Scaling Limit of Successive Approximations for $w'=-w^2$

MR2271236 Hattori, Tetsuya and Ochiai, Hiroyuki Tetsuya HATTORI and Hiroyuki OCHIAI Scaling Limit of Successive Approximations for $w'=-w^2$ Funkcialaj Ekvacioj. Serio Internacia 49 2006 291--319 http://fe.math.kobe-u.ac.jp/FE/FullPapers/49-2/49_291.pdf http://www.ams.org/mathscinet-getitem?mr=MR2271236 We prove existence of scaling limits of sequences of functions defined by the recursion relation $w'_{n+1}(x)=-w_n(x)^2$. which is a successive approximation to $w'(x)=-w(x)^2$, a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as $n\to\infty$ in an asymptotically conformal way, $w_n(x)\asymp q_n\bar{w}(q_nx)$, for a sequence of numbers $\{q_n\}$ and a function $\bar{w}$. We also discuss implication of the results in terms of random sequential bisections of a rod. Scaling limit, Moving singularity, Successive approximation, Random sequential bisections. 34E99, 45G10, 60F05. 49-291 2006 Scaling Limit of Successive Approximations for $w'=-w^2$ HATTORI, Tetsuya and OCHIAI, Hiroyuki Tetsuya HATTORI and Hiroyuki OCHIAI

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