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It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$$_7$$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation.
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This content will become publicly available on April 1, 2026
Asymptotic Properties of Special Function Solutions of the Painlevé III Equation for Fixed Parameters
In this paper, we compute the small and large asymptotics of the special function solutions of the Painlevé‐III equation in the complex plane. We use the representation in terms of Toeplitz determinants of Bessel functions obtained by Masuda. Toeplitz determinants are rewritten as multiple contour integrals using Andrèief's identity. The small and large asymptotics are obtained using elementary asymptotic methods applied to the multiple contour integral. The asymptotics is extended to the whole complex plane using analytic continuation formulas for Bessel functions. The claimed result has not appeared in the literature before. We note that the Toeplitz determinant representation is useful for numerical computations of corresponding solutions of the Painlevé‐III equation.
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- Award ID(s):
- 2103354
- PAR ID:
- 10655511
- Publisher / Repository:
- Wiley Online Library
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- Volume:
- 154
- Issue:
- 4
- ISSN:
- 0022-2526
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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