An interpretation of the presence of both positive and negative nongeneric resonances in the singularity analysis

Physics Letters A 359 (2006) 199–203 www.elsevier.com/locate/pla

An interpretation of the presence of both positive and negative nongeneric resonances in the singularity analysis K. Andriopoulos ∗ , P.G.L. Leach School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, Republic of South Africa Received 16 January 2006; accepted 9 June 2006 Available online 19 June 2006 Communicated by C.R. Doering Dedicated to the Memory of Mark Roy Feix

Abstract We demonstrate that the existence of nongeneric positive and negative resonances found in the performance of the standard singularity analysis can lead to the solution of the differential equation under study in an annulus defined by singularities additional to the one about which the analysis is performed. © 2006 Elsevier B.V. All rights reserved.

1. Introduction The standard approach to the singularity analysis as developed by Painlevé and his coworkers following the pioneering work of Kovalevskaya is the determination of the existence of a Laurent expansion about a polelike singularity in a series of ascending integral powers. In the winter of 1995–1996 in seeking to understand the philosophy behind the mechanism of the ARS algorithm [1–3] Mark Feix initiated an investigation of the whole process underlying the algorithm. In the series of papers which ensued from this enquiry [4–6] Feix and his collaborators explored the heuristic basis of singularity analysis which enabled one to see the specialised position enshrined in the ARS algorithm which, properly, seeks to establish the presence or otherwise of the Painlevé property for a given differential equation. One of the results was a very clear statement of the meaning of the existence of nongeneric negative resonances in terms of the standard interpretation of a Laurent expansion. A Laurent expansion ascending from a negative integral power, termed by Feix a right Painlevé series, represented the solution of the equation within a punctured disc centred on the singular-

ity. An expansion descending from a negative integral power, termed by Feix a left Painlevé series, represented the solution of the equation without a disc. Nongeneric negative resonances had been previously considered [7,8]. Since the presence of nondominant terms in an equation precludes the possibility of the possession of a left Painlevé series, they do not occur with the same frequency as the more usual right Painlevé series. A class of equations which automatically has all terms dominant consists of those equations which possess the two symmetries of invariance under time translation and self-similarity. As this class of equations plays an important role in physical applications, one is not surprised that Feix and his colleagues made particular study of representatives of them.1 Elementary examples of the class are the Riccati equation, x˙ + x 2 = 0, and the Painlevé–Ince equation, x¨ + 3x x˙ + x 3 = 0. In both cases the self-similarity symmetry is Γ2 = −t∂t + x∂x which is consistent with the exponent of the leading order term being −1. In the context of nonlinear evolution equations there is a considerable body of literature devoted to the investigation of integrable hierarchies which can be constructed by means of recursion operators which generate an infinite number of Lie– Bäcklund symmetries [11–13]. Euler et al. [14] provide eight classes of hierarchies which may be linearised. Euler et al. [15]

* Corresponding author.

E-mail addresses: [email protected] (K. Andriopoulos), [email protected] (P.G.L. Leach). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.06.026

1 In addition to the papers cited above see also [9,10].

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recently adapted the concept of recursion operators to ordinary differential equations and demonstrated the construction of hierarchies of ordinary differential equations for one of the eight classes considered by Euler et al. and another class representing a particular type of nonlinearisable equation treated by Petersson et al. [16]. The former hierarchy contains as particular examples of the lower members the above-mentioned Riccati [17] and Painlevé–Ince ([18, (9) p. 33] and [19, p. 332]) equations. The reader seeking a full discussion of these hierarchies and the methods of analysis is referred to these papers. However, in this Letter we provide sufficient information for it to be self-contained. The members of the Riccati hierarchy are characterised by being of maximal symmetry at any given order and, since each is linearisable by means of a Riccati transformation, the solution is trivial. They also have interesting and explicit properties as far as the singularity analysis is concerned. These two features make the members of the Riccati hierarchy ideal for our exposition, the main point of which is to demonstrate by way of precise example that the existence of both positive and negative nongeneric resonances, i.e. excluding −1, is not an hindrance to the possession of the Painlevé property but indeed is to be expected in some equations. We do not make the claim that this is always the case. Rather our point is that it can be the case. In this respect we supplement the question asked by Mark Feix some 10 years ago. 2. The Riccati hierarchy and some properties Of the eight classes of second-order linearisable evolution equations and their recursion operators reported by Euler et al. [14] the class, VIII, which leads to the Riccati hierarchy for ordinary differential equations has the recursion operator R8 [u] = Dx + h8 (u)ux .

