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

The Painleve-III equation with parameters 0 = n + m and 8 = m - n + 1 has a unique rational solution u(x) = un(x; m) with un(8; 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 un(x; m) in the limit n. +8 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 un( 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 Painleve-III equation. In both the interior and exterior domains we provide accurate asymptotic formulae for un(x; m) that we compare with un(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 un(x; m) now accumulate along only one or the other of two "eyebrows," i.e., exterior boundary arcs of E.