A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
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.
Keywords
Cite
@article{arxiv.1303.3434,
title = {A Quasi-Lie Schemes Approach to Second-Order Gambier Equations},
author = {José F. Cariñena and Partha Guha and Javier de Lucas},
journal= {arXiv preprint arXiv:1303.3434},
year = {2013}
}