A connection formula for the second Painlevé transcendent

Abstract

We consider the second Painlevé transcendent

$$\frac{{d^2 y}}{{dx^2 }} = xy + 2y^3 .$$

It is known that if y(x) ∼ k Ai (x) as x → + ∞, where −1<k<1 and Ai (x) denotes Airy's function, then

$$y(x) \sim d|x|^{ - \tfrac{1}{4}} sin\{ \tfrac{2}{3}|x|^{\tfrac{3}{2}} - \tfrac{3}{4}d^2 1n|x| - c\} ,$$

where the constants d, c depend on k. This paper shows that

$$d^2 = \pi ^{ - 1} 1n(1 - k^2 )$$

, which confirms a conjecture by Ablowitz & Segur.