Nonlocal aspects of $λ$-symmetries and ODEs reduction

Details Nonlocal aspects of $λ$-symmetries and ODEs reduction Nonlocal aspects of $λ$-symmetries and ODEs reduction

2007-02-12

arxiv.org/abs/math-ph/0702039v2

Physics

Mathematical Physics

13 pages

Scientific paper

10.1088/1751-8113/40/21/002

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $\lambda$-symmetries (C. Muriel and J. L. Romero, \emph{IMA J. Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE $\mathcal{Y}$ is used here to recover $\lambda$-symmetries of $\mathcal{Y}$ as nonlocal symmetries. In this framework, by embedding $\mathcal{Y}$ into a suitable system $\mathcal{Y}^{\prime}$ determined by the function $\lambda$, any $\lambda$-symmetry of $\mathcal{Y}$ can be recovered by a local symmetry of $\mathcal{Y}^{\prime}$. As a consequence, the reduction method of Muriel and Romero follows from the standard method of reduction by differential invariants applied to $\mathcal{Y}^{\prime}$.

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