Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions - Barhoumi, Ahmad; Lisovyy, Oleg; Miller, Peter D.; Prokhorov, Andrei (Symmetry, Integrability and Geometry: Methods and Applications, 2024)
The third Painlevé equation in its generic form, often referred to as Painlevé-III(₆), is given by d²/d² = 1/(d/d)² − 1/ d/d + (α² + β)/ + 4³ − 4/, α, β ∈ ℂ. Starting from a generic initial solution ₀() corresponding to parameters α, β, denoted as the triple (₀(), α, β), we apply an explicit Bäcklund transformation to generate a family of solutions (ₙ(), α+4, β+4) indexed by ∈ ℕ. We study the large n behavior of the solutions (ₙ(), α + 4, β +4n) under the scaling = / 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 ₙ(/). Our main result is a proof that the limit of solutions ₙ(z/) exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III(₈), d²/d² = 1/(d/dz)² − 1/ d/d + (4²+4)/. A notable application of our result is to rational solutions of Painlevé-III(₆), which are constructed using the seed solution (1, 4,−4) where ∈ 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 = 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 ₆ and ₈, at = 0. We also deduce the large n behavior of the Umemura polynomials in a neighborhood of = 0.