Lie families: theory and applications

TL;DR

This paper studies Lie families of non-autonomous differential equations that admit a common superposition rule, connecting them with Lie and quasi-Lie systems and providing new superposition rules.

Contribution

It introduces the concept of Lie families with time-dependent superposition rules and explores their relation to Lie systems, offering new methods for solving these equations.

Findings

Identified conditions for families of differential equations to admit common superposition rules.

Established connections between Lie families and Lie/quasi-Lie systems.

Provided explicit examples of superposition rules for specific Lie families.

Abstract

We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants. We next study relations of these families, called Lie families, with the theory of Lie and quasi-Lie systems and apply our theory to provide common time-dependent superposition rules for certain Lie families.