Stable transport of information near essentially unstable localized structures

Description

Niveau: Supérieur, Doctorat, Bac+8
Stable transport of information near essentially unstable localized structures THIERRY GALLAY1, GUIDO SCHNEIDER2, HANNES UECKER2 1 Institut Fourier, Universite de Grenoble I, F - 38402 Saint-Martin d'Heres, France 2 Mathematisches Institut I, Universitat Karlsruhe, D - 76128 Karlsruhe, Germany June 23, 2003 Abstract When the steady states at infinity become unstable through a pattern forming bifurca- tion, a travelling wave may bifurcate into a modulated front which is time-periodic in a moving frame. This scenario has been studied by B. Sandstede and A. Scheel for a class of reaction-diffusion systems on the real line. Under general assumptions, they showed that the modulated fronts exist and are spectrally stable near the bifurcation point. Here we consider a model problem for which we can prove the nonlinear stability of these solutions with respect to small localized perturbations. This result does not follow from the spectral stability, because the linearized operator around the modulated front has essential spectrum up to the imaginary axis. The analysis is illustrated by numerical simulations. Contents 1 Introduction 2 2 Main Results 4 3 Existence and stability of Turing patterns 11 3.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Bloch waves .

• ?v ?


• front


• chaffee-infante equation


• generically no modulated


• nonlinear stability


• turing pattern


• modulated front


• fronts has


• bifurcation point

Sujets

• France 2
• Bifurcation theory

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 19
• Langue: English

Extrait

Stabletransportofinformationnearessentially
unstablelocalizedstructures

T
HIERRY
G
ALLAY
1
,G
UIDO
S
CHNEIDER
2
,H
ANNES
U
ECKER
2
1
InstitutFourier,Universite´deGrenobleI,F-38402Saint-Martind'Heres,France
2
MathematischesInstitutI,Universita¨tKarlsruhe,D-76128Karlsruhe,Germany
June23,2003

Abstract
Whenthesteadystatesatinnitybecomeunstablethroughapatternformingbifurca-
tion,atravellingwavemaybifurcateintoamodulatedfrontwhichistime-periodicina
movingframe.ThisscenariohasbeenstudiedbyB.SandstedeandA.Scheelforaclass
ofreaction-diffusionsystemsontherealline.Undergeneralassumptions,theyshowed
thatthemodulatedfrontsexistandarespectrallystablenearthebifurcationpoint.Here
weconsideramodelproblemforwhichwecanprovethenonlinearstabilityofthese
solutionswithrespecttosmalllocalizedperturbations.Thisresultdoesnotfollowfrom
thespectralstability,becausethelinearizedoperatoraroundthemodulatedfronthas
essentialspectrumuptotheimaginaryaxis.Theanalysisisillustratedbynumerical
simulations.

Contents
1Introduction2
2MainResults4
3ExistenceandstabilityofTuringpatterns11
3.1Existence...................................11
3.2Blochwaves..................................13
3.3Linearstability................................14
3.4Nonlinearstability..............................18
4Constructionofmodulatedfronts18
4.1Theidea....................................19
4.2Transversalityinextendedspace.......................21
1

4.3BifurcationofTuringpatterns........................24
4.4Bifurcationofthemodulatedfront......................24
4.5RemarksoncaseIIandcaseIII........................25

5Thenonlinearstabilityproof25
5.1Theidea....................................26
5.2Theunscaledequations............................28
5.3Therenormalizationprocedure........................30
5.4IterationandConclusion...........................38

6ResultsincaseII

04

7Numericalsimulations42
7.1Modulatedfronts...............................42
7.2Modulatedpulses...............................43

