Painlevé-III Monodromy Maps Under the D<inf>6</inf> → D<inf>8</inf> Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

The third Painlevé equation in its generic form, often referred to as Painlevé-III(D6), is given by [Formula Presented] 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), [Formula Presented]. 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 +12) 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.