Embedding algorithms and applications to differential equations

TL;DR

This paper develops algorithms for embedding nilpotent subalgebras in maximal subalgebras using real algebraic group methods, and applies them to find invariant solutions of the wave equation and analyze symmetry algebras.

Contribution

It introduces new algorithms for embedding nilpotent subalgebras and demonstrates their application to differential equations and symmetry analysis.

Findings

Identified non-conjugate subalgebras of the wave equation's symmetry algebra.

Determined a large class of invariant solutions for the wave equation.

Illustrated algorithms on classical systems in contact geometry.

Abstract

Algorithms for embedding certain types of nilpotent subalgebras in maximal subalgebras of the same type are developed, using methods of real algebraic groups. These algorithms are applied to determine non-conjugate subalgebras of the symmetry algebra of the wave equation, which in turn are used to determine a large class of invariant solutions of the wave equation. The algorithms are also illustrated for the symmetry algebra of a classical system of differential equations considered by Cartan in the context of contact geometry.