Nonlocal aspects of $\lambda$-symmetries and ODEs reduction

Abstract

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 ${\mathcal{Y}}^{\prime }$.

13 pages

Cited by in corpus (4)

• Local and nonlocal solvable structures in ODEs reduction
• Noether theorem for mu-symmetries
• Reduction of systems of first-order differential equations via Lambda-symmetries
• Nonlocal interpretation of $\lambda$-variational symmetry-reduction method