Riccati equations revisited: linearization and analytic interpretation of instanton-type solutions (Q2211097)

Asymptotic analysis of the Riccati equations with a large parameter is studied by means of the exact WKB analysis. Instanton-type solutions are represented by asymptotic expansion of the large parameter. By a suitable transformation of the unknown function, the Riccati equation can be reduced to a linear equation. The linearizing transformation is also given by an asymptotic series. Since the linearizing transformation is Borel summable, an analytical interpretation of the instanton-type solution can be established. Since the Riccati equations is a prototype of the Painlevé equations, the method used in this paper is helpful to study instanton-type solutions of the Painlevé equations.