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

José F. Cariñena ; Partha Guha ; Javier de Lucas

Symmetry, integrability and geometry: methods and applications, Tome 9 (2013)

Résumé

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.

Détail

Keywords: Lie system; Kummer–Schwarz equation; Milne–Pinney equation; quasi-Lie scheme; quasi-Lie system; second-order Gambier equation; second-order Riccati equation; superposition rule.

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