True solutions asymptotic to formal WKB solutions of the second Painlevé equation with large parameter (Q2744098)

The author deals with an asymptotic study of the second Painlevé equation (1): \(y''=2y^3 +xy+\alpha\) as \(\alpha\to \infty\). The region \(x\sim \alpha^{2/3}\) is considered. Equation (1) is transformed by the substitution \(y =\alpha^{1/3}u(t)\), \(x=\alpha^{2/3}t\), \(u(t) = u_0(t)+O(\alpha^{-1/2})\) as \(\alpha\to\infty\), where \(u_0(t)\) is a root of the polynomial \(P(u) = 2u^3+tu + 1\), to the form NEWLINE\[NEWLINE\varepsilon^2u''=2u^3+tu=1,NEWLINE\]NEWLINE with \(\varepsilon=\frac 1\alpha\). This equation is studied in the limit \(\varepsilon\to 0\) in a series of papers in which very detailed calculations of formal WKB solutions have been provided.NEWLINENEWLINENEWLINEThe purpose of this paper is to provide a method of proof such that formal WKB solutions correspond to true solutions in appropriate domains.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].