Construction of Single-valued Solutions for Nonintegrable Systems with the Help of the Painleve Test

TL;DR

This paper presents an algorithm leveraging the Painleve test to construct single-valued solutions, including elliptic and trigonometric functions, for nonintegrable systems by transforming differential equations into algebraic systems.

Contribution

It introduces a novel algorithm that uses the Painleve test to find elliptic solutions of nonlinear differential equations, implemented in REDUCE and Maple.

Findings

Algorithm successfully constructs elliptic solutions.

Transformation reduces nonlinear differential equations to algebraic systems.

Implementation in computer algebra systems facilitates solution finding.

Abstract

The Painleve test is very useful to construct not only the Laurent-series solutions but also the elliptic and trigonometric ones. Such single-valued functions are solutions of some polynomial first order differential equations. To find the elliptic solutions we transform an initial nonlinear differential equation in a nonlinear algebraic system in parameters of the Laurent-series solutions of the initial equation. The number of unknowns in the obtained nonlinear system does not depend on number of arbitrary coefficients of the used first order equation. In this paper we describe the corresponding algorithm, which has been realized in REDUCE and Maple.