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

TL;DR

This paper explores the nonlocal nature of $mbda$-symmetries in ordinary differential equations (ODEs) and demonstrates how they can be recovered as local symmetries through an embedding into an extended system, facilitating ODE reduction.

Contribution

It introduces a framework using the concept of coverings to interpret $mbda$-symmetries as nonlocal symmetries and shows how to recover them as local symmetries in an extended system.

Findings

$mbda$-symmetries can be viewed as nonlocal symmetries via coverings.

Embedding an ODE into a suitable extended system allows local symmetry analysis.

The reduction method of Muriel and Romero is derived from standard symmetry reduction techniques.

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