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Riccati Equations Revisited: Linearization and Analytic Interpretation of Instanton-Type Solutions

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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.

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References

  1. Aoki, T., Kawai, T., Takei, Y.: WKB analysis of Painlevé transcendents with a large parameter II. In: Morimoto, M., Kawai, T. (eds.) Structure of Solutions of Differential Equations, pp. 1–49. World Scientific Publishing (1996)

  2. Costin, O., Costin, R.D.: On the formation of singularities of solutions of nonlinear differential systems in antistokes directions. Invent. Math. 145, 425–485 (2001)

    Article  MathSciNet  Google Scholar 

  3. Delabaere, E.: Divergent series, summability and resurgence III: Resurgent methods and the first Painlevé equation. Lecture Notes in Mathematics, vol. 2155. Springer (2016)

  4. Delabaere, E., Pham, F.: Resurgent methods in semi-classical asymptotics. Ann. Inst. Henri Poincaré 71, 1–94 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Garoufalidis, S., Its, A., Kapaev, A.A., Mariño, M.: Asymptotics of the instantons of Painlevé I. Int. Math. Res. Not. 2012, 561–606 (2012)

    Article  Google Scholar 

  6. Kawai, T., Takei, Y.: WKB analysis of Painlevé transcendents with a large parameter. I. Adv. Math. 118, 1–33 (1996)

    Article  MathSciNet  Google Scholar 

  7. Kawai, T., Takei, Y.: WKB analysis of Painlevé transcendents with a large parameter. III. Adv. Math. 134, 178–218 (1998)

    Article  MathSciNet  Google Scholar 

  8. Kawai, T., Takei, Y.: Algebraic analysis of singular perturbation theory. Translations of Mathematical Monographs, Vol. 227. American Mathematical Society (2005)

  9. Koike, T., Schäfke, R.: On the Borel summability of WKB solutions of Schrödinger equations with polynomial potentials and its applications, in preparation

  10. Takei, Y.: Singular-perturbative reduction to Birkhoff normal form and instanton-type formal solutions of Hamiltonian systems. Publ. RIMS Kyoto Univ. 34, 601–627 (1998)

    Article  MathSciNet  Google Scholar 

  11. Takei, Y.: An explicit description of the connection formula for the first Painlevé equation. In: Howls, C.J., Kawai, T., Takei, Y. (eds.) Toward the Exact WKB Analysis of Differential Equations, pp. 271–296. Kyoto University Press, Linear or Non-Linear (2000)

    MATH  Google Scholar 

  12. Takei, Y.: On an exact WKB approach to Ablowitz–Segur’s connection problem for the second Painlevé equation. ANZIAM J. 44, 111–119 (2002)

    Article  MathSciNet  Google Scholar 

  13. Takei, Y.: WKB analysis and Stokes geometry of differential equations. In: Filipuk, G., Haraoka, Y., Michalik, S. (eds.) Analytic, Algebraic and Geometric Aspects of Differential Equations, pp. 263–304. Birkhäuser (2017)

  14. Takei, Y.: On the instanton-type expansions for Painlevé transcendents and elliptic functions. In: Filipuk, G., Lastra, A., Michalik, S., Takei, Y., Żołądek, H. (eds.) Complex Differential and Difference Equations, de Gruyter Proceedings in Mathematics, pp. 365–377. De Gruyter (2019)

  15. Voros, A.: The return of the quartic oscillator. The complex WKB method. Ann. Inst. Henri Poincaré 39, 211–338 (1983)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Doshisha University, Kyotanabe, Kyoto, 610-0394, Japan

    Yoshitsugu Takei

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  1. Yoshitsugu Takei
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Correspondence to Yoshitsugu Takei.

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Communicated by Irene Sabadini.

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Supported by JSPS KAKENHI Grant No. 19H01794.

This article is part of the topical collection “In memory of Carlos A. Berenstein (1944–2019)” edited by Irene Sabadini and Daniele Struppa.

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Takei, Y. Riccati Equations Revisited: Linearization and Analytic Interpretation of Instanton-Type Solutions. Complex Anal. Oper. Theory 14, 78 (2020). https://doi.org/10.1007/s11785-020-01033-y

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  • DOI: https://doi.org/10.1007/s11785-020-01033-y

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