On the relation between standard and $\mu$-symmetries for PDEs

TL;DR

This paper explores the geometric relationship between standard symmetries and $ ext{-}$-symmetries in PDEs, extending the concept of $ ext{-}$-symmetries from ODEs and analyzing their connections to other symmetry types.

Contribution

It provides a geometric interpretation of $ ext{-}$-prolongations and establishes links between $ ext{-}$-symmetries, standard symmetries, and nonlocal symmetries in PDEs.

Findings

Established a geometric framework for $ ext{-}$-symmetries in PDEs.

Demonstrated the relationship between $ ext{-}$-symmetries and standard exact symmetries.

Extended the notion of $ ext{-}$-symmetries to conditional and partial symmetries.

Abstract

We give a geometrical interpretation of the notion of $\mu$-prolongations of vector fields and of the related concept of $\mu$-symmetry for partial differential equations (extending to PDEs the notion of $\lambda$-symmetry for ODEs). We give in particular a result concerning the relationship between $\mu$-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyze the relation between local $\mu$-symmetries and nonlocal standard symmetries.