The Painlevé Approach to Nonlinear Ordinary Differential Equations

Abstract

The "Painlevé analysis" is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory of the (explicit) integration of nonlinear differential equations.To achieve our goal, we will not start the exposition with a more or less precise "Painlevé test". On the contrary, we will finish with it, after a gradual introduction to the rich world of singularities of nonlinear differential equations, so as to remove any cooking recipe.The emphasis is put on embedding each method of the test into the well known theorem of perturbations of Poincaré. A summary can be found at the beginning of each chapter.

Introduction to the Painlevé property, test and analysis

“…This well known method consists in requiring the existence, near every movable singularity, of a Laurent series able to represent the general solution. We will not detail it here because it can be found in many courses, such as [3], but we will rather concentrate on common errors and difficulties. Two errors should not be made: Error 1.…”

Explicit integration of the Hénon-Heiles Hamiltonians 1

“…Quite similarly the canonical transformation to elliptic coordinates [49], 12) maps the Hamilton's equations to the hyperelliptic system (7.8) with…”

Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function

“…First of all one can try to solve a question about the integrability of this equation. For this aim one can apply the Painlevé test to check the integrability of this equation [10]. As a result we can have two variants.…”