Elsevier

Physics Letters A

Volume 301, Issues 3–4, 26 August 2002, Pages 275-280
Physics Letters A

Coalescence limits for higher order Painlevé equations

https://doi.org/10.1016/S0375-9601(02)00972-6Get rights and content

Abstract

It is well-known that the first Painlevé equation arises as a coalescence limit of each of the other five Painlevé equations. This result is important because it shows that, since the solution of the first Painlevé equation cannot be expressed in terms of known functions, then neither can the solutions of the other five Painlevé equations (except possibly for special values of their parameters). Here we derive analogous results for three recently derived higher order ordinary differential equations believed to define new transcendental functions. We show that each of the equations considered has as a coalescence limit a member of the first Painlevé hierarchy. We thus reduce the problem of showing that the solutions of these three cannot be expressed in terms of known functions to that of showing that the same is true for the corresponding first Painlevé equations. This represents the first extension of coalescence results for the Painlevé equations to their higher order analogues.

Introduction

The search for new functions defined as solutions of differential equations led, at the turn of the last century, to the discovery of the six Painlevé equations [1], [2], [3], [4]. One question of particular importance, and indeed of some controversy, was that of whether the solutions of these equations could be expressed in terms of known functions. Since it could be shown that the first Painlevé equation arises as a coalescence limit of the other five (see [4]), this question was reduced to that of showing that the solution of the first Painlevé equation defines a new transcendent. This last problem has in fact only been solved remarkably recently [5], [6], [7].

As an example of a coalescence limit, let us consider that between the second Painlevé equation PII, in v(y) with arbitrary parameter a, vyy−2v3−yv−a=0, and the first Painlevé equation PI, in u(x), uxx+3u2+x=0. Making the change of variables (a rescaled version of that in [4]) v(y)=εu(x)−15,y=2ε2x−332ε10,a=−1128ε15, in (1) leads to the equation uxx+3u2+x−8ε6u2+xu=0, which in the limit ε→0 gives (2). Thus we see that PII contains PI as a coalescence limit. Coalescence limits of the six Painlevé equations may be summarized as [4], [8] and thus we see that PI can be obtained from each of the other Painlevé equations.

It is this limiting process that we seek to explore here for higher order analogues of the Painlevé equations. In particular, we will seek linear transformations of dependent and independent variables, and also of parameters ai (into new parameters A1,A2,…,AN), with coefficients dependent on a parameter ε, v(y)=α(ε)u(x)+β(ε),y=δ(ε)x+γ(ε),ai=j=1Nλj(ε)Aj+μ(ε). We note that all the transformations used in the coalescence processes in (5) are of this form [4] (see, e.g., (3)). It turns out that consideration of such linear transformations is sufficient for the purposes of the present Letter. Our requirement is that the expression obtained by solving the transformed equation for the highest derivative of u be analytic in ε at ε=0; the limit ε→0 then gives our coalescence limit.

There are two reasons why our results are important. First, we extend the analogy between certain Painlevé equations and their higher order analogues. Second, as with Painlevé's results [4], we reduce the number of equations for which it must be shown that their solution cannot be expressed in terms of known functions. This last is of great practical importance in the study of higher order Painlevé equations.

Higher order Painlevé equations may be obtained in a variety of ways. One is by taking similarity reductions of the higher order members of a hierarchy of completely integrable partial differential equations; thus, for example, the modified Korteweg–de Vries hierarchy yields the PII hierarchy [9]. Another is by extending the classification programme of Painlevé to higher order differential equations [10]. A third approach is that developed in [11], [12], [13], [14]. Here we consider coalescence limits for certain higher order analogues of the second Painlevé equation.

Section snippets

Examples from the generalized second Painlevé hierarchy

Here we consider two examples from a generalized version of the PII hierarchy, y−1R̄nvy+∂y−1j=1n−1bjR̄jvy−yv−a=0. Here a and all bj are arbitrary constants, y=d/dy and R̄=∂y2−4v2−4vyy−1v, is the recursion operator of the modified Korteweg–de Vries hierarchy. The hierarchy of equations (7) consists of linear combinations of the members of the PII hierarchy given in [9]; setting n=1 gives PII (1). We note that here we have assumed that the coefficient of the non-autonomous term is non-zero,

A new higher order second Painlevé equation

In this section we consider the second member of an alternative PII hierarchy presented in [14]. This equation was originally derived as a system of equations, 14vyy−3vvy+v3+6vw+cv+g3y−γ2=0,14wyy+3w2+3vwy+3v2w+cw−δ2=0, where g3, c, γ2 and δ2 are all constants. The system (22), (23) has the underlying linear problem Ψy=FΨ,12g3Ψλ=HΨ, where F=−λ+12v1−wλ−12v,H=14−wy−2vw−2λw2w−vy+v2−2g3y−4λ3−4cλ+2γ2+2λv+4λ2+4cvwy−2λwy−wvywy+2vw+2λw+w2+2v2w−2λvw+2g3y+4λ3+4cλ−2γ2−4λ2w−4δ2. Solving (22) for w and

Conclusions

In this Letter we have considered coalescence limits for certain members of two distinct PII hierarchies. We have shown that these equations have as coalescence limits corresponding members of the first Painlevé hierarchy. Thus we at once both extend the analogy between the higher order analogues of the second and first Painlevé equations with those equations themselves, and at the same time reduce the number of equations for which it must be shown that their solutions cannot be expressed in

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