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 small terms are commonly and successfully investigated by means of perturbative techniques. Therefore, it is desirable to combine Lie group methods with perturbation analysis, 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.

A consistent approach to approximate Lie symmetries of differential equations

Di Salvo, Rosa
Primo
Membro del Collaboration Group
;
Gorgone, Matteo
Penultimo
Membro del Collaboration Group
;
Oliveri, Francesco
Ultimo
Membro del Collaboration Group
2018-01-01

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 small terms are commonly and successfully investigated by means of perturbative techniques. Therefore, it is desirable to combine Lie group methods with perturbation analysis, 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.
File in questo prodotto:
File Dimensione Formato  
Di_Salvo_Gorgone_Oliveri_ND_2018.pdf

solo utenti autorizzati

Tipologia: Versione Editoriale (PDF)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 728.93 kB
Formato Adobe PDF
728.93 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11570/3118214
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 6
  • ???jsp.display-item.citation.isi??? 6
social impact