Multi-hump pulses in systems with reflection and phase invariance
type: Item
instance of: scholarly article
title: Multi-hump pulses in systems with reflection and phase invariance
zbMATH Open document ID: 1179.35319
author: Vahagn Manukian
author: B. Sandstede
published in: Journal of Differential Equations
publication date: 2009-09-28T00:00:00Z
review text: First, the authors consider systems of PDEs that have two continuous symmetries, namely, translation and phase invariance. The paradigm for such systems is the cubic-quintic complex Ginzburg-Landau equation. The intention of the authors is to investigate the existence and stability of standing and travelling multi-hump waves in PDEs with reflection and phase symmetries. The considered systems of PDEs respects two continuous symmetries, namely, translation in the spatial derivative variable and a gauge invariance. Therefore, the eigenvalue of the primary pulse at the origin has multiplicity two. \(N\)-pulse will therefore have \(2N\) critical eigenvalues near the origin. The authors also use Lyapunov-Schmidt reduction to relate the existence and stability properties of \(N\)-pulses to the solutions of a reduced set of equations. In fact, the authors extend an known technique from equations with just translation symmetry to systems with translation and phase symmetries. The reduced equations are solved for 2- and 3-pulses. In the last part of the paper, the authors apply the above results to the complex cubic-quintic Ginzburg-Landau equation near the limit \(e=0\) that corresponds to the nonlinear Schrödinger equation.
Mathematics Subject Classification ID: 35Q56
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B06
zbMATH DE Number: 5608031
zbMATH Keywords: hump pulse
zbMATH Keywords: phase symmetries
zbMATH Keywords: \(n\)-pulse solutions
zbMATH Keywords: cubic-quintic
zbMATH Keywords: Ginzburg-Landau equation
Wikidata QID: Q60143893
reviewed by: Marin I. Marin
MaRDI profile type: Publication
full work available at URL: j.jde.2009.06.010
OpenAlex ID: W1995133270
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DOI: 10.1016/J.JDE.2009.06.010
description: scientific article; zbMATH DE number 5608031