Superposition rules and second-order Riccati equations

José F. Cariñena; Javier de Lucas Araujo

doi:10.3934/jgm.2011.3.1

Article Contents

2011, Volume 3, Issue 1: 1-22. Doi: 10.3934/jgm.2011.3.1

This issue Previous Article Next Article Reduction of invariant constrained systems using anholonomic frames

Superposition rules and second-order Riccati equations

José F. Cariñena^1, and
Javier de Lucas Araujo^2,

1.
Departamento de Física Teórica and IUMA, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50.009, Zaragoza
2.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckish 8, P.O. Box 21, 00-956, Warszawa

Received: May 31, 2010

Revised: March 31, 2011

Published: February 2011

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Abstract

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose various generalisations of this notion to second-order differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other second-order differential equations of mathematical and physical interest.

Keywords:

Quasi-Lie scheme,
Lie system,
system of second-order differential equations,
second-order Riccati equation,
superposition rule.

Mathematics Subject Classification: 34A26, 34A34, 53Z05.

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References

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