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International Journal of Differential Equations
Volume 2015, Article ID 340715, 13 pages
Research Article

Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations

Keldysh Institute of Applied Mathematics, Miusskaya Square 4, Moscow 125047, Russia

Received 30 January 2014; Accepted 24 June 2014

Academic Editor: Sining Zheng

Copyright © 2015 Alexander D. Bruno. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 .