Invariant representation and symmetry reduction for differential equations with a small param\-eter.
type: Item
instance of: scholarly article
title: Invariant representation and symmetry reduction for differential equations with a small param\-eter.
zbMATH Open document ID: 1041.35003
DOI: 10.1016/S1007-5704(03)00010-8
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2004-01-06T00:00:00Z
review text: The Lie group analysis of differential equations is one of the effective methods for constructing exact solutions of differential equations under consideration. A wide class of the group methods of integration is based on the theorem of invariant representation of equations. Due to this theorem, if an equation is invariant with respect to the Lie group \(G\), then the equation has an equivalent form defined by invariant functions of the group \(G\). In this paper, a similar theorem for equations with a small parameter is proved. Instead of the Lie transformation groups, the authors consider approximate groups. The features of approximate symmetry reduction are demonstrated with constructing travelling-wave solution for the shallow-water equations \[ \begin{gathered} h_t+ uh_x+ hu_x= 0,\\ u_t+ uu_x+ h_x+ \varepsilon b'(x)+ \varepsilon C_f(h){4^2\over h}= 0.\end{gathered} \]
Mathematics Subject Classification ID: 35A25
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 2021944
zbMATH Keywords: approximate groups
zbMATH Keywords: approximate symmetry reduction
zbMATH Keywords: shallow-water equations
author: Yulia Yu Bagderina
author: R. K. Gazizov
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
cites work: Q4357207
cites work: Q3715336
cites work: Perturbation methods in group analysis
cites work: Representation of general invariants for approximate transformation groups
description: scientific article; zbMATH DE number 2021944