Painlevé-III Monodromy Maps Under the D₆→ D₈ 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 $$ {{ d}^2u}{{ d}x^2} =1/u({{ d}u}{{ d}x} )^2-1/x {{ d}u}{{ d}x} + α u^2 + β/x+4u^3-4/u, α,β ∈ C. $$ 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($D_8$), $$ {{ d}^2U}{{ d}z^2} =1/U({{ d}U}{{ d}z})^2-1/z {{ d}U}{{ d}z} + 4U^2 + 4/z.$$ 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 $z = 0$.