Boutroux's method vs. re-scaling. Lower estimates for the orders of growth of the second and fourth Painlevé transcendents (Q1769837)

This paper gives lower estimates on the growth of transcendental solutions of the second and fourth Painlevé equations \[ \begin{aligned} &w''=2w^3+zw+\alpha, \tag{II} \\ &2ww''=(w')^2+3w^4+8zw^3+4(z^2-\alpha)w^2+2\beta\quad\quad (\,'\,=d/dz). \tag{IV} \end{aligned} \] For every transcendental solution \(w\) of (II), the re-scaling process \(y_n(\mathfrak {z})=p_n^{-1/2}w(p_n+p_n^{-1/2}\mathfrak {z})\) associated with a subsequence of poles [\textit{N. Steinmetz}, Isr. J. Math. 128, 29--52 (2002; Zbl 1016.34091)] leads to the equation \[ \mathfrak{y}''=2\mathfrak{y}^3+\mathfrak{y}\quad\quad (\,'\,=d/d\mathfrak{z}). \] This coincides with the limit of the equation \[ \Theta''=2\Theta^3+\Theta-\frac{\Theta'}{\xi}+\frac{2\alpha} {3\xi}+\frac{\Theta}{9\xi^2}\quad\quad (\,'\,=d/d\xi) \] as \(\xi\to\infty,\) which is derived from (II) by Boutroux's transformation \(\xi=(2/3)z^{3/2},\) \(\Theta=z^{-1/2}w.\) Using these facts, the author constructs a sequence of different poles \(\tilde{p}_n\) of \(w\) such that \(| \tilde{p}_{n+1}| \leq | \tilde{p}_n| +O(| \tilde{p}_n| ^{-1/2}),\) which yields the lower estimate on the growth \(\varrho(w) \geq 3/2.\) For every transcendental solution of (IV), the estimate \(\varrho(w)\geq 2\) is obtained by almost the same argument.