Rational Solutions of the Painlev\'e-III Equation: Large Parameter Asymptotics

The Painlev\'e-III equation with parameters $\Theta_0=n+m$ and $\Theta_\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\infty;m)=1$ whenever $n\in\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in \cite{BothnerMS18}, we study the asymptotic behavior of $u_n(x;m)$ in the limit $n\to+\infty$ with $m\in\mathbb{C}$ held fixed. We isolate an eye-shaped domain $E$ in the $y=n^{-1}x$ plane that asymptotically confines the poles and zeros of $u_n(x;m)$ for all values of the second parameter $m$. We then show that unless $m$ is a half-integer, the interior of $E$ is filled with a locally uniform lattice of poles and zeros, and the density of the poles and zeros is small near the boundary of $E$ but blows up near the origin, which is the only fixed singularity of the Painlev\'e-III equation. In both the interior and exterior domains we provide accurate asymptotic formul\ae\ for $u_n(x;m)$ that we compare with $u_n(x;m)$ itself for finite values of $n$ to illustrate their accuracy. We also consider the exceptional cases where $m$ is a half-integer, showing that the poles and zeros of $u_n(x;m)$ now accumulate along only one or the other of two "eyebrows", i.e., exterior boundary arcs of $E$.