Painlevé-III Monodromy Maps Under the $D_6\to D_8$ 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
$$ \frac{\mathrm{d}^2u}{\mathrm{d}x^2}=\dfrac{1}{u}\left( \frac{\mathrm{d}u}{\mathrm{d}x} \right)^2-\dfrac{1}{x} \frac{\mathrm{d}u}{\mathrm{d}x} + \dfrac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb{C}. $$
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 Bäcklund 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 Painlevé-III equation, known as Painlevé-III($D_8$),
$$ \frac{\mathrm{d}^2U}{\mathrm{d}z^2}=\dfrac{1}{U}\left( \frac{\mathrm{d}U}{\mathrm{d}z}\right)^2-\dfrac{1}{z} \frac{\mathrm{d}U}{\mathrm{d}z} + \dfrac{4U^2 + 4}{z}. $$
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 $m \in \mathbb{C} \setminus \big(\mathbb{Z} + \frac{1}{2}\big)$ 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 $z = 0$.