2-Parameter $$\tau $$-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis

Iwaki, Kohei

doi:10.1007/s00220-020-03769-2

Cited by 24 publications

(32 citation statements)

References 77 publications

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“…These 2-parameter transseries solutions have also appeared in many different guises in other work, e.g. [4,9,10,12,13].…”

Section: From Formal Solutions To Transcendentssupporting

confidence: 53%

“…It is well-known how to obtain a two-parameter family of solutions also on the formal side [3][4][5][6][7][8][9][10]. To achieve this, one extends the concept of a power series to that of a transseries (see e.g.…”

Section: From Formal Solutions To Transcendentsmentioning

confidence: 99%

“…The linear system (B.1) with solution Y = (Y 1 , Y 2 ) T can be reduced to the following set [10] of scalar equations:…”

Section: B Theory Of Isomonodromic Deformationmentioning

confidence: 99%

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Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries

Spaendonck¹,

Vonk²

2022

Preprint

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For more than a century, the Painlevé I equation has played an important role in both physics and mathematics. Its two-parameter family of solutions was studied in many different ways, yet still leads to new surprises and discoveries. Two popular tools in these studies are the theory of isomonodromic deformation that uses the exact WKB method, and the asymptotic description of transcendents in terms of two-parameter transseries. Combining methods from both schools of thought, and following work by Takei and collaborators, we find complete, two-parameter connection formulae for solutions when they cross arbitrary Stokes lines in the complex plane. These formulae allow us to study Stokes phenomenon for the full two-parameter family of transseries solutions. In particular, we recover the exact expressions for the Stokes data that were recently found by Baldino, Schwick, Schiappa and Vega and compare our connection formulae to theirs. We also explain several ambiguities in relating transseries parameter choices to actual Painlevé transcendents, study the monodromy of formal solutions, and provide high-precision numerical tests of our results.

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“…These 2-parameter transseries solutions have also appeared in many different guises in other work, e.g. [4,9,10,12,13].…”

Section: From Formal Solutions To Transcendentssupporting

confidence: 53%

Section: From Formal Solutions To Transcendentsmentioning

confidence: 99%

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Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries

Spaendonck¹,

Vonk²

2022

Preprint

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“…The study of quantization via topological recursion in the unrefined setting has been investigated in varying levels of generality in e.g. [27,34,49,50]. The notion of quantum curve itself first appeared in [51] which discussed a relation between integrable systems and the topological string.…”

Section: Quantum Curvesmentioning

confidence: 99%

Quantum curves from refined topological recursion: the genus 0 case

Kidwai¹,

Osuga²

2022

Preprint

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We formulate geometrically (without reference to physical models) a refined topological recursion applicable to genus zero curves of degree two, inspired by Chekhov-Eynard and Marchal, introducing new degrees of freedom in the process. For such curves, we prove the fundamental properties of the recursion analogous to the unrefined case. We show the quantization of spectral curves due to Iwaki-Koike-Takei can be generalized to this setting and give the explicit formula, which turns out to be related to the unrefined case by a simple transformation. For an important collection of examples, we write down the quantum curves and find that in the Nekrasov-Shatashvili limit, they take an especially simple form.

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“…The elliptic representations for (P II ) and (P III ), nonlinear Stokes phenomena and connection problems are also in the monograph [12] (see also [16]). Concerning the elliptic representation for solutions of (P I ) Iwaki's recent work [17] by the topological recursion is remarkable.…”

Section: Introductionmentioning

confidence: 99%

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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2-Parameter $$\tau $$-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis

Cited by 24 publications

References 77 publications

Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries

Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries

Quantum curves from refined topological recursion: the genus 0 case

Elliptic asymptotic representation of the fifth Painlevé transcendents

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