(2.1)

The basic evolution equation is ut = uxx + λ8 ux + h8 u2x

(2.2)

(here and below we suppress the variable dependence in h8 unless it is necessary for contextual clarity), where λ8 is some parameter. By the t -translation symmetry the corresponding ordinary differential equation is u + h8 u 2 + λ8 u = 0

(2.3)

and successive members of the hierarchy can be obtained by the action of R8 [u] + C on the left-hand side of (2.3). For our purposes we need only λ = 0 = C and h8 = 1 for the construction of the Riccati hierarchy. We write the hierarchy in potential form by replacing u with y so that the first few members of the hierarchy are y  + y 2 = 0, 











(2.4)

y + 3yy + y = 0, 3

2

(2.5) 2 

y + 4yy + 3y + 6y y + y = 0, y

 

4

2 

+ 5yy + 10y y + 10y y + 15yy

(2.6) 2

+ 10y 3 y  + y 5 = 0,

(2.7)

with the recursion operator R = D + y,

(2.8)

where we replace Dx with D since there is just the single independent variable. The results of the application of the Painlevé test in the usual manner are summarised in Table 1 [15]. Of particular interest is the value of the resonances for the different roots of the polynomial equation determining α. From Table 1 we can discern the pattern for the higher-order equations. For the nth-order member of the Riccati hierarchy the nontrivial values of the coefficient of the leading order term can be 1, 2, . . . , n. In the case of the Riccati equation itself we have only the generic resonance. For the Painlevé–Ince equation the nongeneric resonance corresponding to each of the two possible values of α indicates the existence of a left Painlevé series and a right Painlevé series, as is well known [4]. For the higher-order equations the different principal branches indicate quite diverse behaviours. A right Painlevé series exists for the

Table 1 Summary of the results of the Painlevé test applied to the earlier members of the Riccati hierarchy. We commence the numbering of the members from the Riccati equation, (2.4) Member

Characteristic equations for α and r

Roots

I

α2 − α = 0

α = 0, 1 r = −1 α = 0, 1, 2 α = 1: r = −1, 1 α = 2: r = −1, −2 α = 0, 1, 2, 3 α = 1: r = −1, 1, 2 α = 2: r = −1, 1, −2 α = 3: r = −1, −2, −3 α = 0, 1, 2, 3, 4 α = 1: r = −1, 1, 2, 3 α = 2: r = −1, 1, 2, −2 α = 3: r = −1, 1, −2, −3 α = 4: r = −1, −2, −3, −4

II

r +1=0 α 3 − 3α 2 + 2α = 0 r 2 + (3α − 3)r + 3α 2 − 6α + 2 = 0

III

α 4 − 6α 3 + 11α 2 − 6α = 0 r 3 + (4α − 6)r 2 + (6α 2 − 18α + 11)r + 4α 3 − 18α 2 + 22α − 6 = 0

IV

α 5 − 10α 4 + 35α 3 − 50α 2 + 24α = 0 r 4 + 5(α − 2)r 3 + 5(2α 2 − 8α + 7)r 2 + 5(2α 3 − 12α 2 + 21α − 10)r + 5α 4 − 40α 3 + 105α 2 − 100α + 24 = 0

