Singular asymptotics for the Clarkson–McLeod solutions of the fourth Painlevé equation

Xia, Jun; Xu, Shuai‐Xia; Zhao, Yu–Qiu

doi:10.1016/j.physd.2022.133254

Cited by 4 publications

(5 citation statements)

References 9 publications

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“…Furthermore, we accomplish the case b 1 = 0 in (1.18) and (1.21) which is not covered in [18,Equation (1.10)] and [27,Equation (1.5)]. We also show that (1.20) is true for negative half-integer α, a case not considered in our previous work [28]. We also provide a novel proof of (1.19).…”

Section: Asymptotics Of the Clarkson-mcleod Solutionsmentioning

confidence: 63%

“…While α > −1/2, the same numerical results allow us to expect the absence of the real poles of q(x; α, κ). The asymptotic behavior of q(x; α, κ) and the connection formulas in part (c) were recently derived by the current authors in [28]. While, to the best of our knowledge, the asymptotic result in part (b) of the Clarkson-McLeod conjecture has not been confirmed.…”

Section: Conjecture (Clarkson-mcleod [9])mentioning

confidence: 71%

“…When |s * | > 1 and as x → −∞, a detailed RH analysis has been carried out by the current authors in [28] for α ∈ R but α − 1 2 / ∈ Z. In this section, we will briefly review the RH analysis therein, and discuss the exceptional case 1 2 − α ∈ N.…”

Section: Rh Analysis As X → −∞ With |S * | >mentioning

confidence: 99%

See 2 more Smart Citations

Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant

Xia

Zhao

2023

Journal of Differential Equations

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Section: Asymptotics Of the Clarkson-mcleod Solutionsmentioning

confidence: 63%

Section: Conjecture (Clarkson-mcleod [9])mentioning

confidence: 71%

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Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant

Xia

Zhao

2023

Journal of Differential Equations

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“…The potential V is associated to the rank-one case of an extended Calogero system with rational potential [15,16], has an isomonodromic Lax pair [3], and reduces in the β = 0 case to an equation isolated by Halburd and Kecker [10] through Nevanlinna-theoretic methods, as we will see in section 4.1. We also note that this system has been studied by various authors in connection with the fourth Painlevé equation both before [1,2,17] and after Takasaki's paper [38][39][40].…”

Section: Takasaki's Rational Painlevé-calogero System Related To P IVmentioning

confidence: 88%

Takasaki’s rational fourth Painlevé-Calogero system and geometric regularisability of algebro-Painlevé equations

Filipuk,

Stokes

2023

Nonlinearity

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We propose a notion of regularisation which extends Okamoto’s construction of spaces of initial conditions for the Painlevé differential equations to the class of systems with globally finite branching about movable singularities in the sense of the algebro-Painlevé property. We illustrate this regularisation first in the case of a Hamiltonian system obtained by Takasaki as part of the Painlevé-Calogero correspondence, which is related by an algebraic transformation to the fourth Painlevé equation. Through a combination of compactification, blowups and removal of certain curves we obtain a space on which the system is everywhere either regular or regularisable by certain algebraic transformations. We provide an atlas for this space in which the system has a global Hamiltonian structure, with all Hamiltonian functions being polynomial in coordinates just as in the case of the Painlevé equations on Okamoto’s spaces. We also compare the surface associated with the Takasaki system with that of the fourth Painlevé equation, showing that they are related by a combination of blowdowns and a branched double cover. We provide more examples of algebro-Painlevé equations regularised in this way and also discuss applications of this generalised construction of the space of initial conditions to the identification and classification of algebro-Painlevé equations.

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“…We introduce the function V in (5.3) in order to fulfil the matching conditions (5.6) below. Similar techniques have been applied in [43] to study singular asymptotics of the Painlevé IV transcendents, and in [44,45] to derive uniform asymptotic approximations of some orthogonal polynomials.…”

Section: Rh Problem For P (∞)mentioning

confidence: 99%

Asymptotics of the deformed higher order Airy-kernel determinants and applications

Xia

Hao

et al. 2023

Nonlinearity

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We study the one-parameter family of Fredholm determinants det ( I − ρ 2 K n , x ) , ρ ∈ R , where K n , x stands for the integral operator acting on L 2 ( x , + ∞ ) with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy–Widom distribution. Each of the determinants admits an integral representation in terms of a special real solution to the nth member of the Painlevé II hierarchy. Using the Riemann–Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlevé II transcendents as x → − ∞ for 0 < | ρ | < 1 and | ρ | > 1 , respectively. In the case of 0 < | ρ | < 1 , we are able to calculate the constant term in the asymptotic expansion of the determinants, while for | ρ | > 1 , the relevant asymptotics exhibit singular behaviours. Applications of our results are also discussed, which particularly include asymptotic statistical properties of the counting function for the random point process defined by the higher order Airy kernel.

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Singular asymptotics for the Clarkson–McLeod solutions of the fourth Painlevé equation

Cited by 4 publications

References 9 publications

Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant

Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant

Takasaki’s rational fourth Painlevé-Calogero system and geometric regularisability of algebro-Painlevé equations

Asymptotics of the deformed higher order Airy-kernel determinants and applications

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