Analysis of Singular Solutions of Certain Painlevé Equations

Abstract

The six Painlevé equations can be described as the boundary between the non- integrable- and the trivially integrable-systems. Ever since their discovery they have found numerous applications in mathematics and physics. The solutions of the Painlevé equations are, in most cases, highly transcendental and hence cannot be expressed in closed form. Asymptotic methods do better, and can establish the behaviour of some of the solutions of the Painlevé equations in the neighbourhood of a singularity, such as the point at infinity. Although the quantitative nature of these neighbourhoods is not initially implied from the asymptotic analysis, some regularity results exist for some of the Painlevé equations. In this research, we will present such results for some of the remaining Painlevé equations. In particular, we will provide concrete estimates of the intervals of analyticity of a one-parameter family of solutions of the second Painlevé equation, and estimate the domain of analyticity of a “triply-truncated” solution of the fourth Painlevé equation. In addition we will also deduce the existence of solutions with particular asymptotic behaviour for the discrete Painlevé equations, which are discrete integrable nonlinear systems.