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- W2887303828 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 $$nin mathbb {Z}$$. 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 $$nrightarrow +infty $$ with $$min mathbb {C}$$ 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 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." @default.
- W2887303828 created "2018-08-22" @default.
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- W2887303828 date "2019-05-29" @default.
- W2887303828 modified "2023-09-23" @default.
- W2887303828 title "Rational Solutions of the Painlevé-III Equation: Large Parameter Asymptotics" @default.
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- W2887303828 doi "https://doi.org/10.1007/s00365-019-09463-4" @default.
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