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1、JournalofAppliedMathematicsandStochasticAnalysis9,Number1,1996,77-86PAINLEVEANALYSISOFACLASSOFNONLINEARDIFFUSIONEQUATIONSP.CHANDRASEKARANandE.K.RAMASAMIBharathiarUniversityDepartmentofMathematicsCoimbatore5106,INDIA(ReceivedOctober,1994;RevisedOctober,1995)ABSTRACTWestudythePainleveanalysisforacl
2、assofnonlineardiffusionequations.WefindthatinsomecasesithasonlytheconditionalPainlevepropertyandinothercases,justthePainleveproperty.Wealsoobtainedspecialsolutions.Keywords:Nonlinear-DiffusionEquation,PainleveAnalysis,PainleveEquation,SpecialSolutions.AMS(MOS)subjectclassifications:35Q51.1.Introd
3、uctionInrecentyears,muchattentionhasbeenfocusedonhigherordernonlinearpartialdifferentialequations,knownasevolutionequations.Suchnonlinearequationsoftenoccurinthedescriptionofchemicalandbiologicalphenomena.Theiranalyticalstudyhasbeendrawingimmenseinterest.Afundamentalquestionwhendealingwithnonline
4、ardifferentialequationsis"howcanonetellbeforehandwhetherornottheyareintegrable?"Originally,Ablowitzetal[1]conjectur-edthatanonlinearpartialdifferentialequationisintegrableifallitsexactreductionstoordinarydifferentialequationshavethePainleveproperty:thatis,tohavenomovablesingularitiesotherthanpole
5、s.Thisapproachposesanobviousoperationaldifficultyinfindingallexactreductions.ThisdifficultywascircumventedbyWeissetal[10]bypostulatingthatapartialdifferentialequa-tionhasthePainlevepropertyifitssolutionsaresingle-valuedaboutamovablesingularmanifold(z,z2,...,Zn)0,(1.1)whereisanarbitraryfunction.In
6、otherwords,asolutionu(zi)ofapartialdifferentialequationshouldhaveaLaurent-likeexpansionaboutthemovablesingularmanifold0:u(zi)[(zi)]uj(zi)(zi)j,(1.2)2--0whereaisanegativeinteger.Thenumberofarbitraryfunctionsinexpansion(1.2)shouldbeequaltotheorderofthepartialdifferentialequation.Insertingexpansion(
7、1.2)intothetargetedequationyieldsarecurrenceformulathatdeterminesUn(Zi)foralln>0,exceptforafinitenumberofrl,r2,r3,...,rj>0,calledresonances.Forsomeequations,therecurrenceformulasattheresonancevaluesmayresultinconstrain
