Symmetry classification of third-order nonlinear evolution equations. Part I: Semi-simple algebras

TL;DR

This paper classifies third-order nonlinear evolution equations based on their symmetry properties, focusing on those with semi-simple Lie algebra symmetries and their extensions, advancing the understanding of their algebraic structures.

Contribution

It provides a complete point-symmetry classification for third-order evolution equations with semi-simple and extended Lie algebra symmetries, refining existing classification techniques.

Findings

Classified equations with semi-simple symmetry algebras

Extended classification to include solvable algebra extensions

Refined methods for symmetry analysis of evolution equations

Abstract

We give a complete point-symmetry classification of all third-order evolution equations of the form $u_t=F(t,x,u,u_x, u_{xx})u_{xxx}+G(t,x,u,u_x, u_{xx})$ which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.