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

Abstract

The Painlev-III equation with parameters 0 = n + m and = mn + 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 Painlev-III equation. In both the interior and exterior domains we provide accurate asymptotic formulae 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.