On the notion of conditional symmetry of differential equations

TL;DR

This paper systematically explores the concept of conditional symmetry in PDEs, distinguishing true and weak forms, and relates them to other symmetry types, using symmetry-adapted variables and examples like the Boussinesq equation.

Contribution

It provides a unified framework for understanding conditional symmetries, clarifies their distinctions, and relates them to existing symmetry concepts with illustrative examples.

Findings

Defined true and weak conditional symmetries.

Established relationships between conditional and other symmetries.

Applied the framework to examples including the Boussinesq equation.

Abstract

Symmetry properties of PDE's are considered within a systematic and unifying scheme: particular attention is devoted to the notion of conditional symmetry, leading to the distinction and a precise characterization of the notions of ``true'' and ``weak'' conditional symmetry. Their relationship with exact and partial symmetries is also discussed. An extensive use of ``symmetry-adapted'' variables is made; several clarifying examples, including the case of Boussinesq equation, are also provided.