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

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.

23 pages

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• Superposition rules, Lie theorem and partial differential equations
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Cited by in corpus (2)

• Quasi-Lie schemes for PDEs
• Quantum quasi-Lie systems: properties and applications