Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters

TL;DR

This work analyzes the small and large $x$ asymptotics of special function solutions to Painlevé III that are expressible through Toeplitz determinants of cylinder (Bessel) functions. By combining a Riccati–PIII reduction, Bäcklund transformations, and a tau-function/Toda framework, the authors derive explicit determinant representations for the solutions and connect them to the tau-function via Andréief-reduced multiple-contour integrals. They obtain comprehensive zero- and infinity-$x$ asymptotics, including the leading exponents and prefactors in terms of Barnes $G$-functions and Gamma functions, and extend results to the full complex plane using analytic continuation. The results yield both oscillatory and power-law behaviors depending on the parameter regimes, and provide a numerically practical determinant representation for computing Painlevé III special-function solutions. The paper thus fills gaps in the literature by delivering rigorous, piecewise asymptotics for the entire complex plane and by linking Toeplitz/Hankel determinants to Painlevé III via tau-function and Toda structures.

Abstract

In this paper, we compute the small and large $x$ asymptotics of the special function solutions of Painlevé-III equation in the complex plane. We use the representation in terms of Toeplitz determinants of Bessel functions obtained in arXiv:nlin/0302026. Toeplitz determinants are rewritten as multiple contour integrals using Andrèief's identity. The small and large $x$ 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 Toeplitz determinant representation is useful for numerical computations of corresponding solutions of the Painlevé-III equation.

Figures (11)

• Figure 1.1: Leading power in the asymptotics for $n=5$ as a function of $\alpha$. The piecewise expressions for $p_c(\alpha,n)$ and $e(\alpha,n)$ can be found in \ref{['eq:pc']} and \ref{['eq:e']}, respectively.
• Figure 1.2: Complex argument plots of solutions for various values of parameters. The color for each value of the argument can be found in Figure \ref{['illustration0']}
• Figure 4.1: Contour $\widetilde{\Gamma_1}$
• Figure 4.2: Contour $\widetilde{\Gamma_2}$
• Figure A.1: Contour for $J_\nu(x)$.

Theorems & Definitions (55)

• Proposition 1.1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1: DLMF
• Remark 2.1
• Proposition 2.2: DLMF
• Remark 2.2
• Definition 3.1