On the geometry of lambda-symmetries, and PDEs reduction

TL;DR

This paper provides a geometric framework for understanding lambda-symmetries of ODEs and extends these concepts to PDEs using a form-based approach, demonstrating their effectiveness in symmetry reduction and finding invariant solutions.

Contribution

It introduces a geometric characterization of lambda-prolongations and extends the concept to PDEs with a form-based approach, enhancing symmetry analysis tools.

Findings

Lambda-symmetries can be characterized geometrically.

The approach extends to PDEs using a horizontal one-form.

Lambda-symmetries are as effective as standard symmetries for reduction.

Abstract

We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields and $\mu$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions.