Wave Structures and Nonlinear Balances in a Family

arXiv:nlin/0202059v1[nlin.CD]26Feb2002WaveStructuresandNonlinearBalancesinaFamilyof1+1EvolutionaryPDEsDarrylD.HolmandMartinF.StaleyTheoreticalDivisionandCenterforNonlinearStudiesLosAlamosNationalLaboratory,MSB284LosAlamos,NM87545email:[e-mail]26,2002AbstractWeintroducethefollowingfamilyofevolutionary1+1PDEsthatdescribethebalancebetweenconvectionandstretchingforsmallvis-cosityinthedynamicsof1Dnonlinearwavesinfluids:mt+umx|{z}convection+buxm|{z}stretching=νmxx|{z}viscosity,withu=g∗m.Hereu=g∗mdenotesu(x)=R∞−∞g(x−y)m(y)dy.Thisconvolution(orfiltering)relatesvelocityutomomentumdensitymbyintegrationagainstthekernelg(x).Weshallchooseg(x)tobeanevenfunction,sothatuandmhavethesameparityunderspatialreflection.Whenν=0,thisequationisbothreversibleintimeandparityinvariant.Weshallstudytheeffectsofthebalanceparameterbandthekernelg(x)onthesolitarywavestructures,andinvestigatetheirinteractionsanalyticallyforν=0andnumericallyforsmallviscosity,ν6=0.ThisfamilyofequationsadmitstheclassicBurgers“rampsandcliffs”solutionswhicharestablefor−1b1withsmallviscosity.Forb−1,theBurgersrampsandcliffsareunstable.Thestablesolutionforb−1movesleftwardinsteadofrightwardandtendstoastationaryprofile.Whenm=u−α2uxxandν=0,thisprofileis1givenbyu(x)≃sech2(x/(2α))forb=−2,andbyu(x)≃sech(x/α)forb=−3.Forb1,theBurgersrampsandcliffsareagainunstable.Thestablesolitarytravelingwaveforb1andν=0isthe“pulson”u(x,t)=cg(x−ct),whichrestrictstothe“peakon”solutioninthespe-cialcaseg(x)=e−|x|/αwhenm=u−α2uxx.Nonlinearinteractionsamongthesepulsonsorpeakonsaregovernedbythesuperpositionofsolutionsforb1andν=0,u(x,t)=NXi=1pi(t)g(x−qi(t)).Thesesolutionsobeyafinitedimensionaldynamicalsystemforthetime-dependentspeedspi(t)andpositionsqi(t).Westudythepul-sonandpeakoninteractionsanalytically,andwedeterminetheirfatenumericallyunderaddingviscosity.Contents1Introduction41.1Theb-familyoffluidtransportequations............41.2Outlineofthepaper.......................52Historyandgeneralpropertiesoftheb-equation52.1Discretesymmetries:reversibility,parityandsignature....82.2Lagrangianrepresentation....................82.3PreservationofthenormkmkL1/bfor0≤b≤1........92.4Lagrangianrepresentationforintegerb.............102.5ReversibilityandGalileancovariance..............112.6Integralmomentumconservation................113Travelingwavesandgeneralizedfunctions123.1Caseb=0.............................123.1.1Pulsonsforb=0.....................123.1.2Peakons,ramps,andcliffsforb=0...........123.2Caseb6=0.............................143.2.1Specialcasesoftravelingwavesforb6=0........153.3Caseb0.............................1523.3.1Pulsonsforb0.....................153.3.2Peakonsforb1.....................163.4Caseb0.............................163.4.1Caseb=−1/2......................193.4.2Caseb=−1........................203.4.3Caseb=−2stationarysolutions............213.4.4Caseb=−3stationarysolutions............223.4.5Caseb=−4stationarysolutions............233.4.6NumericalResultsforb=−2andb=−3.......234Pulsoninteractionsforb0254.1Pulsoninteractionsforb=2...................274.2Peakoninteractionsforb=2andb=3:numericalresults..274.3Pulson-Pulsoninteractionsforb0andsymmetricg.....314.4Pulson-antiPulsoninteractionsforb1andsymmetricg...344.5SpecializingPulsonstoPeakonsforb=2andb=3......365Peakonsofwidthαforarbitraryb375.1SlopedynamicsforPeakons:inflectionpointsandthesteep-eninglemmawhen1b≤3...................385.2Cases0≤b≤1..........................406Addingviscositytopeakondynamics406.1Burgers−αβequation:analyticalestimates...........426.2Burgers−αβtravelingwavesforβ(3−b)=1&ν=0.....457Thefateofthepeakonsunder(1)addingviscosityand(2)Burgers−αβevolution467.1Thefateofpeakonsunderaddingviscosity...........467.2ThefateofpeakonsunderBurgers−αβevolution.......518Numericalresultsforpeakonscatteringandinitialvalueprob-lems608.1Peakoninitialvalueproblems..................608.1.1Inviscidb-familyofequations..............608.1.2Viscousb-familyofequations..............638.1.3Burgers-αβequation...................648.2Descriptionofournumericalmethods..............6439Conclusions6610Acknowledgements681Introduction1.1Theb-familyoffluidtransportequationsWeshallanalyzeaone-dimensionalversionofactivefluidtransportthatisdescribedbythefollowingfamilyof1+1evolutionaryequations,mt+umx|{z}convection+buxm|{z}stretching=0,withu=g∗m,(1)inindependentvariablestimetandonespatialcoordinatex.Weshallseeksolutionsforthefluidvelocityu(x,t)thataredefinedei-theronthereallineandvanishingatspatialinfinity,oronaperiodicone-dimensionaldomain.Hereu=g∗mdenotestheconvolution(orfiltering),u(x)=Z∞−∞g(x−y)m(y)dy,(2)whichrelatesvelocityutomomentumdensitymbyintegrationagainstkernelg(x)overtherealline.Weshallchooseg(x)tobeanevenfunction,sothatuandmhavethesameparity.Thefamilyofequations(1)ischaracterizedbythekernelgandtherealdimensionlessconstantb,whichistheratioofstretchingtoconvectivetrans-port.Asweshallsee,bisalsothenumberofcovariantdimensionsassociatedwiththemomentumdensitym.Thefunctiong(x)willdeterminethetrav-elingwaveshapeandlengthscaleforequation(1