Painleve I, Coverings of the Sphere and Belyi Functions

Abstract

The theory of poles of solutions of Painleve-I is equivalent to the Nevanlinna problem of constructing a meromorphic function ramified over five points - counting multiplicities - and without critical points. We construct such meromorphic functions as limit of rational ones. In the case of the tritronquee solution these rational functions are Belyi functions.

33 pages, many figures. Version 2: minor corrections and minor changes in the bibliography

References in corpus (5)

• Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of $P_I$
• Universality for the focusing nonlinear Schroedinger equation at the gradient catastrophe point: Rational breathers and poles of the tritronquee solution to Painleve I
• Poles of Integrale Tritronquee and Anharmonic Oscillators. Asymptotic localization from WKB analysis
• Y-System and Deformed Thermodynamic Bethe Ansatz
• On the location of poles for the Ablowitz-Segur family of solutions to the second Painlevé equation

Cited by in corpus (3)

• Poles of Painlevé IV Rationals and their Distribution
• Roots of generalised Hermite polynomials when both parameters are large
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics