear processes in physics (1991), to be published; A perturbative Painlevé approach to nonlinear differential equations, in preparation. [9] A.P. Fordy and A. Pickering, Integrable hierarchies and negative resonances, in preparation. [10] R. Conte, Phys. Lett. A 140 (1989) 383. [ll]J.Chazy,ActaMath.34(19l1)3l7. [12] F.J. Bureau, Ann. Mat. Pura AppI. 66 (1964) 1. [13] H. Yoshida, B. Grammaticos and A. Ramani, Acta AppI. Math. 8 (1987) 75.

354