Construction of solutions for the generalized Henon-Heiles system with the help of the Painleve test

TL;DR

This paper applies the Painleve test to the generalized Henon-Heiles system, deriving new formal solutions as Laurent or Puiseux series, and proves their convergence under certain parameter conditions.

Contribution

It introduces a method to find formal solutions for the nonintegrable Henon-Heiles system using the Painleve test, including convergence proofs.

Findings

Derived new formal Laurent and Puiseux series solutions.

Proved convergence of solutions within specific parameter ranges.

Identified conditions under which solutions match known exact solutions.

Abstract

The Henon-Heiles system in the general form has been considered. In a nonintegrable case with the help of the Painleve test new solutions have been found as formal Laurent or Puiseux series, depending on three parameters. One of parameters determines a location of the singularity point, other parameters determine coefficients of series. It has been proved, that if absolute values of these two parameters are less or equal to unit, then obtained series converge in some ring. For some values of these parameters the obtained Laurent series coincide with the Laurent series of the known exact solutions.