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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2024
• Main Authors: Barhoumi, Ahmad, Lisovyy, Oleg, Miller, Peter D., Prokhorov, Andrei
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2024
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212104
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Painlevé-III Monodromy Maps Under the 𝐷₆ → 𝐷₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions. Ahmad Barhoumi, Oleg Lisovyy, Peter D. Miller and Andrei Prokhorov. SIGMA 20 (2024), 019, 77 pages

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.