Invariant Modules and the Reduction of Nonlinear P

arXiv:solv-int/9904014v115Apr1999InvariantModulesandtheReductionofNonlinearPartialDifferentialEquationstoDynamicalSystemsNikyKamran†DepartmentofMathematicsMcGillUniversityMontr´eal,Qu´[e-mail]‡DepartmentofMathematicsDalhousieUniversityHalifax,[e-mail]§SchoolofMathematicsUniversityofMinnesotaMinneapolis,[e-mail]∼olverAbstract.Wecompletelycharacterizeallnonlinearpartialdifferentialequationsleav-ingagivenfinite-dimensionalvectorspaceofanalyticfunctionsinvariant.Existenceofaninvariantsubspaceleadstoareductionoftheassociateddynamicalpartialdifferentialequa-tionstoasystemofordinarydifferentialequations,andprovideanonlinearcounterparttoquasi-exactlysolvablequantumHamiltonians.TheseresultsrelyonausefulextensionoftheclassicalWronskiandeterminantconditionforlinearindependenceoffunctions.Inaddition,newapproachestothecharacterizationoftheannihilatingdifferentialoperatorsforspacesofanalyticfunctionsarepresented.†SupportedinpartbyanNSERCGrant.‡SupportedinpartbyanNSERCGrant.§SupportedinpartbyNSFGrantDMS98–03154.February4,200811.Introduction.TheconstructionofexplicitsolutionstopartialdifferentialequationsbysymmetryreductiondatesbacktotheoriginalworkofSophusLie,[30].ThereductionofpartialdifferentialequationstoordinarydifferentialequationswasgeneralizedbyClarksonandKruskal,[6],intheirdirectmethod,whichwaslatershown,[35],tobeincludedintheolderBlumanandColenonclassicalsymmetryreductionapproach,[3].Asurveyofthesemethodsasof1992canbefoundin[33].Meanwhile,Galaktionov,[12],introducedthemethodofnonlinearseparationthatreducesapartialdifferentialequationtoasystemofordinarydifferentialequations,developedinfurtherdepthin[12,13,14,15].SimilarideasappearintheworkofKing,[26],andthe“antireduction”methodsintroducedbyFushchychandZhdanov,[10,9].Svirshchevskii,[49,50,51],madetheimportantobser-vationthatonecould,intheone-dimensionalcase,characterizeintermsofhigherorder(orgeneralized)symmetriesthosenonlinearordinarydifferentialoperatorsthatadmitagiveninvariantsubspaceleadingtononlinearseparation.Inquantummechanics,lineardifferentialoperatorswithinvariantsubspacesformthefoundationofthetheoryofquasi-exactlysolvable(QES)quantummodelsasinitiatedbyTurbiner,Shifman,Ushveridze,andcollaborators,[41,42,43,52].ThebasicideaisthataHamiltonianoperatorwhichleavesafinite-dimensionalsubspaceoffunctionsinvariantcanberestrictedtothissubspace,resultinginaneigenvalueproblemwhichcanbesolvedbylinearalgebraictechniques.TheLiealgebraicapproachtoquasi-exactlysolvableproblemsrequiresthatthesubspaceinquestionbeinvariantunderaLiealgebraofdifferentialoperatorsg,inwhichcasetheHamiltonianbelongstotheuniversalenvelopingalgebraofg;see[16,17,37,47,52]fordetailsand[23,29]forapplicationstomolecularspectroscopy,nuclearphysics,andsoon.Zhdanov,[56],indicatedhowonecouldcharacterizequasi-exactlysolvableoperatorsusinghigherordersymmetrymethods.WeshouldalsomentionHel–OrandTeo,[21],whohaveappliedgroup-invariantsubspacesincomputervision,namingtheirelements“steerablefunctions”.MotivatedbyaproblemofBochner,[4],tocharacterizedifferentialoperatorshavingorthogonalpolynomialsolutions,Turbiner,[44],initiatedthestudyofdifferentialoperatorsleavingapolynomialsubspaceinvariant.Inone-dimension,theremarkableresultisthattheoperatorsleavingtheentiresubspaceofdegree≤npolynomialsinvariantarethequasi-exactlysolvableoperatorsconstructedbyLiealgebramethods.Theseresultswerefurtherdevelopedformultidimensionalandmatrixdifferentialoperators,anddifferenceoperatorsbyTurbiner,PostandvandenHijligenberg,[39,40,45,46],andFinkelandKamran,[8].Inthispaper,webroadenthegeneraltheoryofnonlinearseparationtoincludepartialdifferentialoperators,andarguethatitconstitutesthepropernonlineargeneralizationofquasi-exactlysolvablelinearoperators.Ourtheoryprovidesanexplicitcharacterizationofallnonlineardifferentialoperatorsthatleaveagivensubspaceoffunctionsinvariant.Inthetime-independentcase,solutionslyinginthesubspaceareobtainedbysolvingasystemofnonlinearalgebraicequations.Forevolutionequationsandcertainotherdynamicalpartial2differentialequations,weshowhowexplicitsolutionsinthegivensubspaceareobtainedbyreducingthepartialdifferentialequationstoafinite-dimensionaldynamicalsystem.Weillustrateourmethodwithanumberofsignificantexamples.OurmethodscanalsobecomparedandcontrastedwiththemorealgebraictheoryofD-modules,cf.[2,7,31,32].Inparticular,wedescribenewalgorithmsfordeterminingtheannihilatorofagivenfinite-dimensionalsubspace,basedonthestudyofgeneralizedWronskianmatricesandtheirranks.Thesemethodsderivetheirjustificationfromthegeneraltheoryofprolongedgrouptransformationsdevelopedin[36,38].Thefirstsectionofthepaperoutlinesthebasicsettingofourmethods—finite-dimensionalspacesofanalyticfunctionsdefinedonanopensubsetofrealEuclideanspace.Thecaseofanalyticfunctionsofseveralcomplexvariablesismoresubtle,andrequirescohomologicalorgeometricrestrictionsonthedomain.Section3presentsthebasictoolsinourstudy:amulti-dimensionalgeneralizationoftheclassicalWronski