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smallest value of α and a left Painlevé series for the largest value of α. For the intermediate values mixed behaviour occurs. One can have both left Painlevé series and right Painlevé series. 3. Laurent expansions In this section we construct the Laurent expansions from the explicit solutions of the second, third and fourth members of the Riccati hierarchy by use of the linearising transformation which is a standard feature of all members of the hierarchy. As the solution of the first member, the Riccati equation itself, is just one term, we do not consider it, but do recall that the same linearising transformation exists. 3.1. The Painlevé–Ince equation We apply the Riccati transformation, y = αw  /w, to the Painlevé–Ince equation, y  + 3yy  + y 3 = 0,

(3.9)

to obtain the third-order equation 





3

w w + 3(α − 1)w w + (α − 1)(α − 2)w = 0. 2

(3.10)

We note that the values of α which produce a simplification of (3.10) are just those values obtained for the coefficient of the leading order term in the singularity analysis. The degree of simplification differs from the value of α leading to the right Painlevé series to that leading to the left Painlevé series. We choose the former so that (3.10) reduces to the elementary equation with solution w  = 0

and w(x) = A0 + A1 x + A2 x 2 .

(3.11)

The solution to (3.9) is then y(x) =

A1 + 2A2 x . A0 + A1 x + A2 x 2

y(x) =

  1 (x − x1 ) + (x − x2 ) , (x − x1 )(x − x2 )

in the case that |χ| > |x2 − x1 |, which leads to the left Painlevé series. It is easy to see that the coefficient of χ −1 is 1 in the former case and 2 in the latter case. Thus we see the origins of the left and right Painlevé series for the Painlevé–Ince equation. The right series is convergent in the punctured disc centred on x = x1 of radius |x2 −x1 | whereas the left series is convergent on the whole of the complex plane exterior to this disc. 3.2. The third member of the Riccati hierarchy We recall that the third member of the Riccati hierarchy is y  + 4yy  + 3y  2 + 6y 2 y  + y 4 = 0.

(3.13)

where the two roots must be distinct. There is the choice of one of x1 or x2 as the singularity about which the Laurent expansion is to be made. Without loss of generality we may take this as x1 . If we write χ = x − x1 , (3.13) becomes 1 1 + χ χ − (x2 − x1 )   1 1 1 = − χ χ (x2 − x1 ) 1 − (x2 −x 1)

y=

in the case that |χ| < |x2 − x1 |, which leads to the right Painlevé series, and   1 1 1 y= + χ χ 1 − (x2 −x1 ) χ

(3.14)

The analysis follows the same general lines as that for the Painlevé–Ince equation and we simply summarise the details. Under the same Riccati transformation (3.14) becomes w 3 w  + 4(α − 1)w 2 w  w  + 3(α − 1)w 2 w  2 + 6(α − 1)(α − 2)ww  2 w  + (α − 1)(α − 2)(α − 3)w 4 = 0.

(3.15)

We observe that the three values possible for the coefficient of the leading order term appear in (3.15). Obviously the choice α = 1 gives the simplest fourth-order equation. The choices α = 2 and α = 3 are not so simple in that they give nonlinear equations in w. However, the equations are linear in w 2 and w 3 , respectively. This is a general feature for the members of the hierarchy. With α = 1 we obtain the equation and solution w  = 0 and w(x) = A0 + A1 x + A2 x 2 + A3 x 3

(3.16)

so that A1 + 2A2 x + 3A3 x 2 A0 + A1 x + A2 x 2 + A3 x 3 1 1 1 = + + . x − x1 x − x2 x − x 3

y(x) =

(3.12)

We may rewrite the solution in (3.12) as

201

(3.17)

Again the roots of w(x) = 0 must be simple due to the nature of the singularity of the leading order term of y(x). If we suppose that the singularity analysis is about the singularity at x = x1 , i.e. χ = x − x1 , we may rewrite (3.17) as y=

1 1 1 + + χ χ − (x2 − x1 ) χ − (x3 − x1 )

