On the second Painlevé transcendent

Abstract

Numerical and asymptotic approximations to the second Painlevé transcendent, F ± ( z; a ), as determined by the solution of F" – zF ± 2 F 3 = 0 and F ~ a Ai ( z ) ( z ↑ ∞), are presented. The solution for F + is finite for all real z and 0 < a 2 < ∞, but that for F - has at least one pole on the real axis if a 2 > 1. The asymptotic behaviour of F ± in the oscillatory regimé ( – z ± 2F 2 > 0 ), which bears a qualitative resemblance to that of Ai ( z ), is determined for a 2 ≪ 1 and for ± ln (1 ± a 2 ) ≫ 1. The results are relevant for several recent investigations of nonlinear wave motion.