Superposition rules, Lie theorem and partial differential equations

TL;DR

This paper provides a rigorous geometric proof of Lie's Theorem on nonlinear superposition rules for ODEs, introduces an alternative foliation-based definition, and extends the theorem to PDE systems.

Contribution

It offers a complete geometric proof of Lie's Theorem, clarifies the uniqueness of superposition functions, and generalizes the theorem to partial differential equations.

Findings

Established a foliation-based definition of superposition rules.

Proved the uniqueness of the superposition function.

Extended Lie's Theorem to systems of PDEs.

Abstract

A rigorous geometric proof of the Lie's Theorem on nonlinear superposition rules for solutions of non-autonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an alternative but equivalent definition of a superposition rule: it is considered as a foliation with some suitable properties. The problem of uniqueness of the superposition function is solved, the key point being the codimension of the foliation constructed from the given Lie algebra of vector fields. Finally, as a more convincing argument supporting the use of this alternative definition of superposition rule, it is shown that this definition allows an immediate generalization of Lie's Theorem for the case of systems of partial differential equations.