### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 019, 77 pages      arXiv:2307.11217      https://doi.org/10.3842/SIGMA.2024.019
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

### Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

Ahmad Barhoumi ab, Oleg Lisovyy c, Peter D. Miller a and Andrei Prokhorov ad
a) Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI 48109, USA
b) Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 114 28, Stockholm, Sweden
c) Institut Denis-Poisson, Université de Tours, CNRS, Parc de Grandmont, 37200 Tours, France
d) St. Petersburg State University, Universitetskaya emb. 7/9, 199034 St. Petersburg, Russia

Received July 24, 2023, in final form January 23, 2024; Published online March 09, 2024

Abstract
The third Painlevé equation in its generic form, often referred to as Painlevé-III($D_6$), is given by $$\frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left( \frac{{\rm d}u}{{\rm d}x} \right)^2-\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C.$$ 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($D_8$), $$\frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left( \frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z} \frac{{\rm d}U}{{\rm d}z} + \frac{4U^2 + 4}{z}.$$ 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 $z = 0$.

Key words: Painlevé-III equation; Riemann-Hilbert analysis; Umemura polynomials; large-parameter asymptotics.

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