A Quasi-Lie Schemes Approach to Second-Order Gambier Equations

TL;DR

This paper introduces quasi-Lie schemes to analyze second-order Gambier equations, enabling transformations into simpler forms, deriving new constants of motion, and expressing solutions via Riccati equations and harmonic oscillators.

Contribution

It develops a geometric framework using quasi-Lie schemes for second-order Gambier equations, filling gaps in previous methods and providing new solution representations.

Findings

Transformation into canonical forms simplifies analysis.

New constants of motion are derived for Gambier equations.

Solutions are expressed in terms of Riccati equations and harmonic oscillators.

Abstract

A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new constants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators.