This article is cited in 25 papers

Nonlocal Symmetries, Telescopic Vector Fields and $\lambda$-Symmetries of Ordinary Differential Equations

Concepción Muriel, Juan Luis Romero

Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain

Abstract: This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of $\lambda$-symmetries, telescopic vector fields and some nonlocal symmetries obtained by embedding the equation in an auxiliary system. The results let us connect such nonlocal symmetries with approaches that had been previously introduced: the exponential vector fields and the $\lambda$-coverings method. The $\lambda$-symmetry approach let us characterize the nonlocal symmetries that are useful to reduce the order and provides an alternative method of computation that involves less unknowns. The notion of equivalent $\lambda$-symmetries is used to decide whether or not reductions associated to two nonlocal symmetries are strictly different.

Keywords: nonlocal symmetries; $\lambda$-symmetries; telescopic vector fields; order reductions; differential invariants.

MSC: 34A05; 34A34

Received: July 9, 2012; in final form December 19, 2012; Published online December 28, 2012

Language: English

DOI: 10.3842/SIGMA.2012.106

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