Painlev\'e-III Monodromy Maps Under the \(D_6\to D_8\) Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

Abstract

The third Painlev\'e equation in its generic form, often referred to as Painlev\'e-III(\(D_6\)), is given by\[ \dfrac{\mathrm{d}^2u}{\mathrm{d}x^2}=\dfrac{1}{u}\left( \dfrac{\mathrm{d}u}{\mathrm{d}x} \right)^2-\dfrac{1}{x} \dfrac{\mathrm{d}u}{\mathrm{d}x}+\dfrac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \quad \alpha,\beta \in \mathbb{C}.\] Starting from a generic initial solution \(u_0(x)\) corresponding to parameters \(\alpha, \beta\), denoted as the triple \((u_0(x),\alpha,\beta)\), we apply an explicit B\"acklund transformation to generate a family of solutions \((u_n(x),\alpha+4n,\beta+4n)\) indexed by \(n\in \mathbb{N}\). We study the large \(n\) behavior of the solutions \((u_n(x),\alpha+4n,\beta+4n)\) 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 \(u_n(z/n)\). Our main result is a proof that the limit of solutions \(u_n(z/n)\) exists and is given by a solution of the degenerate Painlev\'e-III equation, known as Painlev\'e-III(\(D_8\)), \[\dfrac{\mathrm{d}^2U}{\mathrm{d}z^2}=\dfrac{1}{U}\left(\dfrac{\mathrm{d}U}{\mathrm{d}z}\right)^2-\dfrac{1}{z} \dfrac{\mathrm{d}U}{\mathrm{d}z}+\dfrac{4U^2 + 4}{z}.\] A notable application of our result is to rational solutions of Painlev\'e-III(\(D_6\)), which are constructed using the seed solution \((1,4m,-4m)\) where \(m\in \mathbb{C} \setminus (\mathbb{Z} + \frac{1}{2})\) 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 general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlev\'e-III, both \(D_6\) and \(D_8\) at \(z = 0\). We also deduce the large \(n\) behavior of the Umemura polynomials in a neighborhood of \(z = 0\).