Localized periodic patterns for the non-symmetric

Localizedperiodicpatternsforthenon-symmetricgeneralizedSwift-HohenbergequationC.J.Budd1,andR.Kuske21CentreforNonlinearMechanics,UniversityofBath,UK,BA27AY2DepartmentofMathematics,UniversityofBritishColumbia,CanadaV6T1Z2.SupportedinpartbyNSF-DMS0072311andanNSERCDiscoveryGrant.Correspondingauthor:[e-mail],phone:[phone],fax:[phone]AbstractAnewasymptoticmultiplescaleexpansionisusedtoderiveenvelopeequationsforlocalizedspatiallyperiodicpatternsinthecontextofthegeneralizedSwift-Hohenbergequation.Ananalysisofthisenvelopeequationresultsinparametricconditionsforlocalizedpatterns.Furthermore,ityieldscorrectionsforwavenumberselectionwhichareanorderofmagnitudelargerforasymmetricnonlinearitiesthanforthesymmetriccase.TheanalyticalresultsarecomparedwithnumericalcomputationswhichdemonstratethattheconditionforlocalizedpatternscoincideswithvanishingHamiltonianandLagrangianforperiodicsolutions.OnestrikingfeatureofthechoiceofscalingparametersisthatthederivedconditionforlocalizedpatternsagreeswiththenumericalresultsforasignicantrangeofparameterswhichareanO(1)distancefromthebifurcation,thusprovidinganovelapproachforstudyingtheselocalizedpatterns.Keywords:Asymptoticbalance,Localizedpatterns,Lagrangian,Heteroclinicconnection1IntroductionSpatiallylocalizedoscillatorypatternshavebeenstudiedinavarietyofmodelsandcontexts,includingconvection,chemicalpatterns,elasticity,andoptics(see[1]-[7]andreferencestherein).Inordertostudytheappearanceofthisphenomenon,simpliedmodels,suchastheSwift-Hohenbergequationhavebeenused.TheSwift-Hohenbergequationhasalsobeenusedasacanonicalmodeltostudymanyotherpropertiesofpatterndynamics.Heterogeneityisanobviouscauseoflocalizedoscillations,sinceaspatiallyvaryingmediumcanleadtospatiallyvaryingpatterns,orpatternsappearingonlyinlocalizedregionsofspace;forexample,see[8],[9]and[10].Heterogeneityisnotanecessaryconditionforsuchpatterns;localizedoscillationscanalsooccurinhomogeneoussystems,wheretheyhavebeenreferredtoasstablecoexistingpatterns[11,12,13],orwherepinnedinterfacesorgrainboundariesbetweenrollsandsteadystateshavebeenstudied[14,15].Itistypicallymorediculttoidentifyparametricallytheconditionsforspatiallylocalizedpatternsinthehomogeneouscase,sincetheycanarisethroughdierentmechanisms,dependingonthe1application.Anumberofexperimentalandcomputationalresults[1,2,11,12,13,16],andmanyothers,haveshownthattheselocalizedoscillationscanoccurindierentsystemsofreaction-diusionandSwift-Hohenbergmodels.Theyappearinparameterregionscorrespondingtosubcriticalbifurcations[11,13,16,17]andbistabilityregions[1,2,12]andinthepresenceofbothsymmetricandasymmetricnonlinearities.Inthispaperwelookforspatiallylocalizedcellular(rolltype)patternsinthecontextofthegeneralizedSwift-Hohenberg(SH)equationut=Lu+f(u)q2c+@2@x2!2u+ru+f(u)(1.1)f(u)=b2u2+b3u3+b4u4+b5u5+::::(1.2)AtypicalpatternofthistypeisillustratedinFigure1.1andcomprisesasequenceofnearidenticalcellsclosetox=0withrapiddecaytozeroasjxj!1inthenalcell.Intheequation(1.2)uplaysthesameroleasthevariablefortemperatureinconvectionproblemsorconcentrationinchemicalreactions,andrplaystheroleofacontrolorbifurcationparameterwithacriticalvalueofr=0.Itiswellknownthatthezerosolutionisstabletoperiodicperturbationsforr0,andforr0extendedperiodicsolutions(rolls)arestable.Wedescribelocalizedoscillatorysolutionsof(1.1)byderivinganequationfortheenvelopeoramplitudeofthepatterns.Anasymptoticexpansionforuthatbalancesrelativesizesofthelinearandnonlineareectsisthebasisfortheconstructionoftheamplitudeequation,whichprovidesconditionsforthelocalizedsolutionsintermsofthemodelparameters.Inparticular,weareabletoconstructheteroclinicconnectionsbetweenthesteady(zero)stateandoscillatorypatterns.Theparametervaluesatwhichtheseheteroclinicconnectionsoccurlieatthecentreoftherangeofparametersatwhichweobserverollpatternswithanarbitrarynumberofrolls.Thenwehaveadescriptionoftheinterfaceorgrainboundaryforthecoexistingsteadystateandrolls.Theconditionsfortheseheteroclinicconnectionsbetweenzeroandtheoscillatorypatternscanbewrittenintermsofarelationshipbetweenthebifurcationparameterrandthecoecientsbjofthenonlineartermin(1.2)whichtakestheform2(r)=342(bj;r)g(bj;r):(1.3)Here,theparameters2,andgareobtainedfromthecoecientsintheamplitudeequation,andweomittheirargumentsintheremainderofthepaper.Theparameter1=1representsthelengthscaleoftheslowevolutionofthepatternandptheratiobetweentheamplitudescaleand.Theexpression(1.3)representsacurvewhichliesinthecenterofaregionforobservingtheselocalizedpatterns;theregioncanbedescribedbyincludingexponentiallysmallcorrectionstotheresult(1.3),butwedonotcomputetheseinthispaper.ThesearediscussedfurtherinSection6andin[14,15,18].Muchoftheanalysisofthispaperismotivatedby[18]whichgivesanasymptoticanalysisofthesteadystateequationforastrutonaWinklerfoundationu+Pu+u+a2u2+a3u3+a4u4+a5u5+::::=0;(1.4)withtheloadPasbifurcationparameter.Instudies[7]and[18]of(1.4)andinourconsiderationofthegeneralizedSwift-Hohenbergequation(1.1),thefocusisontwo2classesofnonlinearity.Intermsofthecoec