1Introduction

Localizedstructuressuchaspulsesandfrontsplayanimportantroleinthemathematical
theoryofinformationtransport.Typicalsituationswheresuchnonlinearphenomenaarise
arethepropagationofelectromagneticwavesinwiresorbers[AA97,NM92],andthe
motionofelectricpulsesalongnerveaxons[Hux52].Animportantissue,bothfromatheo-
reticalandapracticalpointofview,istherobustnessofthesesolutionswithrespecttosmall
inhomogeneitiesofthepropagationmedium.
Inaremarkablepaper[SS99],B.SandstedeandA.Scheelstudiedanewbifurcationscena-
riofortravelingpulsesinreaction-diffusionsystemsontherealline.Theyinvestigatedthe
situationwherethehomogeneoussteadystateatinnitybecomesunstableandbifurcatesto
aspatiallyperiodicTuringpattern.Theoriginallystablepulsethusundergoesanessential
instability,inthesensethattheessentialspectrumofthelinearizedoperatorcrossesthe
imaginaryaxisatthebifurcationpoint.Undergeneralassumptions,theauthorsshowedthat
theoriginalpulsebifurcatestoamodulatedpulsewhichsitime-periodicinauniformly
translatingframe.Theyalsoprovedthatthisbifurcatingsolutionisspectrallystable[SS00].
However,sincethespectrumofthelinearizationextendsallthewaytotheimaginaryaxis
(withoutgap),thislastresultdoesnotimmediatelyimplythenonlinearstabilityofthemo-
dulatedpulse.Theanalysisof[SS99,SS00]canbegeneralizedtofrontsolutionsconnecting
twodifferentstableequilibria[SS01a,SS01b].Inthiscase,modulatedfrontsmaybifurcate
fromanexistingtravelingwaveifoneorbothofthereststatesatinnitybecomeunstable.
Inthispaper,wegobeyondthelinearstabilityanalysisof[SS00,SS01b]andweshow,
atleastonaspecicexample,thatmodulatedfrontsare
nonlinearly
stablewithrespectto
spatiallylocalizedperturbations.Insimpleterms,thisresultimpliesthatinformationcanbe
transportedinastablemannerevenifthepropagationmediumbecomesunstablethrougha

2

Turingbifurcation.Tokeeptheanalysisassimpleaspossible,wedonotconsiderabstract
reaction-diffusionsystemsasin[SS99],butweprefertoconcentrateonamodelproblem
thatexhibitsallfeaturesofthegeneralcase.Althoughthishasnotbeenprovedsofar,we
certainlyexpectthatallresultsbelowholdtrueforgeneralreaction-diffusionsystemsunder
thesameassumptionsasin[SS00](forpulses)or[SS01b](forfronts).
OurmodelproblemisaChaffee-Infanteequationfortherstvariable
u
coupledtoaSwift-
Hohenbergequationforthesecondvariable
v
,namely:
∂
t
u
=
∂
x
2
u
+
1
(
u
−
c
0
)(1
−
u
2
)+
v,
2)1(∂
t
v
=
−
(1+
∂
x
2
)
2
v
+
αv
−
v
3
−
γvF
(
u
)
,
where
u
(
x,t
)
,v
(
x,t
)
∈
R
,
x
∈
R
,and
t
≥
0
.Thissystemisespeciallyconvenientto
analyze,becauseitcouplestwoscalarequationswhichareratherwellunderstood.Inwhat
follows,thespeedparameter
c
0
andthecouplingparameter
γ
willbexed,with
0
<c
0
<
1
and
γ>
0
nottoobig(seeTheorem2.3below).Ourbifurcationparameter
α
willthenvary
inaneighborhoodofthebifurcationpoint
α
=0
.Tocoverallinterestingcases,weshall
considerthreedifferentfunctions
F
,namely
I)
F
(
u
)=1
−
u
2
,
II)
F
(
u
)=1
−
u,
III)
F
(
u
)=1+
u.
Forallchoicesof
F
,system(1)possessestwospatiallyhomogeneousequilibria
(
u,v
)=
(
±
1
,
0)
andaone-parameterfamilyoffrontsolutions
(
u,v
)=(tanh((
x
−
c
0
t
−
x
0
)
/
2)
,
0)
,x
0
∈
R
,
(2)
connectingtheseequilibria.For
α<
0
,theequilibriaandthefamilyoffrontsolutions
areasymptoticallystablewithsomeexponentialrate.When
α
crossestheoriginfromleft
toright,someoftheequilibriabecomeunstable,dependingontheparticularchoiceof
F
.
IncaseI,thesteadystatesaheadofandbehindthefrontundergoaTuringbifurcationand
spatiallyperiodicequilibriaarecreated.IncaseII,thishappensonlyforthesteadystate
(
u,v
)=(1
,
0)
aheadofthefront,andincaseIIIonlyforthesteadystate
(
u,v
)=(
−
1
,
0)
behindthefront.Inthisrespect,caseIisclosetothecaseofapulse.

