Meromorphic Solution of the Degenerate Third Painlev\'e Equation Vanishing at the Origin

TL;DR

This paper proves the existence and uniqueness of a specific odd meromorphic solution to the degenerate third Painlevé equation that vanishes at the origin, analyzing its local and asymptotic properties.

Contribution

It establishes the unique odd meromorphic solution of dP3 with zero at the origin and investigates its Taylor expansion and asymptotic behavior.

Findings

Existence and uniqueness of the solution u(τ) with u(0)=0

Explicit description of Taylor coefficients at the origin

Asymptotic behavior of the solution as τ→+∞

Abstract

We prove that there exists the unique odd meromorphic solution of dP3, $u(\tau)$ such that $u(0)=0$, and study some of its properties, mainly: the coefficients of its Taylor expansion at the origin and asymptotic behaviour as $\tau\to+\infty$.