Differential Equations for Approximate Solutions of Painlev\'e Equations: Application to the Algebraic Solutions of the Painlev\'e-III $({\rm D}_7)$ Equation

TL;DR

This paper develops a method to rigorously connect approximate solutions of Painlevé-III (D7) equations, derived via Riemann-Hilbert techniques, to their limiting algebraic differential equations, simplifying the analysis of their asymptotic behavior.

Contribution

It introduces a novel approach to directly derive limiting differential equations from approximate solutions of Painlevé equations, bypassing complex algebro-geometric computations.

Findings

Established a direct link between approximate solutions and the Weierstrass equation.

Applied the method to algebraic solutions of Painlevé-III (D7).

Demonstrated the approach's effectiveness in asymptotic analysis.

Abstract

It is well known that the Painlev\'e 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 Painlev\'e-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.