A connection formula for the Second Painlevé Transcendent

Abstract

We consider a particular case of the Second Painlevé Transcendent

$$y^{''} = xy + 2y^3$$

It is known that if y(x) ∼ kAi(x) as x → +∞, then if 0<k<1,

$$y(x) \sim d\left| x \right|^{ - {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 4$}}} sin\{ \frac{2}{3}\left| x \right|3/2 - \frac{3}{4}d2\ell n\left| x \right| - c\} asx \to - \infty$$

where d(k) and c(k) are the connection formulae for this nonlinear ordinary differential equation.

The lecture shows that

$$d2(k) = - \pi - 1\ell n(1 - k2)$$

which confirms the numerical estimates of Abtowitz and Segar.