On some special solutions of the fifth Painlevé equation

Andreev, F. V.

doi:10.1007/bf02673589

Cited by 5 publications

(25 citation statements)

References 4 publications

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“…Asymptotic expansions as x → 0 and as x → ∞ for various solutions to the fifth Painlevé equation and the system (1.44)-(1.45) were obtained in several works, see e.g. [1,2,3,25,31,33]. The solution v which is of interest to us decays exponentially at +∞, is integrable near 0 if Re α > − 1 2 , and it has no poles on (0, +∞) if α > − 1 2 ∈ R and β ∈ iR.…”

Section: Rh Problem Formentioning

confidence: 98%

Emergence of a singularity for Toeplitz determinants and Painlevé V

Claeys¹,

Its²,

Krasovsky³

2011

Duke Math. J.

136

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We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter t. For t positive, the symbols are regular so that the determinants obey Szegő's strong limit theorem. If t = 0, the symbol possesses a Fisher-Hartwig singularity. Letting t → 0 we analyze the emergence of a Fisher-Hartwig singularity and a transition between the two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a special Painlevé V transcendent. A particular case of our result complements the classical description of Wu, McCoy, Tracy, and Barouch of the behavior of a 2-spin correlation function for a large distance between spins in the two-dimensional Ising model as the phase transition occurs.Using the RH properties of S, N , and P , we obtain the following. RH problem for R(a) R is analytic in C \ Σ R , where Σ R is the union of ∂U and the parts of Σ 1 , Σ 2 lying outside U (see Figure 4). (3.29) and J k (z), k = 1, 2 are the jump matrices (3.3), (3.5) of S. (c) As z → ∞, R(z) = I + O(z −1 ). Using (3.3), (3.5), and (3.23), we observe a crucial fact: J R (z) = I + O(n −1 ), for z ∈ ∂U , (3.30) J R (z) = I + O(e −cn ), for z ∈ Σ R \ ∂U , c > 0 (3.31)

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Section: Rh Problem Formentioning

confidence: 98%

Emergence of a singularity for Toeplitz determinants and Painlevé V

Claeys¹,

Its²,

Krasovsky³

2011

Duke Math. J.

136

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“…For the truncated or classical solutions, the condition m 0 11 m 1 11 m 0 21 m 1 12 = 0 is not fulfilled (cf. [5, §5], [2], [4]).…”

Section: Resultsmentioning

confidence: 99%

“…By this fact combined with the surjectivity of the Riemann-Hilbert correspondence [7], [35] and the uniqueness [5, Propositions 2.1 and 2.2] (see also [12,Proposition 5.9 and Theorem 5.5], [13], [5, § §3, 4, 5], [2], [3] and [4]) we have Proposition 3.1. Let Y(P V ) be the family of solutions of (P V ).…”

Section: Basic Factsmentioning

confidence: 99%

“…For y(x) corresponding to each (M 0 , M 1 ) ∈ M * the asymptotic representations along the positive real axis are found in [5, § §3, 4, 5], [2]. In [5, Theorem 4.1], for example, the expression of the general singular solution is valid in the strip Re x = t ≥ t 0 , |Im x| ≪ 1 except for neighbourhoods of poles and 1-points which are disjoint and have a uniform radius.…”

Section: Basic Factsmentioning

confidence: 99%

“…12 = 0 or m 0 21 = 0, truncated solutions such that a φ (t) ≪ t −1 for |φ| < π/2 are given by [2]. Remark 3.6.…”

Section: )) With the Propertiesmentioning

confidence: 99%

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Elliptic asymptotic representation of the fifth Painlevé transcendents

Shimomura¹

2020

Preprint

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For the fifth Painlevé transcendents an asymptotic representation by the Jacobi sn-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrised by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the sn-function and the ϑ-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the sn-function satisfying the Boutroux equations and to construct deductively the elliptic representation.

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Connection formulae for asymptotics of the fifth Painlevé transcendent on the real axis

2000

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Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as t → ı∞ are found. These asymptotics are parameterized by monodromy data of the associated linear ODE,The parametrization allows one to derive connection formulas for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulas are also considered.

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On some special solutions of the fifth Painlevé equation

Abstract: Four kinds of special solutions of the fifth Painlevd equation are described. Their asymptotic expansions for t --* +oc are given. The corresponding monodromy data are calculated. This gives the possibility of obtaining connection formulas. Bibliography: 7 titles.

Cited by 5 publications

References 4 publications

Emergence of a singularity for Toeplitz determinants and Painlevé V

Emergence of a singularity for Toeplitz determinants and Painlevé V

Elliptic asymptotic representation of the fifth Painlevé transcendents

Connection formulae for asymptotics of the fifth Painlevé transcendent on the real axis

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