1

0

1

0

1-1--50050-50050
Figure1:Modulatedfrontsfor(1)incasesI(left)andII(right).Thesnapshotsshowthe
u
-componentobtainedfromgenericinitialdataatsomelargetime;seealsosection7.
Atthebifurcationpoint
α
=0
,thefrontsolutions(2)becomeessentiallyunstableand,in
casesIandII,afamilyofmodulatedfrontsiscreated.Thesesolutionsaretime-periodicin
3

amovingframewithspeed
c
≈
c
0
,andtheyconnectaspatiallyperiodicTuringpatternat
x
=+
∞
toanotherTuringpattern(caseI)ortotheuniformsteadystate
(
−
1
,
0)
(caseII)at
x
=
−∞
,seeFig.1.WeshallnotconsidercaseIIIanylonger,sincetheanalysisin[SS01a]
showsthatatleastgenericallynomodulatedfrontsexistinthatcase;typicallythepatternis
outrunbythefront,seeg.6onpage44foranillustration.Asforthestability,itturnsout
thatincaseIIthefamilyofmodulatedfrontsisasymptoticallystablewithexponentialrate.
Thiscanbeprovedrathereasilyusingweightedspaces,seesection6.Thusthechallenging
caseinprovingstabilityiscaseI.Inthissituation,thelinearizationaroundthemodulating
frontshascontinuousspectrumuptotheimaginaryaxis.Itisthepurposeofthispaperto
explainhowneverthelessthenonlinearstabilityofthesesolutionscanbeshown.
Remark1.1
IncaseI,themodelproblem(1)seemsnon-generic,sincebothhomogeneous
equilibria
(
±
1
,
0)
undergoaTuringbifurcationatthesamevalueoftheparameter
α
.Infact,
weimplicitlyrestrictouranalysistosystemsforwhichthedestabilizationofbothequilibria
hasthesameorigin
(inourexample,thisisthecouplingofthebistable
u
-equationtoa
single
Swift-Hohenbergequation).Thisalsoexplainswhythewavelengthsofthebifurcating
patternsaheadofandbehindthefrontcoincide.Aswasobservedbyoneofthereferees
ofthispaper,itwouldthenbemorenaturaltoconsiderthecaseofapulseinsteadofa
front.Butthenwewouldhavetoreplacethescalar
u
-equationin(1)bya
2
×
2
system,
whichmakestheanalysisevenmoreintricate.Also,wefounditinterestingtoencompassall
possiblecases(I,II,andIII)inasingle,relativelysimplemodel.
Acknowledgments:
ThisworkwassupportedbytheFrench-Germancooperationproject
PROCOPE00307TKentitledAttractorsforextendedsystems.TheauthorsalsothankB.
SandstedeandA.Scheelforstimulatingdiscussions,andbothrefereesforusefulcomments
andsuggestions.

∂2uu1(uc)(1u2)+v2MainResults
Inthissection,wegiveourresultsinthemostinterestingcase,i.e.when
F
(
u
)=1
−
u
2
in(1).
Tosimplifythenotation,werewrite(1)intheform
∂
t
U
=
L
(
∂
x
)
U
+
N