Towards nonlinear stability of sources via a modified Burgers equation

Margaret Beck, Toan Nguyen, Björn Sandstede, Kevin Zumbrun

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Coherent structures are solutions to reactiondiffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at x=±∞ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a Burgers-type equation as a model problem that captures some of the essential features of sources, we show how this phenomenon can be analysed and asymptotic nonlinear stability be established in this simpler context.

Original languageEnglish (US)
Pages (from-to)382-392
Number of pages11
JournalPhysica D: Nonlinear Phenomena
Volume241
Issue number4
DOIs
StatePublished - Feb 15 2012

Fingerprint

Burger equation
group velocity
traveling waves
far fields
perturbation

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

Cite this

Beck, Margaret ; Nguyen, Toan ; Sandstede, Björn ; Zumbrun, Kevin. / Towards nonlinear stability of sources via a modified Burgers equation. In: Physica D: Nonlinear Phenomena. 2012 ; Vol. 241, No. 4. pp. 382-392.
@article{b65261cbb37346fc855bce6e28231512,
title = "Towards nonlinear stability of sources via a modified Burgers equation",
abstract = "Coherent structures are solutions to reactiondiffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at x=±∞ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a Burgers-type equation as a model problem that captures some of the essential features of sources, we show how this phenomenon can be analysed and asymptotic nonlinear stability be established in this simpler context.",
author = "Margaret Beck and Toan Nguyen and Bj{\"o}rn Sandstede and Kevin Zumbrun",
year = "2012",
month = "2",
day = "15",
doi = "10.1016/j.physd.2011.10.018",
language = "English (US)",
volume = "241",
pages = "382--392",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "4",

}

Towards nonlinear stability of sources via a modified Burgers equation. / Beck, Margaret; Nguyen, Toan; Sandstede, Björn; Zumbrun, Kevin.

In: Physica D: Nonlinear Phenomena, Vol. 241, No. 4, 15.02.2012, p. 382-392.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Towards nonlinear stability of sources via a modified Burgers equation

AU - Beck, Margaret

AU - Nguyen, Toan

AU - Sandstede, Björn

AU - Zumbrun, Kevin

PY - 2012/2/15

Y1 - 2012/2/15

N2 - Coherent structures are solutions to reactiondiffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at x=±∞ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a Burgers-type equation as a model problem that captures some of the essential features of sources, we show how this phenomenon can be analysed and asymptotic nonlinear stability be established in this simpler context.

AB - Coherent structures are solutions to reactiondiffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at x=±∞ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a Burgers-type equation as a model problem that captures some of the essential features of sources, we show how this phenomenon can be analysed and asymptotic nonlinear stability be established in this simpler context.

UR - http://www.scopus.com/inward/record.url?scp=84855245909&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855245909&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2011.10.018

DO - 10.1016/j.physd.2011.10.018

M3 - Article

AN - SCOPUS:84855245909

VL - 241

SP - 382

EP - 392

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 4

ER -