arXiv subjects

Shun Shimomura

Publications and source records attributed to Shun Shimomura.

9 recordsLinked to original sources

arXiv2026

Full characterisation of Painlevé V asymptotics and nonlinear monodromy-Stokes structure

For a generic Painlevé V equation we characterise all the asymptotics in a right half plane near the point at infinity, that is, we find classified explicit solutions that are, by the Riemann-Hilbert correspondence, labelled with monodromy data filling up the whole monodromy manifold. To do so, in addition to the asymptotics by Andreev and Kitaev along the positive real axis, we require elliptic asymptotics along generic directions and newly provided truncated solutions arising from a general solution along the imaginary axes. To know analytic continuations outside this region we formulate a nonlinear monodromy-Stokes structure, which is observed as changes of monodromy data contained in the explicit expressions of solutions.

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arXiv2026

Elliptic asymptotic representation of the fifth Painlevé transcendents

For the fifth Painlevé transcendents an asymptotic representation by the Jacobi $\mathrm{sn}$-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part may be understood to depend on the phase shift as a single integration constant, which is parametrised by monodromy data for the associated isomonodromy deformation. The other integration constant is contained in the error term or in a correction function. This paper contains corrections of the Stokes graph and of the related results in the early version.

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arXiv2025

Two error bounds of the elliptic asymptotics for the fifth Painlevé transcendents

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 an estimate for its magnitude as was conjectured. Analogous formula is obtained for the error term of the correction function associated with the Lagrangian.

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arXiv2025

Elliptic asymptotics for the complete third Painlevé transcendents

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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arXiv2024

Kapaev's global asymptotics of the fourth Painlevé transcendents. Elliptic asymptotics

For the fourth Painlevé transcendents we derive elliptic asymptotic representations, which were announced by late Professor Kapaev without proofs. Then we newly obtain related results including the correction function.

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arXiv2024

Explicit error term of the elliptic asymptotics for the first Painlevé transcendents

For the first Painlevé transcendents Kitaev established an asymptotic representation in terms of the Weierstrass pe-function in cheese-like strips near the point at infinity. We present an explicit error bound of this asymptotic expression, which leads to the order estimate of exponent $-1$.

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arXiv2022

Boutroux ansatz for the degenerate third Painlevé transcendents

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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arXiv2018

Three-Parameter Solutions of the PV Schlesinger-Type Equation near the Point at Infinity and the Monodromy Data

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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arXiv2016

Critical behaviours of the fifth Painlevé transcendents and the monodromy data

For the fifth Painlevé equation, we present families of convergent series solutions near the origin and the corresponding monodromy data for the associated isomonodromy linear system. These solutions are of complex power type, of inverse logarithmic type and of Taylor series type. It is also possible to compute the monodromy data in non-generic cases. Solutions of logarithmic type are derived from those of inverse logarithmic type through a Bäcklund transformation found by Gromak. In a special case the complex power type of solutions have relatively simple oscillatory expressions. For the complex power type of solutions in the generic case, we clarify the structure of the analytic continuation on the universal covering around the origin, and examine the distribution of zeros, poles and 1-points. It is shown that two kinds of spiral domains including a sector as a special case are alternately arrayed; the domains of one kind contain sequences both of zeros and of poles, and those of the other kind sequences of 1-points.

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