Quasi-Lie families, schemes, invariants and their applications to Abel equations

TL;DR

This paper introduces quasi-Lie families and invariants, providing new geometric insights and tools for analyzing Abel equations, including invariants and superposition rules, advancing the understanding of these differential equations.

Contribution

It develops methods to identify quasi-Lie invariants and characterizes Abel equations through quasi-Lie schemes, offering new geometric and algebraic insights.

Findings

Retrieved the Liouville invariant for Abel equations

Identified new quasi-Lie invariants of Abel equations

Characterized Abel equations via quasi-Lie schemes

Abstract

This work analyses types of group actions on families of $t$-dependent vector fields of a particular class, the hereby called quasi-Lie families. We devise methods to obtain the defined here quasi-Lie invariants, namely a kind of functions constant along the orbits of the above-mentioned actions. Our techniques lead to a deep geometrical understanding of quasi-Lie schemes and quasi-Lie systems giving rise to several new results. Our achievements are illustrated by studying Abel and Riccati equations. We retrieve the Liouville invariant and study other new quasi-Lie invariants of Abel equations. Several Abel equations with a superposition rule are described and we characterise Abel equations via quasi-Lie schemes.