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This article is cited in 8 scientific papers (total in 8 papers)
A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
José F. Cariñenaa, Partha Guhab, Javier de Lucasc 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
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:
Lie system; Kummer–Schwarz equation; Milne–Pinney equation; quasi-Lie scheme; quasi-Lie system; second-order Gambier equation; second-order Riccati equation; superposition rule.
Received: September 26, 2012; in final form March 14, 2013; Published online March 26, 2013
Citation:
José F. Cariñena, Partha Guha, Javier de Lucas, “A Quasi-Lie Schemes Approach to Second-Order Gambier Equations”, SIGMA, 9 (2013), 026, 23 pp.
Linking options:
https://www.mathnet.ru/eng/sigma809 https://www.mathnet.ru/eng/sigma/v9/p26
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