Painlevé-III Monodromy Maps Under the D_6→ D_8 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(D_6), is given by d^2u/ dx^2 =1/u( du/ dx)^2-1/x du/ dx+α u^2+β/x+4u^3-4/u, α,β∈ℂ. Starting from a generic initial solution u_0(x) corresponding to parameters α, β, denoted as the triple (u_0(x),α,β), we apply an explicit Bäcklund transformation to generate a family of solutions (u_n(x),α+4n,β+4n) indexed by n ∈ℕ. We study the large n behavior of the solutions (u_n(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 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 Painlevé-III equation, known as Painlevé-III(D_8), d^2U/ dz^2 =1/U( dU/ dz)^2-1/z dU/ dz+4U^2+4/z. A notable application of our result is to rational solutions of Painlevé-III(D_6), 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 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.