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SIGMA 9 (2013), 026, 23 pages arXiv:1303.3434
https://doi.org/10.3842/SIGMA.2013.026
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”
A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
José F. Cariñena a, Partha Guha b and Javier de Lucas c
a) Department of Theoretical Physics and IUMA, University of Zaragoza, Pedro Cerbuna 12, 50.009, Zaragoza, Spain
b) S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata - 700.098, India
c) Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszyński University, Wóy-cickiego 1/3, 01-938, Warsaw, Poland
Received September 26, 2012, in final form March 14, 2013; Published online March 26, 2013
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.
Key words:
Lie system; Kummer-Schwarz equation; Milne-Pinney equation; quasi-Lie scheme; quasi-Lie system;
second-order Gambier equation; second-order Riccati equation; superposition rule.
pdf (484 kb)
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