Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations

الملخص EN

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 anODE.

Among them we select regular power-elliptic expansions and give a survey of all such expansions in solutions of the Painlevé equations P 1 , … , P 6 .