(3.18)

for which a series expansion is possible in three ways in contrast to the two ways for the Painlevé–Ince equation. When |χ| < |x2 − x1 | and |χ| < |x3 − x1 |, we write (3.18) as     1 1 1 1 1 y= − − χ χ χ (x2 − x1 ) 1 − x2 −x (x3 − x1 ) 1 − x3 −x 1 1 so that we have an expansion in ascending powers of χ , i.e. the right Painlevé series. When |χ| > |x2 − x1 | and |χ| > |x3 − x1 |, (3.18) takes the form     1 1 1 1 1 + y= + 1 1 χ χ 1 − x2 −x χ 1 − x3 −x χ χ

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which is an expansion in descending powers of χ , i.e. the left Painlevé series. When |χ| > |x2 − x1 | and |χ| < |x3 − x1 | (there is no difference if the inequalities are reversed apart from some trivial relabelling), it is necessary to write (3.18) as     1 1 1 1 1 y= + − χ 1 χ χ 1 − x2 −x (x3 − x1 ) 1 − x3 −x χ 1 so that the second term is expanded as a series of decreasing powers and the third term is expanded as a series of increasing powers. This is a (full) Laurent expansion, i.e. the indexing of the elements of the series ranges from minus infinity to plus infinity. For this member of the Riccati hierarchy there are three patterns for the nongeneric resonances. In the case that α = 1 the nongeneric resonances are both positive and lead to the right Painlevé series indicated above. In the case that α = 3 the two nongeneric resonances are negative and we have the left Painlevé series. However, when α = 2, one of the nongeneric resonances is positive and the other is negative. The usual interpretation of this is that one must choose just one branch to make sense of the concept of leading order behaviour. Yet here we have demonstrated that the explicit solution obtained naturally contains a series going to the left and a series going to the right. The right series has a finite radius of convergence. The left series is not valid on and within a disk centred on the singularity, but is valid beyond that to infinity. The (full) Laurent expansion converges on the interior of an annulus. Consequently the mix of nongeneric positive and negative resonances makes sense. We do not have sufficient evidence to claim that this is the interpretation to be given in every instance of the occurrence of mixed signs in the nongeneric resonances, but we do have an effective counterexample to the claim that nongeneric resonances must all be positive or all be negative. 3.3. The fourth member of the Riccati hierarchy We have made our point concerning the admissibility of nongeneric resonances of different sign in terms of the possibility to express the solution in terms of a convergent Laurent series. To emphasize our point we consider just one more example. The fourth member of the Riccati hierarchy is y  + 5yy  + 10y  y  + 10y 2 y  + 15yy  2 + 10y 3 y  + y 5 = 0. (3.19) Under the Riccati transformation (3.19) becomes the fifth-order equation w 4 w  + 5(α − 1)w 3 w  w  + 10(α − 1)w 3 w  w  + 20(α − 1)(α − 2)w 2 w  2 w  + 15(α − 1)(α − 2)w 2 w  w  2 + 10(α − 1)(α − 2)(α − 3)ww  3 w  + (α − 1)(α − 2)(α − 3)(α − 4)w 5 = 0

(3.20)

in which it is evident that the four possible values of the coefficient of the leading order term lead to a simplification of the equation. As we observed in the case of the third member, there is no essential difference in the final result since all values lead

to a linear equation and so we use the simplest value, α = 1, to obtain w  = 0 and w(x) = A0 + A1 x + A2 x 2 + A3 x 3 + A4 x 4 . (3.21) The solution of (3.19) may then be written in the form y(x) =

1 1 1 1 + + + . x − x 1 x − x2 x − x3 x − x4

(3.22)

