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

Abstract

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

References in corpus (5)

• Level-Spacing Distributions and the Bessel Kernel
• On the Umemura Polynomials for the Painlevé III equation
• Zeros of the Jimbo, Miwa, Ueno tau function
• Rational Solutions of the Fifth Painlevé Equation. Generalised Laguerre Polynomials
• A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation

Cited by in corpus (1)

• Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters