A connection formula for the second Painlevé transcendent
We consider the second Painlevé transcendent \(d^ 2y/dx^ 2=xy+2y^ 3.\) It is known that if y(x)\(\sim k Ai(x)\) as \(x\to +\infty\), where \(- 1<k<1\) and Ai(x) denotes Airy's function, then \[ y(x)\quad \sim \quad d| x|^{-1/4}\sin \{\frac{2}{3}| x|^{3/2}- \frac{3}{2}d^ 2\ell n| x| -c\}, \] as \(x\to -\infty\), where the constants d, c depend on k. This paper shows that \(d^ 2(k)=-\pi^{- 1}\ell n(1-k^ 2),\) which confirms a conjecture by Ablowitz and Segur.
scientific article; zbMATH DE number 4065451