Traveling Wave Solutions for Bistable Differential-Difference Equations with Periodic Diffusion

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1、SIAMJ.APPL.MATH.c2001SocietyforIndustrialandAppliedMathematicsVol.61,No.5,pp.1648–1679TRAVELINGWAVESOLUTIONSFORBISTABLEDIFFERENTIAL-DIFFERENCEEQUATIONSWITHPERIODICDIFFUSION∗CHRISTOPHERE.ELMER†ANDERIKS.VANVLECK‡Abstract.Weconsidertravelingwavesolutionstospatiallydiscretereaction-diffusio

2、nequationswithnonlocalvariablediffusionandbistablenonlinearities.Tofindthetravelingwavesolutionsweintroduceanansatzinwhichthewavespeeddependsontheunderlyinglatticeaswellasontime.Forthecaseofspatiallyperiodicdiffusionweobtainanalyticsolutionsforthetravelingwaveproblemusingapiecewiselinearno

3、nlinearity.Theformulaforthewaveformsisimplicitlydefinedinthegeneralperiodiccaseandweprovideanexplicitformulaforthecaseofperiodtwodiffusion.Wepresentnumericalstudiesfortimet=0fixedandforthetimeevolutionofthetravelingwaves.Whent=0westudythecasesofhomogeneous,periodtwo,andperiodfourdiffusionco

4、efficientsusingacubicnonlinearity,anduncover,numerically,aperioddoublingbifurcationinthewavespeedversusdetuningparameterrelation.Forthetimeevolutioncasewealsodiscoveradetuningparameterdependentbifurcationinobservedphenomena,whichisaproductofboththenonlocaldiffusionoperatorandthespinodaleffe

5、ctsofthenonlinearity.Keywords.travelingwaves,propogationfailure,bistableequation,periodicdiffusionAMSsubjectclassifications.35K57,74N99PII.S00361399993571131.Introduction.Problemsthataremodeledbynonlinearspatiallydiscretedif-fusionequationsoccurinmanyareasofscience.Inthematerialssciencese

6、xamplesincludecrystalgrowth,interfacemotionincrystallinematerials,spinodaldecompo-sition,andgrainboundarymovementinthinfilms.Spatiallydiscretediffusionop-eratorsallowthematerialbeingmodeledtobedescribedintermsofitsunderlyingcrystallinelattice.Phenomenasuchascrystallographicpinningandlatti

7、ceanisotropy,phenomenathatareoftenaddedtospatiallycontinuousmodelsinan“adhoc”man-ner,occurnaturallyinspatiallydiscretematerialsmodels.Inbiology,examplesofapplicationsofspatiallydiscretemodelsincludethebidomainmodelforcardiactissue(defibrillation),tissuefiltration,gasexchangeinlun