Details

Autor(en) / Beteiligte

• Bothner, Thomas
• Miller, Peter D.

Titel

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

Ist Teil von

• Constructive approximation, 2020-02, Vol.51 (1), p.123-224

Ort / Verlag

New York: Springer US

Erscheinungsjahr

2020

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• The Painlevé-III equation with parameters Θ 0 = 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 .