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Riccati equations revisited: linearization and analytic interpretation of instanton-type solutions. (English) Zbl 07273472

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.

MSC:

34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
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