Papers - Shimomura, Shun

Explicit Error Term of the Elliptic Asymptotics for the First Painlevé Transcendents

Shimomura S.

Computational Methods and Function Theory  2025

Research paper (scientific journal), Single Work, Lead author, Corresponding author, Accepted,  ISSN  16179447

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For the first Painlevé transcendents Kitaev established an asymptotic representation in terms of the Weierstrass pe-function in cheese-like strips tending along generic directions near the point at infinity. For this elliptic expression we present an explicit asymptotic formula of the error term, which leads to the error bound of exponent.

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Elliptic Asymptotics for the Complete Third Painlevé Transcendents

Shimomura S.

Funkcialaj Ekvacioj 68 ( 1 ) 69 - 117 2025

Single Work, Lead author, Corresponding author, Accepted,  ISSN  05328721

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For a general solution of the third Painlevé equation of complete type we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Jacobi sn-function in cheese-like strips along generic directions. The expression is derived by using isomonodromy deformation of a linear system governed by the third Painlevé equation of this type. In our calculation of the WKB analysis, the treated Stokes curve ranges on both upper and lower sheets of the two sheeted Riemann surface.

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TWO ERROR BOUNDS OF THE ELLIPTIC ASYMPTOTICS FOR THE FIFTH PAINLEVÉ TRANSCENDENTS

Shimomura S.

Kyushu Journal of Mathematics 78 ( 2 ) 487 - 502 2024

Single Work, Lead author, Corresponding author, Accepted,  ISSN  13406116

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For the fifth Painlevé equation it is known that a general solution is represented asymptotically by an elliptic function in cheese-like strips near the point at infinity. We present an explicit asymptotic formula for the error term of this expression, which leads to the error bound as was conjectured. An analogous formula with its bound is obtained for the error term of the correction function associated with the Lagrangian.

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Boutroux Ansatz for the Degenerate Third Painlevé Transcendents

Shimomura S.

Publications of the Research Institute for Mathematical Sciences 60 ( 4 ) 651 - 698 2024

Research paper (bulletin of university, research institution), Lead author, Corresponding author, Accepted,  ISSN  00345318

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For a general solution of the degenerate third Painlevé equation we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Weierstrass pe-function in cheese-like strips along generic directions. The expression is obtained by using isomonodromy deformation of a linear system governed by the degenerate third Painlevé equation.

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Elliptic asymptotic representation of the fifth Painleve transcendents. Corrected and revised version

Shun Shimomura

arXiv:2012.07321, math.CA, math-ph (arXiv:2012.07321, math.CA, math-ph)   2023.12

Research paper (other academic), Single Work, Lead author, Corresponding author

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Corrected the errors of the paper published in
Kyushu Journal of Mathematics Vol.76 no.1 (2022),
and extensively revised for readability.


corrected and revised version

CORRIGENDUM: ELLIPTIC ASYMPTOTIC REPRESENTATION OF THE FIFTH PAINLEVÉ TRANSCENDENTS (Kyushu J. Math. 76(1) (2022), 43–99)

Shimomura S.

Kyushu Journal of Mathematics 77 ( 1 ) 191 - 202 2023

Research paper (bulletin of university, research institution), Lead author, Corresponding author, Accepted,  ISSN  13406116

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The paper [S. Shimomura, Kyushu J. Math. 76 (2022), 43–99] contains an incorrect Stokes graph. The amendment to the Stokes graph replaces the phase shifts of asymptotic solutions in the main theorems.

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ELLIPTIC ASYMPTOTIC REPRESENTATION OF THE FIFTH PAINLEVÉ TRANSCENDENTS

Shimomura S.

Kyushu Journal of Mathematics 76 ( 1 ) 43 - 99 2022

Research paper (bulletin of university, research institution), Lead author, Corresponding author,  ISSN  13406116

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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 parametrized 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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On a General Singular Solution of the Fifth Painlevé Equation Along the Positive Real Axis

Shimomura S.

