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 Hm,n and generalised Okamoto Qm,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.
D.M. is an FCT Researcher supported by the FCT Investigator Grant IF/00069/2015. D.M. is
also partially supported by the FCT Research Project PTDC/MAT-STA/0975/2014. The
present work began in December 2015 while D.M. was a Visiting Scholar at the University
of Sydney funded by the ARC Discovery Project DP130100967.
P.R. is a research associate at the University of Sydney, supported by Nalini Joshi’s ARC
Laureate Fellowship Project FL120100094.
Received:July 20, 2017; in final form December 18, 2017; Published online January 6, 2018
\Bibitem{MasRof18}
\by Davide~Masoero, Pieter~Roffelsen
\paper Poles of Painlev\'e IV Rationals and their Distribution
\jour SIGMA
\yr 2018
\vol 14
\papernumber 002
\totalpages 49
\mathnet{http://mi.mathnet.ru/sigma1301}
\crossref{https://doi.org/10.3842/SIGMA.2018.002}
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This publication is cited in the following 12 articles:
Nalini Joshi, Pieter Roffelsen, “On symmetric solutions of the fourth q-Painlevé equation”, J. Phys. A: Math. Theor., 56:18 (2023), 185201
V. Yu. Novokshenov, “Approximation of Zeros of Generalized Hermite Polynomials by Modulated Elliptic Function”, J Math Sci, 264:3 (2022), 353
Mikhail Bershtein, Pavlo Gavrylenko, Alba Grassi, “Quantum Spectral Problems and Isomonodromic Deformations”, Commun. Math. Phys., 393:1 (2022), 347
R. Conti, D. Masoero, “Counting monster potentials”, J. High Energy Phys., 2021, no. 2, 59
Xia J. Xu Sh.-X. Zhao Yu.-Q., “Isomonodromy Sets of Accessory Parameters For Heun Class Equations”, Stud. Appl. Math., 146:4 (2021), 901–952
D. Masoero, P. Roffelsen, “Roots of generalised Hermite polynomials when both parameters are large”, Nonlinearity, 34:3 (2021), 1663–1732
D. Gomez-Ullate, Y. Grandati, R. Milson, “Complete classification of rational solutions of a(2n)-Painleve systems”, Adv. Math., 385 (2021), 1077707
N. Bonneux, “Asymptotic behavior of wronskian polynomials that are factorized viap-cores andp-quotients”, Math. Phys. Anal. Geom., 23:4 (2020), 36
T. Bothner, P. D. Miller, “Rational solutions of the Painleve-iii equation: large parameter asymptotics”, Constr. Approx., 51:1 (2020), 123–224
Clarkson P.A. Gomez-Ullate D. Grandati Y. Milson R., “Cyclic Maya Diagrams and Rational Solutions of Higher Order Painleve Systems”, Stud. Appl. Math., 144:3 (2020), 357–385
N. Bonneux, C. Dunning, M. Stevens, “Coefficients of wronskian Hermite polynomials”, Stud. Appl. Math., 144:3 (2020), 245–288
Victor Yu. Novokshenov, “Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators”, SIGMA, 14 (2018), 106, 13 pp.
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