## International Journal of Differential Equations

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

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

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#### 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**

Received: 30 January 2014

Accepted: 24 June 2014

First available in Project Euclid: 20 January 2017

**Permanent link to this document**

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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