A direct algorithm of one-dimensional optimal system for the group invariant solutions

TL;DR

This paper introduces a direct, systematic algorithm for constructing one-dimensional optimal systems of group-invariant solutions by classifying one-dimensional Lie algebras, demonstrated through examples like the KdV and heat equations.

Contribution

It presents a novel, comprehensive method for deriving one-dimensional optimal systems that ensures inequivalence without additional proofs.

Findings

Algorithm guarantees completeness of the optimal system.

Method effectively classifies Lie algebra invariants.

Illustrated with classical PDE examples.

Abstract

A direct and systematic algorithm is proposed to find one-dimensional optimal system for the group invariant solutions, which is attributed to the classification of its corresponding one-dimensional Lie algebra. Since the method is based on different values of all the invariants, the process itself can both guarantee the comprehensiveness and demonstrate the inequivalence of the optimal system, with no further proof. To illustrate our method more clearly , we give a couple of well-known examples: the Korteweg-de Vries (KdV) equation and the heat equation.