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

References in corpus (2)

• On the geometry of lambda-symmetries, and PDEs reduction
• On the relation between standard and $\mu$-symmetries for PDEs

Cited by in corpus (13)

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• Reduction of systems of first-order differential equations via Lambda-symmetries
• A generalization of λ-symmetry reduction for systems of ODEs: σ-symmetries
• ${\mathcal{C}}^{\infty }$-symmetries of distributions and integrability
• A gauge-theoretic description of $\mu$-prolongations, and $\mu$-symmetries of differential equations
• Nonlocal interpretation of $\lambda$-variational symmetry-reduction method
• On the Geometry of Twisted Symmetries: Gauging and Coverings
• Nonlocal pseudosymmetries and Bäcklund transformations as $\mathcal{C}$-morphisms
• Twisted symmetries of differential equations