, Volume 103, Issue 2, pp 97-138

A connection formula for the second Painlevé transcendent

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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.