A Consistent Approach to Approximate Lie Symmetries of Differential Equations

TL;DR

This paper introduces a new, consistent method for applying Lie symmetry analysis to differential equations with small perturbations, extending classical symmetry techniques to approximate solutions.

Contribution

It proposes a novel approach that integrates Lie symmetry methods with perturbation theory, improving the analysis of perturbed differential equations.

Findings

The new method extends Lie symmetry analysis to approximate solutions.

Applications demonstrate the effectiveness of the approach.

The approach simplifies computations compared to previous methods.

Abstract

Lie theory of continuous transformations provides a unified and powerful approach for handling differential equations. Unfortunately, any small perturbation of an equation usually destroys some important symmetries, and this reduces the applicability of Lie group methods to differential equations arising in concrete applications. On the other hand, differential equations containing \emph{small terms} are commonly and successfully investigated by means of perturbative techniques. Therefore, it is desirable to combine Lie group methods with perturbation analysis, \emph{i.e.}, to establish an approximate symmetry theory. There are two widely used approaches to approximate symmetries: the one proposed in 1988 by Baikov, Gazizov and Ibragimov, and the one introduced in 1989 by Fushchich and Shtelen. Moreover, some variations of the Fushchich--Shtelen method have been proposed with the aim of reducing the length of computations. Here, we propose a new approach that is consistent with perturbation theory and allows to extend all the relevant features of Lie group analysis to an approximate context. Some applications are also presented.