Computational Methods and Function Theory (Computational Methods and Function Theory)  21 ( 4 ) 633 - 651 2021

Research paper (scientific journal), Single Work, Accepted,  ISSN  16179447

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We propose a system of non-linear equations equivalent to the fifth Painlevé equation, which enables us to examine the general singular solution given by Andreev and Kitaev along the positive real axis. We present a two-parameter family of asymptotic solutions corresponding to this general singular solution, and pose a conjecture.

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Three-parameter solutions of the PV Schlesinger-type equation near the point at infinity and the monodromy data

Shimomura S.

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) (Symmetry, Integrability and Geometry: Methods and Applications (SIGMA))  14 2018

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© 2018, Institute of Mathematics. All rights reserved. For the Schlesinger-type equation related to the fifth Painlevé equation (V) via isomonodromy deformation, we present a three-parameter family of matrix solutions along the imaginary axis near the point at infinity, and also the corresponding monodromy data. Two-parameter solutions of (V) with their monodromy data immediately follow from our results. Under certain conditions, these solutions of (V) admit sequences of zeros and of poles along the imaginary axis. The monodromy data are obtained by matching techniques for a perturbed linear system.

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Critical behaviours of the fifth Painleve transcendents and the monodromy data

SHIMOMURA SHUN

Kyushu J. Math. 71   139 - 185 2017

Research paper (bulletin of university, research institution), Single Work, Accepted

Logarithmic solutions of the fifth Painleve equation near the origin

SHIMOMURA SHUN

Tokyo J. Math. 39 ( 3 ) 797 - 825 2017

Research paper (scientific journal), Single Work, Accepted

The sixth Painleve transcendents and the associated Schlesinger equation

SHIMOMURA SHUN

Publ. RIMS Kyoto Univ. 51   417 - 463 2015

Research paper (scientific journal), Single Work, Accepted

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Series expansions of Painleve transcendents near the point at infinity

SHIMOMURA SHUN

Funkcial. Ekvac. 58   277 - 319 2015

Research paper (scientific journal), Single Work, Accepted

Fourth moment of the Riemann zeta-function with a shift along the real line

SHIMOMURA SHUN

Tokyo J. Math. 36   355 - 377 2013

Research paper (scientific journal), Single Work, Accepted

Frequency of a-points for the fifth and the third Painleve transcendents in a sector

SHIMOMURA SHUN

Tohoku Math. J. 65   591 - 605 2013

Research paper (scientific journal), Single Work, Accepted

Continuous limit of the difference second Painlevé equation and its asymptotic solutions

SHIMOMURA SHUN

J. Math. Soc. Japan 64 ( 3 ) 733–781 2012

Research paper (scientific journal), Single Work, Accepted

Rational solutions of difference Painlevé equations

SHIMOMURA SHUN

Tokyo J. Math 35 ( 1 ) 85–95 2012

Research paper (scientific journal), Single Work, Accepted

Algebraic independence of certain numbers related to modular functions

Elsner, Carsten; SHIMOMURA SHUN; Shiokawa, Iekata

Funct. Approx. Comment. Math. (Adam Mickiewicz University Press)  47 ( 1 ) 121–141 - 141 2012

Research paper (scientific journal), Joint Work, Accepted,  ISSN  2080-9433

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Shifted fourth moment of the Riemann zeta-function

SHIMOMURA SHUN

Acta Math. Hungar. 137 ( 1-2 ) 104–129 2012

Research paper (scientific journal), Single Work, Accepted

Exceptional algebraic relations for reciprocal sums of Fibonacci and Lucas numbers

Elsner, Carsten; SHIMOMURA SHUN; Shiokawa, Iekata

Diophantine Analysis and Related Fields DARF 2011, AIP Conference Proceedings 1385, American Institute of Physics (Melville, New York)  1385   17-31 2011

Research paper (scientific journal), Joint Work