On Complicated Expansions of Solutions to ODES

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Abstract

Polynomial ordinary differential equations are studied by asymptotic methods. The truncated equation associated with a vertex or a nonhorizontal edge of their polygon of the initial equation is assumed to have a solution containing the logarithm of the independent variable. It is shown that, under very weak constraints, this nonpower asymptotic form of solutions to the original equation can be extended to an asymptotic expansion of these solutions. This is an expansion in powers of the independent variable with coefficients being Laurent series in decreasing powers of the logarithm. Such expansions are sometimes called psi-series. Algorithms for such computations are described. Six examples are given. Four of them are concern with Painlevé equations. An unexpected property of these expansions is revealed.

Keywords

ordinary differential equation asymptotic expansion solution with logarithms Painlevé equation 

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References

  1. 1.
    A. D. Bruno, “Asymptotics and expansions of solutions to an ordinary differential equation,” Russ. Math. Surv. 59 (3), 429–480 (2004).MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. D. Bruno, Preprint No. 36, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2011).Google Scholar
  3. 3.
    A. D. Bruno, “Exponential expansions of solutions to an ordinary differential equation,” Dokl. Math. 85 (2), 259–264 (2012).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. D. Bruno, “Power-elliptic expansions of solutions to an ordinary differential equation,” Comput. Math. Math. Phys. 52 (12), 1650–1661 (2012).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. D. Bruno, “Power geometry and elliptic expansions of solutions to the Painlevé equations,” Int. J. Differ. Equations 2015, Article ID 340715. http://dx.doi.org/. doi 10.1155/2015/340715Google Scholar
  6. 6.
    A. D. Bruno, “Elements of nonlinear analysis,” Mathematical Forum, Ser. Itogi Nauki Yug Rossii (Yuzhn. Mat. Inst., Vladikavkaz. Nauchn. Tsentra, Vladikavkaz, 2015), pp. 13–33 [in Russian].Google Scholar
  7. 7.
    A. D. Bruno, “Asymptotic solution of nonlinear algebraic and differential equations,” Int. Math. Forum 10 (11), 535–564 (2015). http://dx.doi.org/. doi 10.12988/imf.2015.5974CrossRefGoogle Scholar
  8. 8.
    A. D. Bruno, Preprint No. 36, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2005).Google Scholar
  9. 9.
    A. D. Bruno, “Complicated expansions of solutions to an ordinary differential equation,” Dokl. Math. 73 (1), 117–120 (2006).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    A. D. Bruno and I. V. Goryuchkina, “Asymptotic expansions of solutions of the sixth Painlevé equation,” Trans. Moscow Math. Soc. 71, 1–104 (2010).CrossRefMATHGoogle Scholar
  11. 11.
    A. D. Bruno, Preprint No. 81, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2006).Google Scholar
  12. 12.
    A. D. Bruno, “Complicated expansions of solutions to a system of ordinary differential equations,” Dokl. Math. 78 (1), 477–480 (2008).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    E. L. Hille, Differential Equations in the Complex Domain (Wiley-Interscience, New York, 1976).MATHGoogle Scholar
  14. 14.
    M. Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, New York, 1989).MATHGoogle Scholar
  15. 15.
    V. I. Gromak, I. Laine, and S. Shimomura, Painlevé Differential Equations in the Complex Plane (Walter de Gruyter, New York, 2002).CrossRefMATHGoogle Scholar
  16. 16.
    A. D. Bruno and A. V. Gridnev, Preprint No. 10, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2010).Google Scholar
  17. 17.
    A. D. Bruno and A. V. Parusnikova, Preprint No. 72, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2010).Google Scholar
  18. 18.
    A. D. Bruno and A. V. Parusnikova, “Local expansions of solutions to the fifth Painlevé equation,” Dokl. Math. 83 (3), 348–352 (2011).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    A. D. Bruno, Preprint No. 15, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2011).Google Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia

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