## International Journal of Differential Equations

### Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations

Alexander D. Bruno

#### Abstract

We consider an ordinary differential equation (ODE) which can be written as a polynomial in variables and derivatives. Several types of asymptotic expansions of its solutions can be found by algorithms of 2D Power Geometry. They are power, power-logarithmic, exotic, and complicated expansions. Here we develop 3D Power Geometry and apply it for calculation power-elliptic expansions of solutions to an ODE. Among them we select regular power-elliptic expansions and give a survey of all such expansions in solutions of the Painlevé equations ${P}_{\mathrm{1}},\dots ,{P}_{\mathrm{6}}$.

#### Article information

Source
Int. J. Differ. Equ., Volume 2015 (2015), Article ID 340715, 13 pages.

Dates
Accepted: 24 June 2014
First available in Project Euclid: 20 January 2017

https://projecteuclid.org/euclid.ijde/1484881428

Digital Object Identifier
doi:10.1155/2015/340715

Mathematical Reviews number (MathSciNet)
MR3312759

Zentralblatt MATH identifier
1339.34063

#### Citation

Bruno, Alexander D. Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations. Int. J. Differ. Equ. 2015 (2015), Article ID 340715, 13 pages. doi:10.1155/2015/340715. https://projecteuclid.org/euclid.ijde/1484881428

#### References

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• A. D. Bruno, Power Geometry in Algebraic and Differential Equations, Elsevier, Amsterdam, The Netherlands, 2000, translated from Fizmatlit, Moscow, Russia, 1998 (Russian).
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• A. D. Bruno and A. V. Parusnikova, “Elliptic and periodic asymptotic forms of solutions to P$_{5}$,” in Painlevé Equations and Related Topics, A. D. Bruno and A. B. Batkhin, Eds., pp. 53–65, De Gruyter, Berlin, Germany, 2012.
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• I. V. Goryuchkina, “Three-dimensional analysis of asymptotic forms of the solutions to the sixth Painlevé equation,” Preprint of KIAM no. 56, pp. 24, Moscow, Russia, 2010, (Russian). \endinput