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 $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2-\frac{1}{x}\frac{{\rm d}u}{{\rm d}x}+\frac{αu^2+β}{x}+4u^3-\frac{4}{u}, \qquad α,β\in \mathbb C.$$ 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 $n \in \mathbb N$. 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($D_8$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z}\frac{{\rm d}U}{{\rm d}z}+\frac{4U^2+4}{z}.$$ 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 $z=0$.

Figures (12)

• Figure 1: Left: the rational solution $u_{10}(x;0.25)$. Right: the limiting solution $U(z;0.25)$, where we recall the notation $z=nx$ for $n=10$. All poles of $u_{10}(x;0.25)$ are simple with residue $\frac{1}{2}$/$-\frac{1}{2}$, indicated in the plot with red/yellow circles. Likewise, all zeros of $u_{10}(x;0.25)$ are simple with derivative $2$/$-2$, indicated in the plot with pink/green squares. On the other hand, all poles and zeros of $U(z;0.25)$ have multiplicity $2$ and are marked with red circles and green squares respectively.
• Figure 2: Left: the solution $u_{10}(x)$ of \ref{['eq:PIII-$D_6$']} generated by ten iterations of \ref{['eq:Gromak-transformation']} with seed $u_0(x)$ corresponding to monodromy data $\mu=0.23+0.39\mathrm{i}$ (see \ref{['eq:e1-e2-mu-eta']}), $e_2=-0.45-0.96\mathrm{i}$ and $\alpha=40.5+0.63\mathrm{i}$, $\beta=40.98+0.59\mathrm{i}$. Right: the limiting solution $U(z)$ of \ref{['eq:PIII-$D_8$']}. The labeling of poles and zeros is the same as in Figure \ref{['fig:rational']}. Note that both $u_{10}(x)$ and $U(z)$ are branched at the origin.
• Figure 3: The jump contour $L^{(6)}$ for $\mathbf{\Psi}(\lambda, x)$ and definition of $\mathbf J_{\mathbf \Psi}(\lambda)$ when $x>0$.
• Figure 4: The analogue of the contour $L^{(6)}$ in Figure \ref{['fig:1']} when $|{\operatorname{Arg}}(x)| \neq 0$.
• Figure 5: The jump contour $L^{(8)}$ and definition of $\mathbf J_{{\mathbf{\Omega}}}(\lambda)$ when $z>0$.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3: see Section \ref{['sec:monodromy-rep-$D_6$']} for details
• Theorem 1.4
• Proposition 1.5
• Proposition 1.6
• Theorem 1.7: FIKNINN
• Theorem 2.1
• Lemma 2.2: clarkson2018constructive
• Lemma 2.3