Optimal systems and invariant solutions for the curve shortening problem
type: Item
instance of: scholarly article
title: Optimal systems and invariant solutions for the curve shortening problem
zbMATH Open document ID: 1029.53080
author: Kai-Seng Chou
author: Guan-Xin Li
published in: Communications in Analysis and Geometry
publication date: 2003-04-07T00:00:00Z
review text: The authors give a systematic investigation of the group invariant solutions of the generalized curve shortening problem described by the following non-uniformly parabolic equation \[ u_t=\left|\frac{u_{xx}}{(1+u^{2}_{x})^{3/2}}\right|^{\sigma-1} \frac{u_{xx}}{(1+u^{2}_{x})}, \quad \sigma>0 \] For this equation they determine the Lie algebra of all infinitesimal symmetries and show that they form a basis of all symmetries if \(\sigma\not =1/3\). When \(\sigma=1/3\) the Lie algebra of infinitesimal symmetries is of dimension seven, two more than for the non-affine case \(\sigma=1/3\). To describe all group invariant solutions an optimal system of subalgebras is determined.
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 22E60
zbMATH DE Number: 1893845
zbMATH Keywords: curve shortening problem
zbMATH Keywords: optimal system of subalgebras
zbMATH Keywords: invariant solutions
zbMATH Keywords: group analysis of differential equations
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
DOI: 10.4310/CAG.2002.V10.N2.A1
description: scientific article; zbMATH DE number 1893845