On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials

Roffelsen, Pieter

doi:10.3842/sigma.2012.099

Cited by 4 publications

(4 citation statements)

References 6 publications

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“…We notice that the same question for Painlevé II rationals has recently been settled by many authors with a wealth of different methods[2,5,6,41,52,53].…”

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confidence: 77%

Poles of Painlevé IV Rationals and their Distribution

Masoero¹,

Roffelsen²

2018

SIGMA

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Abstract. We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite H m,n and generalised Okamoto Q m,n polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalised Hermite polynomials in the asymptotic regime when m is large and n fixed. To the memory of Andrei A. Kapaev, a master of Painlevé equations, untiring author, referee and reviewer.

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“…We notice that the same question for Painlevé II rationals has recently been settled by many authors with a wealth of different methods[2,5,6,41,52,53].…”

mentioning

confidence: 77%

Poles of Painlevé IV Rationals and their Distribution

Masoero¹,

Roffelsen²

2018

SIGMA

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“…It has been shown that the polynomials tQ n pzqu 8 n"0 all have simple roots and that Q m and Q m´1 can have no roots in common [20], facts that are consistent via (1.8) with the fact that all poles of ppyq are simple with residues ˘1. Real roots of the Yablonskii-Vorob'ev polynomials correspond to real poles of ppyq, and these have been studied extensively by Roffelsen who has shown that all nonzero real roots are all irrational [34] and that there are precisely tpn `1q{3u negative roots of Q n and tpn `1q{2u total real roots of Q n , and Q n p0q " 0 if and only if n " 1 pmod 3q [35]. Also, the real roots of Q n`1 and Q n´1 interlace, as was proven by Clarkson [10].…”

Section: Representation In Terms Of Special Polynomialsmentioning

confidence: 99%

Rational Solutions of the Painlevé-II Equation Revisited

Miller¹,

Sheng

2017

SIGMA

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Abstract. The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlevé-II functions, and then we describe three different Riemann-Hilbert representations of them that have appeared in the literature: a representation by means of the isomonodromy theory of the Flaschka-Newell Lax pair, a second representation by means of the isomonodromy theory of the Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and BertolaBothner Riemann-Hilbert representations of the rational Painlevé-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlevé-II functions obtained from these Riemann-Hilbert representations by means of the steepest descent method.

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“…Generalizing further to y ∈ C, the idea that the rational Painlevé functions considered here may have a particularly interesting structure in the complex y-plane when m is large is indicated by two previous results. Numerically generated plots [9,17,19,24,30] of the m(m + 1)/2 complex zeros of the related mth Yablonskii-Vorob'ev polynomial (see [34,35]) form a highly regular pattern in a triangular-type region with curved sides of size proportional to m 2/3 as m → ∞ (see also the work of Roffelsen [32,33] for other recent results on the Yablonskii-Vorob'ev polynomials, including interlacing properties of the real roots). The rational Painlevé-II functions P m are the logarithmic derivatives of ratios of successive Yablonskii-Vorob'ev The zeros (open circles) and poles (filled circles) of U 13 (y) (left) and P 13 (y) (right) in the complex x-plane, where x = (13 − 1 2 ) −2/3 y, along with the m-independent boundary of the region T .…”

Section: Solutions Of the Inhomogeneousmentioning

confidence: 99%

“…Generalizing further to y ∈ C, the idea that the rational Painlevé functions considered here may have a particularly interesting structure in the complex y-plane when m is large is indicated by two previous results. Clarkson and Mansfield [8] noted from numerically-generated plots that the m(m + 1)/2 complex zeros of the related m th Yablonskii-Vorob'ev polynomial form a highly regular pattern in a triangular-type region with curved sides of size proportional to m 2/3 as m → ∞ (see also the work of Roffelson [25,26] for other recent results on the Yablonskii-Vorob'ev polynomials, including interlacing properties of the roots). The rational Painlevé-II functions P m (y) are the logarithmic derivatives of ratios of successive Yablonskii-Vorob'ev polynomials (or equivalently, the Painlevé-II functions U m (y) are themselves such ratios), and thus the zeros and poles of U m and P m exhibit the same qualitative behavior (see Figure 1, in which we present figures similar to those in [8] but in a rescaled independent variable).…”

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confidence: 99%

Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour

2014

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Abstract. This paper is a continuation of our analysis, begun in [7], of the rational solutions of the inhomogeneous Painlevé-II equation and associated rational solutions of the homogeneous coupled Painlevé-II system in the limit of large degree. In this paper we establish asymptotic formulae valid near a certain curvilinear triangle in the complex plane that was previously shown to separate two distinct types of asymptotic behavior. Our results display both a trigonometric degeneration of the rational Painlevé-II functions and also a degeneration to the tritronquée solution of the Painlevé-I equation. Our rigorous analysis is based on the steepest descent method applied to a Riemann-Hilbert representation of the rational Painlevé-II functions, and supplies leading-order formulae as well as error estimates.

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On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials

Cited by 4 publications

References 6 publications

Poles of Painlevé IV Rationals and their Distribution

Poles of Painlevé IV Rationals and their Distribution

Rational Solutions of the Painlevé-II Equation Revisited

Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour

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