Asymptotic behaviour of the fifth Painlevé transcendents in the space of initial values

Abstract

Abstract. We study the asymptotic behaviour of the solutions of the fifth Painlevé equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and connected and, moreover, that any solution with an essential singularity at zero has an infinite number of poles and zeroes, and any solution with an essential singularity at infinity has infinite number of poles and, moreover, takes the value unity infinitely many times.

Global asymptotics of the sixth Painlevé equation in Okamoto's space

“…Our main purpose is to describe the significant features of the flow in the singular limit x → 0. In similar studies of the first, second, and fourth Painlevé equations [10,11,12] in singular limits, we showed that successive resolutions of the Painlevé vector field at base points terminates after nine blow ups of CP 2 , while for the fifth and third Painlevé equations we showed that the construction consists of eleven blow ups and two blow downs [13,14]. The initial value space in each case is then obtained by removing the infinity set, denoted I, which are blow-ups of points not reached by any solution.…”

Global Asymptotics of the Sixth Painlevé Equation in Okamoto’s Space

“…Our main purpose is to describe the significant features of the flow in the singular limit . In similar studies of the first, second and fourth Painlevé equations [4, 16, 17] in singular limits, we showed that successive resolutions of the Painlevé vector field at base points terminate after nine blow-ups of , while for the fifth and third Painlevé equations, we showed that the construction consists of 11 blow-ups and two blow-downs [18, 19]. The initial value space in each case is then obtained by removing the infinity set, denoted , which is blow-ups of points not reached by any solution.…”

Connection formulae for asymptotics of the fifth Painlevé transcendent on the imaginary axis: I

“…It is important to mention that during this time some interesting papers devoted to the study of asymptotics of the fifth Painlevé functions have been published. [16][17][18] The paper is organized as follows. In Section 2, we define the monodromy data for Equation (1).…”