A Discussion on the Different Notions of Symmetry of Differential Equations

TL;DR

This paper explores various notions of symmetry in partial differential equations, introducing and characterizing standard and weak conditional symmetries, and examining their relationships with exact and partial symmetries using symmetry-adapted variables.

Contribution

It provides a unified framework for understanding different symmetry types in PDEs, including precise definitions and relationships among them.

Findings

Introduction of standard and weak conditional symmetries

Use of symmetry-adapted variables for analysis

Clarifying examples illustrating symmetry concepts

Abstract

A discussion is presented, within a simple unifying scheme, about different types of symmetry of PDE's, with the introduction and a precise characterization of the notions of "standard" and "weak" conditional symmetries, together with their relationship with exact and partial symmetries. An extensive use of "symmetry-adapted" variables will be made, and some clarifying examples are provided.