Lie remarkable partial differential equations characterized by Lie algebras of point symmetries

TL;DR

This paper explores the inverse Lie problem by providing examples of coupled Lie remarkable PDEs, uniquely characterized by their Lie symmetry algebras, including hierarchies of second order and specific third order systems.

Contribution

It introduces new classes of Lie remarkable PDEs linked to affine and projective symmetry groups, advancing the understanding of their symmetry-characterization.

Findings

Hierarchies of second order PDEs characterized by affine transformations

A system of two third order PDEs characterized by projective transformations

Explicit examples of coupled Lie remarkable equations

Abstract

Within the framework of inverse Lie problem, we give some non-trivial examples of coupled Lie remarkable equations, \textit{i.e.}, classes of differential equations that are in correspondence with their Lie point symmetries. In particular, we determine hierarchies of second order partial differential equations uniquely characterized by affine transformations of $\mathbb{R}^{n+m}$, and a system of two third order partial differential equations in two independent variables uniquely determined by the Lie algebra of projective transformations of $\mathbb{R}^4$.