Evans function and blow up methods in critical eigenvalue problems

Arnd Scheel

Description

Niveau: Supérieur, Doctorat, Bac+8
Evans function and blow-up methods in critical eigenvalue problems Bjorn Sandstede Department of Mathematics The Ohio State University Columbus, USA Arnd Scheel Department of Mathematics University of Minnesota Minneapolis, USA October 18, 2002 Abstract Contact defects are among the modulated waves in oscillatory media that arise generically in reaction-diffusion systems. An interesting property of these defects is that the asymptotic spatial wavenumber is approached only with algebraic order O(1/x) (the associated phase diverges logarithmically). The essential spectrum of the PDE linearization about a contact defect always has a branch point at the origin. We show that the Evans function can be extended across this branch point and discuss the smoothness properties of this extension. The construction utilizes blow-up techniques and is quite general in nature. We comment on known relations between roots of the Evans function and the temporal asymptotics of Green's functions, and discuss applications to algebraically decaying solitons. 1 Introduction The goal of this paper is to investigate the stability properties of a class of nonlinear waves that arises in dissipative, pattern-forming partial differential equations (PDEs). Consider a reaction- diffusion system Ut = DUxx + F (U), (1.1) posed on the real line x ? R, where U ? RN and D is a diagonal positive diffusion matrix.

• chart given


• since z˙2


• reaction- diffusion systems


• blow-up techniques


• contact defects


• diffusion system


• chart


• limiting wave

Sujets

• Ohio State University
• Scheel

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 22
• Langue: English

Extrait

Evansfunctionandblow-upmethodsincriticaleigenvalueproblems Bjo¨rnSandstedeArndScheel DepartmentofMathematicsDepartmentofMathematics TheOhioStateUniversityUniversityofMinnesota Columbus,USAMinneapolis,USA October18,2002 Abstract Contactdefectsareamongthemodulatedwavesinoscillatorymediathatarisegenericallyin reaction-diffusionsystems.Aninterestingpropertyofthesedefectsisthattheasymptoticspatial wavenumberisapproachedonlywithalgebraicorderO(1/x)(theassociatedphasediverges logarithmically).TheessentialspectrumofthePDElinearizationaboutacontactdefectalways hasabranchpointattheorigin.WeshowthattheEvansfunctioncanbeextendedacrossthis branchpointanddiscussthesmoothnesspropertiesofthisextension.Theconstructionutilizes blow-uptechniquesandisquitegeneralinnature.Wecommentonknownrelationsbetween rootsoftheEvansfunctionandthetemporalasymptoticsofGreen’sfunctions,anddiscuss applicationstoalgebraicallydecayingsolitons. 1Introduction Thegoalofthispaperistoinvestigatethestabilitypropertiesofaclassofnonlinearwavesthat arisesindissipative,pattern-formingpartialdifferentialequations(PDEs).Considerareaction- diffusionsystem U t =DU xx +F(U),(1.1) posedonthereallinex ∈ R ,whereU ∈ R N andDisadiagonalpositivediffusionmatrix.We assumethat(1.1)supportsafamilyofwave-trainsolutionsU(x,t)=U(kx − ωt;k)thatare2π- periodicinξ=kx − ωt.Weassumethatadispersionrelationω=ω dr (k)relatesthetemporal frequencyωtothespatialwavenumberk.Ofinterestarethenmodulatedwavesordefects,i.e. wavesthataretimeperiodicinanappropriatemovingcoordinateframe,thatconvergetotwo, possiblydifferent,wavetrainsasx →±∞ .Inanaccompanyingpaper[17],wegivealistofdefects thatoccurgenericallyinsuchamedium.Aparticularlyinteresting,butpoorlyunderstood,robust typeofdefectconsistsofamodulatedwavethatisasymptoticatx= ±∞ totwowavetrainswhich havethesamewavenumberk ∗ sothatthedefecttravelswiththegroupvelocityc g :=ω 0 dr (k ∗ )of thesebackgroundwaves: U ∗ (x − c g t,ω ∗ t)=U ∗ (x − c g t,ω ∗ t+2π)(x,t) ∈ R × R + (1.2) | U ∗ (x − c g t,ω ∗ t) − U(k ∗ x − ω ∗ t − θ ± (x − c g t);k ∗ ) |→ 0forx →±∞ 1

Figure1:Theleftpictureshowsacontourplotofaplanarspiralwavethatarisesafter theasymptoticwavetrainsexhibitedaperiod-doublingbifurcation.Toaccommodatethe geometry,acontactdefectformsalongastraightline(amagnificationofthedefectisshown intherightplot). whereweassumethatω ∗ :=ω dr (k ∗ ) − c g k ∗ 6 =0(thisassumptionsimplymeansthatthegroupand phasevelocitiesoftheselectedwavetrainU(k ∗ x − ω dr (k ∗ )t;k ∗ )donotcoincide).Wecalledthis typeofsolutionsto(1.1)contactdefectssinceinformation,transportedwiththegroupvelocities oftheasymptoticwavetrains,neitherentersnorleavesthelocalizeddefect.Acharacteristic commonfeatureofcontactdefectsisthealgebraicrelaxationofthewavenumberθ 0 ± (ξ)=O(1/ | ξ | ) for | ξ |→∞ togetherwiththelogarithmicdivergenceoftheasymptoticphaseθ ± (ξ).Contact defectsturnouttobelimitsofshock-typedefectswherethegroupvelocityc g, ± ofthetwodifferent limitingwavetrainsatx= ±∞ isdirectedtowardsthedefectsothatc g, − >c defect >c g,+ .In spiteofthischaracterizationasalimit,contactdefectsappearinanopenandnon-emptyclass ofreaction-diffusionsystems,foropensetsofasymptoticwavenumbersk,andhaveactuallybeen observedexperimentallyinseveraldifferentcontexts(see[17]fordetails).Oneexampleareline defectsthatoccuratperiod-doublingbifurcationsofspiralwaves(seeFigure1). Ourgoalistoobtainaproperdescriptionofthereaction-diffusionsystem,linearizedatcontact defects,andthespectrumoftheassociatedperiodmap;recallthatdefectsaretime-periodicsolu- tionsof(1.1).Thekeyissueistotraceeigenvaluesofthelinearizedperiodmapintotheessential spectrumbyusingavariantoftheEvansfunctionthatwasoriginallyintroducedin[1].Themain obstacleisthesimultaneouspresenceofabranchpointinthecontinuousspectrumandtheweak algebraicdecayofthex-dependentcoefficientsinthelinearizedequationwhichappearsincethe profileofthecontactdefectdependsonx.Webelievethataproperunderstandingofthelinearized problemwillproveusefulinfutureworkonnonlinearstabilityandinteractionpropertiesofcontact defects.Furthermore,thetechniquesthatwedevelophereenableustoassignlinearstabilityprop- ertiesinnonlinearbifurcationdiagramssuchasthosearisinginlockingandunlockingbifurcations ofdifferenttypesofdefects[17].Weremarkthatitwasshownin[14],seealso[9],thattheEvans functioncanbeconstructedforalgebraicallydecayingcoefficientsaslongasonestaysawayfrom theabsolutespectrumthatwasintroducedin[15]. 2