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

The Painlevé-III equation with parameters Θ = 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 ^{- 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 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.