On the internal approach to differential equations 3. Infinitesimal symmetries

TL;DR

This paper develops a geometric framework for partial differential equations that allows for higher-order infinitesimal symmetries, extending classical Lie symmetry methods without relying on traditional variable hierarchies.

Contribution

It introduces a geometric approach to PDEs that accommodates higher-order symmetries and relaxes the preservation of variable hierarchies, adapting Lie's classical method.

Findings

Higher-order infinitesimal symmetries are characterized geometrically.

The approach simplifies the analysis of symmetry groups of PDEs.

Classical Lie methods are extended to a more general geometric setting.

Abstract

The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher--order infinitesimal symmetries which provide a simplifying "linear aproximation" of general groups of higher--order symmetries. The classical Lie's approach is appropriately adapted.