Self-similar planar curves related to modified Korteweg-de Vries equation
type: Item
instance of: scholarly article
title: Self-similar planar curves related to modified Korteweg-de Vries equation
zbMATH Open document ID: 1124.35076
author: Galina Perelman
author: L. Vega
published in: Journal of Differential Equations
publication date: 2007-04-19T00:00:00Z
review text: The planar curve \(z(s,t)= x(s,t)+ iy(s,t)\) with \(s\in\mathbb{R}\) the arclength and \(t\in\mathbb{R}\) the time evolution parameter is subjected to the flow equation \(z_t= -z_{sss}+ {3\over 2}\overline z_s z^2_{ss}\) and then the curvature \(k\) satisfies the modified Korteweg-de Vries equation \(k_t+ k_{sss}+{3\over 2} k^2 k_s= 0.\) The authors prove the existence of a solution satisfying the initial condition \(k(x, 0)= a\delta+\mu\text{ p.v. }{1\over x},\) where \(a\), \(\mu\) are constants close to zero and \(\delta\) is the Dirac function. The selfsimilar solutions \[ k(s, t)= {2\over (3t)^{1/3}}\,u\Biggl({s\over (3t)^{1/3}}\Biggr) \] are found. Then the problem is reduced to a thorough study of the ordinary differential equation \(u_{xx}- xu+ 2u^3=\mu\) in the main part of the article.
reviewed by: Jan Chrastina
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K25
zbMATH DE Number: 5144841
zbMATH Keywords: planar curve
zbMATH Keywords: selfsimilar solution
zbMATH Keywords: modified Korteweg-de Vries equation
zbMATH Keywords: curvature
zbMATH Keywords: existence
MaRDI profile type: Publication
full work available at URL: j.jde.2006.12.018
OpenAlex ID: W2144382444
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DOI: 10.1016/J.JDE.2006.12.018
description: scientific article; zbMATH DE number 5144841