Riccati Equations Revisited: Linearization and Analytic Interpretation of Instanton-Type Solutions
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Complex Analysis and Operator Theory
Riccati Equations Revisited: Linearization and Analytic Interpretation of Instanton-Type Solutions Yoshitsugu Takei1 Received: 5 May 2020 / Accepted: 29 August 2020 © Springer Nature Switzerland AG 2020
Abstract As a prototype of Painlevé equations, we discuss instanton-type solutions of the Riccati equation with a large parameter in this paper. Based on the linearization and its modified version, we provide an analytic interpretation of instanton-type solutions through the Borel resummation technique. Keywords Riccati equations · Exact WKB analysis · Instanton-type solutions · Linearization
1 Introduction Painlevé equations are second order nonlinear ordinary differential equations that Painlevé and his student Gambier discovered in their attempts to find new transcendental functions characterized by nonlinear differential equations. Painlevé equations have several remarkable properties such as the Painlevé property (i.e., the meromorphy of solutions), Weyl group symmetry, the relationship with isomonodromic deformations of linear differential equations, etc. and now attract interests of many mathematicians in various fields. In a series of papers ([1,6,7,11]; see also [10,12]) Aoki, Kawai and the author of this paper have developed the exact WKB analysis (i.e., WKB analysis based on the Borel resummation technique; cf., e.g., [4,8,15]) for Painlevé equations
Communicated by Irene Sabadini. This article is part of the topical collection “In memory of Carlos A. Berenstein (1944–2019)” edited by Irene Sabadini and Daniele Struppa. Data sharing not applicable to this article as no datasets were generated or analysed during this study. Supported by JSPS KAKENHI Grant No. 19H01794.
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Yoshitsugu Takei [email protected] Department of Mathematical Sciences, Doshisha University, Kyotanabe, Kyoto 610-0394, Japan 0123456789().: V,-vol
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Y. Takei
with a large parameter to analyze the global behavior of their solutions. One of the main purposes of this series of papers is the analysis of Stokes phenomena for formal solutions of Painlevé equations and there what we call instanton-type solutions play a crucially important role. Here, for example, in the case of the first Painlevé equation η−2
d 2λ = 6λ2 + t, dt 2
(PI)
that is, the simplest Painlevé equation, an instanton-type solution means a formal solution of the following form: λI (t, η; α, β) = λ0 + η−1/2 (12λ0 )−1/4 αeηI + βe−ηI + η−1
2
(1/2)
b2−2k (t; α, β)e((2−2k)ηI ) + · · · ,
(1)
k=0
where t is an independent variable, η is a large parameter, α and β are free complex √ ( j/2) parameters, λ0 = λ0 (t) = −t/6, bl (t; α, β) are some appropriate functions that are recursively determined, and I =
t
12λ0 dt + η−1 αβ log(η2 (12λ0 )5 ).
(2)
Such an instanton-type solution appears after some Stokes phenomena occur with the formal power series solution λ(0) = λ0 (t)+η−2 λ2 (t)+· · · of (PI) (cf. [3]). Although instanton-type solutions are indispensable in describing the Stokes phenom
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