Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 106, 21 pages      arXiv:1212.6475
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Nonlocal Symmetries, Telescopic Vector Fields and λ-Symmetries of Ordinary Differential Equations

Concepción Muriel and Juan Luis Romero
Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain

Received July 09, 2012, in final form December 19, 2012; Published online December 28, 2012

This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of λ-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 λ-coverings method. The λ-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 λ-symmetries is used to decide whether or not reductions associated to two nonlocal symmetries are strictly different.

Key words: nonlocal symmetries; λ-symmetries; telescopic vector fields; order reductions; differential invariants.

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