Construction of Special Solutions for Nonintegrable Dynamical Systems with the help of the Painleve Analysis

TL;DR

This paper uses Painleve analysis to construct special solutions for the nonintegrable Henon-Heiles system, revealing new Laurent series solutions and their convergence properties, and demonstrating the method's ability to find elliptic solutions.

Contribution

It introduces a novel application of Painleve test to derive Laurent series solutions for nonintegrable systems, including elliptic solutions, expanding solution techniques.

Findings

Found new Laurent series solutions depending on three parameters.

Demonstrated convergence of the series in a specific ring.

Showed that some series match known exact solutions.

Abstract

The generalized Henon-Heiles system has been considered. In two nonintegrable cases with the help of the Painleve test new special solutions have been found as Laurent series, depending on three parameters. The obtained series converge in some ring. One of parameters determines the singularity point location, other parameters determine coefficients of series. For some values of these parameters the obtained Laurent series coincide with the Laurent series of the known exact solutions. The Painleve test can be used not only to construct local solutions as the Laurent series but also to find elliptic solutions.