Isomonodromy for the Degenerate Fifth Painlev\'e Equation

TL;DR

This paper explicitly constructs the moduli spaces and Riemann-Hilbert correspondence for the degenerate fifth Painlevé equation, proving it has the Painlevé property and identifying its geometric structure.

Contribution

It explicitly computes the moduli spaces for connections and monodromy, proving the Riemann-Hilbert morphism is an isomorphism for this equation.

Findings

Extended Riemann-Hilbert morphism is an isomorphism

The equation has the Painlevé property

Formulas for the equation and Bäcklund transformations are derived

Abstract

This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlev\'e equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann-Hilbert morphism is an isomorphism. As a consequence these equations have the Painlev\'e property and the Okamoto-Painlev\'e space is identified with a moduli space of connections. Using MAPLE computations, one obtains formulas for the degenerate fifth Painlev\'e equation, for the B\"acklund transformations.