Rational Solutions of the Painlevé-III Equation: Large Parameter Asymptotics
Thomas Bothner and Peter D. Miller
Department of Mathematics, University of Michigan
Abstract:
The Painlevé-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 [4], we study the asymptotic behavior \(u_n(x;m)\) 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 Painlevé-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 \(E\).