Painlevé-III Monodromy Maps Under the 𝐷₆ → 𝐷₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

Abstract

The third Painlevé equation in its generic form, often referred to as Painlevé-III(𝐷₆), is given by d²𝑢/d𝑥² = 1/𝑢(d𝑢/d𝑥)² − 1/𝑥 d𝑢/d𝑥 + (α𝑢² + β)/𝑥 + 4𝑢³ − 4/𝑢, α, β ∈ ℂ. Starting from a generic initial solution 𝑢₀(𝑥) corresponding to parameters α, β, denoted as the triple (𝑢₀(𝑥), α, β), we apply an explicit Bäcklund transformation to generate a family of solutions (𝑢ₙ(𝑥), α+4𝑛, β+4𝑛) indexed by 𝑛 ∈ ℕ. We study the large n behavior of the solutions (𝑢ₙ(𝑥), α + 4𝑛, β +4n) under the scaling 𝑥 = 𝓏/𝑛 in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution 𝑢ₙ(𝓏/𝑛). Our main result is a proof that the limit of solutions 𝑢ₙ(z/𝑛) exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III(𝐷₈), d²𝑈/d𝓏² = 1/𝑈(d𝑈/dz𝓏)² − 1/𝓏 d𝑈/d𝓏 + (4𝑈²+4)/𝓏. A notable application of our result is to rational solutions of Painlevé-III(𝐷₆), which are constructed using the seed solution (1, 4𝑚,−4𝑚) where 𝑚 ∈ ℂ∖(ℤ + 1/2) and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at 𝓏 = 0 when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both 𝐷₆ and 𝐷₈, at 𝓏 = 0. We also deduce the large n behavior of the Umemura polynomials in a neighborhood of 𝓏 = 0.