On extending the Painlev\'e test to the one-dimensional Vlasov equation

TL;DR

This paper extends the Painlevé test to analyze the one-dimensional Vlasov equation, classifying solutions with the Painlevé property and deriving constraints that describe plasma evolution in electric fields.

Contribution

It introduces a nontrivial generalization of the Painlevé test for the Vlasov equation and provides a detailed classification of solutions based on pole surfaces.

Findings

Classification of solutions with pole surfaces

Derived compatibility conditions for Laurent series

Explicit solutions for plasma evolution in electric fields

Abstract

An analysis of possible extension of the Painlev\'e test, to encompass the one-dimensional Vlasov equation, is performed. The extending requires a nontrivial generalization of the test. The proposed singularity analysis provides classification of the solutions possessing the Painlev\'e property by the order and number of pole surfaces. The compatibility conditions for the Laurent series have the form of an overdetermined system of 1st order differential equations, which themselves need a compatibility condition. This eventually leads to constraints which implicitly yield a family of solutions. The complete calculation is provided for the case of one simple order pole. The solutions describe evolution of plasmas in a uniform electric field.