Truncated Solutions of the Fifth Painlevé Equation

Abstract

Abstract. The fifth Painlevé equation admits several families of solutions behaving exponentially in their proper sectors near infinity, which are called truncated solutions. For these truncated solutions, we discuss the frequency of a-points including poles outside the corresponding sectors. Except for some special cases, all the values are equally distributed, and for each a there exist infinitely many a-points.

Series Expansions of Painlevé Transcendents near the Point at Infinity

“…its proof in Section 4). For Theorems 2.6 through 2.14 similar facts are valid, in which the corresponding sectors are as follows: y II; 0; n ðc; xÞ: jarg x À ð2n þ 1Þp=3j < Y 0 < 2p=3 for n ¼ G1; y II; y; n ðc; xÞ: jarg x À 2np=3j < Y 0 < 2p=3 for n ¼ G1; y IV; y; 1; n ðc; xÞ: jarg x À ð2n þ 1Þp=4j < Y 0 < p=2 for n ¼ G1; 2; y IV; y; 2; n ðc; xÞ: jarg x À np=2j < Y 0 < p=2 for n ¼ G1; 2; y IV; 0; Ã; n ðc; xÞ, y IV; 0; n ðc; xÞ: jarg x À ð2n þ 1Þp=4j < Y 0 < p=2 for n ¼ G1; 2; y V; À1; n ðc; xÞ: jarg x À ð2n þ 1Þp=2j < Y 0 < p for n A Znf0g; y V; 0; Ã; n ðc; xÞ, y V; 0;n ðc; xÞ: jarg x À npj < Y 0 < p for n A Znf0g; y V; 0; 0; n ðc; xÞ: jarg x À 2npj < Y 0 < 2p for n A Znf0g; y III; n ðc; xÞ: jarg x À ð2n þ 1Þp=2j < Y 0 < p for n A Znf0g; y III; Ã; n ðc; xÞ: jarg x À 3ð2n þ 1Þp=4j < Y 0 < 3p=2 for n A Znf0g; y III; 0; n ðc; xÞ: of the tronqueé type, for which movable poles and quasi-linear Stokes phenomena are studied in [6], [15], [13], [18] (see also [10]); and tronqueé solutions of (V) or (V 0 ) are found in [1], [33]. A kind of formal solutions obtained by power geometry (cf.…”

Truncated Solutions of Painlevé Equation PV

“…It is known that these five families represent asymptotic behaviors of truncated solutions, analytic in (almost) a half plane for large |x|; the position of the half plane is determined by the exponentially small terms [2,34].…”

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).…”