論文

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

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.

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.

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.

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.

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.

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.

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.

Φ２のみたす方程式系の解について無限遠特異集合の交わる点の近くでの漸近的挙動を調べた。（22頁）