Rational Solutions of the Painlevé-III Equation: Large Parameter Asymptotics

The Painlevé-III equation with parameters \(Θ _0=n+m\) and \(Θ _∞ =m-n+1\) has a unique rational solution \(u(x)=u_n(x;m)\) with \(u_n(∞ ;m)=1\) whenever \(n∈ Z\) . Using a Riemann–Hilbert representation proposed in Bothner et al. (Stud Appl Math 141:626–679, 2018 ), we study the asymptotic behavior of \(u_n(x;m)\) in the limit \(n→ +∞ \) with \(m∈ C\) held fixed. We isolate an eye-shaped domain E in the \(y=n⁻¹x\) 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 Painlevé-III equation. In both the interior and exterior domains we provide accurate asymptotic formulæ 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 .