The situation should now be clear. We assume that an additional expansion is to be about the singularity at x = x1 and that the additional singularities are ordered in distance from x1 as they stand. Then in a punctured disc centred on x1 and of radius of |x2 − x1 | there is a right Painlevé series. The left Painlevé series exists outside the disc centred on x1 and of radius |x4 − x1 |. The region between these two discs is divided into two annuli. Obviously both are centred on x1 . The first is defined by |x2 − x1 | < |x| < |x3 − x1 | and the second by |x3 − x1 | < |x| < |x4 − x1 |. The first corresponds to α = 2 with the resonances r = −2, −1, 1, 2 and the second to α = 3 with the resonances r = −3, −2, −1, 1. In each of these four regions (3.19) is analytic so that the equation possesses the Painlevé property as the explicit solution simply demonstrates. The vagaries of the signs of the resonances are a consequence of the nature of the Laurent expansion appropriate for different regions of the complex plane as a local representation of the solution of this fourth-order nonlinear ordinary differential equation. It should be evident from the examples of the Riccati hierarchy considered that higher members of the hierarchy display similar properties with simply an increase in the number of annuli. Each of the possible values for the coefficient of the leading order term leads to a Laurent expansion which is appropriate for a particular portion of the complex plane. In essence the complex plane is covered except for a sequence of circles of increasing radius centred upon the singularity about which the analysis has been undertaken. The interior region is covered by a right Painlevé series and the exterior region by a left Painlevé series. Between these two regions there is a succession of annuli each of which has its own Laurent expansion with the summation ranging from minus infinity to plus infinity. The leading order term is dominant—apart from the value of its coefficient—for these different regions of the complex plane. 4. Conclusion The Painlevé–Ince equation has a long history of appearance in theoretical aspects and divers applications of differential equations. This is in a variety of contexts including singularity analysis, symmetry analysis and linearisation. Euler et al. [15] added a new dimension to its properties by their demonstration of its relationship to a particular class of linearisable and integrable (in the sense of possessing an infinite number of Lie–Bäcklund symmetries) nonlinear evolution partial differential equations. In their generation of the Riccati hierarchy of linearisable nonlinear ordinary differential equations Euler

K. Andriopoulos, P.G.L. Leach / Physics Letters A 359 (2006) 199–203

et al. have provided a rich testbed for properties of nonlinear ordinary differential equations. In this Letter we have used the hierarchy to investigate the question of the meaning of positive and negative nongeneric resonances. We found that as one ascends the hierarchy the number of the possible coefficients of the leading order term increases. Indeed it is equal to the order of the equation. Each of the admissible coefficients produces its own set of nongeneric resonances. For third and higher-order equations in the hierarchy there are three possible arrangements of the nongeneric resonances. One is to produce a set of positive resonances which is in the realm of the traditional Painlevé analysis. One is to produce a set of negative resonances which has in the last decade or so been recognised as being an equally valid analysis even though it yields a solution which is not in the neighbourhood of the polelike singularity. The positive resonances occur for the lowest value of the coefficient of the leading order term and negative for the highest value. For the values of the coefficient between the lowest and the highest one can see a transition from all positive nongeneric resonances to all negative nongeneric resonances. Since we have an explicit solution for every member of the Riccati hierarchy,2 it is possible to relate the pattern of resonances to the different ways of expressing the solution of the equation in terms of Laurent series. It is clear that for the members of the Riccati hierarchy each set of mixed resonances corresponds to a Laurent series valid over an annulus. With increasing order of the equation the number of annuli increases. The completeness of the properties of the members of the Riccati hierarchy make them excellent vehicles for the illustration of points such as we have discussed in this Letter. Generally speaking the differential equation possessing the Painlevé property does not come with equally complete properties. Usually there exists just the right Painlevé series. In the case of equations such as the Painlevé–Ince equation one finds both left and right series. This is unusual. Sometimes one finds an equation which has simply the left series. This then brings us to a ques-

2 We have provided only the first few in this Letter, but the general result is found in Euler et al. [15] and does not require much imagination to infer from the solutions given above.

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tion as yet unanswered. Can an equation with only a mixture of positive and negative nongeneric resonances in fact possess the Painlevé property? Acknowledgements P.G.L.L. thanks Mark Feix for many illuminating discussions on the nature of symmetry and singularity over a period of long and fruitful collaboration and the University of KwaZuluNatal for its continued support. K.A. thanks the State Scholarship Foundation (Hellenic Democracy). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16] [17] [18] [19]

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