The asymptotic behavior of solutions of the sine-Gordon equation with singularities zero
type: Item
instance of: scholarly article
title: The asymptotic behavior of solutions of the sine-Gordon equation with singularities zero
zbMATH Open document ID: 0985.37084
DOI: 10.1007/BF02676663
author: V. L. Vereshchagin
published in: Mathematical Notes
publication date: 2000-11-16T00:00:00Z
review text: The author deals with the case of a radially symmetric equation of the form \(u_{xx}+{1\over x} u_x+ \sin u= 0\) and studies the asymptotic behaviour of its solution. It turns out that the solutions with singularity of the form \[ u\sim \alpha\log{1\over x}+ \beta+ O(1),\quad x^2= y^2+ t^2\to 0,\quad \alpha,\beta\in \mathbb{R} \] at zero have an asymptotic limit at infinity: \[ u(x)= 2\pi k+ \gamma\sqrt{{\pi\over 2x}} e^{-x}[1+ O(x^{-1})],\quad x\to\infty, \] where \(k\) is an integer. Moreover, the corresponding modulation equation and phase shift are stated.
Mathematics Subject Classification ID: 37K40
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1531333
zbMATH Keywords: sine-Gordon equation
zbMATH Keywords: radially symmetric equation
zbMATH Keywords: modulation equation
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1531333